On regular shock refraction in hydro- and magnetohydrodynamics€¦ · Arenberg Doctoral School of Science, Engineering & Technology Faculty of Science Department of Mathematics Centre

$Page 1: On regular shock refraction in hydro- and magnetohydrodynamics€¦ · Arenberg Doctoral School of Science, Engineering & Technology Faculty of Science Department of Mathematics Centre$
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, may 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. M. Goossens, president in the SciencesProf. Dr. S. PoedtsProf. Dr. Ir. G. LapentaProf. Dr. W. Van AsscheProf. Dr. Ir. T. BaelmansProf. Dr. Ir. B. Koren

(CWI, Amsterdam)

U.D.C. numbermay 2010

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

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Preface

Four years ago, Tom Van Doorsselaere invited me for his public PhD defense at theArenberg castle in Heverlee. He gave a presentation which I didn’t understand atall, but which made me conclude that plasma physics is a misterious but interestingdiscipline. A few months later, I applied for a PhD position myself, with thisdissertation as a result. I would like to thank everybody who contributed to it inany way. More explicitly, my gratefullness 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 main interests. Also he has shown patience with myspecific talent of 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 as a pleasant working environment. I appreciatethe nice conversations with my former office mates Prof. Andria Rogava, Dr.Emannuel Chané, Prof. Giga Gogoberidze, Giorgi Dalakishili, Andrey Divin andAlkis Vlasis. I thank the Alma/Coffee Team for the agreeable lunch/coffee breaks.Further, the CPA deserves a word of gratefullness for the stimulating internationalenvironment.

I also enjoyed being a teaching assistent for the courses "Hogere Wiskunde I" and"Hogere Wiskunde II" in the dydactical team led by Prof. Johan Quaegebeur.I appreciate the dydactical insights i gained through this experience. It was apleasure teaching mathematics to the bachelor students in economical engeneering.

Of course I thank the members of the examination committee for investing timeto read this work and giving usefull comments and feedback.

i

ii Contents

Since the bow should not be bent too often, I really appreciated the time I’ve spendwith my friends in various ways. I think about the kot dinners, late evening pseudo-philosophical discussions, playing darts, soccer and table games, or creating chaosat Naamsevest 80. 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 (sometimes nerdy)approach. Let’s move on to our next adventure.

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 Electromagnetism . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.4 Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . 5

1.1.5 Characteristic speeds and shocks . . . . . . . . . . . . . . . 8

1.2 The Richtmyer-Meshkov instability . . . . . . . . . . . . . . . . . . 13

1.2.1 Shock refraction . . . . . . . . . . . . . . . . . . . . . . . . 14

1.2.2 Shock refraction: a 1D example . . . . . . . . . . . . . . . . 15

1.2.3 Richtmyer-Meshkov instability . . . . . . . . . . . . . . . . 21

1.2.4 Vorticity evolution equation for inviscid compressible MHDflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1.3 AMRVAC and the VTK file format . . . . . . . . . . . . . . . . . . . 25

1.3.1 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3.2 Adaptive Mesh Refinement . . . . . . . . . . . . . . . . . . 26

1.3.3 Structure of AMRVAC . . . . . . . . . . . . . . . . . . . . . 27

1.3.4 Compilation . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

iii

iv CONTENTS

1.3.5 The VTK file format . . . . . . . . . . . . . . . . . . . . . . 29

1.4 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2 An exact Riemann solver for regular shock refraction in HD 31

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2 Configuration and governing equations . . . . . . . . . . . . . . . . 33

2.2.1 Problem setup . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.2.2 Stationary MHD equations . . . . . . . . . . . . . . . . . . 34

2.2.3 Planar stationary Rankine-Hugoniot condition . . . . . . . 35

2.3 Riemann Solver based solution strategy . . . . . . . . . . . . . . . 36

2.3.1 Dimensionless representation . . . . . . . . . . . . . . . . . 36

2.3.2 Relations across a contact discontinuity and an expansion fan 38

2.3.3 Relations across a shock . . . . . . . . . . . . . . . . . . . . 40

2.3.4 Shock refraction as a Riemann problem . . . . . . . . . . . 41

2.3.5 Solution inside of an expansion fan . . . . . . . . . . . . . . 42

2.4 Implementation and numerical details . . . . . . . . . . . . . . . . 43

2.4.1 Details on the Newton-Raphson iteration . . . . . . . . . . 43

2.4.2 Details on AMRVAC . . . . . . . . . . . . . . . . . . . . . . 44

2.4.3 Following an interface numerically . . . . . . . . . . . . . . 44

2.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.5.1 Fast-Slow example solution . . . . . . . . . . . . . . . . . . 45

2.5.2 Slow-Fast example . . . . . . . . . . . . . . . . . . . . . . . 47

2.5.3 Tracing the critical angle for regular shock refraction . . . . 47

2.5.4 Abd-El-Fattah and Hendersons experiment . . . . . . . . . 48

2.5.5 Connecting slow-fast to fast-slow refraction . . . . . . . . . 50

2.5.6 Effect of a perpendicular magnetic field . . . . . . . . . . . 52

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

CONTENTS v

3 Shock Refraction in ideal MHD 57

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.3 MHD shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.4 Riemann Solver based solution strategy . . . . . . . . . . . . . . . 66

3.4.1 Dimensionless representation . . . . . . . . . . . . . . . . . 66

3.4.2 Path variables . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4.3 Tackling the signals . . . . . . . . . . . . . . . . . . . . . . 69

3.5 Demonstration of result . . . . . . . . . . . . . . . . . . . . . . . . 70

3.6 Suppression of the Richtmyer-Meshkov Instability . . . . . . . . . . 71

3.7 Conclusion and Future Work . . . . . . . . . . . . . . . . . . . . . 72

4 Parameter ranges for intermediate shocks 75

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1.1 Intermediate shocks in MHD . . . . . . . . . . . . . . . . . 75

4.1.2 The Rankine-Hugoniot jump conditions . . . . . . . . . . . 77

4.1.3 MHD shock types: classical 1 − 2 − 3 − 4 classification . . . 78

4.2 Solution to the Rankine-Hugoniot conditions . . . . . . . . . . . . 79

4.3 Physical meaning of Ω . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.4.1 (θ,M)-diagrams at fixed β . . . . . . . . . . . . . . . . . . 84

4.4.2 Equivalence classes introduced by the RH conditions . . . . 87

4.4.3 Positive pressure requirement . . . . . . . . . . . . . . . . . 89

4.4.4 Switch-on shocks and switch-off shocks . . . . . . . . . . . . 92

4.4.5 Parameter ranges for 1 → 3 shocks. . . . . . . . . . . . . . . 94

4.4.6 Parameter ranges for 2 → 3 shocks. . . . . . . . . . . . . . . 97

4.4.7 Parameter ranges for 1 → 4 shocks . . . . . . . . . . . . . . 98

4.4.8 Parameter ranges for 2 → 4 shocks . . . . . . . . . . . . . . 99

vi CONTENTS

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5 Nederlandstalige samenvatting 101

