Thesis

Arenberg Doctoral School of Science, Engineering & Technology

Faculty of Science

Department of Mathematics

Centre for Plasma Astrophysics

On regular shock refraction in hydro-and magnetohydrodynamics

Peter Delmont

Dissertation presented in partialfulfillment of the requirements forthe degree of Doctorin the Sciences

Leuven, September 2010

On regular shock refraction in hydro-and magnetohydrodynamics

Peter Delmont

Supervisor: Dissertation presented in partialProf. Dr. R. Keppens fulfillment of the requirements forJury: the degree of Doctor

Prof. Dr. Ir. G. Lapenta (Chairman) in the SciencesProf. Dr. S. Poedts (Secretary)Prof. Dr. M. GoossensProf. Dr. W. Van AsscheProf. Dr. Ir. T. BaelmansProf. Dr. Ir. B. Koren (CWI, Amsterdam)

Leuven, September 2010

© Katholieke Universiteit Leuven – Faculty of ScienceCelestĳnenlaan 200 B, B-3001 Leuven (Belgium)

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Wettelĳk depot D/2010/10.705/49ISBN 978-90-8649-356-2

Preface

Four years ago, Tom Van Doorsselaere invited me for his public PhD defense atthe Arenberg castle in Heverlee. He gave a presentation which I didn’t understandat all, but which made me conclude that plasma physics is a mysterious butinteresting discipline. A few months later, I applied for a PhD position myself,with this dissertation as a result. I would like to thank everybody who contributedto it in any way. More explicitly, my gratefulness goes out to the following people.

First of all, I sincerely thank Prof. Rony Keppens. He has always been availablewith good advice and critical suggestions, but at the same time he gave me thefreedom to focus on my own interests. He has shown patience with my specifictalent for creating chaos. I am sure that his guidance has led me to scientificcontributions and insights, which would never have been possible without it.

Also I thank Dr. Tom Van Doorsselaere for being my helpdesk. In the first twoyears of this project, I have often annoyed him with technical computer and physicsproblems. He has been very patient and was (almost) always willing to help. Nextto him, also Dr. Dries Kimpe, Dr. Bart van der Holst, Dr. Zakaria Meliani andDr. Allard Jan van Marle have offered me good technical support.

I have experienced the CPA, and the department of mathematics, as a pleasantworking environment. I appreciate the nice conversations with my former officemates Prof. Andria Rogava, Dr. Emmanuel Chané, Prof. Giga Gogoberidze,Giorgi Dalakishvili, Andrey Divin and Alkis Vlasis. I thank the Alma/CoffeeTeam for the agreeable lunch/coffee breaks. Further, the CPA deserves a wordof gratefulness for the stimulating international atmosphere.

Next to the research, I have enjoyed being a teaching assistant for thecourses "Hogere Wiskunde I", "Hogere Wiskunde II" and "Wiskunde voorbedrĳfseconomen" in the didactic team led by Prof. Johan Quaegebeur. Iappreciate the didactic insights I gained through this experience. It was a pleasureteaching mathematics to the bachelor students in economical engineering.

i

ii Contents

Of course I thank the members of the examination committee for investing timeto read this work and giving useful comments, interesting suggestions and criticalfeedback. I hope they are pleased by the way I have processed their input.

Since the bow should not be bent too often, I like spending time with my friendsin various ways. I think about the many times we went out drinking and the manytimes we shared our thoughts, about the walks and the dinners we shared and thedarts and table games we played.

I thank my family for the mental support along the way.

Finally, Char, you complete this list. I am grateful for the inspiration and energyyou’ve given me during this period by my side. I truly appreciate your way oflooking at life, and I especially love how it balances my own approach.

Contents

Contents ii

1 Introduction 1

1.1 The equations of magnetohydrodynamics . . . . . . . . . . . . . . 1

1.1.1 Plasma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 Fluid mechanics . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.3 Adiabatic equation of state . . . . . . . . . . . . . . . . . . 5

1.1.4 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . 6

1.1.5 Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . 7

1.1.6 Scale independence of the MHD equations . . . . . . . . . . 10

1.2 Shock waves in (M)HD . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.2.1 The Rankine-Hugoniot jump conditions . . . . . . . . . . . 10

1.2.2 Characteristic speeds . . . . . . . . . . . . . . . . . . . . . . 13

1.2.3 Characteristic speeds in HD . . . . . . . . . . . . . . . . . . 15

1.2.4 Characteristic speeds in ideal MHD . . . . . . . . . . . . . 16

1.2.5 Wave steepening: an HD example . . . . . . . . . . . . . . 17

1.2.6 Wave steepening: an MHD example . . . . . . . . . . . . . 18

1.3 The Richtmyer-Meshkov instability . . . . . . . . . . . . . . . . . . 20

1.3.1 Shock tube problems . . . . . . . . . . . . . . . . . . . . . . 23

1.3.2 Shock refraction: a 1D example . . . . . . . . . . . . . . . . 25

iii

iv CONTENTS

1.3.3 Richtmyer-Meshkov instability . . . . . . . . . . . . . . . . 31

1.3.4 Vorticity evolution equation for inviscid compressible MHDflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.4 AMRVAC and the VTK file format . . . . . . . . . . . . . . . . . . . 35

1.4.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.4.2 Adaptive Mesh Refinement . . . . . . . . . . . . . . . . . . 36

1.4.3 The VTK file format . . . . . . . . . . . . . . . . . . . . . . 37

1.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2 An exact Riemann solver for regular shock refraction 41

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.2 Configuration and governing equations . . . . . . . . . . . . . . . . 43

2.2.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.2.2 Stationary MHD equations . . . . . . . . . . . . . . . . . . 44

2.2.3 Planar stationary Rankine-Hugoniot condition . . . . . . . 45

2.3 Riemann Solver based solution strategy . . . . . . . . . . . . . . . 46

2.3.1 Dimensionless representation . . . . . . . . . . . . . . . . . 46

2.3.2 Relations across a contact discontinuity and an expansion fan 48

2.3.3 Relations across a shock . . . . . . . . . . . . . . . . . . . . 51

2.3.4 Shock refraction as a Riemann problem . . . . . . . . . . . 51

2.3.5 Solution inside of an expansion fan . . . . . . . . . . . . . . 52

2.4 Implementation and numerical details . . . . . . . . . . . . . . . . 53

2.4.1 Details on the Newton-Raphson iteration . . . . . . . . . . 53

2.4.2 Details on AMRVAC . . . . . . . . . . . . . . . . . . . . . . 53

2.4.3 Following an interface numerically . . . . . . . . . . . . . . 54

2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.5.1 Fast-Slow example . . . . . . . . . . . . . . . . . . . . . . . 55

2.5.2 Slow-Fast example . . . . . . . . . . . . . . . . . . . . . . . 57

CONTENTS v

2.5.3 Tracing the critical angle for regular shock refraction . . . . 57

2.5.4 Abd-El-Fattah and Hendersons experiment . . . . . . . . . 59

2.5.5 Connecting slow-fast to fast-slow refraction . . . . . . . . . 60

2.5.6 Effect of a perpendicular magnetic field . . . . . . . . . . . 61

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3 Shock Refraction in ideal MHD 67

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . 70

3.3 MHD shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.4 Riemann Solver based solution strategy . . . . . . . . . . . . . . . 74

3.4.1 Dimensionless representation . . . . . . . . . . . . . . . . . 74

3.4.2 Path variables . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.4.3 Tackling the signals . . . . . . . . . . . . . . . . . . . . . . 80

3.5 Demonstration of result . . . . . . . . . . . . . . . . . . . . . . . . 80

3.6 Suppression of the Richtmyer-Meshkov Instability . . . . . . . . . . 82

3.7 Wave configuration transitions . . . . . . . . . . . . . . . . . . . . 83

3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4 Parameter ranges for intermediate shocks 87

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.1.1 Intermediate shocks in MHD . . . . . . . . . . . . . . . . . 87

4.1.2 The Rankine-Hugoniot jump conditions . . . . . . . . . . . 89

4.1.3 MHD shock types: classical 1 − 2 − 3 − 4 classification . . . 90

4.2 Solution to the Rankine-Hugoniot conditions . . . . . . . . . . . . 91

4.3 Physical meaning of Ω . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.4.1 (θ,M)-diagrams at fixed β . . . . . . . . . . . . . . . . . . 96

vi CONTENTS

4.4.2 Equivalence classes introduced by the RH conditions . . . . 99

4.4.3 Positive pressure requirement . . . . . . . . . . . . . . . . . 101

4.4.4 Switch-on shocks and switch-off shocks . . . . . . . . . . . . 104

4.4.5 Parameter ranges for 1 → 3 shocks. . . . . . . . . . . . . . . 106

4.4.6 Parameter ranges for 2 → 3 shocks. . . . . . . . . . . . . . . 109

4.4.7 Parameter ranges for 1 → 4 shocks . . . . . . . . . . . . . . 110

4.4.8 Parameter ranges for 2 → 4 shocks . . . . . . . . . . . . . . 111

4.4.9 Relationship between pre- and post-shock magnetic field . . 112

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5 Nederlandstalige samenvatting 115

6 Conclusions 121

A Structure of AMRVAC 123

B Compilation 125

C USR-file 127

D Stationary planar Rankine-Hugoniot conditions 129

E Relations across a shock 131

F Solving the cubic analytically 135

G Integration across rarefaction waves 141

Bibliography 145

Curriculum Vitae 153

Education and Research History . . . . . . . . . . . . . . . . . . . . . . 153

List of Scientific Contributions . . . . . . . . . . . . . . . . . . . . . . . 153

Chapter 1

Introduction

1.1 The equations of magnetohydrodynamics

1.1.1 Plasma

In 1927, the term plasma was introduced by Langmuir & Jones [52] to refer toionized gases, which are characterized by their two-fluid nature consisting of freelymoving electrons and ions. They were probably inspired by the similarity of thisfluid with the blood plasma, consisting of red and white corpuscles. At first,we define a plasma as a completely ionized gas. Since the freely moving ionsand electrons are charged particles allowing for electric currents, electromagneticforces will play a central role in the description of a plasma, which is thereforefundamentally different from the description of ordinary gases. It leads us toconsider plasma as a fourth phase, next to the well-known phases of solid, liquidand gas.

As shown in figure 1.1, plasmas do exist in the physical world under a widerange of temperatures and number densities. In fact, as Goedbloed & Poedts[32] put it: "Astronomers agree that, ignoring the more speculative nature ofdark matter, matter in the Universe consists more than 90% of plasma. Henceplasma is the ordinary state of matter in the Universe." The environment in whichplasmas occur allows us to classify them into two major categories, namely (i)astrophysical or (ii) laboratory plasmas. At the Centre for Plasma Astrophysics,research is done on, e.g., (relativistic) astrophysical jets, coronal mass ejections(CME’s) or space weather. Figure 1.2 shows an example of those typical plasmaenvironments. The left frame shows a satellite (TRACE) observation of the solaratmosphere (taken from [6]). The picture was taken on November 9th, 2000, one

1

2 INTRODUCTION

Figure 1.1: Plasmas exist in the physical world under a wide range of plasmaparameters. They mostly occur in astrophysical or laboratory environments(Taken from [67]).

day after a solar storm. The right panel shows a laboratory plasma, namelya confined plasma created within the TEXTOR tokamak (taken from [28]). Insuch tokamaks, research on controlled nuclear fusion is performed. The typicalvalues for characteristic parameters in these different environments can differ byorders of magnitude, but the theory of magnetohydrodynamics (MHD) describesthe dynamical behaviour of any plasma, satisfying certain conditions which wedescribe later on.

The Sun is by astrophysicists sometimes refered to as an excellent naturalplasma laboratory. Since its distance to the earth is only one astronomicalunit, it is the star best observed. This human interest in the sun isnothing recent: next to worshiping it, many cultures before ours have madeobservations of the sun and have studied it. Ghezzi & Ruggles [34] discoverede.g. a 2300-year old solar observatory in coastal Peru. Despite this oldinterest in the sun, which mainly focused on large scale phenomena such asgravity, plasma physics is a relatively young discipline. This of course is not

THE EQUATIONS OF MAGNETOHYDRODYNAMICS 3

Figure 1.2: Two typical plasma environments. Left: A TRACE observation of acoronal loop on the solar surface. (taken from [6]) Right: A picture taken in thetokamak TEXTOR in Jülich. (taken from [28])

surprising, since in order to describe the large scale motion of a plasma, oneuses the equations of ideal magnetohydrodynamics. Ideal MHD is in essencea generalization of hydrodynamics (HD), where one allows for non-vanishingmagnetic fields. Therefore MHD is basically founded on (i) fluid mechanics and(ii) electromagnetism.

1.1.2 Fluid mechanics

Fluid mechanics is a discipline with a long history. While in the ancient Greek era,Archimedes mainly focused on equilibria, medieval Arab physicists like Al-Bırunıand Al-Khazini augmented this work with dynamical aspects. Fluid mechanicsreached the west through da Vinci, and finally, it was Bernoulli who introduceda mathematical description in his Hydrodynamica (1738). This led Euler topublish his "Principes generaux du mouvement des fluides" in 1757, where heintroduced one of the first published systems of partial differential equations,nowadays referred to as the Euler equations. The Euler equations in their originalform describe conservation of mass and momentum, and are therefore not fullydetermined. Indeed, there are only two equations in three independent variables.We therefore need a third equation to close the system. This equation is called theequation of state. At first, one added the equation of incompressibility to close thesystem, but later on also more realistic equations of state for gaseous environmentswere introduced. One often exploit the so-called adiabatic equation of state, whichwe will briefly describe in the next section. A derivation of these Euler equationscan be found in any textbook on fluid mechanics. When the adiabatic equation is

4 INTRODUCTION

used as an equation of state, the Euler equations take the following form:

∂U

∂t+∂F

∂x+∂G

∂y+∂H

∂z= 0, (1.1)

where the vector of conserved variables

U =

(

ρ, ρvx, ρvy, ρvz ,ρ(v2

x + v2y + v2

z)

2+

p

γ − 1

)t

(1.2)

and we introduced flux terms :

F =

ρvxρv2x + pρvxvyρvxvz

vx

(

ρ(v2x+v2y+v2z)

2 + γγ−1p

)

, (1.3)

G =

ρvyρvxvyρv2y + pρvyvz

vy

(

ρ(v2x+v2y+v2z)

2 + γγ−1p

)

, (1.4)

and

H =

ρvzρvxvzρvyvzρv2z + p

vz

(

ρ(v2x+v2y+v2z)

2 + γγ−1p

)

. (1.5)

These equations should hold at any time over the complete domain. t representstime and the ratio of specific heat, γ, is a dimensionless equation parameter(explained in a little more detail in the next section), but the meaning of ρ, v

and p are not that straightforward. All those quantities are defined statisticallyby the following reasoning in phase space. Every particle at any time t has anexact position r = (x, y, z) and velocity w = (wx, wy, wz) (in any arbitrarychosen Cartesian frame). In the classical case, it is usual to define the phasespace now as R6, such that it contains a combination of locations and velocities(x, y, z, wx, wy, wz). One then defines a distribution function f : R7 → R+, such


that in any infinitesimal volume element, drdv, in phase space, the probablenumber of particles, N , in that volume element at time t is given by

N(t) ≡∫ ∫

f(r,w, t)drdw. (1.6)

The exact definition of mass density ρ, velocity v and thermal pressure p are nowgiven by averaging only over the velocity dimensions of phase space:

ρ(r, t) ≡ Mp

∫

f(r,w, t)dw, (1.7)

v(r, t) ≡∫

wf(r,w, t)dw∫

f(r,w, t)dw, (1.8)

p(r, t) ≡ Mp

3

∫

(w − v)2f(r,w, t)dw, (1.9)

where Mp is the mass of a single particle. When the gas consists of multipledifferent particles (as is the case in MHD) a generalization has to be made. Forthis we refer to any textbook on fluid mechanics. The Euler equations thus expressthe conservation of mass (i.e. the first line), the conservation of momentum(mx,my,mz) ≡ m ≡ ρv = (ρvx, ρvy, ρvz) (i.e. the next three lines) and the

conservation of energy e ≡ ρ(v2x+v2y+v2z)

2 + pγ−1 (i.e. the fifth line).

1.1.3 Adiabatic equation of state

In 1834, Clapeyron was the first to formulate the ideal gas law, which is validunder relatively mild conditions. The equation, as it reads nowadays, is given by

pV = nRT. (1.10)

Here p denotes the thermal pressure as before, V is the volume of the gas element,n is the amount of gas, R is the universal gas constant and T is the absolutetemperature. It follows that under isothermal conditions pV is constant. Now,taking derivatives of equation (1.10) leads to

pdV + V dp = nRdT. (1.11)

When a gas element expands, it decreases its internal energy and performs work.Now, the adiabatic equation of state concerns, as the name suggests, adiabaticprocesses where no heat is absorbed. The first law of thermodynamics is thusvalid for any volume element of gas, stating that the internal energy of a gas

6 INTRODUCTION

element balances out with the work done on it. The loss of internal energy in thegas element is proportional to the amount of gas and the change in temperature.The proportionality constant cv is called the specific heat for constant volume.The work done by the gas element equals pdV . Therefore the first law ofthermodynamics reads

ncvdT + pdV = dQ = 0. (1.12)

Combining equations (1.10) and (1.11) gives now

cv +R

cv

dV

V+dp

p= 0. (1.13)

The proportional constant cv+Rcv

is by definition nothing else than the ratio ofspecific heats, γ, which we introduced in the previous section.

We are now ready to define the entropy S. Consider a gas element with initialpressure p0 and initial volume V0. Say the gas has expanded and forms a state ofpressure p1 and volume V1. Then we define the entropy S

S ≡ cv lnp1

p0+ (cv +R) ln

V1

V0(1.14)

It is clear that the adiabatic equation of state now reduces as dS = 0.

Anyhow, this law should clearly only hold when the expansion is continuous.This text is also concerned about discontinuous expansion, where the second lawof thermodynamics should hold: dS ≥ 0. This last expression is completelyequivalent with d

dtpργ ≥ 0, and from now on, we will refer to p

ργ as being theentropy S.

1.1.4 Electromagnetism

According to Stringari & Wilson [79], one of the first published experimentsthat proved the relation between electricity and magnetism was performed byRomagnosi in 1802 (and thus not by Oersted, who experimented in the 1820’s).This led to a cascade of strong discoveries throughout the nineteenth century,which in 1865 eventually led to Maxwell’s theory of electromagnetism. Maxwell’s


equations, in their most exploited form, are given by

∇ · E =τ

ǫ0, (1.15)

∇ ·B = 0, (1.16)

∇× E = −∂B∂t, (1.17)

∇× B = µ0ǫ0∂E

∂t+ µ0j. (1.18)

In these equations the constants ǫ0 and µ0 respectively are the permittivity ofvacuum (ǫ0 = 8.85·10−12F ·m−1) and the permeability of vacuum (µ0 = 4π10−7T ·m ·A−1). τ denotes the charge density, E is the electric field, j the current density,and finally B is the magnetic field. The first equation is called Poisson’s law,while the second equation implies that no magnetic monopoles exist. The thirdequation is called Faraday’s law, and the last equation is Ampère’s law, extendedwith the displacement current distribution. Since µ0ǫ0 = c−2

l << 1(s2m−2) (wherecl denotes the light speed), one often drops the displacement current ǫ0

∂E∂t from

Ampère’s law, which then becomes ∇× B = µ0j.

1.1.5 Magnetohydrodynamics

Since a plasma is according to our first definition defined as an ionized gas,magnetic fields play an important role in plasmas and should not be neglected. Inorder to describe the dynamical behavior of such plasmas, one needs to generalizethe Euler equations by including a magnetic field. The mathematical relationbetween the magnetic field and the conserved HD variables is given by the idealMHD equations: after noting that a current is in direct relation with the speedof the electrons, one can eliminate the current j from Ampère’s law. ExtendingEuler’s equations with the Lorentz force and Maxwell’s equations leads to theequations of ideal MHD. For details we refer to Goedbloed & Poedts [32], Goossens[36] or any introductory textbook on MHD. In this dissertation we will consistentlyuse the conservational form of the ideal MHD equations, which in Cartesian vectornotation becomes:

∂U

∂t+∂F

∂x+∂G

∂y+∂H

∂z= 0, (1.19)

8 INTRODUCTION

where the vector of conserved variables

U =

(

ρ, ρvx, ρvy, ρvz,ρ(v2

x + v2y + v2

z)

2+

p

γ − 1+B2x +B2

y +B2z

2µ0, Bx, By, Bz

)t

(1.20)

and we introduced flux terms :

F =

ρvx

ρv2x + p− B2

x

2µ0+

B2y+B2

z

2µ0

ρvxvy − BxBy

µ0

ρvxvz − BxBz

µ0

(ρ(v2x+v2y+v2z)

2 + γγ−1p+

B2y+B2

z

2µ0)vx − (vyBy+vzBz)Bx

µ0

0vyBx − vxByvzBx − vxBz

, (1.21)

G =

ρvyρvxvy − BxBy

µ0

ρv2y + p− B2

y

2µ0+

B2x+B2

z

2µ0

ρvyvz − ByBz

µ0

(ρ(v2x+v2y+v2z)

2 + γγ−1p+

B2x+B2

z

2µ0)vy − (vxBx+vzBz)By

µ0

vxBy − vyBx0

vzBy − vyBz

, (1.22)

and

H =

ρvzρvxvz − BxBz

µ0

ρvyvz − ByBz

µ0

ρv2z + p− B2

z

2µ0+

B2x+B2

y

2µ0

(ρ(v2x+v2y+v2z)

2 + γγ−1p+

B2y+B2

z

2µ0)vz − (vxBx+vyBy)Bz

µ0

vxBz − vzBxvyBz − vzBy

0

. (1.23)

As before, ρ is the mass density, v = (vx, vy, vz) is the velocity, p is the thermalpressure and B = (Bx, By, Bz) denotes the magnetic field. Since these conserved


variables are often used, they have a particular name and notation. As before, the

momentumm = ρv and the total energy density e ≡ ρ(v2x+v2y+v2z)

2 + pγ−1+

B2x+B2

y+B2z

2 .The first line of the MHD equations expresses the conservation of mass. The nextthree lines express the conservation of momentum, taking the Lorentz force intoaccount. The fifth line expresses the conservation of energy, which includes themagnetic energy, and the last three lines finally express Faraday’s law, where theelectric field is eliminated by use of Ohm’s law, which under ideal (non-resistive)MHD conditions reduces to E = −v×B. The ideal MHD system consists now ofeight partial differential equations in the eight primitive variables ρ,v, p,B. Oncethe values of these variables are found, the pre-Maxwell equations are used tosolve for the electric field E and the current j. Note that when the magnetic fieldvanishes, the MHD equations reduce to the Euler equations. Finally also note thatas in the Maxwell equations, the no-monopole equation is still an initial condition.Thus

∇ · B = 0 (1.24)

completes the ideal MHD equations.

Finally note that some conditions need to be satisfied, in order to validate thismacroscopic plasma description, namely

• The interaction with neutrals should be small compared to the interactionof charged particles. Therefore we conclude that typical time scales shouldbe negligible in comparison with the time scales of collisions with neutralsor

long-range Coulomb interactions must dominate over short-rangeinteractions;

• the plasma should be quasi-neutral. When using for the typical length scales

of particle interaction the Debye length, which is defined as λD ≡√

ǫ0 Tne e2

(where ne is the electron density, T is the plasma temperature and e is theelectron charge), we conclude that

the typical length scales should be much larger than the Debye length;

• and finally, the particles should be close enough to each other such thatthere are enough charged particle interactions. Defining a Debye sphere asa sphere with radius λD, this means that

there have to be enough charged particles in a Debye sphere.

10 INTRODUCTION

1.1.6 Scale independence of the MHD equations

Since the MHD equations are scale independent, one can give a dimensionlessrepresentation. We will do so, following Goedbloed & Poedts [32]. They arguethat one needs to make three independent scalar scalings. One can e.g. scale themagnetic field components to a typical value B, the length to a typical length l,and the density to a typical density ρ, i.e.:

ρ ≡ ρ

ρ, (1.25)

B ≡ B

B, (1.26)

l ≡ l

l. (1.27)

In order to get rid of the inconvenient constant µ0, we now scale the velocities

to v ≡ B√µ0ρ

, such that the times are scaled to t ≡ lv . We arbitrarily scale p to

p ≡ ρv2. Finally, from the scaling of length and time, it is now trivial that thethe differential operators ∇ and ∂t are respectively scaled to ∇ ≡ l∇ and t∂t. Inother words: if one defines the dimensionless variables p ≡ p

p and vι ≡ vι

v , theMHD equations can now be re-written in exactly the same form as 1.19, but nowevery symbol ♦ in equations (1.19)-(1.23) should now be replaced by ♦, and allthe µ0’s should be dropped. From now on we will exploit this dimensionless form,but we will consistently drop the bars from the notation, and exploit the newlyintroduced dimensionless variables. Recovering the physical values from the set ofdimensionless variables is now straightforward (taking into account that t = tt).

1.2 Shock waves in (M)HD

1.2.1 The Rankine-Hugoniot jump conditions

Our research deals with shocks in HD and ideal MHD. Let us briefly discussthe concept of characteristic speeds of a (hyperbolic) system of conservation laws(such as HD and ideal MHD), and its connection to so-called weak solutions tothe system. For simplicity, we restrict our analysis to the one-dimensional case.But first let us mention that shock waves are relevant. Figure 1.3 shows twoshocks. The left frame (taken from [73]) shows the HD shock wave which causedthe Sailor Hat Explosion crater on the Hawaiian island Kahoolawe, in a US armyexperiment in 1965. A movie of the experiment can be seen on [74]. The middleand right pictures of figure 1.3 are taken by the SOHO satellite observatory (taken

SHOCK WAVES IN (M)HD 11

Figure 1.3: The left picture shows a hydrodynamical shock wave during a US armyexperiment (taken from [73]). The middle and right picture show a solar MHDshock wave (taken from [6]).

from [6]) and shows an MHD shock wave on the solar surface. The picture istaken on September 24, 1997. Solar shock waves are accompanied by violentevents like solar flares or CME’s, which can affect Earth’s magnetosphere and itsorbiting satellites. These are two examples which are meant to justify (M)HDshock research. However, in this dissertation, we strife for a more mathematicaldescription of shock waves.

Consider a solution U(x, t0) of a hyperbolic system of conservation laws Ut+Fx =0 at a certain time t0. Suppose U is constant at both sides of a continuoustransition layer (as shown in the left frame of figure 1.4), i.e.:

U =

Ul, −δ/2 > xU(x), −δ/2 < x < δ/2Ur, x > δ/2

(1.28)

In the limit of δ → 0, we call the solution a weak solution to the hyperbolic system.Note that at |x| > δ, the conservation law is trivially satisfied. in the limit case forδ → 0, we derive the so-called Rankine-Hugoniot jump conditions (RH) at x = 0,which describe solutions to the integral form of the MHD equations.

Let ui be a i-th conserved variable, and let Fi be the corresponding flux term.Take an infinitesimal volume element [x0, x1]× [t0, t1] in the (x, t)-plane such thatat time t0 the shock is located at x0, and at time t1 the shock is located at x1, asshown in the right frame of figure 1.4. The i-th line of the one dimensional MHDequations now reads as ∂Ui

∂t+ ∂Fi

∂x = 0. Now: denoting x = (t, x), Fi = (Ui, Fi)

and ∇ = (∂t,∇), the integral form of these equations becomes

∫∫

(x,t)

∇Fidx = 0, (1.29)

12 INTRODUCTION

x

U

δ/2−δ/2

Ur

Ul

Figure 1.4: Left: Two constant states connected through a transition layer ofthickness δ. In the limit case δ → 0, a discontinuity appears. Right: Integration onan infinitesimal volume element [x0, x1]× [t0, t1] leads to the RH jump conditions.

According to Gauss’ theorem, this equation reduces as∮

Fi · ndS = 0. (1.30)

Here S is the boundary of the volume element and n is the normal vector onS pointing outwards. It is now straightforward to evaluate the left hand side ofequation (1.30), and find that

(x1 − x0)(Ui,r − Ui,l) − (t1 − t0)(Fi,r − Fi,l) = 0, (1.31)

where the index r refers to the right region and the index l refers to the left region.Rewriting equation (1.31) and collecting all the values of i, we arrive at

[[F]] = s [[U]] , (1.32)

where the jump [[·]] = ·r − ·l, and s ≡ x1−x0

t1−t0 is the shock speed.

For now, it is sufficient to note that in (M)HD, these discontinuities are calledshocks whenever the normal velocity jumps across the discontinuity do not vanish:[[ux]] 6= 0. Otherwise, the discontinuity is called a linear discontinuity. An exampleof such a linear discontinuity, both in HD and ideal MHD, is a contact discontinuity(CD), where the only primitive variable which jumps across it is the density ρ.

Discontinuous solutions, which thus only satisfy the RH conditions, are called weaksolutions.


1.2.2 Characteristic speeds

Let Ut + Fx = 0 be a n-dimensional hyperbolic system of conservation laws. Wecan rewrite this system in quasilinear form as

Ut + FUUx = 0. (1.33)

One can introduce the variable ξ ≡ xt and suppose that U : R × R+ → Rn :

(x, t) 7→ U(x, t) is a homogeneous function, i.e. ∀µ ≥ 0 : U(µx, µt) = U(x, t).these solutions are in fluid mechanics often referred to as self-similar solutions:the vector of conserved variables U(ξ) is a function of ξ only.

we can rewrite this quasilinear form again, now as an eigenvalue problem

FUUξ = ξUξ, (1.34)

from which it is clear that wherever U is not locally constant (Uξ 6= 0), ξ mustbe an eigenvalue of the flux Jacobian FU. Since the system is hyperbolic, all neigenvalues must be real, by definition of hyperbolicity.

