c Copyright by Jonathan C. McKinney, 2004research.physics.illinois.edu/Publications/theses/copies/McKinney_Jonathan.pdf · BY JONATHAN C. MCKINNEY B.S., Texas A&M University of College

c© Copyright by Jonathan C. McKinney, 2004

BLACK HOLE ACCRETION SYSTEMS

BY

JONATHAN C. MCKINNEY

B.S., Texas A&M University of College Station, 1996M.S., University of Illinois at Urbana-Champaign, 1999

DISSERTATION

Submitted in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy in Physics

in the Graduate College of theUniversity of Illinois at Urbana-Champaign, 2004

Urbana, Illinois

Abstract

Accretion onto a black hole is the most efficient known process to convert gravitational energyinto radiation. Some gamma-ray bursts (GRBs), some X-ray binaries, and all active galacticnuclei (AGN) are likely powered by accretion onto a rotating black hole. I model the global, time-dependent accretion flow around a black hole using the nonradiative viscous hydrodynamic (VHD),Newtonian magnetohydrodynamic (MHD), and general relativistic MHD (GRMHD) equations ofmotion, which are integrated numerically. I wrote the VHD and nonrelativistic MHD numericalcodes, and coauthored the GRMHD code (Gammie et al., 2003).

I first studied VHD accretion disk models, such as those studied by Igumenshchev et al. (1999,2000) and Stone et al. (1999), hereafter IA and SPB. IA and SPB use the VHD model of accretion,but they gave qualitatively different measurements of the energy per baryon accreted, angularmomentum per baryon accreted, and the radial scaling law for various quantities. While therewas concern in the community that the different results were due to numerical error, I foundthat the different results could be reproduced using my single VHD code to model the accretionflow (McKinney and Gammie, 2002). The differences in their results were due to differences intheir experimental designs. Seemingly small changes to the VHD model introduced nonnegligiblechanges to the results. This suggests that a self-consistent MHD, rather than phenomenologicalVHD, model for turbulence is required to study accretion flow.

First discovered during the VHD study described above, I found that VHD, nonrelativisticMHD, and GRMHD numerical accretion disk models can produce significant numerical artifactsunless the flow near the inner radial boundary condition, at rin, is out of causal contact with theflow at r > rin. For the VHD model, this corresponds to setting rin so the flow there is alwaysingoing at supersonic speeds. For the MHD models, this corresponds to setting rin so the flow thereis always ingoing at superfast speeds. For the GRMHD model, this is easily constructed by usingKerr-Schild (horizon-penetrating) coordinates. In this case, rin is chosen to be inside the horizon,where all waves are ingoing.

I next studied MHD and GRMHD numerical accretion disk models to test the results of previ-ously studied phenomenological disk models. These models are based on the Shakura & Sunyaevα-disk model and suggest the disk should terminate at the innermost stable circular orbit (ISCO) ofa black hole. Simplified GRMHD models predict super-efficient accretion due to energy extractionfrom a rotating black hole. I found that MHD and GRMHD numerical models show that the diskdoes not terminate at the ISCO, and magnetic fields continue to exert a torque on the disk insidethe ISCO. The disk will likely continue to emit radiation inside the ISCO, altering the predictedspectra of accretion disks. GRMHD numerical models of thick and thin disks show that the energy

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per baryon accreted closely follows the thin disk efficiency, so super-efficient accretion does not seemto be a generic property of thick magnetized relativistic disks (McKinney and Gammie, 2004).

The Blandford-Znajek (BZ) effect, describing the extraction of spin energy of a rotating blackhole by the magnetosphere, plausibly powers the jet in some GRBs, some microquasars, and allAGN. I found that GRMHD numerical models of thick disks around a rotating black hole showthat an evacuated, nearly force-free magnetosphere develops as predicted by BZ (McKinney andGammie, 2004). The BZ solution for the energy extracted is remarkably accurate in this regionfor a black hole with a/M . 0.5 and qualitatively accurate for all a/M , where a is the Kerr spinparameter and M is the mass of the black hole. GRMHD numerical models with a/M & 0.5 showa mildly relativistic (Lorentz factor Γ ∼ 1.5 − 3) collimated Poynting jet around the polar axis.Currently, no self-consistent MHD model of the accretion flow around a black hole shows a jet withΓ & 3. Additional physics is likely required to obtain Γ ∼ 100 as models predict in GRBs, and toobtain Γ ∼ 3− 10 as seen in some microquasars and AGN.

I studied the VHD, MHD, and GRMHD accretion models by performing numerical simulationson our group’s Beowulf computer clusters, which I designed and constructed. For about $50,000,one can buy a private cluster of computers that will provide as much computing power as today’stypical time-shared “supercomputer.” I give an account of the procedures necessary to design,build, and test a Beowulf cluster. The main conclusion is obvious: test one’s code on test nodesbefore purchasing the entire cluster in order to confirm the performance and reliability of the chosencomponents (CPUs, motherboard, network, etc.).

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To my mom.

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Acknowledgments

I strongly express my gratitude to my supervisor, Prof. Charles Forbes Gammie, whose expertise,understanding, and patience added considerably to my graduate experience. Charles demonstratesa broad-minded, razor-sharp knowledge of astrophysics, and his example has driven me to becomea better scientist. Charles has tirelessly reviewed many of my written works, and with his help, Ihave gradually improved my skills as a writer.

I express a special thanks to Scott Noble and Ruben Krasnopolsky, who both proofread theunpublished parts of my thesis. They gave excellent comments that greatly improved the focus,clarity, and readability of the thesis. Throughout their careers as postdocs for Charles, they haveprovided me with many stimulating conversations and uncountable helpful pointers. I thank PaulRicker for excellent comments about the Beowulf cluster appendix.

I thank the members of my thesis committee: Charles Gammie, Stu Shapiro, Susan Lamb, andJen-Chieh Peng. Their time is precious and I am grateful for all the comments on my writtenthesis and oral defense. I particularly thank Stu Shapiro for being a great source of inspiration andguidance in the study of relativity and compact objects.

I thank Charles Gammie, Stu Shapiro, Bill Watson, David Campbell, and Peter Anninos forwriting recommendation letters for my future career as a postdoc at Harvard with Ramesh Narayan,who I thank for hiring me and being patient while I finish my thesis.

I thank and love my family for the support they provided me through my entire life and in par-ticular, I must acknowledge my fiance and best friend, Elena, without whose love, encouragement,and editing assistance I would not have finished this thesis.

My financial support was largely provided by a NASA GSRP Fellowship Grant NGT5-50343(S01-GSRP-044) and partially supported by a GE fellowship. During my thesis work, Charleswas supported by an NCSA Faculty Fellowship, the UIUC Research Board, NSF ITR grant PHY02-05155, and NSF PECASE grant AST 00-93091. Computations were done in part under NCSAgrants AST010012N and AST010009N using the Origin 2000 and Posic Linux cluster at NCSA.Some computations were performed on Platinum.

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Table of Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv

List of Abbreviations and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

1 Observations, Theory, and Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Summary of Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Introduction to Black Hole Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 Formation of Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.2 Gamma-Ray Bursts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.3 Black Hole X-ray Binaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.2.4 Normal and Active Galactic Nuclei and Quasars . . . . . . . . . . . . . . . . 12

1.3 Basic Accretion Disk Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.1 Accretion Luminosity and Mass of the Compact Object . . . . . . . . . . . . 151.3.2 Some Accretion-Based Arguments . . . . . . . . . . . . . . . . . . . . . . . . 19

1.4 Models of Accretion Disks and GRBs . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.4.1 Angular Momentum Transport Models . . . . . . . . . . . . . . . . . . . . . . 251.4.2 Radiative Disk Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.4.3 Gamma-Ray Bursts Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.5 Characteristic Quantities and Model Validity Estimates . . . . . . . . . . . . . . . . 381.5.1 Estimated State and Structure of Accretion Flow . . . . . . . . . . . . . . . . 381.5.2 Validity of the Fluid, MHD, and ideal MHD Approximations . . . . . . . . . 46

1.6 Summary of Motivation for a GRMHD Model and Open Questions . . . . . . . . . . 491.7 Summary of Dissertation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

1.7.1 Viscous Hydrodynamics Summary . . . . . . . . . . . . . . . . . . . . . . . . 521.7.2 Global 2D/3D MHD Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 541.7.3 HARM / GRMHD Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 561.7.4 BZ Effect Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 571.7.5 BZ/Inflow Solution Comparison Summary . . . . . . . . . . . . . . . . . . . . 58

2 Numerical Models of Viscous Accretion Flows Near Black Holes . . . . . . . . 602.1 Summary of Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.3 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 622.4 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.4.1 Numerical Treatment of Low Density Regions . . . . . . . . . . . . . . . . . . 662.4.2 Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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2.4.3 Code Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.5.1 Fiducial Model Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.5.2 Dependence on Inner Boundary Location and Gravitational Potential . . . . 742.5.3 Comparison of Torus and Injection Models . . . . . . . . . . . . . . . . . . . 762.5.4 Other Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.6 VHD Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 782.7 Global 2D MHD Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792.8 Global 3D MHD Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3 HARM: A Numerical Scheme for General Relativistic Magnetohydrodynamics 823.1 Summary of Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 823.3 A GRMHD Primer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.4 Numerical Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.4.1 Constrained Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 883.4.2 Wave Speeds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.4.3 Implementation Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.5 Code Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.5.1 Linear Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 923.5.2 Nonlinear Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963.5.3 Transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.5.4 Orszag-Tang Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1003.5.5 Bondi Flow in Schwarzschild Geometry . . . . . . . . . . . . . . . . . . . . . 1053.5.6 Magnetized Bondi Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1053.5.7 Magnetized Equatorial Inflow in Kerr Geometry . . . . . . . . . . . . . . . . 1073.5.8 Equilibrium Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

3.6 Magnetized Torus Near Rotating Black Hole . . . . . . . . . . . . . . . . . . . . . . . 1113.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4 A Measurement of the Electromagnetic Luminosity of a Kerr Black Hole . . . 1164.1 Summary of Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.3 Review of Analytic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.3.1 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1194.3.2 Governing Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.3.3 Blandford-Znajek Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1214.3.4 Equatorial MHD Inflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

4.4 Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1284.4.1 Fiducial Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.4.2 Comparison with BZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1364.4.3 Comparison to Inflow Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 141

4.5 Parameter Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1444.5.1 Black Hole Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1444.5.2 Field Geometry and Strength . . . . . . . . . . . . . . . . . . . . . . . . . . . 1484.5.3 Numerical Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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5 Future Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.1 Thin Disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555.2 Relativistic Jet-Disk Connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1565.3 Gamma-Ray Bursts: GRMHD Collapsar Model . . . . . . . . . . . . . . . . . . . . . 156

A Model of Plasmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159A.1 Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

A.1.1 EOS for GRBs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160A.1.2 EOS for X-ray binaries and AGN . . . . . . . . . . . . . . . . . . . . . . . . . 162

A.2 Validity of the Fluid Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . 162A.3 Validity of the Ideal Fluid Approximation . . . . . . . . . . . . . . . . . . . . . . . . 164A.4 Validity of the Plasma and MHD Approximations . . . . . . . . . . . . . . . . . . . . 165A.5 Validity of the Ideal MHD Approximation . . . . . . . . . . . . . . . . . . . . . . . . 167

B Beowulf cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172B.1 History of Design Decisions for our Clusters . . . . . . . . . . . . . . . . . . . . . . . 172B.2 Cluster Performance and Advanced Network Drivers . . . . . . . . . . . . . . . . . . 179B.3 Cluster access to Internet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183B.4 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184B.5 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

B.5.1 Choices for OS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189B.5.2 OS Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190B.5.3 Software Installation and Usage Notes . . . . . . . . . . . . . . . . . . . . . . 191B.5.4 MPI Implementation in Fluid Codes . . . . . . . . . . . . . . . . . . . . . . . 194B.5.5 Running MPI Jobs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

B.6 Testing Cluster Reliability and Performance . . . . . . . . . . . . . . . . . . . . . . . 197B.6.1 Reliability and Performance Issues . . . . . . . . . . . . . . . . . . . . . . . . 197B.6.2 Bandwidth and Latency of Network . . . . . . . . . . . . . . . . . . . . . . . 199B.6.3 Code Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

B.7 Beowulf Cluster Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

Curriculum Vitae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

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List of Figures

1.1 Cartoon plot of one hemisphere of the axisymmetric accretion disk, corona, fun-nel/wind, and plunging regions. The outer radius of the GRMHD simulation is40GM/c2. The fiducial radius (r = 12L, where L = GM/c2) is where all “state”quantities are evaluated. These state quantities are then used to estimate the validityof the fluid, MHD, and ideal MHD approximation at this location. . . . . . . . . . . 39

2.1 Time-averaged spatial structure of fiducial run (Run A; α = 0.1, rin = 2.7GM/c2,and rout = 600GM/c2). Shown are the density (upper left), Bernoulli parame-ter (Be = (1/2)v2 + c2

s/(γ − 1) + Ψ) (upper right; dotted line is a negative con-tour), scaled mass flux r2 sin θ(ρ0v) (lower left), and scaled angular momentum fluxr3 sin2 θ(ρ0vvφ + Π · φ) (lower right). The flow is not symmetric about the equa-tor because the flow exhibits long timescale antisymmetric variations. Convectivebubbles form at the interface between positive and negative Bernoulli parameter(i.e. unbound and bound matter). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2.2 The evolution of M/Minj , e = E/(Mc2), and l = L c/(GMM) in the fiducial run(Run A). The dotted line indicates the thin disk value. The run has clearly entereda quasi-steady state. The evolution is relatively smooth with a small variation on atimescale τ ≈ 4×104. This is the timescale for convective bubble formation (the lowpoint in rest-mass accretion rate is when bubble forms). For this model the bubbleforms at alternate poles. A full cycle requires of order one rotation period at theinjection radius. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.3 The radial run of θ and time averaged quantities from the fiducial run (Run A).Shown are the density (upper left), squared sound speed (upper right), radial velocity(lower left), specific angular momentum (lower right; solid line), and circular orbitspecific angular momentum (lower right; dashed line). Crudely speaking, the innerflow is consistent with a radial power law. The best fits to a power law are: ρ0 ∝ r−0.6,cs ∝ r−0.5, |vr| ∝ r−2, and vφ ∝ r−0.8. The plots are averaged over θ = π/2± π/6. . 73

2.4 The effect of moving the inner boundary on the accretion rates of mass, angularmomentum, and energy (Run A vs. Run B). The top panel shows M/Minj , themiddle panel E/(Mc2), and the bottom panel L c/(GMM). The solid curve is RunA, which has rin = 2.7GM/c2. The dashed curve is Run B, which has rin = 6GM/c2.Evidently Run B has a different variability structure and different time averagedvalues for the accretion rates. The relatively rapid and high-amplitude variations inRun B are due to nonphysical interactions with the inner radial boundary. Only byensuring a supersonic flow (as in Run A) can one avoid these nonphysical effects. . 75

2.5 Field line snapshot after 8 orbits (at the t = 0 torus density maximum at r =9.4GM/c2) for a global 2D MHD simulation. Full non-linear turbulence drives theaccretion process. Note that the polar field is essentially radial while the accretiondisk is dominated by turbulence generated by the MRI. . . . . . . . . . . . . . . . . 80

x

3.1 The L1 norm of the error in u for a slow wave as a function of Nx for both themonotonized central (MC) and minmod limiter. The straight lines show the slopeexpected for second order convergence. . . . . . . . . . . . . . . . . . . . . . . . . . 93

3.2 The L1 norm of the error in the single nonzero component of the velocity for anAlfven wave as a function of Nx for both the monotonized central (MC) and minmodlimiter. The straight lines show the slope expected for second order convergence. . . 94

3.3 The L1 norm of the error in u for a fast wave as a function of Nx for both themonotonized central (MC) and minmod limiter. The straight lines show the slopeexpected for second order convergence. . . . . . . . . . . . . . . . . . . . . . . . . . 95

3.4 The run of density in the Komissarov nonlinear wave tests. . . . . . . . . . . . . . . 973.5 The run of ux in the Komissarov nonlinear wave tests. . . . . . . . . . . . . . . . . 983.6 Snapshot of the final state in HARM’s integration of Ryu & Jones test 5A (a version

of the Brio & Wu shock tube) but with c = 100. The figure shows primitive variablevalues at t = 0.15. Quantitative agreement is found to within ≈ 1%, as expected. . 99

3.7 Snapshot of the final state in HARM’s integration of Ryu & Jones test 2A, withc = 100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

3.8 Convergence results for the transport test. . . . . . . . . . . . . . . . . . . . . . . . 1023.9 A cut through the density in the nonrelativistic Orszag-Tang vortex solution from

HARM (solid line, with c = 100), from VAC (dashed line), and 4× the difference(lower solid line) at a resolution of 6402. . . . . . . . . . . . . . . . . . . . . . . . . 103

3.10 Comparison of results from HARM and the nonrelativistic MHD code VAC for theOrszag-Tang vortex. The plot shows the L1 norm of the difference between thetwo results as a function of resolution for the primitive variables ρ0 (squares) and u(triangles). The straight line shows the slope expected for first order convergence.The errors are large because they are an integral over an area of (2π)2. . . . . . . . 104

3.11 Convergence results for the unmagnetized Bondi accretion test onto a Schwarzschildblack hole. The straight line shows the slope expected for second order convergence. 106

3.12 Convergence results for the magnetized Bondi accretion test onto a Schwarzschildblack hole. The straight line shows the slope expected for second order convergence. 108

3.13 The equatorial inflow solution in the Kerr metric for a/M = 0.5 and magnetizationparameter Fθφ = 0.5. The panels show density, radial component of the four-velocityin Boyer-Lindquist coordinates (with the square showing the location of the fastpoint), the φ component of the four-velocity, and the toroidal magnetic field Bφ =Fφt. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

3.14 Convergence results for the magnetized inflow solution in a Kerr metric with a/M =0.5. Parameters for the initial, quasi-analytic solution are given in the text. Thestraight line shows the slope expected for second order convergence. The L1 errornorm for each of the nontrivial variables are shown. The small deviation from secondorder convergence at high resolution is due to numerical errors in the quasi-analyticsolution used to initialize the solution. . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.15 Convergence results for the Fishbone and Moncrief equilibrium disk around ana/M = 0.95 black hole. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3.16 Density field, for a magnetized torus around a Kerr black hole with a/M = 0.5 att = 0 (left) and at t = 2000M (right). The color is mapped from the logarithm ofthe density; black is low and dark red is high. The resolution is 3002. . . . . . . . . 113

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3.17 Evolution of the rest-mass accretion rate (top), the specific energy of the accretedmatter (middle), and the specific angular momentum of the accreted matter (bottom)for a black hole with a/M = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.1 Initial (left) and final (right) distribution of log ρ0 in the fiducial model on the r sin θ−r cos θ plane. At t = 0 black corresponds to ρ0 ≈ 4× 10−7 and dark red correspondsto ρ0 = 1. For t = 2000, black corresponds to ρ0 ≈ 4×10−7 and dark red correspondsto ρ0 = 0.57. The black half circle at the left edge is the black hole. . . . . . . . . . 130

4.2 (a) The distribution of β, b2/ρ0, and ut in the fiducial run, based on time and hemi-spherically averaged data. Starting from the axis and moving toward the equator:(1) ut = −1 contour shown as a solid black line; (2) b2/ρ0 = 1 contour shown as ared line; (3) β = 1 contour shown as a magenta line that nearly matches part of theut = −1 contour line; and (4) β = 3 contour is shown as cyan line. (b) Motivatedby the left panel, the right panel indicates the location of the five main subregionsof the black hole magnetosphere. They are (1) the disk: a matter dominated regionwhere b2/ρ0 ¿ 1; (2) the funnel: a magnetically dominated region around the poleswhere b2/ρ0 À 1 where the magnetic field is collimated and twists around and upthe axis into an outflow; (3) the corona: a region in the relatively low density upperlayers of the disk with weak time-averaged poloidal field; (4) the plunging region;and (5) the wind, which straddles the corona-funnel boundary. See Section 4.4.1 fora discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

4.3 Initial (left) and final (right) distribution of Aφ. Level surfaces coincide with mag-netic field lines and field line density corresponds to poloidal field strength. In theinitial state field lines follow density contours if ρ0 > 0.2ρ0,max. . . . . . . . . . . . . 133

4.4 Contour plot of the time and hemispheric average of Aφ. Level surfaces coincidewith magnetic field lines and field line density corresponds to poloidal field strength. 134

4.5 Evolution of rest-mass, energy, and angular momentum accretion rate for our fiducialrun of a weakly magnetized tori around a black hole with spin a = 0.938. For500 < t < 2000 the time average of these values is M0 ' 0.35, E/M0 ' 0.87, andL/M0 ' 1.46 as shown by the dashed lines. The dotted lines show the classical thindisk values (E/M0 ' 0.82 and L/M0 ' 1.95). See Section 4.4.1 for a discussion. . . . 135

4.6 Electromagnetic energy flux density F(EM)E (θ) on the horizon for the fiducial run,

based on time and hemisphere averaged data. The mean electromagnetic energy fluxis directed outward. See Section 4.4.1 for a discussion. . . . . . . . . . . . . . . . . . 137

4.7 The run of the force-free parameter ζ for the a = 0.5 run; when ζ ¿ 1 the field isapproximately force-free. The parameter has been time and hemisphere averaged.The contours show (beginning from the pole and moving toward the equator) ζ =10−3, 10−2, 10−1. The small closed contours at large radius and close to the axis haveζ = 10−2. The small closed contours from the equator to θ ∼ π/4 have ζ = 10−1.See Section 4.4.2 for a discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

4.8 Left panel: Magnetic field angular frequency on the horizon relative to black holerotation ω(θ)/ΩH . The solid line indicates time and hemisphere averaged data fromour a = 0.5 MHD integration. The middle dotted line is the prediction of the BZmodel (ω/ΩH = 1/2). The dashed line (top) is the value predicted by the inflowmodel. Right panel: the run of field rotation frequency ω with radius along a singlefield line that intersects the horizon at θ = 0.2. ω is constant to within 3%, asexpected for a steady flow. See sections 4.4.2 and 4.4.3 for a discussion. . . . . . . . 139

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4.9 (a) Square of radial field ((Br(θ))2) on the horizon in the a = 0.5 MHD integration,from time and hemisphere averaged data. Solid line is the field for our numericalmodel. The dotted line shows the Blandford and Znajek (1977) perturbed monopolesolution with the field strength normalized to the numerical solution at the pole.The dashed line is the inflow solution. (b) Electromagnetic energy flux F

(EM)E (θ) on

the horizon in the a = 0.5 MHD integration, from time and hemisphere averageddata. The solid line shows the numerical model, the dotted line shows BZ’s spun-upmonopole solution, and the dashed line shows the inflow solution. See sections 4.4.2and 4.4.3 for a discussion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

4.10 A comparison of the time-averaged fiducial model near the equator (within θ =π/2 ± 0.3) with the inflow solution of Gammie (1999). In the right two panels theblack dotted line is the thin disk value. In all cases the red vertical line is the locationof the ISCO. The black line for the upper left panel is the numerical result. For theother three panels, the particle term is shown in cyan, the internal energy term isshown in magenta, and the electromagnetic term is shown in green. The blue linein each plot represents the inflow model result. Notice that the run of density withradius shows no feature at the ISCO. See the Section 4.4.3 for discussion. . . . . . . 142

4.11 The ratio of electromagnetic to matter energy flux on the horizon. The solid lineindicates numerical data while the dotted line indicates a best fit of E(EM)/E(MA) =−0.068(2− r+)2. See Section 4.5.1 for a discussion. . . . . . . . . . . . . . . . . . . . 145

B.1 Astronomy Building computer room electrical layout as of Feb, 2004. BH is locatednorth and center. This excludes the addition of another new node for BH and thecollaborating group’s cluster of 12 nodes. . . . . . . . . . . . . . . . . . . . . . . . . 178

B.2 BH Beowulf cluster primary elements. . . . . . . . . . . . . . . . . . . . . . . . . . . 180B.3 Digital photograph of BH cluster. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181B.4 Cluster block diagram of BH cluster. Figure shows bandwidth between elements on

motherboard and switch. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185B.5 Bandwidth (left) and latency (right) for gigabit Ethernet connection on BH cluster. 200

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List of Tables

1.1 Fiducial Black Hole Accretion Systems 1 . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Fiducial Black Hole Accretion Systems 2 . . . . . . . . . . . . . . . . . . . . . . . . . 181.3 Fiducial Black Hole Accretion Systems 3 . . . . . . . . . . . . . . . . . . . . . . . . 241.4 Accretion Flow Regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401.5 Accretion Disk State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431.6 Validity of Fluid Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461.7 Validity of MHD Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.8 Validity of Ideal MHD Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.1 Parameter List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 692.2 Results List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1 Commonly used symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1184.2 Black Hole Spin Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1464.3 Field Strength and Geometry Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 1484.4 Resolution Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

B.1 Single CPU performance in ZCPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202B.2 Origin 2000 MPI performance in kZCPS for 2D MHD code . . . . . . . . . . . . . . 203B.3 NCSA Platinum MPI performance in kZCPS for 2D & 3D MHD code . . . . . . . . 204B.4 BH Xeon Cluster MPI performance in kZCPS for ZEUS-based 2D MHD code . . . . 205B.5 BH Xeon Cluster MPI performance in kZCPS for 3D MHD code . . . . . . . . . . . 206B.6 BH Xeon BH Cluster MPI performance in kZCPS for 2D HARM code . . . . . . . . 207

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List of Abbreviations and Acronyms

ADAF Advection Dominated Accretion Flow.

AGN Active Galactic Nuclei.

ASCA Advanced Satellite for Cosmology and Astrophysics.

BATSE Burst And Transient Source Experiment.

BeppoSAX Beppo Satellite per Astronomia X, in honor of Giuseppe “Beppo” Occhialini.

BC Boundary Condition.

BH Black Hole.

BHC Black Hole Candidate.

BL Lac BL Lacertae.

BTU British Thermal Unit.

BZ Blandford & Znajek (1977).

CDM Cold Dark Matter.

CPU Central Processing Unit.

CV Cataclysmic Variable.

EM ElectroMagnetic.

EMF Electromotive Force.

EOS Equation Of State.

GM Glenn’s Messages (Myrinet message protocol).

GR General Relativity.

GRB Gamma-Ray Burst.

GRMHD General Relativistic MagnetoHydroDynamics.

GRS Galactic Radio Source.

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HARM High Accuracy Relativistic Magnetohydrodynamics.

HD HydroDynamics.

HDE Henry Draper Extension (catalogue).

HETE High Energy Transient Explorer

HLL Harten-Lax-van Leer

HMXB High-Mass X-ray Binary.

HST Hubble Space Telescope.

IA Igumenshchev & Abramowicz (2000,2001).

IC Index Catalogue (of nebulae).

IMBH Intermediate-Mass Black Hole.

IMF Initial Mass Function.

IR InfraRed.

ISCO Inner-most Stable Circular Orbit.

KS Kerr-Schild.

LF Lax-Friedrich.

LIGO Laser Interferometer Gravitational Wave Observatory.

LISA Laser Interferometer Space Antenna.

LMC Large Magellanic Cloud.

LMXB Low-Mass X-ray Binary.

MC Monotonized Central.

MGC Millennium Galaxy Catalogue.

MHD MagnetoHydroDyanmics.

MKS Modified Kerr-Schild.

MPI Message Passing Interface.

MRI MagnetoRotational Instability.

MTW Misner, Thorne, & Wheeler (1973).

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NGC New General Catalogue.

NS Neutron Star.

NSE Nuclear Statistical Equilibrium.

OS Operating System.

PWF Popham, Woosley, & Fryer (1999).

QED Quantum ElectroDynamics.

QPO Quasi-Periodic Oscillation.

RXTE Rossi X-ray Timing Explorer.

RJ Ryu & Jones (1995).

RMR Relativistic MagnetoRotators.

ROSAT Rontgen Satellite.

SGI Silicon Graphics Incorporated.

SLE Shapiro, Lightman, & Eardley (1976).

SDSS Sloan Digital Sky Survey.

SMBH Super-Massive Black Hole.

SN Supernova.

SPB Stone, Pringle, & Begelman (1999).

SRMHD Special Relativistic MagnetoHydroDynamics.

SS/SS73 Shakura & Sunyaev (1973)

SXT Soft X-ray Transient.

ULX UltraLuminous X-ray (object).

URCA The URCA process involves scattering of a neutrino, proton, neutron, and electron. Coinedby Gamow & Schoenberg in 1941 in humorous reference to the money drain they experiencedgambling in Urca, Brazil.

UV UltraViolet.

VHD Viscous HydroDynamics.

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VLA Very Large Array.

VLBI Very Long Base Interferometry.

VLBA Very Long Baseline Array.

XMM-Newton X-ray Multi-Mirror - Newton.

XSPEC X-Ray Spectral fitting package.

YSO Young Stellar Object.

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Summary

About twice per day, gamma-ray observatories detect a gamma-ray burst (GRB) with a typicalduration of seconds (for a review see Piran 2004). Some GRBs have an associated lower-energyafterglow. Redshift measurements of the afterglow, together with the fireball model of GRBs,suggest GRBs have a luminosity of 1045 − 1053 erg/s. A high luminosity GRB is likely the mostpowerful event in the universe. The known duration of GRBs allows one to estimate that mostGRBs have a total energy of ∼ 1051 erg, which is comparable to the energy of a supernova. Theso-called fireball model suggests a GRB comes from an outflow with a Lorentz factor of Γ ∼ 100,so they have one of the fastest bulk flow velocities in the universe. A GRB likely originates fromthe collapse of a massive star into a neutron star or black hole. The duration of a GRB is muchlonger than the radial free-fall time for the stellar envelope, which suggests the stellar matter hasa nonnegligible angular momentum and an accretion disk likely forms. The GRB is thought to begenerated by a jet from the accretion of stellar matter onto a rapidly spinning neutron star or blackhole. The jet is suspected to penetrate the stellar surface and, according to the standard model,the observed gamma-rays are produced in relativistic shocks within the jet.

About half of all star systems have only one star. Some single-star systems contain a massivestar that collapses into a neutron star or black hole, which eventually accretes all the stellar matterthat could form a disk or material that falls back from the supernova. Some of these may continueto accrete interstellar matter or roam the universe undetected except by gravitational lensing andfaint X-ray emission. However, about half of all star systems are binaries, and the neutron staror black hole can continue to accrete by capturing matter from the companion star. Such systemshave a high X-ray luminosity, so are called X-ray binaries (for a review see Lewin et al. 1995).With a persistent power output up to 1036 − 1038 erg/s, X-ray binaries are the brightest persistentsources of X-rays in the sky. By comparison, the Crab pulsar, otherwise the brightest X-ray sourcein the sky, has an X-ray luminosity of ∼ 1037 erg/s. Many X-ray binaries, called microquasars, havea relativistic jet with Γ ∼ 3 − 10. Some neutron star X-ray binaries unstably burn hydrogen andhelium leading to thermal X-ray bursts with a power output of 1036−1039 erg/s, and some unstablyburn carbon leading to thermal X-ray bursts with a power output of 1043 erg/s. The compact objectin some X-ray binaries has a mass M ∼ 3 − 20 M¯, which suggests that they cannot be neutronstars, so they are dubbed black hole candidates (BHCs). None of these BHC X-ray binaries exhibitthermal X-ray bursts. This suggests they each have no surface and so likely each contain a blackhole with an event horizon. An example black hole X-ray binary is Cygnus X-1, which harbors acompact object with M ∼ 7− 13M¯ (Webster and Murdin, 1972; Bolton, 1972; Gies and Bolton,1986; Herrero et al., 1995).

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Observations of a galaxy’s stellar dynamics, gas kinematics, or the galactic mass to light ratiosuggest that in most galaxies the center harbors a supermassive (M ∼ 105 − 109 M¯) black hole(for a review see Krolik 1999a). Such supermassive black holes likely formed from accretion onto,or merging of, intermediate-mass black holes with M ∼ 102 − 104 M¯. The motions of stars nearour galactic nucleus, SgrA*, provide the best evidence for a black hole with M ∼ 2.6 × 106 M¯(Eckart and Genzel, 1997; Ghez et al., 1998; Schodel et al., 2002). An active galaxy is a galaxyfrom which a significant fraction of the energy output is emitted by the nucleus rather than by thestars, dust, and interstellar gas. Such nuclei are called active galactic nuclei (AGN). AGN havea luminosity of 1042 − 1048 erg/s, which has long been considered to be generated by accretiononto the supermassive black hole. Observations of the active galaxy MCG-6-30-15 show a highlyredshifted Fe Kα line emission profile with significantly different strengths for the red and bluewing of the profile. These features of the Fe Kα line profile are consistent with emission from arapidly rotating v/c ∼ 0.2 accretion disk within a few gravitational radii (GM/c2) of a (possiblyrotating) black hole (Pariev and Bromley, 1998; Tanaka et al., 1995).

Accretion onto a black hole is the most efficient known process to convert gravitational energyinto radiation. Some GRBs, some X-ray binaries, and all AGN are likely powered by accretion ofplasma onto a rotating black hole. Models of the accreting plasma typically use the hydrodynamic(HD) approximation, or when including a magnetic field, the magnetohydrodynamic (MHD) ap-proximation. Since these systems involve relativistic flow around a rotating black hole, the MHDapproximation is solved using general relativity (GR) theory, which gives the GRMHD equationsof motion. This thesis mostly involves the study of nonradiative accretion disks using the GRMHDapproximation. Despite some useful time-independent and simplified HD, MHD, and GRMHD an-alytic solutions, a numerical model is likely required to study the global, time-dependent accretionflow. Typically, the time-dependent solution of accretion flow is only a quasi-steady state, whereflow quantities can have large deviations from the time-averaged value. I developed a viscous HDcode, a nonrelativistic MHD code, and a GRMHD code.

The thin disk solution of Shakura & Sunyaev (SS) uses an unmagnetized viscous model forangular momentum transport, which is likely due to MHD turbulence (Balbus and Hawley, 1991).The SS model predicts that the accretion disk terminates at the innermost stable circular orbit(ISCO) of a black hole. The SS solution is often used to interpret observations of spectra fromaccretion disks, and the termination of the disk leads to a significant spectral feature. Some studiesassume the SS solution to be accurate, and properties such as black hole spin, disk orientation,and the location of the disk inner-edge are derived. I used the nonrelativistic MHD and GRMHDcodes to study magnetized accretion flow. Like others at the time, I found that the disk does notterminate at the ISCO, and magnetic fields continue to exert a torque on the disk inside the ISCO.The disk will continue to emit radiation inside the ISCO, altering the predicted spectra of accretiondisks and altering those derived quantities (Reynolds and Begelman, 1997). For X-ray binaries andAGN, a more accurate spectral fit may suggest that the Fe Kα line emission is from farther out inthe disk and that the black hole is not necessarily rotating.

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Blandford and Znajek (1977) (BZ) argued that accretion flow around a black hole leads toevacuation of the polar regions and the development of a force-free magnetosphere. They found asolution, called the BZ-effect, which describes the extraction of spin energy of a rotating black holeby the magnetosphere. I used the GRMHD code to numerically study thick disks around a rotatingblack hole and found that an evacuated, nearly force-free magnetosphere develops as predicted byBZ. The BZ solution is remarkably accurate in this region for a black hole with a/M . 0.5, wherea is the Kerr spin parameter. For models with a/M & 0.5, I find a mildly relativistic (Γ ∼ 1.5− 3)collimated Poynting jet around the polar axis. However, additional physics is likely required toobtain Γ ∼ 100 as models predict in GRBs, and to obtain Γ ∼ 3− 10 as seen in some microquasarsand AGN.

The Gammie (1999) inflow solution describes a cold, thin accretion flow around a rotating blackhole. The Gammie solution predicts the radial dependence of comoving mass density, comovingmagnetic energy density, and the accretion rate of energy and angular momentum per baryon. TheGammie solution also predicts accretion could occur super-efficiently (i.e. more energy is releasedthan rest mass accreted), and that likely the efficiency is greater than that in a thin disk dueto the magnetic field in the plunging region. I found the numerical GRMHD solution for theradial dependence of those quantities to be marginally consistent with the Gammie inflow solution.Discrepancies are likely due to the numerical solution consisting of hot disk, while the Gammiesolution is for a cold disk. None of the numerical models I studied showed super-efficient accretion.In fact, for most thick and thin disks, the energy per baryon accreted closely followed the thin diskefficiency found by Bardeen (1970).

I designed and constructed a Beowulf cluster of computers to help develop the VHD, MHD,and GRMHD codes and use these codes to perform numerical simulations of accretion flow. Idescribe how to design, build, and test a Beowulf cluster. The main conclusion is that one shouldtest one’s own code on test nodes and construct a test cluster before purchasing the entire cluster.Performance and reliability vary greatly between personal computers (PCs) and server computers(SCs). The main performance difference is due to the PCI bus, on which both add-on and built-in network chips operate. The network chip and the PCI bus are typically the bottleneck inthe performance of parallel calculations. Today’s PCs are all composed of 32-bit 33Mhz-basedmotherboards, while SCs are composed of 64-bit 66/133Mhz-based motherboards. A PC is about1/2 the price of an SC, so PCs are an attractive option. However, for a cluster with more thanonly about 4 nodes, a cluster of PCs costs more per unit performance than a cluster of SCs.

Since some or all black hole accretion systems could harbor a thin disk, I plan to study thindisks and to determine how the BZ luminosity depends on disk thickness. A few groups havestudied GRMHD models of jets, and all fail to show Γ factors as high as in observed jets. I planto study other mechanisms for jet acceleration and collimation, and I plan to study the connectionbetween the disk and jet. I also plan to include the relevant microphysics to the GRMHD modelto study the accretion disk that likely forms during a GRB.

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1 Observations, Theory, and Models

This chapter summarizes the observations, theory, and models of black hole accretion disk systems.Section 1.1 outlines this introduction.

1.1 Summary of Introduction

Against the force of gravity, a spheroidal star is mostly supported by pressure forces, while a thindisk is mostly supported by centrifugal, or even magnetic, forces. The formation of a disk typicallyoccurs because radiative energy is released faster than angular momentum is transported. The diskcan be described as a result of the conservation of mass, energy, and momentum in the gravitationalpotential. As the disk matter falls in the gravitational potential, collisions allow the gravitationalpotential energy to be converted into radiative energy (instead of kinetic energy). Such a disk isknown as an accretion disk. The entire structure of the disk and surrounding dynamically-coupledmedium is called the accretion flow. The gravitating body in some accretion disk systems is aplanet (held up by gas or matter pressure), young stellar object (YSO) (held up by gas or radiationpressure), white dwarf (star held up by electron degeneracy pressure), or neutron star (NS) (starheld up by neutron degeneracy pressure). This thesis is primarily concerned with those accretiondisk systems that contain a black hole. For a review, see Shapiro and Teukolsky (1983).

Accretion disk models often use the fluid approximation to describe the underlying microphysicsof ionized media. The fluid approximation assumes that the smallest region of interest containsa large number of particles and that particle collisions are frequent enough to sustain a statisti-cal equilibrium state. A magnetized fluid is often modeled by the magnetohydrodynamic (MHD)approximation. The typical MHD approximation assumes the fluid is ionized, can be treated asa so-called weakly-coupled plasma, and particles have negligible gyration radii. A weakly-coupledplasma is one that contains both positive and negative charges, but is effectively neutral over adistance that spans many particles. A simplified form of the MHD approximation is the idealMHD approximation, which assumes that the plasma is perfectly conducting, currents arise in-stantaneously from fields, the Hall effect is negligible, electronic pressure gradients are negligible,and the pressure is isotropic (see, e.g., Krall and Trivelpiece 1973). A further simplification is thesingle-component ideal MHD approximation, which assumes all species are in statistical equilib-rium within a fluid element and that there is a well-defined average density, velocity, etc. for thefluid element. For a gas of protons and electrons, the single-component approximation involvesneglecting the electron inertia. For some fluids, some of these approximations are coupled to thesame underlying approximation. For example, if the electron inertia is negligible, then currents

1

arise instantaneously from fields for a gas of protons and electrons. Throughout the thesis, theaccretion flow is assumed to be well-described by the single-component ideal MHD approximation,but this will be tested near the end of this chapter. This thesis considers both the nonrelativisticand general relativistic ideal MHD approximations to describe the black hole accretion flow.

To provide context for the accretion disk theory to be presented later, Section 1.2 gives asummary of some astrophysical objects that probably contain an accreting black hole, such asgamma-ray bursts (GRBs), X-ray binaries, a source in the Galactic nucleus called SgrA*, andactive galactic nuclei (AGN). That section introduces the basic observed and implied features ofthese objects and also introduces 6 fiducial black hole accretion disk systems: 2 GRBs, 1 X-raybinary, SgrA*, and 2 AGN. The purpose of the discussion is to provide context for the accretion disktheory discussion that follows and to motivate general relativistic magnetohydrodynamic (GRMHD)models of these systems by summarizing the evidence for black holes and magnetized relativisticmotion. Later in this chapter, accretion disk models are developed for these 6 fiducial systems.These models are used to determine some accretion flow properties, such as the mass density andtemperature of the disk. Also later, the validity of the fluid, MHD, and ideal MHD approximationsas a model for the accretion flow in these 6 fiducial systems is tested.

Since the inner region of the accretion disk is typically not resolved by current telescopes, thetheory of this part of the accretion disk often rests on a spatially-unresolved flux of photons. Thisflux can be decomposed into a luminosity, time variability, and spectra of the disk system. Sec-tion 1.3 outlines the basic theory of disks around compact stars (such as black holes) by discussingthe luminosity, time variability, and some spectral features of accretion disks. The section showshow the luminosity and time variability independently provide an estimate for the mass of theaccreting compact object. The section discusses how some spectral features, such as the Fe Kα lineprofile, may even provide details about the spatial structure of the inner disk in some X-ray binariesand some AGN (Karas et al., 1992; Tanaka et al., 1995; Pariev and Bromley, 1998; Miller et al.,2004). The purpose of this section is to provide accretion theory-based arguments that suggestsome of the 6 fiducial objects, and those like them, are likely powered by an accretion disk thatextends close to a black hole.

An element of a disk accretes only by losing angular momentum. Since angular momentum isconserved, other elements must gain angular momentum and so move outward. Thus, for a diskto accrete, angular momentum must be transported outward. The first models of accretion disksassumed some form of magnetic or hydrodynamic turbulence drives angular momentum transport(Shakura and Sunyaev, 1973). The turbulence is often phenomenologically modeled as an effectiveshear viscosity that causes differentially-rotating layers to exchange angular momentum (Novikovand Thorne, 1973; Eardley and Lightman, 1975; Pringle, 1981). While a hydrodynamic instabilitymay generate turbulence in a disk, no known hydrodynamic instability drives sufficiently vigorousangular momentum transport to account for observations of astrophysical accretion disks (for areview, see Balbus and Hawley 1998). Later, it was realized that magnetized accretion disks arenaturally unstable, and that the so-called magnetorotational instability (MRI) could drive angular

2

momentum transport and account for accretion in astrophysical disks (Balbus and Hawley, 1991).The first part of Section 1.4 summarizes the effective viscosity and MRI models for the transport ofangular momentum in a highly ionized accretion disk. The purpose of that section is to motivatethe use of a magnetic, rather than viscous, model of the angular momentum transport.

Radiative processes are self-consistently included in accretion disk models that seek to accountfor the observed spectra and luminosity of accretion disk systems. Some radiative processes, suchas Compton scattering or bremsstrahlung, are involved in determining the structure of the diskand in producing a significant observational feature. Other radiative processes, such as Fe Kα linefluorescence, are weakly involved in determining the disk structure, but are important observationalfeatures from disks. Some radiative processes, such as synchrotron radiation from non-thermal par-ticles, require physics beyond the thermodynamic equilibrium approximation. The second part ofSection 1.4 discusses radiative accretion disk models that self-consistently determine disk structure.The purpose of that section is to provide a summary of previous work and to provide estimates ofthe rest-mass accretion rate for the 6 fiducial systems.

Ultimately, radiative models are required to compare with observations, however a nonradiativeGRMHD model may be sufficient to describe the mechanism of jets in black hole accretion systems.This thesis focuses on testing the plausibility of the Blandford-Znajek (BZ) effect (Blandford andZnajek, 1977) as a mechanism to generate relativistic jets, rather than focusing on the mechanismsinvolved in determining observational spectra from the disk. The last part of Section 1.4 discussesGRB models and the likelihood of the BZ-effect powering a GRB. GRBs are discussed more ex-tensively than X-ray binaries and AGN since the nonradiative GRMHD models, as developed inthis thesis, more readily apply to the accretion disk that forms in the so-called “collapsar” modelof a GRB (Woosley, 1993; Paczynski, 1998; MacFadyen and Woosley, 1999). That is, it is latershown that the collapsar GRB model likely does not need a detailed radiative transport model forphotons and neutrinos.

Section 1.5 estimates various quantities, such as mass density, temperature, velocity, magneticfield, and other quantities for the 6 fiducial systems. No radiative GRMHD model, with a realisticequation of state, has been developed or studied numerically. Therefore, a combination of modelsis used and checked for consistency. Section 1.5 uses the results of nonradiative GRMHD numericalmodels (as in Chapter 4), unmagnetized radiative neutrino-cooled analytic models (as in Pophamet al. 1999), and unmagnetized photon-cooled radiative analytic models (as described by Shakuraand Sunyaev 1973; Novikov and Thorne 1973) of accretion disks to make all the estimates. GRBsystems are approximated by a radiative ideal Fermi gas equation of state, while X-ray binary andAGN systems are approximated by an ideal gas + photon radiation pressure equation of state. Asa consistency check of the estimates for the 2 GRB systems, the results of the nonradiative, idealgas GRMHD numerical model are compared to the radiative, ideal Fermi gas model. A summary ofthe radiative Fermi model is provided in the first part of Appendix A. The purpose of the section isto provide estimates of the various quantities that help to provide an intuition about these systems.

Section 1.5 also estimates the validity of the fluid, MHD, and ideal MHD approximation for the

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6 fiducial systems. These estimates are as discussed in any plasma physics book, and summarizedin Appendix A. The purpose of the section is to show that the single-component ideal MHDapproximation is an excellent model1 for the 6 fiducial (and likely all) black hole accretion systems.

Section 1.6 summarizes the properties of GRBs, X-ray binaries, SgrA*, and AGN that suggest aGRMHD model is required to study the accretion disk near the black hole in these systems. Someopen questions are posed about the source of the luminosity and jets in these systems, which maybe resolved by a GRMHD model.

Section 1.7 reviews the main results of chapters 2-5 in this thesis.

1.2 Introduction to Black Hole Systems

Einstein (1916) formulated his general relativity (GR) theory, which has stationary solutions calledblack holes (for a discussion see, e.g., Shapiro and Teukolsky 1983, Chapt. 12). Einstein’s GR theorytoday remains the standard and most fundamental theory of gravity (see, e.g., Misner et al. 1973,and for a historical perspective see Thorne 1994). Schwarzschild (1916) derived an exact, static,and spherically symmetric solution to Einstein’s equations. A generalization of the Schwarzschildsolution for a stationary, charged2, rotating, space-time is the Kerr-Newman metric (see, e.g.,Shapiro and Teukolsky 1983, Chapt. 12, for a discussion). The Schwarzschild and Kerr-Newmansolutions to Einstein’s equations define the properties of stationary, classical black holes as studiedin this thesis. This thesis focuses on black holes as the object at the center of the accretion disk.

The first discussion in this section summarizes the plausible methods for forming a black holeas a remnant of the collapse of a massive gas cloud or a massive star, critical mass build-up ona degenerate star due to accretion, or the collision between two compact objects. The types ofastrophysical systems involved in such black hole formation scenarios include 1) gamma-ray bursts,which each likely contain a black hole with M ∼ 3M¯; 2) black hole X-ray binaries, which eachlikely contain a black hole with M ∼ 3 − 20M¯ ; and 3) galaxies, some of which contain one ormore super-massive black holes (SMBHs) with M ∼ 106 − 109 M¯, where M¯ = 1.989 × 1033 g isthe mass of the Sun.

The second discussion describes the properties of gamma-ray bursts, X-ray binaries, and activegalactic nuclei, and introduces 6 fiducial black hole accretion disk systems: 2 gamma-ray bursts, 1X-ray binary, a radio source at the center of our galaxy called SgrA*, and 2 active galactic nuclei.These 6 fiducial systems are used as examples of black hole accretion systems for several calculations,such as those in Section 1.5. Table 1.1 summarizes the estimated distance to, black hole mass of,and bolometric luminosity of the 6 fiducial objects described in this section. Throughout this thesis,similar tables are presented that cumulate additional quantities as relevant to the discussion in eachsection. A summary description of these objects/events is given below.

1This is shown to be true everywhere except in current sheets.2Charged black holes are likely not astrophysically relevant. Any buildup of charge is quickly neutralized by

accretion and pair production.

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Table 1.1. Fiducial Black Hole Accretion Systems 1

Parameter NS-BH GRB 030329 LMC X-3 SgrA* NGC4258 M87

dL ? 804Mpc 55 kpc 8 kpc 7 Mpc 18 MpcM [M¯] 3 3 10 2.6× 106 4× 107 3× 109

Lbol[erg/s] 1053 3× 1052 1038 1037 3× 1043 2.3× 1042

Note. — dL is the luminosity distance, M is the mass in solar mass units, and Lbol is thebolometric luminosity. These quantities are shown for the 5 fiducial black hole accretiondisk systems. See text for references.

Some of the objects discussed in this section have their distance from the Sun determined bythe observed redshift (z = λobs/λemit − 1, where λ is the photon wavelength) of spectral lines. Forthese objects, the distance is found by integrating the Friedman-Robertson-Walker (FRW) metric todetermine the comoving distance (dC) that gives a luminosity distance of dL = (1+z)dC . The powerlaw ΛCDM model and WMAP (and related) data give a Hubble constant h ∼ 0.71 km s−1 Mpc−1,Ωm ∼ 0.27, and ΩΛ ∼ 0.73 for a flat (k = 0) FRW metric model of the cosmos (Spergel et al.,2003).

1.2.1 Formation of Black Holes

Subrahmanyan Chandrasekhar found that if a white dwarf (an accepted object at the time) issufficiently massive, then the degenerate electrons will become highly relativistic and can drivethe star to collapse (Chandrasekhar, 1931b,a). He predicted that a white dwarf would undergocollapse if the mass reaches Mch ∼ 1.4M¯. The collapse of a white dwarf can lead to a neutronstar supported by neutron degeneracy pressure. Eddington (1935) realized that if Chandrasekharwas correct, then a star with a mass much larger than 1.4M¯ should collapse to a black hole3.

Modern estimates of the upper limit to the mass of a neutron star give M ∼ 1.8−2.2M¯ (Akmalet al., 1998). If one only requires GR and causality to hold, then the upper limit is M ∼ 3.4 M¯(Rhoades and Ruffini, 1974; Hartle, 1978), but see (Yuan et al., 2004; Abramowicz et al., 2002).Regardless of the equation of state, once an object is contained within a light trapping surface, GRtheory predicts that the result must be a black hole (Hawking and Ellis, 1973). A classical blackhole is a simple, conservative model, while alternatives invoke exotic neutron star physics (Bahcallet al., 1990) or modifications to GR in the strong-field limit (see, e.g., Babak and Grishchuk 2003;DeDeo and Psaltis 2003). This thesis considers a compact object with a mass greater than aboutM ∼ 3.4M¯ to be a black hole candidate (BHC).

3The term black hole was coined later by Wheeler in 1968.

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There are two likely ways of creating a “stellar mass black hole” with M ∼ 3−20M¯: 1) stellarcore-collapse; and 2) compact object collisions. Collapse of the iron core of a massive star can leadto a black hole or neutron star remnant and the generation of a supernova (for a review see Woosleyet al. 1993; Wheeler et al. 2000). Black hole formation may also occur during the collision of twoneutron stars (Eichler et al., 1989), or during other collisions involving a compact object (Pophamet al., 1999). Isolated solar-mass black holes have been indirectly detected by micro-lensing events(Bennett et al., 2002; Agol et al., 2002; Agol and Kamionkowski, 2002). Black holes are found withnormal (non-compact) stellar companions in binary systems (for a review see, e.g., Lewin et al.1995; McClintock and Remillard 2003), which are called black hole X-ray binaries due to theirpowerful X-ray emission. The black hole in the binary likely results from the collapse of a massivestar, while the other star remains intact.

The initial mass function (IMF) ξ(M) specifies the distribution in mass of newly formed stellarpopulations, and ξ(M) can be used to estimate the mass and number of black holes compared toother stars. From the observed dependence of bolometric luminosity on star mass, Salpeter (1955)suggested that ξ(M) ∝ M−(1+x), where x = 1.35 and the distribution extends from a lower boundof M1 ∼ 0.1 M¯ to an upper bound of M2 ∼ 125M¯ (see also Miller and Scalo 1979; Scalo 1986).The present-day stellar IMF is now thought to extend up to M ∼ 200M¯ (Larson, 2002) and studiesof population III (low-metallicity) stars suggest the IMF is probably top-heavy (Schneider et al.,2002). Stars with zero metallicity have little mass loss, so massive low-metallicity stars may collapseto massive black holes. Numerical simulations predict there should exist stars with M > 100M¯(Abel et al., 2000, 2002). Other studies suggest the first stars with M < 140M¯ should havemass loss like normal metallicity stars (Fryer, 1999). For stars with 140 M¯ < M < 260M¯, anelectron-positron pair instability during oxygen burning leaves no remnant. For M > 260M¯, thestar collapses directly to a black hole with a remnant mass about half the original stellar mass. ForM ∼ 105 M¯, no stable hydrogen burning occurs and the protostar collapses directly to a blackhole (Baumgarte and Shapiro, 1999; Shibata and Shapiro, 2002).

Cold dark matter (CDM) models suggest the first generation baryonic clouds have M ∼ 106 M¯(Bromm et al., 2002). Such clouds are suggested to fragment into stars. However, a small fractionmay suffer direct collapse to a black hole. If one thousandth of the initial cloud mass collapsesdirectly to a black hole, then there would exist intermediate-mass black holes (IMBHs) with massesM ∼ 102 − 104 M¯ (Balberg and Shapiro, 2002; Shapiro and Shibata, 2002). This is consistentwith the observed correlation between the mass of the central SMBH and the mass of the bulge ofa galaxy, which is MBH/Mbuldge ∼ 10−3 (Kormendy and Gebhardt, 2001). Hierarchical growth ofgalaxies through accretion of satellite galaxies leads to IMBHs in off-center positions, as confirmedby numerical simulations of dense star clusters (Portegies Zwart et al., 1999; Portegies Zwart andMcMillan, 2002; McCrady et al., 2003).

There are so-called ultraluminous X-ray sources (ULXs) that are observationally similar toblack hole X-ray binaries, but ULXs have a higher luminosity of L ∼ 1039 − 1041 erg/s. ROSAT,ASCA, and Chandra observations show that about 50% of spiral galaxies have ULXs that are

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more compact and variable than AGN (Fabbiano et al., 2003), off-center from the galactic center,and associated with star-forming regions. ULXs are also observed in globular clusters in ellipticalgalaxies (Kaaret et al., 2001). A ULX is found in, for example, the center of irregular galaxy M82(Cigar galaxy) (Matsumoto et al., 2001). ULXs could correspond to accreting IMBHs. Simpleluminosity constraints suggest M & 102 − 103 M¯, dynamical friction constraints imply M <

106 M¯, and the velocity cusps in globular clusters imply M ∼ 103 − 104 M¯ (Van Der Marel,2004). However, if the emission is beamed or time-dependent, these ULX sources could be stellar-mass black holes (King et al., 2001). ULXs may be explained as IMBHs confined to the diskof the host galaxy that accrete interstellar matter (Krolik, 2004), although see Rappaport et al.(2004). Whether ULXs are actually accreting IMBHs or stellar mass black holes is an active areaof research.

The formation of a SMBH is a likely result of star formation (Rees, 1984; Genzel et al., 1997).The formation of SMBHs could be due to one or more IMBHs that fall to the galaxy center dueto dynamical friction (Haiman and Loeb, 2001). A scenario for creating a SMBH involving thecollision of multiple IMBHs may be unable to account for all SMBHs today (Islam et al., 2003).This scenario would result in little black hole spin to power radio jets (Hughes and Blandford,2003; Gammie et al., 2004), so if the scenario is accurate then black hole spin would not powerjets. A single IMBH seed could be driven to a supermassive mass by accretion of gas. Thisaccretion scenario generates the observed correlations (Kormendy and Gebhardt, 2001) betweenblack hole mass and bulge mass or bulge velocity dispersion (Van Der Marel, 2004). However,recent evidence of a SMBH binary in radio galaxy 3C66B is consistent with the hierarchical growthof galaxies (Sudou et al., 2003). Thus, both hierarchical (merger dominated) and anti-hierarchical(gas accretion dominated) scenarios may have occurred.

The preceding discussion summarized how black holes form with different masses, ranging fromstellar mass black holes with M ∼ 3 − 20M¯ to SMBHs with M ∼ 106 − 109 M¯, and somebasic observational evidence was given for the existence of these black holes. The following dis-cussion summarizes the evidence for the existence of black holes and accretion disks in specificevents/objects. The purpose of the following discussion is to suggest that some gamma-ray bursts,some X-ray binaries, the centers of some normal galaxies, active galactic nuclei, and quasars arelikely powered by an accreting black hole.

1.2.2 Gamma-Ray Bursts

Gamma-ray bursts (GRBs) are brief bursts of X-rays or gamma-rays that are observed a coupletimes a day, last from seconds to hours, appear randomly distributed in the sky, and do notrepeat (for a review see Piran 1999, 2004). The Vela satellites detected the first GRB in 1967.Not until 1973 were reports of GRBs declassified and published (Klebesadel et al., 1973). In1990, the Burst and Transient Source Experiment (BATSE), onboard the Compton Gamma-RayObservatory, localized several hundreds of bursts. Their random distribution on the sky suggested a

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cosmological origin (Paczynski, 1986; Meegan et al., 1992). However, not until 1997 did BeppoSAXdetect GRB970508 with an optical counterpart (afterglow) with emission lines showing a redshift ofz = 0.835 (dL = 5300Mpc) (Metzger et al., 1997; Frail et al., 1997), demonstrating a cosmologicalorigin.

Approximately two GRB detections occur per day. Possibly related events called X-ray flashes(XRFs) are detected several times a year and exhibit no gamma-ray emission (Heise et al., 2001;Yamazaki et al., 2002). GRB durations form a bimodal distribution of short-duration GRBs lastingan average of δT ∼ 0.3 s and of long-duration GRBs lasting on average of δT ∼ 35 s (Kouveliotouet al., 1993). The shortest GRB currently ever observed lasted for δT ∼ 6ms (Bhat et al., 1992) andhad 200µ s temporal structure, while the longest GRB (actually an XRF) currently ever observedlasted for 2550 s (0.7 hr) (see review by Zand et al. 2003). Short-duration GRBs are observed to havea harder spectrum compared to long-duration GRBs (Paciesas et al., 1999). Some long-durationGRBs have an associated X-ray to radio afterglow emission that contains spectral lines that allowa redshift measurement, which indicates these GRBs occur at cosmological distances. The typicalredshift of a GRB is z ≈ 1 (dL = 6600Mpc) (Meszaros, 2001) and has been measured up to z ∼ 4.5(dL = 43000 Mpc) (Andersen et al., 2000). No afterglow, and hence redshift, has been observed forshort-duration GRBs.

The GRB emission is likely relativistically beamed, but the afterglow redshift can be used toestimate the distance and isotropic rest-frame energy Eiso(γ) = 4πFγd2

L(1 + z)−1 and isotropicrest-frame luminosity Liso(γ) = Eiso(1 + z)/δT = 4πFγd2

L, where Fγ is the observed fluence ofgamma-rays (see, e.g., Frail et al. 2001). For example, GRB 990510 had a 50 − 300 keV fluenceof Fγ = 3 × 10−5 erg/ cm2 over a period of δT ∼ 33 s (Kippen, 1999) and an associated opticalafterglow with FeII and MgII absorption lines with redshift z = 1.619 (Vreeswijk et al., 1999). Thisgives dL ∼ 7600Mpc. Assuming an isotropic emission, the total energy is Eiso(γ) ∼ 5 × 1052 ergand total luminosity is Liso(γ) ∼ 4.2× 1051 erg/s. “Cosmic GRBs” at redshift z & 0.5 are possiblystandard candles with an actual total energy production of about Eγ ∼ 1051 erg (Sari et al., 1999;Frail et al., 2001; Panaitescu and Kumar, 2001; Bloom et al., 2003), although there are significantoutliers with Eγ ∼ 1048 − 1050 erg (Sazonov et al., 2004; Soderberg et al., 2004).

In most GRB models, the GRB is suspected to be produced from a relativistic jet that wasemitted, by some mechanism involving accretion, near the polar axis of a black hole accretiondisk system. The “collapsar” model scenario, used to explain long-duration GRBs, entails thecollapse of a massive star and formation of an accretion disk around a newly-formed black hole ofmass M ∼ 3M¯ (Woosley, 1993; Paczynski, 1998; MacFadyen and Woosley, 1999). Short-durationGRBs may be generated by the collision of a neutron star and black hole, which subsequently formsan accretion disk around the rotating black hole (Narayan et al., 1991, 1992).

This introduction later estimates the properties of a disk that could form during a neutronstar - black hole (NS-BH) collision and during the collapse of a massive star, such as what likelyoccurred in GRB 030329.

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NS-BH Collision Model

There are no afterglows, and so no redshift measurements, associated with short-duration GRBs.Thus, these events have no estimated isotropic luminosity that could provide a clue about the natureof the progenitor. One model for short-duration GRBs is a NS-BH collision, which is estimated tohave an isotropic luminosity of about 1053 erg/s and last less than a second (Narayan et al., 1992;Popham et al., 1999).

GRB 030329

The prompt gamma-ray emission from GRB 030329 lasted for about δT = 25 s (Vanderspek et al.,2003), and within hours was followed by an afterglow observed from X-ray, UV, optical, infrared,and radio emissions. Observations of a supernova light curve (SN2003dh) in the afterglow ofGRB 030329 confirmed that the collapse of a massive star is spatially and temporally connectedto the GRB (Stanek et al., 2003; Kawabata et al., 2003; Uemura et al., 2003; Meszaros, 2003;Hjorth et al., 2003). Emission lines in the afterglow have a redshift of z = 0.169 (dL = 804 Mpc)(Greiner et al., 2003b). The GRB fluence within 30 − 400 keV was Fγ ∼ 10−4 erg cm−2, whichgives Eiso(γ) ∼ 6 × 1051 erg and Liso(γ) ∼ 3 × 1050 erg/s. The likely total beamed energy isEγ ∼ 2 × 1049 erg in a beam with angular width ∼ 3.5 (see, e.g., Greiner et al. 2003a), which isan outlier in a GRB Eγ histogram.

1.2.3 Black Hole X-ray Binaries

Observations indicate that 50-80% of stars are in multiple star systems, with most of these beingbinary stellar systems (for a review see Evans 1999; Tohline 2002). About half of all stars arein binary systems, and some contain an accreting compact object (likely a neutron star or blackhole) and have strong emissions of X-rays that likely originates from an accretion disk (for a reviewsee, e.g., Lewin et al. 1995; McClintock and Remillard 2003). X-rays are emitted mostly fromthe inner region of the accretion disk near the collapsed star (see, e.g., Shapiro and Teukolsky1983, Chapt. 13 for discussion). X-ray binaries have a persistent X-ray luminosity in the rangeof L ∼ 1032 − 1038 erg/s, where the companion star has a bolometric luminosity in the range of1032 − 1038 erg/s.

An X-ray binary is a true binary system, which is resolved as a binary by observing the motionof the normal star due to the gravitational influence of the compact object. The motion is detectedby 1) observing the wobble motion of the normal star on the celestial sphere ; 2) spectroscopicobservations of the light-of-sight velocity ; or 3) the eclipsing of one object by another. The massof the compact object in X-ray binaries is often determined by the mass function

f(Mc,Mn, i) =Mc sin3 i

(1 + Mn/Mc)2=

Porbv3n

2πG, (1.1)

where Mc is the mass of the compact object, Mn is the mass of the normal star, i is the inclination

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of the orbit with respect to the line of sight, Porb is the orbital period of the binary, and vn is theline of sight velocity. A lower limit to Mc can be found by setting Mn = 0 and i = π/2, whileoften other secondary measurements allow higher lower limits on the mass of the compact object.The distribution of the compact object’s estimated mass in X-ray binaries shows a clear bimodaldistribution with about 22 objects with mass 1.1 − 2M¯ and a distribution of about 18 objectswith masses from 3− 16M¯ with a mean and peak of 10M¯ (see, e.g., Zand et al. 2004; Postnovand Cherepashchuk 2004, figure 1, and see McClintock and Remillard 2003).