6 Conclusions 105

A USR-file 107

B Stationary planar Rankine-Hugoniot conditions 109

C Relations across a shock 111

D Solving the cubic analytically 114

E Integration across rarefaction waves 119

Bibliography 123

Curriculum Vitae 129

Education and Research History . . . . . . . . . . . . . . . . . . . . . . 129

List of Scientific Contributions . . . . . . . . . . . . . . . . . . . . . . . 129

Chapter 1

Introduction

1.1 The equations of magnetohydrodynamics

1.1.1 Plasma

In 1927, the term plasma was introduced by Langmuir & Jones [46] 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 wide rangeof temperatures and number densities. In fact, as Goedbloed & Poedts [27] putit: "Astronomers agree that, ignoring the more speculative nature of dark matter,matter in the Universe consists more than 90% of plasma. Hence plasma is theordinary state of matter in the Universe." The environment in which plasmas occurallows 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 plasma environments. The left frameshows a satellite (TRACE) observation of the solar atmosphere (taken from [6]).The picture was taken on November 9th, 2000, one day after a solar storm. The

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 [58]).

right panel shows a laboratory plasma, namely a confined plasma created withinthe TEXTOR tokamak (taken from [23]). In such tokamaks, research on controllednuclear fusion is performed. The typical values for characteristic parameters inthese different environments can differ by orders of magnitude, but the theory ofmagnetohydrodynamics (MHD) describes the dynamical behaviour of any plasma,satisfying certain conditions which we describe later on.

The Sun is by astrophysicists sometimes referred to as an excellent naturalplasma laboratory. Since it’s 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 [29] 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 notsurprising, since in order to describe the large scale motion of a plasma, one

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 [23])

uses 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 heintroduces amongst the first published system 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. At first, one added the equation of incompressibility to close thesystem, but later on also more realistic equations of state for gaseous environmentswere introduced. A derivation of these Euler equations can be found in anytextbook on fluid mechanics. When the adiabatic equation is used as an equation

4 INTRODUCTION

of state, the Euler equations take the following form:

∂ρ

∂t+ ∇ · (ρu) = 0, (1.1)

ρ∂u

∂t+ ρu · ∇u + ∇p = 0, (1.2)

∂p

∂t+ u · ∇p+ γp∇ · u = 0. (1.3)

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, butthe meaning of ρ, u and p are not that straightforward. All those quantitiesare defined statistically in phase space as follows. Every particle at any time thas an exact position r = (x, y, z) and velocity v = (vx, vy, vz) (in any arbitrarychosen Cartesian frame). In the classical case, it is usual to define the phasespace now as R

6, such that it contains combination of locations and velocities(x, y, z, vx, vy, vz). One then defines a distribution function f : R

7 → R+, such

that in any infinitesimal volume element, d3rd3v, in phase space, the probablenumber of particles, N , in that volume element at time t is given by

N(t) ≡∫ ∫

f(r,v, t)d3rd3v. (1.4)

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

ρ(r, t) ≡ M

∫

f(r,v, t)d3v, (1.5)

u(r, t) ≡∫

vf(r,v, t)d3v∫

f(r,v, t)d3v, (1.6)

p(r, t) ≡ M

3

∫

(v − u)2f(r,v, t)d3v, (1.7)

whereM is the mass of a single particle. When the gas consists of multiple differentparticles (as is the case in MHD) a straightforward generalization has to bemade. Herefore we refer to any textbook on fluid mechanics. The Euler equationsthus express the conservation of mass (i.e. 1.1), the conservation of momentum(mx,my,mz) ≡ m ≡ ρu = (ρux, ρuy, ρuz) (i.e. 1.2) and the conservation of energy

e ≡ ρ(u2x+u2

y+u2z)

2 + pγ−1 (i.e. 1.3).


1.1.3 Electromagnetism

According to Stringari & Wilson [69], 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’sequations, in their most exploited form, are given by

∇ · E =ρ

ǫ0, (1.8)

∇ ·B = 0, (1.9)

∇× E = −∂B∂t, (1.10)

∇× B = µ0ǫ0∂E

∂t+ µ0j. (1.11)

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). E is the electric field, j the current density, and finally B is themagnetic field. The first equation is called Poisson’s law, while the second equationimplies that no magnetic monopoles exist. The third equation is called Faraday’slaw, and the latter equation is Ampère’s law, extended with the displacementcurrent distribution. Since µ0ǫ0 = c−2 << 1(s2m−2) (where c denotes the lightspeed), one often drops the displacement current from Ampère’s law, which thenbecomes ∇× B = µ0j.

1.1.4 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 behaviour 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 [27], Goossens[31] or any introductory textbook on MHD. In this dissertation we will consistently

6 INTRODUCTION

use the conservational form of the ideal MHD equations, which in Cartesian vectornotation becomes:

∂U

∂t+∂F

∂x+∂G

∂y+∂H

∂z= 0, (1.12)

where the vector of conserved variables

U =

(

ρ, ρux, ρuy, ρuz,ρ(u2

x + u2y + u2

z)

2+

p

γ − 1+B2x +B2

y +B2z

2, Bx, By, Bz

)t

(1.13)

and we introduced flux terms :

F =

ρux

ρu2x + p− B2

x

2 +B2

y+B2z

2ρuxuy −BxByρuxuz −BxBz

(ρ(u2

x+u2y+u2

z)

2 + γγ−1p+

B2y+B2

z

2 )ux − (uyBy + uzBz)Bx0

uyBx − uxByuzBx − uxBz

, (1.14)

G =

ρuyρuxuy −BxBy

ρu2y + p− B2

y

2 +B2

x+B2z

2

ρuyuz −ByBz

(ρ(u2

x+u2y+u2

z)

2 + γγ−1p+

B2x+B2

z

2 )uy − (uxBx + uzBz)ByuxBy − uyBx

0uzBy − uyBz

, (1.15)

and

H =

ρuzρuxuz −BxBzρuyuz −ByBz

ρu2z + p− B2

z

2 +B2

x+B2y

2

(ρ(u2

x+u2y+u2

z)

2 + γγ−1p+

B2y+B2

z

2 )uz − (uxBx + uyBy)BzuxBz − uzBxuyBz − uzBy

0

. (1.16)


As before, ρ is the mass density, u = (ux, uy, uz) is the velocity, p is the thermalpressure and B = (Bx, By, Bz) denotes the magnetic field. Since these conservedvariables are often used, they have a particular name and notation. We introduce

thus the momentum ρu and the total energy density e ≡ ρ(u2x+u2

y+u2z)

2 + pγ−1 +

B2x+B2

y+B2z

2 . The first line of the MHD equations expresses the conservation ofmass. The following three lines express the conservation of momentum, takingthe Lorentz force into account. The fifth line expresses the conservation ofenergy, which includes the magnetic energy, and the last three lines finally expressFaraday’s law, where the electric field is eliminated by use of Ohm’s law, whichunder ideal MHD conditions reduces to E = −u × B. The ideal MHD systemconsists now of eight partial differential equations in the eight primitive variablesρ,u, p,B. Once the values of these variables are found, the pre-Maxwell equationsare used to solve for the electric field E and the current j. We have also eliminatedthe constants µ0 and ǫ0 from the equations, by appropriate scaling. Finally notethat no-monopole equation is now reduced to an initial condition. Thus

∇ · B = 0 (1.17)

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 Tn e2

(where n 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.