We can find such self-similar solutions if at t = 0, U is constant everywhere, exceptat x = 0. This initial setup is called a Riemann problem. A first observation isthat the eigenvalues λi (1 ≤ i ≤ n, and λ1 ≤ λ2 ≤ ...λi ≤ λi+1 ≤ ... ≤ λn) are afunction of space and time. When solving such a Riemann problem, the appearingsignals are located where ξ = x

t = λi(x, t) and that is why we call the eigenvaluesof the Jacobian of the flux matrix with respect to the conserved variables (or anyother set of independent variables) the characteristic speeds. These characteristicspeeds can be used to define characteristic curves σi,κ : si(x, t) = κ in the (x, t)-plane as follows:

∂si∂t

+ λi∂si∂x

= 0. (1.35)

From the implicit function theorem, it now follows that (x0, t0) ∈ σi,κ can belocally expressed as x(t), such that x(t0) = x0 and

dx

dt(t0) = λi. (1.36)

More compactly, these characteristic curves are also called characteristics, andfrom equation (1.36), these characteristics describe the path at which informationtravels at the i-th characteristic speed λi, while from equation (1.35) we knowthat along this i-th characteristic, the value of si is invariant. si is said to be aRiemann invariant.

The characteristics of the Riemann problem are schematically shown in figure 1.5.The signals separate two regions of constant U, and can (for a non-linear system)

14 INTRODUCTION

λi λnλ1t

x

Figure 1.5: The characteristic speeds of a Riemann problem. Up to n realcharacteristic speeds divide the (x, t)-plane in up to n + 1 regions of constantU. The transitions can both be continuous or discontinuous. The continuous caseis represented by multiple lines, while the discontinuous signals are represented bya single line. The signals travel at speed dx

dt = xt . The regions separated by the

signals have constant speed.

be both continuous or discontinuous, where the discontinuous case can be formedform a continuous case by wave steepening, which we will explain in more detail inSection 1.2.5. Consider now two (x, t)-regions in the space-time plane, separatedby a single (the i-th) signal which is traveling at speed s, it can be shown (and isshown by e.g. Lax [53]) that for discontinuous signals we must have

λi(Ur) ≤ s ≤ λi(Ul). (1.37)

When one of the inequalities is a strict inequality, also the other inequality is astrict inequality. Those discontinuous solutions are called shocks. On the otherhand, when λi(Ul) = s = λi(Ur), we say that the solution a linear discontinuity.Across continuous signals, s varies from λi(Ul) to λi(Ur).

Figure 1.6 shows the characteristics in both the discontinuous and the continuouscase. In the left panel, λi(Ul) > λi(Ur). This means that the characteristics


t

x

t

x

Figure 1.6: Left: The i-th characteristics at both sides of a shock. Sincecharacteristics can not cross, wave steepening leads to shock formation. Right:The i-th characteristics at both sides of an expansion fan.

in Ul and Ur should meet and a shock is formed. The exact value of s is nota priori known from λi(Ul) and λi(Ur), but we do know that it is bounded byλi(Ul) > s > λi(Ur). The right panel shows the situation where λi(Ul) < λi(Ur).Therefore the characteristics never meet. The dotted characteristics do not existat t = 0.

1.2.3 Characteristic speeds in HD

Let us now derive the characteristic speeds of the Euler system. We follow theapproach of Courant & Friedrichs [14], which can also be found in textbooks ase.g. Leveque [54], Gombosi [35] or Batchelor [7]. First of all, the one-dimensionalversion of these equations becomes in conservational form Ut + Fx = 0 with

U = (ρ, ρv, ρ v2

2 + 1γ−1p)

t and the flux term becomes

F =

ρvρv2 + p

v(ρv2

2 + γγ−1p)

. (1.38)

To find the eigenvalues of FU, we solve∣

∣

∣

∣

∣

∣

λ− v −ρ 0−v2 λ− 2ρv −1

− v3

2 − γγ−1p− 3

2ρv2 λ− γ

γ−1v

∣

∣

∣

∣

∣

∣

= 0. (1.39)

and find that the HD system is (strictly) hyperbolic, namely λi ∈ v − c, v, v + c,where we introduced the sound speed

c ≡√

γp

ρ. (1.40)

16 INTRODUCTION

Note that c > 0 such that v − c < v < v + c. Since the system is hyperbolic, wecan diagonalize the Jacobian matrix as FU = RΛR−1, where the i-th column ofR is the right eigenvector belonging to λi, and Λ the diagonal matrix with λi onthe (i, i)-th entry. The quasilinear form simplifies thus as Ut +RΛR−1Ux = 0, orby denoting that the p-th column of R−1 (i.e. the p-th left eigenvector of the FU)as lp, this statement is equivalent to (lp · dU)dx=λpdt = 0. Rewriting this in termsof total derivatives dρ, dv and dp, we find the characteristic equations

(dp− ρcdv)dx=(v−c)dt = 0, (1.41)

(dp− c2dρ)dx=vdt = 0, (1.42)

(dp+ ρcdv)dx=(v+c)dt = 0. (1.43)

This is equivalent to st + Λsx = 0, where the vector of the Riemann invariants s

is given by

s = (s1, s2, s3) =

(

v − 2

γ − 1c, S(=

p

ργ), v +

2

γ − 1c

)

. (1.44)

We have just shown that sp is invariant along the curves dx = λpdt.

Denoting ξ = xt , we can even derive relations across continuous signals. We will

do so in chapter 2, but now it suffices to mention the relations holding acrosscontinuous signals: [li · dU]dx=λjdt = 0, when i 6= j. This means that sj isinvariant across the i-th signal, whenever i 6= j.

1.2.4 Characteristic speeds in ideal MHD

The exact same procedure, performed on the one-dimensional version of equation(1.19) can be performed in one-dimensional MHD (see e.g. Courant & Hilbert[16] or Jeffrey & Taniuti [46]). This leads to the conclusion that the characteristicspeeds of 1D ideal MHD are given by vx± vf,x, vx ± va,x, vx ± vs,x, vx where weintroduced the normal and total Alfvén speed, respectively given by

va,x ≡√

B2x

ρ, (1.45)

va ≡√

B2x +B2

y +B2z

ρ, (1.46)


and the fast and slow magnetosonic speeds, respectively given by

vf,x ≡√

1

2

(

c2 + v2a +

√

(c2 + v2a)

2 − 4c2v2a,x

)

, (1.47)

vs,x ≡√

1

2

(

c2 + v2a −

√

(c2 + v2a)

2 − 4c2v2a,x

)

. (1.48)

Here, again c denotes the sound speed as introduced in equation (1.40). TheMHD system is thus a non-strictly hyperbolic system with three different highlyanisotropic wave speeds. Here the non-strictness of the hyperbolicity means thatthe characteristic speeds are not necessarily distinct. Indeed, when for example(p,Bx, By, Bz) = (γ−1, 1, 0, 0), it follows that vf,x = va,x = vs,x. This anisotropymakes the MHD system much richer than the Euler system, where the only wavespeed is the isotropic sound speed.

Also, if the initial setup of a certain problem is planar, in the sense that thestreamlines and magnetic field lines lay in a single plane, one can again performthe same procedure. Now, the spectrum of this system has only 5 characteristicspeeds, namely vx ± vf,x, vx ± vs,x, vx. Note that this is a subset of the full 1DMHD spectrum.

1.2.5 Wave steepening: an HD example

As mentioned above, the HD equations are nonlinear, and thus allow forlarge amplitude waves. In the wave steepening limit, these waves can becomediscontinuous and a shock can be formed. The necessary conditions are statedin equation (1.37) and shown in the left panel of figure 1.6: since crossingcharacteristics are impossible, they lead to shock formation through wavesteepening.

We illustrate this by a 1-dimensional numerical simulation where we connect twoconstant states continuously through a transition region. We assume that γ = 5

3 .As initial conditions, we set

ρ = γ, (1.49)

v = 3/2, (1.50)

p =

1, ∀x < 1,x2, ∀x ∈ [1, 2],4, ∀x > 2.

(1.51)

18 INTRODUCTION

Now, this leads to the following characteristic speeds:

v − c =

1/2 ∀x < 1,3/2− x ∀x ∈ [1, 2],−1/2 ∀x > 2.

(1.52)

v = 3/2, (1.53)

v + c =

5/2 ∀x < 1,3/2 + x ∀x ∈ [1, 2],

7/2 ∀x > 2.(1.54)

The upper left panel of figure 1.7 shows the initial primitive variables and theupper right panel shows the initial characteristic speeds over the domain. Sincethe characteristic speed u− c is decreasing in function of x, it is clear that it leadsto crossing 1-characteristics, and we expect one shock to be formed. The lower leftand right signals show respectively the primitive variables at t = 0.5 and t = 1. Aspredicted, the most left signal, located at x ≈ 1.5 has steepened, and a stationaryshock will form. Furthermore, notice that the sign changing of v − c is equivalentto a transition from Ms ≡ v/c > 1 to Ms < 1. Here Ms is called the sonic Machnumber. Also, at x ≈ 2.2 a linearly degenerate signal is being formed. Since onlythe mass density jumps across it, it is a CD.

1.2.6 Wave steepening: an MHD example

When a magnetic field is applied, the system has 3 anisotropic characteristic speeds,instead of one single isotropic sound speed. Just as in HD, the MHD equations arenonlinear, and thus allow for large amplitude waves. In the wave steepening limit,these waves can become discontinuous and a shock can be formed. The necessaryconditions are stated in equation (1.37) and shown in the left panel of figure 1.6:since crossing characteristics are impossible, they lead to shock formation throughwave steepening.

We illustrate this by a 1-dimensional numerical simulation where we connect twoconstant states continuously through a transition region. Again, we assume thatγ = 5

3 and as initial conditions we set on the x-axis:


Figure 1.7: Upper Left: The initial configuration for the primitive variables for theHD wave steepening AMRVAC simulation. Note that all variables are continuous.Only the pressure is non-constant in the transition region. Upper Right: Theinitial characteristic speeds. Only v− c is decreasing and will thus lead to crossingcharacteristics. Therefore the left signal will steepen and in the limit case a shockwill be formed. Lower Left: The primitive variables at t = 0.5. Lower Right: Theprimitive variables at t = 1.

20 INTRODUCTION

ρ = γ, (1.55)

vx =

q

4+ 2γ+ 2

γ

√4γ+1+1

2 ∀x < 1,

3−2x+

r

2γ+2

γx2+2+2

q

( γ+1

γ )2x4− γ−1

γx2+1

2 ∀x ∈ [1, 2],r

10+ 8γ+2

q

9+ 40γ

+ 16

γ2 −1

2 ∀x > 2.

(1.56)

vy = 0 (1.57)

p =

1, ∀x < 1,x2, ∀x ∈ [1, 2],4, ∀x > 2.

(1.58)

Bx =√γ (1.59)

By =

1, ∀x < 1,x, ∀x ∈ [1, 2],2, ∀x > 2.

(1.60)

(1.61)

In the upper left frame of figure 1.8, the initial distribution of ρ, vx, p and Byare plotted, while in the upper right frame the initial characteristic speeds of theplanar 1D MHD system are plotted. Note that vx − vf,x is decreasing and willlead to crossing characteristics. Therefore we expect that the most left signal willsteepen and a MHD shock will be formed. This will be a fast MHD shock, whichwe will introduce in Section 3.3.

The two lower panels of figure 1.8 show ρ, vx, p and By at respectively t = 1 andt = 2. We indeed notice that the most left signal has become discontinuous.

1.3 The Richtmyer-Meshkov instability

Any instability of a contact discontinuity (CD) separating two different gases,which is caused by the interaction of this CD with a hydrodynamical shock isreferred to as a Richtmyer-Meshkov instability (RMI). We will give a brief literaturereview of the RMI.

THE RICHTMYER-MESHKOV INSTABILITY 21

Figure 1.8: The initial configuration for the HD wave steepening AMRVACsimulation. Upper Left: The initial configuration for the primitive variables forthe MHD wave steepening AMRVAC simulation. Note that all variables arecontinuous. Upper Right: The initial characteristic speeds. Only vx − vf,x isdecreasing and will thus lead to crossing characteristics. Therefore the left signalwill steepen and in the limit case a shock will be formed. Lower left: The AMRVACsnapshot at t = 1. Lower Right: The AMRVAC snapshot at t = 2.

22 INTRODUCTION

Figure 1.9: Snapshots of the density in an AMRVAC RMI simulation. The firstsnapshot shows the initial setup. The second frame shows the vorticity depositionphase, and the latter snapshot shows the vorticity evolution phase. Here theinterface has become RM-unstable.

(M)HD shocks appear, and are extensively studied, in a wide range of astrophysicalenvironments. van Eerten et al.[25] studied the encounter between the relativisticblast wave from a gamma-ray burster and a stellar wind termination shock,De Sterck et al.[23] performed 3D simulations of stationary slow shocks in themagnetosheath for solar wind conditions, Chané et al.[15], Jacobs et al.[44] andmany others study the interaction of the background wind and interplanetaryshocks. However, astrophysical experiments are very difficult to perform, onehas to restrict oneself to observations. Therefore shock experiments are usuallyperformed in a laboratory environment, such as a shock tube. In this dissertation,we will not perform those kind of experiments ourselves. We will analyse theseexperiments mathematically and perform numerical simulations. Our goal is inthe first place to understand shocks from a more mathematical perspective.

Since the numerical experiments we perform are all shock tube experiments,we will first briefly review the literature on shock tubes and shock refraction.Shock refraction happens whenever a shock interacts with an interface. When ahydrodynamical shock refracts at an oblique density discontinuity, typically threesignals will arise and the interface will become unstable. Physical experiments canbe performed in a so-called shock tube, where a shock is generated and the set-up


is such that it will impinge on density discontinuity. Figure 1.11 shows a schematicrepresentation of such a shock tube. We performed numerical experiments withthe numerical code AMRVAC (see Section 1.4) and figure 1.9 shows the typicalbehavior of the shock refraction process .in three density snapshots Shown is thedensity of a numerical simulation for The first snapshot shows the initial setup ofthe experiment. Left and right sides of the domain are modeled open, while theupper and lower boundaries are rigid walls. On the left side of the domain, thereis a genuine right-moving shock, moving towards an inclined density discontinuity,separating two gases at rest. When the shock and the CD touch each other atthe bottom wall, the shock will refract and disturb the CD. Now the first vorticitydeposition phase has started. Vorticity is deposited on the interface, causing itto roll up. The refraction is typically centered around one single point, calledthe triple point, as indicated in the middle snapshot of figure 1.9. When thispoint reaches the upper wall, it will reflect, and after this reflection the vorticityevolution phase begins. Now the vorticity changes sign, and the interface rolls up.

1.3.1 Shock tube problems

In the HD case, the shock will disturb the CD interface, and refract in a reflected(R) and a transmitted (T) signal. The pattern created by these three signalscan vary, and it is far from trivial to predict which pattern will arise under agiven set of gas parameters. An overview of different refraction patterns at a slow-fast CO2/CH4 interface is given in figure 1.10 (based on Nouragliev et al.[64]).Regular shock refraction means that the three signals meet at a single point, calledthe triple point. Since we can solve those problems exactly, we focus on thosepatterns. A more detailed description of irregular shock refractions is given by,e.g., Henderson [41] or Nouragliev et al.[64]. The upper frame of figure 2.9 shows anumerical simulation Schlieren plot of a regular shock refraction pattern, while thelower frame shown an irregular refraction pattern. Research on hydrodynamicalshock refraction has been performed since the 1940’s. The highly analytical workof von Neumann [62] was pioneering in this field. He was for example the firstone to predict critical angles for regular shock refraction. In 1947, Taub [81]suggests a solution strategy in the same highly analytical tradition: for the regularrefraction with reflected shock (RRR) pattern he rewrites the Rankine-Hugoniotshock conditions as a polynomial of degree 12, and solving this polynomial is thendone numerically. Later on, shock tube experiments started to be performed. Aschematic representation of a shock tube is shown in figure 1.11. In such a shocktube, a shock is generated, often by a piston system, and it propagates throughthe tube at a certain sonic (upstream/downstream) Mach number Ms. Here theshocks (upstream/downstream) sonic Mach number is defined as the ratio of theshock speed s, and (upstream/downstream) sound speed. For shock refractionexperiments, inside of the tube, a membrane mimics a CD separating 2 gases atrest. Such simulations where performed by Jahn [45] to study regular refraction

24 INTRODUCTION

→ Increasing α →Very Weak FNR → FPR→ BPR→ RRR→ RREWeak TNR→ TRR→ FNR→ BPR → RRR→ RREStrong TMR→ BPR → RRE

Notation:RRR Regular Refraction with Reflected shockRRE Regular Refraction with reflected Expansion fanBPR Bound Precursor RefractionFPR Free Precursor RefractionFNR Free precursor von Neumann RefractionTRR Twin Regular RefractionTNR Twin von Neumann Refraction

Figure 1.10: An overview of possible shock refraction patterns at a CD and possiblepattern transitions at a slow-fast CO2/CH4 interface. All these patterns andtransitions have been observed in shock tube experiments by Abd-El Fattah &Henderson [1, 2, 3]. We focus on the regular shock refraction patterns RRR andRRE, since these can be solved exactly. The figure is based on Nouragliev et al.[64].

Figure 1.11: A schematic representation of a shock tube. Taken from [90].

of a plane shock, and later on also by, e.g., Henderson [40], and Abd-El Fattahet al.[1, 2, 3] for irregular shock refraction or by, e.g., Haas en Sturtevant [37] andmore recently by Kreeft & Koren [50] for shock-bubble interaction.


S CD

T

M−10

R CD

0 1

t

x

U0

U1 U4

U2 U3

Figure 1.12: Setup of a 1D shock refraction experiment. At time t = M−10 a

Riemann problem occurs.

1.3.2 Shock refraction: a 1D example

Let us first consider the most trivial interaction of a shock and a CD, and thereforethe most trivial shock tube problem. Consider therefore a 1D domain, with astationary shock located at x = 0. At x = 1, we place a genuine left-moving CD(see figure 1.12). Therefore, we have initially three region, i.e.,

• u1 ≡ (ρ1, v1, p1) at x < 0;

• u0 ≡ (ρ0, v0, p0) at x ∈ [0, 1];

• u4 ≡ (ρ4, v4, p4) at x > 1.

In general, one has three degrees of freedom for nondimensionalization (see e.g.Goedbloed & Poedts [32]) but since we will look for a self-similar solution thenondimensionalization is uniquely determined by fixing two variables in a certainregion. We choose to set

• ρ0 = γ;

• p0 = 1.

Now, the speeds are scaled with respect to the downstream sound speed, i.e. c0 ≡√

γp0ρ0

= 1.

26 INTRODUCTION

The CD is completely determined by the density ratio η ≡ ρ4ρ0

across it. Since allspeeds are scaled with respect to the sound speed in region u0, this region moveswith its sonic Mach number towards the shock. Therefore, we now know that

• (ρ0, v0, p0) = (γ,−M0, 1) and

• (ρ4, v4, p4) = (ηγ,−M0, 1).

At t = 0 the upstream state u1 = (ρ1, v1, p1) is related to the downstream state u0

through the Rankine-Hugoniot conditions. At time t = M−10 , the shock and the

CD interact, and a Riemann problem occurs. The shock will refract in a reflected(R) and a transmitted (T) shock, separated by the shocked contact discontinuity.Across any signal, the 1D HD Rankine-Hugoniot (RH) conditions should hold. Wediscuss these conditions in more depth later on, but for now it suffices to knowthat in any stationary shock frame, these conditions reduce to

ρu =(γ + 1)M2

k

(γ − 1)M2k + 2

ρk, (1.62)

vu =(γ − 1)M2

k + 2

(γ + 1)M2k

uk, (1.63)

pu =2γM2

k + 1 − γ

γ + 1pk. (1.64)

Here the indices u and k refer respectively to the unknown and the known state.Therefore we know that the initial upstream state is given by

(ρ1, v1, p1) =

(

(γ + 1)M20

(γ − 1)M20 + 2

ρ0,−(γ − 1)M2

0 + 2

(γ + 1)M0,2γM2

0 + 1 − γ

γ + 1

)

.

We can express the known sonic Mach number at both the reflected R (Ms,l) andtransmitted T (Ms,r) signals respectively as

Ml =v1 − ξl√

γp1ρ1

, (1.65)

Mr =v4 − ξr√

γp4ρ4

. (1.66)

Here the newly introduced variables ξl and ξr respectively denote the speeds atwhich R and T are traveling. When we apply relations (1.63-1.64) both across thereflected and transmitted signal we arrive at the following solution:


p2 =(4γ2 − 4γ)ξlM

30 +

`

(γ2 − 1) + (2γ2 + 2γ)ξl

´

M20 + 8γξlM0 + 2γ + 2

(γ + 1)`

(γ − 2)M20M2

0+ 2

´ , (1.67)

p3 =2ηγ(M0 + ξr)2 − γ + 1

γ + 1, (1.68)

v2 =(1 − γ2)M3

0 − (γ − 1)2ξlM20 + (2 − 2γ + (1 − γ2)ξ2l )M0 + 4(γ − 1)ξl

(γ + 1)((γ − 1)M20

+ (γ + 1)ξlM0 + 2)+ ξl, (1.69)

v3 =(1 − γ)ηM2

0 + (3 − γ)ηξrM0 + 2ηξ2r − 2

η(γ + 1)(M0 + ξr). (1.70)

Both the pressure and the velocity remain invariant across a CD. Therefore,solving

p2 = p3,v2 = v3.

(1.71)

will lead to a solution for ξl and ξr. For γ = 53 , the solution is given by

ξl =ζ

M0

, (1.72)

ξr =16ζ2M2

0 +M20 + 12ζM0 + 8M4

0 ζ2 − 3M4

0 + 4M30 ζ + 8ξ3l M

30 −M6

0 + 3

M0

`

M20

+ 2ζM0 + 3´ `

M20

+ 3 + 4ζM0

´ , (1.73)

where ζ should satisfy the quartic equation Σ4i=0tiζ

i, where the coefficients aregiven by

t4 = 48η + (16η − 64)M2

0 , (1.74)

t3 = (56η − 96)M4

0 + (208η − 288)M2

0 + 120η, (1.75)

t2 = (60η − 52)M6

0 + (288η − 312)M4

0 + (364η − 468)M2

0 + 120η, (1.76)

t1 = (18η − 12)M8

0 + (132η − 108)M6

0 + (312η − 324)M4

0 + (252η − 324)M2

0 + 54η, (1.77)

t0 = −M10

0 + (9η − 12)M8

0 + (60η − 54)M6

0 + (118η − 108)M4

0 + (60η − 81)M2

0 + 9η.(1.78)

For Ms,0 = 2, the solution for ξl and ξr is plotted in figure 1.13. Also the speed ofthe shocked contact, ξc, is plotted. First of all we notice that the shock will slowdown the CD. The lower the density ratio, the bigger the slowdown of the CD.

28 INTRODUCTION

Figure 1.13: Plotted are ξl, ξr and ξc in function of the density ratio η across theunshocked CD. The sonic Mach number is kept constant, Ms,0 = 2. The blackline shows ξl, the red line represents ξr and the blue line shows ξc, the speed atwhich the shocked contact is traveling.

0

1

2

3

4

5

6

7

8

9

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5

x

ρ

-2.5

-2

-1.5

-1

-0.5

0

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5

x

u

0

1

2

3

4

5

6

-3 -2.5 -2 -1.5 -1 -0.5 0 0.5

x

p

Figure 1.14: The exact solution at t = 3/2 to the simple shock tube problem.Left: ρ(x). Center: v(x). Right: p(x). Note that indeed velocity and pressure areinvariant across the contact, as we claimed before and will show later.

From ξl and ξr, solving the RH conditions across R and T respectively, leads tothe values of U2 and U3. For η = 2, we plot the solution at t = 3 in figure1.14. Note that this solution is exact. The analysis above is performed using themathematical software package MAPLE 11.0.


Figure 1.15: The AMRVAC solution to the introductory shock tube problem. Theeffective resolution in this simulation is 30720. Note that indeed this resultsconfirms our exact solution. The exact solution is overplotted, but not visible.The next figure show zoomed in plots of the mass density profile of the exactsolution, and the mass density profile of the numerical solution.

Let us compare this exact result with the result of a numerical simulationperformed by the code AMRVAC, which is described in more detail in the nextsection. Figure 1.15 shows that this numerical solution agrees with the exactsolution.

Note that, according to figure 1.13 for η < 1 no exact solution exists. Thisis because we only looked for RRR refraction patterns, but the output of thenumerical experiment with η = 0.5 shows that no RRR solution is possible in thiscase. Indeed the reflected signal is then continuous. This kind of signal is calledan expansion fan (see figure 1.17).

Comparing the slow-fast case (η < 1) to the fast-slow case (η > 1), we note thefollowing three transitions at η = 1:

• The initial shock was in both cases located at x = 0. Note that in the fast-slow case the transmitted signal is decelerated (and thus located at x > 0)while in the slow-fast case this signal is accelerated (and thus located atx < 0);

• The nature of the shocked contact remains unchanged. By this we meanthat the shocked contact is slow-fast if the initial contact is slow-fast, andthe shocked contact is fast-slow only if the initial contact is fast-slow. Alsonote that the pressure p and velocity v do not jump across the shockedcontact;

30 INTRODUCTION

Figure 1.16: Upper : The mass density profile of the exact solution compared to themass density profile of the numerical solution across the reflected signal. Middle :The mass density profile of the exact solution compared to the mass density profileof the numerical solution across the CD. Lower : The mass density profile of theexact solution compared to the mass density profile of the numerical solution acrossthe reflected signal.


Figure 1.17: The numerical solution to the introductory shock tube problem withη = 0.5, with the reflected signal becoming an expansion fan. The effectiveresolution in this simulation is again 30720.

• In the fast-slow case, the reflected signal is a shock, and the numericalsolution coincides with our exact solution. In the slow-fast case, on theother hand, the reflected signal is not a shock, but a continuous rarefactionfan. Note that the jumps in p, ρ and v all change sign.

These observations are schematically presented in a (x, t)-plot of the involvedsignals in figure 1.18. The left panel presents the fast-slow case, while the rightpanel presents the slow-fast case. R and T stand for respectively the reflectedand the transmitted shock, while S is the initial shock and CD is the contactdiscontinuity. In Chapter 2 of this dissertation we will generalize these facts tothe case where the CD is inclined, and the problem is thus 2D. In this section wehave solved the 1D shock refraction. We will also need to generalize our solutionstrategy. We will not be able to give a closed form solution, but a numericaliteration will converge to the exact solution of the problem. We will also allow forout-of-plane magnetic fields. In Chapter 3, we will allow for magnetic fields whichare aligned with the shock normal, such that 5 signals will arise.

1.3.3 Richtmyer-Meshkov instability

In the later phase of the shock-interface interaction process, the interface canbecome unstable. This instability is called the Richtmyer-Meshkov instability(RMI), and in contrast to the Rayleigh-Taylor instability (RTI), where an interfacebecomes unstable due to an external force (e.g., gravity), the RMI is caused bythe interaction of an interface with a shock. Nonetheless, in 1960, Richtmyer[69] generalized Taylor’s [82] analysis of the RTI to an impulsive acceleration,

32 INTRODUCTION

S CD

TR CDt

x S CD

TR CDt

x

Figure 1.18: The involved signals in 2 shock refraction cases in an (x, t)-plane.Left: Fast-Slow refraction, Right: Slow-Fast refraction.

as appears in the RMI. He predicted theoretically that the interface wouldbe Richtmyer-Meshkov unstable at all wavelengths, and independent of theorientation of the impulse. Moreover, in contrast to most surface instabilities(as the RTI), the RMI was predicted to grow linearly with time, Atwood number(ρ2−ρ1ρ2+ρ1

, where the indices 1 and 2 refer respectively to the post and the pre-shock

regions), and with wave number k. This means that the interface is unstable forall wavelengths, at least whenever the density ratio η ≡ ρ2

ρ1> 1. By use of shock

tubes, Meshkov [58] experimentally validated Richtmyers predictions and the RMIitself became a topic of extensive theoretical, numerical and experimental research.The first phase of the RMI process can be considered the vorticity (ω ≡ ∇ × v)deposition phase. The magnitude of the deposited vorticity can be found throughshock polar analysis, as extensively described by Henderson [40] in 1966. After theshock has passed the interface, the vorticity evolution phase begins. As Hawley &Zabusky [38] described it in 1989: "The vortex layer diffuses laterally as it rotatesglobally. The ends of the vortex layer begin to roll up. (...) The roll up of the lowerinterface region proceeds as the vorticity binds with its mirror image. This is thedominant mechanism for the formation of the wall vortex". This observation is inagreement with various experimental results. Amongst many others, Sturtevant[80] gives a good overview of these experiments in 1987. Figure 1.19 plots theintegrated vorticity of the AMRVAC RMI simulation presented by van der Holst& Keppens [93] and confirms that the instability grows linearly in the first phase ofthe process, while after reflection of the top wall, the vorticity deposition changessign. In what follows, we derive the MHD generalization of the vorticity evolution


Figure 1.19: The total vorticity evolution during the HD shock refraction processdescribed in van der Holst & Keppens [93]. The vorticity deposition is linear inthe vorticity deposition phase. When the triple point reaches the top wall, thevorticity evolution phase starts and the vorticity deposition changes sign, causingthe interface to become RM-unstable.

equation for 2D inviscid compressible flows.