Classes of X-ray Binaries

There are approximately 240 known X-ray binaries that are classified according to their supposedunderlying physics (Zand et al. 2004 and references therein). These classifications include high-massX-ray binaries (HMXBs), low-mass X-ray binaries (LMXBs), Be X-ray binaries, X-ray bursters,X-ray pulsars, soft X-ray transients (SXTs), and microquasars. A neutron star is the compactobject in some HMXBs, some LMXBs, all Be X-ray binaries, all X-ray bursters, all X-ray pulsars,some SXTs, and some microquasars. A black hole is the compact object in some HMXBs, someLMXBs, some SXTs, and some microquasars. A single object can be in multiple classes.

X-ray binaries are likely powered by the accretion of matter from the normal star onto thecompact star. The formation of the accretion disk can be described by two limiting scenarios asso-ciated with either HMXBs or LMXBs. HMXBs are X-ray binaries with a massive M ∼ 10− 30M¯stellar companion, such as a Be (B-type with prominent emission lines) star or blue supergiant. Anaccretion disk likely forms from the companion’s stellar wind material. LMXBs are X-ray binarieswith a low-mass M . 5M¯ stellar companion, which is typically a main sequence solar mass star.In some cases the companion star is degenerate or evolved (subgiant or red giant). An accretiondisk forms as the companion overfills its Roche lobe (Lagrange gravitational equipotential surfacebetween the compact and normal star) and matter plunges onto the compact object. For a cat-alogue of LMXBs see Liu et al. (2001). The spatial distribution in our galaxy of X-ray binariesshows about 90 LMXBs with a slight concentration in the center, while HMXBs are more evenlydistributed (Grimm et al., 2002).

SXTs are LMXBs that are only discovered after undergoing a so-called accretion outburst(also called an X-ray nova) (Tanaka and Shibazaki, 1996). Typical SXTs are usually faint orunobservable in X-rays during a “quiescent” state. Typical SXTs contain a K-type subgiant ordwarf that is transferring mass to a black hole through an accretion disk. It is thought that during“quiescence,” mass is accumulating in an accretion disk. When an outburst occurs, most of diskfalls into the compact object. The accretion outburst likely occurs due to some type of temperaturedependent disk instability. A similar mechanism operates in dwarf novae (Cataclysmic Variables(CVs)) (Warner, 1995).

Galactic microquasars (Mirabel et al., 1992) are X-ray binaries that are probably powered bya black hole accretion disk system or an exotic mechanism for energy emission from a neutronstar. Galactic microquasars are named for their strikingly similar features to quasars or AGN

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(Mirabel and Rodrıguez, 1994, 1999; Fender and Belloni, 2004), such as apparently superluminal(v/c ∼ 3 − 10) jets, which plausibly originate from a relativistic system with a well-defined axisof rotation. Microquasars have a higher X-ray luminosity than other X-ray binaries. Unlike otherX-ray binaries, microquasars might also be sources of gamma-rays (Romero, 2004). AGN andmicroquasars are likely more than simply morphologically similar (Rees, 1998), since they mayboth contain an accreting rotating black hole.

This thesis later estimates the properties of the disk around the black hole in LMC X-3 describedbelow. A description of the classic galactic microquasar GRS1915+105 and the classic black holeX-ray binary Cygnus X-1 is also given below.

LMC X-3

For a review of LMC X-3 see Cowley (1992). Large Magellanic Cloud (LMC) X-3 is in the LMCabout 55 kpc from the Sun. LMC X-3 likely contains a B5 subgiant with mass 4−8M¯ in a 1.7dayorbit around a compact object, so LMC X-3 is often classified as a HMXB. The normal companionstar’s shape is severely distorted by the compact object, which is likely a ∼ 10M¯ mass black hole(Cowley et al., 1983). The X-ray luminosity of LMC X-3 is Lx ∼ 1038 erg/s (White and Marshall,1984; Treves et al., 1988). Despite being classified as a HMXB, recent XMM-Newton observationssuggest that the disk is likely formed by Roche-lobe overflow (Soria et al., 2001). As with manyX-ray binaries, the observed disk flux of hard or soft X-rays shows periodic behavior (Cowley et al.,1991; Ebisawa et al., 1993) and transitions between hard and soft emission (Wilms et al., 2001a).

GRS1915+105

A classic galactic microquasar is GRS1915+105 (V1487 Aql), which is about 12.5 kpc away fromthe Sun. The companion star is a late-type giant with M ∼ 0.8 ± 0.5M¯, so is also classifiedas a LMXB (Greiner et al., 2001b; Harlaftis and Greiner, 2004). GRS 1915+105 is the mostluminous of all known X-ray binaries (Done et al., 2004), with a highly variably X-ray output of∼ 1038−1040 erg/s and shows emission from apparently superluminal jets (Mirabel and Rodrıguez,1994, 1999; Fender and Belloni, 2004). Radio synchrotron emission from the jet shows plasma blobsmoving at apparently superluminal speeds with an estimated true speed of v/c ∼ 0.92. The blackhole mass is suspected to be about 14M¯ (Greiner et al., 2001a).

Cygnus X-1

Cygnus X-1 (Cyg X-1) is a source about 2.6 kpc away in the Cygnus constellation. The opticalcounterpart is a supergiant called HDE 226868 which is an O9-B0 supergiant with a surface tem-perature of 3.1× 104 K and a mass of 20− 30M¯, so Cygnus X-1 is classified as a HMXB. Cyg X-1is the first object that was broadly agreed to require the existence of a black hole. The compactobject is believed to be a black hole since it has a mass of 7− 13M¯ (Webster and Murdin, 1972;Bolton, 1972; Gies and Bolton, 1986; Herrero et al., 1995).

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1.2.4 Normal and Active Galactic Nuclei and Quasars

All the stars from a normal (non-active) galaxy together have a typical luminosity of 1044 erg/s.Normal galaxies derive their luminosity from an ensemble of stars and so are, to zeroth order,related to a black body spectrum with a small range of strong emission. An active galaxy is agalaxy from which a significant fraction of the energy output is emitted by the nucleus ratherthan by the stars, dust, and interstellar gas. The luminosity of such nuclei is typically between1042 − 1048 erg/s. Some nuclei outshine their host galaxy by 4 orders of magnitude. The nucleiin such active galaxies are called active galactic nuclei (AGN) (see review by Krolik 1999a). Thedistance from the Sun to a typical active galaxy is less than a Gpc (z < 0.2), which is close enoughto resolve the stellar distribution and sometimes the individual stars. AGN come in many differenttypes, which are largely a result of viewing angle (for a discussion of AGN unification models see,e.g. Krolik 1999a).

Unlike normal galaxies, most AGN emit in a broad-band spectrum. For example, NGC 4151is observed to have a flat spectrum from the mid-infrared (1013Hz, 0.04 eV) to the hardest X-raysobserved (1019Hz, 41 keV). The active galaxy Markarian 421, known as a blazar, shows dramaticvariability with flares reaching into the TeV range (Punch et al., 1992). AGN are observed tohave spectra with emission lines such as the Lyα, Balmer lines, and the X-ray Fe Kα line near6.4 keV. These lines are Doppler-broadened from relatively broad lines (several thousand km s−1)to relatively narrow lines (few hundred km s−1). For example, MCG-6-30-15, known as a SeyfertI, exhibits Fe Kα line fluorescence, whose broad, blue-shifted profile likely indicates emission fromdeep within the gravitational potential of a supermassive black hole (Tanaka et al., 1995). SeeKrolik (1999a) for a discussion of AGN spectra features and their likely origins.

Within AGN, there are many intermediate absorbers and emitters between our line of sightand the accretion disk that significantly affect the observations. For example, the broad-line andnarrow-line regions in AGN are an interesting study alone (see, e.g., Krolik 1999a; Elvis 2000, andreferences therein).

Some of the first observations of AGN were of so-called radio galaxies with jets emerging froma point-like source. A typical jet has lobes that extend a few hundred kiloparsecs, a distance thatis comparable to or larger than that of the host galaxy size (see, e.g., the classic review by Bridleand Perley 1984). AGN jets are observed to be relativistic with some having variable structuremoving at an apparent speed of v/c ∼ 3− 10 (see, e.g., West et al. 1998). AGN jets are sources ofsynchrotron radiation (Baade, 1956), indicating the existence of a strong magnetic field. The radiosynchrotron radiation is likely due to internal shocks that produce high-energy electrons in the jetmagnetic field. Recent results also find that X-rays can be produced as the AGN jet collides withexternal matter and produces knots within the jet (Hardcastle et al., 2003; Kraft et al., 2003). Thesestructures are often tracked to measure the apparent velocity, which is typically superluminal. Jetmotion in quasar 3C279 has been followed over a period of several years (Wehrle et al., 2001), andthe jet has apparent velocities of 4.8 − 7.5c. The quasar 3C273 has an apparently superluminal

12

v/c ∼ 10. HST images of the jet in M87 show well-defined knotty structures (Bridle et al., 1994),which move at an apparently superluminal v/c ∼ 6.

AGN are likely powered by accreting SMBHs of mass M ∼ 106 − 109 M¯, as estimated from1) the AGN’s compactness and luminosity ; 2) kinematic models of stellar motion (rotation profileand velocity dispersion) near nuclei ; 3) observations of Doppler shifts or maser emission from disk(see review by Kormendy and Richstone 1995; Richstone et al. 1998) ; 4) reverberation mapping(Blandford and McKee, 1982; Peterson, 1993; Peterson et al., 2004) ; and 5) relativistic effects forsome AGN (Fabian and Vaughan, 2003). Using Hubble Space Telescope (HST) photometry andground-based observations of kinematics, Magorrian et al. (1998) apply dynamical models to 36nearby galaxies. They find that 97% of early-type galaxies have SMBHs, and this suggests SMBHscould be common in (active) galactic nuclei. Similar results are found by van der Marel (1999);Tremaine et al. (2002).

The galactic material near the nucleus provides a source of matter to form an accretion disk.Observations of NGC 4261 show a radio source (3C 720) and jet that likely originate from nearbya black hole, which is estimated to have M ∼ 5×108 M¯ (Ferrarese et al., 1996). Around the radiosource is a dusty torus with a radius 240 pc (∼ 106GM/c2) and mass of Mdisk ∼ 5× 105 M¯. Themass is determined by assuming NGC 4261 has the same ratio of surface density of hydrogen tocolor excess E(B-V) as our Galaxy, as provided by Bohlin et al. (1978). The torus mass distributionis perpendicular to the orientation of the jet (Jaffe et al., 1993), which suggests the disk is feedingthe black hole and producing the jet. The centers of the galaxy, nucleus, and disk do not coincide.This offset between the disk and galaxy suggests that the torus may have formed from an interactingdwarf galaxy, but it must have been captured > 107 years ago (Ferrarese et al., 1996). The offsetbetween the nucleus and galaxy could be accounted for by recoil from a time-dependent jet, assuggested by Shklovski (1982); Rudnick and Edgar (1984). Despite recent advances in observationaltechnology, only instruments currently in the concept phase will have sufficient angular resolutionto spatially resolve the inner (R . 100GM/c2) accretion disk in AGN (Rees, 2001). Therefore,there remain fundamental questions that can only be answered by folding observations throughmodels of AGN disk structure.

A quasi-stellar object (QSO, or quasar) is a distant unresolved luminous object, which is likelythe nucleus of an active galaxy with an unresolved distribution of stars or protostellar gas. Theredshift of quasars ranges from z = 0.06−6.4 (dL = 265Mpc−64000 Mpc), where most quasars areat z ∼ 1.6 (dL = 12000 Mpc) (see Spinard review article in Mason 2004 and see the Sloan DigitalSky Survey (SDSS) derived quasar-redshift histogram in Oguri et al. 2004). The luminosity ofquasars and high-luminosity AGN are comparable. Estimates for the luminosity and compactnessof quasars suggests they are likely powered by accretion onto a SMBH with M ∼ 106 − 109 M¯(Zeldovich, 1964; Salpeter, 1964). High redshift quasars are likely newly forming galaxies, whereall that can be seen is the bright nucleus. Quasars likely evolve into AGN (Lynden-Bell, 1969).For the purposes of this thesis, it is sufficient for AGN and quasars to be referred to collectively asAGN.

13

This thesis later focuses on estimating disk properties within the nucleus of three galaxies thatcontain a SMBH: 1) the normal galaxy SgrA* ; 2) the active galaxy NCG4258 ; and 3) the activegalaxy M87.

SgrA*

The Milky Way is a spiral galaxy about 31 kpc across and has a total luminosity of about 1042 erg/s(from about 400 billion stars) (Mihalas and Binney, 1981). The Galactic nucleus contains a compactvariable radio source called Sgr A* that has a bolometric luminosity of 4× 1037 erg/s (for reviewssee Genzel and Townes 1987; Blitz et al. 1993; Morris and Serabyn 1996; Melia and Falcke 2001)and is about 8 kpc away from the Sun (Reid, 1993). SgrA* may be an X-ray source (Baganoff et al.,2003) and a source for high-energy (∼TeV) gamma-rays (see, e.g., Aharonian and Neronov 2004).As similarly observed in the nucleus of NGC 4261, within 3 pc of SgrA* there is a circumnucleardisk, with T ∼ 7000K, ρ ∼ 107 cm−3, and M ∼ 3 × 105 M¯ (Shukla et al., 2004), where T is thetemperature and ρ is the rest-mass density. This disk is the likely source of matter for the inneraccretion disk.

The strongest case for a SMBH is in our own galaxy. Infra-red imaging of stellar orbits (e.g.S2 with pericenter of 6× 10−4 pc, 124AU, 2100GM/c2) in the central region of our galaxy suggesta dark object lies near the center with M ∼ 2.6 × 106 M¯ (Eckart and Genzel, 1997; Ghez et al.,1998; Schodel et al., 2002). A study of the accelerations and velocities of stars in the central regionshows the central mass must be extremely compact (Ghez et al., 2000; Eckart et al., 2002). Theonly known object that could remain stable for a sufficient amount of time is a SMBH. Alternatives,such as a compact cluster of neutron stars, will collapse on short times scales compared to the ageof a nucleus (Genzel et al., 1997; Maoz, 1998).

NGC4258

The spiral galaxy NGC4258 (M106) is about 7 Mpc (Greenhill et al., 1995b; Herrnstein et al.,1999) away in the constellation Canes Venatici, has a bulge (galaxy without spiral disk) bolometricluminosity of 5 × 1042 erg/s, has a nucleus with a 2 − 10 keV X-ray luminosity of 4 × 1040 erg/s(Makishima et al., 1994), and has an estimated nuclear bolometric luminosity of 3 × 1043 erg/s(see, e.g., Wilkes et al. 1995; Herrnstein et al. 1998; Yuan et al. 2002; Woo and Urry 2002). VLBIobservations of the nucleus show water maser emission from a rotating disk between 0.13 pc−0.26 pc(Greenhill et al., 1995a). The acceleration of the maser spots or the proper motion of the spotsallows a geometric distance measurement (Greenhill et al., 1995a; Herrnstein et al., 1999). Thekinematics of the rotation suggest the nucleus likely harbors a 3.9×107 M¯ mass black hole (Watsonand Wallin, 1994; Miyoshi et al., 1995).

14

M87

The elliptical galaxy M87 (NGC4486,3C 274) is about 30 kpc across, 18 Mpc away in the Virgocluster (Whitmore et al., 1995), has a total luminosity of 2×1044 erg/s, and has a nuclear bolometricluminosity 2.3×1042 erg/s (Ho, 1999). Within 18 pc of the central region of the nucleus is an ionized(T ∼ 104 K) accretion disk whose velocity profile suggests it harbors a 3× 109 M¯ mass black hole(Sargent et al., 1978; Young et al., 1978; Ford et al., 1994; Harms et al., 1994; Macchetto et al.,1997). M87 also shows a well-collimated jet at large distances (Biretta et al., 1991, 1999) and abroad jet near an accretion disk (Junor et al., 1999) with “jet formation” at about .015 pc (3100AU,100GM/c2) from the black hole. HST images of the jet in M87 show a well-defined knotty structure(Bridle et al., 1994), which moves at an apparent speed of v/c ∼ 6.

1.3 Basic Accretion Disk Theory

The previous section introduced GRBs, X-ray binaries, and AGN, and introduced 6 fiducial objectsthat are a reference point for some following discussions. As was discussed, many of these systemslikely harbor a black hole. A black hole by itself possesses no astrophysically significant means togenerate energy. However, a black hole coupled with an accretion disk is possibly the most efficientproducer of large amounts of energy in the universe. Some GRBs, some X-ray binaries, and allAGN are likely powered by gas accretion onto a black hole.

Through dissipative processes, gas tends to bind gravitationally to a compact object. As for aspiral galaxy, the rings of Saturn, protoplanetary/protostar disks, and similar objects, this mattereventually forms a disk if the energy of the matter is lost faster than its angular momentum. In thecase of a highly ionized plasma, the angular momentum is then transported outward by magneticinstabilities and matter is accreted onto the compact object (for a review see Balbus and Hawley1998). For typical astrophysical systems, the disk is centrifugally supported and rotates many timesbefore finally plunging into (or onto) the compact object. During this process a significant amountof energy can be released per baryon rest-mass. The basic theory for how this works is describedin this section.

Below is a description of some basic theoretical concepts used to estimate the mass of thecompact object and the luminosity of the accretion disk. This basic theory is then applied to arguethat most AGN likely harbor SMBHs, GRBs likely originate from stellar objects rather than anAGN or quasars, and many X-ray binaries likely each harbor a black hole.

1.3.1 Accretion Luminosity and Mass of the Compact Object

A black hole is the most compact object known to exist, and this allows black hole accretiondisk systems to release large amounts of gravitational binding energy per baryon accreted. Thecompactness of a star can be estimated as C = GM/Rc2, where R is the surface radius, M isthe mass of the black hole, and G is Newton’s constant. The nonrelativistic gravitational binding

15

energy is proportional to Epot ∼ GMm/R, where m is the baryon mass. The nonrelativisticefficiency (energy released per baryon rest-mass energy) can be estimated as η ∼ Epot/(mc2) = C,which is largest for a black hole at η ∼ 0.5−1 for R at the horizon. A general relativistic calculationdefines this efficiency more precisely.

Nominal Accretion Luminosity

A reasonable estimate for an upper limit to the luminosity of an accretion disk can be found for adisk that is always in Keplerian motion. The matter is assumed to accrete in a tightly wound spiralaround the compact object. The accreting matter slowly drifts inward due to angular momentumexchange between successive rings of matter. The gravitational binding energy is assumed to bereleased as radiation. The final inner radial ring is at the innermost stable circular orbit (ISCO)for a black hole and the surface for other stars (Bardeen, 1970). For the case of a nonrotating blackhole surrounded by a thin disk terminating at the ISCO, the energy per baryon released is η ∼ 6%,while for a maximally rotating black hole η ∼ 42%. Some disk models suggest η . 10−4 near theblack hole (see, e.g., Narayan and Yi 1995), while others suggest η & 1 (see, e.g., Gammie 1999).These efficiencies set an upper limit Lacc, referred to as the nominal accretion luminosity, to theactual luminosity L, with

L . Lacc ≡ ηM0c2, (1.2)

where M0 is the accretion rate of rest-mass. For a star with surface radius R, all the remainingkinetic energy is released as radiation at the surface, where L = 1/2M0v

2ff = GM0M/R is released

by a rest-mass with free fall velocity vff and accretion rate M0 at the surface. The surface efficiencyis thus η = L/M0c

2 = GM/Rc2 = C, which is η ∼ 0.01% for a white dwarf and η ∼ 10% for aneutron star. These efficiencies can be compared to η ∼ 0.7% for hydrogen burning in stars.

Eddington Luminosity and Minimum Mass

Another upper bound to the luminosity of an accretion disk can be estimated by consideringspherical accretion onto a compact object (Bondi, 1952). Assume that the gas flow is composed ofmostly ionized hydrogen, spherical, in steady-state, nonrelativistic, and optically thin (τγ ¿ 1) tophotons. If the gas has a thickness H, then the optical depth for a uniform mass density is

τγ ∼ κρ0H, (1.3)

where κ is the opacity for the appropriate density and temperature regime (Bell and Lin, 1994).For a typical temperature and density of the accretion flow, electron scattering dominates andthen κ ∼ σT Ye/mp, where Ye is the number of electrons per baryon and σT = 8π

3 (e2/mec2)2 ∼

6.65 × 10−25 cm2 is the Thomson scattering cross section. For such an accreting gas, the radialforce balance between radiative scattering of electrons and the gravitational force on protons givesan estimate for the maximum luminosity of the system and the minimum mass of the gravitating

16

object.First, the force of radiation on the gas is found. The radiative energy flux is F = L/(4πr2),

where L is the luminosity of a radially-directed photons from a compact region. From a photonmomentum of p = E/c, the radiative momentum flux is then Prad = L/4πr2c. This is the pressureexerted by photons on a completely absorbing surface. The radiation force on a gas depends onthe opacity, where the outward force on a single free electron is Frad = σT Prad = LσT /4πr2c for agas of completely ionized hydrogen. Second, the inward force of nonrelativistic gravity on the gasis Fgrav = GM(mp +me)/r2 ≈ GMmp/r2. The protons and electrons are assumed to maintain theneutrality of the plasma by electrostatic coupling.

In this spherical, optically thin, nonrelativistic approximation, the force balance between radi-ation and gravity results in a maximum luminosity, called the Eddington luminosity LE , where

L . LE ≡ 4πGMc

κ= 1.3× 1038(M/ M¯) erg/s = 3.2× 104(M/ M¯) L¯, (1.4)

where L¯ = 3.89 × 1033 erg/s is the luminosity of the Sun. A system with mass M with L > LE

would stop accreting due to outward radiative forces. Setting equation 1.2 equal to equation 1.4gives the Eddington rest-mass accretion rate of

M0,E ≡ LE/ηc2. (1.5)

The Eddington argument can be inverted to give the minimum mass (Eddington mass)

M & ME = 3× 10−5(L/L¯)M¯ ∼= (L/L39)(10M¯) (1.6)

for the gravitating body in a system that radiates at the Eddington luminosity, where L39 ≡1039 erg/s. So an object with L = L39 must have a mass M > 10 M¯ to sustain such a luminosity.

The estimate for maximum luminosity LE and minimum mass ME for a system undergoingsteady spherical uniform accretion of an optically thin medium works well to explain the upperlimit to several non-spherical accretion systems (see, e.g., Margon and Ostriker 1973). However,there are magnetized super-Eddington atmospheres that are unstable, but that are quasi-steadyin a time-averaged (statistical) sense (Begelman, 2001). Also, the Eddington luminosity does notgenerally apply to all optically thick mediums. As shown later, the disk in GRB systems is veryoptically thick to photons, so the Eddington luminosity is not expected to be an upper limit. Theluminosity is so high in GRB systems that if the disk were optically thin, then outward radiativeforces would essentially instantaneously halt accretion. Table 1.2 includes the data from Table 1.1with the addition of the ratio of luminosity to Eddington luminosity for the 6 fiducial objects.

Time Variability: Maximum Size and Reverberation Mapping

The maximum size of the emitting region can be estimated from the fluctuations in the frequency in-tegrated luminosity with time scale δT . Assume that the variations in the luminosity of background

17

Table 1.2. Fiducial Black Hole Accretion Systems 2

Parameter NS-BH GRB 030329 LMC X-3 SgrA* NGC4258 M87

dL ? 804 Mpc 55 kpc 8 kpc 7 Mpc 18MpcM [M¯] 3 3 10 2.6× 106 4× 107 3× 109

Lbol[erg/s] 1053 3× 1052 1038 1037 3× 1043 2.3× 1042

Lbol/LE 3× 1014 8× 1013 0.08 3× 10−8 0.006 6× 10−6

objects can be accounted for, and that the variations are from the system of interest. Suppose thatthe entire emitting region is contained within a single point of observation, then fluctuations in theluminosity from causally connected regions must be within R . cδT . There could be historicallyinevitable or coincidental fluctuations over larger scales, but R is the largest causally connectedemitting size due to the limited speed of light.

More information can be gathered from the time-variability of spectral features than the vari-ability of the frequency integrated luminosity. Close to AGN there are many intermediate opticallythick emitting and absorbing clouds (see, e.g., Krolik 1999a; Elvis 2000, and references therein).One can time the emission response of so-called broad-line regions (BLRs) in comparison to thecontinuum emission. It was realized that the time delay between the emission-line variations andcontinuum are due to light travel-time effects within the BLR. The emission lines echo or “rever-berate” the changes in the continuum from the accretion disk. This delay can be used to constrainthe proper motion and size of the BLRs, and then the BLR kinematics can be used to estimate themass of the black hole (Blandford and McKee, 1982; Peterson, 1993; Peterson et al., 2004).

Fe Kα line Profile

An accretion disk is sometimes modeled as having a hot corona (see, e.g., Ostriker 1976), which issuspected to be generated similarly as the Sun’s. In such corona models, the disk is often treated asa cold slab of material. The corona of an accretion disk can generate X-rays, which either Comptonscatter free electrons in the accretion disk or get photoelectrically absorbed by a neutral atom.Photons above the threshold energy for transition cause the ejection of an electron from a neutralatom in the disk. The largest cross section for cosmic abundances is associated with the excitationof a K-shell (n=1) electron into the L-shell (n=2). The de-excitation leads either to fluorescence ofa K-α photon or to the electron exciting another electron (autoionization) that carries the energyaway. The resulting spectrum is a combination of the fluorescent lines, that result from cosmicabundances, and the incident flux that is typically modeled as an X-ray power law (George andFabian, 1991). The fluorescence also includes the Compton reflection bump at higher energies(E ∼ 30 keV) due to reprocessing (Compton scattering or reflection) of the X-ray power law in theoptically thick slab. The observed intensity of this feature in AGN is consistent with the presence

18

of an accretion disk that subtends a solid angle of 2π (see, e.g., Nandra et al. 1991). The Kα linefrom iron has the largest fluorescent emission and has a rest-frame energy of E = 6.40 keV. AGNobservations show that this feature has a much wider profile than expected from simply absorptionalong the line of sight (Makishima, 1986). The true X-ray reflection processes in an accretion diskare typically more complicated than that described above (Ballantyne et al., 2001).

Often reported is the equivalent width (EE.W.) of the emission line. This is equal to the energyat which the cumulative flux at E < EE.W. in the continuum emission has the same energy as thatcontained within the line emission. When viewing the incident plus reflected spectrum, the Fe Kα

line has an equivalent width of E ∼ 180 eV. Other lines have much smaller equivalent widths (Mattet al., 1997).

In summary, in the corona model where the disk is a cold slab, the Fe Kα line emission isproduced by fluorescence when hard X-rays from a hot corona illuminate the cold, optically thickcomponent of an accretion disk (George and Fabian, 1991). However, this model has come under firesince observations show the X-ray continuum and Fe Kα line profile are not temporally correlatedin MCG-6-30-15 (see, e.g., Matsumoto et al. 2003). This suggests the basic model may be wrongor that a detailed model is required.

An important question is whether the properties of the disk and black hole can be measuredwithout directly resolving the accretion disk. Fe Kα line profiles are important since they mayallow a reconstruction of disk properties such as disk radii, the spin of a black hole, inclination, andemissivity (Chen and Halpern, 1989; Fabian et al., 1989; Laor, 1991; Pariev and Bromley, 1998).Observations of the Fe Kα line profile from black hole accretion system are discussed below.

1.3.2 Some Accretion-Based Arguments

The above basic theory shows that an accretion disk could be an efficient producer of large amountsof radiation. It is interesting to give rough arguments that invoke an accretion disk to explain theluminosity and some spectral features in astrophysical systems. The following discussion arguesthat most quasars likely have supermassive black holes, that some X-ray binaries likely each containa black hole, that GRBs are likely related to stars rather than AGN or quasars, that the extractionof rotational energy from a rotating black hole could lead to a large fraction of the luminosity fromGRBs, X-ray binaries, and AGN (or quasars), and that Fe Kα line emission from AGN and X-raybinaries probes the inner radial region of an accretion disk around a (likely rotating) black hole.

Soltan Argument for Rotating Super-Massive Black Holes in Quasars

Black hole accretion has long been considered the most likely power source of quasars (Zeldovich,1964; Salpeter, 1964). The ratio of quasar radiative energy density to supermassive black hole massdensity is ∼ 0.2 (Yu and Tremaine, 2002; Elvis et al., 2002). If the accretion proceeds as a thindisk (Bardeen, 1970) with radiative efficiency η(a) around a black hole with spin parameter a, thenη & 0.2. This suggests that the average black hole spin is a ∼ 0.96. Even an efficiency of η & 0.1

19

gives a & 0.67. This result also applies to nonradiative magnetized thick disk models, where thethick disk efficiency is found to be comparable to the thin disk efficiency (McKinney and Gammie,2004).

Assume that accretion proceeds with a radiative efficiency of ∼ 10%, where the black hole is thespent fuel from the accretion process. One can estimate the remnant black hole mass (Soltan, 1982;Rees, 1984; Cavaliere and Padovani, 1988) from the efficiency η ∼ 0.1, average quasar luminosityL ∼ 1045 erg/s, timescale of quasar activity (age of universe: T ≈ 4.3× 1017 sec), typical distanceto a quasar (redshift z ∼ 1.6 → dC ≈ 1.4×1028 cm), observed number of quasars per square degree(nQ ∼ 100 deg−2 → NQ ≈ 4×106), and average number density of L? galaxies (n? ∼ 2.4×106 Gpc−3,where Gpc = 109 pc) that are capable of harboring a SMBH (Krolik, 1999a). The average accretedmass per L? galaxy is then

M ∼(

NQ

n?

)(LT

ηc2(

43πd3

C

))∼ 107M¯. (1.7)

The left fraction is the number of quasars per galaxy for a given volume of space. The right fractionis the mass accreted per quasar per unit volume of space. The product is the average mass accretedinto the nucleus per galaxy. This suggests that the typical remnant object from accretion shouldbe a SMBH, and the mass is in basic agreement with observations. For a more recent version ofthis argument, see Merloni (2004).

Black Hole Mass of GRBs and AGN from Variability

As mentioned above, the shortest GRB currently ever observed lasted for 6ms (Bhat et al., 1992)with 200µ s temporal structure. This gives a maximum size of 60 km and a maximum black holemass of 40 M¯ for the emitting source, so likely this GRB is (and likely others are) related to acompact star rather than to AGN or quasars.

Intra-day variability is typically observed in all quasars and AGNs, while in MGC-6-30-15 the X-ray luminosity shows rapid variability on order of 100 s (Reynolds et al., 1995; Yaqoob et al., 1997).MCG-6-30-15 in an active galaxy in the constellation of Centaurus with a redshift of z = 0.0078(dL = 37Mpc). For MCG-6-30-15, R . cδT ∼ 170 astronomical units (AU). The compactnessgives M . 2 × 107 M¯ for a black hole. The luminosity of 4 × 1043 erg/s (Reynolds et al., 1997)is limited by the Eddington luminosity, and so M & 3.2 × 105 for a black hole. More advancedtime-series based arguments can be made that estimate the black hole mass to be M ∼ 106 M¯(Reynolds, 2000; Lee et al., 2000; Vaughan et al., 2003).

Within AGN there must exist an object capable of generating power that is comparable to oreven larger than that of an entire galaxy of stars, yet it must be compact enough to fit within∼ 100AU. The current standard model is that the central engine of AGN is an accreting SMBH(for an introductory review see, e.g., Armitage 2004).

20

Dimensional Argument for Blandford-Znajek Effect

The mechanism of jet production is an area of active research. One mechanism that may generate ajet is the extraction of rotational energy from a rotating compact star by the surrounding magneto-sphere or magnetized disk. Compact stars have the largest magnetic fields due to the approximateadvection of magnetic flux during collapse. Compact stars, such as black holes and neutron stars,can hold significant rotational energy that can be released through magnetic braking between thestar and the magnetized part of the accretion flow (such as a magnetosphere) (Goldreich and Ju-lian 1969; Blandford and Znajek 1977; Kim et al. 2004 and references therein). The process ofmagnetically extracting rotational energy from a rotating star is called the Blandford-Znajek (BZ)effect.

For a black hole, the power generated by the BZ-effect can be estimated dimensionally fromthe local energy density of the magnetic field (∼ B2), the volume of the black hole (∼ r3

+), and thelight crossing time (∼ r+/c), giving

LBZ ∼ B2r2+c ∼ 1045

(B

105G

)2 (M

107M¯

)2

erg/s ∼ LE

(B

105G

)2 (M

107M¯

)(1.8)

where B is the magnetic field strength near the black hole, r+ ≡ (GM/c2

) (1 +

√1− a2

)is the

radius of the event horizon, a ≡ J/M2 , and J is the angular momentum of the black hole. Thisdimensional estimate does not behave properly for arbitrary a, but it is accurate within an orderof magnitude for a & 0.5− 1. A more accurate estimate is obtained by invoking the black hole spinangular frequency ΩH = ac/2r+ (see MTW §33.4), which gives LBZ ∼ B2(GM/c2)4Ω2

H/c that hasthe correct a dependence (R. Krasnopolsky, private communication). The density scale can be setby assuming the accretion rate is at the Eddington rate M0,E (e.g., with efficiency, η = 10%) or byestimating the rest-mass accretion rate for a particular object from observations of the luminosityand independent estimates for the mass of the compact object. The magnetic field near the blackhole can then be estimated from MHD simulations or arguments for equipartition between the gasand magnetic pressure near the black hole (i.e. β = pg/(b2/2) ∼ 1), where the pressure is estimatedfrom some model of accretion. For a typical AGN or quasars, MHD simulations give B ∼ 105G andLBZ ∼ 1045 erg/s, which is comparable to the luminosity of jets and radio lobes in the strongest ofAGN. Renormalizing the BZ luminosity for GRBs and X-ray binaries shows that the jets in theseobjects may also be powered by the BZ-effect.

More advanced analytic estimates suggest that the BZ process is too inefficient to account for alarge fraction of power output except in a handful of AGN with low accretion rates and relatively lowefficiency (see, e.g. Ghosh and Abramowicz 1997; Armitage and Natarajan 1999; Livio et al. 1999).However, recently the BZ effect has been determined self-consistently using numerical models offorce-free models (Komissarov, 2001) and, as performed in this thesis, GRMHD models of disks(McKinney and Gammie, 2004). It is found that the BZ luminosity is indeed lower than nominalaccretion luminosity, but that the luminosity due to the BZ effect is focused in a collimated jet at

21

the axis of the black hole accretion disk system. Thus, the BZ effect remains a plausible source ofjet energy.

A similar total luminosity is estimated for electromagnetic disk winds (Blandford and Payne,1982; Krasnopolsky et al., 1999, 2003) and for super-Eddington radiation-driven winds (Lucy andSolomon, 1970; Castor et al., 1975; Abbott, 1982; Vitello and Shlosman, 1988; Warner, 1995; Murrayet al., 1995; Proga et al., 1998). Likely all these effects are important to some degree for somesystems.

X-ray Binary QPOs as Probe of Accretion Disk and Space-Time

Quasi-periodic oscillations (QPOs) with kHz frequencies are observed in the time series of the X-rayemission from some neutron star accretion systems and some black hole accretion systems, such asGRS1915+105. These kHz QPOs provide a plausible means to probe the inner part of an accretiondisk and map the space-time in the vicinity of a black hole. For those black hole systems that havekHz QPOs, the kHz QPOs exhibit a scaling of frequency with black hole mass (f ∝ 1/M) that isconsistent with the kHz QPO being produced in an accretion disk near the black hole (see, e.g.,Abramowicz et al. 2004, and references therein). For a discussion see di Matteo and Psaltis (1999);Strohmayer (2001); Abramowicz et al. (2004); Kato (2004) and references therein.

Event Horizon Argument for Compact Objects in X-ray Binaries

Of the ∼ 240 known X-ray binaries, ∼ 100 show time-dependent phenomena such as pulsations andstrong X-ray bursts that do not correspond to accretion outbursts. X-ray bursters have frequentX-ray bursts with quiescent (lower, persistent) X-ray flux between. The quiescent luminosity isorder 1032 − 1034 erg/s (the companion star has a bolometric luminosity 1032 − 1039 erg/s). X-raybursts are classified as Type I and II, where only Type I bursts are discussed since only they arerelevant to this discussion. Type I X-ray bursts are observed as a burst of thermal emission with aluminosity of 1036 − 1039 erg/s that last for seconds and recurs in hours to days. The burst energyis found to be proportional to the duration of the preceding inactivity period. There are about80 known Type-I X-ray bursts. There are about 8 known so-called superbursts with luminosity1043 erg/s that lasts for hours and recurs in years (Zand et al., 2004). Type-I bursts are thoughtto occur due to unstable nuclear burning of hydrogen or helium, and in the case of superbursts, ofcarbon (see, e.g., Woosley et al. 2004, and references therein).

Type I bursts are believed to be due to unstable nuclear burning on the surface of neutronstars, while the persistent background flux from such objects comes from conversion of gravita-tional energy into radiation. For standard Type-I X-ray bursts, the unstable burning comes from4H → 4He, 3 4He → 12C, and 5 4He + 84 H → 104Pd, which releases 6.7MeV, 0.6MeV, and6.9MeV per baryon, respectively. The latter two processes are the “triple alpha” and the rapidproton (rp) process. The persistent release of gravitational energy is E ∼ GMm/R ∼ 200MeV perbaryon. Thus, the ratio of gravitational to thermonuclear energy is about 30− 40.

22

Evidence for the thermonuclear origin for Type I bursts includes that the ratio of persistentenergy to burst energy is 30 to 40 and one observes Type I behavior: the longer the preceding fuelaccumulation, the more intense the burst. The burning site is argued to be a neutron star sinceonly one normal star is seen in the optical, the Stefan-Boltzmann law L = σ4πR2T 4

eff gives a typicalneutron star radius, and the maximum luminosity is consistent with the Eddington luminosity fora neutron star.

Type I X-ray bursts may provide a useful tool to study X-ray binaries with BHCs. No BHChas ever been observed to have a Type I X-ray burst, which is consistent with the fact that suchbursts can only occur on a surface. Assuming this is correct, observations of Type I bursts on BHCswould reject the event horizon hypothesis and the existence of classical black holes. Thus refiningmodels of Type I X-ray bursts is important in order to predict their occurrence (Narayan andHeyl, 2002, 2003; Woosley et al., 2004). One must be able to predict not only why some neutronstars have Type I bursts, but also why some neutron stars do not have Type I bursts. Preliminarystudies suggest that there is evidence for event horizons (Narayan, 2003a,b; Cornelisse et al., 2003;Tournear et al., 2003; McClintock et al., 2004), but see Abramowicz et al. (2002).

Black Hole Mass/Rotation from Fe Kα line Profiles

Observations of line emission from AGN and X-ray binaries show a broadened profile that may bedue to relativistic effects in the accretion disk near a rotating black hole. One can generate lineprofiles as observed on Earth by mapping photon trajectories in Kerr space-time either analytically(Karas et al., 1992) or numerically (Bromley et al., 1997; Cunningham, 1975). Evidence of anaccretion disk in AGN includes the ASCA observations of the Fe Kα line profile from MCG-6-30-15. The Fe Kα line profile in MCG-6-30-15 is shifted to the red, likely demonstrating generalrelativistic redshifting within a few GM/c2 of the black hole. The red and blue wings of the FeKα line profile have different intensities, which are consistent with the emission coming from partsof the accretion disk that move towards and away from the point of observation. The red andblue wing intensities suggest the accretion disk is moving relativistically with speeds approaching0.2c (Pariev and Bromley, 1998; Tanaka et al., 1995; Fabian et al., 2002; Vaughan and Fabian,2004). The emission profile is consistent with line emission from within a few Schwarzschild radii(2GM/c2) of the black hole (Tanaka et al., 1995; Fabian et al., 2002; Vaughan and Fabian, 2004).X-ray binary Fe Kα line profiles have also been observed and studied (Miller et al., 2004), and theprofiles are consistent with the Fe Kα line being emitted within a few Schwarzschild radii of theblack hole. Thus, the Fe Kα line apparently allows a probe of the innermost part of the accretiondisk.

With Chandra’s and XMM-Newton’s high resolution X-ray imaging, it may be possible to useFe Kα line profiles to test specific aspects of general relativity such as frame dragging (Bromleyet al., 1997). Models of the Fe Kα line from MCG-6-30-15 suggest the emission is ≈ 3 times morelikely to originate from a disk around a rotating black hole than a non-rotating black hole (Bromleyet al., 1997; Dabrowski et al., 1997). Also, observations of the Fe Kα line are consistent with the

23

Table 1.3. Fiducial Black Hole Accretion Systems 3

Parameter NS-BH GRB 030329 LMC X-3 SgrA* NGC4258 M87

dL ? 804 Mpc 55 kpc 8 kpc 7 Mpc 18 MpcM [M¯] 3 3 10 2.6× 106 4× 107 3× 109

Lbol[erg/s] 1053 3× 1052 1038 1037 3× 1043 2.3× 1042

Lbol/LE 3× 1014 8× 1013 0.08 3× 10−8 0.006 6× 10−6

M0 5M¯/ s 0.1M¯/ s 10−8 M¯/ yr 10−5 M¯/ yr 10−2 M¯/ yr 10−2 M¯/ yr

presence of energy extraction from a rotating black hole by the BZ-effect (Wilms et al., 2001b;Miller et al., 2002; Maraschi and Tavecchio, 2003).

Despite these suggestions about black hole rotation and energy extraction, the typical Fe Kα

line profile calculation, such as performed by XSPEC (Speith et al., 1995), assumes the accretiondisk is unmagnetized, viscous, and thin. As is discussed in the next section, such thin disk modelstypically assume that the accretion disk terminates at the ISCO, which marks a sharp transitionwhere matter plunges into the black hole. This sharp transition would lead to specific spectralfeatures in the observations. As discussed below, magnetic fields can torque the disk within theplunging region and keep the matter from simply falling into the black hole at the ISCO. Indeed,numerical GRMHD studies of accretion disks (see, e.g., McKinney and Gammie 2004), suggest thatno sharp transition occurs at the ISCO. Inside the ISCO, the disk should produce weak Fe Kα lineemission that the typical thin disk model might be incorrectly interpreting as evidence for blackhole rotation and energy extraction (Reynolds and Begelman, 1997).

1.4 Models of Accretion Disks and GRBs

The previous section discussed the most basic principles of accretion disk theory and gave someaccretion-based estimates for the properties of these systems. This section goes into more detailby discussing mechanisms for angular momentum transport, some radiative models of accretion,and models for GRBs. First, a discussion of angular momentum transport summarizes why a self-consistent magnetic model is required to study accretion disks. Next, a summary of the thin α-diskmodel (Shakura and Sunyaev, 1973) derivation is given as an example of a radiative viscous HDmodel. The purpose of presenting the derivation summary is to show how such disk solutions arefound and to show the basic structure of the equations. Next, there is a summary of other radiativedisk models. Finally, the section ends with a discussion of models of GRBs. The purpose of thediscussion of GRB models is to motivate a GRMHD model for the study of GRBs and core-collapsesupernovae in general.

The GRB and radiative disk models discussed in this section have been used by others toestimate the rest-mass accretion rate of the 6 fiducial systems. Table 1.3 includes the data from

24

Table 1.2 and adds an estimate for the rest-mass accretion rate. The rest-mass accretion rates forthe NS-BH GRB and Collapsar GRB (GRB 030329) disks come from simulations and estimates inMacFadyen and Woosley (1999); Popham et al. (1999). Estimates for the rest-mass accretion ratefor LMC X-3 are discussed in Ebisawa et al. (1993); Shimura and Takahara (1995); Wilms et al.(2001a); Brocksopp et al. (2001). The rest-mass accretion rate for SgrA* comes from Quataertet al. (1999). The rest-mass accretion rate for NGC4258 comes from Gammie et al. (1999). Therest-mass accretion rate for M87 is from Reynolds et al. (1996); Ho (1999).

1.4.1 Angular Momentum Transport Models

The purpose of this discussion is to summarize why an MHD, rather than HD or viscous HD (VHD),model is required to study ionized accretion disks. First, there is a short discussion of turbulence,which has long believed to be the driver of angular momentum transport in accretion disks. Next,the α-disk model is summarized. Finally, the MHD model is discussed. The discussion of theMHD model includes a summary of the magnetorotational instability (MRI) that likely dominateshydrodynamic instabilities in the generation of turbulence and angular momentum transport inaccretion disks (for a review see Balbus and Hawley 1998). Now, a general discussion of turbulenceis presented.

Turbulent flow can be qualitatively defined as a complex pattern of flow in which microscopicperturbations are enhanced to macroscopic scales as an expression of the flow’s internal degrees offreedom (Kadomtsev, 1965; Tennekes and Lumley, 1972; Eckman, 1981; She and Leveque, 1994;Frisch, 1995). The small-scale pattern of turbulent flow is typically sensitive to the initial conditions(the flow is chaotic), but the average pattern of turbulent flow is insensitive to the initial conditions(the flow is ergodic). Turbulence is 1) stationary if the average values do not vary with time; 2)uniform if the average values do not depend on position; and 3) isotropic if the average values at aspecific point in space do depend on the direction. For a finite system undergoing fully developedturbulence, there is a cascade of energy from large to small scales, at which point the energy isdissipated by kinematic viscosity or resistivity into heat (Tatarski, 1961; Kolmogorov, 1941b,a).Kolmogorov found that an incompressible fluid demonstrating stationary, uniform, isotropic turbu-lence has an “inertial range” of cascading energy, with a wave energy Wk = k−5/3, where k is themagnitude of the wave vector k =

√k2

x + k2y + k2

z . The inertial range is between the largest dimen-sions of a physical system and the dissipation scale. A magnetized, anisotropic, compressible fluidmight not be expected to be described by this simple Kolmogorov spectrum, but paradoxically, as-trophysical measurements are consistent with the Kolmogorov spectra. Two example systems thatfollows the Kolmogorov spectra are the magnetized solar wind (Belcher and Davis, 1971; Goldsteinet al., 1995; Leamon et al., 1998) and the interstellar electron density spectrum on small scales(108 cm− 1015 cm) (Armstrong et al., 1995) and large scales ( pc) (Lazarian et al., 2001).

To study turbulence in accretion disks, one must first identify the source of turbulence and thendetermine the mechanism of dissipation. Since both of these effects have not been well understood

25

for an accretion disk, models of turbulence have been developed. Generally, all black hole accretionmodels assume the approximation of hydrodynamics (HD), or when including a magnetic field,of MHD. Many accretion models have been developed since the pioneering work of Bondi whodescribed a model for inviscid (perfect, ideal) HD spherical accretion (Bondi, 1952). A summary ofsome of the early models by Shakura and Sunyaev (1973); Novikov and Thorne (1973); Pringle andRees (1972); Pringle (1981); Frank et al. (1992) and others can be found in chapter 14 of Shapiroand Teukolsky (1983).

Magnetically-driven winds off the disk are also likely important for angular momentum transport(Blandford and Payne, 1982; Krasnopolsky et al., 1999, 2003) (and see references therein). However,this discussion only considers the process of angular momentum transport in the interior of the disk.

Anomalous Viscosity

See Section 2.3 for the viscous HD (VHD) equations of motion. The so-called α-disk model (Shakuraand Sunyaev, 1973) introduces into the (spherical polar) HD equations of motion an r-φ stress (Πrφ)that is proportional to αP , where α ≡ Πrφ/P is a dimensionless constant and P is the pressure.The addition of a shear stress allows for angular momentum transport by driving an exchangeof momentum between differentially rotating layers of the disk. This anomalous viscosity is anattempt to model the effect that turbulence generates locally within the disk.

While there are global HD instabilities that could initiate the required turbulence, nonlinearstudies show that known HD instabilities saturate at low levels or do not apply to an accretiondisk near a black hole. No local HD linear or nonlinear instabilities are known to exist in Kepleriandisks (Balbus and Hawley, 1998), but this is not a settled issue (see, e.g., Bisnovatyi-Kogan 2004;Kuznetsov et al. 2004). The α-based shear viscosity model can be considered as a zeroth-orderapproximation to what is mostly likely a magnetic effect, which is described in the next section.

The thin disk model of black hole accretion based upon viscosity requires that the inner edge ofthe accretion disk be at the ISCO. Fluid flows through the ISCO and plunges into the black holewithout any effect on the accretion disk or the polar region, as proposed by Page and Thorne (1974).They also realized early on, however, that a magnetic field could significantly alter this picture ofthe plunging region. The magnetic field within the plunging region may torque the accretiondisk outside the ISCO (Krolik, 1999b; Gammie, 1999; Agol and Krolik, 2000). Before discussingmagnetic torques inside the ISCO, a summary of the MHD equations of motion is presented.

MHD Equations of Motion

The governing equations for nonrelativistic ideal MHD are the same as those for viscous HD withoutviscous heating, except for the addition of the Lorentz force to the momentum equation:

ρ0DvDt

= −∇P − ρ0∇φ + (∇×B)×B (1.9)

26

and the addition of the induction equation:

∂B∂t

= ∇× (v ×B) (1.10)

where B is the magnetic field vector (a factor of√

4π is absorbed into the definition of B, as inHeaviside-Lorentz units). The general dispersion relation for linear waves in a uniform backgroundis:

(v2p)(v

2p − (k · va)2)(v4

p − (cmvp)2 + (cs(k · va))2) = 0 (1.11)

where vp = ω/k is the phase velocity, k is the wave vector, k = k/k is the wave unit vector, ω is thefrequency of oscillation, va = B/

√ρ0 is the Alfven velocity, cm =

√c2s + v2

a is the magnetosonicspeed, cs =

√∂P/∂ρ0|S is the sound speed, and S is the entropy. There are clearly 8 wave solutions.

The first term gives two modes: the entropy mode that corresponds to entropy perturbationsat constant pressure ; and the monopole mode that is removed by the solenoidal condition of∇ · B = 0. The second term gives left- and right-going Alfven waves, which are transverse andmove at the speed va cos θ, where θ is the angle between k and va. The third term gives left-and right-going fast and slow magnetosonic waves. For this term, if va · k = 0 the fast wave isa purely compressive, longitudinal wave with phase speed cm. For waves of arbitrary va · k the

phase velocity is√

12c2

m ± 12 [c4

m − 4c2sv

2a cos2 θ]1/2, where the +(−) corresponds to the fast(slow)

magnetosonic wave.Clearly, the limitation of the above nonrelativistic MHD model is the inability to model rel-

ativistic flows. While a pseudo-Newtonian potential (Paczynski and Wiita, 1980) reproduces thepositions of the ISCO, the marginally stable circular orbit, and approximately the binding energyof the last stable orbit, the potential is unable to model a rotating black hole. See Chapters 3and 4 for the general relativistic coordinates, governing equations, and the characteristics of thegeneral relativistic MHD (GRMHD) equations. Both nonrelativistic and relativistic MHD modelsfind similar results that deviate from expectations built on the α-disk model.

MHD Turbulence

Early accretion disk theory considered magnetic stresses as insignificant (Shakura, 1972). Lateraccretion disk studies realized that a magnetic field may provide a type of viscosity (Novikovand Thorne, 1973; Eardley and Lightman, 1975; Pringle, 1981). Indeed, previous experimental(Velikhov, 1959) and theoretical (Chandrasekhar, 1961) work found instabilities in differentiallyrotating magnetized fluids. Balbus and Hawley (1991) rediscovered this magneto-rotational insta-bility (MRI) (or Balbus-Hawley instability) by performing a local stability analysis of a magneticfluid in a differentially rotating Newtonian disk. The MRI operates similarly near a Kerr black hole(Gammie, 2004). The MRI generates angular momentum transport even when the field is weak.

A local stability analysis shows that the MRI grows on a dynamical time of Ω−1 and that thewavelength of the fastest growing mode is λc ∼ va/Ω (Balbus and Hawley, 1991), where Ω is the

27

angular frequency of rotation. Including a magnetic field (and thus the MRI) solves the questionof “what is α” in α-disk models by self-consistently generating the turbulence from the magneticfield’s interaction with the fluid. It is still useful, however, to use α as a dimensionless measure ofthe stress that transports angular momentum. In MHD, α ≡ −δvrδvφ/P and αmag ≡ −δBrδBφ/P

are parameters one can measure anywhere in space (δ is the difference from the mean flow, so α

measures the turbulent part of the flow).Development of the MRI can be understood qualitatively by considering the stability of a

magnetized fluid element in a differentially rotating disk. If a fluid element at radius r0 is threadedby a field line that is parallel to the axis of rotation, then a radial disturbance in the fluid can causethe field line to bend. The initial disturbance causes the fluid element at r < r0 to gain rotationalspeed by conservation of angular momentum. The fluid element at r > r0 loses speed for the samereason. The field line simply flexes back and resists the separation if the field strength is high.However, if the field strength is relatively low, the field line pulls slightly back on the fluid elementsand reduces the angular momentum of the lower fluid element at r < r0 and raises the angularmomentum of the fluid element at r > r0 (i.e. the field transfers angular momentum). This inturn increases their speed and thus separation from the original position resulting in an instability(Balbus and Hawley, 2002). In general, a magnetized accretion disk is unstable to the MRI whendΩ2/dr < 0, where Ω is the frequency of rotation. This criterion is easily satisfied for an accretiondisk around a compact object. By comparison, the Rayleigh (hydrodynamic) criterion is dl/dr < 0for instability, where l = r2Ω is the specific angular momentum. While no accretion disk modelsatisfied the Rayleigh criterion for instability, both theoretical and observed accretion disks havean angular velocity that decreases outwards.

Numerical simulations of the MRI confirm the linear stability analysis (Balbus and Hawley,1991). Subsequent numerical work showed that the MRI is capable of sustaining turbulence in thenonlinear regime (Hawley and Balbus, 1991). The MRI was shown to generate sufficient angularmomentum transport to sustain accretion, and the MRI develops a dynamo in the nonlinear regime.The rediscovery of the MRI renewed hope that accretion disks could be understood by a directcalculation, rather than by using the α-disk model.

One simplified form of the HD or MHD equations of motion is called the local shearing boxapproximation, which is a result of an expansion of the equations at a specific radius with a box sizeδr. The inner and outer “radial” boundaries are treated as periodic with an additional shearing termthat describes the differential rotation. Simulations of accretion disks have been performed in thelocal shearing box approximation (Hawley and Balbus, 1991; Hawley et al., 1995; Stone et al., 1996),which determined the relevance of the MRI instability to, and the vertical magnetized structureof, accretion disks. Accretion disk simulations with 2D axisymmetric or cylindrical symmetries,and full 3D simulations, have been performed to study the global structure of the accretion disknear the black hole (Armitage, 1998; Hawley, 2000, 2001). For an initially 2D symmetric accretiondisk, 3D geometry appears to be required only to sustain a magnetic dynamo. All these simulationshave shown that the MRI self-consistently accounts for the transport of angular in ionized accretion

28

disks.

MHD Effects Inside ISCO and Ergosphere

The previous discussion showed how the MHD model goes beyond the predictive capability of theα-disk model by self-consistently describing the generation of turbulence and angular momentumtransport. Another new result is that the MHD model shows that the disk is magnetically torquedinside the ISCO, as seen in both pseudo-Newtonian numerical models (Hawley, 2000; Hawley andKrolik, 2001, 2002) and GRMHD numerical models (De Villiers et al., 2003a; De Villiers andHawley, 2003b; De Villiers et al., 2003b; McKinney and Gammie, 2004). Although, how applicablethese simulations are to, say, CVs, YSOs, and protoplanetary disks is not certain (Armitage et al.,2001; Reynolds and Armitage, 2001). The α-disk model assumes that the accretion disk terminatesat the ISCO, which marks a sharp transition where matter plunges into the black hole. Thissharp transition leads to specific spectral features in the observations. As mentioned above in thediscussion of the Fe Kα line profile, the presence of a magnetic torques inside the ISCO can affect theconclusions about evidence for black hole rotation and energy extraction. Models of the Fe Kα lineprofile using an α-disk model seemed to be consistent with the presence of an accretion disk arounda rotating black hole (Bromley et al., 1997; Dabrowski et al., 1997; Wilms et al., 2001b; Milleret al., 2002; Maraschi and Tavecchio, 2003). However, inside the ISCO the disk should produceweak Fe Kα line emission that the α-disk model might be incorrectly interpreting as evidence forblack hole rotation and energy extraction (Reynolds and Begelman, 1997).

The magnetic field may also extract angular momentum and energy from a rotating blackhole into the accretion disk through an MHD version of the BZ effect (Blandford and Znajek, 1977;McKinney and Gammie, 2004). Relativistic magnetized steady state accretion disks postulated thata magnetic field might allow a higher accretion efficiency than the thin disk efficiency (includingsuper-efficient η > 1) due to magnetic torques in the plunging region (Gammie, 1999). Remarkably,GRMHD simulations of both thick and thin disks show an efficiency similar to that of a thin disk(McKinney and Gammie, 2004).

1.4.2 Radiative Disk Models

The previous section established that magnetic fields are required to self-consistently describe thegeneration of turbulence and process of angular momentum transport within an accretion disk.MHD models also showed that the disk is magnetically torqued inside the ISCO, and that energycan be magnetically extracted from a rotating black hole. Radiative processes in an accretion diskare also dynamically important because they cool the disk, introduce radiative instabilities, andintroduce a radiative pressure. Ultimately, of course, several different radiative processes may beinvolved in determining observations, but the focus of this section is on those radiative processesthat determine disk structure.

29

Shakura and Sunyaev (1973) constructed an accretion disk solution (SS73 model) that describesthe structure of a disk that steadily accretes due to a local anomalous viscous transport of angu-lar momentum. The anomalous viscosity is parameterized by the α model that was describedin Section 1.4.1. By approximating the thin disk by a surface density using a height-integratedapproximation, including optically thick thermal bremsstrahlung absorption and optically thickelectron scattering, they find a complete solution to the accretion flow and spectrum emitted fromthe disk surface. This model is a useful starting point for estimating the properties of accretiondisks. A relativistic version of the thin disk model is given by Novikov and Thorne (1973); Eard-ley and Lightman (1975), but a pseudo-Newtonian approximation of the gravitational potentialΦ = GM/(r − 2GM/c2) (Paczynski and Wiita, 1980) is sufficient to grasp the salient aspects ofthe solution. A short derivation of this SS73 model is provided in Shapiro and Teukolsky (1983),§14.5, where below is a summary of the main points. Following the SS73 derivation summary is adiscussion of more advanced radiative disk models.

The SS73 model assumes that 1) the disk scale height is much less than the radius (H/R ¿ 1) ;2) an anomalous fluid viscosity introduces a shear stress (Πrφ ∼ αP ) ; 3) rest-mass is conserved ; 4)angular momentum is conserved, where there is a balance between the angular momentum accretedand angular momentum removed by the viscous stress ; 5) vertical momentum is conserved andvertical equilibrium is established, where H/R ≈ cs/vK ; 6) energy is conserved, where there isa balance between kinetic energy, viscous heat generation, and cooling by photon radiation thatdiffuses through the optically thick disk or cools directly in an optically thin disk. The pressure isassumed to be due to radiation and the gas. In the innermost radial region where the temperature ishighest, radiation pressure typically dominates gas pressure. In this thesis, disks in X-ray binariesand AGN are modeled with this photon-cooling model. For such disks, the opacity is typicallydominated by electron scattering, and this is assumed here. These assumptions allow one to derivethe complete state of the disk, where the midplane rest-mass density, midplane temperature, anddimensionless disk height to radius ratio are found to be

ρ0 = (1.5× 10−5 g cm−3)η2(αM)−1m−2r3/2Ξ−2

T = (5× 107 K)(αM)−1/4r−3/8 (1.12)

h/r = 0.95η−1mr−1Ξ,

where Ξ ≡ 1 − (6/r)1/2, m ≡ M0/M0,E , M0,E is the Eddington mass accretion rate given inEquation 1.5, M is measured in units of M¯, and r, h are both measured in units of GM/c2

(i.e. h/r is dimensionless). Notice that the assumption that the disk is thin (h/r ¿ 1) is not validin the inner-radial region when the luminosity is near the Eddington limit.

The standard SS73 disk model predicts a disk midplane temperature of T ∝ M−1/4, and thusAGN would be expected to have much lower temperatures than X-ray binaries. The accretion diskin AGN and X-ray binaries, as described by the SS73 model of accretion, produces a soft quasi-thermal spectrum dominated by UV and optical emission in AGN, and dominated by soft X-rays

30

in black hole X-ray binaries. However, there is greater uniformity in the observed X-ray spectrain these systems, which have hard X-ray emissions in a power law out to E ∼ 0.3MeV. Also, theSS73 model is unstable in the radiation-dominated region of the inner accretion disk, thus violatingthe steady-state assumption (Lightman and Eardley, 1974; Piran, 1978; Abramowicz, 1981).

Since the SS73 model is unstable and does not account for hard X-ray emission, other modelshave been invoked to explain the X-ray emission. These models often introduce a temperaturedifference between the ions and electrons in the flow, known as a two-temperature model. This isplausible since the average energy released per accreted particle is

E ≡ E

M0

∼ ηc2

(GM/c2

R

), (1.13)

where E is the total energy accretion rate. Since R ∝ GM/c2, E is independent of the mass of thecompact object. For η ∼ 10%, this available energy is up to ∼ 51 keV (∼ 6 × 109 K) for electronsand ∼ 94MeV (1012 K) for protons. Thus, it is plausible that the electrons and protons could havedifferent temperatures, with inverse Comptonization cooling the electrons and generating hard X-rays (Thorne and Price, 1975; Sunyaev and Truemper, 1979). Depending upon how thermallycoupled the electrons and protons are, there may be more or less energy available to the electronsto power inverse Comptonization. This process has been considered as a plausible mechanism forgenerating hot coronae in accretion disks (Stern et al., 1995). A discussion of Comptonization canbe found in Rybicki and Lightman (1979); Pozdniakov et al. (1983) and is summarized below.

Compton scattering involves a photon with energy Eγ and an electron with mass me and gastemperature Te. If mec

2 À Eγ À 4kbTe, then the kinetic energy of the electron is negligible andthe photon loses energy. However, if Eγ ¿ 4kbTe, then it can be shown that in the lab frame thephoton gains an energy proportional to Γ2

e, where Γe is the electron’s relativistic Lorentz factor.As long as Eγ ¿ 4kbTe, the photon gains energy until the photon reaches thermal equilibrium withthe electrons. For low enough photon energy and high enough electron energy the photon can behighly energized, a process referred to as unsaturated inverse Comptonization. If there is littleamplification the process is referred to as saturated inverse Comptonization.

The “hot disk” SLE model (Shapiro et al., 1976) is a two temperature model. In this model,in the inner-radial accretion disk, ions reach T ∼ 1011 − 1012 K, while electrons reach T ∼ 109 K.Hard X-rays are produced by inverse Comptonization of soft photons from electrons in the coolerparts of the disk. The two-temperature models suggest that far away from the compact object,where the disk is relatively cold, the disk follows the SS73 model. The disk there is thin, generatingthe UV and optical spectra observed. In the inner radial regions a transition to a two-temperaturethick disk may occur, and then the ions remain hot and heat electrons, which continuously cool togenerate the hard X-rays observed. The transition radius is inversely proportional to the rest-massaccretion rate. At low mass accretion rates, the disk is in the “quiescent” state with much of theinner radial region of the disk forming a thick disk generating low-hard (low luminosity, hard X-ray)emission. At the highest rest-mass accretion rates, the thin disk reaches close to the compact object

31

and disk generates high-soft (high luminosity, soft X-ray or UV) emission.While the SLE model is unstable (Pringle, 1976; Piran, 1978), the two basic considerations may

still be viable. First, accretion flow may be two-temperature. Second, Comptonization of photonsmay provide the hard X-rays even for a single temperature flow. For example, magnetic dissipationhas long been considered a mechanism for heating a hot coronal region (Galeev et al., 1979), andthis is seen in most global simulations of accretion disks. This is consistent with the so-called“hot corona” model (see, e.g., Ostriker 1976), which also models hard X-rays production as due toComptonization.

