ANGULAR MOMENTUM TRANSPORT BY MAGNETOHYDRODYNAMIC TURBULENCE Gordon Ogilvie University of Cambridge TACHOCLINE DYNAMICS 11.11.04
ANGULAR MOMENTUM TRANSPORTBY MAGNETOHYDRODYNAMIC TURBULENCE
Gordon OgilvieUniversity of Cambridge
TACHOCLINE DYNAMICS11.11.04
INTRODUCTIONSOME TACHOCLINE ISSUES (Tobias 2004)► sources of instability : HD and MHD► nonlinear development► turbulence and turbulent transport : HD and MHD
SOME ACCRETION DISC ISSUES► differential rotation and AM transport► HD and MHD instabilities► turbulence and turbulent transport : HD and MHD
COMPARISONTACHOCLINE► thin
ACCRETION DISC► thin
► differentially rotating► magnetized (probably)► turbulent (probably)► large-scale dynamo?
► differentially rotating► magnetized (probably)► turbulent (probably)► large-scale dynamo?
COMPARISONTACHOCLINE► thin
ACCRETION DISC► thin
► differentially rotating► magnetized (probably)► turbulent (probably)► large-scale dynamo?► highly subsonic
► differentially rotating► magnetized (probably)► turbulent (probably)► large-scale dynamo?► highly supersonic
► strong stable stratification?► weak or no stratification?
COMPARISONTACHOCLINE► thin
ACCRETION DISC► thin
► differentially rotating► magnetized (probably)► turbulent (probably)► large-scale dynamo?► highly subsonic
► differentially rotating► magnetized (probably)► turbulent (probably)► large-scale dynamo?► highly supersonic
► strong stable stratification?► difficult to resolve
► weak or no stratification?► difficult to resolve► difficult to simulate► difficult to simulate
ANGULAR MOMENTUM TRANSPORTGENERAL
► spiral arms / shocks► vortices
SMALL-SCALE FEATURES► waves► turbulence
LARGE-SCALE STRUCTURES
► anisotropic magnetic fields (Maxwell stress)► anisotropic motion (Reynolds stress)
► non-axisymmetric gravitational fields
SHEARING SHEET
► local model of a differentially rotating disc► uniform rotation Ω ez plus uniform shear flow –2Ax ey
► appropriate for studies of thin discs
MAGNETOROTATIONAL INSTABILITYOPTIMAL MODE (‘channel flow’)► layer analysis (incompressible ideal fluid, ρ = μ0 = 1)
► exact nonlinear solution but unstable (Goodman & Xu 1994)
u
b
ENERGY AND ANGULAR MOMENTUMENERGY EQUATION (shearing sheet)
► in either growing instability or saturated turbulence,
► AM transport down the gradient of angular velocity► very natural outcome of MHD instabilities► contrast (e.g.) convective instability or forced turbulence
TURBULENCE MODELSEDDY-VISCOSITY MODEL (von Weizsäcker 1948)
VISCOELASTIC MODEL (O 2001; O & Proctor 2003)
REYNOLDS-MAXWELL STRESS MODELS (Kato; O 2003)
SOME CONTROVERSIES
► nonlinear hydrodynamic shear instability
► ‘viscosity’► ‘alpha viscosity’► AM transport by convection
► baroclinic / Rossby-wave instability
CONTINUOUS SPECTRUMINTRODUCTION► cf. Friedlander & Vishik (1995); Terquem & Papaloizou (1996)► problems with a normal-mode approach in shearing media
● modes may require confining boundaries● entirely absent (ky ≠ 0) in the shearing sheet● do not describe parallel shear flow instability
► continuous spectrum and non-modal localized approaches
● contain many of the most important instabilities● derive sufficient conditions for instability
CONTINUOUS SPECTRUMLINEAR THEORY IN IDEAL MHD
► Lagrangian displacement ξ► arbitrary reference state
CONTINUOUS SPECTRUMBASIC STATE► steady and axisymmetric► cylindrical polar coordinates (s,φ,z)► differential rotation► toroidal magnetic field
SOLUTIONS
CONTINUOUS SPECTRUMASYMPTOTIC LOCALIZED SOLUTIONS► envelope localized near a point (s0,z0)► plane-wave form with many wavefronts► finite frequency and vanishing group velocity► ‘frozen wavepacket’
CONTINUOUS SPECTRUMCASE OF ZERO MAGNETIC FIELD
► Høiland (1941) stability criteria► necessary and sufficient for axisymmetric disturbances
CONTINUOUS SPECTRUMLIMIT OF WEAK MAGNETIC FIELD
► Papaloizou & Szuszkiewicz (1992) stability criteria► necessary but not sufficient for stability
CONTINUOUS SPECTRUMCASE OF ZERO ANGULAR VELOCITY
► necessary and sufficient► Tayler (1973) stability criteria
APPLICATION TO ACCRETION DISCS
► allows an understanding of the nonlinear state?
► appropriate ordering scheme for a thin disc reveals● MRI (unavoidable)● magnetic buoyancy instability (possible)
differential rotation
MRI
APPLICATION TO THE TACHOCLINE► appropriate ordering schemes are unclear (to me)► assume overwhelming stable stratification
APPLICATION TO THE TACHOCLINE
► conclusions change under weaker stratification
► appropriate ordering schemes are unclear (to me)► assume overwhelming stable stratification
● weak B: MRI when
● Ω = 0 : Tayler (m = 1) when
● suppressed at the poles if
● cf. Cally (2003) (but not requiring mode confinement)
● sensitivity to radial gradients; magnetic buoyancy
(NB: no MRI in 2D)
REMARKS
PROPER JUSTIFICATION► prove existence of continuous spectrum► asymptotic treatment of non-modal disturbances► justifies ‘local analysis’ for a restricted class of disturbances
ADVANTAGES► algebraic character of eigenvalues and eigenvectors► strictly local character, independent of BCs► deals easily with complicated 2D basic states
REMARKS
► neglects the role of turbulent stresses in the basic state► misses truly global instabilitiesNOTES OF CAUTION
► neglects diffusion (double / triple) in the perturbations● Acheson (1978); Spruit (1999); Menou et al. (2004)
SUMMARY
► methods for analysing linear instabilities
► angular momentum transport and energy arguments
► MRI optimized for AM transport down the gradient of► differences between HD and MHD systems
► analogies are imperfect but of some value
► methods for understanding and modelling turbulent states
angular velocity but of limited applicability in the Sun
► continuous spectrum contains many of the important ones