1.3.4 Vorticity evolution equation for inviscid compressible MHDflows

The evolution of the vorticity in ideal MHD can be quantified as follows. Let ustherefore rewrite the momentum equation as

∂

∂tv + (v · ∇)v +

∇pρ

− (∇× B) × B

ρ= 0. (1.79)

Then taking the curl of this expression leads to

∂

∂t(∇× v) + ∇× ((v · ∇)v) + ∇× ∇p

ρ−∇× (∇× B) × B

ρ= 0, (1.80)

or we find

∂

∂tω + ∇× (∇v · v

2− v × (∇× v))

+∇× ∇pρ

−∇× (∇× B) × B

ρ= 0, (1.81)

which simplifies to

∂

∂tω + ∇× (ω × v) + ∇× ∇p

ρ−∇× (∇× B) × B

ρ= 0. (1.82)

34 INTRODUCTION

This can be rewritten as

− v · ∇ω + ω · ∇v − (∇ · v)ω

−∇× ∇pρ

+ ∇× (∇× B) × B

ρ=

∂

∂tω. (1.83)

The latter two terms of the left hand side (LHS) can be simplified. Indeed,

−∇× ∇pρ

= −∇1

ρ×∇p− 1

ρ∇×∇p

=1

ρ2(∇ρ×∇p) , (1.84)

and

∇× (∇× B) × B

ρ= ∇1

ρ× ((∇× B) × B) +

1

ρ(∇× [(∇× B) × B])

= − 1

ρ2(∇ρ× ((∇× B) × B)) +

1

ρ(∇× [(∇× B) × B])

= − 1

ρ2(∇ρ× ((∇× B) × B)) +

1

ρ(∇× [∇ · BB−∇B2

2])

= − 1

ρ2(∇ρ× ((∇× B) × B)) +

1

ρ(∇× (∇ · BB)). (1.85)

Therefore equation (1.83) simplifies as

∂

∂tω = −v · ∇ω + ω · ∇v − (∇ · v)ω

+1

ρ2(∇ρ× (∇p− (∇× B) × B)) +

1

ρ(∇× (∇ ·BB)). (1.86)

On the other hand, note that when using Lagrangian derivatives

D

Dt

ω

ρ=

1

ρ

D

Dtω − ω

ρ2

D

Dtρ

=1

ρ

∂

∂tω +

1

ρ(v · ∇)ω +

ω

ρ(∇ · v). (1.87)

AMRVAC AND THE VTK FILE FORMAT 35

Combining equations (1.87) and (1.83), together with Dω

Dt = ∂ω

∂t + (v · ∇)ω leadsnow to the vorticity evolution equations for inviscid compressible MHD flows:

D

Dt

ω

ρ=

∇ρ× (∇p− (∇× B) × B)

ρ3+

∇× (∇ ·BB)

ρ2+

(ω · ∇)v

ρ. (1.88)

All shock refraction simulations and analyses performed in this dissertation are

2D. The latter term, (ω·∇)vρ , vanishes in that case, and therefore the vorticity

evolution equation in 2D compressible flows reduces as

D

Dt

ω

ρ=

∇ρ× (∇p− (∇× B) × B)

ρ3+

∇× (∇ ·BB)

ρ2. (1.89)

One could thus conclude that (i) the misalignment of pressure and densitygradients, (ii) the misalignment of the Lorentz force and the density gradientand (iii) the non-uniformity of the magnetic field are responsible for vorticitydeposition in 2D flows.

More information on the RMI can be found in review articles as Rupert [72],Zabusky [104] or Bruillette [10].

1.4 AMRVAC and the VTK file format

Since astrophysical "experiments" are only passively observed and laboratoryexperiments are expensive to perform, numerical codes gained popularity toperform computer experiments. Interpreting and analyzing data produced bya numerical experiment is also easier and accurate. All numerical simulationdiscussed in this dissertation have been performed by (MPI)AMRVAC on theK.U.Leuven HPC cluster VIC. Let us briefly discuss the history and philosophy ofthis code.

1.4.1 History

The Versatile Advection Code (VAC) is a parallel any-dimensional multi-physicsopen source code for solving hyperbolic systems of conservation laws such asadvection, HD and MHD problems, initially developed by Tóth [87]. Oftenastrophysical problems involve super-Alfvénic speeds (as explained later), whichgive rise to discontinuous shock solutions to the MHD equations. A numericalMHD tool should thus exploit appropriate shock capturing schemes. VACdiscretizes the physical domain in a regular recti- or curvilinear grid, and includesshock-capturing schemes such as the Flux Corrected Transport (FCT), a TotalVariation Diminishing (TVD) scheme with a Roe-type approximate Riemann

36 INTRODUCTION

solver (see e.g. Roe [70] or Roe & Balsara [71], a Total Variation DiminishingLax-Friedrich scheme (TVDLF) and implicit and semi-implicit schemes (see Tóthet al.[89], Keppens et al.[48]). VAC is written in Fortran, using a pre-processor,which allows to program in any-dimensional manner. This Loop AnnotationSyntax is called the LASY-syntax (as introduced in Tóth [88]). More informationon VAC can be found in [92].

1.4.2 Adaptive Mesh Refinement

The higher the resolution in a numerical simulation, the more accurate the physicsare represented. Especially when discontinuities are involved, not all regions in thecomplete domain require the same resolution in order to accurately represent thephysics. Berger & Colella [9] proposed the Adaptive Mesh Refinement (AMR)algorithm, and this principle is used in AMRVAC, as presented by Keppenset al.[49]. Initially, the type of AMR exploited in AMRVAC was a patch-basedrefinement. Later on, it has been changed to a hybrid-block based refinement (asexplained in van der Holst & Keppens [93]), and still later evolved to the presentform of the octree based block refinement.

Historically, numerical simulations were first performed on a regular uniformequidistant grid. The main advantage of this approach is that inter- andextrapolations are straightforward to perform. A major disadvantage on the otherhand, is that the resolution is equal on the whole domain, while the main physicsof a certain problem is often centered in certain "interesting regions". The ideaof AMR is now to tackle this by increasing the resolution in these interestingregions. One therefore first chooses a number n of AMR levels. On the first levelone employs a regular grid structure. Every coarsest level grid again consists ofregularly distributed cells. On each of these cells, the conserved variables have acertain value at each time step. One can then exploit various refinement algorithmsto decide whether to flag a certain grid for refinement, but the basic idea is thatlarge gradients should lead to refinement. On the second AMR level, one thendivides every grid in every dimension into 2, such that the grid is divided into 2d

sub-grids, as shown in figure 1.20. Here d is the number of dimensions. Exactlythe same refinement happens to the cells in these flagged grids. An interpolationthen allows to calculate the new values of the conserved variables in these newcells. One recursively repeats this procedure n− 1 times.

In this AMR approach inter- and extrapolation are not as straightforward as on aregular grid. In fact, there is a restriction to this refinement: the proper nestingrequirement should not be violated. The proper nesting requirement states thefollowing. Let grids g1 and g2 be neighboring grids, with level respectively l1 andl2. Then l2 can differ from l1 with at most 1. If this is not the case then, thecoarser of grids g1 and g2 is checked for refinement, until the criterion is satisfied.

OUTLINE OF THE THESIS 37

Figure 1.20: The AMR grids generated at a Cartesian domain. Each of these gridsis divided into a number of uniformly regularly structured cells.

The exploited timestep, ∆t, is constant on each level. Moreover, the Courantcondition should be satisfied on all levels:

∆xi∆t

≥ vmax, (1.90)

where ∆xi is the thickness of a cell in dimension i, ∆t is the timestep, and vmaxis the maximum speed reached in the simulation. The Courant condition ensuresthat information travels at most one cell during one timestep, and is needed forstability of explicit time integration schemes.

In Appendix A we describe the structure of AMRVAC in more detail. In AppendixB we give more information on the compilation procedure, and in Appendix C weshow an example of a USR-file, as introduced in Appendices A and B.

1.4.3 The VTK file format

Astrophysical simulations often lead to huge data files, especially in 3-dimensionalsimulations, and a major issue is the data visualization, which is preferablydone by parallel software. ParaView [66] is exactly such an open source parallelvisualization application. ParaView was developed by Kit ware, inc., and is able toprocess a wide range of data types, amongst which VTK [95], which was especiallydeveloped for ParaView. VTK provides several file formats, which are describedin [96]. We decided to use the VTK file format for unstructured grid, called VTU.Therefore, we developed a Fortran subroutine which converts the AMRVAC outputto VTU output. It can also be read by several other visualization applications,among which VisIt [94]. The output is saved in a binary format.

1.5 Outline of the thesis

The following 3 chapters are peer-reviewed publications with minor modificationsapplied.

38 INTRODUCTION

Chapter 2 is based on Delmont et al.[19]. In this chapter we study the classicalproblem of planar shock refraction at an oblique density discontinuity, separatingtwo gases at rest. When the shock impinges on the density discontinuity, itrefracts and in the hydrodynamical case 3 signals arise. Regular refractionmeans that these signals meet at a single point, called the triple point. Afterreflection from the top wall, the contact discontinuity becomes unstable due tolocal Kelvin-Helmholtz instability, causing the contact surface to roll up anddevelop the Richtmyer-Meshkov instability. We present an exact Riemann solverbased solution strategy to describe the initial self similar refraction phase, bywhich we can quantify the vorticity deposited on the contact interface. Weinvestigate the effect of a perpendicular magnetic field and quantify how addition ofa perpendicular magnetic field increases the deposition of vorticity on the contactinterface slightly under constant Atwood Number. We predict wave patterntransitions, in agreement with experiments, von Neumann shock refraction theory,and numerical simulations performed with the grid-adaptive code AMRVAC. Thesesimulations also describe the later phase of the Richtmyer-Meshkov instability.Early results on the purely HD case were published in Delmont & Keppens [18].

Chapter 3 is based on Delmont et al.[20]. In this chapter we generalize ourresults presented in Chapter 2 to planar ideal MHD. As mentioned earlier, inthe hydrodynamical case, 3 signals arise and the interface becomes Richtmyer-Meshkov unstable due to vorticity deposition on the shocked contact, in idealMHD, on the other hand, when the normal component of the magnetic field doesnot vanish, 5 signals will arise. The interface then typically remains stable, sincethe Rankine-Hugoniot jump conditions in ideal MHD do not allow for vorticitydeposition on a contact discontinuity. Again, we present an exact Riemann solverbased solution strategy to describe the initial self similar refraction phase. Usinggrid-adaptive MHD simulations, we show that after reflection from the top wall,the interface remains stable.

Chapter 4, is based on Delmont & Keppens [21] and focuses on the mathematicaldescription of MHD shocks. Due to the existence of three anisotropic characteristicspeeds, the MHD shock classification is much richer than in hydrodynamics,where only the isotropic sound speed enters. One distinguishes slow, intermediateand fast shocks, where intermediate shocks connect a sub-Alfvénic state to asuper-Alfvénic state. We investigate under which parameter regimes the MHDRankine-Hugoniot conditions, which describe discontinuous solutions to the MHDequations, allow for certain types of intermediate MHD shocks. We derive limitingvalues for the upstream and downstream shock parameters for which shocks of agiven shock type can occur. We revisit this classical topic in nonlinear MHDdynamics, augmenting the recent time reversal duality finding by Goedbloed [33]in the usual shock frame parametrization. Our results generalize known limitingvalues for certain shock types, such as switch-on or switch-off shocks.

OUTLINE OF THE THESIS 39

Chapter 5 finally gives a popularized dutch summary of the work presented in theother chapters.

Chapter 2

An exact Riemann solver forregular shock refraction

In this chapter we analyze the process of regular shock refraction at an inclineddensity discontinuity in hydrodynamics. When a shock refracts, three signalswill arrive. We develop an exact Riemann solver to predict the position of thenew-formed signals, and find the values of the conserved variables in the new-formed regions. We derive critical angles for regular shock refraction, which agreewith numerical AMRVAC simulations and shock tube experiments. Our approachconnects slow-fast and fast-slow refraction in a natural way. Finally, we investigatethe effect of an out-of-plane magnetic field.

After reflection from the top wall, the interface becomes unstable due to localKelvin-Helmholtz instability. This instability is called the Richtmyer-Meshkovinstability.

This work was published in Delmont et al.[19].

2.1 Introduction

We study the classical problem of regular refraction of a shock at an obliquedensity discontinuity. Long ago, von Neumann [62] deduced the critical angles forregularity of the refraction, while Taub [81] found relations between the anglesof refraction. Later on, Henderson [40] extended this work to irregular refractionby use of polar diagrams. An example of an early shock tube experiment wasperformed by Jahn [45]. Amongst many others, Abd-El-Fattah & Henderson [2, 3]performed experiments in which also irregular refraction occurred.

41

42 AN EXACT RIEMANN SOLVER FOR REGULAR SHOCK REFRACTION

In 1960, Richtmyer performed the linear stability analysis of the interaction ofshock waves with density discontinuities, and concluded that the shock-acceleratedcontact is unstable to perturbations of all wavelenghts, for fast-slow interfaces(Richtmyer [69]). In hydrodynamics (HD) an interface is said to be fast-slowif η > 1, and slow-fast otherwise, where η is the density ratio across theinterface (figure 2.1). The instability is not a classical fluid instability in thesense that the perturbations grow linearly and not exponentially. The firstexperimental validation was performed by Meshkov [58]. On the other hand,according to linear analysis the interface remains stable for slow-fast interfaces.This misleading result is only valid in the linear phase of the process and nearthe triple point: a wide range of experimental (e.g. Abd-El-Fattah & Henderson[3]) and numerical (e.g. Nouragliev et al.[64]) results show that also in this casethe interface becomes unstable. The growth rates obtained by linear theorycompare poorly to experimentally determined growth rates (Sturtevant [80]). Thegoverning instability is referred to as the Richtmyer-Meshkov instability (RMI)and is nowadays a topic of research in e.g. inertial confinement fusion ( e.g. Oronet al.[65]), astrophysics (e.g. Kifonidis et al.[51]), and it is a common test problemfor numerical codes (e.g. van der Holst & Keppens [93]).

In essence, the RMI is a local Kelvin-Helmholtz instability, due to the deposition ofvorticity on the shocked contact. Hawley & Zabusky [38] formulate an interestingvortex paradigm, which describes the process of shock refraction, using vorticity asa central concept. Later on, Samtaney et al.[77] performed an extensive analysisof the baroclinic circulation generation on shocked slow-fast interfaces.

A wide range of fields where the RMI occurs, involves ionized, quasi-neutralplasmas, where the magnetic field plays an important role. Therefore, morerecently there has been some research done on the RMI in magnetohydrodynamics(MHD). Samtaney [78] proved by numerical simulations, exploiting Adaptive MeshRefinement (AMR), that the RMI is suppressed in planar MHD, when the initialmagnetic field is normal to the shock. Wheatley et al.[98] solved the problem ofplanar shock refraction analytically, making initial guesses for the refracted angles.The basic idea is that ideal MHD does not allow for a jump in tangential velocity, ifthe magnetic field component normal to the contact discontinuity (CD), does notvanish (see e.g. Goedbloed & Poedts [32]). The solution of the Riemann problemin ideal MHD is well-studied in the literature (e.g. Lax [53]), and due to theexistence of three (slow, Alfvén, fast) wave signals instead of one (sound) signal,it is much richer than the HD case. The Riemann problem usually considers theself similar temporal evolution of an initial discontinuity, while we will considerstationary two dimensional conditions. The interaction of small perturbationswith MHD (switch-on and switch-off) shocks was studied both analytically byTodd [84] and numerically by Chu & Taussig [17]. Later on, the evolutionarity ofintermediate shocks, which cross the Alfvén speed, has been studied extensively.Intermediate shocks are unstable under small perturbations, and are thus not

CONFIGURATION AND GOVERNING EQUATIONS 43

evolutionary. Brio & Wu [11] and De Sterck et al.[22] found intermediate shocksin respectively one and two dimensional simulations. The evolutionary conditionbecame controversial and amongst others Myong & Roe [60, 61] argue that theevolutionary condition is not relevant in dissipative MHD. Chao et al.[12] reporteda 2 → 4 intermediate shock observed by Voyager 1 in 1980 and Feng & Wang [29]recognized a 2 → 3 intermediate shock, which was observed by Voyager 2 in 1979.On the other hand, Barmin et al.[8] argue that if the full set of MHD equationsis used to solve planar MHD, a small tangential disturbance on the magnetic fieldvector splits the rotational jump from the compound wave, transforming it into aslow shock. They investigate the reconstruction process of the non-evolutionarycompound wave into evolutionary shocks. Also Falle & Komissarov [26, 27] donot reject the evolutionary condition, and develop a shock capturing scheme forevolutionary solutions in MHD, However, since all the signals in this paper areessentially hydrodynamical, we do not have to worry about evolutionarity for thesetup considered here.

In this chapter, we solve the problem of regular shock refraction exactly,by developing a stationary two-dimensional Riemann solver. Since a normalcomponent of the magnetic field suppresses the RMI, we investigate the effect of aperpendicular magnetic field. The transition from slow-fast to fast-slow refractionis described in a natural way and the method can predict wave pattern transitions.We also perform numerical simulations using the grid-adaptive code AMRVAC(van der Holst & Keppens [93]; Keppens et al.[49]).

In section 2, we formulate the problem and introduce the governing MHDequations. In section 3, we present our Riemann solver based solution strategyand in section 4, more details on the numerical implementation are described.Finally, in section 5, we present our results, including a case study, the predictionof wave pattern transitions, comparison to experiments and numerical simulations,and the effect of a perpendicular magnetic field on the stability of the CD.

2.2 Configuration and governing equations

2.2.1 Problem setup

As indicated in figure 2.1, the hydrodynamical problem of regular shock refractionis parametrized by 5 independent initial parameters: the angle α between theshock normal and the initial density discontinuity CD, the sonic Mach numberM of the impinging shock, the density ratio η across the CD and the ratios ofspecific heat γl and γr on both sides of the CD. The shock refracts in 3 signals: areflected signal (R), a transmitted signal (T) and a shocked contact discontinuity(CD), where we allow both R and T to be expansion fans or shocks. Adding a


αM

(ρ,v = 0, p,B = (0, 0, Bz), γl) (ηρ,v = 0, p,B = (0, 0, Bz), γr)

Figure 2.1: Initial configuration: a shock moves with shock speed M to an inclineddensity discontinuity. Both the upper and lower boundary are solid walls, whilethe left and the right boundaries are open.

perpendicular magnetic field, B, also introduces the plasma-β in the pre-shockregion,

β =2p

B2, (2.1)

which is in our setup a sixth independent parameter. As argued later, the shockthen still refracts in 3 signals (see figure 2.3): a reflected signal (R), a transmittedsignal (T) and a shocked contact discontinuity (CD), where we allow both R andT to be expansion fans or shocks.

2.2.2 Stationary MHD equations

In order to describe the dynamical behavior of ionized, quasi-neutral plasmas, weuse the framework of ideal MHD. We thereby neglect viscosity and resistivity, andsuppose that the length scales of interest are much larger than the Debye lengthand there are enough particles in a Debye sphere (see e.g. Goedbloed & Poedts[32]). Written out in stationary, conservative form and for our planar problem, theMHD equations (1.19) are reduced to

∂

∂xF +

∂

∂yG = 0, (2.2)

where we introduced the flux terms

F =

(

ρvx, ρv2x + p+

B2

2, ρvxvy, vx(

γ

γ − 1p+ ρ

v2x + v2

y

2+B2), vxB, vxγρ

)t

, (2.3)

CONFIGURATION AND GOVERNING EQUATIONS 45

1

unshocked

shocked 2

n = (sinφ,−cosφ)

x

y

v1

v2

φ

atan(λi)

y

x

Figure 2.2: Left: A stationary shock, separating two constant states across aninclined planar discontinuity. Right: The eigenvalues of the matrix A from (3.6)correspond to the refracted signals.

and

G =

(

ρvy, ρvxvy, ρv2y + p+

B2

2, vy(

γ

γ − 1p+ ρ

v2x + v2

y

2+B2), vyB, vyγρ

)t

. (2.4)

The applied magnetic field B = (0, 0, B) is assumed purely perpendicular to theflow and the velocity v = (vx, vy, 0). Note that the ratio of specific heats, γ, isinterpreted as a variable, rather than as an equation parameter, which is doneto treat gases and plasmas in a simple analytical and numerical way. The latterequation of the system expresses that ∇ · (γρv) = 0. Also note that ∇ · B = 0 istrivially satisfied.

2.2.3 Planar stationary Rankine-Hugoniot condition

We allow weak (i.e. discontinuous) solutions of the system, which are solutionsof the integral form of the MHD equations. The shock occurring in the problemsetup, as well as those that later on may appear as R or T signals obey theRankine-Hugoniot conditions. In the case of two dimensional stationary flows(see figure 2.2), where the shock speed s = 0, the Rankine-Hugoniot conditionsfollow from equation (2.2). When considering a thin continuous transition layerin between the two regions, with thickness δ, solutions of the integral form of

equation (2.2) should satisfy limδ→0

∫ 2

1( ∂∂xF + ∂

∂yG)dl = 0. For vanishing thickness

of the transition layer this yields the Rankine-Hugoniot conditions as


φR

u1

u5u2

u3 u4

RR CD T

Figure 2.3: The wave pattern during interaction of the shock with the CD. Theupper and lower boundaries are rigid walls, while the left and right boundaries areopen.

− limδ→0

∫ 2

1

(

1

sinφ

∂

∂lF− 1

cosφ

∂

∂lG

)

dl = 0 (2.5)

m

[[F]] = ξ [[G]] , (2.6)

where ξ = tanφ and φ is the angle between the x-axis and the shock as indicatedin figure 2.2. The symbol [[ ]] indicates the jump across the interface.

2.3 Riemann Solver based solution strategy

2.3.1 Dimensionless representation

In this section we present how we initialize the problem in a dimensionless manner.In the initial refraction phase, the shock will introduce 3 wave signals (R, CD, T),and 2 new constant states develop, as schematically shown in figure 3.3. We choosea representation in which the initial shock speed s equals its sonic Mach numberM . We determine the value of the primitive variables in the post-shock region byapplying the stationary Rankine-Hugoniot conditions in the shock rest frame. Inabsence of a magnetic field, we use a slightly different way to nondimensionalizethe problem. Note ui = (ρi, vx,i, vy,i, ptot,i, Bi, γi), where the index i refers to the

RIEMANN SOLVER BASED SOLUTION STRATEGY 47

value taken in the i−th region (figure 3.3) and the total pressure

ptot = p+B2

2. (2.7)

In the HD case, we define p = 1 and ρ = γl in u1. Now all velocity componentsare scaled with respect to the sound speed in this region between the impingingshock and the initial CD. Since this region is initially at rest, the sonic Machnumber M of the shock equals its shock speed s. When the shock intersectsthe CD, the triple point follows the unshocked contact slip line. It does so at aspeed vtp = (M,M tanα), therefore we will solve the problem in the frame ofthe stationary triple point. We will look for self-similar solutions in this frame,u = u(φ), where all signals are stationary. We now have that vx = vx −M andvy = vy−M tanα, where v refers to this new frame. From now on we will drop thetilde and only use this new frame. We now have u1 = (γl,−M,−M tanα, 1, 0, γl)

t

and u5 = (ηγl,−M,−M tanα, 1, 0, γr)t. The Rankine-Hugoniot relations now

immediately give a unique solution for u2, namely

u2 =

(

(γ2l + γl)M

2

(γl − 1)M2 + 2,− (γl − 1)M2 + 2

(γl + 1)M,−M tanα,

2γlM2 − γl + 1

γl + 1, 0, γl

)t

.

(2.8)

In MHD, we nondimensionalize by defining B = 1 and ρ = γlβ2 , in region 1. Again

all velocity components are scaled with respect to the sound speed in this region.

We now have that u1 =(

γlβ2 ,−M,−M tanα, β+1

2 , 1, γl

)t

and from the definition

of η, u5 =(

ηγlβ2 ,−M,−M tanα, β+1

2 , 1, γr

)t

. The Rankine-Hugoniot relations

now give the following non-trivial solutions for u2:

u2 =

(−γlβM2ω

, ω,−M tanα, p2 +M2

2ω2,−Mω

, γl

)t

, (2.9)

where

p2 =C1ω + C2

C3ω + C4, (2.10)


is the thermal pressure in the post shock region. We introduced the coefficients

C1 = γl(

β2(4γ2lM

4 − 2γlM2 − γl − 1)

+β(

(γ2l + 4γl − 5)M2 − 2

)

− γl + 2)

, (2.11)

C2 = (γl − 1)M(

β(M2(γ2l + 7γl) − 2γl + 4) − 2γl + 4

)

, (2.12)

C3 = 2γl(γl + 1)(

β((γl − 1)M2 + 2) + 2)

, (2.13)

C4 = 4(γl + 1)(γl − 2)M. (2.14)

The quantity

ω = ω± ≡ −γl(γl − 1)βM2 + 2γl(β + 1) ±√W

2γl(γl + 1)βM, (2.15)

is the normal post-shock velocity relative to the shock, with

W = β2M2(γ3l − γ2

l )(

M2(γl − 1) + 4)

+βγl(4M2(4 + γl − γ2

l ) + 8γl) + 4γ2l . (2.16)

Note that ω must satisfy −M < ω < 0 to represent a genuine right moving shock.We choose the solution where ω = ω+, since the alternative, ω = ω− is a degeneratesolution in the sense that the hydrodynamical limit lim

β→+∞ω− = 0, which does not

represent a right-moving shock.

2.3.2 Relations across a contact discontinuity and an expansionfan

Rewriting equation (2.2) in quasilinear form leads to

ux +(

Fu−1 ·Gu

)

uy = 0. (2.17)

In the frame moving with the triple point, we are searching for self-similar solutionsand we can introduce ξ = y

x = tanφ, so that u = u(ξ). Assuming that ξ 7→u(ξ) is differentiable, manipulating equation (2.17) leads to Auξ = ξuξ. So the


eigenvalues λi of A represent tanφ, where φ is the angle between the refractedsignals and the negative x-axis. The matrix A is given by

A ≡ F−1u Gu =

vy

vx

ρvy

v2x−v2ms− ρvx

v2x−v2ms

vy

vx

1v2x−v2ms

0 0

0vxvy

v2x−v2ms− v2ms

v2x−v2ms− vy

ρ1

v2x−v2ms0 0

0 0vy

vx

1ρvx

0 0

0 − ρv2msvy

v2x−v2ms

ρv2msvx

v2x−v2ms

vxvy

v2x−v2ms0 0

0 − Bvy

v2x−v2ms− Bvx

v2x−v2ms

vy

vx

Bρ

1v2x−v2ms

vy

vx0

0 0 0 0 0vy

vx

. (2.18)

and its eigenvalues are

λ1,2,3,4,5,6 = vxvy + vms√

v2 − v2ms

v2x − v2

ms

,vyvx,vyvx,vyvx,vyvx,vxvy − vms

√

v2 − v2ms

v2x − v2

ms

,

(2.19)

where the magnetosonic speed vms ≡√

c2 + v2a and as before the sound speed

c =√

γpρ and the Alfvén speed va =

√

B2

ρ . Since A has 3 different eigenvalues, 3

different signals will arise. When uξ exists and uξ 6= 0, i.e. inside of expansionfans, uξ is proportional to a right eigenvector ri of A. Derivation of ξ = λi withrespect to ξ gives (∇uλi) · uλ = 1 and thus we find the proportionality constant,giving

uξ =ri

∇uλi · ri. (2.20)

While this result assumed continuous functions, we can also mention relations thathold even across discontinuities like the CD. Denoting the ratio dui

ri= κ, it follows

that [li · du]dx=λjdy= (li · rj)κ = κδi,j , where li and ri are respectively left and

right eigenvectors corresponding to λi. Therefore, if i 6= j,

[li · du]dx=λjdy= 0. (2.21)

From these general considerations the following relations hold across the contactor shear wave where the ratio dy

dx =vy

vx:

vydvx − vxdvy +vms

√v2−v2ms

ρc2 dptot = 0,

vydvx − vxdvy − vms

√v2−v2ms

ρc2 dptot = 0.(2.22)


Since v 6= vms, otherwise all signals would coincide, it follows immediately that thetotal pressure ptot and the direction of the streamlines vy

vxremain constant across

the shocked contact discontinuity.

These relations across the CD allow to solve the full problem using an iterativeprocedure. Inspired by the exact Riemann solver described in Toro [85], we firstguess the total pressure p∗ across the CD. R is a shock when p∗ is larger thanthe post-shock total pressure and T is a shock, only if p∗ is larger than the pre-shock total pressure. Note that the jump in tangential velocity across the CD isa function of p∗ and it must vanish. A simple Newton-Raphson iteration on thisfunction [[

vy

vx]](p∗), finds the correct p∗. We explain further in section 3.5 how we

find the functional expression and iterate to eventually quantify φR, φT , φCD andthe full solution u(x, y, t). From now on p∗ represents the constant total pressureacross the CD.

Similarly, from the general considerations above, equation (2.21) gives that alongdydx =

vxvy±vms

√v2−v2ms

v2x−v2msthe following relations connect two states across expansion

fans:

dρ− 1v2ms

dptot = 0,

vxdvx + vydvy +v2ms

ρc2 dptot = 0,

−ρdptot + ptotρdγ + ptotγdρ = 0,−Bdptot +

(

γp+B2)

dB = 0,

vydvx − vxdvy ± vms

√v2−v2ms

ρc2 dptot = 0.

(2.23)

These can be written in a form which we exploit to numerically integrate thesolution through expansion fans, namely

ρi = ρe +∫ p∗

ptot,e

1v2ms

dptot,

vx,i = vx,e +∫ p∗

ptot,e

±vy

√v2−v2ms−vxvms

ρv2vmsdptot,

vy,i = vy,e +∫ p∗

ptot,e

∓vx

√v2−v2ms−vyvms

ρv2vmsdptot,

Bi = Be +∫ p∗

ptot,e

Bρv2ms

dptot,

pi = pe +∫ p∗

ptot,e

c2

v2msdptot,

γi = γe.