A possibly stable version of the SLE model is the advection dominated accretion flow (ADAF)model. As with the SLE model, the ions remain hot, are unable to cool efficiently, and hold most ofthe accretion energy (Ichimaru, 1977; Rees et al., 1982). The fiducial nonrelativistic ADAF model(Narayan and Yi, 1995) includes synchrotron and bremsstraulung cooling, and Comptonization andsynchrotron self-absorption, where a relativistic version is given by Gammie and Popham (1998);Popham and Gammie (1998). The key result is that the flow is highly inefficient (η ∼ 10−4).This is due to the hot ions holding most of the energy down to the neutron star or black hole,while electrons cool and generate the observed spectra. A black hole absorbs the ADAF with littleradiation from the accretion disk (Narayan and Yi, 1994, 1995; Abramowicz et al., 1995). Suchmodels have been applied fairly successfully to X-ray binaries (Ichimaru, 1977), to AGN (Reeset al., 1982), and to our galactic nucleus (Narayan et al., 1995, 1998; Quataert et al., 1999).

ADAF models have been used to study whether the compact object in an X-ray binary is aneutron star or black hole. The primary difference between these objects is that the neutron starhas a surface, while a black hole does not. A possible method to determine the existence of theevent horizon (or the lack of a surface on a BHC) is by comparing the quiescent luminosity of X-raybinaries with neutron stars to X-ray binaries with BHCs. It is found that X-ray binaries with BHCsare substantially underluminous compared to their neutron star counterparts per unit Eddingtonluminosity, which suggests that the excess luminosity in neutron stars is due to emission from thesurface and that the BHCs have no surface (Narayan et al., 1997; Menou et al., 1999; Garcia et al.,2001; McClintock et al., 2004). In order to confirm this hypothesis one needs to accurately modelthe accretion disk. The radiative models of Narayan et al. basically agree with observations byassuming that the accretion disk takes on either a highly radiative, thin disk state or a nonradiative,thick disk state. In either case, the matter releases its energy upon impacting the surface of theneutron star. If BHCs had a surface, then they too would exhibit this surface luminosity. Sincethey do not, this suggests they have no surface, and given their large mass suggests they are likelyclassical black holes.

The stability of the ADAF model is not certain, and the flow could be unstable to developingoutflows near the black hole (Blandford and Begelman, 1999) or be convectively unstable (Quataertand Gruzinov, 2000). The success of the two-temperatures or hot-corona models is that theypredict the X-ray spectrum better than the SS73 model due to the inclusion of electronic coolingby Comptonization.

32

All of these radiative models assume a viscous model to describe the transport of angularmomentum. The existence of the Balbus & Hawley instability (Balbus and Hawley, 1991) suggestsone should study magnetized accretion disks rather than viscous accretion disks. Only simplifiedanalytic models have appeared that include a magnetic field (Gammie, 1999; Merloni, 2003; Li,2004). Radiative disks have been studied using time-dependent, radiative MHD numerical modelsof accretion flow (see, e.g., Agol et al. 2001; Turner and Stone 2001; Turner et al. 2002; Turner2004). These are effectively SS73 models with magnetic fields. Unlike the SS73 model, thesesimulated disks have a layered vertical structure and are less radiation-dominated for otherwisesimilar parameters as the SS73 model. No numerical or analytic models have been constructedthat include a magnetic field with otherwise similar physics to, say, the two-temperature modelwith ADAF-type radiative processes. No radiative GRMHD models of accretion disks have yetbeen studied numerically.

1.4.3 Gamma-Ray Bursts Models

The standard model for the GRB engine is a hyper-accreting black hole or neutron star (for areview see, e.g., Piran 1999; Meszaros 2002; Piran 2004). The accretion flow is expected to pro-duce large amounts of energy from annihilation of neutrinos or from spin energy extracted from arotating magnetized compact object. Respectively, these two processes are believed to produce anultrarelativistically moving pair plasma that internally shocks to produce gamma-rays, or to pro-duce a Poynting flux jet that develops hydromagnetic instabilities that shocks internally to producegamma-rays. Eventually, this outflow can interact with the surrounding media to produce so-calledexternal shocks.

This section first discusses the prompt gamma-ray and afterglow emission (for a review seePiran 1999, 2004). The purpose of this discussion is to suggest that the GRB progenitor must berelativistic, compact, and likely sometimes occurs during a supernova. Next, this section discussesthe “collapsar” model of the accretion system for long-duration GRBs (Woosley, 1993; Paczynski,1998; MacFadyen and Woosley, 1999). The purpose of this discussion is to suggest that a GRMHDmodel is required to study the collapsar model. Later in Section 1.5, it is shown that the nonra-diative GRMHD models used in this thesis (in Chapters 3 and 4) are a good model for the studyof collapsars.

Models for GRB Emission and Afterglow Emission

The prompt gamma-ray emission from a GRB is found to be non-thermal and highly variablein time. The temporal variability of GRB light curves suggests the GRB progenitor has a sizeR . 60 km for some short-duration bursts, and R . 300 km for some long-duration bursts. However,an isotropic estimate for the number density of gamma-rays nγ ≈ (1053 erg)/((500 keV)R3) is muchlarger than what is required to generate electron-positron pairs (Ee−e+ ∼ 2mec

2). This implies theemitting media would have an optical depth of τ ∼ 1015, and suggests that the emission should

33

be thermal. The prompt gamma-rays are actually non-thermal, and this paradox is called the“compactness problem” (Piran, 1999, 2004). One resolution to the compactness problem is toassume that the flow is moving ultrarelativistically. By assuming the observed radiation is froma source moving toward us with Lorentz factor Γ, blue-shifting of photons and modifications tothe probability of photon collisions require Γ & 100 to obtain τ . 1. However, there is no well-understood mechanism for jet collimation or large Lorentz factors.

If geometric or relativistic beaming occurs, then an estimate of the actual luminosity can beobtained by appropriately reducing the estimate for the isotropic luminosity. The value of theactual luminosity can be useful to determine the nature of the progenitor. If beaming occursgeometrically, then the isotropic luminosity estimate must be reduced by a beaming fraction of∼ θ2

j /2 for θj . 1, where θj is the opening angle of one side of, an assumed, bipolar jet (Rhoads,1999; Sari, 1999). If beamed relativistically, a local observer sees an enhancement of Γ2, so onemust reduce the estimate by 1/Γ2. One expects 3 possible phases if the initial Lorentz factor islarge, and these phases lead to approximately 3 different domains in the light curve (Sari et al.,1999): 1) early relativistic beaming with ∼ (θjΓ)2 causally disconnected patches; 2) a slowdownleads to Γ ∼ θj and thus rapid sideways expansion that leads to a break in the light curve; and 3)at late time Γ ∼ 1 and one observes an exponentially decaying emission.

In this ultrarelativistic fireball model, internal shocks are produced because the outflow hasa range of velocities. These relativistic internal shocks produce gamma rays via synchrotron orsynchrotron self-Comptonization emission. Such a fireball model has a “baryonic contamination”problem (see, e.g., Piran 2004). A flow contaminated with baryons can absorb the radiative energyinto bulk kinetic energy, producing too few photons compared to observations. Internal shocksallow this kinetic energy to be converted back into radiation, but too much conversion too quicklyleads to sub-relativistic bulk motion leading again to a compactness problem.

Another solution to the compactness and baryonic contamination problems is to assume thejet carries nonradiative energy in the form of a Poynting flux, and the flow only creates gamma-rays at sufficiently large distances. For example, a Poynting flux could be converted into radiationfar from the collapsing star by internal dissipation (see, e.g., Thompson 1994; Meszaros and Rees1997; Spruit et al. 2001; Drenkhahn 2002; Drenkhahn and Spruit 2002; Sikora et al. 2003; Lyutikovet al. 2003) such as magnetic reconnection. Preliminary (controversial) evidence for a magneticdominated outflow has been found in GRB 021206 (Coburn and Boggs, 2003), consistent with amagnetic outflow directly from the inner engine (Lyutikov et al., 2003). As discussed below, aPoynting flux jet may be primarily responsible for GRBs.

The GRB afterglow is suspected to be generated by interaction with the interstellar mediumin external shocks (see, e.g., Piran 2004). Optical, radio, and X-ray afterglows have energies of1050 − 1052 erg, generally about 1/10th of the energy in gamma-rays. GRBs, such as GRB 990123and GRB 990510, show breaks in the optical light curve, which is consistent with an external shockmodel. An analysis of the light breaks using an adiabatic synchrotron model for cooling (see, e.g.Piran 1999, 2004), similar to the Blandford-McKee self-similar solution (Blandford and McKee,

34

1976), shows the breaks are consistent with Γ ∼ 100− 500. There is no temporal scaling betweenthe GRB and afterglow lightcurves, and this lack of scaling is consistent with the GRB as generatedin internal shocks and the afterglow as generated in external shocks.

A GRB likely develops from a compact central engine (Piran, 1999; Meszaros, 2002). If oneassumes that the GRB emitting region is causally connected by light propagation effects, then thevariability from GRB light curves gives an estimate of the maximum emitting region (R ∼ cδT ).The typical GRB variability is 1ms − 1 s, giving a maximum emitting region of R ∼ 300 km.The shortest GRB currently ever observed lasted for 6ms (Bhat et al., 1992) with 200µ s temporalstructure. This gives a maximum size of R = 60 km for the emitting source. Energetic and beamingconsiderations suggest a GRB is a collimated jet with an energy of ∼ 1051 erg and a Lorentz factorof 102 − 103 (Sari et al., 1999; Frail et al., 2001; Panaitescu and Kumar, 2001; Bloom et al., 2003).Such supernova-like energies suggest a subclass of massive stars undergoing core-collapse couldgenerate GRBs. There is no clear evidence for whether the GRB is a “standard candle” withstandard energy output and jet beaming characteristics (e.g. see references in Liang et al. 2004).Recent observations suggest that at least some GRBs are not standard candles. For example, GRB031203 had a very low luminosity from an otherwise normally beamed GRB (Sazonov et al., 2004;Soderberg et al., 2004).

The estimated GRB event rate of ∼ 1/(107 yr) per galaxy, or ∼ 1/(105 yr) if there is beaming,also suggests a GRB-SN connection. HST observations show an association between optical after-glows and host galaxies (Sahu et al., 1997; Akerlof et al., 1999), and show a correlation betweenGRBs and star-forming regions in galaxies (Fruchter et al., 1999). Core-collapse supernovae modelshave suggested a GRB-SN connection since GRB 980425 and SN 1998bw at z = 0.008 (Galamaet al., 1998; Iwamoto et al., 1998). The afterglow from long-duration GRB 030329 and supernovaSN2003dh at redshift z = 0.169 occurred essentially simultaneous in time and overlap in spaceconfirming a supernova connection to long duration GRBs (Uemura et al., 2003; Meszaros, 2003;Hjorth et al., 2003; Kawabata et al., 2003; Stanek et al., 2003). Chandra observations of supernovaremnant W49B in our own Milky-Way show the likely remnants of a GRB (Keohane et al., 2004),so a forensic study may reveal details of the GRB event.

Models for the Engine of Long/Short Duration GRBs

The temporal variability of GRB light curves suggests the GRB progenitor is a system with asize R . 60 km, and M . 40M¯ if a compact object, for some short-duration bursts; and witha size R . 300 km, and M . 200M¯ if a compact object, for some long-duration bursts. Theprogenitor must release the binding energy of a solar mass object over an extended period (manylight crossing times), rather than in a single explosion. This suggests an accretion disk origin isplausible. Accretion in other systems is known to produce relativistic jets. The two likely scenariosfor generating a GRB are 1) accretion of a 0.001 − 1M¯ disk around a few solar mass black hole; or 2) gamma-ray dipole or spin-down emission from a highly magnetized neutron star called amagnetar.

35

The matter that forms the accretion disk around the solar-mass black hole or neutron starcould be from core-collapse material, supernova fallback material, a neutron star, a white dwarf,or a Wolf-Rayet star. GRB events have a duration that follows a bimodal distribution, whichcould result from an accretion disk system formed from these different types of secondary materialand their associated angular momentum (and thus circularization radius and accretion time scale)(Popham et al., 1999).

The current leading explanation for a long-duration GRB is the collapse of a massive star(M ∼ 25 − 40M¯). The most attractive model for long-duration GRBs is the collapsar model(Woosley, 1993; Paczynski, 1998; MacFadyen and Woosley, 1999) of which there are two types. AType I collapsar results from a collapsing massive star that has too weak a shock wave to generatea supernova. The collapse of the iron core leads to a neutron star, which then quickly leads to ablack hole and accretion disk from the fall back of the envelope. A Type II collapsar is a successfulsupernova of a massive star that blows away all helium and some heavy elements outside a neutronstar core. Some material fails to reach escape velocity, falls onto the neutron star, and slowly formsa black hole and accretion disk.

The collapsar model as developed by Woosley (1993); MacFadyen and Woosley (1999) includesseveral pieces of microphysics, such as 1) a realistic EOS (Lattimer and Swesty, 1991; Lattimer,1996; Blinnikov et al., 1996; Lattimer and Prakash, 2000); 2) the URCA process for nuclear burning(Bodenheimer and Woosley, 1983); and 3) neutrino cooling and annihilation in the optically thinparts of the flow (Itoh et al., 1989, 1996; MacFadyen and Woosley, 1999). For a discussion ofcollapsar-like GRB accretion disk models, see for example Popham et al. (1999); Narayan et al.(2001); Kohri and Mineshige (2002); Di Matteo et al. (2002). For a discussion of neutrino-drivenexplosions, see for example Fryer and Meszaros (2003). For a discussion of optically thick neutrinoaccretion disks, see for example Lee et al. (2004).

The collapsar model has been studied numerically, including 1) viscous HD models with theabove microphysics, following the evolution of a pre-supernova star to collapse, accretion diskformation, and jet formation (MacFadyen and Woosley, 1999); 2) special relativistic MHD modelsof the jet (MacFadyen et al., 2001); 3) nonrelativistic MHD models with the above microphysics,following the formation of a disk and jet (Proga et al., 2003); and 4) stationary space-time generalrelativistic models, with no microphysics, of a black hole with a pseudo-post supernova star (Mizunoet al., 2004a,b). Numerical simulations show that the penetration of a jet through a stellar surfaceand evolution of the jet through a surrounding pressure gradient can lead to large Lorentz factors ofΓ ∼ 44, close to that required by the standard fireball model (see, e.g., Aloy et al. 2000; Zhang et al.2003). Numerical models have yet to include numerical relativity and MHD to study the collapseof a rotating magnetized massive star to a black hole (see review by Stergioulas 2003, §4.3).

Despite extremely detailed neutrino physics, typically the magnetic field is neglected in core-collapse supernovae calculations that seek to follow the entire collapse of a massive star (for thelatest typical code see, e.g., Liebendorfer et al. 2004). In unraveling the mechanism by which core-collapse supernovae explode, the simulation of a successful supernova has been found to be sensitive

36

to the accuracy of the neutrino transport (Messer et al., 1998; Yamada et al., 1999; Burrows et al.,2000; Rampp and Janka, 2000; Liebendorfer et al., 2001; Mezzacappa et al., 2001). For some timethis has been regarded as implying one requires highly accurate neutrino physics to model core-collapse supernovae (Liebendoerfer, 2004), however this could also be interpreted as suggestingother physics (say, a magnetic field) is required to model core-collapse supernovae.

The GRB is suspected to be due to either (or both) 1) neutrino and anti-neutrino annihilationthat develops into an electron-positron pair fireball at the polar axis of the disk or black holeand eventually produces gamma-rays; or 2) an electromagnetic Poynting flux from the disk orblack hole following field lines at the poles and eventually produces gamma-rays. As discussedabove, Poynting flux generating GRB models solve both the compactness and baryon-contaminationproblems. The energy extracted from a rotating black hole by a magnetic field can easily exceedthe energy for neutrino-mediated energy transport (Meszaros and Rees, 1997). Indeed, all core-collapse events may be powered by MHD processes rather than neutrino processes (Leblanc andWilson, 1970; Bisnovatyi-Kogan and Ruzmaikin, 1974; Bisnovatyi-Kogan et al., 1976; Symbalisty,1984; Woosley and Weaver, 1986; Duncan and Thompson, 1992; Khokhlov et al., 1999; Akiyamaet al., 2003; Thompson et al., 2004a). Core collapse explosions are observed to be significantlypolarized, asymmetric, and often bi-polar, indicating a strong role of rotation and a magnetic field(Wang and Wheeler, 1996; Wheeler et al., 2000; Wang et al., 2001, 2002, 2003). Core-collapse alsoinvolves shearing subject to the Balbus-Hawley instability as in accretion disks (Akiyama et al.,2003). Thus, there is sufficient evidence that long-duration GRB models should include a magneticfield.

There are fewer observational phenomena associated with short-duration GRBs, and so it ismore difficult to model short-duration GRBs than to model long-duration GRBs. Without anafterglow counterpart to the GRB, a redshift measurement is not possible, and correlating a shortGRB with a host galaxy is difficult. The short-duration GRBs are observed to have harder gamma-rays than long-duration bursts. This is believed to be due to a lack of baryons obstructing gamma-ray emission.

Short-duration GRBs could be due to binary NS mergers (Narayan et al., 1992) or NS-BH merg-ers (Narayan et al., 1991). The NS-NS or NS-BH collision processes are currently the best modelfor short-duration GRBs, since these systems produce harder gamma-rays as seen in observations.However, short-duration GRBs could be produced by a relative of the collapsar model, which hasno supernova and little baryon-loading that also leads to harder emission. During NS-NS or NS-BHcollisions, gravitational wave emission is likely significant, and if the GRB is close enough to Earth,gravity waves may be detectable by LIGOII/LISA (van Putten et al., 2004). Thus, gravitationalwaves may help determine the origin of short-duration GRBs.

Some GRBs may also be due to spin energy extraction from a highly magnetized neutron starcalled a magnetar (Wheeler et al., 2000; Rees and Meszaros, 2000; Thompson et al., 2004b). Inthe so-called cannon-ball model, the asymmetric stellar collapse leads to a magnetar moving at∼ 1000 km s−1 (Dado et al., 2002; Huang et al., 2003), where a pair plasma fireball is ejected

37

highly anisotropically (∼ 3− 10), but otherwise as described by Usov (1992).

1.5 Characteristic Quantities and Model Validity Estimates

In Section 1.2, 6 fiducial black hole accretion systems were introduced to establish their distancefrom the Sun, black hole mass, and bolometric luminosity. It was argued that these objects likelyhave accreting black holes. In Section 1.3, the basic theory of accretion was introduced, andallowed a calculation of the ratio of the bolometric to Eddington luminosity. It was found thatthe luminosity, variability, and spectra could be used to deduce basic and precise properties ofblack hole accretion disk systems. In Section 1.4, magnetic and radiative models of accretion diskswere introduced, which have allowed researchers to estimate the rest-mass accretion rate of the 6fiducial systems. It was found that a magnetic model was required for self-consistent generation ofturbulence to drive angular momentum transport, and that radiation is crucial to determine thethickness and structure of an accretion disk. It was also discussed how GRMHD models are likelyrequired to study GRBs and core-collapse supernovae. This section uses this cumulative knowledgeto estimate the state of the accretion disk in the 6 fiducial systems by using an approximate radiativeGRMHD model of the accretion flow.

First, the “state” of the accretion flow is determined. The “state” refers to quantities such asthe density, temperature, and magnetic field strength. In principle, the state should be determinedby a completely self-consistent numerical or analytic solution. However, no such solution hasbeen derived or numerically constructed for a radiating magnetized disk in general relativity. Acompromise is made by using radiative unmagnetized analytic models together with nonradiativeGRMHD numerical solutions of accretion disks as studied in chapter 4. A consistency check isperformed to verify that, indeed, this is a reasonable treatment. Once the complete magnetizedstate of the disk is determined, the validity of the fluid, MHD, and ideal MHD approximations istested for the 6 fiducial systems.

Even though the accretion disk may extend to R & 1000GM/c2, the focus of this study ison the state of the accretion flow, and the validity of the approximations, near the black holewithin R . 40GM/c2 where relativistic effects are important. The density and temperature varywidely within and between systems. While the conditions for the MHD approximation are plausiblysatisfied in the bulk of the disk, they may not be satisfied, for example, in the more vacuous corona(Novikov and Thorne, 1973). While I have tested the validity of the approximations for the entireglobal accretion flow, only the state of the disk and the validity of the approximations in the diskare shown. The disk results shown are representative for the non-disk part of the global solution,and any deviations are discussed.

1.5.1 Estimated State and Structure of Accretion Flow

In order to estimate the state of the GRB disk, and the magnetic field in the other disks, we useresults of nonradiative GRMHD numerical models of accretion flow described in chapter 4. For

38

FUNNEL

CORONA

PLUNGING REGION

DISK

BLACK HOLE

WIN

D

r=40Lr=12L EQUATOR

EL0P

Figure 1.1 Cartoon plot of one hemisphere of the axisymmetric accretion disk, corona, funnel/wind,and plunging regions. The outer radius of the GRMHD simulation is 40GM/c2. The fiducial radius(r = 12L, where L = GM/c2) is where all “state” quantities are evaluated. These state quantitiesare then used to estimate the validity of the fluid, MHD, and ideal MHD approximation at thislocation.

those simulations, the mass density, pressure, and magnetic field are determined self-consistentlyfor a thick disk with scale height to radius ratio of H/R ∼ 0.26. In that study, it was found that theglobal accretion flow can be understood basically as having different regions based upon the ratio ofmagnetic energy to mass energy (B2/ρ0c

2) and the ratio of gas to magnetic pressure (β ≡ pgas/pb,where pb = B2/2). Figure 1.1 shows the regions of the accretion flow. The disk is defined to bewhere β < 1/3, the plunging region where β > 1/3 within a disk scale height, the corona regionwhere 1/3 < β < 1, and the funnel region where B2/ρ0 > 1. The figure shows the fiducial radiusr = rfid = 12GM/c2 and the outer radius r = rout = 40GM/c2 used in the simulation. The fiducialradius rfid is the location where all “state” quantities are evaluated and all “validity parameters”are evaluated that test the validity of the fluid, MHD, and ideal MHD approximations.

These regions can be treated approximately based upon their average properties. We take aspatiotemporal average of quantities during the turbulent period of a global GRMHD simulation.The time average is performed over t = 1000GM/c3 to t = 2000GM/c3, and a volume integral isperformed over each region for each quantity in the table. Table 1.4 shows the average value ofthe dimensionless rest-mass density ρ0/ρ?, where ρ? is some fiducial rest-mass density; the valueof B2/ρ0 ; and the value of β for each of the 4 regions. I have determined the state (density,temperature, magnetic field, optical depth, etc.) for all of these regions for each of the 6 fiducialobjects. I have also determined the validity of the fluid, MHD, and ideal MHD approximations forall these regions for each of the 6 fiducial systems. For brevity, this section only presents the details

39

Table 1.4. Accretion Flow Regions

Region ρ0/ρ? B2/ρ0 β = pgas/pb

Disk 0.072 0.0004 9.5Plunge 0.18 0.018 1.6Corona 0.0068 0.003 1.5Funnel/Jet 0.00023 7.4 0.0047

Note. — Dimensionless quantities for regionsas described in Figure 4.2 for H/R = 0.26GRMHD simulation. See Section 1.5 for a de-tailed description.

for the accretion disk itself.From Table 1.4 one can obtain the dimensionless ideal gas temperature

θ ≡ β/2B2/ρ0

≡ pgas

ρ0c2, (1.14)

which can be converted to kelvin for an ideal gas of protons (where pgas = ρ0kbTp[K, ideal]/mp) toobtain

Tp[K, ideal] ∼= θ

(mpc

2

kb

). (1.15)

The GRMHD simulation of a thick disk, with an initially fixed H/R = 0.26, has a rest-massaccretion rate of

M0,numerical = AM

(ρ?c

(GM

c2

)2)

, (1.16)

where AM = 0.2 from the GRMHD numerical simulation. By setting the numerical mass accretionrate equal to the true rest-mass accretion rate (M0,numerical = M0), the rest-mass density scale (ρ?)and then rest-mass density (ρ0) can be determined from Table 1.4. The true mass accretion rate isdetermined from Table 1.3.

Our GRMHD model is a nonradiative model with an initial (and approximately always) heightto radius ratio of H/R ∼ 0.26. By using the ideal gas law relating density, temperature, and

40

pressure, H/R ∼ cs/vK , and B ∝ √ρ0c, one can derive the following proportionalities

ρ0 ∝(

H

R

)−1

P ∝(

H

R

)

T [K] ∝(

H

R

)2

(1.17)

B ∝(

H

R

)−1/2

,

and from numerical simulations one can estimate that

M0 ∝(

H

R

)−3/5

. (1.18)

These proportionalities are used to scale the simulated solution with H/R = 0.26 to another solutionwith arbitrary H/R. For the sake of consistency, the true rest-mass accretion rate is assumed tovary with H/R similarly as the numerical rest-mass accretion rate. It is assumed that the physicalrest-mass accretion rates are no different for thin or thick disks. The rest-mass accretion rate isdetermined by a few GRMHD simulations that show M0/ρ0 ∝ (H/R)2/5. No detailed GRMHDstudies of thin disks (H/R . 0.01) have been performed that generally verify these scaling laws,but they should be roughly accurate.

The state parameters are estimated for a characteristic length, time, and rest-mass density

L0 =GM

c2

T0 =GM

c3(1.19)

ρ? =

(M0

AM

)(1

cL20

),

where L0 and T0 are the natural length and time scales for a black hole, and ρ? is found from M0 =M0,numerical. The characteristic value of any other quantity can be found from these definitions ofmass, length, and time.

GRB Disk State Determination

Since the GRMHD simulations are nonradiative, the value of H/R is estimated from the results of anunmagnetized, viscous, neutrino-dominated accretion disk model (Popham et al., 1999) (hereafterPWF). PWF determine the balance between the energy loss due to radiation of neutrinos andenergy gain due to turbulent dissipation. The PWF model provides the disk scale height per unitradius (H/R) at a particular radius for a given rest-mass accretion rate. This value of H/R is used

41

to scale, as described above, the state resulting from the nonradiative GRMHD simulation withH/R ∼ 0.26 to the PWF solution for H/R.

X-ray Binary and AGN Disks State Determination

For the X-ray binary and AGN disks, a thin disk state is self-consistently found by solving theequations of mass, angular momentum, and energy conservation for a radiative disk as in the so-called α-disk model (Shakura and Sunyaev, 1973). This typically results in H/R . 0.05. A thickdisk state (studied, but not shown below) is assumed to be an ADAF with H/R ∼ 1 (Narayanand Yi, 1995). The only additional component required for these unmagnetized models is the fieldstrength, which is estimated using the H/R scaling law above. The non-disk regions are assumedto follow Table 1.4 for both thin and thick disks.

The State of the Accretion Disk

The “state” of the accretion disk for the 6 fiducial black hole accretion systems is shown in Table 1.5.The columns of the table correspond to one of the 6 fiducial systems, and the rows give estimatesfor state parameters such as density and field strength. Only the state of the disk region is shown.The first 5 rows of Table 1.5 are from Table 1.3 and shown here for convenience.

State parameters in Table 1.5 are

1. rfid: Characteristic fiducial radius in centimeters in a typical numerical study. Used todetermine accretion and rotation time scales.

2. rres: Characteristic smallest radius in centimeters in a typical numerical study. Used todetermine characteristic smallest scale L.

3. L: Characteristic smallest length scale in centimeters in a typical numerical study. The radialrange is resolved logarithmically dr/r ∝ const. and the height range as H/rres ∝ const..For Nr radial zones dr = rres log (rout/rin)/Nr, where rin and rout are the inner and outerradial ranges of interest. For this study typically rin ∼ GM/c2, near the event horizon, androut ∼ 40− 400GM/c2.

4. T : Characteristic smallest time scale in seconds in a typical numerical study (T = L/c).

5. ρ0: Characteristic rest-mass density in g cm−3.

6. B: Characteristic magnetic field in Gaussian units.

7. B/Bcrit,e: Fraction of QED electronic critical field for which the magnetic field exhibitsparticle modes and photons behave nonlinearly. Bcrit,e ≡ m2

ec3/(e~) ∼= 4 × 1013G. See for

example Heyl and Hernquist (1999). Only GRB disk has a field around the critical value.

8. Tp: Temperature of protons (ionized hydrogen) and other particles in kelvin.

42

Table 1.5. Accretion Disk State

Parameter NS-BH GRB 030329 LMC X-3 SgrA* NGC4258 M87

dL ? 804 Mpc 55 kpc 8 kpc 7 Mpc 18 MpcM [M¯] 3 3 10 2.6× 106 4× 107 3× 109

Lbol[erg/s] 1053 3× 1052 1038 1037 3× 1043 2.3× 1042

Lbol/LE 3× 1014 8× 1013 0.08 3× 10−8 0.006 6× 10−6

M0 5M¯/ s 0.1M¯/ s 10−8 M¯/ yr 10−5 M¯/ yr 10−2 M¯/ yr 10−2 M¯/ yr

rfid[cm] 5.3× 106 5.3× 106 1.8× 107 4.6× 1012 7.1× 1013 5.3× 1015

rres[cm] 4.4× 105 4.4× 105 1.5× 106 3.8× 1011 5.9× 1012 4.4× 1014

L[cm] 5.9× 103 5.9× 103 2× 104 5.1× 109 7.9× 1010 5.9× 1012

T [s] 2× 10−7 2× 10−7 6.6× 10−7 0.17 2.6 2× 102

ρ0[g cm−3] 6.1× 1011 1.6× 1010 0.0072 1.5× 10−6 2.4× 10−7 1.3× 10−8

B[G] 1.7× 1015 2.7× 1014 1.1× 106 1.4× 102 2.8× 102 3.7B/Bcrit,e 38 6.1 − − − −Tp[K] 2.1× 1010 1.2× 1010 1.8× 107 3.5× 105 4.8× 105 7.1× 104

xp 0.0019 0.0011 1.7× 10−6 3.2× 10−8 4.4× 10−8 6.5× 10−9

xe 3.5 2 0.0031 6× 10−5 8.2× 10−5 1.2× 10−5

nn/ndegen,n 0.32 0.0095 − − − −ne/ndegen,e 25 1.6× 102 − − − −Ye 0.01 0.5 0.5 0.5 0.5 0.5Xnuc 0.17 0.17 − − − −T/tNSE 1.3× 104 48 − − − −Xion,H 0.043 0.026 1 1 1 1tacc 0.091 s 0.15 s 32 s 1.4× 102 yr 2.7× 102 yr 8.4× 105 yrtrot 3.9 ms 3.9 ms 13 ms 0.93 hr 0.6 day 0.12 yrH/rfid 0.26 0.2 0.025 0.0011 0.0031 0.00048prad/pgas 0.0031 0.0057 18 0.62 9.7 0.57pgas/pgas,0 0.39 1.6 − − − −τγ 3.4× 1015 3.3× 1015 6.3× 102 1.5× 103 1.1× 104 6.6× 103

τν 1.2 0.0084 − − − −

Note. — First 4 parameters are previously discussed and are based upon astrophysical observations.The value of M0 is based upon both observations and models of the accretion disk (see Section 1.4). Theremaining parameters describe the state of the plasma in the accretion disk for various astrophysicalsystems at a radius of r = rfid = 12GM/c2, close to the black hole. The GRB-type disks (first2 columns) are modeled using a radiative VHD PWF model of the disk thickness (Popham et al.,1999) and nonradiative GRMHD simulations to set all other quantities. The other disks (remaining 4columns) are determined by the SS73 model of a thin accretion disk (Shakura and Sunyaev, 1973) forall quantities except the magnetic field strength, which is determined by scaling the GRMHD solutionto the appropriate H/R.

43

9. xp: Proton relativity parameter, xp = kbTp/(mpc2). Protons are relativistic if greater than 1,

otherwise nonrelativistic.

10. xe: Electron relativity parameter, xe = kbTe/(mec2). Electrons are relativistic if greater than

1, otherwise nonrelativistic. It is assumed that Te ∼ Tp.

11. nn/ndegen,n: Ratio of actual number density of neutrons to number density of neutrons ifdegenerate. Neutrons are degenerate if greater than 1, otherwise nondegenerate. See Kohriand Mineshige (2002). Only the GRB disk has nonnegligible neutron/proton degeneracy.

12. ne/ndegen,e: Ratio of actual number density of electrons to number density of electrons ifdegenerate. Electrons are degenerate if greater than 1, otherwise nondegenerate. See Kohriand Mineshige (2002). Only the GRB disk has nonnegligible electron degeneracy.

13. Ye: Electron fraction: Ye = ne/(np + nn). Assumes charge neutrality and that the positronnumber density is small (true for all the cases studied here). See Kohri and Mineshige (2002).

14. Xnuc: Mass fraction in free nucleons. Rest are mostly α particles. Assumes nuclear statisticalequilibrium. See Woosley and Baron (1992). Not relevant for SS73 model of disk for X-raybinary and AGN. A hot thick disk model may require this.

15. T/τnse: Ratio of the characteristic smallest time to the time to achieve nuclear statisticalequilibrium. See Khokhlov (1989). Not relevant for SS73 model of disk for X-ray binary andAGN. A hot thick disk model may require this.

16. Xion,H : Ionization fraction of electrons in a hydrogen gas. This assumes thermal statisticalequilibrium, and that pressure and radiative ionization are negligible. In GRB disks, theinterparticle spacing ∼ the electron orbit radius, so pressure ionization may increase theionization fraction. See Equation A.10 in Appendix A.

17. τacc: Accretion time scale: τacc ∼ (1/α)√

r3fid/(GM)/(H/rfid)2. Assumes α ∼ 0.1, and the

α model approximately holds.

18. τrot: Rotation time scale: τrot ∼ 2π(r3/2fid + a)

√GM . Assumes a ∼ 0.9375, although at rfid

this is not a significant effect.

19. H/rfid: Scale height of the disk per unit radius at the fiducial radius. Determined self-consistently without a magnetic field for the thin disk state. The SS73 solution is used forAGN and X-ray binaries, and the PWF solution is used for GRBs.

20. prad/pgas: Ratio of radiation to gas pressure, where prad = 13aT 4 for an optically thick medium

(τγ À 1), where a = 4σ/c and σ = 2π5k4b/(15h3c2) is the Stefan-Boltzmann constant. The gas

pressure for X-ray binary or AGN disks is pgas = (kb/µ)ρ0T , where µ is the mean molecularweight per free particle. For a hot fully ionized plasma of cosmic abundances µ ∼= 0.62, while

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for a cold cloud of molecular hydrogen µ ∼= 2.4. The GRB disk pressure is determined as inKohri and Mineshige (2002).

21. pgas/pgas,0: Ratio of GRMHD simulation estimated gas pressure to actual gas pressure asdescribed above. Error results if radiation pressure, degeneracy pressure, or other non-idealgas pressures exceeds the ideal gas pressure used in the GRMHD simulation. Not applicableto X-ray binary and AGN thin disk solution, which is constructed self-consistently as in SS73.

22. τγ : Disk optical depth to photons τγ ∼ κρ0H, where κ is the mean opacity for the given tem-perature and density (see Bell and Lin 1994 for photon scattering and absorption opacities).

23. τν : Disk optical depth to neutrinos. Since this is negligible for accretion flow near X-raybinaries and AGN, this is only shown for GRB disks.

GRB Disk State

See Appendix A for a discussion of GRB equations of state. The field in GRB accretion disksis slightly above the critical magnetic field where QED effects, such as photon splitting γ → γγ

and electron/positron/photon pair generation γ → e+e−γ, occur (see, e.g., Erber 1966). Noticethat the thin GRB disks have sub-relativistic protons and neutrons but have marginally relativisticelectrons. The NS-BH and collapsar disks have degenerate electrons, but marginally non-degenerateprotons and neutrons. For the NS-BH disk, the degeneracy of neutrons leads to a small electronfraction (i.e. high neutronization). The GRB disks are marginally in nuclear statistical equilibrium.At this particular radius, alpha particles dominate free nucleons. The PWF solution at slightlylarger radii gives a disk that is hotter (thicker) and so has more free nucleons.

Gas pressure dominates the thin GRB disks. The ratio of actual gas pressure to simulationgas pressure shows the NS-BH and collapsar disks are well approximated by the simulation gaspressure. Thus, these estimates are self-consistent. Since the GRB disks are very optically thick,the radiation in the disk can be approximated by a radiation pressure term. The collapsar disk hasa low neutrino optical depth, so the optically thin approximation is valid. Thus, a nonradiativeGRMHD model can be easily modified to include optically thin neutrino emission in order to studyradiative GRMHD collapsar models of long-duration GRBs. The NS-BH disk is estimated to havean optical depth of τ ∼ 1, but the effects of electron degeneracy may lower this value (Kohri andMineshige, 2002).

The states of the plunging and coronal regions are similar to the state of the disk. The funnelregion within the NS-BH GRB disk has a magnetic field strength ∼ 300× that of the critical fieldstrength, while the collapsar disk field strength is ∼ 40× the critical field strength. This indicatesa need to account for QED effects in the funnel region.

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Table 1.6. Validity of Fluid Approximation

Parameter NS-BH GRB 030329 LMC X-3 SgrA* NGC4258 M87

L/λmfp,e−p 4.5× 1011 1.7× 1012 1.1× 106 1.6× 1011 2.1× 1011 4× 1013

Tνc,e−p 1.9× 1010 8.2× 1010 2.7× 103 6.3× 107 8.7× 107 7.1× 109

Re 1.5× 1014 2.5× 1014 6.9× 106 7.2× 1012 8.3× 1012 4× 1015

Note. — If a system has all values much greater than unity, then the ideal fluid approxi-mation is valid. If all but Re are much greater than one, then the viscous fluid approximationis valid. The ideal fluid approximation is valid for all these systems in the thin disk stateshown. In a thick disk state (not shown) the X-ray binary and AGN systems do not explicitlybehave as a fluid, but may be forced to behave like a fluid due to plasma instabilities, asoccurs in the solar wind.

X-ray binary and AGN State

In the thin disk state, the X-ray binary and AGN have low relativity parameters and essentiallyno degeneracy. Treating the fluid as a nonrelativistic ideal gas is a good approximation as longas either 1) the gas pressure dominates over radiation pressure in the optically thin regime or 2)the disk is optically thick. Notice that all the disks are optically thick at the fiducial radius, so aradiative diffusion treatment can be used to study these disks at this location. Indeed, treating theradiation as simply a pressure may be sufficient.

The thick disk in the X-ray binary or AGN is estimated to be H/R ∼ 1. These disks are eithermarginally optically thick or completely optically thin. The thick disk (ADAF) models of LMC X-3,Sgr A*, and NGC4258 likely require radiative transport to understand their radiative properties atthe fiducial radius. However, the thick disk (ADAF) model of M87 is very optically thin and theradiation flux may be modeled by an optically thin emissivity. The states of the (thick) plungingand (thick) coronal regions are similar to the state of the thick disk.

1.5.2 Validity of the Fluid, MHD, and ideal MHD Approximations

Table 1.6 shows the estimates that test the validity of the fluid approximation. See Section A.2 fora discussion of the role of these dimensionless parameters. The value of L/λmfp,A−B is the numberof mean free paths, within a distance L, for particle A hitting particle B. The value of Tνc,A−B

is the number of collisions in a characteristic time. The conditions L/λmfp À 1 and Tνc À 1were tested for all colliding species, but for brevity only the electron-proton values are shown. Thevalue of Re is the Reynolds number. If a system has all values much greater than unity, then theideal fluid approximation is valid. If all but Re are much greater than one, then the viscous fluidapproximation is valid.

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Table 1.7. Validity of MHD Approximation

Parameter NS-BH GRB 030329 LMC X-3 SgrA* NGC4258 M87

Tωp 2.1× 1014 3.1× 1014 1.7× 109 6.4× 1012 4× 1013 6.9× 1014

L/λD 5.3× 1013 8.2× 1013 5.4× 109 1.5× 1014 7.8× 1014 3.5× 1016

Λ 3.7× 105 1.3× 105 7.6× 106 1.4× 106 5.6× 106 1.4× 106

Note. — If all rows are much greater than unity, then the plasma approximation is validfor that system. The plasma approximation is valid for all these systems.

All species are highly coupled in the GRB accretion disks. This includes the photons, whichcannot escape the disk. For the hotter regions, the temperature reaches the point where nuclearinteractions dominate over Coulomb interactions. The values shown for electron-proton collisionsare within a few orders of magnitude of the values for the proton-proton and neutron-protoncollisions. The weakest coupling is between protons and photons, but even these together act as afluid.

For the X-ray binary and AGN, the values are shown for the thin disk. The thin X-ray binaryand AGN disks have photons well-coupled to electrons, making the disk optically thick. All speciesare well-coupled except photons and protons.

For the X-ray binary and AGN, thick (ADAF) disk models of the disks may not be treatableby the fluid approximation without further consideration. The most coupled pair in these systemsis the weak coupling between protons and electrons. Despite the possible necessity of Boltzmanntransport in AGN and X-ray binaries with thick disks, likely plasma instabilities operate as in thesolar wind and the plasma can be treated effectively as a fluid. The solar wind too cannot beexplicitly treated as a fluid, yet the MHD approximation is a reasonable approximation (see, e.g.,Usmanov et al. 2000). Plasma instabilities likely force the solar wind to behave as a fluid (Feldmanand Marsch, 1997), and similar phenomena may occur in thick disks. In any event, the single-component fluid approximation serves as an interesting starting point for more realistic plasmacalculations. The plunging, corona, and funnel region regimes are similar.

Table 1.7 shows the estimates that test the validity of the plasma/MHD approximation. SeeSection A.4 for a discussion of the role of these dimensionless parameters. From top to bottom,these parameters are associated with 1) charge separation and a finite current rise time ; 2) chargeseparation; and 3) the coupling of plasma (> 1 implies weakly coupled). If all rows are much greaterthan unity, then the plasma approximation is valid for that system. The plasma approximation isvalid for all the states and regions of all objects. Thus, quasi-neutrality holds.

Table 1.8 shows the estimates that test the validity of the single-component ideal MHD approxi-mation. See Section A.5 for a discussion of the role of these dimensionless parameters. In summary,

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Table 1.8. Validity of Ideal MHD Approximation

Parameter NS-BH GRB 030329 LMC X-3 SgrA* NGC4258 M87

Tωg,p 3.1× 1012 5× 1011 7× 103 2.2× 105 7× 106 7× 106

L/λg,p 5.1× 1013 1.1× 1013 3.8× 106 8.7× 108 2.4× 1010 6.2× 1010

Tω2p/ωg,e 3.4× 1013 3.1× 1014 2.3× 1011 1× 1017 1.2× 1017 3.7× 1019

Lω2pdv/(ωg,ec

2) 6.5× 1010 6× 1011 4.4× 108 1.9× 1014 2.4× 1014 7.1× 1016

Ldvωg,e/v2therm,e 2.6× 1012 6.6× 1011 4× 106 6.6× 109 1.5× 1011 1× 1012

Lωg,p/vtherm,p 5.1× 1013 1.1× 1013 3.8× 106 8.7× 108 2.4× 1010 6.2× 1010

(L/λg,p)2/(Tωg,p) 8.2× 1014 2.3× 1014 2.1× 109 3.4× 1012 7.9× 1013 5.4× 1014

RM 6.7× 1013 1.1× 1014 7.6× 1011 5.3× 1014 1.3× 1016 5.4× 1016

dv/vdrift,p−n - - 2.3× 105 2.4× 1010 2.7× 109 1.2× 1012

dv/vdrift,e−n - - 1.9× 105 1.1× 1017 1.9× 1016 7.3× 1021

dv/vdrift,B 1.5× 1011 1.2× 1012 4.4× 108 1.9× 1014 2.4× 1014 7.1× 1016

Note. — If the term is much greater than unity, then the associated effect is negligible and furthermotivates the ideal-MHD approximation. If all terms are much greater than unity, then the single-component ideal-MHD approximation is valid for that system. The single-component ideal-MHDapproximation is valid for all these systems.

in order from top to bottom, the parameters correspond to the importance of 1) the gyration timescale ; 2) gyration length scale ; 3) charge separation ; 4) the Hall effect (J×B) ; 5) the electronpressure effect ; 6) an anisotropic pressure ; 7) finite gyration radii (FLR vs. MHD order, where > 1implies MHD ordering) ; 8) Ohmic dissipation (RM ≡magnetic Reynolds number) ; 9) ambipolardiffusion drift between protons and neutrons ; 10) ambipolar diffusion drift between electrons andneutrons ; and 11) the drift between electrons and protons that generates the current to sustain themagnetic field. If the term is much greater than unity, then the associated effect is negligible andfurther motivates the ideal-MHD approximation. If all terms are much greater than unity, thenthe ideal-MHD approximation is valid for that system.

The GRB disks (disk thickness set by PWF solution) and both the thin (SS73) and thick(H/R ∼ 1) disk solution for X-ray binary or AGN disks suggest the ideal MHD approximation isvalid. The plunging, corona, and funnel regions are qualitatively similar.

In summary, the fluid approximation is valid for the GRB disks and thin X-ray binary and AGNdisks. However, to maintain fluidity in thick disks in the X-ray binary or AGN, plasma instabilitiesare required to maintain fluidity. This is likely to occur, as in the solar wind (see, e.g., Usmanovet al. 2000; Feldman and Marsch 1997). The plasma approximation is valid for all disks in allregions. If fluidity is sustained, then the plasma and single-component ideal-MHD approximationsare valid for all systems in all regions.

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1.6 Summary of Motivation for a GRMHD Model and Open

Questions

HST, Chandra, and ground-based observations of GRB counterparts/afterglows, such as associatedwith GRB 020813 and GRB 030329, suggest a supernova connection to GRBs (see, e.g., Uemuraet al. 2003; Meszaros 2003). These results support the black hole accretion disk model for generatingGRB jets (see, e.g. Piran 1999; MacFadyen and Woosley 1999; van Putten and Ostriker 2001; Butleret al. 2003; Meszaros 2003). Observations of GRBs will always be folded through models, since theaccretion disk will not be resolved in the foreseeable future. The current most attractive modelfor long duration GRBs is the collapsar model (Woosley, 1993; Paczynski, 1998; MacFadyen andWoosley, 1999). In order to determine the flow’s Lorentz factors and the importance of a magneticfield to jet collimation (Cameron, 2001; Wheeler et al., 2002), a GRMHD model is likely required.

There are currently about 22 known X-ray binaries with a BHC (McClintock and Remillard,2003; Zand et al., 2004), where about 18 of those are confirmed (dynamical) black hole X-raybinaries. Estimates of the mass of the black hole come from observations of oscillations in themotion of the stellar companion. For example, the X-ray binary system Cygnus X-1 was foundto contain a compact object with a mass of M ∼ 7 − 13M¯, which is in excess of any possibleneutron star (Webster and Murdin, 1972; Bolton, 1972; Gies and Bolton, 1986; Herrero et al., 1995).Some of these X-ray binaries, called microquasars since comparable to quasars and related AGN(Mirabel and Rodrıguez, 1994, 1999; Fender and Belloni, 2004), exhibit apparently superluminal(v/c ∼ 3 − 10) jets, which likely originate from a relativistic system with a well-defined axis ofrotation. Microquasars likely form similarly to the jets that form in GRBs, and also likely requirea GRMHD model.

Astronomers have long thought that quasars are likely powered by the accretion of matter ontoa SMBH (Zeldovich, 1964; Salpeter, 1964; Blandford, 1984), and AGN have long been considered tobe related to quasars (Lynden-Bell, 1969). RXTE X-ray observations of AGN show rapid variabilityon the order of days. Observations of MCG-6-15-30 show variability on timescales of order 100 s(Reynolds et al., 1995; Yaqoob et al., 1997), suggesting that the very inner region of an accretiondisk is being probed (Edelson and Vaughan, 2000). Radio interferometry has been able to probewithin a few hundred gravitational radii of a central black hole within Sgr A* and M87 (see, e.g., Loet al. 1998; Junor et al. 1999; Doeleman et al. 2001). Fe Kα line emission shows different intensitiesfor the red and blue wings, which suggests the accretion disk is moving relativistically with speedsapproaching 0.2c (Pariev and Bromley, 1998; Tanaka et al., 1995). Thus, a general relativisticmodel of the accretion disk is required.

Some of the first observations of AGN were of so-called radio galaxies with jets extending a fewhundred kiloparsecs out of a point-like source. AGN jets are observed to be relativistic with somejets having variable structure moving at an apparent speed of 10 times the speed of light, (see,e.g., West et al. 1998). AGN jets are sources of synchrotron radiation (Baade, 1956), indicatingthe existence of a strong magnetic field. HST, VLA, and VLBI all provide high-resolution images

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of jets, including M87’s jet showing detailed large and small scale structure with apparent speedsof v/c ∼ 6 (see, e.g., Biretta et al. 1999). However, only future observatories have any hope ofresolving the inner accretion disk at the source of the jet in AGN (Rees, 2001). AGN jets are likelyproduced similarly to jets in GRBs and microquasars, and also likely require a GRMHD model.

Some GRBs, some X-ray binaries, and all AGN, are likely powered by a black hole accretiondisk system. The single-component ideal-MHD approximation applies to the disk and surroundingregions near the black hole for all these objects. A GRMHD model is likely required to study thisgeneral relativistic, highly magnetized plasma flow. Since the GRMHD equations are non-trivial tostudy analytically, it is necessary to perform numerical studies of these systems. In particular, self-consistent, time-dependent studies of the global solution (i.e. disk-corona or disk-jet connection)require a numerical model.

It has yet to be determined what fraction of the observed luminosity from AGN or X-ray binariesis due to (a) a radiative accretion disk, (b) black hole spin energy extraction, or (c) a disk wind. ForGRBs, photon production due to neutrino annihilation should also be considered as a componentof the luminosity. A radiative GRMHD model is required to determine the disk contribution to theluminosity due to photons or neutrinos, and has yet to be studied self-consistently for the globalaccretion flow. To determine the contribution of energy from black hole spin energy extraction orthe disk near the black hole, one requires a GRMHD model.

Unanswered questions about AGN jets, microquasar jets, and GRB jets include: 1) what isthe energy source for jets? ; 2) what is the connection between the disk and jet? ; 3) what isthe mechanism for jet production, collimation, and variability? ; and 4) is the jet composed ofion/electron or positron/electron pair plasma?

1.7 Summary of Dissertation Results

The introduction summarized the evidence for black hole accretion disk systems and discussedhow black hole accretion disk systems exhibit relativistic magnetized flow near a (likely rotating)black hole. It was discussed how a magnetic field self-consistently generates turbulence and angularmomentum transport in a disk, while unmagnetized models using an α-viscosity model introducead hoc parameters. The observational evidence for relativistic magnetized flow, and the lack of self-consistency in unmagnetized models, demonstrates that a GRMHD model is required. A discussionwas presented of radiative accretion disk models, which find the structure of an accretion disk bysolving the equations of mass, energy, and momentum conservation. At the end of the introduction,the state of the accretion disk in 6 fiducial systems was discussed, and the single-component idealMHD approximation was found to be valid for these 6 fiducial black hole accretion disk systems.It was shown that an optically thin radiative GRMHD collapsar model can accurately and self-consistently model some long-duration GRBs. Finally, a summary was provided for why a GRMHDnumerical model is required to study accretion flow in GRBs, X-ray binaries, and AGN.

Some isolated regions of an accretion flow may be well-described by simplified analytic solutions,

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such as the Blandford-Znajek (BZ) solution in the nearly force-free funnel region around a rotat-ing black hole (Blandford and Znajek, 1977) and the Gammie inflow solution for the high-densityplasma flow in the equatorial plunging region (Gammie, 1999). Typically, in deriving analytic so-lutions, the simplifying assumptions do not allow an understanding of the connection between, anda self-consistent solution for, the black hole, disk, corona, and jet. This is referred to as describingthe “global” accretion flow, rather than the “local” accretion flow. Despite some successful simpli-fied analytic solutions, the VHD, MHD, and GRMHD equations are sufficiently complicated thatnumerical studies would likely yield results not easily (or possibly) derived analytically. Numericalstudies are likely required to understand the global, time-dependent structure of accretion flowsaround rapidly rotating black holes.

In order to study black hole accretion disk systems, I first developed a code that integratesthe 2D global Newtonian VHD equations of motion with a pseudo-Newtonian potential (McKinneyand Gammie, 2002); then I developed a code that integrates the 2D and 3D global nonrelativisticMHD equations of motion with a partially relativistic treatment of Alfven and sound waves and anartificial resistivity to capture heating in current sheets (unpublished); and finally I helped developa code that integrates the GRMHD equations of motion in a stationary space-time (Gammie et al.,2003).

The main results of this thesis are that 1) the inner radial boundary condition must stayout of causal contact with the rest of the flow in order to avoid nonphysical outflows and othernumerically-induced artifacts; and the viscosity models are compromised by the arbitrariness ofmodel parameters (McKinney and Gammie, 2002); 2) the BZ-effect is likely an important sourceof energy due to the development of a strong magnetic field in the funnel region; 3) the BZ modelaccurately predicts the electromagnetic luminosity in the nearly force-free funnel region near blackholes with spin parameter a . 0.5 ; 4) the BZ model is qualitatively accurate for all black holespins; 5) the BZ-effect contributes significantly to the energy content of jets in systems with a thickdisk around a black hole with a spin parameter of a & 0.5; 6) for thick or thin disks, the nominalaccretion efficiency is typically close to the classical thin disk value and generally less than unity(McKinney and Gammie, 2004), while the Gammie inflow model suggested the efficiency may belarger than thin disk (including super-efficient) due to magnetic torques in the plunging region(Gammie, 1999) ; and 7) for thick and thin disks, the angular momentum per baryon accretedis consistent with the predictions of the Gammie inflow model, which suggested that the angularmomentum per baryon accreted should be less than predicted by viscous, unmagnetized thin diskmodels. The following discussion summarizes the results of this thesis to be described in detail inchapters 2 to chapters 4. Following these chapters, in chapter 5, is a summary of possible futurestudies.

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1.7.1 Viscous Hydrodynamics Summary

The turbulent transport of angular momentum has long been suspected as the cause of accretion indisks around stars. Viscosity is a natural model to capture the effective friction due to the mixingaction of turbulent flow (Shakura and Sunyaev, 1973). This viscosity is called an anomalousviscosity, since the true molecular viscosity is too small to drive angular momentum transport.Typically, the anomalous viscosity is estimated as generating a stress proportional to αP , where α

is a constant and P is the pressure. Such a model is referred to as an α-disk model. The actualmechanism of angular momentum transport was unknown until the rediscovery of the MRI (Balbusand Hawley, 1991). The MRI likely dominates the generation of turbulence in ionized accretiondisks (Balbus and Hawley, 1998).

I decided to develop a viscous hydrodynamics (VHD) code to study black hole accretion. TheVHD code uses a numerical method based on ZEUS-2D (Stone and Norman, 1992) with the additionof an explicit scheme for the viscosity. ZEUS uses an operator-split, finite-difference algorithm ona staggered mesh that uses an “artificial viscosity” to capture shocks. This artificial viscosity isonly activated in shocked regions. I was particularly interested in measuring the mass, energy,and angular momentum flux through the inner radial boundary. These diagnostics are useful in adiscussion of accretion disk luminosity and variability.

I used the VHD code to study black hole accretion models, which were similar to previouslystudied models using various experimental designs (Igumenshchev and Abramowicz 1999, 2000;Igumenshchev et al. 2000; all hereafter IA), (Stone et al. 1999, hereafter SPB). IA and SPB useddifferent models for the accretion disk, anomalous viscosity, and gravitational potential. Unfortu-nately, these authors found quite different results using otherwise similar methods. The primarydifferences between IA’s and SPB’s results are in the energy per baryon and angular momentumper baryon accreted. IA and SPB also found different results for the radial scaling power laws ofdensity, pressure, energy per baryon accreted, and angular momentum per baryon accreted. It wasuncertain if these differences were due to the experimental design or due to bugs in the numericalcode. Our goal was to form a bridge between these different results by using an identical code tostudy both types of experiments.

IA chose to form an accretion disk by injecting marginally bound matter (Bernoulli parameter <

0) far from the hole at the rate M0,inj . SPB choose to start with a torus in hydrostatic equilibrium.IA include all components of the (spherical polar) Navier-Stokes viscous stress tensor, while SPBonly include the r − φ and θ − φ components. SPB only include the toroidal shear components.They suggest the differential rotation leading to the MRI causes negligible poloidal shear comparedto the toroidal shear. Also, IA and SPB use different forms for the viscosity coefficient ν (units oflength2/time). IA use ν ∝ c2

s/Ωk, where cs =√

∂P/∂ρ0|S is the sound speed, S is the entropy,and Ωk =

√GM/r3 is the Keplerian angular velocity. SPB typically use ν ∝ ρ0, where ρ0 is the

rest-mass density. IA’s viscosity prescription is chosen to concentrate the stress where the disk ishot, while SPB’s choice is meant to confine the stress to the bulk of the mass. Any dimensionally

52

reasonable form of the viscosity coefficient can be used, but the goal is to choose a form that mimicsthe MRI driven angular momentum transport.

I used a single VHD code to study both IA’s and SPB’s experimental designs, and I was ableto reproduce the results of both IA and SPB. This confirms that the differences in their resultswere not due to a bug, but due to differences in their experimental design. Model choices thatintroduced the most significant differences include the choice of viscosity prescription and how tomodel the source of matter. However, I also found that both IA’s and SPB’s choices for othermodel parameters can lead to numerical artifacts. The model parameters in question are the radiusof the inner boundary condition (rin) and the form of the black hole’s gravitational potential.The gravitational potential can be either purely Newtonian (φ = GM/r) or pseudo-Newtonian(φ = GM/(r − 2GM/c2)) (Paczynski and Wiita, 1980).

There are two competing factors in deciding where to place the inner boundary: 1) the desire toproduce a simulation free of numerical artifacts; and 2) computational cost. The inner boundary’slocation, rin, must be chosen so the boundary condition applied there does not affect the flow atr > rin. In modeling a black hole, one must choose rin so that as rin → 2GM/c2 there are negligiblechanges in the solution. Thus, one must convergence test a solution as rin → 2GM/c2.

Of course, a Newtonian potential has no intrinsic length scale. However, as in Bondi flow (Bondi,1952), the sonic point (the radius for supersonic flow at which −vr/cs = 1) can determine a lengthscale. For inviscid flow, the sonic point determines the physical point at which information cannottravel upstream. In more complicated flows, this supersonic boundary may be time-dependent, butone can define a minimum distance away from rin that always contains a supersonic flow. Indeed,near a black hole there must exist a surface that is always supersonic (Shapiro and Teukolsky,1983).

Moving rin → 2GM/c2 in a nonrelativistic code is expensive (time step, dt ∝ (r − 2GM/c2)).Thus, the inner boundary is usually placed close to the horizon, but far enough away, so thesimulation can be completed in a reasonable time. However, if rin is chosen such that the flow issubsonic at the inner boundary, information can travel back into the flow. Clearly, this violatesthe physical model of a black hole and may lead to spurious measurements for the accretion rateof mass, angular momentum, and energy.

For the VHD numerical models with a pseudo-Newtonian potential, only by choosing rin .2.7GM/c2 was the flow supersonic at rin. In comparison, a numerical model with a pseudo-Newtonian potential with rin = 6GM/c2 has subsonic flow at rin. As studied by IA, a numericalmodel with a Newtonian potential has subsonic flow at all rin.

I found that the accretion of energy and angular momentum per baryon are affected by numericalartifacts for models with subsonic flow at rin. Models with subsonic flow at rin show up to 5 timeslower rest-mass accretion rates, and such models show spurious high-frequency oscillations in theaccretion rates of mass, energy, and angular momentum. Models with a Newtonian potentialnever develop consistent supersonic flow at rin, and the angular momentum per baryon accretedoscillates around zero instead of taking a definite value. Models with subsonic flow at rin showed

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more complicated outflows of matter and energy in the polar regions. Thus, significant numericalartifacts can be avoided by 1) using a pseudo-Newtonian potential; and 2) choosing a suitable rin,such that the flow is supersonic there.

These VHD simulations are limited by the 2D axisymmetric condition, the use of viscosity asa model of turbulence (as opposed to direction simulations of MHD turbulence), and use of thepseudo-Newtonian potential with no relativistic corrections.

1.7.2 Global 2D/3D MHD Summary

The α-disk viscosity model continues, to this day, to be used as a model for the turbulent transportof angular momentum in accretion disks. The thin α-disk model of accretion, based upon thisturbulent viscosity, predicts that the disk plunges radially into the black hole in a sharp transitionat the innermost stable circular orbit (ISCO) (Bardeen, 1970; Shakura and Sunyaev, 1973; Pageand Thorne, 1974; Thorne, 1974; Abramowicz et al., 1978). However, the true kinematic viscosityis much lower than the anomalous viscosity used in α-disk models.

A magnetic field was discovered to develop a magneto-rotational instability (MRI) in accretiondisks with a high ionization fraction (Balbus and Hawley, 1991; Hawley and Balbus, 1991). TheMRI dominates kinematic viscosity as a mechanism for the generation of turbulence and angularmomentum transport in accretion disks (Balbus and Hawley, 1998). The magnetic field may bedynamically important inside the ISCO and generate a non-negligible torque on the disk, and themagnetic field could allow for a super-efficient nominal accretion luminosity (Gammie, 1999; Krolik,1999b). A magnetic field also likely plays a central role in the generation and collimation of a jetnear a compact object (see, e.g., Blandford and Znajek 1977; Blandford and Payne 1982; Begelmanet al. 1984). In particular, the BZ effect may be responsible for producing a jet in gamma-raybursts (GRB), microquasars, and active galactic nuclei (AGN) (Blandford and Znajek, 1977). Themagnetic field could generate and heat corona (see, e.g., Stern et al. 1995).

I was interested in solving the 2D or 3D MHD equations of motion as applied to black holeaccretion systems to determine 1) whether the torque on the disk diminishes near the ISCO ; 2)whether a magnetic field can launch and collimate jets; and 3) whether the polar field is strongenough to make the BZ-effect comparable in magnitude to the accretion disk luminosity. I wasalso interested in whether the polar BZ-effect provides the energy source for jets, but a directmeasurement of the BZ-effect requires a general relativistic calculation.

I developed a ZEUS-type code (Stone and Norman, 1992) that evolves the equations of non-relativistic MHD. This MHD code uses an artificial resistivity (Stone and Pringle, 2001) to avoidthe generation of a class of numerical artifacts called “point shocks” and poor energy conservationin the dissipation of current sheets. The MHD code also includes relativistic corrections that limitthe Alfven and sound speed to the speed of light (Miller and Stone, 2000), as described below.

Typically, MHD numerical studies of a black hole accretion system show the development ofan evacuated funnel region around the poles. In unpublished simulations, I found that this funnel

54

region had Alfven speed va = B/√

ρ0 ∼ 100− 1000c. Since the stability criteria on the time step isdt < dx/va (dx being some grid size), a large Alfven speed sharply reduces the allowed time step. Imodified the equations of motion to limit the speed of Alfven and sound waves to the speed of light ;this is referred to as a wave speed limiter. The method involves including a partial correction for thedisplacement current. The modified equations of motion have Alfven speed va = B/

√ρ0 + B2/c2,

so as ρ0 → 0, va → c. The sound speed is similarly limited.The axisymmetric approximation may lead to unrealistic accretion flow geometries. Global

3D simulations of nonaxisymmetric accretion flow can give rise to a self-sustained dynamo, whileaxisymmetric flow cannot (Cowling, 1934). A numerical method that uses 3D spherical polar co-ordinates, rather than 3D Cartesian coordinates, allows more accurate (less diffusive) simulationsof spherical or cylindrical accretion flow. However, the coordinate singularity in such a coordinatesystem presents at least two problems. First, the coordinate singularity makes it difficult to ac-curately model non-radial flow near the axis. Second, the Alfven wave speed stability criteria onthe time step near the polar axis in 3D spherical polar coordinates gives dt ∝ r sin θ/va for motionin the φ-direction. For a standard rectangular-based grid, the flow is always over-resolved in theφ-direction near the outer radial region of the polar axis, and so simulations require many moretime steps to complete an evolution.