8 INTRODUCTION

Figure 1.3: The left picture shows a hydrodynamical shock wave during a US armyexperiment (taken from [64]). The middle and right picture show a solar MHDshock wave (taken from [6]).

1.1.5 Characteristic speeds and shocks

Shocks and 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 it’s connection to so-called weak solutionsto the system. For simplicity, we restrict our analysis to the one-dimensionalcase. But first let us mention that shock waves are relevant. Figure 1.3 showstwo shocks. The left frame (taken from [64]) shows the HD shock wave whichcaused the Sailor Hat Explosion crater on the Hawaiian island Kahoolawe, in aUS army experiment in 1965. A movie of the experiment can be seen on [65]. Themiddle and right pictures of figure 1.3 are taken by the SOHO satellite observatory(taken from [6]) and shows an MHD shock wave on the solar surface. The pictureis taken 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, i.e.:

U =

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

(1.18)

In the limit of δ → 0, we call the solution a weak solution to the hyperbolic system.Note that at x 6= 0, the conservation law is trivially satisfied, but in order to be


a solution to the system at x = 0, the discontinuous limit of the system shouldbe satisfied. These discontinuous equations are called the Rankine-Hugoniot jumpconditions (RH), and when the discontinuity travels at speed s, the RH conditionsbecome

[[F]] = s [[U]] , (1.19)

where the jump [[·]] = ·r − ·l. For a derivation of these relations for the MHDsystem, we refer to Goedbloed & Poedts [27].

For now, it is sufficient to note that in (M)HD, these discontinuities are calledshocks whenever the normal velocity does not jump across the discontinuity:[[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.

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. (1.20)

When we introduce the variable ξ ≡ xt and suppose self-similarity such that the

vector of conserved variables U(ξ) is a function of ξ only, we can rewrite thisquasilinear form again, now as an eigenvalue problem

FUUξ = ξUξ, (1.21)

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. (1.22)

10 INTRODUCTION

λi λnλ1t

x

Figure 1.4: 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 seperated by the

signals have constant speed.

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. (1.23)

More compactly, these characteristic curves are also called characteristics, andfrom equation 1.23, these characteristics describe the path at which informationtravels at the i-th characteristic speed λi, while from equation 1.22 we know thatalong this i-th characteristic, the value of si is invariant. si is said to be a Riemanninvariant.

The characteristics of the Riemann problem are schematically shown in figure 1.4.The signals seperate two regions of constant U, and can (for a non-linear system)be both continuous or discontinuous, where the discontinuous case can be formedform a continuous case by wave steepening. Consider now two (x, t)-regions in the


t

x

t

x

Figure 1.5: Left: The i-th characteristics at both sides of a shock. Right: The i-thcharacteristics at both sides of an expansion fan.

space-time plane, seperated by a single (the i-th) signal which is traveling at speeds, it can be shown (and is shown by e.g. Lax [47]) that for discontinuous signalswe must have

λi(Ur) ≤ s ≤ λi(Ul). (1.24)

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

Characteristic speeds in HD

Let us now derive the characteristic speeds of the Euler system (1.1 - 1.3). Firstof all, the one-dimensional version of these equations becomes in conservationalform Ut + Fx = 0 with U = (ρ, ρu, ρu2 + 1

γ−1p)t and the flux term becomes

F =

ρuρu2 + p

u(ρu2

2 + γγ−1p)

. (1.25)

12 INTRODUCTION

To find the eigenvalues of FU, we solve

∣

∣

∣

∣

∣

∣

λ− u −ρ 0−u2 λ− 2ρ 0

−u3

2 − γγ−1p− 3

2ρu2 λ− γ

γ−1u

∣

∣

∣

∣

∣

∣

= 0. (1.26)

and find that the HD system is (strictly) hyperbolic, namely λi ∈ u− c, u, u+ c,

where c ≡√

γpρ > 0, such that u− c < u < u + c. Since the system is hyperbolic,

we can diagonalize the Jacobian matrix as FU = RΛR−1, where the i-th columnof R is the right eigenvector belonging to λi, and Λ the diagonal matrix with λion the (i, i)-th entry. The quasilinear form simplifies thus as Ut +RΛR−1Ux = 0,or by denoting that the p-th column of R−1 (i.e. the p-th left eigenvector of theFU) as lp, this statement is equivalent to (lp · dU)dx=λpdt = 0. Rewriting this interms of total derivatives dρ, dv and dp, we find the characteristic equations

(dp− ρcdv)dx=(v−c)dt = 0, (1.27)

(dp− c2dρ)dx=vdt = 0, (1.28)

(dp+ ρcdv)dx=(v+c)dt = 0. (1.29)

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

is given by

s = (s1, s2, s3) =

(

u− 2

γ − 1c,p

ργ≡ S, u+

2

γ − 1c

)

. (1.30)

Here we have introduced the entropy S. We have just shown that sp is invariantacross 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.

Characteristic speeds in ideal MHD

The exact same procedure, performed on the one-dimensional version of equation1.12, leads to the conclusion that the characteristic speeds of 1D ideal MHD aregiven by ux ± vf,x, ux ± va,x, ux ± vs,x, ux where we introduced the normal and

THE RICHTMYER-MESHKOV INSTABILITY 13

total Alfvén speed, respectively given by

va,x ≡√

B2x

ρ, (1.31)

va ≡√

B2x +B2

y +B2z

ρ, (1.32)

and the fast and slow magnetosonic speeds, respectively given by

vf ≡√

1

2

(

c2 + v2a +

√

(c2 + v2a)

2 − 4c2v2a,x

)

, (1.33)

vs ≡√

1

2

(

c2 + v2a −

√

(c2 + v2a)

2 − 4c2v2a,x

)

. (1.34)

(1.35)

The MHD system is thus a non-strictly hyperbolic system with three differenthighly anisotropic wave speeds. This makes the MHD system much richer thanthe Euler system, where the only wave speed is the isotropic sound speed.

1.2 The Richtmyer-Meshkov instability

Any instability of a contact discontinuity (CD) seperating 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. Since the numerical experiments we perform are all shocktube experiments, we will first briefly review the literature on shock tubes andshock refraction. Shock refraction happens whenever a shock interacts with aninterface. Figure 1.6 shows in three snapshots the typical behaviour of the shockrefraction process. The first snapshot shows the initial setup of the experiment.Left and right sides of the domain are modeled open, while the upper and lowerboundaries are rigid walls. On the left side of the domain, there is a genuineright-moving shock, moving towards an inclined density discontinuity, seperatingtwo gases at rest. When the shock and the CD touch eachother at the bottomwall, the shock will refract and disturb the CD. Now the first vorticity depositionphase has started. Vorticity is deposited on the interface, causing it to roll up.The refraction is typically centered around one single point, called the triple point.When this point reaches the upper wall, it will reflect, and after this reflection the

14 INTRODUCTION

Figure 1.6: 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.

vorticity evolution phase begins. Now the vorticity changes sign, and the interfacerolls up.