(2.24)

The indices i and e stand respectively for internal and external, the states at bothsides of the expansion fans. The upper signs hold for reflected expansion fans (i.e.of type R), while the lower sign holds for transmitted expansion fans (i.e. of typeT).


2.3.3 Relations across a shock

Since the system is nonlinear and allows for large-amplitude shock waves, theanalysis given thus far is not sufficient. We must include the possibility of one orboth of the R and T signals to be solutions of the stationary Rankine-Hugoniotconditions (equation (2.6)). The solution is given by

ρi =γ−1

γ+1+ p∗

ptot,e

γ−1

γ+1

p∗

ptot,e+1ρe,

vx,i = vx,e − ξ∓(p∗−ptot,e)ρe(vx,eξ∓−vy,e) ,

vy,i = vy,e +p∗−ptot,e

ρe(vx,eξ∓−vy,e) ,

Bi =γ−1

γ+1+ p∗

ptot,e

γ−1

γ+1

p∗

ptot,e+1Be,

γi = γe,

pi = p∗ − B2i

2 ,φR/T = arctan(ξ+/−),

(2.25)

where

ξ± =ve,xve,y ± ce

√

v2e − c2e

v2e,x − c2e

, (2.26)

and

c2e =(γ − 1)ptot,e + (γ + 1)p∗

2ρe. (2.27)

Again the indices i and e stand respectively for internal and external, the statesat both sides of the shocks. The upper signs holds for reflected shocks, while thelower sign holds for transmitted shocks.

2.3.4 Shock refraction as a Riemann problem

We are now ready to formulate our iterative solution strategy. Since there exist 2invariants across the CD, it follows that we can do an iteration, if we are able toexpress one invariant in function of the other. As mentioned earlier, we choose toiterate on p∗ = ptot,3 = ptot,4. This is the only state variable;e in the solution, andit controls both R and T. We will write φR = φR(u2, p∗) and φT = φT (u5, p

∗),u3 = u3(u2, p

∗) and u4 = u4(u5, p∗). The other invariant should match too,

i.e. vx,3

vy,3− vx,4

vy,4= 0. Since u2 and u5 only depend on the input parameters, this

last expression is a function of p∗. Iteration on p∗ gives p∗ and φR = φR(p∗),φT = φT (p∗), u3 = u3(p

∗) and u4 = u4(p∗) give φCD = arctan

vy,3

vx,3= arctan

vy,4

vx,4,

which solves the problem.


2.3.5 Solution inside of an expansion fan

The only ingredient not yet fully specified by our description above is how todetermine the variation through possible expansion fans. This can be done oncethe solution for p∗ is iteratively found, by integrating equations (2.24) till theappropriate value of ptot. Notice that the location of the tail of the expansion fan

is found by tan(φtail) =vy,ivx,i±ci

√v2i −c2i

v2x,i−c2iand the position of φhead is uniquely

determined by tan(φhead) =vy,evx,e±vms,e

√v2e−v2ms,e

v2x,e−v2ms,e. Inside an expansion fan we

know u(ptot), so now we need to find ptot(φ), in order to find a solution for u(φ).We decompose vectors locally in the normal and tangential directions, which arerespectively referred to with the indices n and t. We denote taking derivativeswith respect to φ as ′. Inside of the expansion fans we have some invariants givenby equations (2.23). The fourth of these immediately leads to p

Bγ as an invariant.Eliminating ptot from dρ− 1

c2 dptot = 0 and −Bdptot+ (γp+B2)dB = 0 yields theinvariant ρ

B , and combining these 2 invariants tells us that the entropy S ≡ pργ

is invariant. The stationary MHD equations (2.2) can then be written in a 4 × 4-system for v′n, v

′t, p

′tot and ρ′ as:

v′n + vt + vnρ′

ρ = 0,

vnvt + vnv′n +

p′tot

ρ = 0,

v2n − vnv

′t = 0,

v2msρ

′ − p′tot = 0,

(2.28)

where we dropped B′ from the system, since it is proportional to ρ′. Note that γ′

vanishes. The system leads to the dispersion relation

v4n − v2

msv2n = 0, (2.29)

which in differential form becomes:

4ρv3nv

′n+v4

nρ′−γv2

np′tot−2γptotvnv

′n−(2−γ)Bv2

nB′−(2−γ)B2vnv

′n = 0. (2.30)

Elimination of v′n, ρ′ and B′ gives

dptotdφ

= 2vtvn

v2ms − 2v2

n

3v2n + (γ − 2)v2

ms

ρv2ms. (2.31)

This expression allows us to then complete the exact solution as a function of φ.

IMPLEMENTATION AND NUMERICAL DETAILS 53

2.4 Implementation and numerical details

2.4.1 Details on the Newton-Raphson iteration

We can generally note that ptot,pre < ptot,post, where the indeces pre and postrefer to respectively the pre- and the post-shock region. This implies that therefraction has 3 possible wave configurations: 2 shocks, a reflected rarefactionfan and a transmitted shock, or 2 expansion fans. Before starting the iterationon [[

vy

vx]](p∗), we determine the governing wave configuration. If [[

vy

vx]](ǫ) and

[[vy

vx]](ptot,5−ǫ) differ in sign, the solution has two rarefaction waves. If [[

vy

vx]](ptot,5+

ǫ) and [[vy

vx]](ptot,2 − ǫ) differ in sign, the solution has a transmitted shock and a

reflected rarefaction wave. In the other case, the solution contains two shocks inits configuration. If R is an expansion fan, we take the guess

p∗0 =min 2ρev

2x,e−(γe−1)ptot,e

γ+1 |e ∈ 2, 5+ ptot,5

2(2.32)

as a starting value of the iteration. This guess is the mean of the critical valueptot,crit , which satisfies

v2e,x − c2(ptot,crit) = 0, (2.33)

and p5, which is the minimal value for a transmitted shock. As we explain in section5.3, v2

2,x − c2(ptot,crit) = 0 is equivalent to v25 − c2 = 0 and v2

5,x − c2(ptot,crit) = 0is equivalent to v2

2 − c2 = 0, and is thus a maximal value for the existence of aregular solution. If R is a shock, we take (1 + ǫ)ppost as a starting value for the

iteration, where ǫ is 10−6. We use a Newton-Raphson iteration: p∗i+1 = p∗i − f(p∗i )f ′(p∗i ) ,

where f ′(p∗) is approximated numerically by f(p∗i +δ)−f(p∗i )δ , where δ = 10−8. The

iteration stops whenp∗i+1−p∗i

p∗i< ǫ, where ǫ = 10−8.

2.4.2 Details on AMRVAC

AMRVAC (van der Holst & Keppens [93]; Keppens et al.[49]) is an AMR code,solving equations of the general form ut+∇·F(u) = S(u,x, t) in any dimensionality.The applications cover multi-dimensional HD, MHD, up to special relativisticmagnetohydrodynamic computations. In regions of interests, the AMR codedynamically refines the grid. The initial grid of our simulation is shown in figure 2.4.The refinement strategy is done by quantifying and comparing gradients. A Löhnerestimator is used, tolerance ǫl = 0.025, with refinement triggered on ρ and auxiliaryvariable ωz, in equal weights.

The AMR in AMRVAC is of a block-based nature, where every refined gridhas 2D children, and D is the dimensionality of the problem. Parallelization is


Figure 2.4: The initial AMR grid at t = 0, for the example in section 5.1.

implemented, using MPI. In all the simulations we use 5 refinement levels, startingwith a resolution of 24 × 120 on the domain [0, 1] × [0, 5], leading to an effectiveresolution of 384 × 1940. The shock is initially located at x = 0.1, while thecontact discontinuity is located at y = (x − 1) tanα. We used the fourth orderRunge-Kutta timestepping, together with a TVDLF-scheme (see Tóth & Odstrcil[86]; Yee [102]) with Woodward-limiter on the primitive variables. The obtainednumerical results were compared to and in agreement with simulations using otherschemes, such as a Roe scheme and the TVD-Muscl scheme. The calculations wereperformed on 4 processors.

2.4.3 Following an interface numerically

The AMRVAC implementation contains slight differences with the theoreticalapproach. Implementing the equations as we introduced them here would leadto excessive numerical diffusion on γ. According to equations 2.24 and 2.25 thevalue of γ does not change through an expansion fan or across a shock. Thereforeγ is a discrete variable in the sense that it only takes a finite number of values (inour case only 2, namely γ ∈ γl, γr), we know γ(x, y, t) exactly, if we are able tofollow the contact discontinuity in time.

Suppose thus that initially a surface, separates 2 regions with different values of γ.Define a function χ : D × R+ → R : (x, y, t) 7→ χ(x, y, t), where D is the physicaldomain of (x, y). Writing χ(x, y) = χ(x, y, 0), we ask χ to vanish on the initialcontact and to be a smooth function obeying

• γ = γl ⇔ χ(x, y) < 0,

• γ = γr ⇔ χ(x, y) > 0.

We take in particular ±χ to quantify the shortest distance from the point (x, y) tothe initial contact, taking the sign into account. Now we only have to note that(χρ)t = χρt + ρχt = −χ∇ · (ρv) − (ρv · ∇)χ = −∇ · (χρv).

RESULTS 55

a)

-6

-4

-2

0

2

4

6

0 2 4 6 8 10 12 14

[[vy/

v x]]

p*

all shock solver

b)

-6

-4

-2

0

2

4

6

0 2 4 6 8 10 12 14

[[vy/

v x]]

p*

right shock solver

c)

-6

-4

-2

0

2

4

6

0 2 4 6 8 10 12 14

[[vy/

v x]]

p*

no shock solver

d)

-6

-4

-2

0

2

4

6

0 2 4 6 8 10 12 14

[[vy/

v x]]

p*

combined solver

Figure 2.5:[[

vy

vx

]]

(p∗) for the reference case from Samtaney [78]: a) all shock

solver; b) right shock solver; c) no shock solver; d) shock ⇔ p∗ > pi. The all shocksolver is selected.

The implemented system is thus (2.2), but the last equation is replaced by(χρvx)x+(χρvy)y = 0. Every timestep we reset the value of γ to γl or γr, dependingon the sign of χ.

It is now straightforward to show that we did not introduce any new signal. Inessence, this is the level set approach, as presented in e.g. Mulder et al.[59].

2.5 Results

2.5.1 Fast-Slow example

As a first hydrodynamical example, we set(

α, β−1, γl, γr, η,M)

=(

π4 , 0,

75 ,

75 , 3, 2

)

,as originally presented in Samtaney [78]. In figure 2.5, the first 3 plots show[[vy

vx]](p∗), when assuming a prescribed wave configuration, for all 3 possible

configurations. The last plot shows the actual function [[vy

vx]](p∗), which consists of


0

2

4

6

8

10

12

14

0 1 2 3 4 5 6

ρ

φ

-2.5

-2

-1.5

-1

-0.5

0

0 1 2 3 4 5 6

v x

φ

-2.5

-2

-1.5

-1

-0.5

0

0 1 2 3 4 5 6

v y

φ

0

2

4

6

8

10

12

14

0 1 2 3 4 5 6

p

φ

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 1 2 3 4 5 6

S

φ

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 1 2 3 4 5 6

v x/v

y

φ

Figure 2.6: Solution to the fast-slow refraction problem, for the reference case fromSamtaney [78]. Notice that p and vx

vyremain constant across the shocked contact.

RESULTS 57

0

20

40

60

80

100

120

0 1 2 3 4 5 6

p

φ

0

10

20

30

40

50

0 1 2 3 4 5 6

S

φ

Figure 2.7: Solution to the slow-fast refraction problem from van der Holst &Keppens [93]. Notice that S remains constant across R.

piecewise copies from the 3 possible configurations in the previous plots. The initialguess is p∗0 = 4.111, the all shock solver is selected, and the iteration convergesafter 6 iterations with p∗ = 6.078. The full solution of the Riemann problem isshown in figure 2.6.

2.5.2 Slow-Fast example

In figure 2.7 we show the full solution of the HD Riemann problem, in which thereflected signal is an expansion fan, connected to the refraction with parameters(

α, β−1, γl, γr, η,M)

=(

π3 , 0,

75 ,

75 ,

110 , 10

)

from van der Holst & Keppens [93]. Therefraction is slow-fast, and R is an expansion fan. Note that p and vy

vxremain

constant across the CD, and the entropy S is an invariant across R.

2.5.3 Tracing the critical angle for regular shock refraction

Let us examine what the effect of the angle of incidence, α, is. Therefore we getback to the example from Section 2.5.1,

(

β−1, γl, γr, η,M)

=(

0, 75 ,

75 , 3, 2

)

and let

α vary: α ∈]

0, π2]

. Note that α = π2 corresponds to a 1-dimensional Riemann

problem. The results are shown in figure 2.8. Note that for regular refraction

v2y,5 > c25. We can understand this by noting that ξ± =

ve,xve,y±ce

√v2e−c2e

v2e,x−c2e=

(

ve,xve,y∓ce

√v2e−c2e

v2e,y−c2e

)−1

= ξ∓, which are the eigenvalues of Gu−1 · Fu = (Fu

−1 ·Gu)−1. Note that we could have started our theory from the quasilinear form uy+(Gu

−1 ·Fu)ux = 0 instead of equation (2.17). If we would have done so, we would

have found eigenvalues ξ, which would correspond to 1arctanφ . Moreover, both the

eigenvalues, ξ+ and ξ−, have 4 singularities, namely c2 ∈ −vx,2, vx,2,−vy,5, vy,5


6

6.05

6.1

6.15

6.2

6.25

6.3

6.35

6.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

p*

α

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

φ

α

φRφCD

φT

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.2 0.4 0.6 0.8 1 1.2 1.4

[[vt]]

α

1

1.2

1.4

1.6

1.8

2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

α

|vy,5|c5

Figure 2.8: Upper Left : p∗(α). Note that for α < 0.61, there are no solutions for p∗:the refraction is irregular; Upper Right : the wave pattern for regular refraction;Lower Left : For α = π

2 , the problem is 1-dimensional and there is no vorticitydeposited on the interface. For decreasing α, the vorticity increases. Lower right:For regular refraction, |vy,5| > c5.

for ξ− and c5 ∈ −vx,5, vx,5,−vy,2, vy,2 for ξ+, where thus c25 = v25,y ⇔ c22 = v2

2

and c22 = v2y,2 ⇔ c25 = v2

5 . It is now clear that it is one of the latter conditionsthat will be met for αcrit. In the example, the transition to irregular refraction

occurs at −vy,5 = c5 and limα→αcrit

p∗ = 2γrηM2 tan2(αcrit)−γl+1

γl+1 = 6.67. Figure 2.9

shows Schlieren plots for density from AMRVAC simulations for the reference caseα = π

4 , and the irregular case where α = 0.3. In the regular case, all signals meetat the triple point, while for α < αcrit = 0.61, the signals do not meet at one triplepoint, the triple point forms a more complex structure and becomes irregular. TheCD, originated at the Mach stem, reaches the triple point through an evanescentwave, which is visible by the contour lines. This pattern is called Mach Reflection-Refraction. Decreasing α even more, the reflected wave transforms in a sequenceof weak wavelets (see e.g. Nouragliev et al.[64]). This pattern, of which the caseα = 0.3 is an example, is called Concave-Forwards irregular Refraction. Theseresults are in agreement with our predictions.

RESULTS 59

Figure 2.9: Schlieren plots of the density for(

β−1, γl, γr, η,M)

=(

0, 75 ,

75 , 3, 2

)

with varying α. Upper: α = π4 : a regular reference case. Lower: α = 0.3: an

irregular case.

2.5.4 Abd-El-Fattah and Hendersons experiment

In 1978, a shock tube experiment was performed by Abd-El-Fattah & Henderson[3]. It became a typical test problem for simulations (see e.g. Nouragliev et al.[64])and refraction theory (see e.g. Henderson [42]). The experiment concerns a slow-fast shock refraction at a CO2/CH4 interface. The gas constants are γCO2

=1.288, γCH4

= 1.303, µCO2= 44.01 and µCH4

= 16.04. Thus η =µCH4

µCO2

= 0.3645.

A very weak shock, M = 1.12 is refracted at the interface under various angles.von Neumann [62] theory predicts the critical angle αcrit = 0.97 and the transitionangle αtrans = 1.01, where the reflected signal is irregular if α < αcrit, a shock ifαcrit < α < αtrans and an expansion fan if αtrans < α. This is in perfect agreementwith the results of our solution strategy as illustrated in figure 2.10. There we show


1.28

1.3

1.32

1.34

1.36

1.38

0.96 0.97 0.98 0.99 1 1.01 1.02 1.03

p

α

p*ppost

0

0.5

1

1.5

2

2.5

3

0.9 1 1.1 1.2 1.3 1.4 1.5

φ

α

φRφCD

φT

Figure 2.10: Exact solution for the Abd-El-Fattah experiment. Left: p∗(α)confirms αcrit = 0.97 and αtrans = 1.01. Right: φ(α).

the pressure p∗ compared to the post shock pressure ppost, as well as the anglesφR, φCD and φT for varying angle of incidence α. Irregular refraction means thatnot all signals meet at a single point. The transition at αcrit is one between aregular shock-shock pattern and an irregular Bound Precursor Refraction, wherethe transmitted signal is ahead of the shocked contact and moves along the contactat nearly the same velocity. This is also confirmed by AMRVAC simulations. Ifthe angle of incidence, α, is decreased even further, the irregular pattern becomesa Free Precursor Refraction, where the transmitted signal moves faster than theshocked contact, and reflects itself, introducing a side-wave, connecting T to CD.When decreasing α even further, another transition to the Free Precursor vonNeumann Refraction occurs.

2.5.5 Connecting slow-fast to fast-slow refraction

Another example of how to trace transitions by the use of our solver is done bychanging the density ratio η across the CD. Let us start from the example givenin section 5.1 and let us vary the value of η.

Here we have(

α, β−1, γl, γr,M)

=(

π4 , 0,

75 ,

75 , 2)

. The results are shown infigure 2.11. Note that, since ppost = 4.5, we have a reflected expansion fan for fast-slow refraction, and a reflected shock for slow-fast refraction. The transmittedsignal plays a crucial role in the nature of the reflected signal: for fast-slowrefraction φT < π

2 , but for slow-fast refraction, φT > π2 and the transmitted

signal bends forwards. We ran our solver for varying values of M and α, and forall HD experiments with γl = γr, we came to the conclusion that a transition fromfast-slow to slow-fast refraction, coincides with a transition from a reflected shockto a reflected expansion fan, with φT = π

2 . This result agrees with AMRVACsimulations. In figure 2.12, a density plot is shown for η = 1.2 and η = 0.8.

RESULTS 61

3

3.5

4

4.5

5

5.5

6

0.6 0.8 1 1.2 1.4 1.6 1.8 2

p

η

p*

ppost

0

0.5

1

1.5

2

0.6 0.8 1 1.2 1.4 1.6 1.8 2

φ

η

φRφCD

φTπ/2

Figure 2.11: Exact solution for(


=(

π4 , 0,

75 ,

75 , 2)

and a varyingrange of the density ratio η. Left: for η < 1 we have p∗ < ppost = 4.5 and thus areflected expansion fan, for η > 1 we have p∗ > ppost = 4.5 and thus a reflectedshock. Right: for η < 1: φT <

π2 and for η > 1: φT >

π2 .

Figure 2.12: Density plots for(


=(

π4 , 0,

75 ,

75 , 2)

. Left: Aslow/fast refraction with η = 0.8. Note that φT > π

2 and R is an expansionfan. Right: A fast/slow refraction with η = 1.2. Note that φT < π

2 and R is ashock.

2.5.6 Effect of a perpendicular magnetic field

So far, none of our results include magnetic fields. We now consider the case wherea purely out-of-plane magnetic field, where the field is perpendicular to the shockfront and thus acts to increase the total pressure and the according flux terms, isadded.

Note that it follows from equation 2.24 and 2.25 that Bρ is invariant across

shocks and rarefaction fans. Therefore, Bρ can only jump across the shocked and

unshocked contact discontinuity and B cannot change sign.

Revisiting the example from section 5.1, we now let the magnetic field vary.


0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.6 0.8 1 1.2 1.4 1.6 1.8 2

[[vt]]

β

[[vt]]

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.6 0.8 1 1.2 1.4 1.6 1.8 2

[[vt]]

β

[[vt]]

Figure 2.13: Left: Solution for the fast-slow problem: strong perpendicularmagnetic fields decrease the instability of the CD. Right: Solution for the slow-fastproblem: strong perpendicular magnetic fields decrease the instability of the CD.

Figure 2.13 shows [[vt]](β) across the CD. Also for η = 0.8, making it a slow-fastproblem, [[vt]](β) is shown. First notice that no shocks are possible for β < 0.476,since ω+ would not satisfy ω+ > −M . Manipulating equation (2.15), we knowthat this is equivalent to

β > βmin ≡ 2

γl(M2 − 1). (2.34)

This relation is also equivalent to c1 > M , which means that the shock issubmagnetosonic, compared to the pre-shock region. Figure 2.14 shows densityplots from AMRVAC simulations at t = 2.0, for (α, γl, γr, η,M) =

(

π4 ,

75 ,

75 , 3, 2

)

with varying β−1. First note that the interface is unstable for the HD case.Increasing β−1 decreases the shock strength. For β−1 = 1 the shock is veryweak: the Atwood number At = 0.17, and the interface remains stable. here weintroduced the Atwood number, which is defined in 2.35.

Shown in figure 2.13, is the vorticity across the CD. In the limit case of this minimalplasma-β the interface is stable, both for fast-slow and slow-fast refraction. Asexpected, in the fast-slow case, the reflected signal is an expansion fan, while it isa shock in the fast-slow case. Also note that the signs of the vorticity differ, causingthe interface to roll up clockwise in the slow-fast regime, and counterclockwise inthe fast-slow regime. When decreasing the magnetic field, the vorticity on theinterface increases in absolute value. This can be understood by noticing that thelimit case of minimal plasma-β is also the limit case of very weak shocks. This canfor example be understood by noting that lim

β→βmin

φCD = α (see figure 2.15). A

convenient way to measure the strength of a shock is by use of its Atwood number.This is often done in the classical HD case (see, e.g., Sadot et al.[75])

At =ρ2 − ρ1

ρ2 + ρ1. (2.35)

RESULTS 63

Figure 2.14: Density plots at t = 2.0 for (α, γl, γr, η,M) =(

π4 ,

75 ,

75 , 3, 2

)

with varying β−1. Upper: β−1 = 0. The hydrodynamical Richtmyer-Meshkov

instability causes the interface to roll up. Center: β−1 = 12 . Although the initial

amount of vorticity deposited on the interface is smaller than in the HD case, thewall reflected signals pass the wall-vortex and interact with the CD, causing theRMI to appear. Lower: β−1 = 1. The shock is very weak and the interfaceremains stable.


0

0.2

0.4

0.6

0.8

1

1.2

1.4

0.6 0.8 1 1.2 1.4 1.6 1.8 2

φ CD

β

1

1.2

1.4

1.6

1.8

2

2.2

2.4

0.6 0.8 1 1.2 1.4 1.6 1.8 2

[[vt]]

/At

β

Figure 2.15: The reference problem from Samtaney [78] with varying β. Left: Thedependence of φCD on β. Note that lim

β→βmin

φCD = π4 = α, since this is the limit

to infinitely weak shocks: limβ→βmin

At = 0. Right: The vorticity deposition in the

shocked contact scales as the Atwood number and limβ→βmin

[[vt]]At = 1.

Figure 2.15 shows the jump across the shocked contact [[vt]], scaled to the shocksAtwood number. Note that in the limit case of very weak shocks the Atwoodnumber equals the jump in tangential velocity across the CD, in dimensionalnotation:

limβ→βmin

[[vt]]c1

At= 1. (2.36)

When keeping the Atwood number constant, the shocks sonic Mach number isgiven by

M =1 +At

1 −At

√

(2 − 2γ − γβ)At2 + (2γβ + 2γ)At− γβ − 2

(γ2β)At2 + (γ2β − γβ)At− γβ(2.37)

=

√

(At+ 1)((γβ + 2γ − 2)At− (γβ + 2))

γβ(1 −At)(γAt+ 1). (2.38)

Note that in the limit for weak shocks

limAt→0

M =

√

γβ + 2

γβ, (2.39)

which is equivalent to equation 2.34, and in the limit for strong shocks, M →∞. Figure 2.16 shows the deposition of vorticity on the shocked contact, for aconstant Atwood number. We conclude that under constant Atwood number, the

RESULTS 65

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2

[[vt]]

β

0

1

2

3

4

5

6

0 0.5 1 1.5 2

M

β

Figure 2.16: Left: The dependence of the vorticity deposition on the shockedcontact in function of β, under constant Atwood number At = 0.17. Right: Theshocks Mach number in function of the plasma-β, under constant Atwood numberAt = 0.17.

Figure 2.17: AMRVAC plots of Bρ for At = 511 , with varying beta. Upper: β = 16.

Lower: β = 0.25.

effect of a perpendicular magnetic field is small: Stronger perpendicular magneticfield increase the deposition of vorticity on the shocked contact slightly. This isconfirmed by AMRVAC simulations (see figure 2.17).


2.6 Conclusions

We developed an exact Riemann solver-based solution strategy for shock refractionat an inclined density discontinuity. Our self-similar solutions agree with the earlystages of nonlinear AMRVAC simulations. We predict the critical angle αcrit forregular refraction, and the results fit with numerical and experimental results.Our solution strategy is complementary to von Neumann theory, and can be usedto predict full solutions of refraction experiments, and we have shown varioustransitions possible through specific parameter variations. For perpendicular fields,the stability of the contact decreases slightly with decreasing β under constantAtwood number. We will generalize our results for planar uniform magnetic fields,where up to 5 signals arise. In this case we search for non-evolutionary solutions,involving intermediate shocks.

Chapter 3

Shock Refraction in ideal MHD

In this chapter we generalize our solution strategy of chapter 2 to regular shockrefraction in ideal MHD. While in the HD case three signals arise, in the planarMHD case five signals arise. Our Riemann solver finds the exact position of thesefive signals, and determines the values of the conserved variables in the new-formedregions.

After reflection from the top wall the interface remains stable since MHD does notallow for vorticity deposition on a contact discontinuity.

This Chapter is published as Delmont & Keppens [20].

3.1 Introduction

The interaction of a shock wave with an inclined density discontinuity is a classicalhydrodynamical (HD) shock tube problem. When the impinging shock refractsat the density discontinuity, 3 signals arise: a reflected signal, a transmittedsignal and a shocked contact in between. Both the reflected and transmittedsignal can be continuous (rarefaction fans) or discontinuous (shocks). Due tolocal Kelvin-Helmholtz instability, the shocked contact becomes unstable, and theRichtmyer-Meshkov instability (RMI) forms. In Chapter 2 (or Delmont et al.[19]), we solved the initial self similar phase of the problem exactly, exploitingexact Riemann solver methods. In this chapter, we extend that study to planarideal magnetohydrodynamics (MHD).

For that planar case, Samtaney [78] showed by numerical grid-adaptive simulationsthat addition of a uniform magnetic field, aligned with the shock normal,

67

68 SHOCK REFRACTION IN IDEAL MHD

suppresses the RMI. The setup of the MHD problem is shown in figure 3.1. Therectangular domain mimics a shock tube, thus the left and the right boundariesare modeled as open, while the upper and the lower boundaries are rigid walls.Completely to the left of the domain, there is a genuine right-moving shock, whichmoves with sonic Mach number M . It moves towards a contact discontinuity,which forms an inclination angle α with the shock normal and separates two gasesat rest, with a density ratio η across it. The ratio of specific heats, γ, is considereda constant equation parameter, and the initially uniform applied magnetic fieldintroduces a plasma-β (in the pre-shock region). These 5 parameters define theproblem uniquely. In the MHD case up to 5 signals arise: a fast (FR) and a slow(SR) reflected signal, a contact discontinuity (CD), and a slow (ST) and a fast(FT) transmitted signal, separating the 4 new formed states, as shown in figure3.3.

Figure 3.2 shows simulation snapshots for the case where (α, γ,M, η) = (π4 ,75 , 2, 3),

after reflection from the top wall. The upper snapshot shows the hydrodynamicalcase (β−1 = 0) and the lower snapshot shows an MHD case where β = 2. In thefirst case the RMI has formed on the interface, while it is clearly suppressed inthe magnetohydrodynamical case. We refer to Delmont et al.[19] for an exactsolution of the HD case. In this paper we generalize the presented solutionstrategy to planar ideal MHD. Our results are in agreement with results fromWheatley et al.[98]. Our approach is inspired by Wheatley et al.[98]. However,our solution strategy for the RH jump conditions is essentially different and theiteration method used is more detailed.

We also compare our results with numerical grid adaptive simulations. Thesesimulations are performed by the Adaptive Mesh Refinement code AMRVAC (seee.g. Keppens et al.[49]; van der Holst & Keppens [93]).

The suppression of the RMI in ideal MHD can be explained as a direct consequenceof the Rankine-Hugoniot jump conditions across a shocked contact. It is well-known that contact discontinuities do not allow for a jump in tangential velocity,when the normal magnetic field component does not vanish (see e.g. Goedbloed& Poedts [32]).

In section 3.2, we introduce the equations employed in the analytical solutionstrategy, namely the planar stationary MHD equations and the Rankine-Hugoniotconditions from their integral form. In section 3.3, we summarize some importantfeatures of MHD shocks. In section 3.4, we describe our solution strategy, whichis based on the Riemann solver for ideal MHD presented by Torrilhon [91] andin section 3.5 we demonstrate the algorithm by solving a case presented firstby Samtaney in [78]. In section 6 finally, we compare our results to AMRVACsimulations.