The MHD code was extended to include a type of 3D Cartesian grid in order to avoid these twoproblems. In this case, there is no coordinate singularity. In order to maintain decent sphericalsymmetry and avoid introducing a strong m = 4 mode due to the Cartesian grid, I model the innerand outer boundaries of the grid as pseudo-spherical (i.e. Cartesian grid approximation of an innerand outer sphere).

Using this MHD code, I reproduced work by Hawley (2000) and Stone and Pringle (2001) whonumerically model a thick disk of magnetized fluid using the MHD approximation. I found thatthere is no sharp transition at the ISCO in any fluid quantities, and the torque does not vanishnear the ISCO. Thin disks studies are required to compare with the thin disk model, but resolvinga thin disk requires much more computer power and has not yet been attempted.

Nonrelativistic or pseudo-Newtonian (Paczynski and Wiita, 1980) MHD models cannot modelblack hole rotation, hence cannot show black hole energy extraction by the BZ-effect. However,dimensional analysis allows one to estimate the BZ luminosity as LBZ ∼ B2r2

+c, where B is themagnetic field and r+ is the radius of the horizon. Since jets form around the polar axis of anaccretion disk near the black hole, I determine this dimensionally estimated BZ luminosity in thepolar region. The numerical model of an accretion disk thus provides the ratio of the BZ luminosityper unit rest-mass accretion rate. If I assume an Eddington rest-mass accretion rate, then I findthat the BZ luminosity is comparable to the energy generation rate observed in jets and radio lobesin the most luminous AGN.

Simulations also show a collimated Poynting flux dominated jet in the funnel region along thepolar axis. However, while the flow appears to be relativistic, there is no way to determine thisaccurately with a (mostly) nonrelativistic numerical model. The modified Newtonian code was

55

unable to properly evolve the relativistic flow in the funnel, since the Alfven and sound speedsreached near the speed of light. For example, I found it impossible to remove the inner radialboundary condition from causal contact with the rest of the flow (i.e. one cannot keep the innerradial boundary inside the fast point in MHD). Also, while I found a large Poynting flux fromthe horizon, the Newtonian model should not be capable of producing such an effect. Most of thePoynting flux energy could be artificially generated by contact between the inner radial boundaryand the flow at larger radii. Thus, confidence was lost in the numerical evolution of the funnelregion. The Lorentz factor of the jet and the dependence of the jet and BZ luminosity on blackhole spin remain undetermined until a general relativistic calculation is performed. Ultimately,these problems with the (unpublished) nonrelativistic MHD results pushed us to develop a generalrelativistic code.

1.7.3 HARM / GRMHD Summary

In collaboration with Charles Gammie and Gabor Toth, I helped developed the HARM (HighAccuracy Relativistic Magnetohydrodynamics) code. HARM directly solves the equations of idealMHD in conservative form using a Godunov method that solves for the conserved fluxes using asimplified Lax-Friedrich (LF) or Harten-Lax-van Leer (HLL) scheme. Conservative methods canaccurately resolve shocks. However, if the various types of energy (rest-mass, internal energy,kinetic energy, and magnetic energy) in the problem become disparate in magnitude, then due tothe form of the conservative equations of motion (the energy scales are sometimes summed together)truncation error in one type of energy can lead to large errors in other energies.

HARM uses 8 primitive quantities: 1) comoving mass density; 2) comoving internal energydensity ; 3,4,5) three velocity components; and 6,7,8) three magnetic field components. HARM usesthe primitive quantities to compute 8 conserved quantities: 1) rest-mass density or, equivalently,particle number ; 2) total energy; 3,4,5) three momentums (one is angular momentum); and 6,7,8)three magnetic fluxes. The physically conserved quantities are total rest-mass (or total particlenumber), energy, angular momentum, and magnetic flux. HARM computes the conserved quantitiesfrom primitive quantities ; evolves the conservative quantities ; and inverts the conserved quantitiesto primitive quantities using a multidimensional Newton-Raphson method.

At the same time as HARM was being written, De Villiers and Hawley (2003a) developed a fullyrelativistic code using a different numerical method. Their code solves the nonconservative form ofthe equations of motion where primitive variables are directly evolved. This has the advantage ofbeing faster than HARM, since it avoids the inversion of conserved to primitive variables. It is alsoless diffusive compared to LF or HLL methods. However, their method has problems with accuracyin strong shocks and it is uncertain whether loss of conservation of particle number, energy, andangular momentum significantly affects the solution.

In practice, HARM and the method by De Villiers and Hawley (2003a) give comparable resultswhen applied in astrophysical contexts. Indeed, their method is better able to handle regions with

56

b2/ρ0 À 1 (magnetic energy density to mass energy density ratio much greater than one), suchas the funnel region. However, both codes have difficulty in this funnel region when evolving anaccretion disk around a rapidly spinning black hole. Their current method uses Boyer-Lindquistcoordinates that requires a high-resolution grid near the event horizon for an accurate measurementof the accretion rate of energy and angular momentum per unit mass. HARM uses Kerr-Schild(horizon-penetrating) coordinates, and this allows HARM to find reasonable accretion rates ofenergy and angular momentum at lower resolution.

HARM could be improved to handle flows with larger b2/ρ0 by using methods that reduce thetruncation error. HARM is limited by the exclusion of dynamic space-time, radiation, and othermicrophysics. Some additional physics will likely continue to be modeled phenomenologically. Oneuseful phenomenological model is a cooling function that maintains a constant disk scale height toradius ratio (H/R). This can be used to study disks with different H/R independent of the originof the scale height, which would naturally be determined by balancing radiative and gas pressureeffects. Other physical processes that do not dynamically couple to the accretion flow, such as thesynchrotron cooling in a jet, can be accurately modeled without significant modification to HARM.

1.7.4 BZ Effect Summary

Some GRBs, microquasars, and AGN may be powered by the electromagnetic braking of a rapidlyrotating black hole. The BZ-effect (here, broadly defined as any electromagnetic means of extractingenergy from a rotating black hole) is the most likely astrophysical means of extracting energy from arapidly rotating black hole. Estimates for the nominal black hole spin in astrophysical environmentsgive a rapid black hole spin of about a ∼ 0.92 (Gammie et al., 2004). Phenomenological estimatesdetermined that the BZ luminosity is likely small compared to the disk luminosity (Ghosh andAbramowicz, 1997; Armitage and Natarajan, 1999; Livio et al., 1999). However, phenomenologicalmodels have only been worked out for simple field geometries. Given the complicated nature of theaccretion flow and its connection to the jet, their estimates may not apply to physical accretionflow or to a global solution of the disk + jet (McKinney and Gammie, 2004).

I investigated the BZ-effect using HARM via axisymmetric numerical simulations of a rapidlyrotating black hole surrounded by a magnetized plasma. The plasma is described by the equationsof GRMHD, and the effects of radiation are neglected. The evolution is followed for 2000GM/c3,and the computational domain extends from inside the event horizon to typically 40GM/c2. Theinitial conditions are similar to the nonrelativistic MHD study. The numerical model starts with aplasma torus, in hydrostatic equilibrium, with an overlaid weak magnetic field. The torus is at aninitially fixed H/R ∼ 0.26, and typically remains at that H/R since no radiation was included.

I found that all models with a & 0.5 have an outward Poynting flux on the horizon in the Kerr-Schild frame, which means energy is extracted from the black hole. None of the models have netoutward energy flux on the horizon. One model, with a net magnetic flux through the disk, showsa net outward angular momentum flux on the horizon. The model with a net magnetic flux in the

57

disk also has the largest black hole energy extraction rate per unit nominal accretion luminosity of80%. Thus, the BZ luminosity can be comparable to the disk luminosity.

The limitations of the numerical models presented include the assumption of axisymmetry,an ideal gas equation of state, and a nonradiative gas. The assumption of axisymmetry is likelynot problematic since I find quantitative agreement with 3D results for the energy and angularmomentum per baryon accreted through the horizon (De Villiers and Hawley, 2003a; De Villierset al., 2003a) and the ratio of electromagnetic energy flux to matter energy flux on the horizon(Krolik, private communication). Models in axisymmetry may overestimate the BZ effect by, forexample, allowing the presence of axisymmetric sheets of magnetic field. In 3D, these sheetswould be disrupted by MHD turbulence. Also, a compressed or tangled axisymmetrically-forcedmagnetic field is less likely to diffuse than the same magnetic field in 3D. Radiative effects arecrucial for comparing with observations, but are only easy to implement in limited form, e.g. weaksynchrotron cooling in jets. Radiative effects are likely dynamically important in accretion disks inX-ray binaries and AGN, such as demonstrated by the photon bubble instability (Gammie, 1998).

1.7.5 BZ/Inflow Solution Comparison Summary

I compared the GRMHD numerical results of black hole accretion to two analytic steady statemodels: 1) the force-free magnetosphere solution of Blandford & Znajek (Blandford and Znajek,1977) and 2) the equatorial inflow solution of Gammie (Gammie, 1999). The BZ solution applies tothe funnel region near the polar axis of the rotating black hole, while the Gammie inflow solutionapplies to the equatorial region inside the ISCO.

I presented a self-contained rederivation of the Blandford-Znajek model in Kerr-Schild (horizonpenetrating) coordinates. Unlike the original BZ derivation, this solution does not require the useof a physical observer to determine the boundary condition on the coordinate singularity at thehorizon, as required in Boyer-Lindquist coordinates. The solution for the far-field radial dependenceof the energy flux only relies on the solution’s separability, rather than requiring the solution tomatch Michel’s (1973) solution, as done by BZ.

I used HARM to evolve an accretion disk around a black hole with a spin parameter of a = 0.5.A nearly force-free region developed in an evacuated funnel region around the poles of the blackhole. Thus, in this region the BZ solution could be compared to the solution from the numericalmodel. In this force-free region, on the horizon, I measured 1) the radial magnetic field (Br), 2) theelectromagnetic rotation frequency (ω), and 3) the electromagnetic energy extracted (E(EM)). Theanalytic BZ solution of these quantities was found to be in excellent agreement with the GRMHDnumerical model solution. Since the BZ solution is a perturbative solution, valid for a ¿ 1, rapidlyrotating black holes are not expected to follow the BZ solution. However, all black hole spins(−0.938 ≤ a ≤ 0.969) and field geometries tested show a nearly maximally efficient BZ-processwith ω ≈ ΩH/2 on the horizon, as expected (Thorne and MacDonald, 1982), where ΩH ≡ a/(2r+)is the rotation frequency of the black hole. Models with arbitrarily large black hole spin show a

58

nearly force-free evacuated funnel region with a BZ-luminosity that is qualitatively similar to theBZ solution. As in the BZ solution, the numerical solution contains field lines that extend to largeradius in the nearly force-free funnel region. As predicted by the BZ solution, along such field linesthe value of ω is nearly constant.

The Gammie inflow solution determines the fully relativistic MHD flow quantities for a sta-tionary, cold (H/R = 0) disk. The solution is completely determined by the rest-mass accretionrate and a parameter that sets the magnetization of the flow. The Gammie inflow solution doesfairly well at predicting the space-time average of many flow quantities. In particular, the Gammieinflow agrees fairly with the numerical solution for the accretion rate of particle energy vs. radius,accretion rate of electromagnetic energy vs. radius, and comoving magnetic energy vs. radius. Theradial velocity and mass density vs. radius do not fit well. This is partially because the Gammieinflow solution does not model hot flow, while the numerical solution is of a hot disk. Future ana-lytic studies may investigate a hot inflow solution, and relax other assumptions (see, e.g., Li 2004),to compare with numerical results. The Gammie inflow model suggests that magnetic torques inthe plunging region may increase the nominal accretion efficiency η (even beyond unity). I foundall models have η < 1. Remarkably, the efficiency closely follows the thin disk efficiency of Bardeen(1970).

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2 Numerical Models of ViscousAccretion Flows Near Black Holes

2.1 Summary of Chapter

We report on a numerical study of viscous fluid accretion onto a black hole. The flow is axisymmetricand uses a pseudo-Newtonian potential to model relativistic effects near the event horizon. Thenumerical method is a variant of the ZEUS code. As a test of our numerical scheme, we are ableto reproduce results from earlier, similar work by Igumenshchev and Abramowicz and Stone etal. We consider models in which mass is injected onto the grid as well as models in which aninitial equilibrium torus is accreted. In each model we measure three “eigenvalues” of the flow: theaccretion rate of mass, angular momentum, and energy. We find that the eigenvalues are sensitiveto rin, the location of the inner radial boundary. Only when the flow is always supersonic onthe inner boundary are the eigenvalues insensitive to small changes in rin. We also report on thesensitivity of the results to other numerical parameters.1

2.2 Introduction

Black hole accretion flows are the most likely central engine for quasars and active galactic nuclei(AGN) (Zeldovich, 1964; Salpeter, 1964). As such they are the subject of intense astrophysicalinterest and speculation. Recent observations from XMM-Newton, Chandra, Hubble, VLBA, andother ground- and space-based observatories have expanded our understanding of the time vari-ability, spectra, and spatial structure of AGN. Radio interferometry, in particular, has been ableto probe within a few hundred gravitational radii (GM/c2) of the central black hole, e.g. Lo et al.(1998); Junor et al. (1999); Doeleman et al. (2001). Despite these observational advances, onlyinstruments now in the concept phase will have sufficient angular resolution to spatially resolve theinner accretion disk (Rees, 2001). And so there remain fundamental questions that we can onlyanswer by folding observations through models of AGN structure.

All black hole accretion flow models require that angular momentum be removed from the flowin some way so that material can flow inwards. In one group of models, angular momentum isremoved directly from the inflow by, e.g., a magneto-centrifugal wind (Blandford and Payne, 1982).Here we will focus on the other group of models in which angular momentum is diffused outward

1Published in ApJ Volume 573, Issue 2, pp. 728-737. Reproduction for this thesis is authorized by the copyrightholder.

60

through the accretion flow.It has long been suspected that the diffusion of angular momentum through an accretion flow is

driven by turbulence. The α model (Shakura and Sunyaev, 1973) introduced a phenomenologicalshear stress into the equations of motion to model the effects of this turbulence. This shear stressis proportional to αP , where α is a dimensionless constant and P is the (gas or gas + radiation)pressure. This shear stress permits an exchange of angular momentum between neighboring, dif-ferentially rotating layers in an accretion disk. In this sense it is analogous to a viscosity (see alsoLynden-Bell and Pringle (1974)) and is often referred to as the “anomalous viscosity.”

The α model artfully avoids the question of the origin and nature of turbulence in accretiondisks. This allows useful estimates to be made absent the solution to a difficult, perhaps in-tractable, problem. Recently, however, significant progress has been made in understanding theorigin of turbulence in accretion flows. It is now known that, in the magnetohydrodynamic (MHD)approximation, an accreting, differentially rotating plasma is destabilized by a weak magnetic field(Balbus and Hawley, 1991; Hawley and Balbus, 1991). This magneto-rotational instability (MRI)generates angular momentum transport under a broad range of conditions. Numerical work hasshown that in a plasma that is fully ionized, which is likely the case for the inner regions of mostblack hole accretion flows, the MRI is capable of sustaining turbulence in the nonlinear regime(Hawley and Balbus, 1991; Hawley et al., 1995; Hawley, 2000; Hawley and Krolik, 2001).

Studies of unmagnetized disks have greatly reduced the probability that a linear or nonlinearhydrodynamic instability drives disk turbulence. While there are known global hydrodynamicinstabilities that could in principle initiate turbulence, these have turned out to saturate at lowlevels or require conditions that are not relevant to an accretion disk near a black hole. As of thiswriting, no local, linear or nonlinear hydrodynamic instabilities that transport angular momentumoutwards are known to exist in Keplerian disks (Balbus and Hawley, 1998).

Work on magnetized disks has now turned to global numerical models. These are possiblethanks to advances in computer hardware and algorithms. Recent work by Hawley (2000); Hawleyand Krolik (2001), Stone and Pringle (2001), and Hawley and Krolik (2001) considers the evolutionof inviscid, nonrelativistic MHD accretion flows in two or three dimensions. Some of this workuses a pseudo-Newtonian, or Paczynski and Wiita (1980), potential as a model for the effects ofstrong-field gravity near the event horizon.

Other work on global models has considered the equations of viscous, compressible fluid dynam-ics as a model for the accreting plasma (Igumenshchev and Abramowicz, 1999; Stone et al., 1999;Igumenshchev and Abramowicz, 2000; Igumenshchev et al., 2000). The viscosity is meant to mockup the effect of small scale turbulence, presumably generated by magnetic fields, on the large scaleflow. In light of work on numerical MHD models, this may seem like a step backwards. The MHDmodels, however, are computationally expensive and introduce new problems with respect to initialand boundary conditions. It therefore seems reasonable to investigate the less expensive α basedviscosity models. In this paper we investigate axisymmetric, numerical, viscous inflow models.

This work was motivated by the earlier work of Igumenshchev and Abramowicz (1999, 2000) and

61

Stone et al. (1999), hereafter referred to as IA99, IA00, and SPB99, respectively (IA99 and IA00are collectively referred to as IA, in which case SPB99 is simply referred to as SPB). These authorsstudied similar viscous inflow models yet found different radial scaling laws for radial velocity,density, and angular momentum. They also found different values for the accretion rate of massand angular momentum. They used different experimental designs, however. We set out to discoverwhether the results from these authors differed due to numerical methods or model parameters.

Along the way, we took a systematic approach to studying numerical parameters and boundaryconditions. One particular point of concern, which will be described in greater detail below, is theinner boundary condition. This lies in the energetically dominant portion of the flow, so errors therecan potentially corrupt the entire model. In this paper we show that aspects of results presented byother researchers are sensitive to model and numerical parameters. These results should be usefulto others contemplating large-scale numerical models of black hole accretion.

The paper is organized as follows. In § 2.3 we discuss our models. In § 2.4 we discuss numericalmethods. In § 2.5 we discuss a fiducial solution and results from a survey of other solutions. In§ 2.6 we summarize our results.

2.3 Model

We are interested in modeling the plasma within a few hundred GM/c2 of a black hole. We willconsider only axisymmetric models (the work of Igumenshchev et al. (2000) suggests that 2D and3D viscous models give similar results). Throughout we use standard spherical polar coordinatesr, θ, and φ.

We solve numerically the axisymmetric, nonrelativistic equations of compressible hydrodynam-ics in the presence of an anomalous stress Π, which is meant to model the effects of small-scaleturbulence on the mean flow. The governing equations then express the conservation of mass

Dρ0

Dt+ ρ0(∇ · v) = 0, (2.1)

momentum,

ρ0DvDt

= −∇P − ρ0∇Ψ−∇ ·Π, (2.2)

and energy,Du

Dt= −(P + u)(∇ · v) + Φ. (2.3)

Here, as usual, D/Dt ≡ ∂/∂t + v · ∇ is the Lagrangian time derivative, ρ0 is the rest-mass density,v is the velocity, u is the internal energy density, P is the pressure, and Ψ is the gravitationalpotential. The dissipation function Φ is given by the product of the anomalous stress tensor Π

with the rate-of-strain tensor e

Φ = Πijeij , (2.4)

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(sum over indices) where the anomalous stress tensor is the term-by-term product

Πij = −2ρ0νeijSij , (2.5)

(no sum over indices) where Sij is a symmetric matrix filled with 0 or 1 that serves as a switch foreach component of the anomalous stress. The rate of strain tensor e is a symmetric tensor that inspherical polar coordinates has

err =∂vr

∂r− 1

3(∇ · v), (2.6)

eθθ =1r

∂vθ

∂θ+

vr

r− 1

3(∇ · v), (2.7)

eφφ =vr

r+

vθ

rcot θ − 1

3(∇ · v) +

1r sin θ

∂vφ

∂φ, (2.8)

erθ =12(r

∂

∂r(vθ

r) +

1r

∂vr

∂θ), (2.9)

erφ =12(r

∂

∂r(vφ

r) +

1r sin θ

∂vr

∂φ), (2.10)

andeθφ =

12(sin θ

r

∂

∂θ(

vφ

sin θ) +

1r sin θ

∂vθ

∂φ). (2.11)

The equation of state isP = (γ − 1)u. (2.12)

For the gravitational potential we use the pseudo-Newtonian potential of Paczynski and Wiita(1980): Ψ = −GM/(r− rg) (here rg ≡ 2GM/c2). This potential reproduces features of the orbitalstructure of a Schwarzschild spacetime, including an innermost stable circular orbit (ISCO) locatedat r = 6GM/c2. In a few cases we use the Newtonian potential Ψ = −GM/r for comparison withothers’ work. IA and SPB describe results exclusively from a Newtonian potential, although IA00report in a footnote that otherwise identical experiments in a pseudo-Newtonian potential showsignificant changes in the flow structure for r ∼ 2GM/c2.

We must now make some choices for the anomalous stress tensor. One might argue on verygeneral grounds for a Navier-Stokes prescription, and indeed IA and we use a prescription whereall elements of S are 1 (the “IA prescription”). SPB, on the other hand, use the Navier-Stokesprescription with all components zero except Srφ, Sθφ, Sφr, and Sφθ (the “SPB prescription”). SPBjustify this choice by arguing that it more appropriately models MHD turbulence; this was latersupported by results presented in Stone and Pringle (2001).

We must also choose a viscosity coefficient. We consider three different prescriptions: onesimilar to those chosen by IA; a second viscosity coefficient similar to that chosen by SPB; and athird, similar form that vanishes rapidly near the poles. Explicitly,

ν = α(c2s/ΩK), (2.13)

63

ν = α(ρ0/ρ?)Ω0r20, (2.14)

ν = α(c2s/ΩK)sin3/2(θ), (2.15)

are the IA, SPB, and MG prescriptions, respectively, where cs =√

γP/ρ0 is the sound speed, and

Ω2K ≡ 1

r

∂Ψ∂r

=GM

r(r − rg)2(2.16)

is the “Keplerian” angular velocity. Here ρ? and Ω0 are values of ρ0 and Ω at a fiducial radius r0.The choice of viscosity coefficient for IA and MG is based, as usual, on dimensional arguments.

SPB99’s choice focuses the viscosity where most of the matter is, a numerical convenience. Our MGprescription is a small modification of the IA prescription to concentrate the viscosity toward theequator. These choices are to a large extent arbitrary, although one might attempt to motivate thechoice by comparison with MHD simulations, as do Stone and Pringle (2001). Nevertheless, somedynamical properties of MHD turbulence, such as the elastic properties that produce magnetictension and hence Alfven waves, can never be modeled with a viscosity.

The model has boundaries at θ = ±π/2 and at r = rin, rout. At the θ boundaries we usethe usual polar axis boundary conditions. At the radial boundaries we use “outflow” boundaryconditions; ideally these boundary conditions should be completely transparent to outgoing waves.

Fuel for the accretion flow must be provided either in the initial conditions or continuously overthe model evolution. Some global numerical accretion flow models have started with an equilibriumtorus. Examples include Hawley (1991), Hawley et al. (1995), Hawley (2000), Hawley and Krolik(2001), Hawley et al. (2001), and Hawley and Krolik (2001). Others have started with an initialconfiguration of matter that is not in equilibrium. For example, matter may be placed in orbitabout the black hole, but with sub-Keplerian angular momentum, so that once the simulationcommences it immediately falls toward the hole. Examples of this approach include Hawley et al.(1984), Koide et al. (1999), Koide et al. (2000), and Meier et al. (2001). This approach may enhancetransients associated with the choice of initial conditions, although it can also be physically wellmotivated, as in studies of core-collapse supernovae. One can also inject fluid continuously ontothe computational grid over the course of the evolution. Examples of this include IA99 and IA00.One might also use an inflow boundary condition at the outer radial edge of the grid, as in Blondinet al. (2001). The main advantage of injection models is that they allow one to achieve a steady,or statistically steady, state. In this paper we will consider only the equilibrium tori and on-gridinjection models.

The equilibrium tori (Papaloizou and Pringle, 1984; Fishbone and Moncrief, 1976; Jaroszynskiet al., 1980) are steady-state solutions to the equations of inviscid hydrodynamics. They assume apolytropic distribution of mass and internal energy, P = Kρ0

γ , and a power-law rotation profile,Ω ∝ (r sin θ)−q. The radial and meridional components of the velocity vanish. There are 5 param-eters that describe the torus: (1) the location of the torus pressure maximum r0 ; (2) the locationof the innermost edge of the torus, rt,in < r0; (3) the maximum value of the density, ρ? = ρ0(r0);

64

(4) the angular velocity gradient q = d ln Ω/d ln R; and (5) the entropy constant K.On-grid injection adds matter to the model at a constant rate. The matter is injected with

a non-zero specific angular momentum and with zero radial or meridional momentum in a steadypattern ρ0(r, θ) which is typically symmetric about the equator. Parameters for this scheme include:(1) a characteristic radius for injection rinj ; (2) the rate of mass injection Minj ; (3) the specificangular momentum of the injected fluid, vφ = f1rΩK . We usually set f1 = 0.95, so that thefluid circularizes near rinj . This restricts transients associated with circularization to the outerportions of the computational domain; (4) the internal energy of the injected fluid, u = f2ρ0Ψ.We always set f2 = 0.2 so that the fluid is marginally bound, i.e. has Bernoulli parameter Be ≡(1/2)v2 + c2

s/(γ − 1) + Ψ < 0.One must also choose the injection pattern ρ0(r, θ). IA99 choose a radially narrow region, but

do not explicitly give ρ0(r, θ). We use a Gaussian, but found that none of the results are sensitiveto the precise profile. The accretion rate of mass, energy, and angular momentum are insensitiveto large changes in the size of the injection region except for the extreme cases of filling the entireθ width or injecting in 2 locations. These extreme cases are sufficiently different to be referredto as a completely different model; they lead to a qualitative change in the flow. For example, afull range θ injection region has matter that will collide with any outflow at the poles. A bipolarinjection leads to an equatorial outflow. Our models have radial width σr = 0.05(rin + rout)/2 andσθ = π/8.

2.4 Numerical Methods

Our numerical method is based on ZEUS-2D (Stone and Norman, 1992) with the addition ofan explicit scheme for the viscosity. ZEUS is an operator-split, finite-difference algorithm on astaggered mesh that uses an artificial viscosity to capture shocks (in addition to the anomalousviscosity in equations [2.2]). This algorithm guarantees that momentum and mass are conserved tomachine precision. Total energy is conserved only to truncation error, so total energy conservationis useful in assessing the accuracy of the evolution.

The inner and outer radial boundary conditions are implemented by copying primitive variablevalues (ρ0, u, and v) from the last zone on the grid into a set of “ghost zones” immediatelyoutside the grid. Inflow from outside the grid is forbidden; we set vr(rin) = 0 if vr(rin) > 0 andvr(rout) = 0 if vr(rout) < 0. Since we expect inflow on the inner boundary, this switch shouldseldom be activated. We have found that frequent activation of the switch is usually an indicationof a numerical problem.

We use a radial grid uniform in log(r− rg). We require that dr(r)/(rin − rg) ≤ 1/4 so that thestructure of the pseudo-Newtonian potential is well resolved. The grid is uniform in θ. The gridhas Nr ×Nθ zones.

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2.4.1 Numerical Treatment of Low Density Regions

Like many schemes for numerical hydrodynamics, ZEUS can tolerate only a limited dynamic rangein density. It is therefore necessary to impose a density minimum ρ0fl to avoid small or negativedensities. Our procedure for imposing the floor is equivalent to adding a small amount of mass tothe grid every time the floor is invoked. Mass is added in such a way that momentum is conserved.To monitor the effect of the density floor, we track the rate of change of total mass and total energy(from kinetic energy change) due to this procedure, Mfl and Efl.

We set ρ0fl = 10−10Minjc3/(GM)2 for injection runs and ρ0fl = 10−5ρ? for torus runs. Lower

values for ρ0fl do not lead to a significant change in the solution. Larger values of ρ0fl giveMfl ∼ M , the accretion rate through the inner boundary. The atmosphere also becomes moremassive and begins to affect torus stability– vertical oscillations are excited in the inner disk by aKelvin-Helmholtz like instability. This should be avoided.

We must also surround the torus in a low density atmosphere in the initial conditions. The den-sity of the atmosphere is ρ0fl and the internal energy density is u = Uoρ0Ψ, where Uo is a constantfraction of order unity (e.g. IA and we choose Uo = 0.2). The addition of the atmosphere has noeffect on the solution since the mass source’s evolution eventually dominates the flow everywhere.SPB99 choose a different method of constructing the initial atmosphere but obtain late-time resultsthat are similar to ours.

It is also necessary to impose a floor ufl on the internal energy density. This we take to be theminimum value of u in the initial atmosphere. As for the mass, we track the rate of change of totalenergy due to the internal energy floor, that along with the kinetic energy is included in Efl.

2.4.2 Diagnostics

Global numerical simulations of accretion flows are complicated; it is possible to measure manyquantities associated with the flow. Some are astrophysically relevant, and some are not. In ourview particular interest attaches to the time-averaged flux of mass, energy, and angular momentumthrough the inner boundary. Physically, these are directly related to the luminosity of the accretionflow and the rate of change of mass and angular momentum of the central black hole. As describedby Narayan and Popham (1993), these are in a sense the nonlinear “eigenvalues” of the model.

The rest-mass accretion rate isM =

∫

S

ρ0v · dS, (2.17)

where S is the inner radial surface of the computational domain. The total energy accretion rate is

E =∫

S

((12v2 + h + Ψ)ρ0v + Π · v) · dS, (2.18)

where h = (u + P )/ρ0 = γu/ρ0 is the specific enthalpy with our equation of state. The angular

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momentum accretion rate is

L =∫

S

r sin θ(ρ0vvφ + Π · φ) · dS. (2.19)

It is also sometimes useful to focus on the reduced eigenvalues l = L/M and e = E/M . Thesevalue of mass, energy, and angular momentum are recorded at about 2 grid zones away from rin.This avoids any error that may occur when evaluating directly on the boundary where the inflowboundary condition is applied.

We also track volume-integrated quantities, the flux of mass, energy, and angular momentumacross all boundaries, and floor added quantities in order to evaluate the consistency of the results.Mass and angular momentum are conserved to machine precision, although “machine precision”implies a surprisingly large random walk in the integrated quantities over the full integration becausethe calculation requires millions of timesteps.

Total energy is conserved to truncation error, not machine precision, and thus is a useful checkon the quality of the simulation. Total energy conservation implies

Eerr = Evol + E + Eout − Efl, (2.20)

where Evol is the rate of change of the volume integrated total energy, Eout is the flux of totalenergy through the outer radial boundary, and Efl is the rate of total energy added due to thekinetic energy change (because of the mass density floor) and internal energy density floor. Ideally,Eerr = 0. Truncation errors can (and do) lead to cumulative, rather than random, changes in thetotal energy. A useful gauge of the magnitude of these errors is Eerr/E. For all runs we performedthe error rate is within 10% of 10−5c3/GM for a torus run and within 10% of 10−4c3/GM for aviscous injection run.

2.4.3 Code Tests

Our version of ZEUS reproduces all hydrodynamic test results from Stone and Norman (1992),including their spherical advection and Sod shock tests. We also find excellent agreement withsteady spherical accretion solutions, i.e. Bondi flow (Bondi, 1952). An inviscid equilibrium torusrun also persists for many dynamical times with insignificant deviations from the initial conditions.

We have parallelized our code using the MPI message passing library. On the Origin 2000 atNCSA we are able to achieve about 2.5×107 zone updates per second using 240 CPUs, or about 35GFLOPs. This is 159 times faster than the single CPU speed, which represents a parallel efficiencyof 66%.

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2.5 Results

The initial motivation for undertaking this calculation was to understand differences between resultsreported in IA and SPB. Using our code, which is based on the same algorithm used in SPB’scalculations, we ran a series of tests attempting to reproduce SPB’s results. These test calculationsused all of SPB’s model choices, including SPB’s viscosity prescription, a Newtonian potential,and a torus for the mass source. We were able to reproduce most quantitative results reported inSPB99’s torus calculations. This includes their radial scaling laws. For example, in a model thatis identical to SPB99’s Run B, we find M ∝ r, ρ0 ∝ r0, c2

s ∝ r−1, vφ ∝ r−1/2, and |vr| ∝ r−1.These power law slopes are identical to those reported by SPB99. As another example, we foundM = 1.23 × 10−3 torus masses per torus orbit at the pressure maximum for a model identical toSPB’s Model A (their fiducial model); SPB report M = 1.0 × 10−3 in the same units. Given thefluctuations in rest-mass accretion rate, our value and SPB’s value are fully consistent. We wereeven able to reproduce certain numerical artifacts associated with the inner radial boundary, suchas a density drop and temperature spike near the inner boundary.

Recall that SPB evolve an initial torus and allow it to accrete; IA use a different experimentaldesign in which matter is steadily injected onto the grid. They also use a different viscosity pre-scription. We ran a second series of test calculations attempting to reproduce IA’s results. Thesetest calculations used all of IA’s model choices, including viscosity prescription, Newtonian poten-tial, etc. We were able to reproduce all of IA99’s calculations except those that include thermalconduction (which we did not attempt to reproduce). In each case we found that the qualitativenature of the flow is similar to that described in IA99. In particular, we agree on which models arestable and unstable and which models exhibit outflows. We also find qualitative agreement withtheir contour plots of, e.g., density pressure, mass flux, and Mach number. We also find qualitativeagreement with their radial run of cs/VK and specific angular momentum. Our results do not agreeprecisely, but this is likely due to small differences in mass injection scheme (because IA99 do notgive their ρ0(r, θ)). Finally, we can also reproduce the radial scalings given in IA00 for their modelA.

The most significant difference between the results of IA and SPB was due to the choice ofanomalous stress prescription, as might have been anticipated. Qualitatively, the stress componentsthat are included in IA and not SPB tend to smooth the flow and suppress turbulence. ThusSPB99’s simulations result in more vigorous convection than simulations performed by IA. Thechoice of mass supply (torus vs. injection) also leads to a significant difference between IA andSPB’s results; this is discussed in more detail below.

The fact that we can reproduce both SPB’s and IA’s results using a single code is consistentwith the hypothesis that differences between their reported results (e.g. the lower degree of con-vection reported in IA than SPB, and the differences in radial scaling laws) is due to differences inexperimental design and viscosity prescription rather than numerical methods. While we cannotcompletely rule out the possibility that SPB, IA, and we have made identical experimental errors,

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Table 2.1. Parameter List

Run Nr Nθ Visc. Potential rin/rg rout/rg Rinj/rg γ α tf (c3/GM)

A 108 50 IA PN 1.35 300 248 3/2 0.1 7.3× 105

B 80 50 IA PN 3 300 248 3/2 0.1 4.4× 105

C 80 50 IA Newt. 3 300 248 3/2 0.1 7.3× 105

D 64 40 MG PN 1.2 76 62 3/2 1.0 3.3× 103

E 128 80 MG PN 1.4 21 17 5/3 0.01 3.0× 104

F 128 80 MG PN 1.4 81 21 5/3 0.01 6.8× 104

Note. — IA and MG are viscosity prescription described in equations 7-9. PN is the pseudo-Newtonianpotential of Paczynski & Wiita (1980). Run B uses Run C as initial conditions. Here rg = 2GM/c2, and inrun F Rinj is the position of the torus density peak ρ0.

Table 2.2. Results List

Run Steady State Time (GM/c3) Max. Mach at rin M0 −(E/(M0c2))× 10−2 L c/(GMM0)

A 4.3× 105 -1.4 5.96× 10−2 2.06 1.75B 2.4× 105 +0.0 1.95× 10−2 3.72 1.29C 2.4× 105 +0.0 9.46× 10−3 6.77 .0746D ≥ 3.3× 103 -0.4 ≥ 3.59× 10−2 4.64 −0.167E 5.5× 103 -3.0 3.48× 10−2 3.01 3.41F 2.0× 104 -3.4 5.03× 10−1 3.11 3.35

Note. — Runs A-E are injection runs with rest-mass accretion rate units in M0,inj and Run F is a torus run with rest-massaccretion rate unit in ρ0(GM)2/c3. Run C’s angular momentum fluctuations are 10 times the average value shown.

this seems unlikely. This comparison thus lends credibility to SPB, IA, and our numerical results.

2.5.1 Fiducial Model Evolution

We now turn from reproducing earlier viscosity models to considering new aspects of our ownmodels. First, consider the evolution of a “fiducial” model (Run A in Table 2.1 and Table 2.2).The fiducial model has rin = 2.7GM/c2, rout = 600GM/c2, rinj = 495GM/c2, γ = 3/2, α = 0.1,Nr = 108, Nθ = 50. It uses a pseudo-Newtonian potential, mass is supplied by injection, and theviscosity prescription follows IA. It was run from t = 0 to t = 7.3× 105GM/c3.

Run A is similar to IA99’s “Model 5”, except that it uses a pseudo-Newtonian potential. In astatistically steady state the flow is characterized by a quasi-periodic outflow. Hot bubbles format the interface between bound (Bernoulli parameter Be = (1/2)v2 + c2

s/(γ − 1) + Ψ < 0) andunbound (Be > 0) material. These hot bubbles are buoyant and move away from the black hole.This appears to be a low-frequency, low wavenumber convective mode (IA refer to it as a “unipolaroutflow”). Higher wavenumber convective modes are evidently suppressed by the viscosity.

Figure 2.1 shows time-averaged plots of various quantities in the fiducial run. The time average

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Figure 2.1 Time-averaged spatial structure of fiducial run (Run A; α = 0.1, rin = 2.7GM/c2, androut = 600GM/c2). Shown are the density (upper left), Bernoulli parameter (Be = (1/2)v2+c2

s/(γ−1) + Ψ) (upper right; dotted line is a negative contour), scaled mass flux r2 sin θ(ρ0v) (lower left),and scaled angular momentum flux r3 sin2 θ(ρ0vvφ +Π · φ) (lower right). The flow is not symmetricabout the equator because the flow exhibits long timescale antisymmetric variations. Convectivebubbles form at the interface between positive and negative Bernoulli parameter (i.e. unbound andbound matter).

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is performed from 200 equally spaced data dumps from t = 4.3×105GM/c3 to t = 7.3×105GM/c3.We show only the region rin < r < 30GM/c2, whereas the computational domain is much larger:rin < r < 600GM/c2. The injection point is located far outside the plotted domain at r =496GM/c2. Because of the strong time-dependence of the flow in the fiducial run, the flow at anyinstant may look very different from these time averaged plots.

The upper left panel in Figure 1 shows the average density; notice that the density is not sym-metric about the equator. This is because the flow involves long-timescale quasi-periodic variationswhich are not quite averaged out over the course of the run. The upper right corner shows theBernoulli parameter Be. Dotted lines are negative; notice that there is a substantial amount offluid near the equator that is unbound in the sense that Be > 0. Nevertheless, this material is stillflowing inward in a nearly laminar fashion. Near the poles, the time-averaged Be < 0, but thisregion experiences large fluctuations. Polar outflows are typically associated with positive fluctua-tions in Be. The lower left panel shows the scaled mass flux r2 sin θ(ρ0v). Notice that much of themass flux is along the surfaces of the inflow rather than along the equator. The lower right panelshows the scaled angular momentum flux r3 sin2 θ(ρ0vvφ + Π · φ). As for the mass flux, most ofthe activity is along the surface of the flow.

Figure 2.2 shows the time series of the reduced eigenvalues: M/Minj , e = E/(Mc2), andl = Lc/(GMM). Also shown as dashed lines are the thin disk values for e and l. These assumea thin, cold disk terminating at r = 6GM/c2. The low value of l is due to two effects. First, thedisk is already sub-Keplerian by the time the flow reaches the innermost stable circular orbit. Inaddition, there are residual viscous torques in the plunging region that lower the specific angularmomentum of the accreted material (see Figure 2.3, below). Notice that Figure 2.2 shows a smoothevolution that varies on a timescale τ ≈ 4 × 104 at late time. The largest variations in rest-massaccretion rate are related to the appearance of large convective bubbles.

Figure 2.3 shows the θ and time averaged run of several quantities with radius. The averagesare taken over 4.3× 105GM/c3 < t < 7.3× 105GM/c3 and |θ − π/2| < π/6 2. The upper left plotshows the run of density. Notice that here, as for the other quantities, there is a spike near rinj ,an intermediate region, and then an inner, roughly power-law region. The upper right plot shows(cs/c)2; the lower left shows |vr|/c. Notice that the radial velocity exceeds the speed of light at theinner boundary. Similarly the azimuthal velocity vφ/c shown in the lower right panel approachesthe speed of light. Also shown in that panel is the circular velocity (dashed line). Evidently theflow is slightly sub-Keplerian at most radii.

The radial run of flow quantities in the inner regions can be fit by power laws, as done by IAand SPB. Our best fit power laws for the fiducial model (Run A) over 2.7GM/c2 < r < 20GM/c2

are ρ0 ∝ r−0.6, cs ∝ r−0.5, |vr| ∝ r−2, and vφ ∝ r−0.8. Between 2.7GM/c2 < r < 6GM/c2, vφ isbest fit by vφ ∝ r−0.9, which is nearly, but not exactly, consistent with conservation of fluid specificangular momentum (vφ ∝ r−1). Angular momentum is not exactly conserved at r < 6GM/c2

because of viscous torques.2Averaging over |θ − π| < π/36 produces nearly identical results, but we have chosen to use IA’s range in θ.

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Figure 2.2 The evolution of M/Minj , e = E/(Mc2), and l = L c/(GMM) in the fiducial run (RunA). The dotted line indicates the thin disk value. The run has clearly entered a quasi-steady state.The evolution is relatively smooth with a small variation on a timescale τ ≈ 4 × 104. This is thetimescale for convective bubble formation (the low point in rest-mass accretion rate is when bubbleforms). For this model the bubble forms at alternate poles. A full cycle requires of order onerotation period at the injection radius.

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Figure 2.3 The radial run of θ and time averaged quantities from the fiducial run (Run A). Shownare the density (upper left), squared sound speed (upper right), radial velocity (lower left), specificangular momentum (lower right; solid line), and circular orbit specific angular momentum (lowerright; dashed line). Crudely speaking, the inner flow is consistent with a radial power law. Thebest fits to a power law are: ρ0 ∝ r−0.6, cs ∝ r−0.5, |vr| ∝ r−2, and vφ ∝ r−0.8. The plots areaveraged over θ = π/2± π/6.

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The careful reader may notice that the power law slopes quoted in the last paragraph are notconsistent with a constant rest-mass accretion rate. This is because the power laws are derivedfrom averages over |θ − π/2| < π/6, following SPB. If one averages over all θ (following IA) andtime, then the resulting profiles are consistent with constant mass, energy, and angular momentumaccretion rates, as they must be for a flow that is steady when averaged over large times.

2.5.2 Dependence on Inner Boundary Location and Gravitational Potential

Having established that the differences between SPB and IA’s models are due to model choicesrather than numerics, we were also interested in studying whether any features of global viscousaccretion models are strongly dependent on numerical parameters. The first parameter we consid-ered was the location of the inner boundary.

In models that use a Newtonian gravitational potential (such as IA and SPB) the location of theinner boundary is not an interesting parameter in the sense that there is no physical lengthscale thatone can compare rin to: it is simply a scaling parameter. In models that use a pseudo-Newtonianpotential, however, there is a feature (a “pit”) in the potential on a lengthscale GM/c2. Startingwith our fiducial model, then, what is the effect of shifting rin?

Our fiducial run has rin = 2.7GM/c2. This may be compared with Run B, which has rin =6GM/c2. Figure 2.4 compares the accretion rates in the two runs. Evidently there are two changesin the solution. First, the time-averaged accretion rates differ by a large factor. The mean rest-mass accretion rate is factor of 3 lower in Run B than Run A. The reduced eigenvalues e and l

also differ by about 50% (see Table 2). Second, the time variation of the accretion rates differs,with Run B showing far more short-timescale variations. The short-timescale variations are due tothe interaction of unstable convective modes with the boundary conditions. Inspection of the runsreveals an enhancement of convection and turbulence near rin in Run B.

The differences between Run B and Run A are caused by the boundary location. Gradualvariation of rin (in models not discussed in detail here) reveals that if the flow on the inner boundaryis everywhere and always supersonic, then the solution is similar to Run A. If the flow is subsonic,then the solution exhibits artifacts like those seen in Run B.

Evidently forcing the flow to be supersonic on the inner boundary causally disconnects the flowfrom the boundary. 3 This eliminates nonphysical reflection of linear and nonlinear waves from theboundary and renders the precise implementation of the numerical boundary conditions irrelevant.

We do not want the reader to think that this problem arises because we happened to choose thewrong numerical implementation of the boundary conditions. Our implementation is the standardZEUS outflow boundary condition, and it is widely used in astrophysical problems. While it maybe possible to implement more transparent boundary conditions in the context of other numericalschemes, a survey of the numerical literature shows that in multiple dimensions this is an area of

3Although the viscous fluid equations of motion are not hyperbolic, and the flow in the supersonic region is inprinciple in causal contact with the rest of the flow, the coupling is exponentially weak.

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Figure 2.4 The effect of moving the inner boundary on the accretion rates of mass, angular momen-tum, and energy (Run A vs. Run B). The top panel shows M/Minj , the middle panel E/(Mc2),and the bottom panel L c/(GMM). The solid curve is Run A, which has rin = 2.7GM/c2. Thedashed curve is Run B, which has rin = 6GM/c2. Evidently Run B has a different variabilitystructure and different time averaged values for the accretion rates. The relatively rapid and high-amplitude variations in Run B are due to nonphysical interactions with the inner radial boundary.Only by ensuring a supersonic flow (as in Run A) can one avoid these nonphysical effects.

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active research (Roe, 1989; Karni, 1991; Dedner et al., 2001; Bruneau and Creus, 2001), and thatno general solution to the problem has been found.

Furthermore, a simple example shows that no local extrapolation scheme can work for allaccretion problems. Consider a numerical model of a steady spherical inflow (Bondi flow) in agravitational potential Ψ(r). Let us suppose that we are primarily interested in accurately mea-suring M . We know from the analytic solution of the problem that M depends on the shape ofthe potential everywhere outside the sonic point. If we place the inner boundary rin outside thesonic point and use a local extrapolation scheme, we won’t always get the correct answer becausethe local extrapolation doesn’t have any information about the shape of the potential between rin

and the sonic point. Put differently, one can’t determine a global solution from local extrapolationat the boundary. The key point is that, while aspects of the solution may be accurate, M (and L

and E) are sensitive to the boundary conditions.It is worth noting that the outer boundary is always in causal contact with the flow, but does

not cause the same type of artifacts as the inner boundary. Experiments show that the flow isqualitatively insensitive to the location and implementation of the outer boundary condition. Thetime averaged M , however, is sensitive to both rout and rinj/rout. The time averaged l and e scaleout this mass dependence and so are qualitatively and quantitatively insensitive to both rout andrinj/rout.

It is also worth noting the effects of changing the gravitational potential. Run C (identical toIA99 Model 5) is identical to Run B except that the potential is now Newtonian. It is qualitativelysimilar to Run B, but M is now a factor of 5 lower than Run A. Run C also has the property thatl oscillates about 0.0. This is a problem if the focus of the simulation is measuring M or L.

To summarize: the location of the inner radial boundary can determine the character of the flow.If the flow is everywhere and always supersonic (or super-fast-magnetosonic in MHD) on the innerboundary then boundary-related corruption of the flow is impossible. Since it is computationallyexpensive to place the inner boundary very deep in the potential (for our model, the time stepdt ∼ (rin − 2GM/c2)), the optimal location for the inner boundary is just inside the radius wherethe radial Mach number always exceeds 1.

The results of IA and SPB do not focus on the time-dependence of the accretion values, somuch of their discussion is unaffected by their treatment of the inner radial boundary. As discussedbelow, there are small changes related to the appearance of outflows.

2.5.3 Comparison of Torus and Injection Models

The torus and injection methods represent sharply different approaches to studying accretion flows.The equilibrium torus presents a physically well-posed problem, but the accretion flow is transient:no steady state can be achieved. The injection method reaches a quasi-steady state, but much ofthe computational domain is wasted on evolving the injection region, which has no astrophysicalanalog: it is nonphysical. It is natural to ask whether these two widely used schemes for supplying

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mass can be made comparable or used to measure any of the same quantities.We selected two runs, E (torus) and F (injection), that had similar mass distributions in an

evolved state. The torus run was studied at a time when M was close to its maximum. Run F’sM is a factor of 10 larger than Run E’s. This difference might have been anticipated from thesensitivity of the injection run to rout and rinj/rout: runs in which mass is concentrated closer tothe outer boundary tend to have lower accretion rates because more of the mass escapes throughthe outer boundary. The time averaged rest-mass accretion rate is therefore strongly dependent onthe method of mass supply.

The energy and angular momentum accretion rates per unit mass are, however, insensitive tothe experimental design. We find that e and l differ by less than 3% in Runs E and F (see Table2.1 and Table 2.2). These quantities are apparently set by conditions near the inner boundary (theISCO for the Pseudo-Newtonian potential), and can be measured in either type of experiment.

2.5.4 Other Parameters

We have varied rinj and rout/rinj , and as reported above, these strongly affect the time-averagedvalue of M . The sense of the effect is that a simulation with a larger rout/rinj loses less matterthrough the outer boundary, and this results in more matter streaming back into the black hole(by up to a factor of 10). The qualitative nature of the flow, however, is roughly independent ofrout/rinj in that, e.g., the temporal power spectrum of M is similar. The qualitative nature of theflow is dependent on rinj . If one fixes rout/rinj and all other parameters, the range in α whereunipolar outflows are observed tends to become smaller and disappears altogether for rinj as smallas 40GM/c2.

The dependence of accretion models similar to ours on α has already been investigated by IA.They find that the flow changes from turbulent to laminar as α is increased and the higher viscositydamps modes of increasing lengthscale. IA find that α . 0.03 the flow is turbulent, and for α & 0.3the flow is laminar. For 0.03 < α < 0.3 the flow exhibits a “unipolar” outflow. Our results are inagreement with IA. However, our models with a pseudo-Newtonian potential and super-sonic flowat the inner radial boundary show a slight shift in the values of α that exhibit unipolar outflows.

We did find a critical value of α ≈ 0.5 above which no supersonic flow at rin could be achieveddue to viscous heating, at least for γ = 3/2 and γ = 5/3 and for rin ≥ 2.1GM/c2. Smaller values ofrin were not computationally practical. This high α is typically associated with a bipolar outflow,as seen by IA. Even in this case, however, a choice of rin = 2.1GM/c2 instead of rin = 6GM/c2

leads to a qualitatively different profile for the flow. The flow with smaller rin = 2.1GM/c2 has abipolar outflow starting at larger radius (10GM/c2) rather than immediately on the boundary aswith rin = 6GM/c2, and the rest-mass accretion rate increases by a factor of 3.

Finally, we studied the dependence of the results on numerical resolution. We find that Nr ×Nθ = 108× 50 is sufficient at α = 0.1 to resolve the shortest wavelength convective mode. Also, wechose our value of r0 to agree with SPB99’s torus models. We experimented with varying r0 and

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find that, all things being equal, smaller r0 gives more laminar flow.

2.6 VHD Summary

Work in this field will shortly focus on global MHD models in pseudo-Newtonian potentials andin full general relativity. In our view it is useful to understand the solution space for physicallyand numerically simpler viscous models before turning to MHD. It is even possible, as Stone andPringle (2001) have claimed, that viscous hydrodynamics provides a crude approximation to theMHD results. In any event, this investigation provides a preview of some of the experimental issuesthat will play a role in most future global numerical investigations of accretion flows.

This investigation was initially motivated by a desire to understand whether the differencesbetween earlier global viscous hydrodynamics simulations performed by IA and SPB were causedby differences in experimental design or numerical method. IA and SPB reported different degreesof convective turbulence in their models and found different radial scalings for vertically averagedquantities such as temperature and density. Using a single code, we were able to reproduce bothsets of results. We conclude that the differences are due to experimental design.

We also found, while reproducing IA and SPB’s results, that some aspects of our solutions weresensitive to the numerical treatment of the region close to the inner boundary in models that usea pseudo-Newtonian potential. In particular, l and e, the specific angular momentum and energyof accreted material, are strongly dependent on rin, the location of the inner boundary. When theflow is supersonic at rin the location of the boundary does not affect l and e. But when the flow issubsonic at rin the flow interacts strongly with the numerical boundary condition. This producesspurious outflow events and makes l and e dependent on rin. Evidently for accurate measurementof these quantities it is necessary to isolate the numerical boundary condition behind a sonictransition that is located within the computational domain; one must place the inner boundarycondition inside a “sound horizon”.

We are not saying that all models that lack a sonic transition in the computational domainare fatally flawed. Whether the treatment of the inner boundary condition is problematic or notdepends on what is being measured. For the nonlinear eigenvalues L, E, and M that we havefocused on here, however, the treatment of the inner boundary condition is crucial. Furthermore,the only guarantee that the inner boundary condition is not governing the solution is to isolate itbehind a sonic transition; this is the only completely safe choice.

The location of the inner boundary may prove even more crucial in MHD models. Much ofthe character of the flow is determined in the turbulent, energetically important region of the flowjust outside the fast magnetosonic transition, just as the region immediately outside the sonictransition determines the nature of the viscous flows described in this paper. We have performedsome preliminary numerical tests and find that, as in the viscous flow, l and e for an MHD floware sensitive to the treatment of the inner boundary. For various reasons it may prove difficultto achieve a fast magnetosonic transition in the computational domain; accurate treatment of the

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inner boundary condition may require fully (general) relativistic MHD.We have also compared two commonly used experimental designs for black hole accretion flow

studies: models that begin with an equilibrium torus, and models that continuously inject fluidonto the grid. The choice between these models is to some degree a matter of taste. We find theequilibrium torus slightly easier to initialize and analyze. Remarkably, the two different approachesproduce indistinguishable measurements for l and e, the specific angular momentum and energy ofthe accreted material.

A parallel, viscous, axisymmetric hydrodynamics code based on that used in this paper can befound at http://kerr.physics.uiuc.edu.

2.7 Global 2D MHD Simulations

Prior analytic calculations find that the magnetic stress −BrBφ → 0 near the innermost stablecircular orbit (ISCO) of a black hole. This has bearing on whether magnetic stresses can exert atorque between the hole and disk. Thus, at least two interesting questions regarding a magneticdisk are: 1) does the magnetic stress approach 0 near the ISCO? and 2) is the polar field strongenough for the BZ effect to be relevant?

With a global 2D MHD code we have been able to reproduce prior work by Hawley (2000)(hereafter H00) and Stone and Pringle (2001) (hereafter SP01). This work involves an equilibriumtorus (in HD) threaded by a field where the vector potential component Aφ ∝ ρ, so that thefield resides totally within the torus and is axisymmetric. H00, SP01, and we all find there isnothing special about the ISCO, contrary to most prior analytic calculations (except in, for example,Gammie (1999)). The magnetic stress continues to grows as r → 2GM/c2 directly through theISCO, and is relatively large compared to the Reynolds stress −vrvφ there as well.

To discover whether the polar field is substantial enough to make the BZ effect a significantsource of energy, we need to simulate both the pole and the disk. This is not possible for H00 sincehis computational grid excises the polar region, and SP01 do not report on this. We find that thepolar field surrounding the black hole is relatively strong compared to that in the disk. This fieldis supported by the disk and hole interaction as the disk continually pumps and compresses fieldinto the black hole.

Figure 3 shows the field lines of a 2D axisymmetric, spherical polar coordinate simulation afterabout 8 orbits (at the t = 0 torus density maximum) once full non-linear MRI turbulence hasdeveloped. We measure the polar magnetic field strength to be 2-4× that of the average disk fieldat the inner radial edge, contrary to estimates by Ghosh and Abramowicz (1997); Livio et al. (1999).With such a strong polar field, it is plausible that the BZ effect generates a significant fraction ofAGN power output.

One can compute the power output of the BZ effect using equation 1.8. I estimate the fieldstrength B by assuming my computer simulation’s mass accretion rate M is equal to the Eddingtonvalue (M ∼ 1.3×1038

(ηc2

)−1 ( MM¯ ) erg s−1), where η is order 10%. In the simulation the accretion

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Figure 2.5 Field line snapshot after 8 orbits (at the t = 0 torus density maximum at r = 9.4GM/c2)for a global 2D MHD simulation. Full non-linear turbulence drives the accretion process. Note thatthe polar field is essentially radial while the accretion disk is dominated by turbulence generatedby the MRI.

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rate scales like M ∝ ρ0(rh/2)3

(GM/c3)and the field strength scales like B ∝

√ρ0c2, where ρ0 is the typical

density of the initial mass source. Thus, M can be used to solve for ρ0 in cgs units which is thenused to solve for B in cgs units. This gives P ∼ 4× 1045 erg s−1.

By limiting the calculation to 2D, the field is forced to avoid twisting around the pole or diffusingaround the pole, so we may be overestimating the polar field. Also, as stated by the anti-dynamotheorem (Cowling, 1934), an isolated 2D axisymmetric system cannot sustain a magnetic dynamo.Thus, the next step in modeling the black hole environment is to perform a global 3D calculation.

2.8 Global 3D MHD Simulation

A global 3D calculation involves about twice the number of computational operations per grid zone,and requires N3 zones instead of N2, so a 3D calculation can be quite expensive. One would preferto use spherical polar coordinates since the black hole is spherical, and because in these coordinatesthe ZEUS algorithm conserves angular momentum well.

We performed global 3D MHD calculations in spherical polar coordinates and found that thesingularity at the poles leads to several problems. The most significant problem is that grid zonesnear the polar axis become smaller since zones have a side with arc length of r sin θdφ. In orderto guarantee a stable algorithm, the Courant condition requires dt ∝ dx/v for a grid size dx andfluid speed v. Since the grid size is dx ∝ θ near the θ = 0 pole, and the polar region is nearly avacuum with potentially strong fields (hence large va), this leads to a requirement that dt ∝ θ/va

and severely limits the time step.We therefore decided to use a Cartesian grid. This presents two problems: 1) angular momentum

is no longer well conserved by the algorithm ; and 2) The grid geometry forces an m = 4 mode(spherical mode for the φ direction) in the solution. We try to diminish both of these problemsby cutting an approximation to a sphere out of the Cartesian mesh at the outer boundary andaround the black hole. Preliminary results show that the time step is well behaved and the m = 4mode is minimal. Angular momentum conservation appears acceptable, but quantitative analysisis pending.

Questions we will attempt to answer with this code are: 1) does an accretion flow generate amagnetized outflow? ; 2) how does Bremsstrahlung cooling affect a magnetized accretion disk? and3) what is the power output due to the BZ effect?

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3 HARM: A Numerical Scheme forGeneral RelativisticMagnetohydrodynamics

3.1 Summary of Chapter

We describe a conservative, shock-capturing scheme for evolving the equations of general relativisticmagnetohydrodynamics. The fluxes are calculated using the Harten, Lax, and van Leer scheme. Avariant of constrained transport, proposed earlier by Toth, is used to maintain a divergence freemagnetic field. Only the covariant form of the metric in a coordinate basis is required to specifythe geometry. We describe code performance on a full suite of test problems in both special andgeneral relativity. On smooth flows we show that it converges at second order. We conclude byshowing some results from the evolution of a magnetized torus near a rotating black hole.1

3.2 Introduction

Quasars, active galactic nuclei (AGN), X-ray binaries, gamma-ray bursts, and core-collapse super-novae are all likely powered by a central engine subject to strong gravity, strong electromagneticfields, and rotation. For convenience, we will refer to this class of objects as relativistic magneto-rotators (RMRs). RMRs are among the most luminous objects in the universe and are therefore thecenter of considerable theoretical attention. Unfortunately the governing physical laws for RMRs,while well known, are nonlinear, time-dependent, and intrinsically multidimensional. This hasstymied development of a first-principles theory for their evolution and observational appearanceand strongly motivates a numerical approach.

To fully understand RMR structure, one must be able to follow the interaction of a non-Maxwellian plasma with a relativistic gravitational field, a strong electromagnetic field, and possiblya strong radiation field as well. This general problem remains beyond the reach of today’s algorithmsand computers. A useful first step, however, might be to study these objects in a nonradiatingmagnetohydrodynamic (MHD) model. In this case the plasma can be treated as a fluid, greatlyreducing the number of degrees of freedom, and the radiation field can be ignored. The relevanceof this approximation must be evaluated in astrophysical context and will not be considered here.

1Published in ApJ Volume 589, Issue 1, pp. 444-457. Reproduction for this thesis is authorized by the copyrightholder. Initial HARM work as in paper primarily performed by CFG, later efforts primarily by JCM, and GT providednumerical advice.

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We were motivated, therefore, to develop a method for integrating the equations of ideal, generalrelativistic MHD (GRMHD), and that method is described in this paper. This is in a sense well-trodden ground: many schemes have already been developed for relativistic fluid dynamics. Whathas not existed until recently is a scheme that: (1) includes magnetic fields; (2) has been fully verifiedand convergence tested; (3) is stable and capable of integrating a flow over many dynamical times.A pioneering GRMHD code has been developed by Koide and collaborators (see, e.g. Koide et al.2002, Koide et al. 1999, Koide et al. 2000, Meier et al. 2001). Our code differs in that we havesubjected it to a fuller series of tests (described in this paper), we can perform longer integrationsthan the rather brief simulations described in the published work of Koide’s group, and our codeexplicitly maintains the divergence-free constraint on the magnetic field. A GRMHD Godunovscheme based on a Roe-type approximate Riemann solver has been developed by Komissarov anddescribed in a conference proceeding (Komissarov, 2001).

A rather complete review of numerical approaches to relativistic fluid dynamics is given byMartı and Muller (2003) and Font (2003). The first numerical GRMHD scheme that we are awareof is by Wilson (1977), who integrated the GRMHD equations in axisymmetry near a Kerr blackhole. While it was recognized throughout the 1970s that relativistic MHD would be relevant toproblems related to black hole accretion (particularly following the seminal work of Blandfordand Znajek (1977) and Phinney (1983); see also Punsly (2001)), no further work appeared untilYokosawa (1993). The discovery by Balbus and Hawley (1991) that magnetic fields play a crucialrole in regulating accretion disk evolution (reviewed in Balbus and Hawley (1998)) and the absenceof purely hydrodynamic means for driving accretion in disks (Balbus and Hawley, 1998) furthermotivated the development of relativistic MHD schemes. More recently there have been severalefforts to develop GRMHD codes, including the already-mentioned work by Koide and collaboratorsand by Komissarov. A ZEUS-like scheme for GRMHD has also been developed and is described ina companion paper (De Villiers and Hawley, 2002).

Special relativistic MHD (SRMHD) is the foundation for any GRMHD scheme, although thereare nontrivial problems in making the transition to full general relativity. SRMHD schemes havebeen developed by vanPutten (1993); Balsara (2001); Koldoba et al. (2002); Komissarov (1999)and Del Zanna et al. (2003). We were particularly influenced by the clear development of thefundamental equations in Komissarov (1999) for his Godunov SRMHD scheme based on a Roe-type approximate Riemann solver, and by the work of Del Zanna and Bucciantini (2002) and DelZanna et al. (2003) who chose to use the simple approximate Riemann solver of Harten et al. (1983)in their special relativistic hydrodynamics and SRMHD schemes, respectively.

Our numerical scheme is called HARM, for High Accuracy Relativistic Magnetohydrodynamics.2

In the next section we develop the basic equations in the form used for numerical integration inHARM (§2). In §3 we describe the basic algorithm. In §4 we describe the performance of the codeon a series of test problems. In §5 we describe a sample evolution of a magnetized torus near arotating black hole.

2Also named in honor of R. Harm, who with M. Schwarzschild was a pioneer of numerical astrophysics.

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3.3 A GRMHD Primer

The equations of general relativistic MHD are well known, but for clarity we will develop them herein the same form used in numerical integration. Unless otherwise noted c = 1 and we follow thenotational conventions of Misner et al. (1973), hereafter MTW. The reader may also find it usefulto consult Anile (1989).