1.2.1 Shock refraction

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 aslow-fast CO2/CH4 interface is given in figure 1.7 (based on [55]). Regular shockrefraction means that the three signals meet at a single point, called the triplepoint. Since we can solve those problems exactly, we focus on those patterns. Amore detailed description of irregular shock refractions is given by e.g. Henderson[36] or Nouragliev et al.[55]. The upper frame of figure 2.9 shows a numericalsimulation Schlieren plot of a regular shock refraction pattern, while the lowerframe shown an irregular refraction pattern. Research on hydrodynamical shockrefraction has been performed since the 1940’s. The highly analytical work ofvon Neumann [54] was pioneering in this field. He was for example the first


→ 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 1.7: 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 [55].

one to predict critical angles for regular shock refraction. In 1947, Taub [71]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 1.8. In such a shocktube, a shock is generated, often by a piston system, and it propagates throughthe tube at a certain Mach number M . For shock refraction experiments, inside ofthe tube, a membrane mimics a CD seperating 2 gases at rest. Such simulationswhere performed by Jahn [39] to study regular refraction of a plane shock, andlater on also by e.g. Henderson [35], and Abd-El Fattah et al.[1, 2, 3] for irregularshock refraction or by e.g. Haas en Sturtevant [32] and more recently by Kreeft &Koren [44] for shock-bubble interaction.

1.2.2 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 1.9), which moves with speed −M0 towards the shock. At t = 0 theupstream state (ρ1, u1, p1) of the stationary shock is located on the negative x-axis, while the downstream state (ρ0, u0, p0) = (γ,−M0, 1) is located at ]0, 1[. Theup- and downstream state are related through the Rankine-Hugoniot conditions.

16 INTRODUCTION

Figure 1.8: Stanford‘s Unique Aerosol Shock Tube Facility. Taken from [80].

S CD

T

M0

R CD

0 1

t

x

U0

U1 U4

U2 U3

Figure 1.9: Setup of a 1D shock refraction experiment. At time t = M0 a Riemannproblem occurs.


The downstream state is connected to the state at the other side of the CD,(ρ4, u4, p4) = (ηγ,−M0, 1), which is located at ]1,+∞[. In fact, when consideringthis set-up in the frame of the stationary CD, it describes a right-moving shock,such that the downstream state has sonic Mach number M0 (≡ u0

c0), and speeds

are scaled to the sound speed c0 ≡√

γp0ρ0

. Note that the problem is completely

determined by that Mach number M0, and the density ratio across the contactη. At time t = M0, the shock and the CD interact, and a Riemann problemoccurs. The shock will refract in a reflected (R) and a transmitted (T) shock,seperated by the shocked contact discontinuity. Across any signal, the 1D HDRankine-Hugoniot (RH) conditions should hold. We discuss these conditions inmore depth later on, but for now it suffices to know that in any stationary shockframe, these conditions reduce to

ρu =(γ + 1)M2

k

(γ − 1)M2k + 2

ρk, (1.36)

uu =(γ − 1)M2

k + 2

(γ + 1)M2k

uk, (1.37)

pu =2γM2

k + 1 − γ

γ + 1pk. (1.38)

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, u1, 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 (Ml) andtransmitted T (Mr) signals respectively as

Ml =u1 − ξl√

γp1ρ1

, (1.39)

Mr =u4 − ξl√

γp4ρ4

. (1.40)

Here the newly introduced variables ξl and ξr respectively denote the speeds atwhich R and T are traveling. When we apply relations (1.37-1.38) both across thereflected and transmitted signal we arrive at the following solution:

18 INTRODUCTION

u2 =(4γ2 − 4γ)ξlM

30 +

`

(γ2 − 1) + (2γ2 + 2γ)ξl

´

M20 + 8γξlM0 + 2γ + 2

(γ + 1)`

(γ − 2)M20M2

0+ 2

´ , (1.41)

u3 =2ηγ(M0 + ξr)2 − γ + 1

γ + 1, (1.42)

p2 =(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, (1.43)

p3 =(1 − γ)ηM2

0 + (3 − γ)ηξrM0 + 2ηξ2r − 2

η(γ + 1)(M0 + ξr). (1.44)

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

p2 = p3,u2 = u3.

(1.45)

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

ξl =ζ

M0

, (1.46)

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

´ , (1.47)

where ζ should satisfy the quartic equation Σ4i=0tiζ

i, where the coefficients aregiven by

t4 = 48η + (16η − 64)M2

0 , (1.48)

t3 = (56η − 96)M4

0 + (208η − 288)M2

0 + 120η, (1.49)

t2 = (60η − 52)M6

0 + (288η − 312)M4

0 + (364η − 468)M2

0 + 120η, (1.50)

t1 = (18η − 12)M8

0 + (132η − 108)M6

0 + (312η − 324)M4

0 + (252η − 324)M2

0 + 54η, (1.51)

t0 = −M10

0 + (9η − 12)M8

0 + (60η − 54)M6

0 + (118η − 108)M4

0 + (60η − 81)M2

0 + 9η.(1.52)

For M0 = 2, the solution for ξl and ξr is plotted in figure 1.10. 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.


Figure 1.10: Plotted are ξl, ξr and ξc in function of the density ratio η across theunshocked CD. The sonic Mach number is kept constant, M0 = 2. The black lineshows ξl, the red line represents ξr and the blue line shows ξc, the speed at whichthe 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 1.11: The exact solution to the simple shock tube problem. Left: ρ(x).Center: u(x). Right: p(x). Note that indeed velocity and pressure are invariantacross 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 figure1.11. Note that this solution is exact. The analysis above is performed using themathematical software package MAPLE 11.0.

20 INTRODUCTION

Figure 1.12: The AMRVAC solution to the introductory shock tube problem.The effective resolution in this simulation is 30720. Note that indeed this resultsconfirms our exact solution.

Figure 1.13: 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.

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 1.12 shows that this numerical solution agrees with the exactsolution. Note that for η < 1 no exact solution is found. This is because we onlylooked for RRR refraction patterns, but the output of the numerical experimentwith η = 0.5 shows that no RRR solution is possible in this case. Indeed thereflected signal is then continuous. This kind of signal is called an expansion fan(see figure 1.13).


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 u do not jump across the shockedcontact;

• 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 u all change sign.

These observations are schematically presented in a (x, t)-plot of the involvedsignals in figure 1.14. 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 be used to solve the problem exactly. We will also allow for out-of-plane magnetic fields. In chapter 3 finally, we will allow for magnetic fields whichare aligned with the shock normal, such that 5 signals will arise.

1.2.3 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[60] generalized Taylor’s [72] analysis of the RTI to an impulsive acceleration,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

22 INTRODUCTION

S CD

TR CDt

x S CD

TR CDt

x

Figure 1.14: The involved signals in 2 shock refraction cases in an (x, t)-plane.Left: Fast-Slow refraction, Right: Slow-Fast refraction.

(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 [50] 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 ≡ ∇ × u)deposition phase. The magnitude of the deposited vorticity can be found throughshock polar analysis, as extensively described by Henderson [35] in 1966. After theshock has passed the interface, the vorticity evolution phase begins. As Hawley &Zabusky [33] 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[70] gives a good overview of these experiments in 1987. Figure 1.15 plots theintegrated vorticity of the AMRVAC RMI simulation presented by van der Holst& Keppens [83] and confirms that the instability grows linearly in the first phaseof the proces, while after reflection of the top wall, the vorticity deposition changessign. In what follows, we derive the MHD generalisation of the vorticity evolutionequation for 2D inviscid compressible flows.