INTRODUCTION 69

αM

(ρ,v = 0, p,B) (ηρ,v = 0, p,B)

Figure 3.1: Initial configuration: a shock moves with shock speed M to an inclineddensity discontinuity. Both the upper and lower boundary are solid walls, whilethe left and the right boundaries are open.

Figure 3.2: In both cases (α, γ,M, η) = (π4 ,75 , 2, 3). Up: β−1 = 0. When

a hydrodynamical shock impinges on a contact discontinuity, the discontinuitybecomes unstable due to local Kelvin-Helmholtz instability. Down: β−1 = 1

2 .

When a normal magnetic field is applied, the Rankine-Hugoniot conditions do notallow for vorticity deposition on a contact discontinuity. Therefore, the interfaceremains stable and the RMI is suppressed.


3.2 Governing equations

In order to describe the dynamical behavior of ionized, quasi-neutral plasmas, weuse the framework of ideal MHD. The initial configuration sketched in figure 3.1will lead to a refraction pattern as sketched in figure 3.3. Regular refraction meansthat all signals meet at a single quintuple point. This quintuple point moves alongthe unshocked contact at speed vqp = (M,M tanα). We will solve the problemin a co-moving frame with the quintuple point. We can then assume that thesolution is self similar and time independent: ∂/∂t = 0. Therefore, we can exploitthe stationary MHD equations, which written out in conservative form and for ourplanar problem, are given by

∂

∂xF +

∂

∂yG = 0, (3.1)

where we introduced the flux terms for the orthogonal x and y directions given by

F =

ρvx

ρv2x + p− B2

x

2 +B2

y

2ρvxvy −BxBy

vx(γγ−1p+ ρ

v2x+v2y2 +B2

y) −BxByvyBx

(3.2)

and

G =

ρvyρvxvy −BxBy

ρv2y + p+

B2x

2 − B2y

2

vy(γγ−1p+ ρ

v2x+v2y2 +B2

x) −BxByvxBy

. (3.3)

This set of equations expresses conservation of mass density, momentum andenergy. The conserved variables and the components of the flux terms are writtenin terms of the mass density ρ, the velocity components vx and vy, the thermalpressure p and the magnetic field components Bx and By. Faraday’s law ensuresconservation of magnetic flux, which in the stationary case becomes ∇ × E = 0.The electric field Ez is therefore a global Riemann invariant:

Ez = vyBx − vxBy. (3.4)

This allows us to eliminate By =vyBx−Ez

vxfrom the system, which is reduced to a

5× 5-system. We hereby exploit the fact that vx(x, t) < 0 in the co-moving frame

GOVERNING EQUATIONS 71

with the quintuple point. In the rest frame this means that the velocity in thex-direction is bounded by the shock speed, during the refraction process.

We allow weak (i.e., discontinuous) solutions of the system, which are solutionsof the integral form of the MHD equations. The discontinuities occurring in theproblem setup (both the impinging shock and the initial contact discontinuity),as well as those that may appear as FR, SR, ST or FT signals later on obeythe Rankine-Hugoniot conditions. In Appendix D we give the planar stationaryRankine-Hugoniot conditions and rewrite them as a 4×4-system in [[Bt]], [[p]], [[ρ]]and [[vt]], where [[Q]] refers to the jump in the quantity Q across the CD, andtangential vector components are referred to by a subscript t.

We now discuss the characteristics of the system (3.1), by writing it out inquasilinear form

ux + Auy = 0, (3.5)

where the matrix A is computed from the flux Jacobian matrices Fu ≡ ∂F∂u and

Gu ≡ ∂G∂u as follows:

A = F−1u

· Gu. (3.6)

Here u can be any arbitrary vector for which Fu is invertible. We set u =(ρ, vx, vy, p, Bx). As argued in Delmont et al.[19], the eigenvalues of A coincidewith the tangent of the refraction angles φFR, φSR, φCD, φST and φFT (see figure3.3). Since the system has 5 distinct eigenvalues, the shock indeed refracts in 5distinct signals.

Computing the eigenvalues λ of A, leads to the characteristic equation

det(A− λI) = (vxλ− vy)(

Σi=0,4tiλi)

= 0. (3.7)

This is a quintic with 5 solutions. One eigenvalue λ =vy

vxis linearly degenerate

and corresponds to the CD. The 4 other eigenvalues are found as roots of a quartic,with coefficients given by

t4 = v4x + v2

x(4c2 − 2v2

a) + 2c2v2a,x, (3.8)

t3 = 4vxvy(c2 + v2

a − v2x) − 8c2va,xva,y, (3.9)

t2 = 6v2xv

2y − 2v2

a(v2 + c2) + 2v2c2, (3.10)

t1 = 4vxvy(c2 + v2

a − v2y) − 8c2va,xva,y, (3.11)

t0 = v4y + v2

y(4c2 − 2v2

a) + 2c2v2a,y. (3.12)


φFR = atan(λFR)

u1

u7u2

u3 u4 u5 u6

Figure 3.3: The wave pattern during interaction of the shock with the CD. Theeigenvalues of the matrix A coincide with the tangent of the refraction angles.Since A has 5 distinct eigenvalues, 5 signals arise.

In these expressions we denote the sound speed c =√

γpρ , the directional Alfvén

speeds va,x =√

B2x

ρ , va,y =√

B2y

ρ and the total Alfvén speed va =√

v2a,x + v2

a,y.

We can rewrite the characteristic equation (3.7) as

v⊥(v2⊥ − v2

f,⊥)(v2⊥ − v2

s,⊥) = 0, (3.13)

where the (squared) normal speed is given by

v2⊥ =

(vxλ− vy)2

1 + λ2, (3.14)

the (squared) fast speed by

v2f,⊥ =

1

2

(

c2 + v2a +

√

(c2 + v2a)

2 − 4c2(Bxλ−By)2

ρ(1 + λ2)

)

, (3.15)

and the (squared) slow speed by

v2s,⊥ =

1

2

(

c2 + v2a −

√

(c2 + v2a)

2 − 4c2(Bxλ−By)2

ρ(1 + λ2)

)

. (3.16)

This then clearly shows that the MHD system has the fast and the slowmagnetosonic speeds in the direction perpendicular to the shock front as its basiccharacteristic speeds.

All the refracted magnetosonic signals can be expansion fans or shocks.

MHD SHOCKS 73

3.3 MHD shocks

MHD shocks are a topic of extensive research (see e.g. Liberman & Velikhovich

[56]; Goedbloed [33]). Introducing the normal Alfvén speed va,n ≡√

B2n

ρ , which

is an eigenvalue of the full three dimensional stationary MHD equations, and

va,t ≡√

B2t

ρ leads to va =√

v2a,n + v2

a,t. The full set of eigenvalues of the full set

of MHD equations is given by ±vf,n,±va,n,±vs,n, 0, where the fast speed vf,nand the slow speed vs,n are respectively defined as in equations (1.48) and (1.48).

Since 0 ≤ vs,n ≤ va,n ≤ vf,n, the full set of MHD equations is non-strictlyhyperbolic. Those speeds define the up- and downstream states into 4 types (and3 transition types plus 1 stationary type for completeness), which in order ofincreasing entropy S ≡ p

ργ :

• (1) superfast: vf,n < |vn|;

• (1=2) fast: |vn| = vf,n;

• (2) subfast: va,n < |vn| < vf,n;

• (2=3) Alfvén: va,n = vn;

• (3) superslow: vs,n < |vn| < va,n;

• (3=4) slow: |vn| = vs,n;

• (4) subslow: |vn| < vs,n;

• (∞) stationary: vn = 0.

The second law of thermodynamics enforces [[S]] > 0 across a shock. If thisinequality is satisfied, we call a shock admissible. When the upstream state is oftype i and the downstream state is of type j, then we define the shock to be oftype i → j. The RH conditions (D.1) now allow for the following types of shocks(see also Falle & Komissarov[27], De Sterck et al.[22], Liberman & Velikhovich [56]and many others):

• (1 → 2)-shocks are called fast shocks. They satisfy Bt,2 > Bt,1 > 0 orBt,2 < Bt,1 < 0, i.e. the magnetic field gets refracted away from the shocknormal.

• (3 → 4)-shocks are called slow shocks. They satisfy Bt,1 > Bt,2 > 0 orBt,1 < Bt,2 < 0, i.e. the magnetic field gets refracted towards the shocknormal.


• (1 → 2 = 3)-shocks are called switch-on shocks, since they satisfy Bt,1 = 0,i.e. the upstream magnetic field is aligned with the shock normal.

• (2 = 3 → 4)-shocks are called switch-off shocks, since they satisfy Bt,2 = 0,i.e. the downstream magnetic field is aligned with the shock normal.

• (1 → 3), (1 → 4), (2 → 3) and (2 → 4)-shocks are called intermediate shocks.They satisfy Bt,2 ≥ 0 ≥ Bt,1 or Bt,2 ≤ 0 ≤ Bt,1, i.e. the upstream magneticfield is aligned with the shock normal.

• (1 → 4)-shocks for which Bt,1 = Bt,2 = 0 also satisfy vt,1 = vt,2 = 0 andare essentially 1-dimensional. In this case both the u- and downstreammagnetic field are aligned with the shock normal. These shocks are calledhydrodynamical shocks;

• (2 = 3 → 2 = 3)-discontinuities are called Alfvén discontinuities (or,alternatively, rotational discontinuities). They satisfy Bt,1 + Bt,2 = 0 suchthat the upstream and the downstream state are equal, except for a signchange of Bt;

• (∞ → ∞)-discontinuities can be contact discontinuities, where only ρ jumpsacross them, or tangential discontinuities, where Bn and vn vanish.

The latter two cases are called linear discontinuities, since [[vn]] vanishes. Theother cases are called MHD shocks. The set of planar MHD equations reducesthe number of possibilities, since we now only have characteristic speeds given by±vf ,±vs, 0, making the planar system strictly hyperbolic.

The existence of intermediate shocks is controversial since they are believed to beunstable under small perturbations. This topic is widely discussed in the literature(theoretically in e.g. Barmin et al.[8]; Falle & Komissarov [26, 27]; Myong & Roe[60, 61], observationally in e.g. Chao et al.[12]; Feng & Wang [29] and numericallyin e.g. Brio & Wu [11], De Sterck et al.[22]). We allow for intermediate shocksin our solution strategy, since they form the central Alfvénic transition for MHDshocks (Goedbloed [33]), and as we show later, they naturally emerge in highresolution numerical simulations.

3.4 Riemann Solver based solution strategy

3.4.1 Dimensionless representation

We now present how we tackle the problem in a dimensionless manner. We choosea representation in which the initial shock speed s equals its sonic Mach numberM . We determine the value of the primitive variables in the post-shock region


vn > vf,n vf,n > vn > va,n

(a)

va,n > vn > vs,n vs,n > vn

(b)

vn > va,n va,n > vn

(c)

vn > vf,n = va,nvn = va,n

(d)

vn = an vn > va,n = vs,n

(e)

vn = an vn = an

(f)


vn = 0 vn = 0

(g)

vn = 0 vn = 0

(h)

vn > vf = an an = vs > vn

(i)

Figure 3.4: The classical 1 − 2 − 3 − 4 classification of MHD discontinuities: a)fast shock; b) slow shock; c) intermediate shock; d) switch-on shock; e) switch-offshock; f)Alfvén discontinuity; g) contact discontinuity; h) tangential discontinuity;i) hydrodynamical shock.

by applying the stationary Rankine-Hugoniot conditions in the shock rest frame.Note ui = (ρi, vx,i, vy,i, pi, Bx,i), where the index i refers to the value taken in thei−th region (figure 3.3). Note that we did not include By,i, since it is completelydetermined by equation (3.4).

We arbitrarily scale by setting p1 = 1 and ρ1 = γ. Now all velocity componentsare scaled with respect to the sound speed in this region between the impingingshock and the initial CD. Since this region is initially at rest, the sonic Machnumber M of the shock equals its shock speed s. When the shock intersectsthe CD, the quintuple point follows the unshocked contact slip line. It does soat a fixed speed vqp = (M,M tanα), therefore we will solve the problem in theframe of the stationary quintuple point. We will look for self similar solutionsin this frame, u = u(φ), where all signals are stationary. We now have thatvx = vx −M and vy = vy −M tanα, where v refers to this new frame. Fromnow on we will drop the tilde and only use this new frame. We now have u1 =


(γ,−M,−M tanα, 1,√

2β )t and u7 = (ηγ,−M,−M tanα, 1,

√

2β )t. The Rankine-

Hugoniot relations now immediately give a unique planar solution for u2, namely

u2 =

(

(γ2 + γ)M2

(γ − 1)M2 + 2,− (γ − 1)M2 + 2

(γ + 1)M,−M tanα,

2γM2 − γ + 1

γ + 1,

√

2

β

)t

.

(3.17)

This then completely determines the initial condition, containing only states u1,u2

and u7 in terms of the dimensionless parameters α, β, γ,M and η.

3.4.2 Path variables

We know that v,B and p are continuous across the CD (see Appendix D). SinceBy is uniquely determined by the other 4 (independent) variables, we have 4independent scalars which should vanish. Hence we can guess 4 path variables,1 across each magneto-acoustic signal, and express the vanishing quantities infunction of those path variables χ ≡ (χFR, χSR, χST , χFT ). The fast magneto-acoustic signals are postulated to be shocks, while the slow magneto-acousticsignals can be both shocks and expansion fans.

In the case of a shock, fast signals are postulated to be fast shocks, while slowsignals can be slow or intermediate shocks. We then select the correct solution tothe Rankine-Hugoniot jump conditions as explained in Appendices D and E. Inthe case of a rarefaction wave, we numerically integrate the MHD equations asexplained in Appendix G. In this manner we reduce the problem to solving

[[vx, vy, Bx, p]](u4(u3(u2, χFR), χSR),u5(u6(u7, χFT ), χST )) = 0. (3.18)

The function χ 7→ [[vx, vy, Bx, p]] is not partially differentiable in all points of itsdomain. The price to pay is that we will postulate the wave configuration.

As said before, every magneto-acoustic signal is controlled by one path variable,such that uu(uk, χsignal), where signal is the signal separating uu from uk. Thepath variables should have a one-to-one relationship with the refraction angles


φsignal. The actually used path variables are given by

χF R = tan(φF R), (3.19)

χF T = tan(φF T ), (3.20)

χSR =

8

>

>

>

<

>

>

>

:

tanh−1

“

2“

tan φSR−tan φa,R

tan φcr,R−tan φa,R

”

− 1”

, ∀φSR ∈ [φcr,R, φa,R]

tanh−1

“

2“

tan φSR−tan φsl,R

tan φa,R−tan φsl,R

”

− 1”

, ∀φSR ∈ [φa,R, φsl,R[

tanh−1

“

2“

tan φSR−tan φst,R

tan φsl,R−tan φst,R

”

− 1”

, ∀φSR ∈ [φsl,R, φst,R[

(3.21)

χST =

8

>

>

>

<

>

>

>

:

tanh−1

“

2“

tan φST −tan φa,T

tan φcr,T −tan φa,T

”

− 1”

, ∀φSR ∈ [φa,T , φcr,T ]

tanh−1

“

2“

tan φST −tan φsl,T

tan φa,T −tan φsl,T

”

− 1”

, ∀φSR ∈ [φsl,T , φa,T [

tanh−1

“

2“

tan φST −tan φst,T

tan φsl,T −tan φst,T

”

− 1”

, ∀φSR ∈ [φst,T , φsl,T [

(3.22)

where we introduced the Alfvénic angles, the critical angles, the slow angles andthe stationary angles. The Alfvénic angles are those for which the upstream stateis exactly Alfvénic and these are defined by

φa,R = arctan

(

B3,y −√ρ3v3,y

B3,x −√ρ3v3,x

)

, (3.23)

φa,T = arctan

(

B6,y +√ρ6v6,y

B6,x +√ρ6v6,x

)

, (3.24)

and the critical angles at which the transition from 1 to 3 real solutions to the RHconditions across the slow signal takes place, by

φcr,R ≡ maxφ < φa,R|(N2 −D3)(φ) = 0, (3.25)

φcr,T ≡ minφ > φa,T |(N2 −D3)(φ) = 0. (3.26)

Finally, the slow angles φsl,R and φsl,T , are those for which the upstream state isexactly slow (and are found numerically) and the stationary angles are defined by

φst,R ≡ arctanvy,3vx,3

, (3.27)

φst,T ≡ arctanvy,6vx,6

. (3.28)

The functions N and D are defined in Appendix F, and these critical angles aredependent on the location of the fast signal.


Figure 3.5: The wave postulation for the slow transmitted signal. Just as inthe HD case, the signal is postulated to be a shock, whenever there exists anentropy increasing shock solution. This is the case whenever vn > vs,n. When theknown state is a 3-state, only a slow shock is possible, and the signal is triviallypostulated to be a slow shock. When the upstream state is super-Alfvénic, thesignal is postulated to be an intermediate shock.

Once the wave configuration is determined, χ determines the solution to theproblem. A secant method iteration solves equation (3.18) and φSC = arctan

vy,4

vx,4

closes the procedure.

Once the wave configuration is postulated, we know if the slow signal angles arebigger or smaller than the Alfvénic angles. The initial values of the path variablescontrolling the fast signals are taken such that the upstream states u2 and u7

are exactly fast. For the slow signals, initially we put χSR = χST = 0. In thecase of an intermediate shock, this means that our starting angles φSR and φSTrespectively satisfy 2φSR = φa,R + φcr,R and 2φST = φa,T + φcr,T . The procedurealso ensures that in every iteration step φa,T < φST < φcr,T and φcr,R < φSR <φa,R. Similarly, in the case of a slow shock, the initial guesses φSR and φSTrespectively satisfy 2φSR = φsl,R + φa,R and 2φST = φsl,T + φa,T . Finally, whenthe slow signal is located at φSR > φsl,R there is no shock solution possible whichsatisfies the entropy condition, and in the case the signal is a slow rarefaction fan.The initial iteration angles φSR and φST respectively satisfy 2φSR = φst,R + φsl,Rand 2φST = φst,T+φsl,T . We hereby postulated that the slow signals are expansionfans when v2

n,3/6(φSR/ST ) < v2s,n,3/6. In this case, the expansion fan is located

between φSR/ST and φsl,R/T . This criterion is equivalent to the criterion that ararefaction fan will only occur at a given position if no shock solutions satisfyingthe entropy condition are possible, and is in this sense a generalization of thecriterion exploited in Delmont et al.[19]. For the slow transmitted signal, thiswave postulation is schematically presented in figure 3.5.


3.4.3 Tackling the signals

The solution to the stationary Rankine-Hugoniot conditions is given in AppendixE. One problem is that the uniqueness of its solution is not guaranteed. We expressthe tangential component of the magnetic field in the downstream state, Bt,u, asthe root of a cubic, which coefficients are expressed in terms of the known upstreamstate. The unknown state is then expressed in terms of Bt,u.

It can be shown that if there is a unique solution to the RH conditions, it is afast or a slow shock. If on the other hand there are 3 solutions, they are all of adifferent shock type. Thus when the shock types of all magneto-acoustic signalsare known, the Rankine-Hugoniot conditions can be solved uniquely. We solve thecubic analytically in Appendix F, noting that roots can be real or complex. Inprinciple this completes the solution algorithm for the RH conditions of a genuineshock. However, the complications are that

• We need to select the appropriate root from the (up to) 3 mathematicalpossibilities, at each magnetosonic signal;

• The analytical expressions contain a square root and a cube root in thecomplex plane. These expressions are discontinuous in the negative realnumbers.

In Appendix F, we also provide additional technical information on howpermutation of the root indices can make the analytical expressions for the roots(F.6-F.8) continuous and differentiable, and hence allows for a secant iterationmethod applied on equation (3.18). The only remaining discontinuities are reached,when the refraction angles cross the critical angles φcr,R or φcr,T , where thetransition from a unique solution to 3 solutions for the RH conditions takes place.Also [[p]] is discontinuous across the Alfvénic angles. This leads to the restrictionthat we cannot cross those critical angles in subsequent iteration steps. This istaken care of by our choice of path variables.

In Appendix G finally, we describe which relations hold across expansion fans andhow the numerical integration through the fan is performed.

3.5 Demonstration of result

We study the case where (α, β, γ, η,M) =(

π4 , 2,

75 , 3, 2

)

, a case which was studiedin detail before by Samtaney [78] and Wheatley et al. [98]. For this case we havethat

DEMONSTRATION OF RESULT 81

u1 = (1.4000,−2.0000,−2.0000, 1.0000, 1.0000), (3.29)

u2 = (3.7333,−0.7500,−2.0000, 4.5000, 1.0000), (3.30)

u7 = (4.2000,−2.0000,−2.0000, 1.0000, 1.0000). (3.31)

The initial guesses for the iteration procedure are χFR, χSR, χST , χFT =(0.5314, 0, 0, 3.7306). The iteration converges to the exact solution for the anglesφFR, φSR, φST , φFT = (0.40569, 0.91702, 1.19426, 1.27673). The correspondingcubics whose roots need to be properly selected are shown in figure 3.6.

Discussing the solution of the refraction pattern in the order of integration, weencounter:

• the fast reflected (FR) signal, located at φFR = 0.40569. The cubic across theinterface is then evaluated and found to be 0.0346B3

t −0.5184B2t +7.2999Bt+

3.9011 = 0, which has, as predicted, only one real solution Bt = −0.5149,such that

u3 = (4.424,−0.834,−1.774, 5.710, 1.159) ,

and By,3 = 0.0685. Note that FR is a fast shock;

• the slow reflected (SR) signal, located at φSR = 0.91702. The cu-bic relation connected to this interface has 3 real solutions: Bt ∈0.13075 10−3, 0.34361,−0.02656. Further physical argumentation isneeded to select the correct root. The first possibility, Bt = 0.13075 10−3,leads to negative pressure. The second option, Bt = 0.34361, corresponds toan intermediate shock, and yields

u4,intermediate = (4.395,−1.290, 2.374, 5.656, 0.198) ,

together with By,4 = 1.186. The entropy condition p4p3<(

ρ4ρ3

)γ

is satisfied.

The last possible solution, Bt = −0.02656 corresponds to a slow shock. Inthis case

u4,slow = (4.596,−1.048,−2.028, 6.025, 0.736),

and By = 0.484, which also satisfies the entropy condition p4p3<(

ρ4ρ3

)γ

;

• the fast transmitted (FT) signal, located at φFT = 1.27673. The cubicrelation across this fast interface again has a unique real solution Bt =−1.10819. This solution corresponds to a fast shock and leads to

u6 = (11.932,−1.213,−2.384, 5.275, 1.237) ,

and By,6 = 0.783;


BK1.0 K0.5 0 0.5 1.0

F(B)

K6

K4

K2

2

4

6

8

10

12

BK0.2 K0.1 0 0.1 0.2 0.3 0.4

F(B)

K0.002

K0.001

0.001

BK0.3 K0.2 K0.1 0 0.1 0.2 0.3 0.4 0.5 0.6

F(B)

K0.010

K0.008

K0.006

K0.004

K0.002

0.002

0.004

0.006

BK2 K1 0 1 2

F(B)

K20

20

40

60

80

Figure 3.6: The cubic for respectively FR, SR, ST, FT.

• the slow transmitted (ST) signal, located at φST = 1.19426. The cubicrelation controlling this interface has 3 different real solutions Bt ∈0.44121,−0.70666 10−3, 0.07837. The first possibility, Bt = 0.44121 canbe eliminated since it does not lead to a solution which satisfies the entropycondition. The second possibility, Bt = −0.70666 10−3 would lead tonegative pressure. The remaining possibility, Bt = 0.07837 corresponds toan intermediate (2 → 4)-shock from u6 to u5 , and yields

u5 = (13.092,−1.048,−2.028, 6.025, 0.736) ,

with By,5 = −0.484, which satisfies the entropy condition.

We now notice that the correct reflected slow signal is the slow shock and uSR =u4,slow, since the shocked contact must satisfy the matching conditions (D.7).

3.6 Suppression of the Richtmyer-Meshkov Instability

Since vn vanishes along a contact discontinuity, the Rankine-Hugoniot jumpconditions simplify as

[[vt, p, Bt]] = 0.

Therefore, they do not allow for vorticity deposition on the contact, and theinterface must remain stable. Figure 3.7 shows snapshots of a numerical simulationof the parameter regime (α, β, γ,M, η) =

(

π4 , 2,

75 , 2, 3

)

performed by the AdaptiveMesh Refinement code AMRVAC. The first frame shows the density, ρ, during theshock refraction. Note that the CD remains stable, since the jump in tangentialvelocity, [[vt]], vanishes. The second frame shows the tangential magnetic fieldcomponent, Bt. The density is discontinuous in every new-formed signal, whilethe CD is not visible in the Bt frame. SR is a slow shock, therefore Bt does notchange sign across this signal, while ST on the other hand is an intermediate shockand Bt changes sign across it.

WAVE CONFIGURATION TRANSITIONS 83

The lower left panel of figure 3.7 shows the streamlines in the stationary shock

frame, over a Schlieren plot of density (exp(

200−|∇ρ|250

)

is shown). The streamlines

are coloured by the density profile. One sees that almost all the vorticity isdeposited on FT. The upper right panel of figure 3.8 Shows the magnetic fieldlines of this reference case. The lines are coloured by the magnetic field strength.One can see that Bt only changes sign at ST, which is the only intermediate signalin the solution. Also one sees that the magnetic field does not jump across theCD: [[B]]CD = 0.

3.7 Wave configuration transitions

We now wonder under which values of the plasma-β the same wave configurationwill be possible. We therefore ran our solver for varying values of the plasma-β, postulating that SR is a slow shock, and ST is an intermediate shock. Thesolver converges for β ∈ [1.0; 2.3]. For β ≈ 1.0, ST will be a switch-off shock, i.e.φST = φa,T , and when we decrease β even more, the slow transmitted signal willbe located even further away from the CD, i.e. φST > φa,T , such that ST nowbecomes a slow shock. On the other hand, when we decrease the magnetic fieldtill β ≈ 2.3, SR will be located at φSR = φa,R, such that SR is a switch-off shock.Decreasing the magnetic field even more such that β > 2.3 will result in a solutionwhere both SR and ST are intermediate shocks.

We compare the reference case ((α, β, γ, η,M) =(

π4 , 2,

75 , 3, 2

)

, see figure 3.7) to acase where β = 0.5, and a case where β = 4.

The magnetic field lines of the case where β = 0.5 are plotted in the left frame offigure 3.8. Here both the reflected and transmitted signals are slow shocks, as Btdoesn’t change sign across them. Finally, the case where β = 4 is shown in theright panel. As predicted, not only ST, but also SR is an intermediate shock now,as Bt changes sign across both the slow signals. In both cases the magnetic field

lines are plotted over a Schlieren plot of density (exp(

200−|∇ρ|250

)

is shown), and

the field lines are coloured by |B|.

3.8 Conclusion

We developed an exact Riemann solver based solution for regular shock refractionin planar ideal MHD. In this case, five signals arise: a CD separates two reflectedsignals from two transmitted signals. We postulate both the fast signals tobe shocks, while the slow signals can be slow shocks, intermediate shocks orrarefaction fans. In the presented case, the transmitted slow signal turns out to


Figure 3.7: Up: Density plot for (α, β, γ, η,M) =(

π4 , 2,

75 , 3, 2

)

. Five signals ariseand their location is found by the iteration procedure. Iterating one more timeyields u(x) and solves the problem. Note that the CD remains stable. Middle: Aplot of the transverse magnetic field, Bt. Note that the transverse magnetic fieldis continuous across the CD: [[Bt]] = 0. Also note that Bt changes sign across ST.Indeed, ST is an intermediate shock. Lower left: Streamlines in the stationaryshock frame, over a Schlieren plot of density. Lower Right: The magnetic fieldlines show ST is an intermediate shock, while SR is a slow shock.

CONCLUSION 85

Figure 3.8: Left: Magnetic field lines fot the case (α, β, γ, η,M) =(

π4 , 0.5,

75 , 3, 2

)

.Both the slow signals are slow shocks. Right: Magnetic field lines for the case

(α, β, γ, η,M) =(

π4 , 4,

75 , 3, 2

)

. Both the slow signals are intermediate shocks.

be an intermediate shock, which agrees with our numerical AMRVAC simulation.In the hydrodynamical case, the Richtmyer-Meshkov instability will occur afterthe shock reflects from the top wall. Since the MHD equations do not allow forvorticity deposition on a contact discontinuity, this instability is suppressed in thepresence of magnetic fields (with a non-vanishing normal component).

Chapter 4

Parameter ranges forintermediate shocks

We investigate under which parameter regimes the MHD Rankine-Hugoniotconditions, which describe discontinuous solutions to the MHD equations, allowfor slow, intermediate and fast shocks. We derive limiting values for the upstreamand downstream shock parameters for which shocks of a given shock type canoccur. We revisit this classical topic in nonlinear MHD dynamics, augmenting therecent time reversal duality finding by Goedbloed [33] in the usual shock frameparametrization.

This work is published in Delmont & Keppens [21].