The first governing equation describes the conservation of particle number:

(nuµ);µ = 0. (3.1)

Here n is the particle number density and uµ is the four-velocity. For numerical purposes we rewritethis in a coordinate basis, replacing n with the “rest-mass density” ρ0 = mn, where m is the meanrest-mass per particle:

1√−g∂µ(

√−g ρ0uµ) = 0. (3.2)

Here g ≡ Det(gµν).The next four equations express conservation of energy-momentum:

Tµν;µ = 0, (3.3)

where Tµν is the stress-energy tensor. In a coordinate basis,

∂t

(√−g T tν

)= −∂i

(√−g T iν

)+√−g T κ

λ Γλνκ, (3.4)

where i denotes a spatial index and Γλνκ is the connection.

The energy-momentum equations have been written with the free index down for a reason.Symmetries of the metric give rise to conserved currents. In the Kerr metric, for example, theaxisymmetry and stationary nature of the metric give rise to conserved angular momentum andenergy currents. In general, for metrics with an ignorable coordinate xµ the source term on theright hand side of the evolution equation for T t

µ vanish. These source terms do not vanish whenthe equation is written with both indices up.

The stress-energy tensor for a system containing only a perfect fluid and an electromagneticfield is the sum of a fluid part,

Tµνfluid = (ρ0 + u + p)uµuν + pgµν , (3.5)

(here u ≡ internal energy and p ≡ pressure) and an electromagnetic part,

TµνEM = FµαF ν

α −14gµνFαβFαβ. (3.6)

Here Fµν is the electromagnetic field tensor (MTW: “Faraday”), and for convenience we have

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absorbed a factor of√

4π into the definition of F .The electromagnetic portion of the stress-energy tensor simplifies if we adopt the ideal MHD

approximation, in which the electric field vanishes in the fluid rest frame due to the high conductivityof the plasma (“E + v×B = 0”). Equivalently the Lorentz force on a charged particle vanishes inthe fluid frame:

uµFµν = 0. (3.7)

It is convenient to define the magnetic field four-vector

bµ ≡ 12εµνκλuνFλκ, (3.8)

where ε is the Levi-Civita tensor. Recall that (following the notation of MTW) εµνλδ = − 1√−g[µνλδ],

where [µνλδ] is the completely antisymmetric symbol and = 0, 1, or −1. These can be combined(with the aid of identity 3.50h of MTW):

Fµν = εµνκλuκbλ. (3.9)

Substitution and some manipulation (using identities 3.50 of MTW and bµuµ = 0; the latter followsfrom the definition of bµ and the antisymmetry of F ) yields

TµνEM = b2uµuν +

12b2gµν − bµbν . (3.10)

Notice that the last two terms are nearly identical to the nonrelativistic MHD stress tensor, whilethe first term is higher order in v/c. To sum up,

TµνMHD = (ρ0 + u + p + b2)uµuν + (p +

12b2)gµν − bµbν (3.11)

is the MHD stress-energy tensor.The electromagnetic field evolution is given by the source-free part of Maxwell’s equations

Fµν,λ + Fλµ,ν + Fνλ,µ = 0. (3.12)

The rest of Maxwell’s equations determine the current

Jµ = Fµν;ν , (3.13)

and are not needed for the evolution, as in nonrelativistic MHD.Maxwell’s equations can be written in conservative form by taking the dual of eq.(3.12):

F ∗µν;ν = 0. (3.14)

Here F ∗µν = 1

2εµνκλF κλ is the dual of the electromagnetic field tensor (MTW: “Maxwell”). In ideal

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MHDF ∗µν = bµuν − bνuµ, (3.15)

which can be proved by taking the dual of eq.(3.9).The components of bµ are not independent, since bµuµ = 0. Following, e.g., Komissarov (1999),

it is useful to define the magnetic field three-vector Bi = F ∗it. In terms of Bi,

bt = Biuµgiµ, (3.16)

bi = (Bi + btui)/ut. (3.17)

The space components of the induction equation then reduce to

∂t(√−gBi) = −∂j(

√−g (bjui − biuj)) (3.18)

and the time component reduces to

1√−g∂i(√−g Bi) = 0, (3.19)

which is the no-monopoles constraint. The appearance of Bi in these last two equations are whatmotivates the introduction of the field three-vector in the first place.

To sum up, the fundamental equations as used in HARM are: the particle number conservationequation (3.2); the four energy-momentum equations (3.4), written in a coordinate basis and usingthe MHD stress-energy tensor of equation (3.11); and the induction equation (3.18), subject to theconstraint (3.19). These hyperbolic 3 equations are written in conservation form, and so can besolved numerically by well-known techniques.

3.4 Numerical Scheme

There are many possible ways to numerically integrate the GRMHD equations. A first, zeroth-order choice is between conservative and nonconservative schemes. Nonconservative schemes suchas ZEUS (Stone and Norman, 1992) have enjoyed wide use in numerical astrophysics. They permitthe integration of an internal energy equation rather than a total energy equation. This can beadvantageous in regions of a flow where the internal energy is small compared to the total energy(highly supersonic flows), which is a common situation in astrophysics. A nonconservative schemefor GRMHD following a ZEUS-like approach has been developed and is described in a companionpaper (De Villiers and Hawley, 2002).

We have decided to write a conservative scheme. One advantage of this choice is that inone dimension, total variation stable schemes are guaranteed to converge to a weak solution of theequations by the Lax-Wendroff theorem (Lax and Wendroff, 1960) and by a theorem due to LeVeque

3The GRMHD equations exhibit the same degeneracies as the nonrelativistic MHD equations.

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(1998). While no such guarantee is available for multidimensional flows, this is a reassuring startingpoint. Furthermore, one is guaranteed that a conservative scheme in any number of dimensionswill satisfy the jump conditions at discontinuities. This is not true in artificial viscosity basednonconservative schemes, which are also known to have trouble in relativistic shocks (Norman andWinkler, 1986). Conservative schemes for GRMHD have also been developed by Komissarov (2001)and by Koide et al. (1999).

A conservative scheme updates a set of “conserved” variables at each timestep. Our vector ofconserved variables is

U ≡ √−g(ρ0ut, T t

t , Tti , B

i). (3.20)

These are updated using fluxes F. We must also choose a set of “primitive” variables, which areinterpolated to model the flow within zones. We use variables with a simple physical interpretation:

P = (ρ0, u, vi, Bi). (3.21)

Here vi = ui/ut is the 3-velocity.4 The functions U(P) and F(P) are analytic, but the inverseoperations (so far as we can determine) are not. 5 There is also no simple expression for F(U).

To evaluate U(P) and F(P) one must find ut and bµ from vi and Bi. To find ut, solve thequadratic equation gµνu

µuν = −1. Next use equations (3.16) and (3.17) to find bµ; these requireonly multiplications and additions. The remainder of the calculation of U(P) and F(P) requiresraising and lowering of indices followed by direct substitution in equation (3.11) to find the com-ponents of the MHD stress-energy tensor.

Since we update U rather than P, we must solve for P(U) at the end of each timestep. Weuse a multidimensional Newton-Raphson routine with the value of P from the last timestep asan initial guess. Since Bi can be obtained analytically, only 5 equations need to be solved. TheNewton-Raphson method requires an expensive evaluation of the Jacobian ∂U/∂P. In practice weevaluate the Jacobian analytically. It is possible to evaluate the Jacobian by numerical derivatives,but this is both expensive and a source of numerical noise.

The evaluation of P(U) is at the heart of our numerical scheme; the procedure must be robust.We have found that it is crucial that the errors (differences between the current and target valuesof U) used to evaluate convergence in the Newton-Raphson scheme be properly normalized. Wenormalized the errors with

√−gρ0ut.

To evaluate F we use a MUSCL type scheme with “HLL” fluxes (Harten et al., 1983). Thefluxes are defined at zone faces. A slope-limited linear extrapolation from the zone center givesPR and PL, the primitive variables at the right and left side of each zone interface. We haveimplemented the monotonized central (“Woodward”, or “MC”) limiter, the van Leer limiter, andthe minmod limiter; unless otherwise stated, the tests described here use the MC limiter, which is

4We initially used ui as primitive variables, but the inversion ut(ui) is not always single-valued, e.g. inside theergosphere of a black hole. That is, there are physical flows with the same value of ui but different values of ut.

5Del Zanna et al. (2003) have found that the inversion P(U) can be reduced analytically to the solution of asingle, nonlinear equation.

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the least diffusive of the three.From PR,PL, calculate the maximum left and right-going wave speeds c±,R, c±,L, and the fluxes

FR = F(PR) and FL = F(PL). Let cmax ≡ MAX(0, c+,R, c+,L) and cmin ≡ −MIN(0, c−,R, c−,L),then the HLL flux is

F =cminFR + cmaxFL − cmaxcmin(UR −UL)

cmax + cmin(3.22)

If cmax = cmin, the HLL flux becomes the so-called local Lax-Friedrichs flux.

3.4.1 Constrained Transport

The pure HLL scheme will not preserve any numerical representation of∇·B = 0. An incomplete listof options for handling this constraint numerically includes: (1) ignore the production of monopolesby truncation error and hope for the best (in our experience this causes the scheme to fail in anycomplex flow); (2) introduce a divergence-cleaning step (this entails solving an elliptic equationat each timestep); (3) use an Evans and Hawley (1988) type constrained transport scheme (thisrequires a staggered mesh, so that the magnetic field components are zone face centered); (4)introduce a diffusion term that causes numerically generated monopoles to diffuse away (Marder(1987); this typically leaves a monopole field with rms value somewhat larger than the truncationerror).

We have chosen a fifth option, a version of constrained transport that can be used with azone-centered scheme. This idea was introduced by one of us in Toth (2000), where it is calledthe flux-interpolated constrained transport (or “flux-CT”) scheme 6. It preserves a numericalrepresentation of ∇ ·B = 0 by smoothing the fluxes with a special operator. The disadvantage ofthis method is that it is more diffusive than the “bare”, unconstrained scheme. The advantage isthat it is extremely simple.

To clarify how zone-centered constrained transport works we now give a specific example fora special relativistic problem in Cartesian coordinates t, x, y. To fix notation, write the inductionequation as

∂tBi = −∂jF

ji , (3.23)

where we have used√−g = 1 and the fluxes F j

i are

F xx = 0

F yy = 0

F yx = byux − bxuy

F xy = bxuy − byux = −F y

x

(3.24)

Notice that the fluxes are centered at different locations on the grid: F x fluxes live on the x face ofeach zone at grid location i− 1/2, j (we use i, j to denote the center of each zone), while F y fluxes

6We have also experimented with Toth’s “flux-CD”, which preserves a different representation of ∇ ·B = 0. Thisappears to be slightly less robust. It also has a larger effective stencil.

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live on the y face at i, j − 1/2. The smoothing operator replaces the numerical (HLL-derived) F ji

with F ji , defined by

F xx (i− 1/2, j) = 0

F yy (i, j − 1/2) = 0

F yx (i, j − 1/2) = 1

8

[2F y

x (i, j − 1/2)

+F yx (i + 1, j − 1/2) + F y

x (i− 1, j − 1/2)−F x

y (i− 1/2, j)− F xy (i + 1/2, j)

−F xy (i− 1/2, j − 1)− F x

y (i + 1/2, j − 1)]

F xy (i− 1/2, j) = 1

8

[2F x

y (i− 1/2, j)

+F xy (i− 1/2, j + 1) + F x

y (i− 1/2, j − 1)−F y

x (i, j − 1/2)− F yx (i, j + 1/2)

−F yx (i− 1, j − 1/2)− F y

x (i− 1, j + 1/2)]

(3.25)

It is a straightforward but tedious exercise to verify that this preserves the following corner-centerednumerical representation of ∇ ·B:

∇ ·B = (Bx(i, j) + Bx(i, j − 1)−Bx(i− 1, j)−Bx(i− 1, j − 1)) /(2∆x)+ (By(i, j) + By(i− 1, j)−By(i, j − 1)−By(i− 1, j − 1)) /(2∆y),

(3.26)

where ∆x and ∆y are the grid spacing.

3.4.2 Wave Speeds

The HLL approximate Riemann solver does not require eigenvectors of the characteristic matrix(as would a Roe-type scheme), but it does require the maximum and minimum wave speed (eigen-values). These wave speeds are also required to fix the timestep via the Courant conditions. Therelevant speed is the phase speed “ω/k” of the wave, and it turns out that only speeds for waveswith wavevectors aligned along coordinate axes are required. Suppose, for example, that one needsto know how rapidly signals propagate in the fluid along the x1 direction. First, find a wavevectorkµ = (−ω, k1, 0, 0), that satisfies the dispersion relation for the mode in question: D(kµ) = 0. Thenthe wave speed is simply ω/k1.

The dispersion relation D(kµ) = 0 for MHD waves has a simple form in a comoving frame.In terms of the relativistic sound speed c2

s = (∂(ρ0 + u)/∂p)−1s = γp/w, (the last holds only if

p = (γ−1)u) and the relativistic Alfven velocity vA = B/√E , where E = b2 +w and w ≡ ρ0 +u+p,

the dispersion relation is

ω(ω2 − (k · vA)2

)×(ω4 − ω2

(k2(v2

A + c2s(1− v2

A/c2)) + c2s(k · vA)2/c2

)+ k2c2

s(k · vA)2)

= 0,(3.27)

Here c is the (temporarily reintroduced) speed of light. The first term is the zero frequency entropy

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mode, the second is the Alfven mode, and the third contains the fast and slow modes. The eighthmode is eliminated by the no-monopoles constraint.

The relativistic sound speed asymptotes to c√

γ − 1 = c/√

3 for γ = 4/3, and the Alfvenspeed asymptotes to c. In the limit that B2/ρ0 À 1 and p/ρ0 6À 1, the GRMHD equationsare a superset of the time-dependent, force-free electrodynamics equations recently discussed byKomissarov (2002c); these contain fast modes and Alfven modes that move with the speed oflight. They are indistinguishable from vacuum electromagnetic modes only when their wavevectoris oriented along the magnetic field.

To find the maximum wave speeds we need to evaluate the comoving-frame dispersion relationfor the fast wave branch from coordinate frame quantities. This is straightforward because thedispersion relation depends on scalars, which can be evaluated in any frame: ω = kµuµ; k2 = KµKµ,where Kµ = (gµν+uµuν)kν is the part of the wavevector normal to the fluid 4-velocity; v2

A = bµbµ/E ;(k · vA) = kµbµ/

√E . The relevant portion of the dispersion relation (for fast and slow modes) isthus a fourth order polynomial in the components of kµ. This can be solved either analytically orby standard numerical methods. The two fast mode speeds are then used as cmax and cmin in theHLL fluxes.

We have found it convenient to replace the full dispersion relation by an approximation:

ω2 = (v2A + c2

s(1− v2A/c2))k2. (3.28)

This overestimates the maximum wavespeed by a factor ≤ 2 in the comoving frame. The maximumerror occurs for k ‖ vA, vA/cs = 1, and vA ¿ c, and it is usually much less, particularly if the fluidis moving super-Alfvenically with respect to the grid. This approximation is convenient because itis quadratic in kµ, and so can be solved more easily.

3.4.3 Implementation Notes

For completeness we now give some details of the implementation of the algorithm.Time Stepping. Our scheme is made second order in time by taking a half-step from tn to

tn+1/2, evaluating F(P(tn+1/2)), and using that to update U(tn) to U(tn+1).Modification of Energy Equation. A direct implementation of the energy equation can be

inaccurate because the magnetic and internal energy density can be orders of magnitude smallerthan the rest mass density. To avoid this we subtract the particle number conservation equationfrom the energy equation, i.e., we evolve

∂t(√−g(T t

t + ρ0ut)) = −∂i(

√−g(T it + ρ0u

i)) +√−gT κ

λ Γλtκ. (3.29)

In the nonrelativistic limit, this procedure subtracts the rest mass energy density from the totalenergy density.

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Specification of Geometric Quantities. In two dimensions we need to evaluate gµν , gµν ,

and√−g at four points in every grid zone (the center, two faces, and one corner) and Γµ

νλ at thezone center. It would be difficult to accurately encode analytic expressions for all these quantities.HARM is coded so that an analytic expression need only be provided for gµν ; all other geomet-ric quantities are calculated numerically. The connection, for example, is obtained to sufficientaccuracy by numerical differentiation of the metric. This minimizes the risk of coding errors inspecifying the geometry. It also minimizes coordinate dependent code, making it relatively easy tochange coordinate systems. Minimal coordinate dependence, besides following the spirit of generalrelativity, enables one to perform a sort of fixed mesh refinement by adapting the coordinates tothe problem at hand. For example, near a Kerr black hole we use log(r) as the radial coordi-nate instead of the usual Boyer-Lindquist r, and this concentrates numerical resolution toward thehorizon, where it is needed.

Density and Internal Energy Floors. Negative densities and internal energies are forbiddenby the GRMHD equations, but numerically nothing prevents their appearance. In fact, negativeinternal energies are common in numerical integrations with large density or pressure contrast.Following common practice, we prevent this by introducing “floor” values for the density andinternal energy. These floors are enforced after the half-step and the full step. They preservevelocity but do not conserve rest mass or energy-momentum.

Outflow Boundary Conditions. In the rotating black hole calculations described below weuse outflow boundary conditions at the inner and outer radial boundaries. The usual implementa-tion of outflow boundary conditions is to simply copy the primitive variables from the boundaryzones into the ghost zones. This can result in unphysical values of the primitive variables in theghost zones– for example, velocities that lie outside the light cone– because of variations in themetric between the boundary and ghost zones.

We have experimented with a variety of schemes for projecting variables into the ghost zones inthe context of black hole accretion flow calculations (described in §5). We find that some are morerobust than others. The most robust extrapolates the density, internal energy, and radial magneticfield according to

P (ghost) = P (boundary)√−g(boundary)/

√−g(ghost), (3.30)

the θ and φ components of the velocity and magnetic field according to

P (ghost) = P (boundary)(1.−∆r/r), (3.31)

and the radial velocity according to

P (ghost) = P (boundary)(1. + ∆r/r). (3.32)

The extrapolation of θ and φ components of magnetic field and velocity results in weak damping

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of these components near the boundary. Slightly different choices of the extrapolation coefficients(i.e. (1.− 2∆r/r)) are much less robust.

Performance. We have implemented both serial and parallel versions of the code. In serialmode the code integrates the black hole accretion problem (described in §5) at ≈ 54, 000 zone cyclesper second on a 2.4 GHz Intel Pentium 4, when compiled using the Intel C compiler. The parallelcode was implemented using MPI.

3.5 Code Verification

Here we present a test suite for verifying a GRMHD code. The tests are nonrelativistic, specialand general relativistic, and one and two dimensional. The list of problems for which there areknown, exact solution is short, since exact solutions of multidimensional GRMHD problems arealgebraically complicated. This list of test problems was developed in collaboration with J. Hawleyand J.-P. de Villiers. Unless otherwise stated we set γ = 4/3 and c = 1.

3.5.1 Linear Modes

This first test considers the evolution of a small amplitude wave in two dimensions. The unperturbedstate is ρ0 = 1, p = 1, ui = 0, By = Bz = 0, Bx = Bx

0 . The basic state is parametrizedby α = (Bx

0 )2/(ρ0c2); our fiducial test runs have α = 1. Onto this basic state we introduce a

perturbation of the form exp(ik · x− iω(k)t), where (kx, ky) = (2π, 2π), and the amplitude is fixedby δBy = 10−4Bx

0 . The computational domain is x, y ∈ [0, 1), [0, 1), and the boundary conditionsare periodic. The wave is either slow, Alfvenic, or fast.

This test exercises almost all terms in the governing equations. The numerical resolution is(Nx, Ny) ≡ N(5, 4) zones, and the integration runs for a single wave period 2π/ω, so that a perfectscheme would return the simulation to its original state. We measure the L1 norm of the differencebetween the final state and the initial state for each primitive variable. For example, we measure

L1(δρ0) =∫

dxdy|ρ0(t = 0)− ρ0(t = 2π/ω)|. (3.33)

for the density.All primitive variables exhibit similar convergence properties (as they must, since with the

exception of the magnetic field, they are tightly coupled together). In Figures 3.1, 3.2, and 3.3we present the L1 norm of the error for runs using the monotonized central limiter and a Courantnumber of 0.8, in addition to the results for the minmod limiter. These runs have α = 1. Figure3.1 shows the results for the slow wave, Figure 3.2 for the Alfven wave, and Figure 3.3 for the fastwave. Evidently the convergence rate asymptotes to second order, although more slowly for theminmod limiter.

The code performs similarly well over a range of α, provided only that δB2/2 ¿ p, which isnecessary for the wave to be in the linear regime. We have been unable to find a value of Bx

0 where

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Figure 3.1 The L1 norm of the error in u for a slow wave as a function of Nx for both the monotonizedcentral (MC) and minmod limiter. The straight lines show the slope expected for second orderconvergence.

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Figure 3.2 The L1 norm of the error in the single nonzero component of the velocity for an Alfvenwave as a function of Nx for both the monotonized central (MC) and minmod limiter. The straightlines show the slope expected for second order convergence.

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Figure 3.3 The L1 norm of the error in u for a fast wave as a function of Nx for both the monotonizedcentral (MC) and minmod limiter. The straight lines show the slope expected for second orderconvergence.

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the code fails completely for a linear amplitude disturbance, although for very large values of Bx0

the evolution becomes inaccurate because of numerical noise in the evaluation of P(U).

3.5.2 Nonlinear Waves

Komissarov (1999) has proposed a suite of one-dimensional nonlinear tests for special relativisticMHD. Komissarov presents a total of 9 tests (see his Table 1). The nonlinear Alfven wave (test5), and the compound wave (test 6) cannot be reconstructed without a separate derivation of theexact analytic solution, and we will not provide that here. For the remaining tests Komissarov’sTable 1 contains several misprints that are corrected in Komissarov (2002b). Our code is ableto integrate each of Komissarov’s remaining 7 tests, although in some cases we must reduce theCourant number (usually 0.8) or resort to the slightly more robust van Leer slope limiter. Teststhat required special treatment are: fast shock (Courant number = 0.5); shock tube 1 (Courantnumber = 0.3, van Leer limiter); shock tube 2 (Courant number = 0.5); collision (Courant number= 0.3; van Leer limiter).

Figures 3.4 and 3.5 show the run of ρ0 and ux, respectively, for all 7 tests. These may becompared with Komissarov’s figures. Notice that, unlike Komissarov, we have in all cases setNx = 400 and x ∈ (−2, 2). We have not obtained the exact solutions used by Komissarov, but thesolutions can still be checked quantitatively. For example, the slow shock wave speed is 0.5; sincethe calculation ends at t = 2 the slow shock front should be, and is, located at x ≈ 1.0. The fastshock speed is 0.2, so at t = 2.5 the fast shock wave front should be, and is, located at x ≈ 0.5.

There are artifacts evident in the figures. In particular there is ringing near the base of theswitch-on and switch-off rarefaction waves. This is common and is seen in Komissarov’s results aswell. In addition the narrow, Lorentz-contracted shell of material behind the shock in shock tube1 is poorly resolved; the correct shell density is ≈ 0.88 but a resolution of Nx > 800 is required tofind this result to an accuracy of a few percent. There is also a transient associated with the fastshock that propagates off the grid and so is not visible in Figures 3.4 and 3.5.

A complete set of nonlinear wave tests for one dimensional nonrelativistic MHD was developedby Ryu and Jones (1995) (hereafter RJ). We can run these under HARM by rescaling the speed oflight to c = 102 in code units, where all velocities in the tests are O(1). This should lead to resultsthat agree with RJ to O(v/c) ≈ 1%. The results can be checked quantitatively by comparison tothe tables provided by RJ.

Figure 3.6 shows our results for RJ test 5A, which is a version of the familiar Brio and Wu(1988) magnetized shock tube test. Like RJ we use 512 zones between x = 0 and x = 1, andwe measure the results at t = 0.15. To compare to RJ quantitatively, consider ux in the regionbehind the fast rarefaction wave, near x = 0.7. RJ report ux = −0.277 here, while we measureux = −0.273, which differs by 1%, as expected. Similar agreement is found for the other variables.The most unsatisfactory feature of the solution is the visible post-shock oscillations. The amplitudeof these features varies, depending on the Courant number (here 0.9) and the choice of slope limiter

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Figure 3.4 The run of density in the Komissarov nonlinear wave tests.

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Figure 3.5 The run of ux in the Komissarov nonlinear wave tests.

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Figure 3.6 Snapshot of the final state in HARM’s integration of Ryu & Jones test 5A (a versionof the Brio & Wu shock tube) but with c = 100. The figure shows primitive variable values att = 0.15. Quantitative agreement is found to within ≈ 1%, as expected.

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(here MC).Figure 3.7 shows our results for RJ test 2A. In the region near x = 0.6, RJ report that By =

1.4126, and we find essentially exact agreement (By = 1.41262) after averaging over a small regionnear x = 0.6. This test has two pairs of closely spaced slow shocks and rotational discontinuitiesthat are difficult to resolve, and our scheme barely obtains the correct peak values of uy and By,even though there are about 20 zones inside the “horns” visible in the By panel of the figure.

3.5.3 Transport

This special relativistic test evolves a disk of enhanced density moving at an angle to the griduntil it returns to its original position. The computation is carried out in a domain x, y ∈[−0.5, 0.5), [−0.5, 0.5) and the boundary conditions are periodic. The initial state has vx = vy = 0.7,or ux = uy ≈ 4.95, corresponding to ut ≈ 7.07. The initial density ρ0 = 1 except in a disk atr < rs = 0.45, where ρ0 = 3/2 + cos(2πr/rs). The initial pressure p = 1, and the initial magneticfield is zero. The test is run until t = 10/7, when the system should return exactly to its initialstate.

Numerically, we use the monotonized central limiter and set the Courant number to 0.8. Theresolution is fixed so that Nx = 5Ny/4. Figure 3.8 shows the L1 norm of the error in ρ0 as afunction of x resolution. The convergence rate asymptotes to second order.

3.5.4 Orszag-Tang Vortex

The Orszag-Tang vortex (OTV) is a classic nonlinear MHD problem (Orszag and Tang, 1979). Herewe compare our code, with the speed of light set to 100 so that that it is effectively nonrelativistic,to the output of VAC (Toth and Odstrcil, 1996), an independent nonrelativistic code developed byone of us. The version of VAC used here is TVD-MUSCL using the monotonized central limiter.It is dimensionally unsplit and uses a scheme similar to HARM to control ∇ ·B. The problem isintegrated in the periodic domain x ∈ (−π, π], y ∈ (−π, π] from t = 0 to t = π. Our version of theOTV has γ = 4/3, but is otherwise identical to the standard problem.

Results are shown in Figure 3.9, which shows ρ0 along a cut through the model at y = π/2 andt = π. The resolution is 6402. The solid line shows the results from HARM; the dashed line showsthe results from VAC. The lower solid line shows the difference between the two multiplied by 4.Evidently our code behaves similarly to VAC on this problem.

We can quantify this by asking how the difference between the HARM and VAC solutionschanges as a function of resolution. Figure 3.10 shows the variation in the L1 norm of the differencebetween the two solutions. Thus the line marked ρ0 shows

∫dxdy|ρ0(HARM;N2)− ρ0(VAC;N2)| (3.34)

evaluated at t = π. The codes converge to one another approximately linearly, as expected for a

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Figure 3.7 Snapshot of the final state in HARM’s integration of Ryu & Jones test 2A, with c = 100.

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Figure 3.8 Convergence results for the transport test.

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Figure 3.9 A cut through the density in the nonrelativistic Orszag-Tang vortex solution from HARM(solid line, with c = 100), from VAC (dashed line), and 4× the difference (lower solid line) at aresolution of 6402.

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Figure 3.10 Comparison of results from HARM and the nonrelativistic MHD code VAC for theOrszag-Tang vortex. The plot shows the L1 norm of the difference between the two results as afunction of resolution for the primitive variables ρ0 (squares) and u (triangles). The straight lineshows the slope expected for first order convergence. The errors are large because they are anintegral over an area of (2π)2.

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flow containing discontinuities. If this study were extended to higher resolution convergence wouldeventually cease because the HARM solution would differ from the VAC solution due to finiterelativistic corrections.

3.5.5 Bondi Flow in Schwarzschild Geometry

Spherically symmetric accretion (Bondi flow) in the Schwarzschild geometry has an analytic solution(see, e.g., Shapiro and Teukolsky 1983) that can be compared with the output of our code. Thisappears to be a one-dimensional test, but for HARM it is actually two dimensional. Althoughthe pressure is independent of the Boyer-Lindquist coordinate θ, the θ acceleration does not vanishidentically. This is because pressure enters the momentum equations through a flux (−∂θ(p sin θ) inthe Newtonian limit) and a source term (p cos θ in the Newtonian limit). Analytically these termscancel; numerically they produce an acceleration that is of order the truncation error.

Our test problem follows that set out in Hawley et al. (1984): we fix the sonic point rs =8GM/c2, M = 4πr2ρ0u

r = −1, and γ = 4/3. The problem is integrated in the domain r ∈(1.9, 20)GM/c2 for ∆t = 100GM/c3. We use coordinates based on the Kerr-Schild system, whoseline element is

ds2 = −(1− 2r/ρ02)dt2 + (4r/ρ0

2)drdt + (1 + 2r/ρ02)dr2 + ρ0

2dθ2+sin2 θ

(ρ0

2 + a2(1 + 2r/ρ02) sin2 θ

)dφ2

−(4ar sin2 θ/ρ02)dtdφ− 2a(1 + 2r/ρ0

2) sin2 θdrdφ,

(3.35)

where we have set GM = c = 1. In (3.35) only ρ02 = r2 +a2 cos2(θ); elsewhere ρ0 is density. In this

test, a = 0. We modify these coordinates by replacing r by x1 = log(r). The new coordinates areimplemented by changing the metric rather than changing the spacing of grid zones. We measurethe L1 norm of the difference between the initial conditions (exact analytic solution) and the finalstate. The difference is taken over the inner 3/4 of the grid in each direction, thus excludingboundary zones where errors may scale differently. This test exercises many terms in the codebecause in Kerr-Schild coordinates only three of the ten independent components of the metric arezero.

The L1 norm of the error in internal energy for the Bondi test is shown in Figure 3.11. Similarresults obtain for the other independent variables. The solution converges at second order.

3.5.6 Magnetized Bondi Flow

The next test considers a Bondi flow containing a spherically symmetric, radial magnetic field. Thesolution to this problem is identical to the Bondi flow described above because the flow is alongthe magnetic field, so all magnetic forces cancel exactly. This is a difficult test, however, becausenumerically the magnetic terms cancel only to truncation error. This causes problems at highmagnetic field strength.

We use the same Bondi solution as in the last subsection and parameterize the magnetic field

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Figure 3.11 Convergence results for the unmagnetized Bondi accretion test onto a Schwarzschildblack hole. The straight line shows the slope expected for second order convergence.

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strength by b2/ρ0 at the inner boundary. Our fiducial run has (b2/ρ0)(r = 1.9GM/c2) = 10.56.The L1 norm of the error in the internal energy is shown in Figure 3.12. Similar results obtain forthe other independent variables. The solution converges at second order.

We have considered models with a range of (b2/ρ0)(rin). Lowering (b2/ρ0)(rin) produces resultssimilar to those in our fiducial test run. Raising (b2/ρ0)(rin) first causes the code to produceinaccurate results (at ∼ 103, where the radial velocity profile is smoothly distorted from the truesolution) and then to fail (at ∼ 104). This is an example of a general problem with conservativeschemes when the basic energy density scales (rest mass, magnetic, and internal) differ by manyorders of magnitude.

3.5.7 Magnetized Equatorial Inflow in Kerr Geometry

This test considers the steady-state, magnetized inflow solutions found by Takahashi et al. (1990), asspecialized to the case of inflow inside the marginally stable orbit by Gammie (1999). This solutionexercises many of the important terms in the governing equations, in particular the interaction ofthe magnetized fluid with the Kerr geometry.

We use Boyer-Lindquist coordinates to specify this problem, but the solution is integrated in themodified Kerr-Schild coordinates described above. The flow is assumed to lie in the neighborhoodof the black hole’s equatorial plane and is thus one dimensional, much like the Weber-Davis modelfor the solar wind. As above, we set GM = c = 1.

The particular inflow solution we consider is for a black hole with spin parameter a/M = 0.5.The model has an accretion rate FM = −1 = 2πρ0r

2ur (adopting the notation of Gammie 1999).The magnetization parameter Fθφ = r2Br = 0.5. The flow is constrained to match to a circularorbit at the marginally stable orbit. This is enough to uniquely specify the flow. It follows that(see Gammie 1999) Ftθ = ΩFθφ, where Ω is the orbital frequency at the marginally stable orbit.For a/M = 0.5,Ω ≈ 0.10859. The solution that is regular at the fast point has angular momentumflux FL = 2πr2(uruφ − brbφ) ≈ −2.8153 and energy flux FE = 2πr2(urut − brbt) ≈ −0.90838.The fast point is located at r ≈ 3.6167, and the radial component of the four-velocity there isur = −0.040547. Figure 3.13 shows the radial run of the solution.

We initialize the flow with a numerical solution that is subject to roundoff error. The near-equatorial nature of the solution is mimicked by using a single zone in the θ direction centered onθ = π/2. The computational domain runs from 1.02× the horizon radius rh to 0.98× the radius ofthe marginally stable orbit rmso. For a/M = 0.5, rh = 1.866, and rmso = 4.233. The analytic flowmodel is cold (zero temperature) but we set the initial internal energy in the code equal to a smallvalue instead. The model is run for ∆t = 15.

Figure 3.14 shows the L1 norm of the error in ρ0, ur, uφ, and Bφ and a function of the total

number of radial gridpoints N . The straight line shows the slope expected for second order con-vergence. The small deviation from second order convergence at high N in several of the variablesis due to numerical errors in the initial solution, which relies on numerical derivatives (Gammie,

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Figure 3.12 Convergence results for the magnetized Bondi accretion test onto a Schwarzschild blackhole. The straight line shows the slope expected for second order convergence.

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Figure 3.13 The equatorial inflow solution in the Kerr metric for a/M = 0.5 and magnetizationparameter Fθφ = 0.5. The panels show density, radial component of the four-velocity in Boyer-Lindquist coordinates (with the square showing the location of the fast point), the φ component ofthe four-velocity, and the toroidal magnetic field Bφ = F φt.

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Figure 3.14 Convergence results for the magnetized inflow solution in a Kerr metric with a/M = 0.5.Parameters for the initial, quasi-analytic solution are given in the text. The straight line showsthe slope expected for second order convergence. The L1 error norm for each of the nontrivialvariables are shown. The small deviation from second order convergence at high resolution is dueto numerical errors in the quasi-analytic solution used to initialize the solution.

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1999).

3.5.8 Equilibrium Torus

Our next test concerns an equilibrium torus. This class of equilibria, found originally by Fishboneand Moncrief (1976) and Abramowicz et al. (1978), consist of a “donut” of plasma surrounding ablack hole. The donut is supported by both centrifugal forces and pressure and is embedded in avacuum. Here we consider a particular instance of the Fishbone & Moncrief solution.

A practical problem with this test is that HARM abhors a vacuum. We have therefore intro-duced floors on the density and internal energy that limit how small these quantities can be. Thefloors are dependent on radius, with ρ0min = 10−4(r/rin)−3/2 and umin = 10−6(r/rin)−5/2. Thismeans that the torus is surrounded by an insubstantial, but dynamic, accreting atmosphere thatinteracts with the torus surface. To minimize the influence of the atmosphere on our convergencetest, we take the L1 norm of the change in variables only over that region where ρ0 > 0.02ρ0max.

The problem is integrated in modified Kerr-Schild coordinates. The Kerr-Schild radius r hasbeen replaced by the logarithmic radial coordinate x1 = ln(r), and the Kerr-Schild latitude θ

has been replaced by x2 such that θ = πx2 + (1/2)(1 − h) sin(2πx2). Clearly 0 ≤ x2 ≤ 1 mapsto 0 ≤ θ ≤ π. This coordinate transformation has a single adjustable parameter h; for h = 1we recover the original coordinate system (the θ coordinate is simply rescaled by π). As h → 0numerical resolution is concentrated near the midplane.

We have integrated a Fishbone-Moncrief disk around a black hole with a/M = 0.95, to maximizegeneral relativistic effects. We set utuφ = const. = 3.85 (this is the defining feature of the Fishbone-Moncrief equilibria) and rin = 3.7. The grid extends radially from rin = 0.98rh = to rout = 20.The coordinate parameter h described in the last paragraph is set to 0.2. The numerical resolutionis N × N , where N = 8, 16, 32, . . . , 512, and the solution is integrated for ∆t = 10. Figure 3.15shows the L1 norm of the error for each variable as a function of N . Second order convergence isobtained.

The sum of the evidence presented in this section strongly suggests that we are solving theequations of GRMHD without significant, compromising errors.

3.6 Magnetized Torus Near Rotating Black Hole

Finally we offer an example of how HARM can be applied to a real astrophysical problem: theevolution of a magnetized torus near a rotating black hole. Again we set GM = c = 1.

The initial conditions contain a Fishbone-Moncrief torus with a/M = 0.5, r(pmax) = 12, andrin = 6. Superposed on this equilibrium is a purely poloidal magnetic field with vector potentialAφ ∝ MAX(ρ0/ρ0max−0.2, 0), where ρ0max is the peak density in the torus. The field is normalizedso that the minimum value of pgas/pmag = 102. The orbital period at the pressure maximum(r = 12), is 264 as measured by an observer at infinity.

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Figure 3.15 Convergence results for the Fishbone and Moncrief equilibrium disk around an a/M =0.95 black hole.

112

Figure 3.16 Density field, for a magnetized torus around a Kerr black hole with a/M = 0.5 at t = 0(left) and at t = 2000M (right). The color is mapped from the logarithm of the density; black islow and dark red is high. The resolution is 3002.

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The integration extends for ∆t = 2000, or about 7.6 orbital periods at the pressure maximum.Figure 3.16 shows the initial and final density states projected on the R = r sin(θ), Z = r cos(θ)plane. Color represents log(ρ0). The coordinate parameter h, which concentrates zones toward themidplane, is set to 0.2. The torus atmosphere is set to the floor values (see above), and the MClimiter is used. The numerical resolution is 3002.

The flux of mass, energy, and angular momentum through the inner boundary are describedin Figure 3.17. Initially the fluxes are small because the initial conditions are near an (unstable)equilibrium. The magnetorotational instability (Balbus and Hawley, 1991) e-folds for just over anorbital period, after which the magnetic field has reached sufficient strength to distort the originaltorus and drop material into the black hole. Later, the torus is turbulent and accretion occurs ata more or less steady rate.

3.7 Conclusion

Like all hydrodynamics codes, HARM has failure modes. We will discuss one that is likely tobe relevant to future astrophysical simulations. When B2/ρ0 À 1 and B2 À u, the magneticenergy is the dominant term in the total energy equation. Because the fields are evolved separately,truncation error in the field evolution can lead to large fractional errors in the velocity and internalenergy. An example of this was discussed in §3.5.6, where the magnetized Bondi flow test fails forlarge values of B2/ρ0.

Another example can be found in the strong cylindrical explosion problem of Komissarov (1999),where an overpressured region embedded in a uniform magnetic field produces a relativistic blastwave. HARM fails on the strong-field version of this problem unless we turn the Courant numberdown to 0.1, use the minmod limiter, and sharply increase the accuracy parameter used in theP(U) inverter. This is a particularly difficult problem, with B2/ρ0 as large as 104. The problemscaused by magnetically dominated regions appears to be generic to conservative relativistic MHDschemes, where small errors in magnetic energy density lead to fractionally large errors in othercomponents of the total energy. At present this is unavoidable, and has motivated the developmentof schemes for the evolution of the electromagnetic field in the force-free limit (Komissarov, 2002c).

Finally, to sum up: we have described and tested a code that evolves the equations of generalrelativistic magnetohydrodynamics. This code, together with the code described in a companionpaper by De Villiers and Hawley (2002), are the first that stably evolve a relativistic plasma ina Kerr spacetime for many light crossing times. The advent of practical, stable GRMHD codesopens the door for the study of many problems in the theory of RMRs. For example, it may bepossible to directly evaluate the importance of magnetic energy extraction from rotating black holesand the importance of black hole spin in determining jet parameters. It may also be possible tocouple these schemes to numerical relativity codes and use them to study dynamical spacetimeswith electromagnetic sources.

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Figure 3.17 Evolution of the rest-mass accretion rate (top), the specific energy of the accretedmatter (middle), and the specific angular momentum of the accreted matter (bottom) for a blackhole with a/M = 0.5.

115

4 A Measurement of theElectromagnetic Luminosity of a KerrBlack Hole

4.1 Summary of Chapter

Some active galactic nuclei, microquasars, and gamma ray bursts may be powered by the electro-magnetic braking of a rapidly rotating black hole. We investigate this possibility via axisymmetricnumerical simulations of a black hole surrounded by a magnetized plasma. The plasma is describedby the equations of general relativistic magnetohydrodynamics, and the effects of radiation areneglected. The evolution is followed for 2000GM/c3, and the computational domain extends frominside the event horizon to typically 40GM/c2. We compare our results to two analytic steady statemodels, including the force-free magnetosphere of Blandford & Znajek. Along the way we presenta self-contained rederivation of the Blandford-Znajek model in Kerr-Schild (horizon penetrating)coordinates. We find that (1) low density polar regions of the numerical models agree well with theBlandford-Znajek model; (2) many of our models have an outward Poynting flux on the horizon inthe Kerr-Schild frame; (3) none of our models have a net outward energy flux on the horizon; and(4) one of our models, in which the initial disk has net magnetic flux, shows a net outward angularmomentum flux on the horizon. We conclude with a discussion of the limitations of our model,astrophysical implications, and problems to be addressed by future numerical experiments.1

4.2 Introduction

A black hole of mass M and angular momentum J = aGM/c, 0 ≤ a/M < 1 has a free energyassociated with its angular momentum (or “spin”). This energy can, in principle, be tapped bymanipulating particle orbits so that negative energy particles are accreted (Penrose, 1969). Spinenergy can also be tapped by superradiant scattering of vacuum electromagnetic waves (Pressand Teukolsky, 1972), gravity waves (Hawking and Hartle, 1972; Teukolsky and Press, 1974), ormagnetohydrodynamic (MHD) waves (Uchida, 1997). It can also be tapped through the action offorce-free electromagnetic fields (Blandford and Znajek, 1977).

The Blandford-Znajek (BZ) effect– broadly used here to mean the extraction of energy fromrotating holes via a magnetized plasma– appears to be the most astrophysically plausible exploita-

1Published in ApJ 20 August 2004 issue. Reproduction for this thesis is authorized by the copyright holder.

116

tion of black hole spin energy. Relativistic jets in active galactic nuclei, galactic microquasars, andgamma-ray bursts (GRBs) may well be powered by the BZ effect. Despite some hints (see, e.g.,Wilms et al. 2001b, Miller et al. 2002, Maraschi and Tavecchio 2003) and the general consistencyof this idea with the data, however, there is no direct observational evidence for black hole energyextraction. In this paper we take an experimental approach and study the BZ effect through directnumerical simulation of a magnetized plasma accreting onto a black hole.

The energy stored in black hole spin is potentially large. If Mirr is the “irreducible mass” ofthe black hole where, in units such that G = c = 1,

M2irr =

12Mr+, (4.1)

and r+ = M(1 +√

1− (a/M)2) is the horizon radius, then the free energy is

Espin = M −Mirr < 5.3× 1061

(M

108 M¯

)erg. (4.2)

or ≈ 30% of the gravitational mass of a maximally rotating hole. This corresponds to a luminosityof . 4× 1010(M/108 M¯) L¯ if released over a Hubble time.

Estimates suggest that black hole accretion is surprisingly efficient, in the sense that the ratio ofquasar radiative energy density to supermassive black hole mass density is ∼ 0.2 (Yu and Tremaine,2002; Elvis et al., 2002). During the accretion process some mass-energy is radiated away and therest is incorporated into the black hole. Through electromagnetic spindown this energy gets asecond chance to escape. A combination of efficient thin disk accretion (in which all radiationis somehow permitted to escape) followed by the Penrose process can in principle extract up to(1 − 1/

√6)c2 = 0.59c2 per gram of accreted rest-mass. In practice, of course, much less energy is

likely to be available. One goal of our investigation is to discover how much less. Part of the answermay lie with the calculations already described in Gammie et al. (2004): if black hole spins arelimited by the equilibrium value found there (a/M ≈ 0.92) then the nominal thin disk efficiency ofthe accretion phase is about ≈ 17%, much less than the 42% expected at a/M = 1.

In this paper we consider the self-consistent evolution of a weakly magnetized torus surroundinga rotating black hole. The evolution is carried out numerically in the axisymmetric ideal MHDapproximation. As the evolution progresses the computational domain develops matter dominatedregions near the equator and electromagnetic field dominated regions near the poles. To fix ex-pectations for the structure of these regions we review two analytic models for the interaction of amagnetized plasma with a black hole in § 4.3. Along the way we develop the relevant notation andcoordinate systems. In § 4.4 we describe our numerical model and give a summary of numericalresults for a high resolution fiducial model. In § 4.5 we consider the dependence of our results onmodel parameters. A discussion and summary may be found in § 4.6. From here on we adopt unitssuch that GM = c = 1. Table 4.1 gives a list of commonly used symbols.

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Table 4.1. Commonly used symbols

Symbol Fiducial Value Description

Model Parametersa 0.938 black hole spin (J/M2)r+ 1.347 radius of the event horizon (r+ = 1 +

√1− a2)

risco 2.044 radius of the ISCO (innermost stable circular orbit)redge 6 radius of inner edge of torusrmax 12 radius of the pressure maximumΩH ≈ 0.3477 spin frequency of zero angular momentum observer at r+

Rin 0.98r+ inner radial grid locationRout 40 outer radial grid locationβ 100 ratio of gas to magnetic pressure (initially pgas,max

pmag,max)

γ 4/3 pgas = (γ − 1)u

DiagnosticsM0 see sections 4.3.2 & 4.4 rest-mass flux into the black holeE see sections 4.3.2 & 4.4 energy flux into the black holeE(EM) see sections 4.3.2 & 4.4 electromagnetic energy fluxE(MA) see sections 4.3.2 & 4.4 matter energy fluxL see sections 4.3.2 & 4.4 angular momentum flux into the black holeL(EM) see sections 4.3.2 & 4.4 electromagnetic angular momentum fluxL(MA) see sections 4.3.2 & 4.4 matter angular momentum fluxL see sections 4.4.1 & 4.5 L = E(EM)/(−εM0) ; ε = 1− E/M0

Variablesb2/2 see section 4.4.3 electromagnetic energy density in the fluid frameBr,Bθ,Bφ see section 4.3.2 magnetic field components. Bi =

∗F

it

Aφ see section 4.4 azimuthal component of electromagnetic vector potentialvr see section 4.4.1 asymptotic radial velocity (i.e. vr at r = ∞)ω see sections 4.4.2 & 4.4.3 spin frequency of electromagnetic fieldΩ see section 4.4.3 spin frequency of fluid (Ω = uφ/ut)

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4.3 Review of Analytic Models

In this section we review two quasi-analytic, steady state models for the interaction of a blackhole with the surrounding plasma. The purpose of this review is to introduce our coordinatesystem and notation and to describe the models in a form suitable for later comparison withnumerical results. Along the way, we give a self-contained derivation of the BZ effect in Kerr-Schild(horizon penetrating) coordinates. To the extent that the analytic and numerical models agree, thecomparison also builds confidence in the numerical models.

4.3.1 Coordinates

Before proceeding it is useful to define three coordinate bases for the Kerr metric.Boyer-Lindquist (BL) coordinates. These are the most familiar coordinates for the Kerr metric.

In BL coordinates t, r, θ, φ

ds2 = −(

1− 2 r

Σ

)dt2 +

Σ∆

dr2 + Σ dθ2 +A sin2 θ

Σdφ2 − 4 a r sin2 θ

Σdφ dt (4.3)

where Σ ≡ r2 +a2 cos2 θ, ∆ ≡ r2− 2r +a2 and A ≡ (r2 +a2)2−a2∆sin2 θ. The determinant of themetric g ≡ Det(gµν) = −Σ2 sin2 θ. In BL coordinates the metric is singular on the event horizonat r = r+ where ∆ = 0.

Kerr-Schild (KS) coordinates. The Kerr-Schild coordinates t, r, θ, φ are regular on the horizon.They are closely related to BL coordinates: r[KS] = r[BL] and θ[KS] = θ[BL]. The line element is

ds2 = −(

1− 2 r

Σ

)dt2 +

(4 r

Σ

)dr dt +

(1 +

2 r

Σ

)dr2 + Σ dθ2

+sin2 θ

(Σ + a2

(1 +

2 r

Σ

)sin2 θ

)dφ2

−(

4 a r sin2 θ

Σ

)dφ dt− 2 a

(1 +

2 r

Σ

)sin2 θ dφ dr, (4.4)

and g = −Σ2 sin2 θ.The transformation matrix from BL to KS is

∂t[KS]∂r[BL]

=2r

∆, (4.5)

and∂φ[KS]∂r[BL]

=a

∆; (4.6)

all other off-diagonal components are 0 and all diagonal components are 1. The inverse transfor-mation matrix is identical, with the signs of the off-diagonal components reversed.

Modified Kerr-Schild (MKS) coordinates. Our numerical integrations are carried out in a mod-

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ified KS coordinates x0, x1, x2, x3, where x0 = t[KS], x3 = φ[KS], and

r = ex1 , (4.7)

θ = πx2 +12(1− h) sin(2πx2). (4.8)

Here h is an adjustable parameter that can be used to concentrate grid zones toward the equatoras h is decreased from 1 to 0. The transformation matrix from KS to MKS is diagonal and triviallyconstructed from the explicit expressions for r and θ in equations 4.7 and 4.8.

4.3.2 Governing Equations

For a magnetized plasma the equations of motion are

Tµν;ν =

(Tµν

MA + TµνEM

);ν

= 0. (4.9)

where Tµν is the stress-energy tensor, which can be split into a matter (MA) and electromagnetic(EM) part. In the fluid approximation

TµνMA = (ρ0 + ε + p)uµuν + pgµν , (4.10)

where ρ0 ≡ rest-mass density, ε ≡ internal energy, p ≡ pressure, uµ is the fluid four-velocity, andwe assume throughout an ideal gas equation of state

p = (γ − 1)ε. (4.11)

In terms of Fµν , the Faraday (or electromagnetic field) tensor,

TµνEM = FµγF ν

γ − 14gµνFαβFαβ, (4.12)

where we have absorbed a factor of√

4π into the definition of Fµν . We assume that particle numberis conserved:

(ρ0uµ);µ = 0. (4.13)

The evolution of the electromagnetic field is given by the space components of the source-freeMaxwell equations

∗F

µν;ν = 0, (4.14)

where∗F is the dual of the Faraday, and the time component gives the no-monopoles constraint.

The inhomogeneous Maxwell equations

Jµ = Fµν;ν (4.15)

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define the current density Jµ but are otherwise not required here. We adopt the ideal MHDapproximation, where

uµFµν = 0, (4.16)

which implies that the electric field vanishes in the rest frame of the fluid.In our numerical models the fundamental (or “primitive”) variables that describe the state of

the plasma are ρ0, ε, Bi ≡ ∗

Fit, plus three variables which describe the motion of the plasma. In

Gammie et al. (2003) we used the plasma three-velocity. Here we use

ui ≡ ui +γβi

α, (4.17)

where γ ≡√

1 + q2, q2 ≡ gij uiuj , βi ≡ gtiα2 is the shift, and α2 = −1/gtt is the lapse. We made

this change to improve numerical stability. Because the three velocity components have a finiterange, truncation error can move the plasma velocity outside the light cone. The variables ui havethe important property that they range over −∞ to ∞, and this makes it impossible for the plasmato step outside the light cone.

To write the electromagnetic quantities in terms of the primitive variables, define the four-vectorbµ with bt ≡ giµBiuµ and bi ≡ (Bi + uibt)/ut. With some manipulation one finds

TµνEM = b2uµuν +

b2

2gµν − bµbν , (4.18)

and∗F

µν= bµuν − bνuµ. (4.19)

The no-monopoles constraint becomes

(√−gBi),i = 0. (4.20)

A more complete account of the relativistic MHD equations can be found in Gammie et al. (2003)or Anile (1989).

4.3.3 Blandford-Znajek Model

BZ studied a rotating black hole surrounded by a stationary, axisymmetric, force-free, magnetizedplasma. They obtain an expression for the energy flux through the event horizon and, given asolution for the field geometry when a = 0, find a perturbative solution when a ¿ 1. Here wepresent a self-contained rederivation, which will be compared to numerical models in Section 4.4.2.Those not interested in the derivation may find a summary set of equations in 4.3.3. A comparisonof the analytic BZ model to our numerical models can be found in Section 4.4.2.

We follow an approach that differs slightly from BZ. We solve Tµν;ν = 0 directly rather than

using JµFµν = 0, which is equivalent in the force-free approximation. Also, because our solution

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is developed in KS coordinates, which are regular on the horizon, we obtain the BZ solution byapplying a regularity condition on the horizon and at large radius, rather than the physicallyequivalent approach of applying a regularity condition on the horizon in the Carter tetrad (Znajek,1977) and then applying the result as a boundary condition in BL coordinates. Finally, if we assumeseparability of the solution then we do not need to require that the solution match the flat-spaceforce-free solution of Michel (1973).

Derivation in KS coordinates

Over the poles of the black hole it is reasonable to expect that the density is low, but the fieldstrength is comparable to that at the equator. In the limit that

b2 À ρ0 + ε + p, (4.21)

where b2 is the field strength in the fluid frame, one may assume that the matter contribution tothe stress energy tensor can be ignored and

Tµν ≈ TµνEM. (4.22)

This is the force-free limit.The ideal MHD condition uµFµν = 0 implies that the electric field vanishes in the rest frame of

the fluid. Therefore the invariant E ·B = 0, or in covariant form∗F

µνFµν = 0. The electromagnetic

field is then said to be degenerate.In the force-free limit the governing equations are then

TµνEM;ν = 0 (4.23)

and∗F

µν;ν = 0. (4.24)

As BZ point out, the same basic set of equations can be derived without assuming that the plasmaobeys the fluid equations.

We now specialize to KS coordinates and write down the Faraday tensor in terms of a vectorpotential Aµ, Fµν = Aν,µ−Aµ,ν . We assume that the field is axisymmetric (∂φ → 0) and stationary(∂t → 0). Evaluating the condition

∗F

µνFµν = 0, one finds

Aφ,θAt,r −At,θAφ,r = 0. (4.25)

It follows that one may writeAt,θ

Aφ,θ=

At,r

Aφ,r≡ −ω(r, θ) (4.26)

where ω(r, θ) is an as-yet-unspecified function. It is usually interpreted as the “rotation frequency”

122

of the electromagnetic field (this is Ferraro’s law of isorotation; see e.g. Frank et al. 2002, §9.7 in anonrelativistic context). This yields Fµν in terms of the free functions ω, Aφ, and Bφ, the toroidalmagnetic field:

Ftr = −Frt = ωAφ,r (4.27)

Ftθ = −Fθt = ωAφ,θ (4.28)

Frθ = −Fθr =√−gBφ (4.29)

Frφ = −Fφr = Aφ,r (4.30)

Fθφ = −Fφθ = Aφ,θ (4.31)

with all other components zero. Written in this form, the electromagnetic field automaticallysatisfies the source-free Maxwell equations. Notice that Aφ,θ =

√−gBr and Aφ,r = −√−gBθ.We want to evaluate the radial energy flux

E ≡ 2π

∫ π

0dθ√−gFE (4.32)

where FE ≡ −T rt . This can be subdivided into a matter F

(MA)E and electromagnetic F

(EM)E part,

although in the force-free limit the matter part vanishes. Similar expressions can be written forthe angular momentum flux L and angular momentum flux density FL, and for the mass flux M0

and mass flux density FM . In the limit of a steady flow these conserved quantities correspond tothe radial flux measured by a stationary observer at large distance from the black hole.

Using the definition of the electromagnetic stress-energy tensor (4.12) and the relations (4.27)-(4.31), it is a straightforward exercise to evaluate

F(EM)E = −2(Br)2ωr(ω − a

2r) sin2 θ −BrBφω∆sin2 θ. (4.33)

The radial angular momentum flux density is F(EM)L = F

(EM)E /ω. One can verify by direct trans-

formation that FE [KS] = FE [BL] and FL[KS] = FL[BL]. On the horizon r = r+ = 1+√

1− a2 and∆ = 0, so the horizon energy flux is

F(EM)E |r=r+ = 2(Br)2ωr+(ΩH − ω) sin2 θ (4.34)

where ΩH ≡ a/(2r+) is the rotation frequency of the black hole (see MTW §33.4). This result,which is identical to BZ’s result, implies that if 0 < ω < ΩH and (Br)2 > 0 then there is an outwarddirected energy flux at the horizon. Because the flux was evaluated in KS coordinates the horizondid not require special treatment as in Znajek (1977).

To finish evaluating E(EM) we need to find Aφ, ω, and Bφ. This requires solving the equationsof motion (4.9). They can be evaluated directly or in the reduced form JµFµν = 0 (as in BZ), inwhich case one must also evaluate the currents using Maxwell’s equations. In either form this is a

123

difficult, nonlinear problem which probably cannot be solved in any general way.To make progress, BZ find solutions to the equations of motion when a = 0, then perturb

them by allowing the black hole to spin slowly with a ¿ 1. If we assume that the initial field hasω = Bφ = 0, then we may expand the vector potential

Aφ = A(0)φ (r, θ) + a2A

(2)φ (r, θ) +O(a4), (4.35)

where A(1)φ = 0 by symmetry (Aφ should be even in a). The field rotation frequency vanishes in

the unperturbed solution, and ω(2) = 0 because ω should be odd in a, so

ω = aω(1)(r, θ) +O(a3) (4.36)

and similarly for the toroidal field

Bφ = aBφ(1)(r, θ) +O(a3). (4.37)

We are now in a position to find the free functions A(2)φ , ω(1), and Bφ(1), given an initial field A

(0)φ

that satisfies the basic equations when a = 0.BZ consider two forms for A

(0)φ : a monopole field and a paraboloidal field. Here we review only

the (possibly split) monopole, where A(0)φ = −C cos θ and C is an arbitrary constant. One may

obtain the perturbed solution by making the following sequence of deductions.(1) The t and φ components of equation (4.9), expanded to lowest nontrivial order in a, require

that F(EM)L and F

(EM)E be independent of radius. Therefore they are functions of θ alone. Since

F(EM)E = aω(1)F

(EM)L , (4.38)

we conclude that ω(1) is a function of θ alone.(2) The r component of equation (4.9), together with the requirement that Bφ(1) be finite on

the horizon (all components of Fµν are well-behaved on the horizon in KS coordinates), yields asingle nontrivial solution:

Bφ(1) = − C

4r2

(1− 4ω(1) +

2r

)(4.39)

This solution is well behaved at the horizon and at large radius as long as ω(1) is finite on thehorizon and grows less rapidly than r2 at large r.

(3) The θ component of equation (4.9), which is the trans-field force balance equation, can nowbe reduced to an equation involving A

(2)φ and ω(1). If we require that A

(2)φ = Cf(r)g(θ), then one

may deduce that (a) ∂θω(1) = 0, i.e. ω(1) = const.; (b) g(θ) = cos θ sin2 θ. Then f(r) must satisfy

f ′′ +2f ′

r(r − 2)− 6f

r(r − 2)+

(r + 2

r3(r − 2)− (ω(1) − 1/8)(r2 + 2r + 4)

r(r − 2)

)= 0 (4.40)

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which is equivalent to BZ’s equation (6.7). This has an exact solution with two constants ofintegration. One of the constants of integration is set by requiring that the solution be finite onthe horizon. Part of the solution can be regularized at large r by fixing the other constant ofintegration, but the remaining divergence can only be zeroed by setting ω(1) = 1/8; this is alreadysuggested by the form of the preceding equation. For r > 2 the regular solution is

f(r) =(

Li2(2r)− ln(1− 2

r) ln

r

2

)r2(2r − 3)

8+

1 + 3r − 6r2

12ln

r

2+

1172

+13r

+r

2− r2

2, (4.41)

where Li2 is the second polylogarithm function:

Li2(x) = −∫ 1

0dt

ln(1− tx)t

. (4.42)

For r < 2 the solution is given by the real part of equation (4.41). In the limit of large r

f(r) ∼ 14r

+O(

ln r

r2

), (4.43)

which agrees with BZ.To sum up, using only the assumption of separability of A

(2)φ and the regularity of physical

quantities in Kerr-Schild coordinates on the horizon and at infinity, we find

ω(1) =18

(4.44)

Bφ(1) = − C

8r2(1 +

4r) (4.45)

andA

(2)φ = Cf(r) cos θ sin2 θ. (4.46)

with f(r) given by equation (4.41). Our solution is identical to BZ’s after transforming to Boyer-Lindquist coordinates and transforming from our Bφ to BZ’s BT , although BZ’s expression for f(r)contains some unclosed parentheses.

BZ Derivation Summary

In Kerr-Schild coordinates, then, the magnetic field components are

Br =C

r2+ a2 C

2r4

(−2 cos θ + r2(1 + 3 cos 2θ)f(r)), (4.47)

Bθ = −a2 C

r2cos θ sin θf ′, (4.48)

125

both accurate through second order in a, and

Bφ = −aC

8r2(1 +

4r), (4.49)

accurate through first order in a. In Boyer-Lindquist coordinates,

Br[BL] = Br[KS], (4.50)

Bθ[BL] = Bθ[KS], (4.51)

Bφ[BL] = Bφ[KS]−Br[KS](a− 2rω)

∆, (4.52)

and BZ’s toroidal fieldBT = ∆ sin2 θBφ[BL] (4.53)

(which is different from BZ’s Bφ).There has been some concern about causality in the application of the force-free approximation

(see, e.g., Punsly 2003, see also Komissarov 2002a, 2004a). The MHD equations are hyperbolicand causal (as are the equations of force-free electrodynamics). Below we show that a numericalevolution of the MHD equations agrees well with the BZ solution in those regions where b2/ρ0 À 1.This is either a remarkable coincidence or else the BZ solution is an accurate representation of thestrong-field limit of ideal MHD.

For comparison with computational models, the most relevant aspects of the BZ theory are that:(1) the field is force-free; (2) the field rotation frequency ω = a/8+O(a3) in the monopole geometrycase and ω = a/8 + O(a3) at the poles (θ = 0, π/2) in the paraboloidal field case considered byBZ;2 (3) if the field geometry is nearly monopolar and a is small enough that the expansion tolowest order in a is accurate, then Br(θ) is given by equation (4.47); and (4) if the field geometryis monopolar and a is small, then the energy flux density FE ∝ sin2 θ on the horizon. We comparethis analytic BZ model to our numerical models in Section 4.4.2.

4.3.4 Equatorial MHD Inflow

Gammie (1999) considered a stationary, axisymmetric MHD inflow in the “plunging region”, be-tween the innermost stable circular orbit (ISCO) and the event horizon. The flow was assumed tobe cold (zero pressure), nearly equatorial, and to proceed along lines of constant latitude θ. Thelatter assumption ignores the requirement of cross-field force-balance. This model is analogous tothe Weber and Davis (1967) model for the solar wind, only turned inside out so that the wind flowsfrom the disk into the black hole. The model builds on earlier work by Takahashi et al. (1990),Phinney (1983), and Camenzind (1986). The analytic model derived here will be used to compareto numerical models in Section 4.4.3.

2According to the numerical results of Komissarov (2001) and the argument of MacDonald and Thorne (1982), ωadjusts to ≈ ΩH/2 hole even at large a.

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The MHD inflow model is stationary (∂t → 0), axisymmetric (∂φ → 0) and nearly equatorial(θ ≈ π/2) so ∂θ → 0 by symmetry. In addition flow proceeds along lines of constant θ. As aresult the model is one dimensional with a single independent variable r. The nontrivial dependentvariables are the radial and azimuthal four-velocity ur and uφ, the radial and azimuthal magneticfield Br and Bφ, and the rest-mass density ρ0.