Figure 1.15: The total vorticity evolution during the HD shock refraction processdescribed in [83]. The vorticity deposition is linear in the vorticity depositionphase. When the triple point reaches the top wall, the vorticity evolution phasestarts and the vorticity deposition changes sign, causing the interface to becomeRM-unstable.

1.2.4 Vorticity evolution equation for inviscid compressible MHD

flows

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

∂

∂tu + (u · ∇)u +

∇pρ

− (∇× B) × B

ρ= 0. (1.53)

Then taking the curl of this expression leads to

∂

∂t(∇× u) + ∇× ((u · ∇)u) + ∇× ∇p

ρ−∇× (∇× B) × B

ρ= 0, (1.54)

or we find

∂

∂tω + ∇× (∇u · u

2− u× (∇× u))

+∇× ∇pρ

−∇× (∇× B) × B

ρ= 0, (1.55)

which simplifies to

∂

∂tω + ∇× (ω × u) + ∇× ∇p

ρ−∇× (∇× B) × B

ρ= 0. (1.56)

24 INTRODUCTION

This can be rewritten as

− u · ∇ω + ω · ∇u − (∇ · u)ω

−∇× ∇pρ

+ ∇× (∇× B) × B

ρ=

∂

∂tω. (1.57)

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) , (1.58)

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)). (1.59)

Therefore equation (1.57) simplifies as

∂

∂tω = −u · ∇ω + ω · ∇u − (∇ · u)ω

+1

ρ2(∇ρ× (∇p− (∇× B) × B)) +

1

ρ(∇× (∇ ·BB)). (1.60)

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(u · ∇)ω +

ω

ρ(∇ · u). (1.61)

AMRVAC AND THE VTK FILE FORMAT 25

Combining equations (1.61) and (1.57), together with Dω

Dt = ∂ω

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

D

Dt

ω

ρ=

∇ρ× (∇p− (∇× B) × B)

ρ3+

∇× (∇ ·BB)

ρ2+

(ω · ∇)u

ρ. (1.62)

All shock refraction simulations and analyses performed in this dissertation are

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

evolution equation in 2D incompressible flows reduces as

D

Dt

ω

ρ=

∇ρ× (∇p− (∇× B) × B)

ρ3+

∇× (∇ ·BB)

ρ2. (1.63)

One could thus conclude that (i) the misalignment of pressure and densitygradients, (ii) the misalignment of the Lorentz force and the density gradient and(iii) the nonuniformity of the magnetic field are responsible for vorticity depositionin 2D flows.

More information on the RMI can be found in review articles as [63, 94, 9].

1.3 AMRVAC and the VTK file format

Since astrophysical "experiments" are only passively obeserved and laboratoryexperiments are expensive to perform, numerical codes gained popularity toperform computer experiments. Interpreting and analysing data produced by anumerical experiment is also easy and accurate. All numerical simulation discussedin this dissertation hae been performed by (MPI)AMRVAC on the K.U.LeuvenHPC cluster VIC. Let us briefly discuss the history and philosophy of this code.

1.3.1 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 [77]. 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 Riemannsolver [61, 62], a Total Variation Diminishing Lax-Friedrich scheme (TVDLF)and implicit and semi-implicit schemes (see Tóth et al.[79], Keppens et al.[42]).

26 INTRODUCTION

VAC is written in Fortran, using a pre-processor, which allows to program in any-dimensional manner (This Loop Annotation Syntax is called the LASY-syntax[78]). More information on VAC can be found in [82].

1.3.2 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 [8] proposed the Adaptive Mesh Refinement (AMR)algorithm, and this principle is used in AMRVAC, as presented by Keppenset al.[43]. 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 [83]), and still later evolved to the present form of the octree basedblock 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 resoluton 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 wether 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

subgrids, as shown in figure 1.16. 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 neighbouring 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 criterium is satisfied.

AMRVAC AND THE VTK FILE FORMAT 27

Figure 1.16: 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, (1.64)

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.

1.3.3 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 seperating two different gases, i.e. gases with differentvalues of γ (as described in Delmont et al.[15]);

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 ([38]), dx([20]),tecplot [73], vtu, ascii are included. I developed the VTU-conversionsubroutine as part of my PhD research.

28 INTRODUCTION

• 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.

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 criterium.

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 refinementcriterium.

The Courant parameter.

• usr: contains the initial conditions. Appendix A gives the usr-file for theshock tube problem presented in [83].

1.3.4 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 default

OUTLINE OF THE THESIS 29

Cartesian- 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 A. Thisfile is related to the RMI problem from Chapter 2.

1.3.5 The VTK file format

Astrophysical simulations often lead to huge data files, especially in 3-dimensionalsimulations, and a major issue is the data visualisation, which is preferablydone by parallel software. ParaView [57] is exactly such an open source parallelvisualisation application. ParaView was developed by Kitware, inc., and is able toprocess a wide range of data types, amongst which VTK [85], which was especiallydeveloped for ParaView. VTK provides several file formats, which are describedin [86]. 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,amongst which VisIt [84]. The output is saved in a binary format.

1.4 Outline of the thesis

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

Chapter 2 is based on Delmont et al.[15]. 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 [14].

Chapter 3 is based on Delmont et al.[16]. In this chapter we generalize ourresults presented in chapter 2 to planar ideal MHD. As mentioned earlier, in the

30 INTRODUCTION

hydrodynamical case, 3 signals arise and the interface becomes Richtmyer-Meshkovunstable due to vorticity deposition on the shocked contact, in ideal MHD, on theother hand, when the normal component of the magnetic field does not vanish, 5signals will arise. The interface then typically remains stable, since the Rankine-Hugoniot jump conditions in ideal MHD do not allow for vorticity deposition on acontact discontinuity. Again, we present an exact Riemann solver based solutionstrategy to describe the initial self similar refraction phase. Using grid-adaptiveMHD simulations, we show that after reflection from the top wall, the interfaceremains stable.

Chapter 4, is based on Delmont & Keppens [17] and focuses on the mathematicaldescription of MHD shocks. Due to the existence of three anisotropicalcharacteristic speeds, the MHD shock classification is much richer than inhydrodynamics, where only the isotropic sound speed enters. One distinguishesslow, intermediate and fast shocks, where intermediate shocks connect a sub-Alfvénic state to a super-Alfvénic state. We investigate under which parameterregimes the MHD Rankine-Hugoniot conditions, which describe discontinuoussolutions to the MHD equations, allow for certain types of intermediate MHDshocks. We derive limiting values for the upstream and downstream shockparameters for which shocks of a given shock type can occur. We revisit thisclassical topic in nonlinear MHD dynamics, augmenting the recent time reversalduality finding by Goedbloed [28] in the usual shock frame parametrization. Ourresults generalize known limiting values for certain shock types, such as switch-onor switch-off shocks.

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

Chapter 2

An exact Riemann solver for

regular shock refraction in HD

In this chapter we analyse 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 thevalues 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.[15].

2.1 Introduction

We study the classical problem of regular refraction of a shock at an obliquedensity discontinuity. Long ago, von Neumann [54] deduced the critical angles forregularity of the refraction, while Taub [71] found relations between the anglesof refraction. Later on, Henderson [35] extended this work to irregular refractionby use of polar diagrams. An example of an early shock tube experiment wasperformed by Jahn [39]. Amongst many others, Abd-El-Fattah & Henderson [2, 3]performed experiments in which also irregular refraction occured.