4.1 Introduction

4.1.1 Intermediate shocks in MHD

The dynamic behavior of plasmas is described by the magnetohydrodynamic(MHD) equations, where a central role is played by the Alfvén speed. Discontinu-ous solutions only satisfy the integral form of the MHD equations, i.e. the Rankine-Hugoniot conditions (RH). These RH conditions have been studied extensively inestablished as well as more recent literature (see e.g. Germain [30]; Anderson [5];Jeffrey & Taniuti [46]; Liberman & Velikhovich [56]; Sturtevant [80]; Gombosi [35];Goedbloed [33]) and just express the basic nonlinear conservation laws across adiscontinuity. Many authors have since paid attention to MHD shock stability aswell (amongst others Akhiezer et al.[4]). The interaction of small perturbations

87

88 PARAMETER RANGES FOR INTERMEDIATE SHOCKS

with MHD (switch-on and switch-off) shocks was studied both analytically byTodd [84] and numerically by Chu & Taussig [17]. Later on, the evolutionarityof intermediate shocks, which cross the Alfvén speed, has been addressed. Adiscontinuity is said to be evolutionary when small perturbations imposed on itlead to an evolution that remains close to the initial discontinuity. According toclassical stability analysis, intermediate shocks are not evolutionary so one shouldnot obtain such shocks in physically realizable situations. On the other hand,Wu [99], De Sterck et al.[22] and others found intermediate shocks in one andtwo dimensional numerical simulations respectively. De Sterck & Poedts [23] werethe first to find intermediate shocks in three dimensional numerical simulations.The evolutionary condition became controversial and Myong & Roe [60, 61],amongst others, argue that the evolutionary condition is not relevant in dissipativeMHD. Wu [100, 101] also argued that intermediate shocks are admissible. Shockobservations in the heliosphere can be found in favor of their existence as well:Chao et al.[12] reported a 2 → 4 intermediate shock observed by Voyager 1 in1980 and Feng & Wang [29] recognized a 2 → 3 intermediate shock, which wasobserved by Voyager 2 in 1979.

Intermediate shocks are not the only way to connect a sub-alfvénic state to asuper-alfvénic state. One can also encounter compound waves in ideal MHD.These compound waves can consist of a slow shock which travels with its maximalpropagation speed and a rarefaction fan directly attached to it. Brio & Wu[11] detected those compound waves in numerical simulations which have becomeclassical test problems for numerical codes. Another type of compound signalconsists of a slow shock layer, immediately followed by a rotational discontinuity(Wheatley et al.[97]). The importance of compound waves is that they can be analternative to intermediate shocks to cross the Alfvén speed. Barmin et al.[8]argue that if the full set of MHD equations is used to solve planar MHD, asmall tangential perturbation on the magnetic field vector splits the rotationaljump from the non-evolutionary compound wave. This transforms the latterone into a slow shock, such that the compound wave is non-evolutionary. Theyinvestigate the reconstruction process of the non-evolutionary compound wave intoevolutionary shocks. Falle & Komissarov [26, 27] also reject intermediate shockson evolutionary grounds, and suggest a shock capturing scheme based on Glimm’smethod (Glimm [31]) for numerically obtaining purely evolutionary solutions inMHD. Our goal in this chapter is to determine in which regions of parameter spaceone might encounter intermediate shocks, which play such a prominent role in allevolutionarity argumentation.

Recently, Goedbloed [33] classified the MHD shocks by rewriting the RH equationsin the de Hoffmann-Teller frame (De Hoffmann & Teller [39]) introducing theexistence of a distinct time reversal duality between entropy-allowed and entropy-forbidden solutions. This work encouraged us to revisit the classical RH conditions

INTRODUCTION 89

and augment these results in terms of the commonly exploited shock parametersin the shock frame. Therefore, another goal in this chapter is to determine inwhich regions of parameter space slow, intermediate and fast shocks can appearand how this relates to Goedbloed’s analysis, in particular regarding the dualitybetween solutions.

4.1.2 The Rankine-Hugoniot jump conditions

The ideal MHD equations are a system of partial differential equations. Whenallowing for large amplitude waves, which in the limit case become discontinuous,these equations are replaced by their (weak) discontinuous form: the RH relations.These RH relations express jump conditions across the discontinuity. In any framewhere the shock is stationary, the MHD RH conditions become

ρvn

ρv2n + p+

B2t

2ρvnvt −BnBt

vn(γγ−1p+ ρ

v2n+v2t2 +B2

n) −BnBtvtvnBt − vtBn

Bn

= 0, (4.1)

where [[G]] expresses the jump G2 − G1 across the shock. In this chapter, index2 refers to the downstream state, and index 1 refers to the upstream state, whileindex n refers to the direction of the shock normal, and index t refers to thetangential vector components in the plane spanned up by B1 and B2. Further, ρ isthe mass density, v is the velocity, p the thermal pressure and B the magnetic field.The ratio of specific heats, γ, is considered a constant parameter, as we will assumean ideal gas equation of state. For a derivation of these well-known expressions, werefer to De Hoffmann & Teller [39]; Jeffrey & Taniuti [46]; Liberman & Velikhovich[56], Goedbloed & Poedts [32]. The eight governing MHD equations have beenreduced to six jump conditions in equation (D.1). Three equations have beendropped from the fact that tangential magnetic field components are forced tolie in the same plane perpendicular to the shock front itself: the conservationof momentum reduced to two equations and Faraday’s law reduces to a singleequation. On the other hand, in the stationary case, the ∇ · B = 0 constraintbecomes a full equation and is added to equation (D.1).

These six jump conditions can be further restricted by noting that Bn is invariantacross the shock. Since the RH conditions are translation invariant, the exactvalues of vt,1 and vt,2 can be eliminated for [[vt]], which is completely determinedby [[Bt]]. Therefore only four quantities really matter. Hence we now supposethat one state, uk = (ρ, vn, p, Bt)k is known, where from now on we consistentlydrop the subscript k from known state quantities. We want to express the other


unknown state, uu, in function of this known state. In order to do so, we introducethree dimensionless parameters connected to the known state. First, the plasma-β,which expresses the ratio of thermal and magnetic pressure. Then, the tangent θof the angle between the shock normal and the magnetic field in the known state.Finally the Alfvén Mach number M , which is the ratio of the normal velocitycomponent in the known state and the normal Alfvén speed in that region, thus

β =2p

B2n +B2

t

, (4.2)

θ ≡ BtBn

, (4.3)

M ≡ |vn|va,n

. (4.4)

Here, the normal Alfvén speed va,n ≡√

B2n

ρ is defined as earlier (in section 3.2).

Analogously, we define the tangential Alfvén speed va,t ≡√

B2t

ρ and the total

Alfvén speed va ≡√

v2a,n + v2

a,t as in section 3.2. The full set of characteristic

speeds of the full set of MHD equations is given by vn±vf,n, vn±va,n, vn±vs,n, vn,where the fast speed vf,n and the slow speed vs,n are respectively defined as inequations (1.48) and (1.48). On the other hand, the restricted set of planar MHDequations, describing MHD dynamics with all vector quantities restricted to liein the same plane, has only five characteristic speeds vn ± vf,n, vn ± vs,n, vn.Ignoring cold MHD, where the thermal pressure p vanishes, and assuming thatthe normal component of the magnetic field Bn (which can be seen as an equationparameter now) does not vanish, we conclude that the planar system is strictlyhyperbolic, since vf,n = vs,n would imply that both va and c would vanish.Obviously c cannot vanish, and for the same reason also vs,n 6= 0.

4.1.3 MHD shock types: classical 1 − 2 − 3 − 4 classification

Since the RH conditions do not necessarily lead to a unique solution, it is commonto introduce another discrete characterization: the shock type i → j. It is well-known (see e.g. Liberman & Velikhovich [56]) that once the shock type and onestate is given, if a solution exists, it must be unique. Since 0 ≤ vs,n ≤ va,n ≤ vf,n,the full set of MHD equations is hyperbolic, but non-strictly hyperbolic. Thosecharacteristic speeds categorize the up- and downstream states into the classical1 − 2 − 3 − 4-classification as explained in section 3.3.

We call a shock admissible if it satisfies the second law of thermodynamics: [[S]] >0 and admissible versus inadmissible shocks can be related through the time duality

SOLUTION TO THE RANKINE-HUGONIOT CONDITIONS 91

principle from Goedbloed [33]. When the upstream state is of type i and thedownstream state is of type j, then the shock type is i → j. Furthermore, interms of these shock types, the admissibility condition translates as i < j. TheRH conditions (D.1) together with the admissibility condition (i < j) now lead tothe MHD shock classification presented in section 3.3. All these different shocktypes are shown in figure 3.4.

The classification given in section 3.3 is well-known and features in many classicaltextbooks, such as Liberman & Velikhovich [56]. Recently, a contribution to thisclassical theory was described by Goedbloed [33], where the RH conditions forMHD shocks were rewritten in an insightful, symmetric form by changing to thede Hoffmann-Teller frame, where [[vt]] = [[Bt]] (see De Hoffmann & Teller [39]). Inthat work, the central role played by intermediate shocks was emphasized, and timereversal duality arguments were introduced, linking admissible and inadmissibleshocks. Here we augment this recent study by paying attention to how the MHDshocks appear in the shock reference frame, and analyzing under which conditionsintermediate shocks occur.

4.2 Solution to the Rankine-Hugoniot conditions

We scale densities to the known density ρ and magnetic fields to Bn (Bn is constantand equal for known and unknown state, as seen in equation (D.1)). Since nodistances or times are involved, the problem is now uniquely scaled. Note thatvelocities are scaled to the known normal Alfvén speed va,n. The known state isnow given by

u ≡ (ρ, vn, p, Bt) =

(

1, σM,β(1 + θ2)

2, θ

)

, (4.5)

where σ ≡ −1 when the known state is upstream and σ ≡ 1 when the known stateis downstream, and σ just gives the proper sign to the known normal velocitycomponent.


Under the assumption that M 6= 1 and θ 6= 0, solving the RH equations leads tothe unknown state quantities

ρu =M2ψ

θ(M2 − 1) + ψ, (4.6)

vn,u = σθ(M2 − 1) + ψ

Mψ, (4.7)

pu =

(

2

γ + 1+ψ2 + θψ + 2

M2 − 1

)

M2 − γ − 1

γ + 1

β(1 + θ2) + (ψ − θ)2

2, (4.8)

Bt,u = ψ, (4.9)

where ψ satisfies the cubic equation

C(ψ) ≡ ψ3 + τ2ψ2 + τ1ψ + τ0 = 0, (4.10)

and its coefficients are given by

τ2 = −θ(

(γ − 1)(M2 − 1) −M2)

, (4.11)

τ1 = (M2 − 1)(

(γ − 1)(M2 − 1) + γ(β(θ2 + 1) + θ2) − 2)

, (4.12)

τ0 = −(γ + 1)θ(M2 − 1)2. (4.13)

The tangential velocity is then found from

[[vt]] =θ − ψ

M. (4.14)

We can now define the dimensionless parameters referring to the unknown statein function of those in the known state. In our first view, we will keep ψ in theexpressions, noting that the cubic equation (4.10) can be seen as a continuouslypartially differentiable function of the known dimensionless parameters and ψ, thusC(M, θ, β, ψ), such that the implicit function theorem ensures that locally ψ can beseen as a ψ(M, θ, β). Moreover ψ(M, θ, β) is continuously partially differentiablewhenever ∂C

∂ψ (M, θ, β, ψ(M, θ, β)) does not vanish. This latter restriction meansthat ψ cannot be a double root of the cubic. As we will show later, this condition isequivalent to the condition that none of the characteristic speeds in the unknownregion vanishes.

PHYSICAL MEANING OF Ω 93

After straightforward algebra, the expressions for (Mu, θu, βu) now become

Mu =

s

(M2 − 1)θ + ψ

ψ, (4.15)

θu = ψ, (4.16)

βu =

`

(γ − 1)((θ − ψ)2 + (1 + θ2)β) − 4M2´

(M2− 1) + 2M2(ψθ + ψ2)

(M2 − 1) (γ + 1) (1 + ψ2).(4.17)

In terms of the dimensionless parameters, these equations lead to the followingwell-known (see e.g. Goedbloed [33]) invariants across a shock:

[[(M2 − 1)θ]] = 0, (4.18)

[[

2M2 + β(1 + θ2) + θ2]]

= 0, (4.19)

[[

(γ

γ − 1β +M2)(1 + θ2)M2

]]

= 0. (4.20)

When we define the new quantity Ω, computed from the coefficients of thegoverning cubic equation (4.10), as

Ω ≡ 27τ20 + 4τ3

1 + 4τ22 τ0 − τ2

2 τ21 − 18τ2τ1τ0, (4.21)

it is well-known that the cubic C(ψ) has three real solutions when

Ω < 0. (4.22)

Note that this criterion is equivalent to equation (F.3).

Mathematically speaking, the RH conditions are thus governed by the existenceand multiplicity of the real roots of a cubic, hence Ω = 0 will play a crucialrole, as will be explained in section 3. It should be noted, however, that mostanalytical expressions in this choice of frame and parametrization are complicatedexpressions, e.g. one can note that Ω is a polynomial of degree 6 in M2, where(M2 − 1) appears as a double factor. The same governing equations, as expressedin the de Hoffmann-Teller frame exploited by Goedbloed [33] give a much morecompact algebraic form along with an easier classification.

4.3 Physical meaning of Ω

Note that Ω = 0 is exactly the condition for having a real solution with multiplicitytwo to C(ψ) = 0. Therefore, whenever Ω 6= 0, (Mu, θu, βu) is a smooth function of


Figure 4.1: Left: The (θ,M) state plane for β = 110 and γ = 5

3 . Shown are thecurves fast: vn = vf,n, Alfvén: vn = va,n and slow: vn = vs,n. These curvesseparate the (θ,M) plane in the classical 1− 2− 3− 4 state regions. Right: Shownis the curve Ω = 0. Note that states with M = 1 appear as a double zero of Ω.Where Ω > 0, only one real solution to the RH conditions exist. On the otherhand, where Ω < 0, there exist three real solutions to the RH conditions. OnΩ = 0, the RH solutions allow for 2 distinct real solutions.

(M, θ, β). A related observation is that equation (4.22) is the analytical conditionfor the existence of an intermediate shock, since when only one solution exists tothe RH jump conditions it must be a fast or a slow shock (see e.g. Liberman &Velikhovich [56]).

We will argue that all the states, which can be connected to a state which isexactly fast, slow or Alfvénic, satisfy Ω = 0. In other words: the surfaceω ≡ (M, θ, β) |Ω(M, θ, β) = 0 in our 3D parameter space will be shown tocorrespond to all states which can be connected by the RH conditions to knownstates satisfying (vn − va,n)(v

2n − v2

f,n)(v2n − v2

s,n) = 0. One can visualize theseconcepts by drawing the corresponding curves in the (θ,M)-plane, for fixed β andγ. This is done in figure (4.1), where the plot at left concentrates on the 1−2−3−4state regions, and the panel at right shows Ω = 0 curves.

First, we make some general observations. 1-states can be connected to a 2-state, a3-state and a 4-state. Therefore, they can only appear as an upstream state. Thereare at most three different real solutions with a 1-state as an known upstreamstate. When there is only one solution, this solution is a fast 1 → 2 shock. In thetransition case where two different real solutions exist, these solutions include onefast (1 → 2) shock, and two coinciding intermediate 1 → 3 = 4 shocks. Finally,when there are three different real solutions, one of these is a fast (1 → 2) shock and

PHYSICAL MEANING OF Ω 95

the 2 other solutions are respectively an intermediate 1 → 3 and an intermediate1 → 4 shock.

Similarly, 2-states can be connected to a 1-state, a 3-state and a 4-state. Therefore,the admissibility condition which demands that i < j for an i → j shock, ensuresthat a 2-state can, at most, occur in one downstream state solution and twoupstream state solutions to the RH conditions. Again, when only one real solutionexists, it is a fast (1 → 2) shock. In the transition case where two differentreal solutions exist, these solutions include one fast shock and two coincidingintermediate 2 → 3 = 4 shocks. Finally, when there are three different realsolutions, one of these is a fast shock, and the 2 other solutions are respectivelyan intermediate 2 → 3 and an intermediate 2 → 4 shock.

The same reasoning states that generally speaking, 3-states can be connected toa 1-state, a 2-state and a 4-state. The admissibility condition ensures that theycan, at most, occur in two downstream state solutions and one upstream statesolution to the RH conditions. Now, when only one real solution exists, it is aslow (3 → 4) shock. In the transition case where two different real solutions exist,these solutions include one slow (3 → 4) shock and two coinciding intermediate1 = 2 → 3 shocks. Finally, when there are three different real solutions, one ofthese is a slow shock, and the 2 other solutions are respectively an intermediate1 → 3 and an intermediate 2 → 3 shock.

Generally speaking, 4-states can be connected to a 1-state, a 2-state and a 3-state.Therefore, the admissibility condition ensures that they can, at most, occur inthree downstream state solutions to the RH conditions. Again, when only onereal solution exists, it is a slow (3 → 4) shock. In the transition case where twodifferent real solutions exist, these solutions include one slow (3 → 4) shock andtwo coinciding intermediate 1 = 2 → 4 shocks. Finally, when there are threedifferent real solutions, one of these is a slow (3 → 4) shock and the 2 othersolutions are respectively an intermediate 1 → 4 and an intermediate 2 → 4 shock.

These general observations now allow us to consider the transition cases. A 1 = 2-state, where vn = vf,n or a 3 = 4-state, where vn = vs,n, will always have thetrivial solution. Therefore they can only appear in none or two real non-trivialintermediate solutions, depending on the sign of Ω. Both these solutions, labeledas unknown state a and b, have uk as a double solution, thereby satisfying Ωu,a =Ωu,b = 0, and as we will show in section 4.2 mutually satisfy the RH conditions.This is argued in what follows.

Consider an upstream state which satisfies vn = vf,n. Straightforward algebrashows that C(θ) = 0, so one of the solutions to the RH jump conditions is the

trivial one. By solving C(ψ)ψ−θ = 0, one finds 2 solutions. Those solutions are real

only when Ω ≤ 0. In this case they lead to a 1 = 2 → 3 and a 1 = 2 → 4 solution,which we respectively call uu,a and uu,b. These uu,a and uu,b mutually satisfy the


RH conditions and can therefore be separated by a slow 3 → 4 shock. Both thesesolutions have our specific known upstream state uk, where vn = vf,n, as a doublesolution, and therefore satisfy Ωu,a = Ωu,b = 0.

Completely analogously, consider now a downstream state with vn = vs,n.Straightforward algebra shows that C(θ) = 0, so one of the solutions to the RH

jump conditions is the trivial one. By solving C(ψ)ψ−θ = 0, one finds 2 solutions.

Those solutions are real only when Ω ≤ 0. In this case they lead to a 1 → 3 = 4 anda 2 → 3 = 4 solution, which we respectively call uu,a and uu,b. We again find thatuu,a and uu,b mutually satisfy the RH conditions and can therefore be separatedby a fast shock. Both these solutions have our specific known downstream stateuk, where vn = vs,n, as a double solution, and therefore satisfy Ωu,a = Ωu,b = 0.

Finally, consider a state where vn = va,n. Now C(ψ) simplifies as ψ3 + θiψ2 =

0, where i = 1, 2 selects respectively the up- or the downstream solution. Thesolutions to the RH conditions are now a switch-on shock, a switch-off shock anda rotational wave. The switch-on and the switch-off solution also mutually satisfythe RH jump condition. This can be easily checked by straightforward algebra.Note that in this case Ωu = 0 automatically.

4.4 Results

Since the governing expressions in the shock frame are complicated, all calculationsreported here are performed by a symbolic computational software package,namely MAPLE 11.0. We search for critical values for the parameters in bothup- and downstream states, for which different shock types can occur. Thesecritical values are found by simple geometrical arguments using the knowledgeobtained so far: namely that in the (θ,M)-plane, 2 families of curves as shownin figure 4.1 split up the state plane in various regions. Unless explicitly statedotherwise, we assume from now on that γ = 5

3 .

4.4.1 (θ, M)-diagrams at fixed β

For varying β, we plot both the curves where the known upstream normal velocitycomponent vn is exactly fast, Alfvénic or slow (i.e. vn = vf,n, vn = va,n, vn =vs,n) and the curves ω : Ω = 0 in a (θ,M)-diagram in figure 4.2. Those curvesdivide the (θ,M)-parameter space in regions, where certain types of shocks aremathematically possible. An important transition occurs at β = 2

γ . Here thesound speed equals the Alfvén speed and the thermal pressure equals the magneticpressure. When β < 2

γ , we call the plasma magnetically dominated. In this case,

the curves vn = vf,n and vn = va,n have a common point at (θ,M) = (0, 1). When

RESULTS 97

the plasma is thermally dominated, β > 2γ , the curves vn = vs,n and vn = va,n

have a common point at (θ,M) = (0, 1).

Every region is then coded with a latin number code for (θ,M), indicative of anupstream state, i.e. (θ1,M1). This means:

• (I) One fast shock and two intermediate shocks of type 1 → 3 and 1 → 4 arepossible;

• (II) Only a fast shock is possible;

• (III) Two intermediate shocks of type 2 → 3 and 2 → 4 are possible;

• (IV) No shocks are possible;

• (V) Only a slow shock is possible.

We can interpret the graphs in terms of the downstream state in a similar manner,where (θ,M) = (θ2,M2). Every region is also coded with a letter code, meaningthe following:

• (A) One slow shock and two intermediate shocks of type 2 → 4 and 3 → 4are possible;

• (B) Only a slow shock is possible;

• (C) Two intermediate shocks of type 1 → 4 and 2 → 4 are possible;

• (D) No shocks are possible;

• (E) Only a fast shock is possible.

These regions are shown in figure 4.2, where the axes labels must be interpretedas (θ2,M2) instead of (θ1,M1) from above.

The 1 − 2 − 3 − 4 classification divides the (β,M, θ)-space into four regions. Thesurface ω : Ω = 0 divides each of those regions into two: one region where Ω > 0and one where Ω < 0, such that the (β,M, θ) space is divided into eight regionsby these curves.

The time reversal duality principle from Goedbloed [33] is hereby made visiblein the fact that every region corresponds to 1 or 3 mathematical solutions, butthe coding tells whether the state can appear as an upstream or as a downstreamregion.


(a) (b)

(c) (d)

RESULTS 99

(a) (b)

Figure 4.2: Parameter space divided into various regions. Panels differ in theirβ-value: a) β = 0.1, b) β = 1.0, c) β = 1.2 = 2

γ , d) β = 1.3, e) β = 1.3637, f)β = 1.5. The latin cypher-letter code is explained in the text, and related to thetime reversal duality argument put forth by Goedbloed [33].

4.4.2 Equivalence classes introduced by the RH conditions

The RH conditions are equivalent to equations (4.18-4.20), which express theexistence of three shock invariants. Therefore two states can be connectedthrough the stationary RH conditions if and only if they have the same valuefor the expression ζ1 ≡ (M2 − 1)θ, ζ2 ≡ 2M2 + β(1 + θ2) + θ2 and ζ3 ≡( γγ−1β + M2)(1 + θ2)M2. Denoting the relation "state A can be connected to

state B through the stationary RH conditions" as A 7→RH B, this relation 7→RH isan equivalence. Indeed:

• 7→RH is reflexive: A 7→RH A. Every state can be connected to itself throughthe stationary RH conditions.

• 7→RH is symmetric: A 7→RH B ⇒ B 7→RH A. If state A can be connectedto state B through the stationary RH conditions, then also state B canbe connected to state A by these conditions. Of course only one of theseconnections satisfies the entropy condition.

• 7→RH is transitive: A 7→RH B ∧ B 7→RH C ⇒ A 7→RH C. Indeed: if A 7→RH B,then ζi(A) = ζi(B), and if B 7→RH C, then ζi(B) = ζi(C). Hence A 7→RH

B ∧B 7→RH C implies ζi(A) = ζi(B) = ζi(C), which means that A 7→RH C.


We conclude that (ζ1, ζ2, ζ3) defines equivalence classes on the parameter space.All these equivalence classes contain one, two, three or four states.

A state A is in an equivalence class with one element in the following cases:

• If M = 1 and θ = 0. In this degenerate case the switch-on solution, theswitch off-solution and the rotational solution all coincide.

• If Ω > 0, and (v2n − v2

s,n)(v2n − v2

f,n) = 0. In this case, the only real solutionto the RH conditions is A itself.

A state A is in an equivalence class with two elements, in the following cases.

• If Ω > 0 and (v2n − v2

s,n)(v2n − v2

f,n) 6= 0. In this case the equivalence classcontains A itself, and the state introduced by the unique (non-trivial) realsolution to the cubic equation (4.10).

• If Ω = 0 and (v2n − v2

s,n)(v2n − v2

f,n) = 0. Since Ω = 0, the solutions to theRH conditions crossing the Alfvén speed are coinciding. Since vn is exactlyfast or exactly slow, it appears itself as a solution to the cubic.

A state A is in an equivalence class with three elements in the following cases.

• if Ω = 0, but (v2n−v2

s,n)(v2n−v2

f,n) 6= 0. As explained earlier, in this case theRH conditions have an exactly slow or fast state as a double solution, and asingle solution which also satisfies Ω = 0.

• if Ω < 0 and vn = vf,n or vn = vs,n. As explained earlier, in this case the RHconditions have 2 different solutions with Ω = 0, and the considered stateitself as a third solution.

A state is in an equivalence class with four elements in the following case:

• if Ω < 0 and vn 6= vf,n and vn 6= vs,n. As explained above, in this case the RHconditions have three different (non-trivial) solutions. Also, the consideredstate itself is in the equivalence class.

• if vn = va,n. The RH conditions now have a switch-on shock, a switch-off shock and a rotational wave as a solution. The considered state itselfcompletes the equivalence class.

Hence, when A is a superfast state, with Ω > 0, it is in an equivalence class withexactly one other state. This state should be subfast, since we know that if only

RESULTS 101

one solution exists, it does not cross the Alfvén speed. Also, the solution statehas only one solution, therefore it also satisfies Ω > 0. Interpreting this resulton figure 4.2, we conclude that the II − C-region can only be connected to theIV − E-region and the other way around.

Completely analogously we find that a subslow state with Ω > 0 can only beconnected with a superslow state, also satisfying Ω > 0. Interpreting this againon figure 4.2, we conclude that the IV − B-region can only be connected to theV −D-region, and vice versa. Further, we conclude that for any state in the regionsI−D, III−E, V −C and IV −A, there exist states in the regions I−D, III−E,V − C and IV − A respectively, which can be mutually connected through theRH jump condition. The initial eight regions of parameter space have thus beendivided into two groups of two mutually connectable regions and one group of fourmutually connectable regions.

If we do not take into account that βu should be positive, these connections aremappings. We know the entropy condition in an i → j shock reduces to j > i.Thus, when we consider one equivalence class we have four or two states, such thatthe entropy increases with state type. Therefore, we know that when one of themathematical solutions has negative pressure, it must be the 1-states. When twoof those states have negative pressure, it must be the 1-state and the 2-state, andso on. In the next section, we will additionally consider the physical restrictionthat all pressures should be positive.

4.4.3 Positive pressure requirement

Until now we have only used the nonlinear relations expressed by RH, as wellas the admissibility condition, to count the number of solutions in the (θ,M, β)state space. The admissibility restriction is that [[S]] > 0, which is satisfied for ani → j-shock if and only if j > i. Further restrictions are that the solution shouldhave positive thermal pressure, pu > 0 and density ρu > 0. The latter restrictionis trivially satisfied. In fact, we can only encounter problems when the knownstate is a downstream state, because when the upstream pressure is positive theupstream entropy is also positive. Hence, the downstream entropy is positive andthe downstream pressure too. Therefore, we can expect that for a given 1-state,the positive pressure requirement is trivially satisfied. When the known state isa 2-state, we expect to encounter one critical surface dividing the 2-state regionsinto sub-regions where the fast (1 → 2) shock solution has positive versus negativethermal pressure. When the given state is a 3-state, we expect to encounter two ofsuch critical surfaces. And when the given state is a 4-state we expect to encounterthree of such critical surfaces. Therefore, the parameter space is now divided in16 regions, as summarized in table (4.1).


Ω < 0 Ω > 01-state 1 12-state 2 23-state 3 14-state 4 2

Table 4.1: The positive pressure requirement divides the eight regions in parameterspace in even more regions. The number of sub-regions in which the original regionsare divided are shown above. Since the region is only important when the givenstate is downstream, it is not surprising that the total number of regions is doubledfrom 8 to 16.

Figure 4.3 shows the regions in which pu > 0 for β = 0.1 and β = 2. The figurenow plots: (i) the three curves defining vn = vs,n, vn = va,n and vn = vf,n; (ii) thecurves where Ω = 0; and (iii) the lines defining pu = 0. Finding these regions isstraightforward: pick the correct root of the cubic, fill it out in the expression forpu and make it vanish. The governing expressions can even be found analytically.As mentioned above, the addition of these pu = 0 curves now divides the parameterspace into 16 regions, namely:

• (i) This state can occur as the upstream state of a fast shock;

• (ii) This state can occur as the downstream state of a fast shock;

• (iii) This state cannot occur, since the only solution has negative pressure;

• (iv) This state can occur as the upstream state of a fast shock, and as theupstream state of an intermediate 1 → 3 and 1 → 4 shock;

• (v) This state can occur as the upstream state of an intermediate 2 → 3 and2 → 4 shock, and as the downstream state of a fast shock;

• (vi) This state can occur as the upstream state of an intermediate 2 → 3and 2 → 4 shock, but not as the downstream state of a fast shock, since thesolution would have a negative thermal pressure;

• (vii) This state can occur as the upstream state of a slow shock, or as thedownstream state of an intermediate 1 → 3 and 2 → 3 shock;

• (viii) In this region, one of the three solutions has negative pressure. Thiswill always be the 1 → 3-solution. Therefore this region can occur as theupstream state of a slow shock, or as a downstream state of an intermediate2 → 3 shock;

RESULTS 103

Figure 4.3: The 16 regions in parameter space where pu > 0, respectively forβ = 0.1 and β = 2.


• (ix) This region can occur as the upstream state of a slow shock;

• (x) This state has two solutions with negative pressure: namely bothintermediate solutions. Therefore it can only occur as the upstream state ofa slow shock;

• (xi) This state can occur as the downstream region of a slow shock;

• (xii) This state has two solutions with negative pressure: namely bothintermediate solutions. Therefore it can only occur as the downstream stateof a slow shock;

• (xiii) This state cannot occur, since the single real solution has negativepressure.