With these assumptions the equations of general relativistic MHD can be integrated completely.The constancy of energy flux

−√−gT rt = const., (4.54)

and angular momentum flux √−gT rφ = const., (4.55)

follow from Tµν;µ = 0. The source-free Maxwell equations imply

√−gBr = const., (4.56)

which expresses the constraint ∇ ·B = 0, and the relativistic “isorotation law”,

√−g∗F

rφ=√−g(urbφ − uφbr) = const. (4.57)

where bµ is the magnetic field four-vector (defined above). Finally, conservation of particle numberimplies √−gρ0u

r = const. (4.58)

These five constants yield five constraints on the five nontrivial fundamental variables ur, uφ, Br,Bφ, and ρ0. Given the constants, and using the constitutive relations that relate the constants andfundamental variables, one can solve the resulting set of nonlinear equations for the fundamentalvariables at each radius.

The next step is to determine the constants. The radial magnetic flux and the rest-massflux are determined by conditions in the disk and can be left as free parameters. The remainingthree degrees of freedom are fixed by imposing boundary conditions. Gammie (1999) imposed thefollowing conditions: (1) the flow is regular at the fast point (the flow is automatically regular atthe Alfven point– see Phinney (1983) for a discussion– and the slow point is absent because theflow is cold) ; and (2,3) the four-velocity components ur and uφ match onto a cold disk at theISCO.

Energy can be extracted from the black hole if the Alfven point lies inside the ergosphere(Takahashi et al., 1990). Gammie (1999) calculated E and L as a function of a and Br and showedthat for even modest magnetic field strength these were modified from the values anticipated inclassical thin disk theory. The implications of these modified fluxes for the structure– particularlythe surface brightness– of a thin disk were explored by Agol and Krolik (2000).

For comparison with numerical models, the key predictions of the inflow model are: (1) the

127

constancy of the conserved quantities with radius; (2) matching of the flow velocity to circularorbits at the ISCO; (3) modification of the angular momentum and energy fluxes from their thindisk values; and (4) the run of all the fluid variables with radius in the plunging region.

4.4 Numerical Experiments

All our experiments evolve a weakly magnetized torus around a Kerr black hole in axisymmetry.The focus of our numerical investigation is to study a high resolution model (4.4.1), compare withthe BZ model (4.4.2), and compare to the Gammie inflow model (4.4.3). In § 4.5 we investigate howvarious parameters affect the results. Any dimensional quantity can be recovered from a numericalquantity since we set GM = c = ρ? = 1, where ρ? is some rest mass density set to unity in asimulation.

The initial conditions consist of an equilibrium torus (Fishbone and Moncrief 1976 ; Abramowiczet al. 1978) which is a “donut” of plasma with a black hole at the center. The donut is supportedagainst gravity by centrifugal and pressure forces, and is embedded in a vacuum. We considera particular instance of the Fishbone and Moncrief (1976) solutions, which are defined by thecondition utuφ = const. We normalize the peak density ρ0,max = ρ? to 1 and fix the inner edge ofthe torus at redge = 6. We also set γ = 4/3.3 Absent a magnetic field, the initial torus is a stableequilibrium.4

Into the initial torus we introduce a purely poloidal magnetic field. The field can be describedusing a vector potential with a single nonzero component Aφ ∝ MAX(ρ0/ρ0,max − 0.2, 0) The fieldis therefore restricted to regions with ρ0/ρ0,max > 0.2. The field is normalized so that the minimumratio of gas to magnetic pressure is 100. The equilibrium is therefore only weakly perturbed by themagnetic field. It is, however, no longer stable (Balbus and Hawley, 1991; Gammie, 2004).

Our numerical scheme is HARM (Gammie et al., 2003), a conservative, shock-capturing schemefor evolving the equations of general relativistic MHD. HARM uses constrained transport to main-tain a divergence-free magnetic field (Evans and Hawley, 1988; Toth, 2000). The inversion of con-served quantities to primitive variables is performed by solving a single non-linear equation (DelZanna and Bucciantini, 2002) or by a slower but more robust multi-dimensional Newton-Raphsonmethod. Unless otherwise stated we use modified Kerr-Schild (MKS) coordinates with h = 0.3.The computational domain is axisymmetric, with a grid that typically extends from rin = 0.98r+

to rout = 40, and from θ = 0 to θ = π/2.HARM is unable to evolve a vacuum, so we are forced to introduce “floors” on the density and

internal energy. When the density or internal energy drop below these values they are immedi-ately reset. This sacrifices exact conservation of energy, particle number, and angular momentum,although it is reasonable to assume that when the floors are small enough the true solution isrecovered. The floors are position dependent, with ρ0,min = 10−4r−3/2 and εmin = 10−6r−5/2. We

3We have run a limited number of γ = 5/3 models and find results essentially identical to those discussed below.4In axisymmetry. The torus is unstable to global nonaxisymmetric modes (Papaloizou and Pringle, 1983).

128

discuss the effect of varying the floor in Section 4.5.3.At the outer boundary we use an “outflow” boundary condition. This means we project all

primitive variables into the ghost zones while forbidding inflow. The inner boundary condition isidentical except that, because the boundary is inside the event horizon, we never need to worryabout backflow into the computational domain. At the poles we use a reflection boundary conditionwhere we impose appropriate symmetries for each variable across the axis.

4.4.1 Fiducial Model

First consider the evolution of a high resolution fiducial model with a = 0.938. This is close to thespin equilibrium value (where d(a/M)/dt = 0) found by Gammie et al. (2004) for a series of similarFishbone-Moncrief tori.

The fiducial model has utuφ = 4.281, the pressure maximum is located at rmax = 12, the inneredge at (r, θ) = (6, π/2), and the outer edge at (r, θ) = (42, π/2). The orbital period at the pressuremaximum 2π(a + r

3/2max) ' 267, as measured by an observer at infinity.

The numerical resolution of the fiducial model is 4562. The zones are equally spaced in modifiedKerr-Schild coordinates x1 and x2, with coordinate parameters h = 0.3. Small perturbations areintroduced in the velocity field, and the model is run for ∆t = 2000, or about 7.6 orbital periodsat the pressure maximum.

The initial state is Balbus-Hawley unstable. The inner edge of the disk quickly makes a tran-sition to turbulence. Transport of angular momentum by the magnetic field causes material toplunge from the inner edge of the disk into the black hole. The turbulent region gradually expandsoutward to involve the entire disk. The disk relaxes toward a “Keplerian” velocity profile, meaningthat the orbital frequency along the equator is close to the circular orbit frequency. The disk en-ters a long, quasi-steady phase in which the accretion rates of rest-mass, angular momentum, andenergy onto the black hole fluctuate around a well-defined mean.

Figure 4.1 shows the initial and final density states projected on the (R, z = r sin θ, r cos θ)-plane. Color represents log(ρ0). The initial density maximum is 1 and the minimum is ≈ 4× 10−7.The final state contains shocks driven by the interaction with the magnetic field, outflows near thesurface of the disk, and an evacuated “funnel” region near the poles.

The left panel in Figure 4.2 indicates the relative densities of internal, magnetic, and rest-mass energy. The magenta and cyan contours show the ratio of the average pressure to averagemagnetic pressure, β ≡ 2p/b2. The overbar indicates an average taken over 1000 < t < 2000 andover both hemispheres. The cyan contour indicates β = 3 and encircles most of the high density,approximately Keplerian disk. The magenta contour indicates β = 1. The red contour indicateswhere b2/ρ0 = 1. Between the pole and this contour the magnetic energy density exceeds theinternal and rest-mass energy density. The black contour surrounds a region, extending to largeradius, where −ut > 1 and the flow is directed outward (at large radius −ut asymptotes to theLorentz factor). That is, the particle energy-at-infinity is larger than the rest-mass density: so

129

Figure 4.1 Initial (left) and final (right) distribution of log ρ0 in the fiducial model on the r sin θ −r cos θ plane. At t = 0 black corresponds to ρ0 ≈ 4× 10−7 and dark red corresponds to ρ0 = 1. Fort = 2000, black corresponds to ρ0 ≈ 4 × 10−7 and dark red corresponds to ρ0 = 0.57. The blackhalf circle at the left edge is the black hole.

130

FUNNEL

CORONA

PLUNGING REGION

DISK

BLACK HOLE

WIN

D

Figure 4.2 (a) The distribution of β, b2/ρ0, and ut in the fiducial run, based on time and hemi-spherically averaged data. Starting from the axis and moving toward the equator: (1) ut = −1contour shown as a solid black line; (2) b2/ρ0 = 1 contour shown as a red line; (3) β = 1 contourshown as a magenta line that nearly matches part of the ut = −1 contour line; and (4) β = 3contour is shown as cyan line. (b) Motivated by the left panel, the right panel indicates the loca-tion of the five main subregions of the black hole magnetosphere. They are (1) the disk: a matterdominated region where b2/ρ0 ¿ 1; (2) the funnel: a magnetically dominated region around thepoles where b2/ρ0 À 1 where the magnetic field is collimated and twists around and up the axisinto an outflow; (3) the corona: a region in the relatively low density upper layers of the disk withweak time-averaged poloidal field; (4) the plunging region; and (5) the wind, which straddles thecorona-funnel boundary. See Section 4.4.1 for a discussion.

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the fluid is in a sense, unbound. We use the value of ut to estimate the radial component of the3-velocity at infinity (vr), which is independent of the coordinate system.

The right panel in Figure 4.2 defines some useful terminology inspired by the left panel, followingDe Villiers and Hawley (2003a) and Hirose et al. (2003). Moving from the axis to the equator, the“funnel” is the nearly evacuated, strongly magnetized region (b2 À ρ0 + ε + p), that develops overthe poles. The “wind” consists of a cone of material near the edge of the funnel that is flowingoutward with an asymptotic radial velocity of vr ∼ 0.75c. Near the outer edge of our computationaldomain the wind becomes marginally superfast. The “corona” lies between the funnel and the diskand has b2/2 ∼ p except in strongly magnetized filaments. In the “disk” b2/2 < p and the plasmafollows nearly Keplerian orbits. Finally, the “plunging” region, which lies between the disk andthe event horizon, contains accreting material moving on magnetic field and pressure modifiedgeodesics.

Figure 4.3 shows the evolution of the poloidal magnetic field. The panels show contours ofconstant Aφ, so the density of contours is directly related to the poloidal field strength, and thecontours follow magnetic field lines. The contours are projected on the (R = r sin θ, z = r cos θ)-plane, and show the initial and final state. The initial field is confined to a region much smallerthan the torus as a whole because field is introduced only in those portions of the disk that haveρ0 > 0.2ρ0,max. Notice that by the end of the simulation the field has mixed in to the funnel regionand has a regular geometry there. In the disk and at the surface of the disk the field is curved onthe scale of the disk scale height. The field strengths and geometries we see are consistent withHirose et al. (2003). This includes the absence of disk to disk field loops, and that the funnelfield collimates instead of connecting back into the disk (thus providing a means for the outflow toescape to large radii).

Figure 4.4 shows contours of time and hemisphere averaged Aφ. The time averaged field is evenmore regular in the funnel than the snapshot in Figure 4.3. Time averaging tends to sharply reducethe field strength in the corona and disk because the field fluctuates in magnitude and directionthere.

Figure 4.5 shows the accretion rate of rest-mass (M0), energy per unit rest-mass (E/M0), andangular momentum per unit rest-mass (E/M0) evaluated inside the horizon at the inner boundaryof the computational domain. For 500 < t < 2000 the time average values are M0 ≈ 0.35,E/M0 ≈ 0.87, and L/M0 ≈ 1.46. These average values are shown as dashed lines. The dottedlines show the classical thin disk values E/M0 ≈ 0.82 and L/M0 ≈ 1.95 obtained by setting theseratios equal to respectively the specific energy and angular momentum of particles on the ISCO.The energy per baryon is therefore slightly above the thin disk value, but the angular momentumper baryon is significantly below the thin disk value.

It may be useful to recast the energy flux in terms of a nominal “radiative efficiency”5 ε =1− E/M0. For the fiducial run ε = 13%, which is slightly lower than the thin disk with ε = 18%.This is likely due to the high temperature of the flow. On the horizon about 20% of the energy flux

5Our evolution is nonradiative, so the true radiative efficiency is zero.

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Figure 4.3 Initial (left) and final (right) distribution of Aφ. Level surfaces coincide with magneticfield lines and field line density corresponds to poloidal field strength. In the initial state field linesfollow density contours if ρ0 > 0.2ρ0,max.

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Figure 4.4 Contour plot of the time and hemispheric average of Aφ. Level surfaces coincide withmagnetic field lines and field line density corresponds to poloidal field strength.

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Figure 4.5 Evolution of rest-mass, energy, and angular momentum accretion rate for our fiducialrun of a weakly magnetized tori around a black hole with spin a = 0.938. For 500 < t < 2000 thetime average of these values is M0 ' 0.35, E/M0 ' 0.87, and L/M0 ' 1.46 as shown by the dashedlines. The dotted lines show the classical thin disk values (E/M0 ' 0.82 and L/M0 ' 1.95). SeeSection 4.4.1 for a discussion.

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would vanish if we set the internal energy to zero. The corresponding zero-temperature efficiency(1 + ut) would be 32%.

The chief object of our study is to measure the electromagnetic luminosity of the hole. The timeand hemisphere averaged electromagnetic energy flux on the horizon is shown in Figure 4.6. In thefunnel region the energy flux density is outward, as predicted by the force-free model of BZ. Wecompute other interesting quantities by integrating over the horizon and taking a time average (fortechnical reasons we are using a less resolved time sampling here than used to make Figure 4.5, butthe time averages have fractional differences of only 10%). We find E(EM)/E(MA) = −2.3%, wherethe energies per baryon are E(EM)/M0 = −0.018 and E(MA)/M0 = 0.77. It is useful to define theratio of electromagnetic luminosity to nominal accretion luminosity L = E(EM)/(−εM0). We findL = 16%. Thus while the electromagnetic energy flux is outward, it is a small fraction of the inwardmaterial energy flux and the BZ luminosity is small compared to the nominal accretion luminosity.

The BZ luminosity from the polar regions of the horizon provides about 1/2 of the radialelectromagnetic energy flux (E(EM)) in the wind, while another 1/2 comes from the surface of thecorona but is mostly dissipated into kinetic energy of the wind. Therefore, the BZ-effect is animportant source of Poynting flux in the wind at large distances since it flows freely along highlyordered poloidal field lines connected to unbound plasma.

A control calculation at a = 0 and a resolution of 2562 gives E(EM)/E(MA) = 0.33% andL = −6.5%, where the energies per baryon are E(EM)/M0 = 0.0032 and E(MA)/M0 = 0.95.E(EM)/M0 > 0 and L < 0 are as expected, since the outward energy flux must vanish for anonrotating hole (i.e. the BZ effect is not operating). For our sequence of models the BZ effectdoes not operate for a . 0.5 (see Section 4.5.1). The matter energy flux ratio may be compared tothe thin disk value of E(MA)/M0 = 0.94.

4.4.2 Comparison with BZ

The BZ solution was reviewed in Section 4.3.3. BZ were able to find steady force-free field solutionsin the limit that a ¿ 1. Since the fiducial run has a = 0.938, we ran a special a = 0.5 model forcomparison with BZ.

The BZ solution was found in the force-free limit, so the first question one might ask is whetherany region of the model is force-free. To measure this we recall that in the force-free limit

Tµν;ν = FµνJν = 0. (4.59)

So when the field is force-free the parameter

ζ =∣∣∣∣FµνJνFµκJκ

JµJµFκλF κλ

∣∣∣∣ (4.60)

is small compared to 1.Figure 4.7 shows the time and hemispherical averaged ζ(r, θ) from t = 1000 to t = 2000 for

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Figure 4.6 Electromagnetic energy flux density F(EM)E (θ) on the horizon for the fiducial run, based

on time and hemisphere averaged data. The mean electromagnetic energy flux is directed outward.See Section 4.4.1 for a discussion.

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Figure 4.7 The run of the force-free parameter ζ for the a = 0.5 run; when ζ ¿ 1 the field isapproximately force-free. The parameter has been time and hemisphere averaged. The contoursshow (beginning from the pole and moving toward the equator) ζ = 10−3, 10−2, 10−1. The smallclosed contours at large radius and close to the axis have ζ = 10−2. The small closed contours fromthe equator to θ ∼ π/4 have ζ = 10−1. See Section 4.4.2 for a discussion.

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Figure 4.8 Left panel: Magnetic field angular frequency on the horizon relative to black holerotation ω(θ)/ΩH . The solid line indicates time and hemisphere averaged data from our a = 0.5MHD integration. The middle dotted line is the prediction of the BZ model (ω/ΩH = 1/2). Thedashed line (top) is the value predicted by the inflow model. Right panel: the run of field rotationfrequency ω with radius along a single field line that intersects the horizon at θ = 0.2. ω is constantto within 3%, as expected for a steady flow. See sections 4.4.2 and 4.4.3 for a discussion.

the a = 0.5 model. The contours show (beginning from the pole and moving toward the equator)ζ = 10−3, 10−2, 10−1. The entire funnel region has ζ < 10−2 and is therefore effectively force-free.This is true in both a time-averaged and instantaneous sense in the funnel for all our runs. Thisopens the possibility that the BZ solution describes the funnel.

A key feature of the BZ model is that the field rotation frequency ω ≈ ΩH/2 for a ¿ 1 if thefield has a monopole geometry. Figure 4.8a shows the ratio ω/ΩH on the horizon. Within theforce-free region, which runs from 0 < θ < 0.4 on the horizon, the average ω/ΩH ≈ 0.45. The smalldifference from the BZ could be due to higher order terms in the expansion in a, but Komissarov(2001) has integrated the equations of force-free electrodynamics for a monopolar field geometryand at a = 0.5 finds that ω rises from ≈ 0.495ΩH at the pole to ≈ 0.51ΩH at the equator, so thisseems unlikely. The difference is more likely due to small deviations from force-free behavior (massloading of field lines by the numerical “floor” on the density).

In an axisymmetric steady state both the force-free equations and the MHD equations predictthat the rotation frequency ω (and other quantities) are constant along field lines. Figure 4.8bshows the variation of ω with radius along a field line that intersects the horizon at θ = 0.33. Asexpected ω ≈ const., with a variation of less than 3% from maximum to minimum.

BZ’s spun-up monopole model makes definite predictions about the variation of Br and FE on

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Figure 4.9 (a) Square of radial field ((Br(θ))2) on the horizon in the a = 0.5 MHD integration, fromtime and hemisphere averaged data. Solid line is the field for our numerical model. The dottedline shows the Blandford and Znajek (1977) perturbed monopole solution with the field strengthnormalized to the numerical solution at the pole. The dashed line is the inflow solution. (b)Electromagnetic energy flux F

(EM)E (θ) on the horizon in the a = 0.5 MHD integration, from time

and hemisphere averaged data. The solid line shows the numerical model, the dotted line showsBZ’s spun-up monopole solution, and the dashed line shows the inflow solution. See sections 4.4.2and 4.4.3 for a discussion.

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the horizon. Figure 4.9a shows the variation in time and hemisphere averaged (Br)2 and comparesto BZ’s monopole field calculation. The single adjustable parameter of the model normalizes thefield strength. We have set this normalization by requiring that (Br)2 match at the pole. Evidentlythe variation matches the BZ prediction closely even well outside the force-free region at θ ≈ 1.1.Figure 4.9b shows the variation in radial energy flux on the horizon as predicted by the BZ modelusing the pole-normalized field. Here the match is quite close out to θ ≈ π/4. It is slightly surprisingthat the BZ solution does so well even in regions that are not force-free. This is likely a result oftrans-field force balance and geometry controlling the distribution of field on the horizon and hencethe radial energy flux.

To summarize: in our low spin numerical experiment the funnel is approximately force-freewithin the funnel. It is approximately in a steady state and hence ω is approximately constantalong field lines. Furthermore, ω, Br, and the radial electromagnetic energy flux are all in goodagreement with the spun-up monopole force-free model on the horizon. We have not comparedthe entire funnel region with the monopole model because the field is collimated there and notwell-described by the monopole solution.

4.4.3 Comparison to Inflow Solution

The inflow solution of Gammie (1999) considers a near-equatorial stationary MHD inflow in theplunging region, reviewed in Section 4.3.4. Here we compare the inflow models with the fiducialmodel. Unlike the funnel, the plunging region is rapidly fluctuating, so we expect the inflow modelto match only the time-averaged data from the simulation.

The inflow model has two free parameters: the field strength and the accretion rate. The fieldstrength we match by finding the parameter that gives the best fit to the mean magnetic energydensity between the ISCO and the event horizon. The rest-mass flux is chosen to agree with thetime-averaged data from the simulation. The ratio of the field strength to the square root of theaccretion rate is a dimensionless parameter that controls the solution; in the units of Gammie(1999), where 2πρ0u

r√−g = −1, we use Fθφ = 1.09 for the comparison model.Figure 4.10 shows a comparison of ur, L/M0, comoving energy densities (ρ0, b2/2, and ε), and

energy fluxes (E/M0) in the inflow solution. The comparison data from the fiducial run has beenaveraged over |θ − π/2| < 0.3 and 500 < t < 2000. Each panel in the figure contains a vertical lineat the ISCO.

The upper left panel compares the radial component of the four-velocity (in KS and BL co-ordinates) in the inflow and numerical solutions. The substantial differences are due to the finitetemperature of the flow; the inflow solution is cold by assumption. Radial pressure gradients inthe numerical model (which are absent in the inflow solution) begin to accelerate material inwardoutside the ISCO, and the flow becomes supersonic near the ISCO.

The upper right and lower right panels show components of the energy and angular momentumflux from the simulation and inflow solutions. The dashed horizontal line in each case shows the

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Figure 4.10 A comparison of the time-averaged fiducial model near the equator (within θ = π/2±0.3) with the inflow solution of Gammie (1999). In the right two panels the black dotted line isthe thin disk value. In all cases the red vertical line is the location of the ISCO. The black line forthe upper left panel is the numerical result. For the other three panels, the particle term is shownin cyan, the internal energy term is shown in magenta, and the electromagnetic term is shown ingreen. The blue line in each plot represents the inflow model result. Notice that the run of densitywith radius shows no feature at the ISCO. See the Section 4.4.3 for discussion.

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values expected for a thin disk; the inflow solution is constrained to match the thin disk at theISCO. The cyan lines show uφ (upper panel) and ut (lower panel) from the simulation, whilethe blue lines show the prediction from the inflow model. The energy flux matches rather well(although notice that this is only a small fraction of the energy flux), while the angular momentumis overestimated; in the simulation the plasma has sub-Keplerian angular momentum by the timeit reaches the ISCO.

The electromagnetic components of the per unit mass flux of angular momentum flux (b2uφ/ρ0−brbφ/(ρ0u

r)) and energy flux (b2ut/ρ0 − brbt/(ρ0ur)) are also shown in the upper and lower right

panels of Figure 4.10 (green line ≡ simulation, blue line ≡ inflow solution). The inflow solutionmatches well, although it tends to overestimate the magnitude of the outward directed energy flux.

The magenta lines in the upper and lower right panels show the internal energy componentof the normalized angular momentum flux ((ε + p)uφ/ρ0) and energy flux (−(ε + p)ut/ρ0). Thiscomponent of the fluxes is zero by assumption in the inflow solution, and it is evidently an importantcomponent of the fluxes in our thick disk simulations. This leads to large corrections to the angularmomentum and energy fluxes; the total normalized angular momentum flux is significantly smallerthan the thin disk prediction, while the energy flux is, seemingly by conspiracy, very close to thethin disk.

The lower left panel shows the rest-mass density from the inflow solution (upper blue line)and from the simulation (cyan line). The mass flux in the inflow solution is normalized so that itmatches the simulation mass flux. Since mass flux is approximately constant with radius, the runof density is directly related to the run of ur. What is remarkable here is that there is no featurein the simulation ρ0 near the ISCO. In fact it is nearly constant from well outside the ISCO into the event horizon. The surface density varies smoothly as well. This confirms the point madeby Krolik and Hawley (2002) in their pseudo-Newtonian solution: there is no sharp feature at theISCO. This has implications for iron line profiles, as discussed by Reynolds and Begelman (1997).

The lower left panel also shows the run of internal energy density in the simulation (it is zero byassumption in the inflow solution). Again, there is no sharp feature at the ISCO, just a gentle riseinward toward the event horizon. Because the density is nearly constant with radius this impliesthat entropy is increasing inward. Therefore there is some dissipation of kinetic or magnetic energyinto internal energy in the inflow region.

The lower left panel of Figure 4.10 shows the run of magnetic energy density b2/2 in the inflowsolution (lower blue line) and simulation (green line). The normalization of the inflow magneticenergy is a parameter, but its radial slope is not.

Finally, the inflow solution predicts that ω = ΩISCO. Figure 4.8a shows the run of ω/ΩH onthe horizon for the a = 0.5 model. The dashed line shows the ISCO value of ω/ΩH . At the equatorthe time-averaged numerical value lies within about 10% of the ISCO value: the numerical averageω/ΩH = 0.685, while the ΩISCO/ΩH = 0.8136 at the ISCO. In the a = 0.938 run the numericalaverage ω/ΩH = 0.681, while ΩISCO/ΩH = 0.745 at the ISCO.

To sum up, the inflow model does a surprisingly good job of matching some aspects of the

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time-averaged simulation. It does not match the profile or boundary condition at the ISCO for theradial velocity or the total angular momentum and energy fluxes, because the simulation flow ishot, while the inflow solution has zero temperature by assumption.

What is most surprising is that the energy per baryon accreted in the numerical model matchesthe thin disk prediction. The inflow model predicts that the energy per baryon accreted shouldbe lower than the thin disk prediction, enhancing the nominal accretion efficiency (Gammie, 1999;Krolik, 1999a; Agol and Krolik, 2000). The difference is apparently due to the finite temperatureof the numerical model and the consequent change in boundary conditions at the ISCO. Theseboundary conditions evidently adjust themselves to maintain the energy flux at the thin disk value.The angular momentum flux is affected by the field, however, with the specific angular momentumof the accreted material in the fiducial run about 25% lower than the thin disk.

4.5 Parameter Study

Our numerical model has a number of physical and numerical parameters. Here we check thesensitivity of the model to: (1) black hole spin parameter a; (2) initial magnetic field geometryand initial magnetic field strength; and (3) numerical parameters such as (a) location of the in-ner boundary (rin); (b) outer radial (rout) boundary; (c) radial and θ resolution, including thecoordinate parameter h; and (d) parameters describing the density and internal energy floors.

4.5.1 Black Hole Spin

The fiducial run has a rather low outgoing electromagnetic energy flux compared to the ingoingmatter energy flux. It is possible that this varies sharply with black hole spin and that more rapidlyrotating holes exhibit much larger electromagnetic luminosity. We have performed a survey over a,keeping all parameters identical to those in the fiducial run, except that the resolution is loweredto 2562 and the location of the pressure maximum is adjusted to keep H/R ≈ const.

The results are shown in Figure 4.11 and described in Table 4.2. The figure shows the measuredratio of electromagnetic to rest-mass energy flux; the dashed line shows a fit

E(EM)

E(MA)≈ −0.068(2− r+)2. (4.61)

This fit applies only to this particular sequence of models; models with different initial field geome-tries give different results, as we shall see below. For all a > 0 we find E(EM) > 0 in the funnel.For a < 0.5 this outward funnel flux is balanced by an inward electromagnetic energy flux near theequator. For our most extreme run with a = 0.969 the outward electromagnetic flux is still dom-inated by the inward particle flux. The ratio of electromagnetic luminosity to nominal accretionluminosity is L = 27%, so the nominal accretion luminosity dominates over the BZ luminosity.

The accretion rate of angular momentum is also a strong function of spin. As discussed inGammie et al. (2004), accretion flows around rapidly spinning holes have da/dt < 0. Our fiducial

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Figure 4.11 The ratio of electromagnetic to matter energy flux on the horizon. The solid lineindicates numerical data while the dotted line indicates a best fit of E(EM)/E(MA) = −0.068(2 −r+)2. See Section 4.5.1 for a discussion.

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Table 4.2. Black Hole Spin Study

a 104 × E(EM)/E(MA) E/M0 L/M0 a/M0 M0 L

-0.938 105 0.958 3.806 -5.583 -0.908 -0.240.000 34.4 0.950 3.068 -3.049 -0.870 -0.0650.050 31.2 0.952 3.025 -2.921 -0.709 -0.0620.100 35.8 0.948 2.896 -2.713 -0.767 -0.0660.150 29.7 0.949 2.881 -2.597 -0.796 -0.0550.200 26.9 0.948 2.817 -2.439 -0.776 -0.0500.250 9.17 0.946 2.749 -2.302 -0.747 -0.0160.300 3.30 0.937 2.759 -2.217 -0.571 -0.00490.350 1.32 0.933 2.605 -1.975 -0.620 -0.00180.400 1.15 0.937 2.763 -1.986 -0.241 -0.00170.500 -9.85 0.933 2.583 -1.665 -0.252 0.0140.600 -28.5 0.929 2.489 -1.347 -0.318 0.0370.750 -81.8 0.908 2.150 -0.808 -0.276 0.0830.875 -291 0.852 1.440 -0.152 -0.170 0.170.895 -254 0.891 1.723 -0.204 -0.215 0.200.900 -315 0.882 1.674 -0.118 -0.193 0.240.938 -318 0.856 1.396 0.067 -0.203 0.230.969 -410 0.869 1.374 0.217 -0.172 0.27

Note. — All models same as fiducial except at a resolution of 2562 and rmax

is used to keep H/R ∼ constant. These values can be compared to Tables 3and 4. The efficiency is 1−E/M0. A positive a/M0 corresponds to a spindownof the black hole since M0 < 0.

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model, in fact, is spinning down. Previous estimates suggested that spin equilibrium is reached ata ∼ 0.998 (Thorne, 1974). Our models reach spin equilibrium at a ∼ 0.92.

The variation of field strength and geometry with black hole spin is also of interest. To measurevariation of field strength, we probe the flow near four locations: 1) in the funnel near the horizon(“funnel/horizon”); 2) in the plunging region near the horizon (“plunging/horizon”); 3) at theISCO; and 4) at the pressure maximum. We then take a time and spatial average of the comovingelectromagnetic energy density b2/2 over a small region near each of these locations. The ratio ofb2(funnel/horizon) to b2(plunging/horizon) changes from 0.43 at a = 0 to 0.74 at a = 0.938. Theratio of b2(funnel/horizon) to b2(ISCO) varies from 2.53 at a = 0 to 2.14 at a = 0.938. The ratiob2(funnel/horizon) to pressure maximum varies from 4.8 at a = 0 to 15.7 at a = 0.938. In summary,the field strength increases from the ISCO to the horizon by a factor of ∼ 3 at a = 0 and by afactor of ∼ 6 at a = 0.938, and on the horizon is slightly larger at the equator than at the poles bya factor of ∼ 2. Only the ratio of b2(pressure maximum) to other locations in the plunging regionor at the horizon depends strongly on black hole spin.

Our observed increase in horizon field strength with black hole spin agrees with results reportedby De Villiers et al. (2003a). (Livio et al., 1999) suggest that 1) there is no reason the field strengthnear the black hole horizon should be stronger than in the central regions of the disk, which theysuggest implies the electromagnetic output from the disk (corona) dominates the BZ luminosity ;and 2) the spin of the hole is probably irrelevant to the electromagnetic output in the wind. We havefound that the field strength from the pressure maximum of the disk to the horizon increases by afactor ∼ 5− 16 depending on the black hole spin. Also, for a = 0.5 we find that the BZ luminositycontributes about 1/5 to the total E(EM) in the wind, and for a = 0.938 about 1/2. The rest ofE(EM) in the wind is provided by the disk corona, which is mostly dissipated into kinetic energyof the wind. This suggests that black hole spin is important in determining the electromagneticenergy flux in the wind, and that the BZ luminosity in the funnel can be a significant contributionto Poynting flux at large distances.

We see no sign of the expulsion of flux from the horizon reported by Bicak and Janis (1985), whofind that the flux through one hemisphere of the horizon, due to external sources and calculated inaxisymmetry using vacuum electrodynamics, vanishes when the spin of the hole is maximal. It ispossible that we have not gone close enough to a = 1 to observe this effect.

To investigate the variation of field geometry in the funnel region with a we trace field linesfrom θin on the horizon to θout on the outer boundary and define a collimation factor θin/θout. Thecollimation factor is similar for all field lines in the funnel region. It reaches a minimum of ≈ 5/2for the fiducial run, and rises to nearly 2 for a = 0 and again to nearly 2 for a ∼ 1. The collimationfactor depends on the location of the outer boundary; for models with rout = 400 the collimationfactor is 10 and the field lines are nearly cylindrical at the outer boundary.

We have also studied the variation of the field rotation frequency ω in the funnel. ω/ΩH variesweakly with a, from 0.53 at a = 0.25 to 0.45 at a = 0.938, consistent with the hypothesis advancedby Thorne et al. (1986) that ω/ΩH ≈ 1/2.

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Table 4.3. Field Strength and Geometry Study

Field Geometry Aφ β E(EM)/E(MA) E/M0 L/M0 a/M0 M0 L

A0φ 100 −0.0312 0.856 1.40 0.0674 −0.203 0.21

A0φ 500 −0.0115 0.879 1.94 −0.293 −0.0474 0.085

A0φ sin (log (r/h)) 100 −0.0355 0.892 1.24 0.278 −0.541 0.42

A0φ| sin(2θ)| 100 −0.0112 0.888 1.91 −0.299 −0.0746 0.083

r sin θ 100 −0.147 0.773 −0.997 1.807 −1.769 0.79r sin θ 400 −0.157 0.813 0.0617 1.184 −0.715 0.67

Note. — A0φ is the fiducial model field geometry and β = 100 is the fiducial ratio of

gas to magnetic pressure. The r sin θ field geometry is a uniform vertical field model with βset by disk values at the equator. All other model and numerical parameters are as in thefiducial model except that the resolution is 2562. The efficiency is 1− E/M0. A positive a/M0

corresponds to a spindown of the black hole because M0 < 0.

4.5.2 Field Geometry and Strength

The outcome of the simulation may also depend on the field geometry and strength in the initialconditions. This seems more likely for axisymmetric models such as ours where the evolution mayretain a stronger memory of the initial conditions than comparable three dimensional models.

We begin by investigating the dependence of outcome on initial field strength, parameterizedby β ≡ pgas,max/pmag,max (notice that the two maxima never occur at the same location in space,so this ratio varies over a wide range when evaluated at individual locations in the disk). Weconsider models with β = (100, 500) and find a weak dependence on β. For the β = 100 model(the fiducial model at a resolution of 2562) we find ω/ΩH ≈ 0.45, E(EM)/E(MA) ≈ −3.1%, andL = 21%. β = 500 leads to ω/ΩH ≈ 0.42, E(EM)/E(MA) = −1.2%, and L = 8.5%. Notice thata higher spatial resolution is required to fully resolve weak field models, although all runs in thiscomparison were done at 2562; the decrease in electromagnetic energy extracted at β = 500 maytherefore be due to resolution.

We also vary the field geometry from the single loop used in our fiducial model, which hasvector potential Aφ ∝ MAX(P/Pmax − 0.2, 0). We do this by multiplying the vector potential bysin(log(r/h)) or | sin(2θ)|. The former decompresses the field lines at the inner radial edge giving afield strength that is more uniform around the loop (for an extended disk this would yield a sequenceof field loops centered at the midplane with alternating sense of circulation). The latter yields twoloops, one centered above the equator and the other below, with the same sense of circulation.The sin(log(r/h)) modulation gives ω/ΩH ≈ 0.44, E(EM)/E(MA) ≈ −3.6%, and L = 42%. The| sin(2θ)| modulation gives ω/ΩH ≈ 0.40, E(EM)/E(MA) ≈ −1.1%, and L = 8.3%. Increasing the

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number of initial field loops therefore leads to a weak (factor of 2− 3) decrease in E(EM)/E(MA),while making the field strength more uniform around the loop increases L by a factor of 2 with anearly constant E(EM)/E(MA). Higher resolution studies may better resolve these simulations andshow weaker dependence on field geometry.

We have also considered a purely vertical field geometry: Aφ ∝ r sin θ. In a Newtonian contextthis would correspond to a uniform z field in cylindrical coordinates. The field is normalized sothat β = pgas,max/pmag,max = 100 and 400 in the equator of the torus. The outcome is differentfrom any of the other models.

The funnel field in the vertical field run is strong compared to the disk field. The accretion rateis larger, by a factor of 5, than the fiducial run. In the early stages there is a brief net outflow ofenergy from the black hole (although the total energy released from the hole is negligible comparedto the energy gained at later times). The β = 100 model has a high mean efficiency; E/M0 = 0.77,compared to 0.82 expected for a thin disk. There is also a net outflow of angular momentum fromthe black hole, with L/M0 = −1.00, compared to 1.95 expected for a thin disk. The wind has a peakasymptotic radial velocity vr = 0.94c, attained near the outer boundary, compared to vr = 0.75cfor the fiducial run. Finally, the model has ω/ΩH ≈ 0.41, E(EM)/E(MA) ≈ −15%, and L = 79%.The β = 400 vertical field model has very similar properties, which suggests that we are resolvingthe β = 100 model. Table 4.3 summarizes measurements from the varying field geometry models.

The models with net vertical field exhibit markedly different behavior from the fiducial model.It seems likely that some of this difference is due to the axisymmetric nature of the model; in 3Dmatter can accrete between the vertical field lines without having to push them into the hole. Thatis, in 3D, it would be easier for the hole to rid itself of the dipole moment that it acquires in thenet vertical field calculation. But we cannot say with any confidence what the outcome is until afull 3D experiment on a disk with nonnegligible magnetic dipole moment.

4.5.3 Numerical Parameters

We have run the fiducial model at resolutions of 642, 1282, 128 × 64, 2562, and 4562. There is aweak dependence on resolution in the sense that E(EM)/E(MA) is smaller at higher resolutions.Lower resolution models do not sustain turbulence for as long as high resolution models, so weaverage over 500 < t < 1000, when all models are turbulent. Table 4.4 gives a summary of resultsfrom the resolution study. In every case the nominal radiative efficiency is close to the thin diskvalue.

Resolution of the near-horizon region, where the energy density is large, is also a concern,because our accretion rates are measured there. We have checked dependence on radial numericalresolution of the near-horizon region by modifying the coordinate definition in equation (7) to readr = R0 + ex1 rather than r = ex1 . Increasing R0 from 0 to the horizon radius increases the numberof grid zones located near the horizon. We ran a model with R0 = 0.5 and found no significantdifference from a comparable model with R0 = 0. This suggests that we are adequately resolving

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Table 4.4. Resolution Study

Resolution E(EM)/E(MA) E/M0 L/M0 a/M0 M0 L

642 -0.0528 0.914 1.630 0.036 -0.159 0.55128× 64 -0.0438 0.841 1.420 0.121 -0.165 0.231282 -0.0447 0.887 1.518 0.087 -0.167 0.382562 -0.0316 0.874 1.274 0.198 -0.186 0.274562 -0.0261 0.865 1.381 0.216 -0.299 0.18

Note. — Numerator and denominators are separately time averagedfrom 500 < t < 1000 at the horizon. This interval is chosen so that allmodels are turbulent (in the lowest resolution model turbulence decaysshortly after t = 1000). The 4562 model is the fiducial model. The nom-inal radiative efficiency is 1 − E/M0. A positive a/M0 corresponds to aspindown of the black hole because M0 < 0.

the near-horizon region.We also varied rin and rout and found no measurable difference in E(EM), E(MA), (Br)2, and ω

on the horizon. We have moved rout from 40 to 400 and rin from 0.7r+ to 0.98r+ and find negligibledifferences in these quantities on the horizon. The solution is not sensitive to the location of theinner or outer boundary. Moving the inner boundary of the computational domain outside thehorizon (e.g. 1.05r+) leads to strong reflections from the boundary conditions and, ultimately,failure of the run. It is possible that better inner boundary conditions or higher resolutions couldovercome this difficulty, but it seems cleaner to simply leave the boundary inside the event horizonat r = 0.98r+, out of causal contact with the rest of the simulation.

The model with larger rout = 400 does exhibit some new features. The magnetic field lines inthe funnel region have a collimation factor of 10 by the time they reach the outer boundary. AtR = 40, however, both the rout = 400 model and the fiducial model have a collimation factor of5/2. By rout = 400 the field lines are nearly cylindrical. The peak of the radial component of theasymptotic 3-velocity in the wind is identical to the fiducial run with vr = 0.75c, indicating littleacceleration between R = 40 and R = 400.

The main numerical uncertainty in our experiments arise from the floor on the density andinternal energy. We varied the floor scaling from ρ0,min = 10−4r−3/2 and umin = 10−6r−5/2 toρ0,min = 10−4r−2.7 and umin = 10−6r−3.7 (we chose these scalings so that b2/ρ0 would be nearlyconstant with radius in the funnel). While this significantly affects b2/ρ0, it does not otherwiseaffect E(EM) and E(MA) or the mean values of Br and ω measured on the horizon.

We varied the floor normalization at r = 1 from the fiducial values (ρ0,min, εmin) = (10−4, 10−6)to (10−5, 10−7), and (10−6, 10−8). This causes almost no change in the flow near the horizon. In

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the funnel, however, we are at the limit of our ability to integrate the MHD equations (b2/ρ0 À 1).Our integration fails when we attempt to use a mass density floor ρ0,min(r = 1) . 10−5 whenrout À the outer edge of the initial torus. Lower floors lead to faster outflows vr À 0.99c in thefunnel region, which are more likely to be numerically unstable. These results hint that low densitymodels will produce fast outflows, but a confirmation awaits a more stable GRMHD algorithm.

The funnel region is difficult to integrate reliably, because when b2/ρ0 À 1 small fractional errorsin field evolution lead to large fractional errors in the evolution of other flow variables. This is aconsequence of our conservative scheme, in which all the dependent variables are coupled togetherby the interconversion of primitive and conserved variables. Evolution of the MHD equations innonconservative form (e.g. using an internal, rather than total, energy equation), as in De Villiersand Hawley (2003a), may be slightly more robust, although De Villiers and Hawley eventuallyexperience similar problems in the funnel. In any event, the close correspondence between thenumerical experiment and the BZ model raises confidence in the results and suggests that themagnetic field, if not the mass density and internal energy density, is being evolved reliably.

4.6 Discussion

We have used a general relativistic MHD code, HARM, to evolve a weakly magnetized thick diskaround a Kerr black hole. Our main result is that we find an outward electromagnetic energy flux onthe event horizon, as anticipated by Blandford and Znajek (1977). The funnel region near the polaraxis of the black hole is consistent with the Blandford-Znajek model. The outward electromagneticenergy flux is, however, overwhelmed by the inward flux of energy associated with the rest-massand internal energy of the accreting plasma. This result essentially confirms work by Ghosh andAbramowicz (1997) that suggested the BZ luminosity should be small or comparable to the nominalaccretion luminosity (L . 1).

One of our models discussed here, however, begins with a vertical field threading the torus,exhibits a brief episode of outward net energy flux. This appears to be a transient associated withthe initial conditions. The same model exhibits a steady net outflow of angular momentum from theblack hole. Of all our models, the vertical field model has the largest negative −E(EM)/E(MA) ≈15% (ratio of the electromagnetic energy flux to ingoing matter energy flux) and largest L =E(EM)/(−εM0) ≈ 80% (ratio of electromagnetic luminosity to nominal accretion luminosity). Thissuggests that the BZ effect could play a significant role if the disk has a net dipole moment andaccumulates magnetic flux that crosses the horizon. This possibility will be considered in futurework.

Consistent with the results found earlier by De Villiers et al. (2003a), we find that our modelscan be divided into four regions: (1) a “funnel” region with b2/ρ0 & 1 and β ¿ 1; (2) a coronawith 1 . β . 3; (3); an equatorial disk with β > 3; and (4) a plunging region between the diskand event horizon with β ∼ 1 and a nearly laminar inflow from the disk to the black hole. We alsofind no feature in the surface density at or near the ISCO (see Figure 4.10), which agrees with the

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results by Krolik and Hawley (2002); De Villiers et al. (2003a) and consistent with Reynolds andBegelman (1997). This is contrary to the sharp transition predicted by thin disk models and usedby XSPEC to fit X-ray spectra.

We have shown that the funnel region is nearly force-free, and is well-described by the stationaryforce-free magnetosphere model of Blandford and Znajek (1977), for which we have presented a self-contained derivation in Kerr-Schild coordinates. We find agreement between the BZ model and oursimulations in measurements of energy flux, magnetic field, efficiency of accretion, and spindownpower output. In all cases we find that in the force-free region the field rotation frequency is abouthalf the black hole spin frequency, ΩH ≡ a/(2r+). This spin frequency maximizes electromagneticenergy output from the hole. This result is consistent with expectations of MacDonald and Thorne(1982) and the force-free numerical results of Komissarov (2001).

We have also compared the time-average of the plunging region in our fiducial model with thestationary MHD inflow model of Gammie (1999), which assumes that the flow matches a cold disk atthe ISCO. The inflow model matches the simulated rest-mass flux and electromagnetic flux of energyand angular momentum surprisingly well, particularly considering the strongly variable nature ofthe simulated flow in the plunging region. The inflow model fails to match other aspects of theflow, such as the radial component of the four-velocity. This is mainly due to the finite temperatureof the simulated flow; the inflow solution assumes zero temperature. It is slightly surprising thatthe total angular momentum flux is close to the value predicted by the zero temperature inflowsolution, and 20% less than what is predicted by the thin disk, yet the total energy flux is almostexactly what is predicted by the thin disk. It is as yet unclear whether this is due to coincidenceor conspiracy.

For a set of models similar to the fiducial model, the ratio of electromagnetic to matter energyfluxes is sensitive to the black hole spin, reaching −7% for a ∼ 1. The evolution is sensitive to theinitial field geometry. Models with a net vertical field are more efficient, and more electromagneti-cally active than models with comparable field strength but zero net vertical field. Our models havea weak dependence on resolution in the sense that as resolution increases the relative importanceof electromagnetic energy fluxes on the horizon diminishes.

With an rout = 400 model we demonstrate that an outgoing electromagnetic energy flux canreach large radii. The field in the funnel region does not connect back into the disk. Rather thepoloidal components lie parallel to the polar axis. The field lines are collimated by a factor of 5/2at r = 40 and by a factor of 10 at r = 400. An outflow along the boundaries of the funnel reachesa maximum vr ≈ 0.75c, but this is sensitive to the value of our artificial density “floor”: a modelwith lower density reaches even larger radial velocities at the outer boundary of the computationaldomain.

Koide et al. (2002) have evolved a cold, highly magnetized uniform density plasma (ρ0/p =0.06, b2/ρ0 = 10) as it falls into a rapidly spinning (a = 0.99995) black hole in Boyer-Lindquistcoordinates for a time ≈ 14GM/c3 using the MHD approximation. This initial state does notcorrespond to an accretion disk system. They demonstrated, however, that a transient net energy

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extraction is possible from a spinning black hole. Because of the short evolution time they areunable to say whether the energy extraction process is possible in steady state. Koide (2003) givesan expanded discussion of the above system.

In contrast to the results of Koide et al. (2002) and Koide (2003), we model a disk with aninitially hydrodynamic equilibrium fluid that is weakly magnetized. We also use a Kerr-Schild (hori-zon penetrating) coordinate system that avoids potential problems associated with the treatmentof inner boundary condition in Boyer-Lindquist coordinates. In our simulation the Balbus-Hawleyinstability drives turbulence and accretion in a steady state where we evolve for a time 2000GM/c3.We measure a sustained outward electromagnetic energy flux that is smaller than the inward matterenergy flux (i.e. net inward energy flux). Their model is evolved for too short a time to observe theunbound mass outflow in the funnel region as seen by us and De Villiers et al. (2003a, 2004).

De Villiers et al. (2004)(hereafter DH) have also considered the numerical evolution of weaklymagnetized tori around rotating black holes. Their models are quite similar to ours in manyrespects, although they differ in that: (1) their models are three dimensional while our models areaxisymmetric; (2) they use a nonconservative numerical method (De Villiers and Hawley, 2003a);(3) DH use Boyer-Lindquist while we use Kerr-Schild coordinates; (4) DH choose γ = 5/3 whilewe use γ = 4/3; (5) DH’s initial pressure maximum is located at 25M , while ours are typically at12M . Our results for the energy and angular momentum per baryon accreted from Table 4.2 can becompared to Table 1 of DH by computing E/M0 = ∆Ei/∆Mi and L/M0 = ∆Li/∆Mi. For modelswith a = (0, 0.5, 0.9) DH find E/M0 = (0.91, 0.91, 0.84) while we find E/M0 = (0.96, 0.93, 0.88).For the same models DH find L/M0 = (3.1, 2.6, 1.9), while we find L/M0 = (3.1, 2.6, 1.7). Giventhe differences in the models and numerical methods, this quantitative agreement is remarkable.Our models and De Villier and Hawley’s models also agree qualitatively in the sense that both showa similar geometry of disk, corona, and funnel and both imply that spin equilibrium is achieved ata ∼ 0.9 (see Gammie et al. 2004).

Komissarov (2001) finds the BZ solution to be stable in force-free electrodynamics, and Komis-sarov (2002a, 2004a) find the BZ solution to be causal, but inconsistent with the membraneparadigm. We find our numerical solutions to be consistent with the BZ solution in the low-densityfunnel region around the black hole. A numerical general relativistic MHD study of strongly mag-netized (monopole magnetic field) accretion by Komissarov (2004b) is also consistent with the BZsolution. For the strong field chosen he finds a considerably faster outflow (Lorentz factors of ≈ 14)than found in our models (Lorentz factors of ≈ 1.5− 3.0). Komissarov’s model does not contain adisk.

The limitations of the numerical models presented here include the assumption of axisymmetryand a nonradiative gas. The effect of axisymmetry can be tested by comparing our models withthe three dimensional models of De Villiers et al. (2004); the angular momentum and energy peraccreted baryon in the two models differs by only a few percent. In addition the jet structureobserved in De Villiers et al. (2004) is nearly axisymmetric. This is encouraging, although it isunlikely that an axisymmetric calculation can capture the full range of possible dynamical behavior

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in the accretion flow.The radiation field, which we have completely neglected here, is likely to play a significant role

in the flow dynamics, through radiation force on the outflowing plasma in the wind and throughphoton bubbles in the disk Gammie (1998); Socrates and Blaes (2002). It will also, of course, playa significant role in heating and cooling the plasma. This is clearly the most significant limitationof our calculation– particularly from the standpoint of comparison with observations– and clearlythe most numerically difficult problem to overcome.

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5 Future Studies

For future work, I plan to perform simulations that will 1) establish the power of the BZ effect as afunction of disk thickness ; 2) determine the connection between the disk and the jet; 3) determinethe relationship between the black hole spin and jet; and 4) establish the requirement of GRMHDto jet speed, collimation, and stability. I also plan to specifically study the collapsar model of GRBsusing a GRMHD model with additional GRB-type microphysics.

5.1 Thin Disks

Chapter 4 presented a study of the BZ luminosity of a rotating black hole surrounded by a thickdisk, for which the scale height (H) is comparable to the radius (R). I plan to extend thoseresults to the thin disk regime, which may be applicable to some black hole accretion disk systems.Prior analytic work suggests that thin disks may have an insignificant BZ luminosity (Ghosh andAbramowicz, 1997; Livio et al., 1999). If thin disks have a weak BZ-effect and if the energy fromthe BZ-effect dominates the energy content of a jet in black hole systems, then those systems witha black hole and jet most certainly have a thick accretion disk near a rapidly rotating black hole.

A self-consistent treatment of radiation generally leads to a disk with a radially dependentH/R, especially for disks near the black hole event horizon. While radiative effects are important,a general relativistic treatment is non-trivial (see, e.g., Cardall and Mezzacappa 2003; Cardall2004). I plan to model thin disks with an ad hoc cooling function that sustains a constant H/R

for all radii. The proposed H/R study may be sufficient to obtain an estimate for the BZ poweroutput as a function of H/R without resorting to radiative transport.

Compared to an otherwise equivalent thick disk simulation, the planned thin disk simulationrequires a larger radial numerical resolution in order to resolve the MRI. Resolving the MRI, withabout 6 zones per fastest growing wavelength, is crucial in sustaining turbulence for a sufficientperiod of time to reach steady state. For example, a typical thick (H/R ∼ 0.26) disk simulationrequires a resolution of 256 × 256. A thin disk model with a = 0 and H/R = 0.01 requires aresolution of 6500 × 1200 and a thin disk model with a = 0.938 and H/R = 0.02 requires aresolution of 9800× 920. Also, in order to avoid artificially loading the (larger) funnel region withbaryons, the thin disk model requires a lower density floor than the thick disk model. Also, a lowerdensity floor is required for convergence of the magnetic field in the funnel region. Using a lowerdensity floor (and thus increasing b2/ρ0) may require improving the algorithms used in HARM.

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5.2 Relativistic Jet-Disk Connection

GRMHD computational studies are on the verge of understanding the mechanism of jet formationfrom an accretion disk around a rotating black hole (McKinney and Gammie, 2004; De Villiers et al.,2004). These numerical models are beginning to identify 1) the mechanism of jet acceleration ; 2)the relative contributions of electromagnetic and kinetic energy to the jet; and 3) the baryon loadingin the jet – i.e. the amount of rest-mass in the jet. The resolution of these issues are one focus ofmy future study, and some work has begun as described below.

The source of energy generating a jet could arise from the BZ mechanism (an electromagneticblack hole wind), from the Blandford-Payne mechanism (a matter disk wind), from an electromag-netic disk-corona wind, radiation driving, or some combination of all these processes. I performedGRMHD simulations of rapidly rotating (a & 0.5) black holes that are spun-down by the BZ-effect,which generates an outgoing electromagnetic energy flux in a mildly relativistic (v/c ∼ 0.3), colli-mated outflow that reaches large distances. The black hole spin energy is extracted directly fromthe rotating black hole and flows along collimated field lines frozen within unbound, baryon-poormatter. For the particular parameter space I explored, the Poynting flux always dominates thematter-energy flux in the jet.

There is a small matter-energy jet at the edge between the corona and funnel in our models withhigh black hole spin. The electromagnetic component of this “disk wind” appears to only providea source of acceleration for the outgoing matter in the funnel region, since the fluid is tied-up inthe magnetic field within the corona. Hawley’s group at Virginia, and our group, have performedsimulations that show that black hole spin plays a significant role in determining the existence andpower of a jet, but the details are only beginning to be studied (De Villiers et al., 2004).

5.3 Gamma-Ray Bursts: GRMHD Collapsar Model

I plan to use a GRMHD model to study the accretion disk that likely forms during a GRB, inthe framework of the collapsar model (Woosley, 1993; Paczynski, 1998; MacFadyen and Woosley,1999). Some interesting unresolved questions include 1) what fraction of the observed luminosity ofGRBs is due to power by black hole spin energy extraction, accretion disk luminosity, or neutrinoannihilation, 2) what is the mechanism for jet production, collimation, and GRB variability ; 3)what is the efficiency of neutrino annihilation and magnetic field reconnection ; and 4) is the jetcomposed of ion/electron or positron/electron pair plasma and what particle/radiation physicsdominates GRB radiation.

Numerical studies of the Type I collapsar model have been performed using Newtonian hydro-dynamic (HD) (MacFadyen and Woosley, 1999) and special relativistic HD models (Aloy et al.,2000; Zhang et al., 2003). Type II collapsars have been studied using Newtonian HD numericalmodels (MacFadyen et al., 2001). All of these models include a realistic EOS, neutrino cooling, andsome estimates of neutrino annihilation. These unmagnetized numerical models include an ad hoc

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alpha-viscosity model of angular momentum transport (Shakura and Sunyaev, 1973). However, themagnetic field is likely explicitly required to launch and collimate jets (Cameron, 2001; Wheeleret al., 2002), and the angular momentum transport is likely driven by the MRI (Balbus and Hawley,1991).

Recent numerical studies of non-self-gravitating accretion disks use the nonrelativistic MHDapproximation (Proga et al., 2003) and use a code based on ZEUS (Stone and Norman, 1992).They include a realistic EOS, photodisintegration of helium, and neutrino cooling. They observea neutrino luminosity of ∼ 1052 erg/s, which is close to the expected GRB luminosity. However,their model does not include neutrino annihilation, so their jet contains no energy from annihilatingneutrinos that may alter the jet structure. Also, they cannot estimate what fraction of the energyin the jet comes from neutrino annihilation compared to electromagnetic luminosity from the disk.Neutrino-generated jets are likely important (see, e.g., Fryer and Meszaros 2003). They observe ajet that is dominated by Poynting flux, rather than matter-energy flux. However, since their modelis Newtonian, they are unable to measure the Lorentz factor of the jet and cannot model a spinningblack hole, which may be required to generate a Poynting flux via the BZ-effect.

As discussed above, I used a similar ZEUS-based MHD code with a pseudo-Newtonian potential(Paczynski and Wiita, 1980) to study similar flows without microphysics. My results suggest thattheir studies of the jet could be dominated by numerical artifacts. For example, I found it impossibleto remove the inner radial boundary condition from causal contact with the rest of the flow (i.e. onecannot keep the inner radial boundary inside the fast point in MHD). Also, while I found a largePoynting flux from the horizon, the Newtonian model should not be capable of producing such aneffect. The results of Proga et al. (2003) are likely similar. Most of the Poynting flux energy couldbe artificially generated by contact between the inner radial boundary and the flow at larger radii.Indeed, GRMHD simulations show that accreting black holes with spin parameter a = 0 generateweak outflows and much smaller Poynting flux at large distances (McKinney and Gammie, 2004).

GRMHD numerical models suggest Poynting flux jets are a natural result of accretion arounda rapidly rotating black hole (De Villiers et al., 2003a; Koide, 2003; McKinney and Gammie,2004). However, such simulations ignore possibly important sources of jet energy, such as neutrinoannihilation, and lack accurate models of reconnection (see, e.g., Sikora et al. 2003). RecentGRMHD collapsar simulations have been performed that fail to generate the estimated observedGRB energy and Lorentz factors (Mizuno et al., 2004a,b). However, their model includes nomicrophysics; they use a shock model, rather than the core of a massive star, as their initialconditions; and the simulation lasts for only a small number of dynamical times (∼ 200GM/c3).The black hole accretion disk system itself may not generate the large Lorentz factors. For example,a radial pressure gradient in the star and external atmosphere can accelerate the jet to Γ ∼ 44,close to that required by observations (see, e.g., Aloy et al. 2000; Zhang et al. 2003).

The current GRMHD numerical model has an ideal-gas equation of state and includes noradiation of neutrinos. To the current GRMHD numerical model, I plan to add a realistic equationof state (EOS) and neutrino cooling and annihilation in the optically thin parts of the flow. I plan to

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first include an approximate EOS (Popham et al., 1999; Kohri and Mineshige, 2002), which includeselectrons, positrons, protons, neutrons, alpha particles, and photons as a single-component fluid. Allparticles are assumed to be in thermodynamic and nuclear statistical equilibrium. This approximateEOS allows the electrons, positrons, protons, and neutrons to have (fairly) arbitrary relativity anddegeneracy. I later plan to include an more accurate EOS solver that allows for any relativityand degeneracy (Lattimer and Swesty, 1991; Lattimer, 1996; Blinnikov et al., 1996; Lattimer andPrakash, 2000). The disk is optically thick to all particles except potentially neutrinos, so thesingle-component fluid approximation is not applicable to neutrinos. Since photons and positronsare in thermal equilibrium, they will be treated as an additional source of radiative pressure. Thephotodisintegration of alpha particles and recombination of free nuclei into alpha particles will alsobe included.

Neutrinos are to be included as part of the fluid for parts of the disk that are optically thickto neutrinos and otherwise stream freely for the optically thin parts. The neutrinos can radiate inthe neutrino “optically” thin parts of the disk (see, e.g., Itoh et al. 1996; MacFadyen and Woosley1999), while neutrinos provide a pressure in the neutrino “optically” thick parts of the flow. Thisneutrino pressure can be estimated as if optically thick, but with an extinction factor in the radiativeneutrino emissivity. As a result, the pressure is reduced by a corresponding factor (Lee et al., 2004).A diffusion scheme for the radiation of neutrinos in the optically thick regime will then be applied.If required, a full Boltzmann transport scheme will be used (see, e.g., Cardall and Mezzacappa 2003;Cardall 2004). Initially, I do not plan to include nuclear burning, since it is likely only relevant tothe supernova component and not to the GRB jet.

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A Model of Plasmas

This appendix discusses the assumptions of the fluid, MHD, and ideal MHD approximations ofa plasma (for a classic review see, e.g. Landau and Lifshitz 1959, 1980). The purpose of thisappendix is to present an overview of the parameters estimated in Section 1.5, which shows thatthe single-component ideal MHD approximation is valid for the accretion flow likely to be present inGRBs, X-ray binaries, and AGN. As described there, only thick accretion disks, with their diffuseplasmas, are not explicitly treatable by the fluid approximation. However, the solar wind toocannot be explicitly treated as a fluid, yet the MHD approximation is a reasonable approximation(see, e.g., Usmanov et al. 2000). Plasma instabilities likely force the solar wind to behave as a fluid(Feldman and Marsch, 1997), and a similar phenomena may occur in thick disks. A discussion ofplasma instabilities is beyond the scope of this thesis, so the discussion below is restricted to thefluid, MHD, and ideal MHD approximations. First, a general overview of these approximations isgiven.

If a gas predominantly collides elastically and contains many particles moving randomly withina characteristic length scale (L), then the gas can be approximated as conforming to a space-velocitydistribution function. Liouville’s theorem and expressions for the body-forces are used to derivethe distribution evolution equation. This results in the so-called Vlasov or collisionless Boltzmannequation. Collisions can be introduced as a source or sink of particles within the distribution,resulting in the collisional Boltzmann equation. The fluid approximation further assumes thatthese collisions drive the system through a series of statistical equilibrium states.

The fluid approximation can be derived as a series of mass-weighted velocity moments of theBoltzmann equation. The MHD approximation can be derived by including a series of charge-weighted velocity moments of the Boltzmann equation. In general, the fluid has many species,but these can often be reduced into a single-component fluid or MHD approximation. The single-component approximation is obtained by summing each of the equations over species and by appro-priately defining the average quantities. For perfect (ideal) fluids, collisions only drive the systemto an equilibrium state, and other effects on the distribution function are ignored. For imperfectfluids, a closure scheme, such as the asymptotic closure scheme of Chapman-Enskog (Chapman andCowling, 1953), is used to relate higher order (say, in ε = λmfp/L) collisional terms as functionsof fluid quantities obtained from the velocity moments, where λmfp is the mean free path betweenparticle collisions.

The discussion that follows does not present a text-book derivation of the fluid, MHD, and idealMHD approximations, but rather presents a summary of the salient features of the approximationsand gives parameters for estimating the validity of the approximations. Note, however, a general

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plasma is sufficiently complicated that no set of dimensionless parameters can absolutely determinewhether each part of the approximations, described below, are valid. Specific configurations ofmatter and magnetic field can develop macroscopic instabilities, for example as with the magne-torotational instability (Balbus and Hawley, 1991). Also, microscopic particle effects can build upto become macroscopically relevant, for example during reconnection (see, e.g., Parker 1963, 1979;Syrovatskii 1981; Biskamp 1986; Taylor 1986; Parker 1988; Masuda et al. 1994; Low 1996) or dueto plasma instabilities (see reviews by, e.g., Melrose 1986; Hasegawa 1975; Begelman and Chiueh1988). Only a full nonlinear study can verify the dimensional arguments presented below.

A.1 Equation of State

An equation of state (EOS) is needed to close the fluid or MHD equations of motion. The EOSof matter near the black hole depends mostly on the density and temperature if the system isin nuclear and thermodynamic statistical equilibrium. Thermodynamic statistical equilibrium ismaintained if all species are in thermal contact through collisions. Thus, one needs to know theelastic and inelastic scattering and absorption cross sections.

The elastic scattering cross sections can be used to estimate not only the validity of the fluidapproximation, but also to estimate the thermodynamic coupling between species. If some speciesare thermally coupled faster than the accretion time scale, then they will equilibrate to similartemperatures. However, if they do not transfer energy by collisions faster than they are accreted,then they could decouple, resulting in a two-temperature flow for a gas of protons and electrons. Forexample, it is plausible that in X-ray binaries and AGN a hot two-temperature disk forms (Shapiroet al., 1976; Narayan and Yi, 1995). In two-temperature or disk-corona models, the electrons cancool by Comptonization, while the protons and nuclei have no mechanism to cool efficiently beforebeing accreted by the black hole. This effect is neglected for the EOS discussed here.