31

32 AN EXACT RIEMANN SOLVER FOR REGULAR SHOCK REFRACTION IN HD

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 [60]). 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 3.1). 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 [50]. 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.[55]) results show that also in this casethe interface becomes unstable. The growth rates obtained by linear theorycompare poorly to experimentally determined growth rates (Sturtevant [70]). 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.[56]), astrophysics (e.g. Kifonidis et al.[45]), and it is a common test problemfor numerical codes (e.g. van der Holst & Keppens [83]).

In essence, the RMI is a local Kelvin-Helmholtz instability, due to the deposition ofvorticity on the shocked contact. Hawley & Zabusky [33] formulate an interestingvortex paradigm, which describes the process of shock refraction, using vorticity asa central concept. Later on, Samtaney et al.[67] 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 [68] 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.[88] 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 [27]). The solution of the Riemann problemin ideal MHD is well-studied in the literature (e.g. Lax [47]), 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 [74] and numerically by Chu & Taussig [13]. 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 33

evolutionary. Brio & Wu [10] and De Sterck et al.[18] found intermediate shocksin respectively one and two dimensional simulations. The evolutionary conditionbecame controversial and amongst others Myong & Roe [52, 53] argue that theevolutionary condition is not relevant in dissipative MHD. Chao et al.[11] reporteda 2 → 4 intermediate shock observed by Voyager 1 in 1980 and Feng & Wang [24]recognised a 2 → 3 intermediate shock, which was observed by Voyager 2 in 1979.On the other hand, Barmin et al.[7] 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 [21, 22] 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 [83]; Keppens et al.[43]).

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 3.1, the hydrodynamical problem of regular shock refractionis parametrised 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 3.3): 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 behaviour 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[27]). Written out in stationary, conservative form and for our planar problem, theMHD equations 1.12 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 35

1

unshocked

shocked 2

n = (sinφ,−cosφ)

x

y

v1

v2

φ

atan(λi)

y

x

Figure 2.2: Left: A stationary shock, seperating 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 solutions of the system, which are solutions of the integral formof the MHD equations. The shock occuring in the problem setup, as well asthose that later on may appear as R or T signals obey the Rankine-Hugoniotconditions. In the case of two dimensional stationary flows (see figure 2.2), wherethe shock speed s = 0, the Rankine-Hugoniot conditions follow from equation(3.1). When considering a thin continuous transition layer in between the tworegions, with thickness δ, solutions of 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 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 initialise the problem in a dimensionless manner.In the initial refraction phase, the shock wil 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 nondimensionalisethe problem. Note ui = (ρi, vx,i, vy,i, ptot,i, Bi, γi), where the index i refers to the

RIEMANN SOLVER BASED SOLUTION STRATEGY 37

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 components arescaled with respect to the sound speed in this region between the impinging shockand the initial CD. Since this region is initially at rest, the sonic Mach numberM of the shock equals its shock speed s. When the shock intersects the CD, thetriple point follows the unshocked contact slip line. It does so at a speed vtp =(M,M tanα), therefore we will solve the problem in the frame of the stationarytriple point. We will look for selfsimilar solutions in this frame, u = u(φ), whereall signals are stationary. We now have that vx = vx −M and vy = vy −M tanα,where v refers to this new frame. From now on we will drop the tilde and onlyuse this new frame. We now have u1 = (γl,−M,−M tanα, 1, 0, γl)

t and u5 =(ηγl,−M,−Mtanα, 1, 0, γr)

t. The Rankine-Hugoniot relations now immediatelygive a unique solution for u2, namely

u2 =

(

(γ2l + γl)M

2

(γl − 1)M2 + 2,− (γl − 1)M2 + 2

(γl + 1)M,−Mtanα,

2γlM2 − γl + 1

γl + 1, 0, γl

)t

.

(2.8)

In MHD, we nondimensionalise by definining 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,−Mtanα, β+1

2 , 1, γl

)t

and from the

definition of η, u5 =(

ηγlβ2 ,−M,−Mtanα, β+1

2 , 1, γr

)t

. The Rankine-Hugoniot

relations now give the following non-trivial solutions for u2:

u2 =

(−γlβM2ω

, ω,−Mtanα, p2 +M2

2ω2,−Mω

, γl

)t

, (2.9)

where

p2 =Aω +B

Cω +D, (2.10)


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

A = γl(

β2(4γ2lM

4 − 2γlM2 − γl − 1)

+β(

(γ2l + 4γl − 5)M2 − 2

)

− γl + 2)

, (2.11)

B = (γl − 1)M(

β(M2(γ2l + 7γl) − 2γl + 4) − 2γl + 4

)

, (2.12)

C = 2γl(γl + 1)(

β((γl − 1)M2 + 2) + 2)

, (2.13)

D = 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 rightmoving shock.

2.3.2 Relations across a contact discontinuity and an expansion

fan

Rewriting equation (3.1) 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 selfsimilar solutionsand we can introduce ξ = y

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


of A represent tanφ, where φ is the angle between the refracted signals and thenegative x-axis. The matrix A is given by

A ≡ F−1u Gu =

vy

vx

ρvy

v2x−c2− ρvx

v2x−c2vy

vx

1v2x−c2

0 0

0vxvy

v2x−c2− c2

v2x−c2− vy

ρ1

v2x−c20 0

0 0vy

vx

1ρvx

0 0

0 − ρc2vy

v2x−c2ρc2vx

v2x−c2vxvy

v2x−c20 0

0 − Bvy

v2x−c2− Bvx

v2x−c2vy

vx

Bρ

1v2x−c2

vy

vx0

0 0 0 0 0vy

vx

. (2.18)

and its eigenvalues are

λ1,2,3,4,5,6 = vxvy + c√v2 − c2

v2x − c2

,vyvx,vyvx,vyvx,vyvx,vxvy − c

√v2 − c2

v2x − c2

, (2.19)

where the magnetosonic speed c ≡√

v2s + v2

a and the sound speed vs =√

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 expansion fans, uξ isproportional to a right eigenvector ri of A. Derivation of ξ = λi with respect to ξgives (∇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 + c√v2−c2ρv2s

dptot = 0,

vydvx − vxdvy − c√v2−c2ρv2s

dptot = 0.(2.22)

Since v 6= c, 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 [75], 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 aross 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±c√v2−c2

v2x−c2the following relations connect two states across expansion

fans:

dρ− 1c2 dptot = 0,

vxdvx + vydvy + c2

ρv2sdptot = 0,

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

(

γp+B2)

dB = 0,

vydvx − vxdvy ± c√v2−c2ρv2s

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

1c2 dptot,

vx,i = vx,e +∫ p∗

ptot,e

±vy

√v2−c2−vxcρv2c dptot,

vy,i = vy,e +∫ p∗

ptot,e

∓vx

√v2−c2−vycρv2c dptot,

Bi = Be +∫ p∗

ptot,e

Bρc2 dptot,

pi = pe +∫ p∗

ptot,e

v2sc2 dptot,

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 or


both 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 = atan(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 variab;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 = atan

vy,3

vx,3= atan

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±ce

√v2e−c2e

v2x,e−c2e. 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). Wedecompose 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 (3.1) 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,

c2ρ′ − 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 − c2v2

n = 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

c2 − 2v2n

3v2n + (γ − 2)c2

ρc2. (2.31)