• (xiv) This state cannot occur, since all 3 solutions have negative pressure.

• (xv) This state can occur as the downstream state of a slow shock, andas the downstream state of an intermediate 2 → 4 shock, but not as thedownstream state of an intermediate 1 → 4 shock, since the solution wouldhave a negative thermal pressure;

• (xvi) This state can occur as the downstream state of a slow shock, and asthe downstream state of an intermediate 1 → 4 and 2 → 4 shock;

The combination of graphing the surface defined by Ω = 0, together with thesurfaces defined by vn = vs,n, vn = va,n, vn = vf,n and the positive pressurerequirement hence provides a complete, but admittedly non-trivial, graphicalmeans to the many possibilities for the MHD shock transitions. We now continueto exploit this knowledge to delimit possible parameter ranges for certain shocktypes. As a matter of fact, in what follows, we will use figure 4.3 and how thisfigure varies with β, to find limiting values of the parameters at which differentshock types can occur. By doing so, we will actually find the equations of thecurves traced out in (θ,M, β) parameter space that are labeled with P, Q, R, S, T,U and V as indicated in these plots. These points identify the intersections of thevarious regions, and thus act to delimit realizable parameter ranges for shocks.

4.4.4 Switch-on shocks and switch-off shocks

Switch-on shocks are possible where θ = 0. An extra condition for switch-onshocks to be possible is Ω < 0, since otherwise only fast shocks are possible. Notethat switch-on shocks are only possible when β < 2

γ . In this case the plasma issaid to be magnetically dominated. Therefore the maximum value at which we can

RESULTS 105

find switch-on shocks, is found by filling θ = 0 out in the expression for Ω. Thisworks out to be

1 < Ms-on <

√

γ(1 − β) + 1

γ − 1. (4.23)

For the same reason, switch-off shocks can be found in a completely similar manner,yielding

√

γ(1 − β) + 1

γ − 1< Ms-off < 1, (4.24)

whenever γ(1−β)+1γ−1 > 0, i.e. when β < γ+1

γ . This agrees with the well-known

expressions for these degenerate shock cases (e.g. Kennel et al.[47]).

When solving the RH conditions for θ = 0, when β < γ+1γ , two switch-on or

switch-off solutions exist. For γ = 53 , these solutions are given by

βu =1 − 2M2 − β

(γ − 1)M4 + (γ(β − 2))M2 + γ(1 − β)− 1 (4.25)

Mu = 1 (4.26)

θu = ±√

((γβ − 2) − (γ − 1)(M2 − 1))(M2 − 1). (4.27)

Finally, there is also another solution, which is a hydrodynamical shock. Thissolution is given by

(βu,Mu, θu) =

(

6M2 − β

4,

√

2M2 + 5β

8, 0

)

, (4.28)

which only has positive pressure for β < 6M2.

Figure 4.4 shows the plane given by θ = 0 in parameter space. The graphs plottedare (i) the curves given by Ω = 0; (ii) the curve vn = va,n; (iii) the curvegiven by pHD ≡ 6M2 − β = 0, which is the limiting curve for the existence ofa hydrodynamical shock solution; and (iv) the curve given by ps-off ≡ −4M2 −2 β+2+5M2β+2M4 = 0, which is a limiting curve for the existence of a switch-off solution. The entropy condition ensures that there is no equivalent limitingcurve for switch-on solutions. Note that the known state of a switch-on shock,where θ = 0 is always magnetically dominated, whereas the known state of aswitch-off shock, where θ = 0 is always thermally dominated. Also shown are thecurves where the hydrodynamical shock solutions and the switch-on and switch-off


solution have pu = 0. Again, the intersection point, separating different regions,are labeled with A,B,C and D and used in what follows.

We conclude that the maximum Mach number at which switch-on solutions exist,is reached in point A, for which (θ,M, β) = (0, 2, 0), as indicated in figure 4.4,while the maximum plasma-β for switch-on shocks is reached in point B, where(θ,M, β) = (0, 1, 2

γ = 1.2). Therefore the Mach number for the existence of aswitch on-shock is bounded by 1 < M < 2 and the plasma-β at which theseshocks can occur must satisfy 0 < β < 2

γ = 1.2. For switch-off shocks, the

minimum plasma-β is reached in point B and is therefore β = 65 = 1.2. The

minimum value of the Mach number M is reached in point C whose coordinatesare found by solving

pu = 0,√

γ(1−β)+1γ−1 = M.

(4.29)

Therefore C satisfies (θ,M, β) = (0,√

γ−1γ+1 ,

4γ+1 ) = (0, 0.5, 1.5), and switch-off

shocks satisfy

1

2=

√

γ − 1

γ + 1≤Ms-off < 1. (4.30)

In fact, it is straightforward to show that in C, both the curves pHD = 0 andps-off = 0 touch, while ω : Ω = 0 also contains C.

Regarding HD shocks, the maximum plasma-β at which they can exist is reachedin D, where (θ,M, β) = (0, 1, 4

γ−1) = (0, 1, 6), such that all HD shocks satisfyβ < 6.

Finally note that ps-off = 0 has M =√

γ−1γ = 0.6325 as a horizontal asymptotic,

such that for θ = 0, the RH conditions always lead to at least one solution wheneverM > 0.6325.

4.4.5 Parameter ranges for 1 → 3 shocks.

The upstream state.

We now search for critical values for the upstream parameters, for which the RHconditions allow for 1 → 3 shocks. First, note that no 1 → 3 shocks are possiblefor β > 2

γ , since region (iv), as shown in figure 4.3 then no longer exists.

When, at fixed β, implicitly taking derivatives of M to θ on the boundary Ω = 0,it can be shown that ∂M

∂θ < 0, for θ > 0 and M > 1, and the other way around:∂M∂θ < 0, for θ < 0 and M > 1. Therefore the maximum value of M on Ω = 0

RESULTS 107

Figure 4.4: The regions where switch-on or switch-off shocks can occur are coloredin greyscale. By finding the coordinates of points A, B and C, we know thelimiting values for which these shocks can occur. More details are given in thetext.

is reached when θ = 0 (which in figure 4.3 coincides with point P). Hence themaximum value of the upstream Mach number at which intermediate 1 → 3 shockscan be found is reached at θ = 0. For varying β, the left panel of figure 4.5 showsthe maximum value of M1,1-3. Therefore 1 < M1,1-3 < 2.

The maximum value of θ1 for which intermediate 1 → 3 shocks can be found, isreached on the curve traced out by point Q for varying β in figure 4.3, and thusrequires solving the system

vn = vf,n,Ω = 0.

(4.31)

The analytical expression of the solution is again complicated, but the right panelof figure 4.5 plots the solution in function of β. Since this critical value is decreasingfor increasing β, and the limit value for β = 0, equals 0.65633, we conclude that−0.65633 < θ1,1→3 < 0.65633.


b0.0 0.2 0.4 0.6 0.8 1.0 1.2

M

1.0

1.2

1.4

1.6

1.8

2.0

Figure 4.5: Left: The critical upstream Alfvén Mach number for the existence ofintermediate 1 → 3 shocks in function of the upstream β. Right: The criticalupstream θ number for the existence of intermediate 1 → 3 shocks in function ofthe upstream β.

Figure 4.6: The critical downstream θ for which 1 → 3 shocks can occur, infunction of the downstream β.

The downstream state.

We now search for critical values for the downstream parameters, for which theRH conditions allow for 1 → 3 shocks.

RESULTS 109

The downstream region in which 1 → 3 shocks can occur, is region (vii) (as alsoshown in figure 4.3. Hence, we find the maximum downstream θ for 1 → 3 shocksto be reached on the curve traced out by point S for varying β (as also shown infigure 4.3). Therefore this value can be found by solving

M = 1,pu = 0.

(4.32)

We find this critical θ to be located at

θ =

√

(γ − 1)β2 + (γ − 3)β − 2√

β(γβ + γ − 1)

(γ − 1)(β + 1)2,

for β ∈]0, 4γ−1 ]. Note that the maximum value 1√

γ−1is reached for β = γ−1.

Therefore no 1 → 3 shocks are possible for θ > (γ − 1)−1/2 = 0.77460.

As a bonus, we derived that for β > 4γ−1 = 0, no 1 → 3 shocks can occur. Figure

4.6 plots this critical value of θ in function of β.

We find the maximum downstream M for 1 → 3 shocks to be reached on thecurve related to the intersections labeled as T or U (as also shown in figure 4.3),depending on the value of β. At β = 0.1 this maximum downstream value is foundto be located at M = 0.94943, as seen in the top panel of figure 4.3..

4.4.6 Parameter ranges for 2 → 3 shocks.

The upstream state.

The maximum value of M at which the RH conditions allow for 2 → 3 shocks, isreached on the curve related to point Q, as shown in figure 4.3, and can thus befound by solving equations (4.31). The left panel of figure 4.7 also shows a plotof the Alfvén Mach number on Q for varying β. Since this value is decreasingfor increasing β, and the limit value for β = 0, equals 1.19615, we find that1 < M1,2→3 < 1.19615.

The maximum upstream value of θ at which 2 → 3 shocks can occur is reached onthe curve related to point R, as shown in figure 4.3. Hence, we need to solve thefollowing system:

pu = 0,Ω = 0.

(4.33)

A straightforward iteration on β shows that the maximum value is reached atβ = 0.44, and equals θ2 = 1.34283, hence −1.34283 < θ2,2→3 < 1.34283. The


0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.2 0.4 0.6 0.8 1 1.2θ

β

Figure 4.7: Left: The critical upstream Alfvén Mach number for the existence ofintermediate 2 → 3 shocks in function of the upstream β. Right: The critical valueof θ, for the upstream state of an intermediate 2 → 3 shock for varying β.

variation with β is shown in the right panel of figure 4.7.


Since the downstream state of a 2 → 3 shock is located in region (vii) or (viii),as also shown in figure 4.3, and the lower branch of the pu = 0 surface, startingat T does not cross M = 1, it follows that there are no limiting values for thedownstream θ of an intermediate 2 → 3 shock.

We find the maximum downstream M for 2 → 3 shocks to be reached on the curveT or V (as also shown in figure 4.3), depending on the value of β. Therefore, atβ = 0.1 this critical value is found to be located at M = 0.94943 again.

4.4.7 Parameter ranges for 1 → 4 shocks

The upstream state.

The exact same reasoning we made for the upstream state of an intermediate 1 → 3shock can be repeated, thus the limiting values for the upstream state of a 1 → 4shock are exactly the same.


We search for critical values for the downstream parameters, for which the Rankine-Hugoniot conditions allow for 1 → 4 shocks. The downstream state of a 1 → 4

RESULTS 111

shock, must be located in region (xvi).

We first find the minimal value of the Mach number M at which 1 → 4 can occur.A first important observation is that for all β > 1.2, the (θ,M)-parameter spacecontains a region (xiv), since it can be shown that for all β, the pu = 0 curvecrosses the vn = vs,n.

At β = 32 , the pu = 0 curve crosses the Ω = 0 curve in θ = 0. Therefore, when

β > 1.5, the minimum value of M in region (xiv) is reached at θ = 0. When1.2 < β < 1.5, this minimum value can be located at point T , as labeled in thebottom panel of figure 4.3).

When β > 1.5, this value is reached on curve T, hence we need to solve

pu = 0;θ = 0,

(4.34)

in order to find the minimum Mach number for 1 → 4 intermediate shocks. For

fixed β, this minimum value is found to be M =

√

4−5β+√

25β2−24β

4 . This functionreaches its minimum at β = 1.5, and the minimal value is 0.5. It can be shownthat for β < 1.5, this minimum value of M is bigger. Therefore 0.5 < M2,1→4 < 1.

For fixed β2, we can also find the maximum value for θ2. Therefore we solve thesystem

pu = 0;vn = vs,n.

(4.35)

For β = 2, we find this critical value to be θ = 0.75604, as seen in figure 4.3.

4.4.8 Parameter ranges for 2 → 4 shocks

The upstream state.

The upstream state for an intermediate 2 → 4 shock should be located in region(v). The limiting upstream values for 2 → 3 shocks are exactly the same as thelimiting upstream values for 2 → 3 shocks.


The downstream state of a 2 → 4 shock, must be located in region (xv) or (xvi).Since on pu = 0, it can be shown that, for fixed β, ∂M

∂θ > 0. Therefore M2,2,→4

will reach its minimum value in region (xvi), and it equals the minimum value forM2,1→4.

The maximum value of θ2,2→4 is reached on the curve corresponding to point Vin figure 4.3, and for β = 2, θ2,2→4 = 1.65654, as also shown in figure 4.3.


4.4.9 Relationship between pre- and post-shock magnetic field

In hydrodynamical shock refraction, von Neumann [63], Courant & Friedrichs[14], Liepmann & Roshko [55], Anderson [5] and many others have studied therelationship between the pre- and the post-shock streamlines in several initialsetups. Since in the MHD case, these are both dependent on the pre- andpost-shock magnetic field, we will generalize these studies by investigating therelationship between the pre- and the post-shock magnetic fields.

Therefore we plot (θ, ψ)-diagrams for fixed (β,M). We assume that the knownstate is the upstream state. Figure 4.8 plots these relations for β = 0.1 and varyingM . In the first plot, we have set M ∈ 1.1; 1.2; 1.3; 1.4. In this range, both fastand intermediate solutions are possible. In the second panel of figure 4.8, we showthe (θ, ψ)-diagram for M = 1.1, and divide it into certain regions. These regionsmean the following:

• Region A: The solution is a fast shock.

• Region B: There are no solutions with (M,β, θ) = (1.1, 0.1, θ) as upstreamstate.

• Region C: There are no solutions with (M,β, θ) = (1.1, 0.1, θ) as upstreamstate. The solution with (M,β, θ) = (1.1, 0.1, θ) even has negative pressure.

• Region D: There is an intermediate 1 → 4 solution.

• Region E: There is an intermediate 2 → 4 solution.

• Region F: There is an intermediate 2 → 3 solution.

• Region G: There is an intermediate 1 → 3 solution.

On the other hand, we can also assume that at fixed (β,M), the known state is thedownstream state. The third panel of figure 4.8 plots for β = 2 and varying M , thecurves connecting the known and unknown magnetic fields. The curves are shownfor M ∈ 0.3, 0.5, 0.7, 0.9. Here we did not worry about the admissibility of thesolution. In the fourth panel of figure 4.8, we only plot the physically admissibleparts of these curves.

4.5 Conclusion

Magnetohydrodynamical shocks are governed by the Rankine-Hugoniot jumpconditions. These equations can be solved analytically, and doing so essentiallyreduces to solving a cubic equation. This solution can have one or three real

CONCLUSION 113

Figure 4.8: The relation between up- and downstream Bt. In the upper panels,the known state corresponds to the upstream state, and in the lower panels, theknown state corresponds to the downstream state. Upper left: β = 0.1 and M ∈1.1, 1.2, 1.3, 1.4. According to figure 4.3, three solutions exist in this case. Foreach M , the curve reaches its maximum value at Ω = 0. Upper right: Transitionsbetween certain shock types. In the fast branch, only region A leads to an upstreamstate solution. In the intermediate branch, the solution changes shock type: regionD contains 1 → 4 solutions, region E contains 2 → 4 solutions, region F contains2 → 3 solutions and finally region G leads to 1 → 3 shocks. Lower left: β = 2.For M ∈ 0.3, 0.5, 0.7, 0.9, the relation between the known and unknown Btare plotted. The physical admissibility is not taken into account. Lower right:β = 2. For M ∈ 0.3, 0.5, 0.7, 0.9, the relation between the known and unknownBt are plotted. The physical admissibility is taken into account now. The figurecorresponds to the lower panel of figure 4.3.


solutions. When there is only one real solution, it corresponds to a fast or a slowmagneto-acoustic shock, but when there are three real solutions, also intermediateshock solutions, which cross the Alfvén speed, can be found. Inspired by the timereversal principle from Goedbloed [33], we revisited the RH shock relations in thefrequently employed shock frame, and made the duality visible in the (θ,M, β)state space. Using the thus obtained graphical classification of the state space,augmented with a positive pressure requirement, we derived limiting values forparameters in the shock rest frame at which intermediate shocks can be found.

Chapter 5

Nederlandstalige samenvatting

"Over schokbreking in hydro- en magnetohydrodynamica", dat is de letterlĳkevertaling van de titel van dit proefschrift. Eerst leggen we de begrippenhydrodynamica (HD) en magnetohydrodynamica (MHD) uit, daarna kunnen wedieper ingaan op schokken, en schokbreking.

HD, ook wel vloeistofmechanica genoemd, is een wiskundig model dat dient omhet gedrag van fluïda te beschrĳven. Hoewel zowel gassen als vloeistoffen fluïdazĳn, gebruiken we in deze thesis HD enkel om het gedrag van gassen te beschrĳven.Fluïda bestaan uit vele neutrale deeltjes die onderling interageren. In principe ishet mogelĳk om Newtoniaanse mechanica te gebruiken om de snelheid en positievan ieder deeltje te berekenen doorheen de tĳd. Dit zou echter leiden tot eenenorm stelsel van vergelĳkingen. Zelfs voor zeer kleine systemen is het oplossenvan zo’n stelsel onbegonnen werk. We moeten dus verstandiger te werk gaan, willenwe het gedrag van fluïda op een bruikbare manier beschrĳven. In plaats van debeweging van elk deeltje afzonderlĳk te beschrĳven, wordt het fluïdum beschouwdals een continuüm. In deze continuümbeschrĳving spreken we over massadichtheid,momentum en energiedichtheid, wat essentieel statistische begrippen zĳn. Deaannames die HD dan mogelĳk maken zĳn het behoud van massa, het behoudvan impuls en het behoud van energie.

Een fenomeen dat geïntroduceerd wordt door deze continuümbeschrĳving vangassen is de golf. Een golf is een verstoring die zich voortplant in tĳd enruimte. Een verstoring is hier gedefinieerd als een kleine afwĳking van eenachtergrondstoestand, in om het even welke grootheden (zoals dichtheid, snelheid,temperatuur). Hierbĳ is het belangrĳk op te merken dat enkel de verstoring zichverplaatst, niet noodzakelĳk de individuele deeltjes. Denk daarbĳ bĳvoorbeeld aaneen watergolf, een microgolf, een radiogolf of een geluidsgolf, of zelfs een mexicanwave.

115

116 NEDERLANDSTALIGE SAMENVATTING

Elk gas heeft een karakteristieke snelheid. Deze karakteristieke snelheid wordtde geluidssnelheid van het gas genoemd. Het karakteristieke aan deze snelheidis dat er een schokgolf onstaat wanneer het medium lokaal de geluidssnelheidoverschrĳdt. Dit is bĳvoorbeeld het geval wanneer een vliegtuig door degeluidsmuur vliegt. Exacter uitgedrukt: een stationaire HD schok scheidt eensupersonische toestand van een subsonische toestand. Deze eigenschap wordt dePrandtl-Meyer -eigenschap genoemd. Wiskundig valt dit te begrĳpen door op temerken dat de HD vergelĳkingen in dit geval enkel discontinue oplossingen hebben,die exact deze schokgolven beschrĳven. Verder kan aangetoond worden dat degeluidssnelheid isotroop is: deze snelheid is namelĳk even groot in elke richting.Daarom verplaatst een golf zich sferisch.

De temperatuur van een gas wordt feitelĳk bepaald door de random snelheid vande elektronen: hoe hoger die snelheid, hoe hoger de temperatuur. Wanneer eengas opgewarmd wordt, stĳgt de snelheid van de elektronen rond de protonen.Wordt het gas voldoende opgewarmd, zodat de elektronen genoeg kinetischeenergie bezitten opdat hun momentum de aantrekkingskracht van de protonenkan overwinnen, dan komen de elektronen los van de ionen, zodat het gas bestaatuit ionen en vrĳ bewegende elektronen. Een gas in deze toestand wordt een plasmagenoemd. Gezien elektronen en ionen respectievelĳk een negatieve en een positieveelektrische lading hebben, induceren ze een elektrisch veld, dat op zĳn beurt eenmagneetveld creëert. Dit magnetisch veld induceert een magnetische druk zodat defysica van het probleem essentieel verandert. Het wiskundig systeem dat plasma’sbeschrĳft heet MHD. Dit wiskundig systeem is nu zevendimensionaal, waar het HDsysteem driedimensionaal was. Dit heeft verregaande gevolgen. Terwĳl HD enkelde geluidssnelheid als karakteristieke snelheid heeft, heeft MHD 3 karakteristiekesnelheden: de trage magnetosonische snelheid, de Alfvénsnelheid en de snellemagnetosonische snelheid. Verder kan aangetoond worden dat geen van dezedrie snelheden isotroop is. Informatie plant zich dus niet in elke richting evensnel voort. De veralgemening van de Prandtl-Meyer-eigenschap is nu verre vantriviaal. In plaats van een sub- en een supersonische toestand, kan men met dedrie karakteristieke MHD snelheden nu vier verschillende toestanden definiëren.Men noemt deze toestanden respectievelĳk supersnel (1), subsnel (2), supertraag(3) of subtraag (4). Een schok die een supersnelle toestand van een subsnelletoestand scheidt wordt een snelle MHD-schok genoemd. Een schok die eensubtrage toestand van een supertrage toestand scheidt wordt een trage MHD-schok genoemd, en alle andere schokken worden intermediaire schokken genoemd.Intermediaire schokken zĳn zeker en vast oplossingen van de vergelĳkingen dieschokgolven beschrĳven (de Rankine-Hugoniot sprongvoorwaarden), maar er isgeen eensgezindheid over het bestaan van deze schokken in de fysische wereld.

Alleszins, in hoofdstuk 4 van deze thesis leiden we af welke wiskundige voorwaardenhet bestaan van zo’n intermediaire schokken toelaten. Ik beschrĳf kort onzeaanpak. De Rankine-Hugoniot sprongvoorwaarden (RHV) moeten op elk punt

NEDERLANDSTALIGE SAMENVATTING 117

in de ruimte en op elk tĳdstip gelden. Dit zĳn feitelĳk 8 vergelĳkingen in 8onbekenden, maar gezien een MHD-schok lokaal bekeken een 2D fenomeen is,is het mogelĳk om de RHV te reduceren tot 6 vergelĳkingen, in 6 onbekenden:de massadichtheid ρ, de thermische druk p, twee componenten van de snelheid(vn, vt) en de twee componenten van het magneetveld (Bn, Bt). Nu kunnenwe een schok volledig bepalen door drie dimensieloze parameters, zodat we deRHV verder willen vereenvoudigen in essentieel drie vergelĳkingen. Uit één vande Maxwell vergelĳkingen, ∇ · B = 0, weten we dat Bn constant is aan beidekanten van de schok, zodat we Bn van de vergelĳkingen kunnen elimineren.Uit het behoud van momentum kunnen we ρ van de vergelĳkingen elimineren,en tenslotte: gezien de RHV translatieinvariant zĳn kunnen we vt van devergelĳkingen elimineren zodat we inderdaad tot drie vergelĳkingen komen indrie onbekenden. We zetten dit stelsel nu om tot een stelsel in de dimensielozeparameters, en versimpelen het verder tot 1 vergelĳking in 1 onbekende. Dit iseen derdegraadsvergelĳking. Verder wordt de redenering eenvoudig. Het oplossenvan een derdegraadsvergelĳking is gekend. Het wordt dikwĳls verkeerdelĳktoegeschreven aan Cardano, een Italiaanse wiskundige uit de Renaissance, maarfeitelĳk was de Perzische wiskundige Khayyam (1048 - 1131) hem voor. In elk geval,een derdegraadsvergelĳking kan één, twee of drie verschillende (reële) oplossingenhebben, en we hebben een nodige en voldoende voorwaarde voor de uniciteit van deoplossing. We gaan dat zelf ook na in Appendix F. Daarnaast is het niet zo moeilĳkaan te tonen dat een oplossing van deze vergelĳking die tot een intermediaire schokleidt enkel mogelĳk is wanneer de derdegraadsvergelĳking geen unieke oplossingheeft. Dit leidt ons tot een wiskundige voorwaarde voor de mogelĳkheid totintermediaire schokken. Uit deze voorwaarde kunnen we bĳvoorbeeld afleidendat geen intermediaire schokken mogelĳk zĳn wanneer de normale snelheid groteris dan tweemaal de normale Alfvénsnelheid. Tot zover hoofdstuk 4.

Hoofdstuk 2 en 3 behandelen schokbreking in respectievelĳk het HD- en hetMHD-geval. Een contactdiscontinuïteit (CD) is het oppervlak dat twee gassenof plasmas scheidt. In schoktubes worden CD’s nagebootst met behulp van eenheel dun membraan. In deze schoktubes kan men schokken creëren, en deze lateninterageren met een CD. Onze studie is echter niet experimenteel, maar semi-analytisch. Dit betekent dat de oplossing exact is, maar er zĳn iteraties nodig omdeze exacte oplossing te benaderen. We voeren ook computerexperimenten uit metde numerieke code AMRVAC en vergelĳken de exacte oplossing met de numeriekeoplossing. De oplossingen komen goed overeen.

In het HD-geval breekt de schok op de CD en drie signalen onstaan: eengereflecteerd signaal, een overgebracht signaal en het geschokte contact daartussen.In het MHD geval onstaan (in het pure planaire geval) vĳf signalen: tweegereflecteerde signalen, twee overgebrachte signalen, en de geschokte CD ertussenin. Onze bedoeling is de exacte locatie van deze nieuwe signalen tevoorspellen. Onze methode steunt feitelĳk op een methode die vooral in

118 NEDERLANDSTALIGE SAMENVATTING

computersimulaties gebruikt wordt, namelĳk een exacte Riemannoplosser. Ik legeerst het idee uit in het HD geval, en daarna leg ik kort uit welke veralgemeningennodig zĳn om de Riemannoplosser te laten werken in MHD.

Eerst en vooral moeten we een assenstelsel kiezen waarin we het probleem zullenoplossen. Als we dit slim doen, en een assenstelsel kiezen dat mee met de schokreist, wordt het probleem sterk vereenvoudigd: Alle signalen zĳn nu immersstationair. Tenminste wanneer we veronderstellen dat het meest eenvoudigerefractiepatroon gevormd wordt, namelĳk dat waarbĳ alle signalen samenkomenin één enkel punt. Dit wordt reguliere refractie genoemd. Gezien de CD tweeverschillende gassen scheidt, moet de snelheid er lokaal gelĳk zĳn aan 0. Ook hetdrukverschil tussen de twee kanten van de CD moet er 0 zĳn. Het idee is nu datwe een gok wagen voor deze druk rond de CD. Na wat niet-triviale algebra wetenwe tot welke fout op de snelheden rond de CD onze gok voor de druk leidt, en metdeze kennis kunnen we onze gok verbeteren, tot op de gewenste nauwkeurigheid.Eens we de exacte druk kennen, kunnen we de posities van de signalen bepalen.

Wanneer de schok van een licht naar een zwaar medium reist, zal het overgebrachtesignaal trager bewegen dat de initiële schok. In dit geval zal het gereflecteerdesignaal tevens een schok zĳn. Het geschokte contact zal instabiel worden enoprollen naar rechts. Deze instabiliteit wordt de Richtmyer-Meshkov instabiliteitgenoemd. Wanneer de schok daarentegen van een zwaar naar een licht mediumreist, zal het overgebrachte signaal sneller bewegen dan de initiële schok. Hetgereflecteerde signaal zal in dit geval geen schok meer zĳn, maar een continusignaal dat een verdunningsgolf genoemd wordt. De geschokte CD zal ook in ditgeval Richtmyer-Meshkov instabiel worden, maar nu zal het contact oprollen naarlinks. Figuur 2.12 toont een AMRVAC snapshot van deze beide gevallen.

Niet alle parameterkeuzes leiden tot reguliere schokrefractie. We tonen aan datwanneer de hoek tussen het schokfront en de CD een bepaalde kritische hoekoverschreidt, zo’n regulier refractiepatroon onmogelĳk is. Onze Riemannoplossermaakt het mogelĳk deze kritische hoek te voorspellen, en dus op voorhand teweten of de refractie al dan niet regulier zal zĳn. Figuur 2.9 toont een reguliereen een irreguliere schokrefractie.

Het basisidee waarop de MHD Riemannoplosser steunt is in essentie hetzelfde. Erzĳn enkel enkele technische complicaties die de implementatie van deze oplossermoeilĳker maken. Waar we in het HD geval enkel een gok voor de druk rondde CD maken, maken we hier gokken voor de magneetveldcomponenten, desnelheidscomponenten en de thermische druk. Daardoor zal de numerieke iteratienu op een vierdimensionele functie uitgevoerd moeten worden. Bovendien is dezefunctie niet overal continu, zodat het verbeteren van onze gokken niet triviaal is.Erger nog: zoals we in hoofdstuk vier aantonen kunnen de RH sprongvoorwaardenin MHD tot drie verschillende (reële) oplossingen hebben. Gezien we dezevergelĳkingen tot viermaal oplossen leidt dit in elke tĳdstap tot hoogstens 81

NEDERLANDSTALIGE SAMENVATTING 119

verschillende oplossingen. We verhelpen dit euvel door een kleine toegeving tedoen: we postuleren de aard van de verschillende signalen vooraleer we de oplosseraan het werk laten. Hierbĳ postuleren we steeds dat de snelle signalen snelleschokken zĳn. De trage signalen daarentegen kunnen zowel verdunningsgolven,trage schokken als intermediaire schokken zĳn. Tevens kunnen we transitiesvaststellen tussen verschillende gevallen. We hebben parameterregimes gevondenwaaronder de oplosser zowel een overgebrachte als gereflecteerde intermediaire golfvoorspelt. De AMRVAC simulaties bevestigen dit resultaat. Figuur 3.8 toont demagneetvelden in verschillende MHD gevallen.