The particle distributions need not be thermal, nor are observations constrained to only observethe most average electrons within the thermal distribution. Low-frequency radio observations ofSgrA* could be explained if a small fraction of electrons reside in a power-law tail of energies (Yuanet al., 2003). Non-thermal electrons may be generated in reconnection events, such as has beenobserved in tokamak experiments of high density plasmas generated by MHD modes (Savrukhin,2002) or in solar flares (Priest, 1981; Kahler, 1992; Zank and Gaisser, 1992; Miller et al., 1997;Priest and Forbes, 2002).

A.1.1 EOS for GRBs

Assuming that GRBs arise from a stellar core-collapse scenario (Woosley, 1993; Paczynski, 1998;MacFadyen and Woosley, 1999), the accretion disk formed during a GRB likely has mostly freeneutrons, protons, alpha particles, electrons, positrons, various heavy nuclei, and neutrino andphoton radiation (see, e.g., MacFadyen and Woosley 1999). In thermodynamic and nuclear sta-tistical equilibrium, the GRB disk state can be treated using an ideal Fermi EOS with arbitrary

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degeneracy for all species and arbitrary relativity parameter (x) for all species, where x ≡ kbT/mc2

for a species with mass m (Blinnikov et al., 1996). Specifically, the approximations of Kohri andMineshige (2002) are used in this thesis. A modified form of equation (26) in Kohri and Mineshige(2002) is used to improve the accuracy of the results near the transition between degenerate andnon-degenerate electrons (Q = mn −me is reintroduced into that equation, as required for consis-tency). A Fermi EOS leads to important deviations from an ideal gas EOS due to neutronizationin thin disks that may form during a GRB. The Fermi EOS determines the fraction of electronsper baryon (Ye).

Fully ionized helium (alpha particles) can be dynamically important in the accretion shock thatdevelops during GRB disk formation. The accretion shock, where nuclei are photodisintegratedinto free nucleons, is located at r ∼ 120GM/c2 in the collapsar model (MacFadyen and Woosley,1999). Also, within the neutrino-cooled disk or after the free nucleons are ejected out in a wind,they recombine to form heavier nuclei (e.g., alpha particles and 56Ni). Indeed, nucleosynthesisis important for understanding the generation of supernova and the GRB-supernova connection.These estimates ignore the supernova component and focus on the GRB event itself, which likelydoes not require following nucleosynthesis (MacFadyen and Woosley, 1999). This will at mostrequire understanding the photodisintegration and recombination of alpha particles. If nuclearstatistical equilibrium (NSE) holds (Khokhlov, 1989), then a Saha-like equation can be used todetermine the fraction of free nucleons from

Xnuc = 30.97(

T

1010

)9/8 ( ρ

1010

)−3/4exp

(−6.096( T1010 )

)(A.1)

(Woosley and Baron, 1992). The remainder of nucleons are bound in α particles.The gas pressure (pgas) is defined as the sum of each species’ (proton, electron, neutron, alpha

particles, nuclei, radiation, and neutrino) partial pressure, with baryons or electrons being eithercompletely nondegenerate or completely degenerate, and completely relativistic or completely non-relativistic (Kohri and Mineshige, 2002). It has been found that such approximate equations of stateagree to within 10% (Popham et al., 1999) of more general calculations by Blinnikov et al. (1996).In this approximation, if the disk is optically-thick to photons, then the photons are treated as aradiation pressure. If the disk is optically thin to photons, then photons stream out with a pressureextinction factor of about 1− exp(−τγ), where τγ is the photon optical depth given in equation 1.3.The local cooling rate would be multiplied by exp(−τγ). This does not model radiative transportprocesses that may lead to radiative instabilities, such as the photon-bubble instability (Gammie,1998).

The same radiation pressure extinction prescription is applied to the neutrinos (Lee et al., 2004)with a neutrino optical depth of

τν ∼ τs,ν + τa,eN + τa,νν , (A.2)

where τs,ν = 2.7× 10−7(

Tp

1011

)2 ( ρ1010

)H, τa,eN = 4.5× 10−7Xnuc

(Tp

1011

)2 ( ρ1010

)H,

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τa,νν = 2.5 × 10−7(

Tp

1011

)5H, Tp is the proton temperature, and H is a typical disk scale height

(Di Matteo et al., 2002). These three terms respectively correspond to 1) neutrino-neutron/protonscattering ; 2) neutrino absorption onto free protons or neutrons ; and 3) inverse electron-positronpair annihilation (i.e. inverse URCA). These processes dominate the inverses of the bremsstrahlungand plasmon neutrino emission processes, which are thus neglected. For an overview, see alsoShapiro and Teukolsky (1983) §18.5. In the electron degeneracy regime, the neutrino emissionis slightly reduced due to a reduction of Ye (Kohri and Mineshige, 2002). The effect of electrondegeneracy on the optical depth is not accounted for in this thesis, so the above estimated τν shouldbe considered a reasonable estimate, but strictly only an upper bound.

Some numerical studies of simplistic pseudo-analytic viscous (unmagnetized) Fermi gas modelsof GRB disks have been performed (Popham et al., 1999; Narayan et al., 2001; Kohri and Mineshige,2002; Di Matteo et al., 2002). GRMHD (or even magnetized) numerical models of accretion disksthat incorporate a self-consistent Fermi gas EOS have yet to be performed by anyone. In Section 1.5,where Ye, pgas, and other quantities are reported for the GRB disks, the relation between massdensity, pressure, and magnetic field is actually from the GRMHD numerical model, rather thanfrom a self-consistent model that includes a Fermi EOS. Section 1.5 reports the ratio of Fermigas pressure to ideal gas pressure to check the consistency of this actually inconsistent approach.It turns out that this approach is fairly consistent, and certainly good enough for estimating thevalidity of the fluid, MHD, and ideal MHD approximations.

A.1.2 EOS for X-ray binaries and AGN

The species’ abundances in accretion disks in X-ray binary and AGN are most likely similar tocosmic abundances. A disk with cosmic abundances would be composed of mostly ionized hydrogen,some ionized helium, few ionized metals, electrons, and photon radiation (Fabian, 1998). For thetemperatures and densities of these accretion disks, the appropriate approximate EOS is simplyan ideal gamma-law gas + radiation pressure (which is simply a reduced form of the GRB EOSdescribed above). This EOS is used to compute the actual structure of the accretion disk for theseobjects using SS73 thin disk (Shakura and Sunyaev, 1973) or ADAF thick disk (Narayan and Yi,1995) models.

A.2 Validity of the Fluid Approximation

This section summarizes the parameters used to estimate the validity of the fluid approximation.The fluid approximation is valid only if the characteristic time (T ) is much longer than the effectivecollision time τ . Also, the effective mean free path for particles λmfp must be much less thanthe length scale (L) of the problem. An “effective” collision is one that has any nonzero crosssection but generates a 90 change in the direction of motion. Essentially all particles interact atsome density and energy, but this section only considers the dominant, typically elastic, scattering

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mechanisms. Most elastic scattering processes dominate inelastic scattering when energies are lessthan the mass energy of a particle.

In order to determine the mean free path, we should consider the total scattering cross sectionσ = σ(E), where E is the center of mass energy. The effective mean free path for particle type Ato collide with particle type B is

λmfp,A−B =

õAB/mA

σABnB, (A.3)

where nB is the number density of particle type B, and µAB = mAmB/(mA + mB) is the reducedmass. Notice that the mean free path for particle A to hit particle B is not necessarily the same asthe mean free path for particle B to hit particle A.

To determine the collisional rate between each species, the relativistic relative velocity

vrel,A−B =vA + vB

1 + (vAvB/c2)(A.4)

is used to estimate the mean relative velocity within a thermal distribution. This expression for therelative velocity assumes that each species is an ideal gas moving with relativistic thermal velocity

vtherm = c

√x(2 + x)1 + x

, (A.5)

where x = kbT [K]/(mc2) is the relativity parameter. This is simply the relativistic extension ofthe nonrelativistic thermal velocity vtherm,non−rel. ∼

√KE/m. The relative velocity can be used

to find the binary collision rateνc,A−B =

vrel,A−B

λmfp,A−B. (A.6)

The proton-electron scattering cross section σpe is determined by the Coulomb interaction. Thenonrelativistic Coulomb energy U(r, v) = 1

2mv2−Ze2/r of one charged particle in the electrostaticfield of another vanishes U(r, v) = 0 at a radius

λc =Ze2

kbTp, (A.7)

where Z is the atomic number. This is the characteristic radius for interaction and thus σZe ∼ λ2c .

Relativistic corrections, of order unity, give σZe ∼ λ2c

(1+x

x(2+x)

)2csc4(θ/2), where x = kbTp/(mec

2)and θ is the scattering angle. There are additional corrections of order unity not considered (see,e.g., McKinley and Feshbach 1948). A more accurate cross section accounting for Debye screeningis discussed below. The Thomson scattering cross section for photon-electron scattering is σT,e =0.665× 10−24 cm2. For photon-proton scattering σT,p = 1.967× 10−31 cm2.

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Additional Scattering Cross Sections for GRB Disks

The proton-proton elastic scattering cross section is also dominated by the Coulomb interaction,except at high energies that occur in hot (thick) disks. For example, strong nuclear forces at50 MeV, corresponding to Tp ∼ 6 × 1011 K, give σpp ∼ 0.110−24 cm2 (Berdoz et al., 1986). Noticethat at such energies the Coulomb cross section is only σc,pp = 8 × 10−30 cm2, which is muchsmaller than the total scattering cross section. The Coulomb term dominates the nuclear termat low energies. At low energies the nuclear neutron-proton scattering cross section reaches up toσpp,nuc.max = 36× 10−24 cm2 (see Table 3-3 in Wong 1990). The transition between Coulomb andstrong interaction domination is at Tp ∼ 108 K.

Proton-proton inelastic scattering becomes important above energy production for mesons suchas pions. The lightest meson is π± with E ∼ 140MeV, which corresponds to a temperature ofTp ∼ 1.6 × 1012 K. This corresponds to twice the upper limit of the typical temperatures in aGRB accretion disk, although such processes may be important in short moments when/where theplasma may be significantly hotter. The cross section for pion production in pp-scattering aboveE ∼ 140 MeV is σpp,π . 0.0110−24 cm2 (see fig 3-4 in Wong 1990). Our own galactic nucleus mayemit gamma-rays due to proton-proton collisions in a thick disk due to neutral pions decaying into∼ 70MeV gamma-rays (Mahadevan et al., 1997).

Nucleons in GRB disks likely have kinetic energies of order 9 MeV. Neutron-proton cross sec-tions for 9 MeV are σnp ∼ 0.4× 10−24 cm2 (see, e.g., Peterson et al. 1960, and references therein).At Tn,p ∼ 1013 K ∼ 860MeV, σnp ∼ 0.4× 10−25 cm2. As energies reach zero, the nuclear neutron-proton scattering cross section reaches up to σnp,nuc.max = 70× 10−24 cm2 (see Table 3-3 in Wong1990). Scattering cross sections between the dipole moment of a neutron and electron can be foundin Yakovlev and Shalybkov (1990).

A.3 Validity of the Ideal Fluid Approximation

This section summarizes the parameters used to estimate the validity of the ideal fluid approxi-mation. In the approximation of the Boltzmann equation, higher order (ε = λmfp/L ¿ 1) termsare associated with, for example, shear viscosity, kinematic viscosity, and thermal conductivity. Inthe MHD approximation, higher order terms are associated with Ohmic losses and the diffusionof magnetic field. All these effects are generally non-adiabatic and so alter the entropy of a fluidelement. See Shu (1992) for more detailed discussions.

A standard measure of the importance of viscosity is the ratio (Re), known as the Reynoldsnumber, of the inertial term to the viscous term in the momentum equation, where

Re =v L

νvisc(A.8)

and νvisc ≡ µ/ρ is the kinematic viscosity associated with diffusion of vorticity, v is the relative shearvelocity, L is the characteristic length scale, and the shear viscosity coefficient is µ ∼ mvthermal/σ.

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Thus νvisc ∼ vthermalλmfp gives Re ∼ vdr/(vthermalλmfp). The Reynolds number can also beconsidered as the ratio of the viscous drag time on the largest scales (L2/νvisc) to the eddy turnovertime of a fluid element (L/v). The Reynolds number should be measured relative to the comovingframe of the fluid to be consistent with the scattering cross section measurement, and so the onlyvelocity that matters is a relative (or differential) velocity. If the flow is supersonic and λmfp ¿ L,then Re À 1 and so viscous forces are much less important than inertial effects.

In order to motivate an inviscid (ideal) approximation for the fluid, the Reynolds number mustsatisfy Re À 1. As shown in Equation A.8, the limit to such an approximation is the smallestcharacteristic differential velocity that needs to be resolved over a characteristic length scale. In anaccretion disk, one smallest characteristic differential velocity desired to be resolved is the radialdifferential shear velocity

v = dv ≡(

drdvK

dr

)∣∣∣∣(r=rfid)

, (A.9)

where dr = L is the smallest differential distance to be resolved at r = rfid, vK ≡ rΩK is theKeplerian velocity, and dv is evaluated at fiducial outer radius of rfid = 40GM/c2, within whichthis thesis is focused. The nonrelativistic Keplerian velocity is vK =

√GM/r, while the general

relativistic Keplerian velocity is vK = r√

GM/(r3/2 + a) for black hole spin parameter a for orbitsbeyond the ISCO. Another useful characteristic velocity is the sound speed v = cs and characteristiclength scale corresponding to the disk scale height L = H or L = (H/R)R.

A.4 Validity of the Plasma and MHD Approximations

This section summarizes the quantities used to estimate the validity of the plasma approximation.This discussion of the plasma, MHD, and ideal MHD approximations follows any plasma (Chap-man and Cowling, 1953; Spitzer, 1956; Stix, 1962; Boyd and Sanderson, 1969; Biskamp, 1993;Baumjohann and Treumann, 1996; Treumann and Baumjohann, 1997), astrophysics (Shu, 1992),or electrodynamics (Jackson, 1962) book. The standard plasma approximation assumes that 1)the fluid is ionized; 2) the time scale of interest (T ) is larger than the time scale for electric fieldoscillations, and the smallest length scale of interest (L) is larger than the electric field screeninglength (λD) ; and 3) the plasma is weakly coupled. The MHD approximation in addition assumesthat the particles behave like a fluid.

A plasma is assumed to be at least partially ionized, since otherwise the fluid approximationis sufficient. In thermal statistical equilibrium (and ignoring radiative ionization), the number ofionized atoms can be determined from the fractional ionization by the Saha equation. The thermalionization fraction (X) for nonrelativistic (i.e. x ¿ 1) hydrogen (H) is found from the Saha equation

X2

1−X=

1nh3

(2πmekbTH)3/2 exp(− IH

kbTH

), (A.10)

where IH ∼ 13.6 eV is the ionization potential energy. Metals, such as iron, have an ionization

165

energy of Ipe ∼ 7.9024 eV for the outermost electron. At the temperatures of typical accretiondisks, the resulting ionization fractions are similar for hydrogen and metals. The disks in all of thesystems studied in this thesis are estimated to have high ionization fractions.

Macroscopic electric fields are present in otherwise neutral plasmas due to charge separation,which can lead to plasma oscillations. The square of the plasma oscillation frequency is given by

ω2p = ω2

pe + ω2pp = ω2

pe

√1 + R ≈ ω2

pe, (A.11)

where R = me/mp, ω2pe = 4πe2ne/me, and ω2

pp = 4πe2ne/mp. The oscillation frequency of electronsand protons is identical, but the oscillation (velocity) amplitude of the protons is R times that ofthe electrons. Thus, the protons are relatively immobile for plasma oscillations. Note that onlya warm plasma with proton-acoustic modes oscillates at ωpp. Different geometries for the plasmaoscillations (such as a spherical charge compression oscillation) give a plasma frequency to withinfactors of 10 of ωp. Relativistic corrections are difficult to derive analytically, but a numericalsolution is easily found. For electronic oscillations reaching Lorentz factor Γ ∼ 18 (estimated for aGRB-type accretion disk), the fully relativistic plasma frequency is ∼ ωp/4. If Tωp/(2π) À 1, thenplasma oscillations can be ignored on time scales longer than T , the shortest time scale of interestby definition. This is estimated to be true for all the black hole accretion systems studied in thisthesis.

While ωp is the plasma frequency for a cold plasma, a hot plasma has a range of allowedfrequencies, with a lower limit of ωp. No electric waves can propagate with frequency ω < ωp,and waves with wavelength λ < λD are Landau damped (see below). A relativistic treatment ofthe plasma waves is necessary if the maximum nonrelativistic velocity of the electrons (vmax,e =vmax,p/R = x02πωpe/

√1 + R ∼ x02πωpe ∼ x02πωp) gives ∼ x02πωp & c. One might expect that

the typical displacement of x0 would be the Debye length that is described below, which then simplysuggests that a relativistic correction is needed if vtherm,e & c or kbTe & mec

2. In the ultrarelativisticlimit of a hot plasma of electrons and nonrelativistic protons, ωp ∼ 4πe2nec

2/(3kbTe) (see, e.g.Medvedev 1999). Thus, relativistic corrections always lead to a lower plasma frequency. Sincethe plasma approximation is only valid for large ωp, relativistic corrections should be includedto validate whether Tωp/(2π) À 1. Even for the most relativistic flows studied in this thesis,Tωp/(2π) À 1, so the plasma approximation is valid.

For a hot plasma with electrons moving at speed vtherm,e, a characteristic length scale is theDebye length

λD =ωp

vtherm,e. (A.12)

If there are many electrons within the Debye length scale of a proton, then one can show thatthe electric field of a proton is screened by the electrons on a length scale of λD (see, e.g., Shu1992). This sets the outer scale for the Coulomb interaction, while the inner scale is set by theradius, at r = λc, at which particle thermal energy is equal to the potential energy. By symmetry,if electrons screen the field of protons, then protons have screened the field of electrons. At scales

166

much larger than λD the plasma acts collectively, while at scales much smaller than λD the plasmaacts as individual charges. For scales less than λD, plasma oscillations are Landau damped dueto resonant wave-particle interactions (for a physical interpretation of Landau damping see, e.g.,Shu 1992, p.401). Thus, if λD ¿ L, then the plasma acts collectively on scales larger than L, thesmallest scale of interest by definition. All the black hole accretion systems studied in this thesisare estimated to have λD ¿ L, so the plasmas in these systems act collectively.

The plasma parameter is defined as the reciprocal of the Coulomb logarithm, which is thenumber of particles Λ with number density ne = np within the screening sphere of radius r = 2λD.This gives Λ = 4/3πne(2λD)3 ∼ 10πneλ

3D. A more careful derivation for a hydrogen plasma gives

Λ = 24πneλ3D. (A.13)

One can show that 4πΛ ∼ (λn/λc)3/2 ∼ (λD/λc) ∼ ωp/νc, where λn = n−1/3 is the interparticlespacing and νc is the Coulomb collision frequency described below. A weakly coupled plasma isdefined as having Λ À 1, such that Debye screening is effective and charge neutrality (ne− −ne+ ∼ np) holds. In a weakly coupled plasma, the particle kinetic energies are large comparedto potential energies. This is what is typically meant by a plasma. A strongly coupled plasma isdefined as Λ ¿ 1, which implies the particle potential energies are large compared to the kineticenergies. A strongly coupled plasma is irrelevant for the plasmas studied in this thesis. Noticefrom Λ = (3/(4πn))1/2(1/e3)(kbTp)3/2 that strongly coupled plasmas tend to be cold and dense,whereas weakly coupled plasmas are diffuse and hot. All black hole systems studied in this thesishave Λ À 1, so they are weakly coupled plasmas.

For weakly coupled plasmas in which the Coulomb interaction is screened on the Debye length,binary collisions are well-defined. Since Coulomb collisions involve many small angle scatteringevents, νc must account for the inner and outer Coulomb interaction scales. The typical collisionfrequency due to binary Coulomb interactions is νc ∼ ν0 ∼ vthermalnλ2

c . A more accurate accountingof Debye screening gives

νc = ν0 log(

λD

λc

). (A.14)

In summary, a medium is a weakly coupled plasma if Xion ∼ 1, λD ¿ L, Tωp/(2π) À 1, andΛ À 1. As first discussed for the fluid approximation, the MHD approximation of a plasma assumesthat collisions sustain a statistical equilibrium space-velocity distribution function of particles ontime scales much shorter than T , the shortest time scale of interest by definition (λmfp ¿ L andTνc À 1).

A.5 Validity of the Ideal MHD Approximation

This section discusses non-ideal MHD effects and the parameters used to estimate whether they arenegligible. Non-ideal MHD effects are due to particle collisions. Particle collisions can be modeled

167

via plasma transport coefficients, which result from higher-order expansions of the Boltzmannequation and from a closure relationship between the higher-order terms and the lower-order terms.It is assumed that any higher-order Boltzmann transport properties can be treated as diffusive termsin a relaxation theory (see, e.g., §19.31 in Chapman and Cowling 1953). Below we discuss some ofthese higher-order effects and the validity of approximations that neglect them.

The general multi-component MHD approximation makes no explicit assumptions about thenature of the electromagnetic field or current and charge sources on scales of L or T for L À λD

and T À 2π/ωp. However, for a specific derivation of the MHD approximation, the current (J) andcharge (ρe) densities must be chosen to be consistent with the electric and magnetic fields as definedby Maxwell’s equations. Several reasonable simplifications can be made, such as to 1) model theprotons and electrons (or any other particles) as one component — thereby ignoring drift amongspecies and the inertia of electrons ; 2) treat the plasma as ideal – thereby ignoring the effects ofcollisions (apart from the fluid approximation itself) ; and 3) neglect the displacement current (asdone in the nonrelativistic approximation). These are referred to as the single-component, ideal,and nonrelativistic MHD approximations, respectively.

If np = ne, then the current distribution for protons and electrons is J = enpup − eneue ≡−eneve, where ve is the drift velocity of electrons relative to the protons. Ampere’s law then givesthat the drift velocity of electrons relative to protons to sustain the magnetic field is

vdrift,B ∼ cB

4πeneL(A.15)

(Shu, 1992). For all the black hole accretion flows considered in this thesis, the drift velocity ismuch smaller than any shear velocity. This further supports the single-component ideal MHDapproximation.

In weakly coupled plasmas, the magnetic field can effectively play the role of collisions byconstraining charged particles to magnetic field lines within a Larmor radius. Indeed, there is sucha thing as a “collisionless shock” (Tidman and Krall, 1971). A charged particle gyrates around afield line until interrupted by collisions. The gyration frequency for a constant speed around a fieldline is

ωg =eB

Γmc, (A.16)

which assumes that the field is created by an external current. The gyration (Larmor) radius is

λg =ωg

vthermal. (A.17)

The ideal MHD approximation assumes that the gyration radii and gyration time scale of particlesare negligible, such that L À λg and T À 2π/ωg. If the magnetic field is nonuniform over thelength . λg, then the gyration itself produces a net current and the electrons can drift significantlyrelative to the protons. One can show that gyration only produces a net current if the electroninertia is non-negligible, which leads to a break-down in the single-component approximation. In

168

current sheets, the gyration of particles and electron inertia must be accounted for. Ignoringcurrent sheets, the gyration must be considered if Tωgp À (L/λgp)2 À 1, which is the so-called“finite Larmor radius” (FLR) ordering associated with the so-called FLR-MHD approximation.The finite size of the gyration can be ignored if (L/λgp)2 À Tωgp À 1, which is the so-calledMHD ordering associated with the common MHD approximation. For the black hole accretionflows studied in this thesis, the MHD ordering is estimated to hold. This further supports thesingle-component ideal MHD approximation.

In the following it is assumed that charge neutrality holds, the flow can be modeled as a single-component, and the gyration of particles can be ignored. All that remains to close the equationis a charge current density J. In the most trivial form J = σ0E, as given by Ohm’s law. A so-called generalized nonrelativistic Ohm’s law (see Krall and Trivelpiece 1973, Eq. 3.5.9) includesthe effects of a finite current-rise time (∂J/∂t term), Hall-effect (J ×B term), and an anisotropiccharge pressure – the so-called pressure effect (∇ · P term, where P is the pressure tensor). Ageneral relativistic generalized Ohm’s law in addition accounts for relativistic thermal velocities,relativistic beaming of particles, and is covariant (Meier, 2004). The MHD approximation is oftenused in a form that ignores these effects, where J×B can be neglected if Lω2

p/ωg,edv/c2 À 1, ∂J/∂t

can be neglected if Tωp À 1, off-diagonal pressure terms can be neglected if Lωg,p/vtherm,p À 1,and ∇pe can be neglected if Ldvωg,e/v2

thermal,e À 1, where pe is the electron pressure (Krall andTrivelpiece, 1973). By ignoring ∂J/∂t, currents are generated by fields instantly, which is acausal(if only negligibly so in practice). The off-diagonal pressure terms are related to the neglect ofthe finite Larmor radius, since the protons and electrons feel a different average force due to theirdifferent orbital radii. All these effects are estimated to be negligible in the black hole accretionsystems studied in this thesis, so the rest of this section discusses a less generalized Ohm’s law.

From a single-component negligible-drift mostly-ideal (ideal except for standard Ohmic dissi-pation) nonrelativistic MHD approximation, one can show that the conduction electron current inthe ion rest frame (primed) quantities is

J′ = −enev′e = σ0E′, (A.18)

or in terms of lab frame (unprimed) quantities is

J = σ0(E + v ×B/c), (A.19)

where σ0 = nee2/(meνc) is the electrical conductivity. One can show that the evolution of the

magnetic field, given by Faraday’s law of induction, contains a purely inductive term called theLorentz effect and a purely diffusive term associated with an electrical resistivity η = c2/(4πσ0) =c2νc/ω2

p. Associated with the resistivity is the conversion of fluid motion into heat energy calledJoule or Ohmic dissipation, which is a similar phenomena as found for viscous fluids. The timescale for diffusion is tD ∼ L2/η, so typically in astrophysical (large L) systems, with an otherwisesimilar resistivity, tD À t. The resistive term ηJ in the generalized Ohm’s law can be neglected if

169

the so-called magnetic Reynolds number

RM ≡ Ldv

η(A.20)

gives RM À 1, where RM is derived as the ratio of the EMF (Lorentz) to diffusion terms in theinduction equation. The magnetic Reynolds number can also be considered as the ratio of themagnetic field decay time (L2/η) to the eddy turnover time (L/v). The black hole accretion flowsstudied in this thesis have an estimated RM À 1. This further supports the single-component idealMHD approximation.

This simple picture of resistivity is inaccurate in the presence of current sheets, which areconsidered to be present in, for example, solar flares. In current sheets, the resistivity is suspectedto be driven by more complicated processes, such as by plasma or MHD instabilities. Theseprocesses are sometimes modeled by a larger anomalous resistivity, generically referred to as Bohmdiffusion. Some Bohm models suggest an effective resistivity with ηeff ∼ v2

therm/ωg. Some self-consistent mechanisms have been proposed (see, e.g. Drake et al. 1994), but the study of currentsheets is an active area of research.

A nonuniform magnetic field leads to E × B-like curvature, gradient, and pitch drifts. Thepitch drift can lead to the reflection of charged particles due to magnetic mirrors, such as in amagnetic bottle or the Earth’s van Allen belts. Gravity acts as an effective electric field to producea gravitational drift. The curvature and gradient drift speeds are . vtherm(λg/L), which can becompared to typical flow speed v or typical differential flow speed dv. One can derive so-calledgyrokinetic-MHD hybrid models to allow for such effects. These magnetic drift speeds are small inall systems of interest in this thesis since λg/L ¿ 1 and vtherm ∼ v, dv. This further supports thesingle-component ideal MHD approximation.

A partially ionized plasma has neutrals. The drift velocity between neutrals and charged par-ticles due to ambipolar diffusion can be compared to the characteristic smallest velocity (Draineet al., 1983), or it can be estimated by considering the ratio of the ambipolar diffusion to theinductive term in the induction equation (Balbus and Terquem, 2001). By setting the drag forcefd = γdrag,i−nρnρivdrift,i−n equal to the nonrelativistic Lorentz force fl = 1

4π (∇×B)×B, the driftvelocity can be estimated as

vdrift,i−n ∼ B2

4πγdrag,i−nρnρiL, (A.21)

where γdrag,i−n = 〈vrel,i−nσin〉/(mi + µ), vrel,i−n is the speed of the charged particle i as seen inthe rest frame of the neutrals n, σin is the scattering cross section between i and n, µ = ρ/nn

is the mean mass per particle, ρ is the mass density, and nn is the number density of neutralparticles (see, e.g., Shu 1992). For slow velocities, neutral-ion or neutral-electron coupling oc-curs due to an induced dipole moment in the neutral particle, and then σin ∝ v−1

rel,i−n, so thatγdrag,i−n ≈ Const. ∼ 3 × 1013 cm3 g−1 s−1. At large enough drift velocities this effect is dimin-ished and eventually the scattering cross section is dominated by the particle geometry, giving

170

σin ∼ 4π(

(rn+ri)2

mn+mi

)(|vdrift,i−n|). The drift of electrons relative to neutrals can also be estimated

(see, e.g., Balbus and Terquem 2001). Despite the fact that some black hole accretion disks studiedin this thesis have low ionization fractions, most have negligible drift velocities compared to thecharacteristic smallest velocity, dv. Only thick (hot) disks modeled for an X-ray binary and someAGN give nonnegligible drift velocities between protons and neutrals, but they are fully ionized sothis is not relevant. This further supports the single-component ideal MHD approximation.

At Tn,p & 1011 K, which may occur in a GRB disk, the nuclear interactions dominate theproton-neutron scattering cross section. At all relevant temperatures, the GRB disk has a largeneutron-proton collision frequency and so neutrals do not likely drift. Estimating the GRB diskneutral drift is nontrivial, so this is not estimated precisely in this thesis. Rough estimates showthat the protons and neutrons do not drift, although direct electron-neutron drift could occur wereit not for the proton-electron electrostatic attraction. Some relevant estimates are provided inGoldreich and Reisenegger (1992), where the neutron star matter cross sections are taken fromYakovlev and Shalybkov (1990).

For (Re, RM ) →∞, Ohm’s law gives

E + v ×B/c = 0, (A.22)

which if combined with the equations of motion is referred to as the ideal MHD approximation.The ideal MHD approximation assumes there is no electric field in the frame comoving with theprotons. In this case, dissipation of magnetic energy does not occur, and the magnetic field lines arefrozen into the fluid (so-called “flux-freezing”). The single-component ideal MHD approximationis estimated to be valid for the black hole accretion disk systems studied in this thesis.

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B Beowulf cluster

In this appendix, we discuss clustering of computers to achieve an efficient, low cost way of doingmassive calculations through parallel processing. Such a computing cluster is referred to as aBeowulf cluster. A Beowulf cluster consists of otherwise independent computers, called nodes,connected by a network. In the simplest setup, all nodes connect to a single switch. For thepurposes of access from the outside, another switch with Internet access is used. The parallelcomputing power of a Beowulf cluster falls somewhere between that of a massively parallel machine,such as the Cray T3D or SGI Origin 2000, and a simple network of workstations with no collectiveprocessing.

We give a synoptic history of our cluster design in § B.1, discuss performance issues in § B.2,and Internet accessibility in § B.3. In § B.4, we discuss the hardware that comprises a cluster,such as the motherboard, CPU, memory, network, and hard drive (HD). In § B.5 we discuss theoperating system (OS), message passing interface (MPI), and how to make a hyperbolic physicscode a parallel program. In § B.6 we discuss important tests of a cluster, such as testing CPUperformance, reliability, code compiler, memory, network interface, and how our code performs.Finally in § B.7 we summarize the results and conclusions of this appendix.

B.1 History of Design Decisions for our Clusters

A short history of Beowulf clusters can be found at the Beowulf website1. Several factors influencedthe desire to build clusters of computers for parallel computation using commercial off-the-shelf(COTS) products, which includes: 1) in the early 1990s, the performance per unit price of individualcomputers became comparable to classical supercomputers, for which the CPUs obtain each other’sdata via proprietary communication technology rather than what is classified as network technology;2) in the mid-1990s, computer hardware and the Linux OS became highly reliable; 3) in the mid-1990s, committees formed to promote and develop parallel communication technology, such as theMPI2 (Message Passing Interface) Forum; and 4) developers at Argonne National Laboratory3

(ANL) and Mississippi State University (MSU) developed MPICH4, a portable implementation ofMPI.

Parallel processing on a cluster of computers became both practical and more affordable thansupercomputers. Computer centers such as NCSA began experimenting with Beowulf clusters in

1http://www.beowulf.org/beowulf/history.html2http://www.mpi-forum.org/3http://www.anl.gov/4http://www-unix.mcs.anl.gov/mpi/mpich/

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the year 2000 with a cluster called Posic consisting of 550Mhz Intel Pentium III systems. In 2001,NCSA created a production cluster called Platinum consisting of 1Ghz Intel Pentium III systems(now retired by Tungsten)5. The typical lifetime of a Beowulf cluster has been about 2 years.

NCSA offers a large amount of resources that can be obtained by submitting a proposal throughan allocations board6. Based upon one’s request for computer time, they decide how much computertime one should be given. For example, in 2002 we obtained 100,000 CPU-hours of computer timeon NCSA’s Platinum cluster of 1GHz Intel P3s, and in 2003-2004 we obtained 300,000 CPU-hours on NCSA’s Tungsten cluster of 3Ghz P4 Xeons (each Tungsten time unit is worth about 5Xthat of a Platinum time unit). An NCSA allocation expires after about a year from the point ofactivation. Using an NCSA cluster removes the technical hassle associated with computer hardwareand OS details. The major problem with supercomputer centers such as NCSA is they often explorebleeding edge computer science and make it difficult to perform bleeding edge science. For example,the NCSA Tungsten cluster was scheduled to be ready October 2003, but it was not in productionmode until June 2004. The primary reason for the delay was their attempt to use a new file systemcalled Lustre7, which turned out to have bugs. The purpose of Lustre is to replace NFS (networkfile system)-type technologies for distributed file systems.

Since 1999 we have investigated Beowulf clusters as platforms for high-resolution astrophysicalsimulations. Our ultimate goal was to perform general relativistic accretion flow simulations usingthe to-be developed HARM (see chapter 3). In comparison to computer time on an NCSA cluster, alocal cluster is more easily accessible, allows us complete control over what programs are installed,and is self-managed. Self-managing a cluster gives the user “root” privileges, allowing the user tocontrol the basic Linux environment, such as the version of compiler, MPICH, and other utilitiesthat cannot be installed by a user on an NCSA cluster. We also considered a local cluster usefulfor test simulations before using NCSA computer time to perform production simulations. NCSAcomputer time can be used rather quickly when using many CPUs, and mistakes can be costly.

With a small budget one can create a cluster that rivals a large NCSA cluster in the quantityof CPU-hours over the period of a year available for our private use. The number of hours in ayear of computing time as allocated by NCSA is 8760. For an equivalent of 100,000 CPU-hours onPlatinum, we would require 6 dual-CPU 1Ghz P3s. This is quite inexpensive. For an equivalent of300,000 CPU-hours on Tungsten, we would require about 17 3.06Ghz dual-CPU nodes.

The largest expense of a Beowulf cluster is the initial cost of hardware and associated envi-ronmental expenses (such as cooling units and power circuits). Much time can be spent installingthe OS. As we discuss below, some Linux distributions essentially eliminate the complications andtime needed to install and upgrade the OS. The primary expense of money and time after thecluster is built is replacing failed node components. In particular, HD failure is the limiting factorin determining the number of nodes to purchase. In our experience, HDs fail the fastest of any

5http://www.ncsa.uiuc.edu/UserInfo/Resources/6http://www.ncsa.uiuc.edu/UserInfo/Allocations/7http://www.lustre.org/

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hardware component, at a rate of about 1%8 per year for mechanical failure, and based on ourexperience, the failure rate is about 5% per year for partial data loss requiring reinstalling the OS.If we assume it would be reasonable to reinstall a node’s hard drive about once per month and eachnode has one HD, then the largest cluster one should purchase is 240 nodes. If one is only limitedby one mechanical failure per month, then the largest cluster one should purchase is 1200 nodes.This gives an operational cost of about $200/month for the new HD, plus time and effort to installthe new HD.

Below we discuss the purchase of a 16 node 2.4Ghz P4 Xeon cluster, that together with ourcollaborators will soon (as of this writing) expand into a 28 node Xeon cluster. We should expectabout one HD failure per year resulting in loss of data that requires reinstalling the OS. The currentcost per node is about $1600 and network equipment about $5000. This will give us a total numberof Tungsten-equivalent CPU-hours of about 192,000 for about $50,000 of NSF funds (equivalentto about 3 graduate students for one year). This equates to $0.26 per Tungsten CPU-hour. Wewould require about $78,000 to obtain about 46 nodes to have 300,000 Tungsten CPU-hours peryear. Before discussing our cluster in more detail, we first discuss the history of our older clusters.We then discuss the history of our decision making for our Xeon-based cluster.

Our first cluster was based on 2 dual-CPU Alpha-based nodes. At the time, Alpha CPUs wereconsiderably faster than Intel CPUs (Pentium III at 550Mhz). Despite the high price of the Alphasystems, 2 nodes are much easier to manage than the equivalent 6-8 dual-CPU Intel nodes needed.Fewer nodes also allows for higher performance for network intense operations. At the time wechose a Myrinet9 network technology running at 1000Mbit/s, which was significantly faster than therelatively inexpensive Ethernet technology that ran at 100Mbit/s. The cluster operated as expected,so we added an additional node when funding became available. By choosing the most expensivecluster components, the cluster was shielded from the performance uncertainties associated withlow-end technology. However, the Alphas were sufficiently high-end to be cumbersome in Linux.For example, they were difficult to upgrade to new Linux versions and incompatible with severalprograms. This 3-node Alpha cluster served us for about 2 years.

In 2001, we received additional funding to purchase a new cluster consisting of 3 2.0Ghz dual-CPU Intel nodes connected by Myrinet and gigabit Ethernet, where the Myrinet boards from theAlphas were used in the Intel systems. The nodes were named Alphadog, Gravitas, and Horizon,so the cluster was called the AGH cluster. The Intel-based systems were easier to manage than theAlpha systems due to the lack of full Linux support for Alphas, although the 2-year old Alphashad comparable performance. This system served us for about a year as a 3-node cluster, and forthe past 1.5 years has been the backbone of our Xeon cluster described below.

In 2002, our group and collaborators received $50,000 in NSF funds to purchase a computercluster to perform simulations of relativistic magnetized gravitating flows. Given our group’s priorcluster experience we decided to split the $50,000 between the 2 groups. Our group forged ahead

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and developed a cluster while our collaborators researched how their code performs on such systems.This gave us $25,000 to purchase a computer cluster. We originally considered simply purchasingsystems as in the AGH cluster, but we first decided to study what systems would give the highestperformance per unit cost. We thought it plausible to test whether single-CPU Intel Pentium 4(P4) systems with either gigabit or Myrinet would perform as well as the AGH nodes. This wouldbe advantageous since the P4 nodes were about 2-3X lower in cost than each node in the AGHcluster. The P4 test cluster could also be tested against the AGH cluster, and any gigabit orMyrinet board can be swapped between the P4 and AGH cluster for comparison.

We purchased a test-cluster of 4 P4 nodes with the hope that they would eventually form part ofa larger cluster, where in the worst case scenario these nodes would become new workstations. Wetested the P4 nodes with 32-bit 33Mhz 100Mbit Ethernet onboard (82801 BA/BAM/CA/CAM),the AGH cluster nodes with 64-bit 133Mhz gigabit Ethernet onboard (82544EI rev2), P4 and AGHnodes with Myrinet boards (LANai 7 version PCI64A and M2L-PCI64B-2, 2MB boards, 66Mhz),and 4-5 different gigabit boards provided by our supplier, SWT10. These gigabit boards includedthe 32-bit 33Mhz Intel Pro/1000 MT Desktop (Intel 82545EM rev01), 64-bit 66Mhz Intel Pro/1000MT Server (Intel 82540EM rev02), and a 64-bit 66Mhz 3com 3C996B-T (Broadcom NetXtremeBCM5701 rev 21). We did not really consider buying expensive gigabit boards (>$200), but wewanted to understand how such boards performed.

Chips on the motherboard are generally much less expensive than add-on boards. Onboard chipscan often provide increased performance due to an efficient design and because they are integratedwith the latest features included on the motherboard. Myrinet at the time cost $1200 per board,and a Myrinet switch for 16 hosts cost $5000. Today the Myrinet boards are 50% cheaper, but theswitches cost about the same. The gigabit boards range from $50 for the Intel Desktop to $500 forthe 3COM board. We purchased an 8-port non-blocking switch for $700 in order to test gigabit, aLinksys EG000811.

We compared all permutations of boards, systems, and number of processors involved in acalculation using MPI bandwidth/latency benchmarks, a ZEUS-based code, and HARM in orderto test system+network performance. We found that the AGH cluster using its onboard gigabitperformed the best. The AGH cluster using onboard gigabit even outperformed Myrinet in dual-CPU HARM calculations using all 3 nodes and 6 CPUs. We found that the P4 cluster with anyboard gives about 45MB/s peak bandwidth, whereas the AGH cluster gives about 90MB/s with any64-bit board. The P4 systems with any gigabit board obtained a latency of ∼ 70µ s for transferringsmall message sizes, Myrinet obtained ∼ 9µ s, the add-on 32/64-bit 66Mhz gigabit obtained ∼ 50µ son AGH, and the AGH onboard gigabit obtained ∼ 50µ s. HARM performed best on the AGHonboard gigabit by passing relatively large blocks per unit of computation. In such a case thelatency of ∼ 50µ s vs. ∼ 9µ s for Myrinet is indistinguishable. The ZEUS-based code with manymore small communications per computation performed best on Myrinet. Note that MPI tests

10http://www.swt.com/11http://www.linksys.com/products/product.asp?grid=29&prid=407

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on 64-bit 66Mhz systems show year 2004 Myrinet technology to achieve about 225MB/s and smallmessage size latency of about 8µ s12, so its plausible that Myrinet could offer a significant advantageover gigabit today. However, today’s 10Gigabit technology may counter that peak bandwidth13.Despite the likely higher peak bandwidth of 10Gigabit, the latency will still be bounded by theTCP overhead of standard Linux drivers.

The lower P4 performance is likely due to the 32-bit 33Mhz PCI bus14,15. As of this writing nosingle-CPU Intel workstation motherboard has a faster bus than 32-bit 33Mhz. The 32-bit 33MhzIntel Desktop gigabit performed poorly in an AGH node compared to 64-bit 66Mhz gigabit boards,indicating that the 32-bit board, rather than the rest of the system, was limiting the communication.The onboard AGH gigabit likely performed better than the add-on 64-bit 66Mhz gigabit boardsbecause the AGH cluster nodes have a 64-bit 133Mhz PCI-X bus connection to the gigabit, andthe built-in design allows motherboard manufactures to design an optimal interface that allows theonboard gigabit to operate as efficiently as possible at the full speed of PCI-X.

Based upon these results we considered purchasing systems similar to AGH as we had originallyintended. However, around this time in early December 2002, we learned of a soon-to-be-releasedtype of motherboard that would have all the technical benefits of a server type motherboard thatis typically an ATX-E (extended) form factor16, but would be compact in ATX form factor likeworkstation motherboards. The so-called Value Server motherboard17 (at the time the Tyan Tigeri7501) costs about the same as a high-end workstation motherboard ($250), while costing about1/2-1/3 as much as the typical server motherboard18(at the time the Tyan Thunder i7501). Thereis always an uncertainty about performance for untested systems, so we had doubts that thiswas a proper choice. The AGH motherboard was from Tyan and worked great, the Value Servermotherboard was also from Tyan, the Value Server motherboard is actually a similar model to thefull server motherboard, and both contain very similar components. In particular, the onboardIntel gigabit chip was only slightly different and supposedly improved from the AGH gigabit chip.Thus, we decided to take the risk. This allowed us to obtain two more nodes due to the lower costper node. In the worst case scenario these new nodes would be turned into a computer farm19. Wedecided to obtain about 12 nodes to add to the AGH cluster for a total of 15 nodes20. The deliverytime for providing the systems was delayed a few times due to the lack of product availability, butwe finally had the systems by the beginning of February 2003.

A gigabit network switch, as opposed to a network HUB, is required for fast communicationbetween nodes. A switch communicates to a network board only that information meant for it

12http://www.myrinet.com/myrinet/performance/MPICH-GM/index.html13http://www.10gea.org/index.htm14http://www.compute-aid.com/64bitpci.html15http://www.intel.com/design/bridge/16“Form factor” denotes the style and size of the way in which the motherboard attaches to the computer case.17http://www.tyan.com/products/html/tigeri7501.html18http://www.tyan.com/products/html/thunderi7501.html19Most of the cluster’s computer cycles have been used for single CPU processes since its activation.20At the time (Aug 2004) of this writing we have added 3 more new nodes, and we have removed 1 older AGH-type

node to serve as a web server

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based upon its IP address. A HUB performs no such routing and every network board receivesdata meant for all nodes. High-end switches are typically non-blocking, which means that thecommunication is never limited by the switch and all network boards can achieve maximum speed.We originally planned on purchasing an HP 4108gl 21, but we required a gigabit switch that followedthe always changing University of Illinois regulations as determined by the Network Design Office 22

(NDO). The typical function of the NDO is to design a network that suites the needs of Universitystaff that fits with the NDO’s idea of what constitutes appropriate networking equipment. Theyknew of problems with the HP 4108gl that could cause network failures for certain devices on theUniversity network. After providing the NDO with the desired purpose and design of our network,the NDO agreed on the HP 5308xl 23 switch, which we ultimately purchased for the new cluster.This switch is composed of a chassis and up to 8 modules. The chassis holds the modules, containsthe technology to allow all the modules to communicate, and powers and cools the entire switch.The modules contain the technology to handle a specific type of network connector, such as copperor optical fiber. The cost of the switch chassis (model J4819A) was $2036. The gigabit coppermodules with 4 ports each (HP Procurve Switch XL 100/1000-T module (J4821A)) were $746 each.For 15 nodes and one Internet connection we required 4 modules, but purchased 5 modules sincewe would likely add more nodes later.

The final cluster element was the rack to hold the system. The purpose of the rack is to savefloor space by stacking vertically while still allowing easy access to each node. We decided to buy ametal rack from Bed Bath and Beyond from InterMETRO24. Each unit supports 800lbs and eachof the 4 shelves (including top) supports about 200lbs. We purchased a unit that would contain 12new nodes and the switch on one shelf, while the Alpha cluster and 3 older AGH nodes would beon another shelf. Each new node is about 30lbs, each Alpha about 70lbs, old AGH nodes about50lbs, and the switch is about 30lbs. This gives 360lbs for each shelf, so the shelving is more thansufficient.

With the equipment to be purchased settled, we set out to determine whether there was sufficientpower and cooling for the new cluster in the Astronomy building’s computer room. We needed tosupport our old Alpha cluster as well. One can determine the peak power required for a giventype of equipment using the APC calculators25 online or estimate it from individual specificationsheets. A typical new or AGH-type node consumes 270Watts of power, the HP 5308xl gigabitswitch consumes about 620Watts, and the 3 old Alpha systems consume about 370 Watts each.With an efficiency factor of 1.33 for Watts to VA, we obtain ∼77A for a 120V line. Thus we requireda total of about 4 20A electrical circuits.

During this investigation it was important to determine the electrical layout and total poweravailable for our system. The electrical outlet and hardware components in the Astronomy Building

21http://www.hp.com/rnd/products/switches/switch4100glseries/overview.htm22http://www.cites.uiuc.edu/commtech/ndo.html23http://www.hp.com/rnd/products/switches/switch5300xlseries/overview.htm24http://www.metro.com/consumer/index.cfm25http://www.apcc.com/template/size/apc/index.cfm

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Figure B.1 Astronomy Building computer room electrical layout as of Feb, 2004. BH is locatednorth and center. This excludes the addition of another new node for BH and the collaboratinggroup’s cluster of 12 nodes.

computer room as of early year 2004 are shown in Figure B.1. We determined that the additionof our computers required rewiring 2 240V 30A circuits into 3 120V 20A circuits. Often oldercomputers used 240V since higher power consumption is more efficient at higher voltages. Beforemaking these rewiring changes we tried putting the cluster into operation by using any availableoutlets. In particular, 1 20A outlet was connected to a set of 8 nodes, and we experienced powerfailures when fully using the cluster. The peak requirement for 8 nodes is 24A, so we were simplyunderestimating the likelihood of the peak power use. This was temporarily corrected by furtherdistributing the power over the outlets with remaining capacity until the 240V 30A outlets wereconverted to 120V 20A.

The power in Watts consumed also gives the heat generated, where the typical cooling unit isquantified in how many tons of cooling it provides, where 1 ton is 12000 BTU/hr and 1 Watt is3.413 BTU/hr. Thus we required about 1.6 tons of cooling. The items already in the machine roomincluded 7 300Watt machines, 1 switch at 1600 Watts, and lighting at about 300 Watts for a totalof 3700 Watts. Thus the total peak cooling required was about 2.7 tons, while the cooling unit atthe time was only 2 tons provided by a Liebert Challenger 226. Thus in principle we did not havesufficient cooling. In practice these are peak cooling requirements not often necessary except for

26http://www.liebert.com/support/training/env/manuals_obsolete.asp

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extended periods of maximum use of all systems. Essentially the heating of electrical componentsis much more extended in time than the electrical demand. Thus, while electrical requirements arestrict for any moment in time, only prolonged periods of maximum use require maximum cooling.The cooling unit handled our needs, and we have never had an overheating incident since the clusterwas installed.

Later additions by us and our collaborators in mid-2004 increased the need for power andcooling. After accounting for the addition of 3 new machines by us and a future addition of 12nodes (attached to our switch) by our collaborating group, the total required cooling is 4 tons forthe entire machine room. This required the purchase of an additional 5 tons of cooling providedby a Liebert Challenger 300027. The new 5 ton unit will supply the entire computer room, whilethe older 2 ton unit will operate only when the 5 ton unit fails. This allows time to shut downcomputers properly. These additional computers also required 3 more 20A circuits. The totalestimated cost for this cooling+electric upgrade is $28,500.

Our new cluster is called BH, an acronym for Black Hole. A basic schematic of the BH clusteris shown in Figure B.2. A digital photograph taken soon after the OS, software, and MPI wereinstalled is shown in Figure B.3. BH currently has 17 nodes where one is used primarily forcompilation of code, initiating parallel programs, and services such as a web server.

B.2 Cluster Performance and Advanced Network Drivers

Soon after BH was set up, we measured the latency and peak bandwidth for the new nodes and weredisappointed that the peak bandwidth was only 60MB/s and the latency was 60µ s. This realizedour worst case scenario that the nodes would be turned into a computer farm. This outcome was aresult of impatience and the desire to avoiding wasting money by purchasing more test units thatwe would not use in a cluster.

We considered methods to optimize the network performance. We hypothesized that the defaultIntel drivers were not properly dealing with the new chip in the new BH nodes. We could usespecialized network drivers or specialized Linux OSs with modified kernels to improve performance.Some of the projects or companies out there that offer related services include GAMMA28, EMP29,M-VIA30, Scyld31, Score32, MPI/Pro33, and Scali34. Some of these projects or companies offerspecific Linux distributions with modified kernels, some offer highly specific Linux distributionswith management software and no modified kernel or drivers, and others offer specialized Linuxdrivers to be used with any Linux distribution. As far as speed is concerned, the specialized Linux

27http://www.liebert.com/dynamic/displayproduct.asp?id=542&cycles=60Hz28http://www.disi.unige.it/project/gamma/index.html29http://www.osc.edu/~pw/emp/30http://www.nersc.gov/research/ftg/via/31http://www.scyld.com/32http://www.pccluster.org/33http://www.mpi-softtech.com/34http://www.scali.com

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NODES

BH00-BH16

CLUSTER SWITCH

17 NETWORK CABLES

INTERNET SWITCH

1 NETWORK CABLE

Figure B.2 BH Beowulf cluster primary elements.

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Figure B.3 Digital photograph of BH cluster.

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drivers offer improvements by including an OS-bypass or user-level communication drivers. Thetypical Linux network driver requires support of TCP and other archaic OS-level protocols that arenot optimal for intensive communications. These OS-bypass drivers separate their activity from theLinux kernel and only provide special services such as a protocol for MPI communication. Anotherfeature that improves network efficiency is called zero-copy. This feature allows the network torun without interaction with the CPU in processing memory buffers. Today, many companies,such as Intel, are including the zero-copy feature in Linux network drivers. These features areincluded in the commercial product Myrinet when using their GM drivers. The most interestingdriver modification is GAMMA, which obtains a latency of 11µ s and 113MB/s on a Server-basedmotherboard 1Ghz P3 cluster, exceeding the performance of Myrinet at the time. Donald Becker,master of the networking universe, father of the Beowulf cluster, and chief technical officer &founder of Scyld, considered GAMMA the most promising technology at the time.

However, this seemed an unreliable option since many projects such as GAMMA have a veryshort life (e.g. Giganet-VIA, ServernetII-VIA, InfiniBand, U-Net, AM II, LPC, PM, FM, GigaE-PM, and BIP are all dead) and were no longer being worked on at the time of our investigations.Specifically, GAMMA supported different devices and obtained quite excellent performance on anoptical-based Netgear GA621, but by the time of this investigation we already purchased a copper-based gigabit switch. The successor to the GA621 with a copper connection called Netgear GA622actually includes an unrelated chip that performs poorly by comparison. The M-VIA project onlysupported one still-available expensive $500 board (SysKonnect35). The EMP project seemed tobe quite interesting from a performance perspective, however few chips were supported and thenetwork boards were unavailable or expensive.

The typical problem for these programmers is that by the time their specialized driver is writtenfor a chip, the manufacturer no longer produces the board with that chip. Often a single boardwill have many revisions of the same chip with entirely different, incompatible functions that haveto be reprogrammed (or re-hacked). Support from manufacturers for Linux driver developers hasalways typically been difficult to obtain due to the secrecy associated with corporate technology.Typically, chips are hacked by programmers until the drivers work, but driver development hasnot kept up pace with chip revisions. Clearly this process leads to unreliable support, where theobsolescence of a chip can mean the death of an associated project. Only companies that developboth the hardware and drivers with OS-bypass (such as Myricom) can offer reasonable reliabilityand stability to the user.

Another problem is that these OS-bypass related drivers and OSs are generally non-trivialto install and only come with obscure instructions. Drivers such as GAMMA come with manyrestrictions on how the network can be used due to limitations in the state of the developmentof the specialized driver or OS. Discussions with Donald Becker regarding OS-bypass provided ahistorical insight. He says it took about 20 years to “get TCP right”, so likely it will take a whilebefore OS-bypass or a related technology is cheap and reliable.

35http://www.syskonnect.com/syskonnect/performance/gig-over-copper.htm

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We did communicate with Marc Ehlert who designed GAMMA and he used our ZEUS-basedNewtonian 2D MHD code to test GAMMA on his 1Ghz P3 cluster. GAMMA obtained no betterperformance than default Linux drivers with our code for a typical problem and resolution. Itis possible that a faster CPU would make better use of the higher bandwidth and lower latencyGAMMA offers. Ultimately, the problem with GAMMA was that it only supported single-CPUnodes. Donald Becker, who has been a network driver programmer for several network chips, reportsto us that the SMP (Shared Memory Processor) issue is a critical reason why OS-bypass has notbeen successfully implemented (except at the time by Myricom with Myrinet). Apparently thedriver must be much more sophisticated to be compatible with SMP. Since most Beowulf clusterstake advantage of SMP nodes, this is a large conflict. At the time of this writing the GAMMAproject does support dual-CPU nodes, but still only supports the NetGear GA621 for gigabit.

Ultimately we got “lucky” and found a large increase in performance on BH by using the adviceof Intel and Tyan regarding our performance problems. Tyan explained that the newer gigabitchips were actually not malfunctioning, but the Intel driver was flawed for the newer chips, whichwas a known issue to Intel. They suggested 2 trivial changes to the “/etc/modules.conf” file to passparameters to the Intel driver. This includes the “InterruptThrottleRate=0” and “TxIntDelay=0”lines. We are currently using Intel driver “e1000-4.4.19”. After incorporating this change the AGH-type node latency improved from 50µ s to 30µ s, and the peak bandwidth changed from 90MB/sto 83MB/s. The new Value Server system latency improved from 60µ s to 27µ s, and the peakbandwidth improved from 60MB/s to 100MB/s.

B.3 Cluster access to Internet

The University of Illinois at Urbana-Champaign (UIUC) provided BH with a range of private IPaddresses. A private IP address can access the UIUC network, but not the Internet outside. Privateaddresses were provided due to the scarcity of addresses available on the UIUC domain. Privateaddresses also enhance security by not providing “Internet intruders” with any means to connect.To use the private IP address we connect our gigabit switch to the Astronomy Department networkswitch, which connects to the UIUC network.

In order for part of BH to be visible to the Internet we also obtained a single public IP ad-dress. One computer (BH00) is connected to the Internet by directly connecting one of its networkinterfaces to the Astronomy Department network switch. This single computer can thus provideweb, ftp, ssh, and any other service to computers on the Internet. Actually, the Internet relatedservices are not required to run on BH and certain programs can be moved to other nodes to obtaina higher level of security for BH. Security must be maintained by updating the OS when patchesthat remove vulnerabilities are available.

The final basic set of connections made in a cluster is between every node in the cluster andthe cluster switch. For BH, we have chosen a simple star-type network topology that providesmaximum performance per unit cost.

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B.4 Hardware

The hardware in a cluster must be reliable, relatively stable against new products, high performance,and all the equipment must work together seamlessly. A typical system primarily consists of amotherboard, CPU, memory, network interface, and hard drive. Figure B.4 is a cluster blockdiagram for the BH cluster and includes all the important elements of a cluster that we discussin this section, along with the size and bandwidth of the pipe between each chip and the networkcomponents.

The motherboard forms the backbone of each node by providing a means for communicationbetween various types of hardware. Chips on the motherboard communicate through many differentprotocols at various speeds to other chips on the motherboard. A motherboard typically containsthose hardware chips that are relatively inexpensive (compared to additional hardware), commonlyin use, and have no need to be upgraded. The primary chips on the motherboard are called thechipset. This is comprised of some number of chips that coordinate activity between any secondarychips on the motherboard, the CPU, memory, and other additional hardware. The BH nodes usethe E7501 chipset36.

The chipset is primarily composed of 3 chips: 1) Memory Controller Hub (MCH) ; 2) I/OController Hub (ICH) ; and 3) PCI Controller Hub (PCH). The MCH is the spinal cord of thenervous system and coordinates activity between the CPU, ICH, and PCH. Secondary chips couldinclude hard drive controllers, hardware monitoring, external I/O controller, PCI controller, andBIOS. Today’s motherboards often incorporate other chips that would otherwise be added as aPCI board. This includes a video controller or one or more SCSI (advanced hard drive) controllers.The motherboard used for the new BH nodes is the Tyan Tiger 750137.

The brain of a computer is the CPU, which interprets and executes program instructions thatcan be operated on program information. The new BH nodes have 2.4Ghz Intel P4 Xeon CPUs38

with a 512KB L2 cache. In today’s typical system, the set of CPUs communicate on a single buscalled the front side bus (FSB). Each CPU in a dual-CPU system communicates to other CPUsor the MCH using this bus that operates at 400 - 533Mhz for the BH cluster systems. The FSBprotocol is 64-bit and thus the peak bandwidth is 3.2GB/s - 4.27 GB/s, respectively (each periodof the cycle contains 64-bits of data).

The CPU itself has memory units called L1, L2, L3 caches (L stands for level). The L1 cacheis the smallest and fastest cache. Progressively higher numbered L caches are larger and slower.The purpose of a CPU cache is to temporarily store program instructions and information. Inthe BH cluster the L1 cache is a mere 8K, but operates at 16.4GB/s. The L2 cache is 512K andoperates at 14GB/s. The size and speed of the CPU caches are determined by cost effectivenessfor a given amount of performance as predicted by market prices for memory. There are severaladvanced features included in various CPUs. The Intel and AMD lines of CPUs include SIMD (sin-

36http://www.intel.com/design/chipsets/e7501/index.htm37http://www.tyan.com/products/html/tigeri7501_spec.html38http://www.intel.com/products/server/processors/server/xeon/index.htm

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Figure B.4 Cluster block diagram of BH cluster. Figure shows bandwidth between elements onmotherboard and switch.

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gle instruction, multiple data) type instructions that allow for a single instruction, which executesevery fixed number of CPU clock cycles, to operate on multiple data. This reduces the overheadassociated with instructions and allows integer and floating point operations to be performed 2Xfaster for instructions operating on 2X the standard data size. Another feature of relevance forclusters is Hyper-Threading. This feature is meant to increase CPU efficiency by having a singlephysical processor support up to 2 thread (or processes). This improves performance when mul-tiple programs compete for CPU time in a multi-tasking environment. However, typically in acluster environment one only wants to have 1 process per CPU, so this is not typically beneficial.Some benchmarks with 1 process per CPU show that Hyper-Threading actually slows down per-formance39. We find no difference in performance for our codes running 1 process per CPU withor without Hyper-Threading.

The main memory of a computer serves as a relatively fast repository of program data. Themain memory is essentially an extension of the CPU caches but operates over the FSB bus throughthe MCH. A typical Intel server based motherboard has dual-channel memory40, which means thebus connecting the memory to the MCH has 2 independent channels for 2 sets of memory chips.This allows the 2 CPUs (or any other device) to independently operate on those 2 sets of memorychips. Each channel of memory in each node of the BH cluster has memory that communicates tothe MCH in 64-bit chunks at 266Mhz for an effective total of 128-bit memory. The bus betweenthe MCH and memory is actually 144-bit, which includes 16 bits of error correction encoding toguarantee the data is accurate. The memory used in the BH nodes is called PC2100 DDR. The PCrefers to personal computer and the 2100 refers to 2100MB/s, which is the approximate effectivepeak bandwidth in MB/s per channel.

Memory will continue to increase in bandwidth, but historically the large increases in memorybandwidth often do little for overall computational performance. Most scientific codes do well tomake optimal use of the CPU caches.

The network provides the means to transfer data between all the nodes. Some classes of problemssolvable by a computer can be solved on a cluster of networked nodes in order to model the functionof a single computer. Any problem that can be broken into a collection of localized problems, whichcan each be efficiently solved on one node, can benefit from a cluster of nodes in a network.

While there are many ways of connecting a network that can be specialized for some classes ofproblems, the most general topology for one connection per node is simply to connect the node to anetwork switch. In this case it is best if the network switch is non-blocking. A non-blocking switchallows every port to communicate to every other port simultaneously, bidirectionally at maximumspeed. Current network technology is fastest per unit cost for gigabit networks. Gigabit networkchips are often incorporated into server-type motherboards which greatly lowers the cost. A gigabitnetwork chip has a peak bidirectional bandwidth of 0.25GB/s.

One can potentially increase the bandwidth between any 2 nodes by giving a node another39http://www.Linuxclustersinstitute.org/Linux-HPC-Revolution/Archive/PDF02/11-Leng_T.pdf40http://www.cpuplanet.com/features/article.php/1587771

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network connection. However, one has to determine if the CPU, memory, and PCI bus havesufficient bandwidth to make this worthwhile. Generically, the CPU and memory are much fasterthan the PCI bus, and thus the PCI bus is the bottleneck. However, the true measured bandwidthcan significantly differ from theoretical estimates, so in practice one should do experiments onvarious hardware setups. For example, Intel P4 motherboards with a 32-bit 33Mhz PCI bus havea peak bandwidth of 133MB/s, while on that bus the gigabit network performance is typically45MB/s independent of the hardware. Older server systems have a 33Mhz 64-bit PCI bus whichgives a peak bandwidth of 264MB/s, while peak gigabit performance is about 90MB/s. NewerMyrinet and 10Gigabit technologies can reach near the theoretical limits of a 66Mhz 64-bit busoperating at a bidirectional total of 440MB/s.