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

IMPLEMENTATION AND NUMERICAL DETAILS 43

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

2.4 Implementation and numerical details

2.4.1 Details on the Newton-Raphson iteration

We can generally note that ptot,pre < ptot,post. This implies that the refractionhas 3 possible wave configurations: 2 shocks, a reflected rarefaction fan and atransmitted shock, or 2 expansion fans. Before starting the iteration on [[

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 in itsconfiguration. 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 interation: 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 [83]; Keppens et al.[43]) 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 infigure 2.4. The refinement strategy is done by quantifying and comparing gradients.The AMR in AMRVAC is of a block-based nature, where every refined gridhas 2D children, and D is the dimensionality of the problem. Parallelisation isimplemented, 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 the contactdiscontinuity is located at y = (x−1)tanα. We used the fourth order Runge-Kuttatimestepping, together with a TVDLF-scheme (see Tóth & Odstrcil [76]; Yee [92])with Woodward-limiter on the primitive variables. The obtained numerical resultswere compared to and in agreement with simulations using other schemes, such asa Roe scheme and the TVD-Muscl scheme. The calculations were performed on 4processors.

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 lead toexcessive numerical diffusion on γ. Since γ is a discrete variable we know γ(x, y, t)exactly, if we are able to follow the contact discontinuity in time. Suppose thusthat initially a surface, seperates 2 regions with different values of γ. Define afunction χ : D×R

+ → R : (x, y, t) 7→ χ(x, y, t), where D is the physical domain of(x, y). Writing χ(x, y) = χ(x, y, 0), we ask χ to vanish on the initial contact andto 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). The implementedsystem is thus ( 3.1), but the last equation is replaced by (χρvx)x + (χρvy)y = 0.It is now straightforward to show that we did not introduce any new signal. Inessence, this is the approach presented in Mulder et al.[51].

RESULTS 45

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 [68]: a) all shock

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

2.5 Results

2.5.1 Fast-Slow example solution

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 [68]. 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

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.


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 [68]. Notice that p and vx

vyremain constant across the shocked contact.

RESULTS 47

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 [83]. Notice that S remains constant across R.

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 [83]. 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 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 1atanφ . Moreover, both the

eigenvalues, ξ+ and ξ−, have 4 singularities, namely c2 ∈ −vx,2, vx,2,−vy,5, vy,5for ξ− and c5 ∈ −vx,5, vx,5,−vy,2, vy,2 for ξ+, where thus c25 = v2

5,y ⇔ c22 = v22

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∗ = 2γrηM2tan2(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 and α = 0.3. In the regular case, all signals meet at


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.

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 contourlines. 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.. [55]). This pattern, of which the caseα = 0.3 is an example, is called Concave-Forwards irregular Refraction. Theseresults are in agreement with our predictions.

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.[55])and refraction theory (see e.g. Henderson [37]). 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.

RESULTS 49

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.

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).


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(

α, β−1, γl, γr,M)

=(

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 .

A very weak shock, M = 1.12 is refracted at the interface under various angles.von Neumann [54] 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 showthe 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(


=(

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-slow

RESULTS 51

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.

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-fast problem: strong perpendicular magnetic fields decrease the instability of theCD.

refraction φT < π2 , but for slow-fast refraction, φT > π

2 and the transmittedsignal 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.


2.5.6 Effect of a perpendicular magnetic field

In general, the MHD equations result in the following jump conditions across acontact discontinuity

p+B2

t

2BnBnBtvtBn

= 0. (2.34)

It follows, that if the component Bn of the magnetic field, normal to the shockfront is non-vanishing, a case we did not consider so far, the MHD equationsdo not allow for vorticity deposition on a contact discontinuity and the RMI issuppressed (Wheatley et al.[88]). The remaining question is what the effect of apurely tangential magnetic field is, where the field is perpendicular to the shockfront and thus acts to increase the total pressure and the according flux terms.

Also note that it follows from equations 3.18 and 3.19 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.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 know thatthis is equivalent to

β > βmin ≡ 2

γl(M2 − 1). (2.35)

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 instable for the HD case.Increasing β−1 decreases the shock strength. For β−1 = 1 the shock is very weak:the Atwood number At = 0.17, and the interface remains stable.

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 can

RESULTS 53

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-Meshkovinstability causes the interface to roll up. Center : β−1 = 1

2 . Although the initialamount 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 [68] 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.

for example be understood by noting that limβ→βmin

φCD = α (see figure 2.15 ). A

convenient way to measure the strength of a shock is by use of its Atwood number

At =ρ2 − ρ1

ρ2 + ρ1. (2.36)

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]]vs,1

At= 1. (2.37)

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(2.38)

=

√

(At+ 1)((γβ + 2γ − 2)At− (γβ + 2))

γβ(1 −At)(γAt+ 1). (2.39)

Note that in the limit for weak shocks

limAt→0

M =

√

γβ + 2

γβ, (2.40)

CONCLUSIONS 55

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 : Solution for the fast-slow problem: strong perpendicularmagnetic fields decrease the instability of the CD. Right : Solution for the slow-fast problem: strong perpendicular magnetic fields decrease the instability of theCD.

which is equivalent to 2.35, and in the limit for strong shocks, M → ∞. Figure2.16 shows the deposition of vorticity on the shocked contact, for a constantAtwood number. We conclude that under constant Atwood number, the effectof a perpendicular magnetic field is small: Stronger perpendicular magnetic fieldincrease the deposition of vorticity on the shocked contact slightly. This isconfirmed by AMRVAC sumulations (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 generalise our results for planar uniform magnetic fields,where up to 5 signals arise. In this case we will search for non-evolutionarysolutions, involving intermediate shocks.


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

lower : β = 0.25.

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 accepted for publication (Delmont & Keppens [16]).

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 1 (or Delmont et al. [15]),we solved the initial self similar phase of the problem exactly, exploiting exactRiemann solver methods. In this chapter, we extend that study to planar idealmagnetohydrodynamics (MHD).

For that planar case, Samtaney [68] showed by numerical grid-adaptive simulationsthat addition of a uniform magnetic field, aligned with the shock normal,

57

58 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 modelled 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 ≡ 2p

B2 (in the pre-shock region). These 5 parameters definethe problem uniquely. In the MHD case up to 5 signals arise: a fast (FR) anda slow (SR) reflected signal, a contact discontinuity (CD), and a slow (ST) anda fast (FT) transmitted signal, separating the 4 new formed states, as shown infigure 3.3.

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.[15] for an exactsolution of the HD case. In this paper we generalise the presented solutionstrategy to planar ideal MHD. Our results are in agreement with results fromWheatley et al.[88]. Our approach is inspired by Wheatley et al.[88]. 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.[43]; van der Holst & Keppens [83]).

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. Goedbloeds& Poedts [27]).

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 [81] andin section 3.5 we demonstrate the algorithm by solving a case presented firstby Samtaney in [68]. In section 6 finally, we compare our results to AMRVACsimulations.