Verder kan beargumenteerd worden dat de Richtmyer-Meshkov instabiliteit in hetMHD geval onderdrukt is, en de CD stabiel blĳft. Figuur 3.2 toont het verschiltussen het HD en het MHD geval.

Chapter 6

Conclusions

We developed an exact Riemann solver-based solution strategy for shock refractionat an inclined contact discontinuity (CD) in HD and ideal MHD.

In the HD case, our self-similar solutions agree with the early stages of nonlinearAMRVAC simulations. We predict the critical angle αcrit for regular refraction,and the results fit with numerical and experimental results. Our solution strategyis complementary to von Neumann theory, and can be used to predict fullsolutions of refraction experiments, and we have shown various transitions possiblethrough specific parameter variations. After reflection from the top wall, the CDbecomes RM-unstable. Adding perpendicular magnetic fields leaves the contactRM-unstable.

In planar ideal MHD, the shock refracts in five signals instead of three. Afterreflection from the top wall, the CD remains RM-stable, since the ideal MHDequations do not allow for vorticity deposition on a CD. We are able to reproduceresults from the literature and results by numerical simulations performed byAMRVAC.

Magnetohydrodynamical shocks are governed by the Rankine-Hugoniot jumpconditions. These equations can be solved analytically, and doing so essentiallyreduces to solving a cubic equation. This solution can have one or three realsolutions. When there is only one real solution, it corresponds to a fast or a slowmagnetoacoustic shock, but when there are three real solutions, also intermediateshock solutions, which cross the Alfvén speed, can be found. Inspired by the timereversal principle from Goedbloed [33], we revisited the RH shock relations in thefrequently employed shock frame, and made the duality visible in the (θ,M, β)state space. Using the thus obtained graphical classification of the state space,augmented with a positive pressure requirement, we derived limiting values for

121

122 CONCLUSIONS

parameters in the shock rest frame at which intermediate shocks can be found.

We are generalizing this approach to shock refraction in relativistic hydrodynamics.

Next to this shock refraction research, we have also added a data conversionsubroutine to AMRVAC, such that the scientific data produced by the numericalcode AMRVAC (van der Holst & Keppens [93]; Keppens et al.[49]) can be read inby visualization software as ParaView and VisIt.

Appendix A

Structure of AMRVAC

The AMRVAC code exists of several directories:

• src: contains the source code, namely:

the main program: after compilation, this program is adapted to thephysics, numerical methods, dimensionality, etc.;

the physics subroutines : AMRVAC contains physics modules foradvection (for testing purposes) and classical and (special) relativistic hydro-and magnetohydrodynamics. There is also a radiative cooling routineimplemented. We have developed classical HD and MHD modules whichallow for an interface separating two different gases, i.e. gases with differentvalues of γ (as described in Delmont et al.[19]);

the numerical methods : including TVD, TVDLF, HLLC, TVDMU,approximate Roe solvers etc. and limiters including minmod, Woodward,etc.;

IO subroutines;

Conversion subroutines: Conversion subroutines to idl [43], dx [24],tecplot [83], vtu, ascii are included. I developed the VTU-conversionsubroutine as part of my PhD research.

• par: contains simulation dependent choices such as:

Input and output files. AMRVAC produces two kinds of output files.The OUT-file saves all the conserved variables, and optionally also otheruser defined variables, at certain times. The log file saves global integratedvalues of conserved variables at certain times, plus the number of cells ateach AMR-level.

123

124 STRUCTURE OF AMRVAC

Saving times. The user can decide at which moment an OUT file isproduced, and at which moment global integrated data are saved in the logfile.

The stop criterion.

The numerical methods. The user decides which numerical methodsshould be used. This choice can be level dependent. Also the limiters and∇ · B-control algorithm is chosen by the user.

Boundary conditions. The number of ghost cells is defined here. Alsothe physics dependent boundaries are defined, for every boundary, for everyconserved variable. Pre-defined choices include continuous to mimic openboundaries, symmetric or antisymmetric to mimic rigid walls or periodicfor periodic problems (e.g., in cylindrical coordinates). The user has thepossibility to define other boundaries too.

AMR related choices, such as the number of AMR levels, the resolutionon the coarsest level, the size of the domain, the tolerance for the refinementcriterion.

The Courant parameter.

• usr: contains the initial conditions. Appendix C gives the usr-file for theHD shock tube problem presented in Delmont et al.[19], and in Chapter 2.

Appendix B

Compilation

Before creating the usr- and the par-file, the AMRVAC user knows the geometry,dimensionality and physics of the problem. He also needs to define the values cithat define the numbers of cells (per dimension i) in one grid block. Once this isdone, the compilation can be done as follows:

make c l eansetamrvac −p=mhd −d=23 −g=16 ,16 −u=rimmhd23problem1make amrvac

The setamrvac command selects the correct physics, dimensionality and ci. Inthe example given above, the MHD module will be used for a 2.5D problem,which means that vectors have three components on a two-dimensional -by defaultCartesian- domain. Every grid will consist of 16 × 16 cells, including ghost cells.The usr-file used will be rimmhd23problem1.t, and is given in Appendix C. Thisfile is related to the RMI problem from Chapter 2.

125

Appendix C

USR-file

This appendix shows the USR file used for the simulation of the shock tube problempresented in Delmont et al.[19], or in Chapter 2. It is written in LASY syntax, asintroduced in Tóth [88].

subrout ine in i tonegr id _us r (w, ixG^L , x )

! i n i t i a l i z e one gr id

inc lude ’ amrvacdef . f ’

! s c a l a r si n t e g e r : : ixG^L

! ar ray sdouble p r e c i s i o n : : w( ixG^S , 1 : nw) , x ( ixG^T, 1 : ndim)

! l o c a l s c a l a r sdouble p r e c i s i o n : : xpi , vpost , rhopost , ppost , xshock , xpi1 , xbound , tang

!−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

ok te s t = index ( t e s t s t r , ’ in i t onegr id_usr ’)>=1i f ( ok te s t ) wr i t e ( unitterm , ∗ ) ’−> in i tonegr id_u sr ( in ) : ’ ,&

’ ixG^L : ’ , ixG^L

ÎFONED stop ’ This i s not a 1D problem ’ ÎFTHREED stop ’ This i s not a 3D problem ’ ÎFTWOD

! parameters

127

128 USR-FILE

eqpar (gamma_)=1.4 d0M=10.0d0eta =0.1d0tang=one

! l o c a t i o n shock and CDxshock =0.1 d0xbound=1.5 d0

vpost=M−(( eqpar (gamma_)−one )∗M∗∗2+2.0 d0 ) / ( ( eqpar (gamma_)+one )∗M)rhopost=−one ∗( vpost+M)∗ eqpar (gamma_)∗Mppost=(2∗eqpar (gamma_)∗M∗∗2−eqpar (gamma_)+one )/( eqpar (gamma_)+one )

where (x ( ixG^S,1) > xshock . and . ( x ( ixG^S,1) >x ( ixG^S , 2 ) / tang+xbound ) )! pre shock reg ionw( ixG^S , rho_)=eqpar (gamma_)∗ etaw( ixG^S ,m1_)=zerow( ixG^S ,m2_)=zerow( ixG^S , e_)=one /( eqpar (gamma_)−one )

endwherewhere (x ( ixG^S,1) > xshock . and . ( x ( ixG^S,1)<=x ( ixG^S , 2 ) / tang+xbound ) )

! pre shock reg ionw( ixG^S , rho_)=eqpar (gamma_)w( ixG^S ,m1_)=zerow( ixG^S ,m2_)=zerow( ixG^S , e_)=one /( eqpar (gamma_)−one )

endwherewhere (x ( ixG^S,1)<= xshock )

! post shock reg ionw( ixG^S , rho_)=rhopostw( ixG^S ,m1_)=vpost ∗ rhopostw( ixG^S ,m2_)=zerow( ixG^S , e_)=one /( eqpar (gamma_)−one )+0.5 d0∗ rhopost ∗ vpost ∗∗2

endwhere

re tu rnend subrout ine in i tonegr id_us r

Appendix D

Stationary planarRankine-Hugoniot conditions

We allow weak solutions of the system, which are solutions of the integral formof the MHD equations, that may contain discontinuities. The shock occuring inthe problem setup, as well as those that may appear as FR, SR, ST or FTsignals later on obey the Rankine-Hugoniot conditions. In the case where theshock speed s = 0, the Rankine-Hugoniot conditions follow from equation (3.1).When considering a thin continuous transition layer in between the two regions,with thickness δ, solutions for the integral form of equation (3.1) should satisfy

limδ→0

∫ 2

1 ( ∂∂xF + ∂∂yG)dl = 0. For vanishing thickness of the transition layer, this

yields

ρvn

ρv2n + p− B2

n

2 +B2

t

2ρvnvt −BnBt

vn(γγ−1p+ ρ

v2n+v2t2 +B2

n) −BnBtvtvnBt − vtBn

Bn

= 0, (D.1)

where the index n refers to the direction normal to the shock front and the index trefers to the direction tangential to the shock front (see e.g. Goedbloed & Poedts[32]). From equations (D.1) we know that m ≡ ρvn and Bn are constant across a

129

130 STATIONARY PLANAR RANKINE-HUGONIOT CONDITIONS

shock. Inserting these constants in the other equations of (D.1) yields

m2[[1

ρ]] + [[p]] +

1

2[[B2

t ]] = 0, (D.2)

m[[vt]] −Bn[[Bt]] = 0, (D.3)

m[[Btρ

]] −Bn[[vt]] = 0, (D.4)

γ

γ − 1[[p

ρ]] +

m2

2[[

1

ρ2]] + [[

B2t

ρ]] − B2

n

2m2[[B2

t ]] = 0, (D.5)

where we used equation (D.4) to eliminate vt from [[vn( γγ−1p + ρ

v2n+v2t2 + B2

n) −BnBtvt]] = 0 to arrive at

γ

γ − 1m[[

p

ρ]] +

m3

2[[

1

ρ2]] +m[[

B2t

ρ]] − B2

n

2m[[B2

t ]] = 0, (D.6)

which is equivalent to equation (D.5), under the assumption that m 6= 0, which istrue for all magnetoacoustic signals.

Since the signal located at dydx =

vy

vxis a contact discontinuity, it clearly obeys

m = 0. This simplifies equations (D.2-D.4) drastically:

[[(vt, Bt, p)t]] = 0. (D.7)

Hence, in terms of the primitive variables, this means that vx, vy, p and Bx, andthus also By, remain constant across a contact discontinuity.

Appendix E

Relations across a shock

Suppose we know the primitive variables uk at one side of a stationary shock, anddenote the unknown primitive variables at the other side of the shock uu.

We will discuss the consequences of the RH condition relating up- and downstreamstates, for the case where Bt,k 6= 0. After elimination of [[vt]] from equation (D.4),we arrive at the following 3 × 3-system (see also Torrilhon [91]):

p− 1 + C(1

ρ− 1) +

1

2(B2

t −A2) = 0, (E.1)

C(Btρ

−A) −B2(Bt −A) = 0, (E.2)

1

γ − 1(p

ρ− 1) +

1

2(1

ρ− 1)(p+ 1) +

1

4(1

ρ− 1)(Bt −A)2 = 0, (E.3)

where we introduced the dimensionless quantities connecting the up-and down-stream values given by ρ = ρu

ρk, p = pu

pk, vt =

vt,u

ckand Bt =

Bt,u√pk

and the

dimensionless parameters, quantifying the known values in state uk by A ≡ Bt,k√pk

,

B ≡ Bn,k√pk

and C =ρkv

2n,k

pk. These parameters allow for simple criteria for fast,

slow or intermediate shocks: fast shocks are characterised by Bt

A > 1, slow shocks

are characterised by 0 < Bt

A < 1 and intermediate shocks satisfy Bt

A < 0, sincethe known state is the upstream state and the unknown state is the downstreamstate.

131

132 RELATIONS ACROSS A SHOCK

Note that super-Alfvénic flow in the direction normal to the shock front in thestate uk now implies

C > B2 ⇔ v2n,k > a2

n,k. (E.4)

The solution to equations (E.1-E.3) is given by

ρ =CBt

AC +B2(Bt − A), (E.5)

p =2C

γ + 1− γ − 1

γ + 1

(

(A− Bt)2

2+ 1

)

− BtC(A+ Bt)

(γ + 1)(C −B2), (E.6)

where Bt must satisfy a cubic relation

Σi=0,3τiBit = 0, (E.7)

with its coefficients given by

τ3 = B2, (E.8)

τ2 =(

(γ − 1)(B2 − C) + C)

A, (E.9)

τ1 = ((γ + 1)B2 − (γ − 1)C − (2 +A2)γ)(B2 − C), (E.10)

τ0 = −(γ + 1)A(B2 − C)2. (E.11)

Once the solution Bt from equation (E.7) is determined, and used to calculatep and ρ from equations (E.5)-(E.6), we find the upstream state from ρu = ρρk,pu = ppk and Bt,u = Bt

√pk. Finally equation (D.1) delivers vn,u and Bn,u, while

from equation (D.4), we know vt,u:

vn,u =m

ρρk, (E.12)

vt,u = vt,k +Bn,kρkvn,k

(Bt,n −Bt,u), (E.13)

Bn,u = Bn,k. (E.14)

Finally, note that the regular solution discussed above is only valid when B2 6= C,i.e., when the normal velocity in the known region does not equal the Alfvén

RELATIONS ACROSS A SHOCK 133

velocity, or equivalently when Bt,k does not vanish. In this special case the system(E.1-E.3) has four mathematical solutions possible, one of which is the trivialsolution, one of which is a rotational shock (which is the limit case of solution(E.5-E.6)) and finally the following non-trivial switch-off shock:

Bt = 0, (E.15)

ρ =2(γ + 1)C

γ(A2 + 2C + 2) ±√

(γ(A2 + 2))2

+ 4C(C +A2 − 2γ), (E.16)

p =2 + 2C +A2 ∓

√

(γ(A2 + 2))2

+ 4C(C +A2 − 2γ)

2(γ + 1). (E.17)

Appendix F

Solving the cubic analytically

The cubic (E.7) is solved analytically by the following procedure. Defining thereal quantities

D = 4(τ22 − 3τ3τ1), (F.1)

N = 4(2τ32 − 9τ3τ2τ1 + 27τ2

3 τ0), (F.2)

one notes that D is the discriminant of the derivative of the cubic. When D > 0,the inequality N2 −D3 < 0 gives limiting values for τ0, for which the 2 extremaof the cubic have opposite signs. Since N2 −D3 < 0 implies D > 0, the criteriumon the coefficient of the cubic to have 3 different real solutions is

N2 −D3 < 0. (F.3)

From these real-valued quantities, we define the following, possibly complex,quantities:

H =(

√

N2 −D3 −N)

13

, (F.4)

J =D

H. (F.5)

135

136 SOLVING THE CUBIC ANALYTICALLY

In terms of these introduced quantities, the 3 solutions of the cubic equation (E.7)are given by

Bt,0 =J +H − 2τ2

6τ3, (F.6)

Bt,1 = −J +H + 4τ2 −√

3(H − J)i

12τ3, (F.7)

Bt,2 = −J +H + 4τ2 +√

3(H − J)i

12τ3. (F.8)

The evaluation of expressions (F.6 - F.8) requires the evaluation of√N2 −D3 and

(√N2 −D3 −N

)

13 . The principal square root √

: R → R is continuous, but theprincipal cube root 3

√: C → C has a branch cut along the negative real axis. It

follows that h : R2 → C : (N,D) 7→(√N2 −D3 −N

)

13 is discontinuous when√

N2 −D3 −N is a negative (and thus real) number.

Before we explain how to permute the indices of the roots (F.6-F.8), let us firstrewrite these expressions as

Bι = Hι + Jι, (F.9)

where Hι = Hι − τ26τ3

, Jι = Jι − τ26τ3

, Hι =

“

− 12+

√3

2i”ιH

6τ3, Jι =

“

− 12+

√3

2i”−ι

J

6τ3and

ι ∈ 0, 1, 2.

Let us make some small technical remarks.

• Note that((

− 12 +

√3

2 i)ι)3

= 1, for each integer value of ι;

• Note that τ3 > 0. Therefore, equation (F.9) is well-defined. Also, when weknow that one of those expressions is real, then its sign equals the sign of itsnumerator;

• It turns out that dividing the complex plane into 6 sextants is useful for ouranalysis. We therefore define the sextants

Sj ≡ r(cosϕ+ i sinϕ)|(j − 1)π

3< ϕ < j

π

3, r > 0, (F.10)

and Li, the lines separating those sextants, i.e.,

Lj ≡ r(cosϕ+ i sinϕ)|ϕ = jπ

3, r > 0. (F.11)

These regions are shown in figure F.1.

SOLVING THE CUBIC ANALYTICALLY 137

Figure F.1: The sextants Si and their separators, Li.

One can distinguish the following cases.

• Case I: D > 0, N > 0 and N2 −D3 > 0.

Note that H ∈ L1 and since D > 0, J ∈ L5. Therefore, H1, J1 ∈ L3, J2 ∈ L1

and H2 ∈ L5. It follows that Bt,1 is a real number. Note that√N2 −D3 <

N , thus 2(N2 − D3) < 2N√N2 −D3 or

(√N2 −D3 −N

)2< D3, thus

H <√D or |H | < |J |. Therefore, Bt,2 ∈ S1 and B0 ∈ S6.

• Case II: D > 0, N < 0 and N2 −D3 > 0.

Note that H ∈ L6 and since D > 0 also J ∈ L6. Therefore, H2, J1 ∈ L4 andH1, J2 ∈ L2. It follows that B0 is a real number. Note that

√N2 −D3 > N ,

thus 2(N2−D3) > 2N√N2 −D3 or

(√N2 −D3 −N

)2> D3, thusH >

√D

or |H | > |J |. Therefore, Bt,1 ∈ S3 and Bt,2 ∈ S4.

• Case III: D > 0, N > 0 and N2 −D3 < 0.

Note that H ∈ S1 and since D > 0, J ∈ S6. Therefore, H1 ∈ S3, H2 ∈ S5,J1 ∈ S4 and J2 ∈ S2. Since |

√N2 −D3 − N |2 = D3, |H | =

√D and

|H | = |J |. Hence all of the roots are real. After some algebra, we know that

Bt,0 > − τ23τ3

+√D2 , Bt,1 < − τ2

3τ3−

√D2 and − τ2

3τ3−

√D2 < Bt,2 < − τ2

3τ3−

√D2 .

138 SOLVING THE CUBIC ANALYTICALLY

• Case IV: D > 0, N < 0 and N2 −D3 < 0.

Note that H ∈ S6 and since D > 0, J ∈ S1. Therefore, H1 ∈ S2, H2 ∈ S4,J1 ∈ S3 and J2 ∈ S5. Since |

√N2 −D3 − N |2 = D3, |H | =

√D and

|H | = |J |. Hence all the roots are real. We deduce that Bt,0 > − τ23τ3

+√D2 ,

Bt,2 < − τ23τ3

−√D2 and − τ2

3τ3−

√D2 < Bt,1 < − τ2

3τ3−

√D2 .

• Case V: D < 0 ⇒ N2 −D3 > 0.

Note that H ∈ L6 and thus J ∈ L3. Hence Bt,0 is a real number. Since D isthe discriminant of the derivative of the cubic, we know that the cubic onlyhas one real root. Note that J1 ∈ L1, H1 ∈ L2, H2 ∈ L4 and J2 ∈ L5, whichtells us that ImBt,1 > 0 and ImBt,2 < 0.

The Rankine-Hugoniot conditions indeed allow for a unique real solution if andonly if N2 −D3 > 0.

When N , D or N2 −D3 change sign, we might need to permute the indices. Wewill not describe all the possible permutations in detail, but instead will illustratethis in one example. Suppose that in 2 subsequent iteration steps, D > 0, but inthe first iteration step N > 0, while in the following step N < 0. Comparing case1 and case 2, we come to the conclusion that we need to permute the indices ofBt,0 and Bt,1.

So far, we considered Bt,ι just as a finction of the cubics coefficients τκ. Thesecoefficients are actually functions of the known state parameters, which are ontheir turn functions of the guessed wave refraction angles. The question remainsat which angles Bt,ι(φguess) are continuous (and differentiable). Here φguess refersto the guessed wave refraction angle.

Let us illustrate these findings by a simplified example. Take the initial guessφFT = 1.27678, which is the exact solution, and let the guess for φST vary. In figureF.2 we show the real part of the unpermuted roots of the cubic (E.7), togetherwith N2 −D3 and B2 − C across the corresponding signal.

Note that the Alfvénic angle φa,T = 1.1936, since at this position B2−C vanishes.Therefore, when φST < φa,T , the slow transmitted signal is a slow shock. On theother hand, when φST > φa,T , the signal is an intermediate shock. Also note that

φa,T is a double root of N2 −D3. Since B2 − C changes sign, the roots Bt,1 and

Bt,2 are permuted.

The critical angle φcr,T = 1.1948 is actually the smallest root of N2 −D3, biggerthan φa,T . For φ > φcr,T , the cubic (E.7) has only one real solution and three realsolutions for φ < φCr,T .

SOLVING THE CUBIC ANALYTICALLY 139

-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

1.193 1.1935 1.194 1.1945 1.195 1.1955 1.196

Re(

Bi)/

A

φ

Re(B1)Re(B2)Re(B3)

-0.1

-0.05

0

0.05

0.1

0.15

1.193 1.1935 1.194 1.1945 1.195 1.1955 1.196

para

met

ers

φ

N2-D3

B2-C

Figure F.2: Left: the real parts of the roots of the cubic (E.7) for ST. Right: The

graphs of (B2 − C)(φ) and (N2 −D3)(φ). Note that the Alfvénic angle is a rootof B2 − C and a double root of N2 −D3.

N changes sign at φ = 1.1953, therefore, Bt,0 and Bt,1 are permuted here, sinceD > 0. The exact solution is located at φ = 1.19283, and the root we select isBt,1.

Appendix G

Integration across rarefactionwaves

When the slow signal is located at a position where no shock solution is possiblewhich satisfies the entropy condition, we postulate the solution to be a rarefactionfan. In this case, we rewrite the stationary MHD equations (3.1) in cylindrical form,by local decomposition in normal and tangential components. We additionallyassume self similarity, i.e. ∂

∂t = 0, where the index t again refers to the tangentialdirection. Since the entropy S can be shown to be invariant in expansion fans(see e.g. De Sterck et al.[22]), we replace the energy equation by the isentropicequation. Doing so, we obtain

vn ρ 0 0 00 ρv4

n − (vtBn − Ez)2 vnBn(vtBn − Ez) v3

n 00 −Bn(vtBn − Ez) −ρv3

n + vnB2n 0 0

γp 0 0 ρ 00 0 0 0 vn

ρ′

v′nv′tp′

B′n

=

−ρvtvt(vtBn − Ez)

2

2v2nB

2n − ρv4

n + vtBn(vtBn − Ez)0

−vtBn + Ez

(G.1)

where the prime ′ denotes derivation in the normal direction. Also note that Bt iseliminated from the system since it is completely determined by the other variables.

141

142 INTEGRATION ACROSS RAREFACTION WAVES

Only when the system (G.1) is singular it has a solution, in other words

vn

(

v4n −

(

B2n

ρ+

(vtBn − Ez)2

ρvn+γp

ρ

)

v2n +

γp

ρ

Bnρ

)

= 0, (G.2)

m

v⊥(v2⊥ − v2

f,⊥)(v2⊥ − v2

s,⊥) = 0, (G.3)

which is equivalent to characteristic equation (3.7). This relation should hold insideof the expansion fans, since u is differentiable there. The solution to equations(G.1) is given by

ρ′(φ) = 2ρvnvt(2v

2n − a2 − c2) + c2anat

(4 + 2γ)v4n − ((2γ + 1)a2 + (3 + γ)c2)v2

n + γa2nc

2, (G.4)

p′(φ) = 2γpvnvt(2v

2n − a2 − c2) + c2anat

(4 + 2γ)v4n − ((2γ + 1)a2 + (3 + γ)c2)v2

n + γa2nc

2, (G.5)

B′n(φ) = −Bt, (G.6)

v′n(φ) =vt(v

2n((2γ − 1)a2 + (γ + 1)c2 − 2γv2

n) − γa2nc

2) + 2vnc2anat

(4 + 2γ)v4n − ((2γ + 1)a2 + (3 + γ)c2)v2

n + γa2nc

2,(G.7)

v′t(φ) =Σ6i=0τiv

in

((4 + 2γ)v4n − ((2γ + 1)a2 + (3 + γ)c2)v2

n + γa2nc

2)vnat. (G.8)

Here we introduced the symbols an ≡ Bn√2p

and at ≡ Bt√2p

, and the coefficients τiare given by

τ0 = −γa4nc

3, (G.9)

τ1 = 0, (G.10)

τ2 = ((γ − 2)anat + (γ + 3)c2 + (2γ + 1)a2)anc2, (G.11)

τ3 = 2(a2 + c2)anvt, (G.12)

τ4 = −(2γ + 1)ata2 − ((γ + 3)at + (2γ + 4)an)c

2, (G.13)

τ5 = −4anvt, (G.14)

τ6 = (2γ + 4)at. (G.15)

INTEGRATION ACROSS RAREFACTION WAVES 143

These expressions are valid for slow rarefaction fans. The integration has to beperformed, starting from φsl,R/T till φSR/ST .

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Curriculum Vitae

Education

• 1991-1997Wetenschappen-wiskunde (8u), VIA Tienen

• 1997-2001Licentiaat Wiskunde, K.U.Leuven

List of Scientific Contributions

Publications

• P. Delmont & R. Keppens, ‘An exact solution strategy for regular shockrefraction at a density discontinuity’, 2008, ECA 32D, D2.008

• P. Delmont, R. Keppens, & B. van der Holst, ‘An exact Riemann solverbased solution for regular shock refraction’, 2009, J. Fluid Mech. 627, 33-53.

• P. Delmont & R. Keppens, ’Shock refraction in ideal MHD’, 2010, Journalof Physics: Conference Series, 216, 012007

• P. Delmont & R. Keppens, ’Parameter regimes for slow, intermediate andfast MHD shocks’, 2010, J. Plasma Phys., doi:10.1017/S0022377810000115

• P. Delmont & R. Keppens, ’Limiting parameter values for switch-onand switch-off shocks in ideal magnetohydrodynamics’, 2010, acta technica,accepted

• P. Delmont & R. Keppens, ’Rankine-Hugoniot conditions and intermdiateMHD shocks’, 2010, ECA, submitted

153

154 SCIENTIFIC CONTRIBUTIONS

Poster Contributions

• P. Delmont, R. Keppens, B. van der Holst & Z. Meliani, ’Grid-adaptiveapproaches for computing magnetized plasma dynamics’, poster at ‘JETSETschool on Numerical MHD and instabilities’, Torino, Italy, 8-13 January2007.

• P. Delmont & R. Keppens, ’Planar Richtmyer Meshkov instabilities inMHD’, poster at ’32th conference of the Dutch and Flemish NumericalAnalysis Communities’, Woudschoten, The Netherlands, 3-5 october 2007.

• P. Delmont & R. Keppens, ’Supression of the Richtmeyer-Meshkov instabil-ity in MHD’, poster at ’internal kick-off meeting for the Leuven MathematicalModeling and Computational Science Centre (LMCC)’, Leuven, Belgium, 24april 2008.

• P. Delmont & R. Keppens, ’An exact Riemann solver based solution forregular shock refraction’, poster at ’35th EPS plasma physics conference’,Hersonissos, Crete, Greece, 9-13 june 2008.

• P. Delmont & R. Keppens, ’Supression of the Richtmeyer-Meshkovinstability in MHD’, poster at ’33th conference of the Dutch and FlemishNumerical Analysis Communities’, Woudschoten, The Netherlands, 8-10october 2008. (Winner of the Poster Prize)

• R. Keppens, Z. Meliani, A. J. van Marle, P. Delmont & A. Vlasis, ’Multi-scale simulations with MPIAMRVAC’, Poster at ’LMCC workshop onModeling and simulations of multi-scale and multi-physics systems’, Leuven,Belgium, 8-9 september 2009.

• P. Delmont & R. Keppens, ’Parameter ranges for intermediate MHDshocks’, poster at ’34th conference of the Dutch and Flemish NumericalAnalysis Communities’, Woudschoten, The Netherlands, 7-9 october 2009

• P. Delmont & R. Keppens, ’Parameter regimes for intermediate MHDshocks’, poster at ’37th EPS plasma physics conference’, Dublin, Ireland,21-25 june 2010

Lectures & Seminars

• P. Delmont, ’The Richtmyer-Meshkov Instability in 2D hydrodynamicalflows’, seminar at Centre for Plasma Astrophysics, Leuven, Belgium, 13december 2007

• P. Delmont & R Keppens, ’An exact Riemann-solver-based solution forregular shock refraction’, oral at Belgian Physical Society General scientificmeeting 2009, Hasselt, Belgium, 1 april 2009.

BIBLIOGRAPHY 155

• P. Delmont & R. Keppens, ’An exact Riemann-solver strategy for regularshock refraction’, oral at BIFD 2009, Nottingham, UK, 10-13 augustus 2009.

• P. Delmont, ’Parameter ranges for intermediate MHD shocks’, seminar atCentre for Plasma Astrophysics, Leuven, Belgium, 16 december 2009.

• P. Delmont & R. Keppens, ’Parameter regimes for intermediate MHDshocks’, oral at ’24th Symposium on Plasma Physics and Technology’,Prague, Czech Republic, 14-17 june 2010.

Thesis

Documents

van druk

van marle

van assche

tom van doorsselaere

mhd example

regular shock refraction

sciences leuven

ideal mhd