For motherboards that have PCI-X technology, which is a standard 64-bit bus at 133Mhz, thepeak bandwidth is about 1GB/s. Thus raw estimates would conclude that the PCI-X bus hassufficient capability to handle up to 4 gigabit network chips. The actual measured bandwidth ofa gigabit network chip on the PCI-X bus is not measurably different from the 64-bit PCI bus at33Mhz at 90MB/s. Actual hardware never operates at peak bandwidth for a variety of reasons,such as network drivers and true hardware operating conditions.

As described in Section B.1, we did consider advanced drivers that eliminate the overheadassociated with the CPU interaction and OS. The GAMMA based driver can achieve a much lowerlatency and a bit more peak bandwidth than default Linux drivers. However, we found this optionto be unreliable and impractical due to several limitations of the current technology and scarcesupport for existing hardware.

A gigabit switch connects all the nodes in a cluster. A network switch should be non-blockingfor maximum performance. A non-blocking switch never limits the traffic between each node byprocessing all routing at sufficiently high rate so data is never stalled. Other features include hard-ware support for jumbo-frames. The typical mean transfer unit (MTU), or smallest message size,is no more than 1500 bytes. Jumbo-frames support typically allows up to 9000 bytes. This valueis the maximum at which the standard TCP 32-bit CRC error correction check loses effectivenessagainst data loss. However, this is only useful for very long message size data transfers not typicalof all computing done on clusters. The BH cluster uses an HP 5308xl non-blocking switch andobtains up to 9.6GB/s bandwidth on the so-called “backplane”. This describes the connectionsbehind all the ports. Each port input can go to 1 port output, and each pair of input/outputsuses 0.125GB/s of bandwidth. Thus, theoretically the switch could handle up to 76 ports beforeperformance is inhibited. Currently the BH cluster uses 17 ports and our collaborating group willsoon require 12 more ports, yet this will still be far from over-utilizing the switch. Unlike computerOSs, network switches are very efficient at network processing due to their proprietary, optimizedOSs.

Older forms of the gigabit network technology exist, such as the 10Mbps Ethernet that theoret-ically has a peak bidirectional bandwidth of 2.5MB/s, but actually only obtains (under any OS ordriver known to the author) about 0.5MB/s. Likewise, a 100Mbps Ethernet network theoretically

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could obtain a bandwidth of 25MB/s, but achieves about a peak of 10MB/s under a broad numberof OSs, hardware, and drivers.

We use hard drives for both immediate simulation diagnostics and storage of old simulations.Each node has the capability to hold 4 HDs on the EIDE channel, and by distributing the HDsamong the many nodes the effective bandwidth is parallelized and the reliability is enhanced com-pared to a single large hard drive system.

The hard drive used in computer clusters can be a weak point in terms of reliability, and can becritical since failure can mean loss of data – generally the most important item in any cluster. Thetypical HD manufacturer specification sheets report a rate of failure using the mean time betweenfailure (MTBF), the number of bits before 1 bit is read or written with errors, the annualized failurerate (AFR), and the “minimum” design lifetime of a drive. At the time of this writing, the typicalhigh performance EIDE HD has a MTBF of 500,000 to 1,000,000 hours, a bit error rate of 1 out of1013 − 1015, an annualized failure rate of about 1%, and a “minimum” design lifetime of 5 years.The MTBF and bit error rate only indicate HD error in reading or writing and assumes the drivedoes not otherwise fail catastrophically, such as from bearing failure (often associated with clickingsounds). The AFR is most representative in that it reports the actual rate of return due to failureby customers.

Over the 4 years of dealing with approximately 30 HDs (in an arbitrary number of computers),we have had about 6 HD failures, where about 2 have been terminal (i.e. not restorable afterattempting a reformat). If this were representative of HD failure rates, this represents a 5% chancethat in a year a HD will fail with loss of data. Our experience of terminal failures is a ∼2% annualfailure rate. The reported failure rate by manufacturers for actual used drives and returned dueto any problem is 1% per year. Perhaps our luck is 2-5X worse than average, but more likely ourproblems are associated with obtaining the latest technology since we are interested in purchasingthe new, larger hard drives. Also, we likely accessed our HDs more frequently than the average user.In summary, we suggest replacing drives that are still alive every 5 years and expect about 2-5%to die each year. Making backups to other HDs or having HDs perform some form of redundancyof data is critical.

We have often considered getting a tape backup system, but have always ended up buying aHD due to their simplicity, general robustness, and low price. Tape backups are generally morerobust than hard drives, but are of course sensitive to a magnetic field. The author has hadpersonal experience with consumer tape backups before CDRs became available, and often hadreliability problems with only 2 year old tape backups that were thought to be in a safe locationand environment.

We do have a CD burner and recently added a DVD burner to store small data sets or homedirectories containing code and other source material. CDs and DVDs are not immune to dataloss problems. Typical CDs or DVDs, especially those one burns using consumer devices, onlyhave a guaranteed lifetime of about 5-10 years, with some perhaps having as little as 2 years of

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life 41,42. CD-R manufacturers performing media longevity studies using industry defined tests andmathematical modeling, claim an theoretical longevity from 70 years to over 200 years. We suggestthat any data on CDs or CDRs, for which one needs to guarantee reliability, be refreshed every 5years.

B.5 Software

In this section we discuss the software required to build, operate, and maintain a cluster usingLinux. We discuss the MPI library and how a fluid code uses MPI to perform parallel calculations.

B.5.1 Choices for OS

Linux43 offers a reliable, stable, and high performance OS for a cluster of computers. Compared toWindows and other OSs that have come and gone (such as IBM’s OS/2), Linux is a programmer’sdream. Linux is highly configurable, with open source44 access to the entire system45. Sincethe author started using Linux in 1994 several versions of Linux have come and gone. TodayRedHat Linux dominates the commercial market, with the free version of RedHat no longer beingsupported except through the Fedora project. Other versions worth mentioning are Mandrake,Knoppix, Debian, SUSE, Gentoo, and Slackware. The author has used all of these versions andhas found RedHat to be the easiest to deal with. RedHat keeps to the simplicity of the oldLinux versions, thus avoiding confusing configuration issues introduced by versions such as SUSE.Essentially SUSE adds all the problems of a Windows-like configuration without the benefits ofWindows’ seamlessness. For example, several parts of SUSE’s configuration program “lock-up”and can corrupt the configuration files. The SUSE configuration files are more complicated thanother distributions, and are generally not human readable. Most problems are due to lack of broadsupport by hardware companies for Linux. Since the demise of a free RedHat version, Fedoraand Mandrake are the biggest contenders that are vying for the non-commercial market, which isapplicable to private university computer clusters. We are currently evaluating whether Fedora isa reasonable migration path from our current installation of RedHat.

Also, there are several “cluster” Linux distributions46 that include the means to install a clusterfrom one master node by simply having the slave nodes boot off a CD or floppy. The master nodedistributes and installs Linux on the entire cluster. Probably the most promising is the Scyld47 dis-tribution that provides a single system image cluster using specially developed tools and libraries.Scyld not only makes installation trivial, but the entire cluster can be managed and operated like

41http://news.independent.co.uk/world/science_technology/story.jsp?story=51348642http://www.pctechguide.com/09cdr-rw.htm43http://www.Linux.org/info/44http://www.gnu.org/45Several commercial products and drivers exist for Linux that are closed source, but these have never been used

for this cluster.46http://lcic.org/distros.html\#cluster47http://www.scyld.com/

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a single stand-alone computer. Scyld also modifies the kernel to achieve optimal performance fora broad range of network hardware. The price is typically $200 per node. Another commercialproduct, Scali48, competes with Scyld without the installation and management benefits, but withthe performance benefits. The performance of Scyld and Scali can be better than of a standardLinux distribution, although generally they require advanced network hardware to achieve higherperformance than standard Linux distributions. Scali and Scyld on Myrinet achieve a drop inlatency from 7-8µ s to 4µ s and an increase in peak bandwidth from 220MB/s to 260MB/s. OS-CAR49,50 is a free cluster distribution that offers the same installation and management benefitsas Scyld without the performance enhancements. Note that none of these products attempts todevelop OS-bypass network drivers for specific hardware, such as the GAMMA project discussedin Section B.2.

B.5.2 OS Installation

For our relatively small 15-17 node BH cluster, we decided to install RedHat and manage thecluster ourselves. We wanted to avoid the complications associated with learning a new OS, specificconfiguration issues, and we had a fear of losing support due to the death of several cluster projects.The performance of our cluster using standard Intel drivers with modified parameters was excellentcompared to any cluster with such cheap technology. Below we describe the steps we followed toinstall the cluster by “hand”, after the cluster hardware installation has been completed. Duringthis discussion we mention places where we learned of problems with the system.

First, download the distribution. If you plan to install from CD, the best advice we can giveis download an ISO51 distribution from the fastest server you can find. The availability of onlineISO images to burn a CD to install Linux is certainly becoming more common today. In the past,most distributions online were set up for direct installation. This is generally an unreliable meansto install Linux due to network slowdowns or outages.

If you plan to do a network installation, download the distribution in any form (e.g. extract theISO images) and place the distribution on a local directory that has an ftp server. All computerscan use ftp to do a network installation from your own ftp server. For RedHat the floppy is usedto boot each node to install in network mode.

Second, install Linux on each node if you are not using a second-generation Beowulf Linuxdistribution that installs the OS automatically. We simply installed Linux on each node manually,fairly simultaneously, by doing a network installation on each node. It takes about 5 minutesto start the installation off the floppy. This requires switching a keyboard and monitor to eachnode, but is fairly efficient. For RedHat we chose a “custom installation” with “install everything”selected so that all libraries and applications are installed. Each BH node took about 30 minutes

48http://www.scali.com/49http://sourceforge.net/projects/oscar/50http://www.openclustergroup.org/51http://www.Linuxiso.org/

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to install Linux.The time it takes for each computer to install Linux is a good first measure of whether the

systems are independently performing properly. We had 3 systems take 45-60 minutes to install. Wefound them to have slower than average hard drive performance by using “hdparm -tT /dev/hda”.In particular the fast computers showed a buffered read bandwidth of 45MB/s, buffered-cache readof ∼ 700MB/s, while the slower computers had a large variation in HD speeds from 15MB/s to35MB/s and no different buffered-cache read performance. This result was based upon repeatedtests52. Swapping hard drives, cables, CPU or chipset heat sinks, etc. lead to no differences. Onlyafter replacing the motherboard did the slow systems reach the normal performance of all the othernodes.

Third, recompile the kernel for a streamlined Linux. We compiled our own kernel to trimdown unnecessary modules and features. The only clear general statement that can be made aboutcompiling a kernel is to avoid unneeded hardware drivers. One should go through every optionavailable and attempt to fully understand whether it is needed or not. Generally new kernelschange only slightly so this is not a difficult process when learned once. We installed Linux 2.4.20,and the kernel config file53 is available online. Linux 2.654 will likely be more desirable than Linux2.4, since it includes many network related optimizations that may reduce network latency andincrease peak bandwidth.

B.5.3 Software Installation and Usage Notes

Here we discuss the installation of MPI, Kerberos, Nagios, the Intel compiler, and group software.We discuss how we installed this on one node and then distributed that single modified state of thefile system to the other nodes.

The installation procedure for MPI is well documented55. We found that compiling with theSMP option “-comm=shared” leads to significant loss in performance, despite the CPUs apparentlybeing fully utilized. Using the standard options results in good performance, even if the CPUs areunder-utilized due to communication time taking longer than CPU time for a given calculation.

Kerberized56 versions of several programs are available. Kerberos offers the same benefits asssh but with higher performance. The default settings for Kerberos connection are encryptedpassword exchange, but unencrypted connections. This results in faster communications. Forbuffered transfers, we find that a typical “scp” copy using OpenSSH, or any other ssh program,uses 80-130% of the CPU(s) on the client side and < 1% on the server side for transfers on gigabit,while only obtaining about 13MB/s. Unencrypted or Kerberized versions of “ftp” or “rcp” giveabout 100MB/s transfer bandwidth and 30% CPU utilization on the server side and 10% on theclient side.

52http://rainman.astro.uiuc.edu/cluster/scripts/hdparm.sh53http://rainman.astro.uiuc.edu/cluster/scripts/jon2.4.20smp54http://www.kernel.org/55http://www-unix.mcs.anl.gov/mpi/mpich/56http://web.mit.edu/kerberos/www/

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We have also found Kerberos is very useful for scripting between nodes. We find that sshcommands can timeout after generating greater than about 10 simultaneous ssh commands froma single node. This means that a scripted run through many computers may or may not reachall computers, an intolerable unreliability. We have never encountered such problems with “rsh”,so we use this for scripting when generating simultaneous commands to the cluster. The “rsh”program is also much faster than ssh for starting MPI jobs.

On a related note, we found “ncftp” on RedHat 8.0 to be slow. We obtain about 4-6MB/s ongigabit and the server side uses 100% CPU (“wu-ftpd” or “in.ftpd”) and 13-20% CPU on the clientside. Using normal ftp gives 90-100MB/s at 100% CPU. For unknown reasons “ncftp” is causingthe CPU to be under-utilized. No such problems were encountered under SUSE.

We found it useful to install a cluster monitoring program called Nagios57. Nagios can be usedfor many advanced purposes and is easily and highly configurable. We use it to monitor each nodein the BH cluster and all our workstations for 1) ping, 2) network services availability (SMTP, ftp,ssh, HTTP), 3) number of users, 4) total processes, 5) too high CPU load, temperature problemsusing the lm sensors58 package (currently version “lm sensors-2.7.0” which requires “i2c-2.7.0”),and 6) too small hard drive space available. It “nags” a specified user or group about problems viaemail, which is why SMTP (sendmail) is run on the cluster. It reports the status of the computersand allows changes to be made through a well designed web interface, which is why HTTP (apache)is run on the cluster. Nagios can be run on any computer that can access the relevant serviceson the remote computers. Other Nagios-like programs are available, such as Big Brother59 andGanglia60.

The compiler of choice for Intel-compatible nodes is “icc” for C and C++ and “ifc” for Fortran.These offer about 2-3X faster performance than “gcc” for C or g77 (f77) for Fortran, for our codes.A free non-commercial license is available online for both the C/C++ compiler61 and the Fortrancompiler62. The standard “gcc” remains necessary for several programs that are incompatible with“icc” due to Intel’s lack of full Linux support. For scientific codes this is often a moot pointbecause they make limited use of complicated libraries. However, several programs and librariescannot be compiled with “icc”. Attempts will either generate compiler errors (that can sometimesbe “fixed”), or the code will simply not function properly (sometimes generating insidious failuresof apparently unrelated origin). We suggest only compiling something with “icc” if the code doesnot use complicated routines or external libraries. Intel’s “icc” packages include a debugger andone can purchase a profiler, but the standard gnu debugger “gdb” and gnu profiler “gprof” can beused with “icc”. The most accurate profile report from gprof on a code compiled with “icc” is withno optimizations, which can alter the profile significantly compared to an optimized compilation.

57http://www.nagios.org/58http://secure.netroedge.com/~lm78/59http://www.bb4.org/60http://ganglia.sourceforge.net/61http://www.intel.com/software/products/compilers/clin/noncom.htm62http://www.intel.com/software/products/compilers/flin/noncom.htm

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When compiling with optimizations, gprof can fail to create a reasonable report, so we only usegprof as a guide.

Compiling MPI, especially the Myrinet version of MPICH over GM, is often problematic with“icc”. GM is Myrinet’s network device driver for accessing Myrinet hardware using MPI functions.Also, compiling Fortran 90 support for MPI with simultaneous support for C/C++ can be trickyand has unresolved problems due to Intel’s incomplete C and Fortran support. A script we used tocompile the standard MPI is available online63.

Group software should be placed in “/usr/local” so the software can be easily copied to anycomputer. Any additional software that is installed from source (in “/usr/src”) should be installedto “/usr/local” and “encapped” with EPKG64 or a similar program. EPKG allows one to easilyinstall and uninstall any installed package without worrying about corrupting the “/usr/local” pathwith old programs. One simply makes a link from your program directory to “/usr/local/encap”and run inside that directory “epkg -i <dirname>”.

Many of the installation issues, such as what services to install, can be worked out on onecomputer and distributed to all other computers. See the script online for a guide to how thiswas done for BH65. All BH nodes have ssh, Kerberized (ftp,telnet,rsh,klogin,kshell,eklogin), whilecurrently the master node (BH00) has also “httpd”, “smtp”, and “kerberos-adm” installed.

We found it useful to create a cluster equivalent to Linux programs “top” and “df” that wouldpost to a website. This required updating the procps version to “procps-3.1.6”, which includes,for example, improved “top” and “ps” programs. The new “top” program has many options, andwe set these options by including a “.toprc”66 file in the home directory of root. These optionsdetermine the final output of running “top” as root. We then run a cron job every 3 minutes forthe cluster “top” and cluster “df” by adding the line to the “/etc/crontab/” list67. We also runa script every hour that shows the user usage of disk space. The scripts for the cluster “top” and“df” are available online68. The list of available top and df websites is available online69. As anaside, we suggest using the C language for complicated scripts. C scripts can be powerful, fast, andincorporated with pure scripting languages like bash. C scripts have the advantage of being easy tounderstand, easy to make, and are easy to read. Bash70, perl, and other scripting languages havemany exceptions to the basic rules of syntax, and this syntax complexity makes it difficult to writemarginally complicated tasks.

Prior to the BH cluster we had been using NFS (network file system) to allow every computerto see the disk of every computer with a large hard drive for general storage. While this mayhave security concerns, the primary practical problem with this is a lack of stability. We found

63http://rainman.astro.uiuc.edu/cluster/scripts/mpich.make.intel.p464http://www.encap.org/epkg/65http://rainman.astro.uiuc.edu/cluster/scripts/clusterinstall.sh66http://rainman.astro.uiuc.edu/cluster/scripts/.toprc67http://rainman.astro.uiuc.edu/cluster/cronstuff/crontab68http://rainman.astro.uiuc.edu/cluster/cronstuff69http://rainman.astro.uiuc.edu/cluster/computers.html70http://www.tldp.org/LDP/abs/html/

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that if any single computer with a storage drive went down or became unavailable, all computersmapping that drive locally would lockup. This destroys the usefulness of NFS by eliminating thedistributed nature of a cluster of workstations. For this reason we do not use NFS, and we simplyassume that each person can copy information using “scp” or “rcp”, and that MPI programs willmanage their diagnostics to write to disk in parallel or collect the data to one drive manually. Thatis, the MPI program should not assume that the disk is one large disk for all nodes. This losesthe transparency of disk storage provided by NFS, but is worth the increased stability and parallelperformance. Lustre71 may provide an NFS-like distributed file system, but with improved stability,performance, and security. NFS is fairly insecure and has stability and performance problems thatLustre seeks to eliminate. Lustre removes the NFS stability problems by avoiding any single pointthat can bring total failure of the distributed file system. Lustre allows up to 10,000 nodes withnearly 90% efficiency of total hard drive bandwidth for parallel file I/O.

We maintain security by periodically updating packages from RedHat. Since the demise ofthe free RedHat, we may attempt to move to Fedora, Mandrake, or some commercial clusterdistribution like Scyld or Scali. We maintain security by using “tcp wrappers”. A “tcp wrapper” isa program that checks all incoming connections for certain services and only allows computers withpre-specified IP addresses to connect. This is controlled by the “hosts.allow” and “hosts.deny” filesin “/etc/”. For the website we control access either through standard entries in “httpd.conf”, or byusing “.htaccess” files and assigned passwords. The former is used to control access to somewhatsensitive web pages by computers outside our group, and the latter is used to control Nagios access.Mistakes can be made in setting up which services are running. One can check what services arerunning on a computer using “nmap <hostname>”.

B.5.4 MPI Implementation in Fluid Codes

MPI is a library specification for message-passing, proposed as a standard by a committee ofvendors, implementers, and users. MPICH72 is a freely available, portable implementation of MPI.Historically, for those who develop their own physics codes, there have been 2 distinct approachesto parallelizing a code. One approach is to use a pre-existing library suite that permits a simplified,but prescribed method of parallelization. The other approach involves directly using an MPI library,such as MPICH, and developing an interface. Interfaces to the MPI libraries, such as DAGH73,74,Kelp75, and CHOMBO76, invite the programmer to ignore the details of MPI and write good physicscode. They often offer very complicated services that only take a handful of functions to use andso are easy to implement, such as parallel adaptive mesh support in CHOMBO.

Problems with such MPI interfaces include: (1) a lack of continued support (DAGH, Kelp), (2)71http://www.Lustre.org/documentation.html72http://www-unix.mcs.anl.gov/mpi/mpich/73http://www.caip.rutgers.edu/~parashar/DAGH/74http://www.cs.utexas.edu/users/dagh/75http://www.cs.ucsd.edu/groups/hpcl/scg/kelp/76http://seesar.lbl.gov/anag/chombo/

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over-generalization and slowness (DAGH), and (3) integration into code is fixed and very dependenton that library. As with any Linux project, support is often only guaranteed when supported by alarge set of corporations or universities. Projects like DAGH and Kelp were only supported by afew universities and later collapsed as funding was not available and key programmers moved on.

We wished to avoid performance problems, being dependent on a high-level library, and stabilityuncertainties. We decided to write our own MPI interfaces for intra-nodal setup, all inter-nodalcommunications, and for parallel file I/O using the MPICH library. We only use about 10 MPIfunctions to do all our parallel processing related tasks. These functions include 1) initializationtype functions: MPI Init, MPI Comm size, MPI Comm rank, MPI Get processor name, which aretrivially used and involve about 5 lines of code; 2) management functions: MPI Barrier, used tofully synchronize all CPUs at 1 point in the code, which is rarely used; 3) Transfer functions:MPI Bcast, MPI Reduce, MPI Allreduce, MPI Wait, MPI Irecv, MPI Isend; and 4) finishing func-tions: MPI Abort, MPI Finalize, which are trivially used at the end of either an aborted or suc-cessful simulation, using about 5 lines of code. The transfer functions involve the most complicatedpart of MPI by 1) setting up the communications between nodes (about 40 lines of code); 2) thecommunication of boundary values shared to nodes, which is about 100 lines of code; and 3) thediagnostics of parallel file output and other collective operations to obtain integrated quantities,which involves about 2000 lines of code.

Clearly the diagnostics are the most difficult and time consuming part of making a code parallelusing MPI, despite our effort to modularize the procedures. Once the modularization is done it istrivial to output new diagnostics by following examples in the code. Several MPI-277 (MPI version2) parallel I/O functions can be used, as we have done. However, support for MPI-2 featureswas non-existent in 1999 when we were developing our code to use MPI. Even in 2002, NCSA’sPlatinum cluster had no support for parallel I/O. In order to use NCSA clusters as soon as possiblewe implemented our own file I/O procedures. Our file I/O routines use a fixed small memorystorage buffer regardless of the number of nodes or number of columns of data per zone. Theseroutines can write to disk on a single master node or write in parallel to each node’s hard drive.The second most complicated MPI coding involves the transfer functions.

Ideal fluid codes operate on a grid of zones with typically hyperbolic equations of motion. Sucha code can be decomposed (“domain decomposition”) into smaller per-CPU grids and the boundaryvalues on each CPU can be exchanged to the neighboring CPUs when necessary. The routine thattransfers the boundary values between nodes is quite trivial and was developed and debugged ina week by the author for a ZEUS-based code in 1999. There are at least 2 methods to improvetransfer efficiency for a simple homogeneous domain decomposition: 1) use non-blocking routines;and 2) use multi-layered boundary value zones. A non-blocking routine allows the CPU to continueprocessing while the network hardware is transferring data to other CPUs. This compared to“standard” blocking routines that halt all involved CPUs until the communication is complete.Blocking routines are a bit more trivial to implement, which is why they are more common. Non-

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blocking routines alone might allow a small gain in efficiency by parallelizing transfers between allnodes when transferring boundary values between all computers each time step. Optimal use ofnon-blocking routines is obtained by using 2 layers of boundary zones per CPU. One boundarylayer is the standard boundary conditions layer, while the next layer is a 2-zone wide ring ofnon-boundary zones just inside the boundary conditions layer. This assumes the stencil size is2-zone each direction which is typical of ZEUS and HARM. The procedure to follow is simple:1) start computations on all zones; 2) exchange boundary values using non-blocking routine; 3)continue calculation on the remaining subset of non-boundary layer zones not including the 2-zone wide ring; 4) once the subset calculation is complete, complete the non-blocking call withan MPI Wait() call; and 5) calculate the 2-zone ring that required the updated boundary values.This procedure continues indefinitely. This procedure allows the CPU to compute while boundaryvalues were being transferred, allowing maximum CPU utilization.

The new time step is computed by first computing the minimum time step required for eachCPU, then using an MPI Allreduce() call. This communicates to every CPU the minimum of allthe CPUs values for the time step. When a algorithmic failure occurs in MPI, its often difficultto track down the error without careful MPI considerations. Failure checks can be organized byhaving every function that could return a 1 on failure and 0 on success, and checking every functioncall’s return value. Upon success the code would continue. Upon failure the code would return a 1for failure. As long as the failure checks are placed before each MPI-type call, then all the CPUswill fail synchronously. Ignoring this issue will leave hung jobs and no clear record of the cause ofthe failure.

B.5.5 Running MPI Jobs

For a large cluster, or for a large number of users for a relatively small cluster, preexisting schedulersand batch processing applications are necessary. The most widely used program to help organizeall the requests of users to perform computations is the OpenPBS78,79 batch system (PBS standsfor portable batch system). This system accepts user requests to run MPI programs on a cluster,runs the program, and can even schedule when the program should run. However, the most widelyused scheduler for determining when someone’s program should run is the Maui80 scheduler. Theseare popular programs, and NCSA uses both PBS and Maui. These programs are not trivial toset up. This author set up these systems on the BH cluster, but we decided that they were toorestrictive for our small group.

For relatively small clusters or for clusters with few people, its easy to rely on the users to checkwhat processors are used and how much hard drive space is available to ultimately decide whatnodes the parallel job will run on. In order to facilitate this process scripts are useful to generateeasy to access lists of this information. As described previously, we have a script that generates a

78http://www.openpbs.org/79http://www-unix.mcs.anl.gov/openpbs/80http://www.supercluster.org/maui/

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webpage showing the “cluster top” and “cluster df”. These show available CPUs, what processesare on each CPU, CPU usage, and hard drive space available on each node. This is useful for bothparallel and single CPU runs.

We leave it up to the user to run MPI programs or single-CPU programs. The standard“mpirun” script that is included with MPICH has several seemingly complicated restrictions onstarting an MPI program. For example, the job must be started on the first node that will performthe calculations. There are some ways around this particular problem, but other problems remain.The author has written a simple script that makes it easier to start a batch job from one node andthen run the job on other nodes. This “mympirun.sh” script will also create a working directory onevery node and copy the binary file to that directory on every node. This is useful in a non-NFS filesystem environment where each hard drive space on each node is invisible to the other nodes. Thescript uses either “ssh” or “rsh” and “scp” or “rcp” to perform the startup tasks. The programs“rsh” and “rcp” are much faster for starting jobs on many nodes. The script is available online81.

B.6 Testing Cluster Reliability and Performance

Benchmarks can provide insight into how different software, hardware, and configurations of each ofthese modify performance. Ultimately one should test the code to be run on a system, but syntheticbenchmarks can be useful for more general performance characteristics. A synthetic benchmarkis simply a test that measures an isolated hardware component or feature. Synthetic benchmarksalso allow one to compare all nodes to each other to verify that each node’s hardware componentsoperate nominally.

B.6.1 Reliability and Performance Issues

Chips on the motherboard are unlikely to fail if cooled properly. However, the heat sinks on theCPUs can be unreliable. The fans can stop working or the thermal contact between the CPUand the heat sink can be poor. One can use an IR (infrared) thermometer (available at, e.g.,Radio Shack) to obtain a precise temperature measurement for all the chips. We obtained an IRthermometer to measure the CPU temperature to be sure the “lm sensors” package was measuringthe temperature accurately since some CPU temperatures were reported as being quite high. Wefound those temperatures to be accurate, and so we decided to remove the sides from the cases.This dropped the CPU temperatures from 50C to 25-40C depending upon the vertical location ofthe computer in the computer room (higher systems are hotter). During this process we found thatthe gigabit chip runs at about 70C, the hottest of all the chips on the motherboard. Tyan discussedthis with Intel and it was determined that this chip is known to run this hot.

An efficient way to find problems in hardware and software is to stress test, or burn-in, thecluster. Burning-in a computer cluster involves stress-testing the cluster to maximum levels of

81http://rainman.astro.uiuc.edu/cluster/scripts/mympirun.sh

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CPU, memory use, hard drive use, and network communication for long periods of time. This testswhether there is sufficient power and cooling available and tests whether any hardware componentsmight easily malfunction. CPUs from AMD and Intel are extremely reliable and generally operateas specified. We test CPU performance using lapack, specfp, stream82, our own codes, and a fewother interesting benchmarks. The Benchmark HQ website83 and sourceforge84 are good resourcesfor Linux benchmarking software.

Despite the reliability of AMD and Intel CPUs, the author’s experience is that motherboardsmade for AMD CPUs (AMD does not make motherboards), and chipsets not made by AMD orIntel for AMD or Intel CPUs, are much less reliable than those chipsets and motherboards designeddirectly by AMD or Intel. There are several computer companies85,86,87 that do not offer anythingexcept Intel products for chipsets, motherboards, and CPUs due to reliability problems with AMDmotherboards and chipsets or non-Intel chipsets for Intel CPUs. Assuring compatibility with theCPU and other motherboard components is a complex task. The CPU has generally been the mostcomplex piece of equipment on a motherboard88, so a motherboard and chipset designed by a CPUmanufacturer is more likely to be reliable and compatible with existing technology. We chose anIntel 2.4Ghz Xeon CPU with an Intel chipset and a Tyan motherboard. Tyan is a well-respectedmotherboard manufacturer for Intel CPUs. We chose a CPU speed that was not overpriced byIntel, so was comparably priced to AMD processors of the same speed. It is possible that over thepast couple years that AMD related chipsets and motherboards have become more robust. It is yetto be seen how reliable the AMD Opteron motherboards and chipsets are.

Memory performance can often be a bottleneck in physics computations by having (1) limitedbandwidth and high latency to/from the CPU, (2) limited bandwidth and high latency to/from thesystem bus, (3) limited bandwidth to/from the interface boards. Typically ECC (error correctioncoding) is used in memory for cluster computers to prevent errors in memory affecting the rest ofthe system. Most server-based motherboards require ECC memory. Generally memory in today’smarket is reliable assuming one chooses a memory module that is a brand-name rather than generic.A special memory feature of server motherboards is dual-channel memory. Dual-channel memoryallows interleaved access to the memory by the 2 CPUs, which allows each CPU in dual-CPUnodes to access the memory at nearly the peak memory bandwidth. Dual-channel memory reducesso-called “memory contention” that can occur when both CPUs access the memory simultaneously.

We have found that specific memory speeds have little impact on the speed of our codes. Thatis, within the range of memory speeds available, the fastest memory seems to be little different fromthe standard memory. We purchased DDR PC2100 memory, where DDR stands for double data

82http://www.cs.virginia.edu/stream/83http://www.benchmarkhq.ru/english.html84http://lbs.sourceforge.net/85http://www.dell.com/86http://www.gateway.com87http://www.simplifiedcomputers.com/88Actually, for the past 2 years 3D graphics chips have been more complicated than even CPUs

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rate, and the memory operates at 2100MB/s peak bandwidth. We use “memtest86”89 to checksingle-CPU memory speed and check for memory errors during a reboot of a system. Memtest86acts as an entire OS that loads like any Linux kernel. One simply adds another entry to the“/etc/grub.conf” or “/etc/lilo.conf” files. “Grub” and “lilo” are programs that run at boot timethat load the primary OS component and pass a few key arguments to the OS. One can also testthe memory using “stream”.

B.6.2 Bandwidth and Latency of Network

Two basic measures of network performance are (1) peak bandwidth for large messages and (2)latency for small messages. These 2 measures characterize a full analysis of the amount of time ittakes to transfer a message of arbitrary size. In comparing various onboard gigabit chips and PCIboards in 64-bit 66Mhz and 133Mhz PCI slots, we realized that the onboard chips operate faster orequal to add-on boards. In sections B.1 and B.2 we gave the performance results for all our testson gigabit and Myrinet. These benchmarks used the “mpptest” program included with MPICHin the “examples/perftest” directory, and an example is available online90. Figure B.5 shows the1-way bandwidth and latency for MPI communications between 2 2.4Ghz BH nodes for a rangeof message sizes. Notice that gigabit only achieves peak bandwidth for message sizes greater thanabout 20kB reaching a peak bandwidth of about 100MB/s. Also, notice that the latency for smallmessage sizes is about 27µ s. These performance figures can be compared to similar tests withmodern Myrinet boards on MPICH over GM91 that achieve a peak bandwidth of about 250MB/sand a latency of about 9µ s. NetPIPE92 can also be used to measure network performance, but itdoes not account for overhead due to MPI.

One can measure the data overhead associated with MPI on the network by comparing the“tcpdump” results for the number of bytes transferred on a device with the known number oftransfers in, say, a fluid code. From a fluid code one can count the total number of bytes transferredas boundary values per time step as N = NvNBCNxNb/vNt, where Nv is the number of namedsingle unit variables transferred, NBC is the number of boundary zones per named variable, Nx

is the typical dimension of the edge between CPUs, Nb/v is the number of bytes per variable,and Nt is the number of unidirectional transfers per time step per CPU to CPU communication.For our ZEUS-based MHD code, we have Nv = 36, NBC = 2, Nb/v = 8, and Nt = 2. Thisgives N = 1152Nx, or about 1kB per zone edge. The typical message passed has Nv = 1 andNt = 1, so gives N = 16Nx. For a typical value of Nx = 128 the typical message size is 2kB pertime step. This is a small message size, and from Figure B.5 we see that the bandwidth is only20MB/s. Often when running on hundreds of CPUs (e.g. 200 = 25× 8) Nx = 64 leads to a morepractical total resolution. In this case the bandwidth on gigabit is only about 10MB/s, 1/10 the

89http://www.memtest86.com/90http://rainman.astro.uiuc.edu/cluster/scripts/mpptest.sh91http://www.myri.com/myrinet/performance/MPICH-GM/index.html92http://www.scl.ameslab.gov/netpipe/

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Figure B.5 Bandwidth (left) and latency (right) for gigabit Ethernet connection on BH cluster.

peak bandwidth. For a 3D simulation no such limits occur since then Nx = N1N2. A practicalresolution is N1 = N2 = 32. In this case the smallest message size is N = 16kB giving 85MB/s ongigabit, near the peak bandwidth. HARM operates by always transferring Nv = 8, so N = 128Nx,and for Nx = 64 we have the smallest message size is N = 8kB for a bandwidth of 60MB/s, whichis reasonably near the peak bandwidth. HARM in 3D will easily reach the peak bandwidth ofgigabit, and consequently require a faster network.

One can also check for whether the communication is completing faster or slower than the CPUprocessing by checking the percent of CPU usage in “top” (assuming MPICH is compiled without“-comm=shared”). For example, our HARM code with 1282 zones per node achieves about 80%CPU efficiency when using 16 CPUs. We advise not running single processes on the nodes runningMPI jobs with lower than 100% efficiency, since we find that the actual speed of the MPI processesslows down nonlinearly in proportion to their CPU usage when the CPU is oversubscribed. Thiscould be due to memory contention. Simply using “nice” to “renice” the other processes to a nicelevel of 19 does not completely remove the cache inefficiency produced by oversubscribed CPUs,but leads to performance more proportional to the CPU usage shown by “top”.

B.6.3 Code Performance

Older NCSA systems, such as the Origin 2000 (now replaced by Copper), included utilities such as“perfex” that gave detailed information about cache use (percent miss), MFLOPs, and other vitalstatistics useful for profiling a code. There are some dead open source projects with “perfex”-like

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utilities93. The current prevailing and actively supported “perfex”-like utility is PAPI94. For 32-bitIntel-compatible CPUs, PAPI requires the kernel be recompiled after a patch is installed.

We assume all our codes are well designed for the cache and follow high performance codingetiquette95,96. We use the well-known performance metric of “zone cycles per second” (ZCPS) formeasuring fluid code performance. This refers to the number of grid elements that are calculatedper real time second on a given system. The code efficiency for a given number of CPUs is definedas % Eff= ZCPS/(NCPU ∗ZCPS1CPU )∗100, where NCPU is the number of CPUs and ZCPS1CPU isthe ZCPS for 1 CPU. In general, testing of a fluid code should be performed both with and withoutdiagnostics that are written to file. This allows one to test both raw processing speed and normalprocessing speed, which allows us to evaluate how the HD is limiting raw performance. Below weonly report the raw processing speed in ZCPS.

We consider the performance of our viscous hydrodynamics (VHD), 2D magnetohydrodynamics(MHD), 3D MHD, and HARM codes on some of the systems available to our group over the years.All these codes can be run in parallel using MPI. The systems tested include the NCSA SGI Origin2000 composed of 256 R10000 250Mhz CPUs networked internally by SGI’s proprietary hardwareconnecting each CPU’s memory, the NCSA Platinum cluster composed of 1024 1Ghz Intel Pentium3 (P3) CPUs on 512 dual-CPU nodes networked with Myrinet, the NCSA Itanium cluster composedof 268 Intel 800Mhz Itanium I CPUs on 134 dual-CPU nodes networked with Myrinet, and ourBH cluster composed of 28 2.4Ghz P4 Xeon CPUs on 14 dual-CPU nodes networked with gigabit(we exclude the 2.0Ghz P4 Xeons from these tests). We expect NCSA’s Tungsten 3.06Ghz P4Xeon cluster to show similar performance to BH, with Tungsten using Myrinet showing improvedefficiency for a large number of processes. We provide tables of the number of ZCPS and efficiencyof some of our codes on each system. We run all codes in double precision. Single precision offerssome speed advantages, but often is insufficient to preserve important mathematical constraints.For example, after the number of time steps needed to study accretion flows, a single-precisionbased solenoidal constraint leads to errors larger than truncation error. All the codes are run withsimilar initial conditions that model the accretion flow near a black hole.

As a baseline for parallel performance comparison, the number of ZCPS for each of our codes fora single CPU run is shown in Table B.1. The HARM code tests shown are from our original HARMcode that had no failure checks, so is faster than our latest version of HARM with many failurechecks. These checks slow our new HARM down by about a factor of 2X. We plan to profile thenew HARM to optimize these failure checks. Notice that for our codes the Intel Itanium I processoris about a factor of 2X slower than the older 1Ghz Intel P3 processor, which was unexpected basedupon synthetic benchmarks. Our collaborating group performing numerical relativity simulationsfound Itanium I CPUs to be about 2X faster than the 1Ghz Intel P3 processor. This demonstrates

93http://www.osc.edu/~troy/lperfex/94http://icl.cs.utk.edu/papi/95http://www.intel.com/cd/ids/developer/asmo-na/eng/microprocessors/ia32/pentium4/optimization/

43896.htm96http://rainman.astro.uiuc.edu/cluster/sgiperf.pdf

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Table B.1. Single CPU performance in ZCPS

CPU VHD 2D MHD 3D MHD HARM

R10000 110 100Alpha 21264 210 204 110 301GHz P3 155 141 70 17800Mhz Itanium I 82 75 30 82.4Ghz P4 Xeon 300 282 144 40

Note. — ZCPS from actual measurements. The resolution perCPU is 642 for 2D and 403 for 3D. Tungsten performance can beestimated by simply multiplying the Xeon performance by ∼ 1.3.

the necessity to profile one’s own code rather than relying on anecdotes or synthetic benchmarks.Notice also that the 4-5 year old Alpha CPUs are nearly as fast as the 2.4Ghz Intel P4 Xeon CPUs.

We briefly mention our Alpha cluster with Myrinet that has only 6 CPUs. For a 4-CPU MPItest of the 2D MHD we find an efficiency of 80%, while for a 6-CPU test we find an efficiency of70%.

When the NCSA Origin 2000 was available, we performed MPI tests using our 2D MHD codewith artificial resistivity and the Alfven -limiter enabled. File writing of diagnostics was disabled.The results of these tests are shown in Table B.2. We found the Origin 2000 to have disk performanceproblems for more than about 64 CPUs, which showed up when turning on file writing of ourdiagnostics. File writing with all diagnostics using 256 CPUs drops the efficiency to 37%, whilewith half the diagnostics drops to 55-65%. Otherwise the performance levels off at about 70%efficiency even up to 256 CPUs, which is excellent for the 2D MHD code.

The tables of 2D & 3D MHD and HARM performance on Platinum and BH show columnsfor 1) the number of cpus; 2) the number of CPUs per node used; 3) the number of nodes used;4) the CPU tile geometry; 5) the number of ZCPS per unit 1000; and 6) the parallel efficiency.Note that the 2D and 3D MHD code’s inner loop stride is the “i” or “x”-direction. Thus, theCPU tile geometry is denoted as Tx × Ty, where Tx is the number of tiles in the x-direction andTy is the number of tiles in the y-direction. Alternatively, HARM’s inner loop stride is the “j” or“y”-direction. For comparison purposes the CPU tile geometry in the table is flipped to show “y× x”. This allows direct table-to-table comparisons.

We performed nonrelativistic MHD simulations using NCSA’s Platinum cluster using the 2Dand 3D MHD codes. The results of the 2D and 3D MHD tests are shown in Table B.3. Noticethat Platinum operates about 20% faster on 1cpu/node than 2cpus/node. Our collaborating groupperforming numerical relativity simulations found a decrease in performance between 30-60% when

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Table B.2. Origin 2000 MPI performance in kZCPS for 2D MHD code

# CPUs Speedup % Eff kZCPS

1 1.0 100 1104 3.7 93 4109 8.7 90 95516 11.2 70 123536 24.2 67 266249 34.1 70 375164 36.5 57 4015121 92.7 77 10201256 173.9 72 19129

Note. — The resolution per CPU is 642

in 2D.

using 2cpus/node compared to 1cpu/node, so they avoid using 2cpus/node. NCSA jobs are chargedper node rather than per CPU, so a 30% performance decrease is reasonable to avoid the factorof 2X charge in CPU-hours. We have underlined those tests that show notably low efficienciescompared to similar runs. No simple reason could be found for these low efficiencies.

See Table B.4 for MPI tests of the 2D MHD code on our BH cluster. All these tests are run onthe 2.4Ghz Xeon nodes only, thus we test up to only 14 nodes (rather than 17 – the total numberof nodes). These simulations are in 2D spherical polar coordinates. We test both 642 and 2562 tilesizes per CPU. Artificial resistivity and the Alfven -limiter are enabled. File writing is disabled.Timing is based upon the wall clock time with nodes otherwise free of processes. The code runsfor a fixed number of time steps (about 1500 for 642 and about 90 for 2562). We have underlinedthose tests that show notably low efficiencies compared to similar runs. No simple reason could befound for these low efficiencies.

See Table B.5 for MPI tests of the 3D MHD code. These simulations use a tile size perCPU of 403. The grid is based upon a 3D Cartesian coordinates with inner and outer Cartesianapproximations to spherical shells. Artificial resistivity and the Alfven -limiter are enabled. Filewriting is disabled. Timing is based upon the wall clock time with nodes otherwise free of processes.The code is run for a fixed number of time steps (about 120).

See Table B.6 for MPI tests of HARM. These simulations are in 2D spherical polar coordinatesin Kerr-Schild coordinates. We test both 642 and 2562 tile sizes per CPU. The timing is basedupon the wall clock time with nodes otherwise free of processes. The code runs for a fixed numberof time steps (about 100 for 642 tile and about 10 for 2562). We use the latest HARM code runninga fiducial simulation as in McKinney and Gammie (2004) with a fixed 3 iterations per zone for

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Table B.3. NCSA Platinum MPI performance in kZCPS for 2D & 3D MHD code

# CPUs cpus/node # nodes geom kZCPS % Eff

2D MHD1 1 1 1× 1 141 1002 2 1 1× 2 218 772 1 2 1× 2 246 873 1 3 1× 3 351 834 2 2 2× 2 190 344 1 4 2× 2 437 776 2 3 2× 3 267 326 1 6 2× 3 589 7032 2 16 4× 8 2146 4832 1 32 4× 8 2800 623D MHD1 1 1 1× 1× 1 70 1002 2 1 1× 1× 2 117 842 1 2 1× 1× 2 137 984 2 2 1× 2× 2 220 794 1 4 1× 2× 2 265 956 2 3 1× 2× 3 308 736 1 6 1× 2× 3 377 908 2 4 2× 2× 2 418 758 1 8 2× 2× 2 514 9216 2 8 2× 2× 4 775 6916 1 16 2× 2× 4 964 8624 2 12 2× 3× 4 985 5924 1 24 2× 3× 4 1212 7232 2 16 2× 4× 4 1321 5932 1 32 2× 4× 4 1582 70

Note. — The resolution per CPU is 642 in 2D and 403 in 3D.Using 2562 gives a bit lower per CPU performance and a bit betterMPI efficiency.

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Table B.4. BH Xeon Cluster MPI performance in kZCPS for ZEUS-based 2D MHD code

# CPUs cpus/node # nodes geom 1kZCPS 1%Eff 2kZCPS 2%Eff

1 1 1 1× 1 282 100 305 1002 2 1 1× 2 511 91 513 842 1 2 1× 2 513 91 572 943 1 3 1× 3 696 82 753 824 2 2 2× 2 720 64 843 704 1 4 2× 2 787 70 950 784 1 4 4× 1 638 57 848 704 1 4 1× 4 650 58 1007 836 2 3 2× 3 872 52 1110 616 1 6 1× 6 1297 77 1428 786 1 6 6× 1 954 56 1195 656 1 6 2× 3 976 57 1218 6612 2 6 2× 6 1587 47 2176 6012 1 12 2× 6 1626 48 2108 5812 1 12 1× 12 2069 61 2375 6512 1 12 12× 1 1990 59 2000 5514 2 7 2× 7 1767 45 2493 5814 1 14 2× 7 1762 45 2297 5414 1 14 7× 2 1951 50 2335 5514 1 14 14× 1 2352 60 2359 5514 1 14 1× 14 2354 60 2680 6328 2 14 4× 7 2245 28 4431 5228 2 14 1× 28 3849 49 5094 6028 2 14 28× 1 4191 53 4187 49

Note. — (1)642 per CPU tile size. (2)2562 per CPU tile size.

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Table B.5. BH Xeon Cluster MPI performance in kZCPS for 3D MHD code

# CPUs cpus/node # nodes geom kZCPS % Eff

1 1 1 1× 1× 1 161 1002 2 1 1× 1× 2 282 882 1 2 1× 1× 2 307 954 2 2 1× 2× 2 528 824 1 4 1× 2× 2 600 936 2 3 1× 2× 3 750 786 1 6 1× 2× 3 854 888 2 4 2× 2× 2 931 728 1 8 2× 2× 2 1117 8712 2 6 2× 2× 3 1317 6812 1 12 2× 2× 3 1323 6816 1 16 2× 2× 4 1673 6516 1 16 1× 4× 4 1603 6216 1 16 1× 1× 16 1784 6916 1 16 16× 1× 1 1609 6624 2 12 2× 3× 4 2247 5828 2 14 2× 2× 7 2740 61

Note. — Per CPU tile size is 403.

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Table B.6. BH Xeon BH Cluster MPI performance in kZCPS for 2D HARM code

# CPUs cpus/node # nodes geom 1kZCPS 1%Eff 2kZCPS 2%Eff

1 1 1 1× 1 17 100 18 1002 2 1 1× 2 33 97 32 892 2 1 2× 1 33 97 33 922 1 2 2× 1 35 100 34 942 1 2 1× 2 35 100 34 943 1 3 3× 1 46 90 49 914 2 2 2× 2 61 90 67 934 1 4 2× 2 61 90 62 864 1 4 1× 4 57 84 62 864 1 4 4× 1 61 90 62 866 2 3 2× 3 91 89 96 896 2 3 3× 2 86 84 97 906 1 6 6× 1 88 86 93 866 1 6 1× 6 86 84 101 946 1 6 3× 2 92 90 100 9312 2 6 2× 6 175 86 152 7012 2 6 6× 2 174 85 192 8912 1 12 6× 2 164 80 184 8512 1 12 12× 1 169 83 185 8612 1 12 1× 12 172 84 184 8514 2 7 2× 7 209 88 169 6714 2 7 7× 2 207 87 219 8714 1 14 7× 2 198 83 191 7614 1 14 2× 7 200 84 168 6714 1 14 1× 14 203 85 205 8114 1 14 14× 1 209 88 213 8528 2 14 7× 4 348 73 430 8528 2 14 4× 7 369 78 377 7528 2 14 1× 28 335 70 427 8528 2 14 28× 1 396 83 408 81

Note. — (1) 642 per CPU tile size. (2) 2562 per CPU tile size. Tile geometry isflipped for direct comparison to the 2D & 3D MHD code tables. Actual tile geometryshown is Ty × Tx, where Tx is the number of CPU tiles in the “x-direction” and Ty isthe number of CPU tiles in the “y-direction”.

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variable inversion, which is typical. Some simulations with many failure corrections use up to 15iterations per zone, reducing per CPU performance by about 50%. The results of this table maydiffer from other results obtained using older versions of HARM. These differences are due to howthe code was configured when tested and a lack of optimizations in the latest version of HARM. Wedisable file writing for these tests. All these tests are run on the 2.4Ghz Xeon nodes (i.e. not the2.0Ghz Xeon nodes). We have underlined those tests that show notably low efficiencies comparedto similar runs with different tile geometries. In these poorly performing cases, the CPU geometryis extended in the inner loop stride direction.

Similar tests performed on the BH cluster using 100Mbps Ethernet show similar performanceto gigabit for a small number (≤ 8) of CPUs for HARM and 3D MHD tests, but 2D MHD testresults for any CPU number & 4 results in poor performance due to the higher latency of Ethernet.

HARM is not functional on the Tungsten cluster due to issues that are not yet understood.Tungsten started operating in production mode with Myrinet on June 23, 2004, about the timethis appendix was written. We expect Tungsten to operate better than BH due to the Myrinetinterface, and we expect Tungsten using gigabit will operate similarly to BH after taking the CPUspeed difference into account. We expect that for a large number of CPUs (say 200 operating on25× 8 CPU tile geometry), Tungsten with Myrinet will achieve an efficiency of about 70%.

We find that for HARM on the BH cluster, using only 1cpu/node gives faster performancethan 2cpus/node by 10-17%. The older Alpha cluster shows no change in performance between1cpu/node and 2cpus/node. Some Platinum results show 2cpu/node performance problems, es-pecially for 2D MHD simulations that show up to 50% performance degradation on 2cpus/nodecompared to 1cpu/node. HARM performs quite similarly on both 1 and 2 cpus/node and on anytile geometries. Our collaborating group performing numerical relativity calculations using theirlatest codes finds an even larger performance drop of 50-70% using 2cpus/node from 1cpu/node.Despite this effect they still plan to purchase dual-CPU nodes due to the cost effectiveness of dual-CPU nodes. They might plausibly modify their code or develop new code to improve dual-CPUefficiency. Our collaborating group finds overall higher MPI efficiencies due to a large number ofcomputations per zone, although they may improve MPI performance by changing from DAGH(an antiquated MPI library) to directly using MPI functions.

Notice also that the CPU tile geometry can significantly impact performance. For a fixedtile size, the number of bytes transferred is smallest for an equal number of tiles per dimension.However, for codes that require many small messages such as the 2D MHD code, extended CPUgeometries can be more efficient by increasing the bandwidth of message passing and making theCPU to do more work per unit of communication. Otherwise, the CPU is under-utilized due tothe system waiting for the messages to complete passing. For example, the 2D MHD code using28 CPUs and 2cpus/node with 4× 7 CPU tile geometry achieves only 40% CPU utilization, whilewith 1 × 28 CPU tile geometry achieves 80% CPU utilization. Assuming the code uses the CPUcache efficiently on the inner loop stride, elongating along the inner loop stride has no benefits. Infact, we find a slightly larger performance for tile geometries elongated along the outer loop stride

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for 28 CPUs with 2cpus/node for the 2D MHD code. However, for 4 CPUs and 1cpus/node a 2× 2geometry is optimal.

Increasing the tile size can also affect performance for codes that operate on small message sizes.By comparing the 642 and 2562 cases in Table B.4, notice that the larger 2562 per CPU tile sizetypically runs faster per CPU and is typically more efficient in MPI due to larger message sizes.

B.7 Beowulf Cluster Summary

We have described the method used to design, build, test, and use a Beowulf cluster from startto finish. The design process involves testing plausible configurations of hardware and software bysetting up small test clusters. This process is critical in verifying a specific configuration will besuccessful. Building a Beowulf cluster involves mounting the nodes on a rack or shelf, consideringcooling and electrical issues, and setting up the physical network connections and physical compo-nents of the network switch. We found that a simple, cheap rack suffices for the number of nodes(15-28) we purchased. Modifying the electric outlets and circuits in the computer room took timeto authorize and complete, and in the meantime we learned that stress-testing a system is key toverifying that the power supplied to the cluster is sufficient. The network has been transparentand working at 100MB/s peak bandwidth and 27µ s latency using standard Intel drivers with somemodified parameters.

We found that synthetic benchmarks tell only a small part of the performance story, and thattesting codes expected to run on the cluster (or similar test cluster) is the only clean way to estimatecluster performance. We generally perform hyperbolic, fluid dynamic simulations of accretionflows around black holes. Our collaborators perform numerical relativistic dynamical space-timesimulations. They find very different results on nearly every system we tested, including the finalBH cluster. For example, they find their code performs 2X faster on Itanium I CPUs than onPlatinum 1Ghz Intel P3 processors, while we find our code performs 2X slower.

We have provided tables of single CPU and parallel performance of all our accretion flow codes,including our VHD, 2D MHD, 3D MHD, and GRMHD (HARM) codes. We have tested these codeson several systems, including the SGI Origin 2000, the NCSA Platinum cluster, the NCSA Itanium,and our BH cluster. We expect NCSA’s latest cluster, Tungsten, to show similar performance toBH. Tungsten using Myrinet should show improved efficiency for a large number of processes.

We found that tile size, tile geometry, and whether one uses 1cpu/node or 2cpus/node cansignificantly affect parallel performance. Our HARM code is least affected by such considerations,with 10X the operations per CPU per zone compared to the 2D MHD code. This results in a muchlonger time between communications, leading naturally to a higher parallel code efficiency.

HARM and the 2D MHD codes perform marginally better at 2562 tile size than at 642 tile size.HARM typically performs best with a larger number of tiles along the outer loop stride, especiallyfor the 2562 tile size per CPU. However, the 2D and 3D MHD codes show no consistent preferencefor tile geometry for their worst case efficiencies. HARM and the 3D MHD codes essentially run

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the same on 1cpu/node and 2cpus/node, while the 2D MHD code clearly operates more efficientlywhen using 1cpu/node.

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Curriculum Vitae

Jonathan C. McKinney

Address

Physics DepartmentUniversity of Illinois at Urbana-Champaign1110 West Green StreetUrbana, IL 61801

Phone: (217) 766-6555fax: (217) 244-7638email: [email protected]: rainman.astro.uiuc.edu/ jon/

Personal

Birth November 22, 1973 in Methuen, MA, USA

Education

2004 Ph.D. in Physics, Theoretical Astrophysics, UIUC

Dissertation: Black Hole Accretion Disk, Jet, and Corona (Prof. Gammie, advisor)

1999 M.S. in Physics, UIUC

1996 B.S. in Physics (Magna Cum Laude), Texas A&M Univ., College Station

Sr. Thesis: 2-D Wavefunction Time Evolution using Wavelets/Fourier transforms investigatingPeriodic Potentials (Prof. Siu Ah Chin, advisor)

Jr. Thesis: Atmospheric Polarization due to Incoherent Light from Sun: A Layer Model of theAtmosphere (Prof. George W. Kattawar, advisor)

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Fellowships, Awards, & Contributions

2004 Institute for Theory and Computation (ITC) fellowship, CfA, Harvard College Observatory

2003 Coauthored NCSA NRAC proposal awarded 300,000 Tungsten SUs, with a significant portionfor my research. PI: Charles F. Gammie

2002 Contributed to NSF ITR Program Award 0205155, MHD Simulations in Full General Rela-tivity . PI: Charles F. Gammie

2001 Contributed to NSF Award 0093091, Theory of Black Hole Accretion Flows . PI: Charles F.Gammie.

2001-2004 NASA GSRP Fellow (S01-GSRP-044), annual proposals, sole author, source of currentfunding

1997, 2000 General Electric Fellow (for scholastic excellence as graduate)

1996-1997 Faculty Achievement Award (for leadership in the College of Science at Texas A&M)

1996 Summer Research Fellowship, University of Illinios at U-C

1994-2003 Golden Key National Honor Award: National Honor Society

Employment and Training

2004- ITC Postdoctoral Fellow, CfA, Harvard College Observatory, Harvard University

2001-2004 NASA Fellow, UIUC

1999-2000 Research Assistant, UIUCComputational and theoretical study of black hole accretion disks.Advisor: Prof. Charles GammieMolecular Clouds, Galactic Dynamics, Accretion Disks

2000 Teaching Assistant, UIUC, Graduate level, The Physics of Compact Objects , Prof. Stuart L.Shapiro

1998 Research Assistant, UIUCGeneral relativistic hydrodynamic processes involving shocks as applied to cosmological sheets.Advisor: Prof. Mike Norman, Sr. res. scientist, NCSA (now UCSD)Numerical methods to model astrophysical fluid dynamical systems

1997-1999 Teaching Assistant, UIUC, ENGR. level: Q.M., E&M, Mech., and Stat. Mech.

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1996-1997 Teaching Assistant, Texas A&M, Graduate level: Stat. Mech.

1997Sum Electrical Engineer Assistant for Turnkey in Plano, TX for Mr. Steve Williams

1996Sum Research Assistant (REU), UIUCSymmetry properties and chaos in electron transport in semiconductor superlattices.Advisor: Prof. David K. Campbell (now Dean Boston University)Nonlinear Dynamics of Electrons in Mesoscopic Nanostructures

1996 Research Assistant, Texas A&M University1-D and 2-D wavefunction time evolution using wavelet/Fourier transform. Periodic reflectionlessquantum waveguides.Advisor: Prof. Siu Ah ChinTheoretical nuclear physics; high-density matter; lattice calculations; Monte Carlo methods

1995Sum Repair and Maintenance Technician for Dallas Semiconductor in Dallas, TX for Mr.David Massey

1995 Research Assistant, Texas A&M UniversityAtmospheric polarization due to incoherent light from Sun interacting with the layers of the sky.Light from Moon and Sun that create the green and blue shock on sunsets and sunrises.Advisor: Prof. George W. KattawarAtmospheric/oceanic optics; radiative transfer w/ elastic and inelastic scattering in the atmosphere-ocean system

1994 Computer Technician and Autocad design for B.L. & P. Engineers in Dallas, TX for ScottBrady

Publications

McKinney, J.C. and Gammie, C.F., A measurement of the hydromagnetic luminosity of a Kerrblack hole , 2004, ApJ, 611 , 977M

Watson, W. D., Wiebe, D. S., McKinney, J. C., and Gammie, C. F., Anisotropy of magnetohydro-dynamic turbulence and the polarized spectra of OH masers , 2004, ApJ, 604 , 707W

Gammie, C.F., Shapiro, S.L., and McKinney, J.C., Black hole spin evolution , 2004, ApJ, 602 ,312G

Gammie, Charles F., McKinney, Jonathan C., and Toth, Gabor, HARM: A numerical scheme forgeneral relativistic magnetohydrodynamics , 2003, ApJ, 589 , 444G

McKinney, J. C. and Gammie, C. F., Numerical models of viscous accretion flows near black holes, 2002, ApJ, 573 , 728M

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Anninos, P. and McKinney, J. Relativistic hydrodynamics of cosmological sheets , 1999, Phys. Rev.D 60 , 064011

K. N. Alekseev, E. H. Cannon, J. C. McKinney, F. V. Kusmartsev, and D. K. Campbell. Symmetry-breaking and chaos in electron transport in semiconductor superlattices , 1998, Physica D. 113 ,129-133

K. N. Alekseev, E. H. Cannon, J. C. McKinney, F. V. Kuzmartsev, and D. K. Campbell. Sponta-neous dc current generation in a resistively shunted semiconductor superlattice driven by a terahertzfield , 1998, Phys. Rev. Lett. 80 , 2669-2672

McKinney, J. C., Alekseev, K. N., Cannon, E. H., and Campbell, D. K., Dissipative chaos andsymmetry-breaking in semiconductor superlattices, 1996, REU Thesis, U. of Illinois, Urbana

Publications in Preparation

McKinney, J.C. and Gammie, C.F., General relativistic MHD simulations of thin disks , 2004, inpreparation

Refereed Journals

2002 - Astrophysical Journal, Astrophysical Journal Letters

Invited Talks

CTA Seminar on Theoretical Astrophysics & General Relativityhttp://www.physics.uiuc.edu/Research/CTA/seminars/

2003Spr Intermediate-Mass Black Holes: Formation Theories & Observational Constraints

2002Spr Efficient Acceleration and Radiation in Poynting Flux Powered GRB Outflows

2002Fal High-Energy Gamma Rays from AGN, GRBs, and Plerions

2001Fal Black Hole accretion in Active Galactic Nuclei (also prelim. exam)

2001Spr Bar-Driven Dark Matter Halo Evolution: A Resolution of the Cusp-Core Controversy

2000Fal Gamma-Ray bursts: Magnetized Collapsars and duration of GRBs

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2000Spr Planet Formation: The Effects of Thermal Energetics on 3-D Hydrodynamic Instabilitiesin Massive Protostellar Disks

1999Fal Discussion of Numerical Methods on the study of AGN: GRMHD

1999Spr Global Magnetohydrodynamical Simulations of Accretion Tori

REU (Research Experience for Undergraduates)http://www.physics.uiuc.edu/education/undergrad/reu/

1996Sum Semiconductor Superlattices

Professional Memberships

1999- American Astronomical Society (AAS)

1999- American Physical Society (APS)

Undergraduate Memberships

1994-1997 Society of Physics students (SPS)

1992-1996 Texas A&M Physics ClubMeets every Tuesday with guest speakersOngoing Projects such as the high school laser showParticipated in all these and weekend shows to the public

1993-1997 Texas A&M Astronomy ClubMeets every Friday NightObserve and photograph different phenomena, such as the Shoemaker-Levy Comet impact onJupiterhttp://www.physics.sfasu.edu/astro/sl9.html

Computational Experience

Beowulf Cluster Principle designer, builder, and manager of a 32 CPU Linux gigabit & Myrinetcluster for testing, development, and up to medium scale simulations. Design and associated pub-lications:http://rainman.astro.uiuc.edu/cluster/

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Digital Demo Room Helped create a web portal for astrophysical simulations: http://ddr.astro.uiuc.edu

Software Written Developed ZEUS-like parallel 3D MHD code from scratchContributed significantly to writing HARM

Parallelized all our groups codes using MPI

Workshops NCSA Microprocessor Performance Tuning, Jan 2002NCSA Linux Clusters Institute Workshop, Oct 2001NCSA MPI Workshop, Mar 2001

Science Applications Expert: Mathematica, Supermongo, MatlabBasic: Maple

Operating Systems Expert: All forms of Linux, DOS, and WindowsBasic: VMS, Solaris, SUN

Programming Expert: C, FORTRAN77 & 90, C++, Visual C++, Bash

References

Prof. Charles F. GammieDept. of Physics, MC-704University of Illinoisat Urbana-Champaign1110 West Green StreetUrbana, IL 61801-3080(217) 333-8646 (office)(217) 244-7638 (fax)[email protected]

Prof. Stuart L. ShapiroDept of Physics, MC-704University of Illinoisat Urbana-Champaign1110 West Green StreetUrbana, IL 61801-3080(217) 333-5427 (office)(217) 333-9783 (lab)[email protected]

Prof. William D. WatsonDept. of Physics, MC-704University of Illinois

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at Urbana-Champaign1110 West Green StreetUrbana, IL 61801-3080(217) 333-7240 (office)[email protected]

Prof. & Dean David K. CampbellBoston UniversityCollege of Engineering44 Cummington StreetBoston, MA 02215(617) 353-2800 (office)(617) 353-5929 (fax)[email protected]

Dr. Peter AnninosUniversity of CaliforniaLawrence LivermoreNational LaboratoryLivermore, CA [email protected]

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