INTRODUCTION 59

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 behaviour 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 problem ina comoving frame with the quintuple point. We can then assume that the solutionis self similar and time independent: ∂/∂t = 0. Therefore, we can exploit thestationary 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 assume that vx(x, t) < 0 in the comoving frame with the

GOVERNING EQUATIONS 61

quintuple point. In the rest frame this means that the velocity in the x-directionis bounded by the shock speed, during the refraction process.

We allow weak solutions of the system, which are solutions of the integral formof the MHD equations. The discontinuities occuring in the problem setup (boththe impinging shock and the initial contact discontinuity), as well as those thatmay appear as FR, SR, ST or FT signals later on obey the Rankine-Hugoniotconditions. In appendix A we give the planar stationary Rankine-Hugoniotconditions 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, and tangential vectorcomponents 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.[15], 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 − 2a2) + 2c2a2

x, (3.8)

t3 = 4vxvy(c2 + a2 − v2

x) − 8c2axay, (3.9)

t2 = 6v2xv

2y − 2a2(v2 + c2) + 2v2c2, (3.10)

t1 = 4vxvy(c2 + a2 − v2

y) − 8c2axay, (3.11)

t0 = v4y + v2

y(4c2 − 2a2) + 2c2a2

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 ax =√

B2x

ρ , ay =√

B2y

ρ and the total Alfvén speed a =√

a2x + a2

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 + a2 +

√

(c2 + a2)2 − 4c2

(Bxλ−By)2

ρ(1 + λ2)

)

, (3.15)

and the (squared) slow speed by

v2s,⊥ =

1

2

(

c2 + a2 −√

(c2 + a2)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 63

3.3 MHD shocks

MHD shocks are a topic of extensive research (see e.g. [48, 28]). Introducing

the normal Alfvén speed an ≡√

B2n

ρ , which is an eigenvalue of the full three

dimensional stationary MHD equations, and at ≡√

B2t

ρ leads to a =√

a2n + a2

t .

The full set of eigenvalues of the full set of MHD equations is given by±vf ,±an,±vs, 0, where the fast speed vf and the slow speed vs are definedas

v2f ≡ 1

2

(

c2 + a2 +

√

(c2 + a2)2 − 4c2a2n

)

; (3.17)

v2s ≡ 1

2

(

c2 + a2 −√

(c2 + a2)2 − 4c2a2

n

)

. (3.18)

Since 0 ≤ vs ≤ an ≤ vf , the full set of MHD equations is non-strictly hyperbolic.Those speeds define the up- and downstream states into 4 types (and 3 transitiontypes plus 1 stationary type for completeness), which in order of increasing entropyS ≡ p

ργ :

• (1) superfast: vf < |vn|;

• (1=2) fast: |vn| = vf ;

• (2) subfast: an < |vn| < vf ;

• (2=3) Alfvén: an = vn;

• (3) superslow: vs < |vn| < an;

• (3=4) slow: |vn| = vs;

• (4) subslow: vs < |vn| < an;

• (∞) stationary: vx = 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 (B.1) now allow for the following types of shocks:

• (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.[7]; Falle & Komissarov [21, 22]; Myong & Roe[52, 53], observationally in e.g. Chao et al.[11]; Feng & Wang [24] and numericallyin e.g. Brio & Wu [10], De Sterck et al.[18]). We allow for intermediate shocksin our solution strategy, since they form the central Alfvénic transition for MHDshocks (Goedbloed [28]), and as we show later, they naturally emerge in highresolution numerical simulations.

MHD SHOCKS 65

vn > vf vf > vn > an

(a)

an > vn > vs vs > vn

(b)

vn > an an > vn

(c)

vf < an vn = an

(d)

vn = an vs > an

(e)

vn = an vn = an

(f)


vn = 0 vn = 0

(a)

vn = 0 vn = 0

(b)

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.

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 regionby 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,Mtanα), 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,−Mtanα, 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,−Mtanα,

2γM2 − γ + 1

γ + 1,

√

2

β

)t

.

(3.19)

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 A).Since By is uniquely determined by the other 4 (independent) variables, wehave 4 independent scalars which should vanish. Hence we can guess 4 pathvariables, 1 across each magnetoacoustic signal, and express the vanishingquantities in function of those path variables χ ≡ (χFR, χSR, χST , χFT ). Thefast magnetoacoustic signals are postulated to be shocks, while the slowmagnetoacoustic signals 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 B and C. Inthe case of a rarefaction wave, we numerically integrate the MHD equations asexplained in E. In this manner we reduce the problem to solving

[[vx, vy, Bx, p]](u4(u3(u2, χFR), χSR),u5(u6(u7, χFT ), χST )) = 0. (3.20)

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 magnetoacoustic 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), (3.21)

χF T = tan(φF T ), (3.22)

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[

(3.23)

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 [

(3.24)

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 = atan

(

B3,y −√ρ3v3,y

B3,x −√ρ3v3,x

)

, (3.25)

φa,T = atan

(

B6,y +√ρ6v6,y

B6,x +√ρ6v6,x

)

, (3.26)

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, (3.27)

φcr,T ≡ minφ > φa,T |(N2 −D3)(φ) = 0. (3.28)

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 ≡ atanvy,3vx,3

, (3.29)

φst,T ≡ atanvy,6vx,6

. (3.30)

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


Once the wave configuration is determined, χ determines the solution to theproblem. A secant method iteration solves equation (3.20) and φSC = atan

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 areexactly fast. For the slow signals, initially we put χSR = χST = 0. In the case of anintermediate shock, this means that our starting angles φSR and φST respectivelysatisfy 2φSR = φa,R + φcr,R and 2φST = φa,T + φcr,T . The procedure also ensuresthat 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 φST respectively satisfy2φSR = φsl,R + φa,R and 2φST = φsl,T + φa,T . Finally, when the slow signalis located at φSR > φsl,R there is no shock solution possible which satisfies theentropy condition, and in the case the signal is a slow rarefaction fan. The initialiteration angles φSR and φST respectively satisfy 2φSR = φst,R+φsl,R and 2φST =φst,T +φsl,T . We hereby postulated that the slow signals are expansion fans whenv2n,3/6(φSR/ST ) < v2

sl,3/6. In this case, the expansion fan is located between φSR/STand φsl,R/T . This criterion is equivalent to the criterion that a rarefaction fan willonly occur at a given position if no shock solutions satisfying the entropy conditionare possible, and is in this sense a generalisation of the criterion exploited in [15].

3.4.3 Tackling the signals

The solution to the stationary Rankine-Hugoniot conditions is given in appendix B.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 ofa different shock type. Thus when the shock types of all magnetoacoustic signalsare known, the Rankine-Hugoniot conditions can be solved uniquely. We solve thecubic analytically in Appendix C, 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 C, we also provide additional technical information on how permuta-tion of the root indices can make the analytical expressions for the roots (D.8)continuous and differentiable, and hence allows for a secant iteration methodapplied on equation (3.20). The only remaining discontinuities are reached, whenthe refraction angles cross the critical angles φcr,R or φcr,T , where the transitionfrom a unique solution to 3 solutions for the RH conditions takes place. Also [[p]]is discontinuous across the Alfvénic angles. This leads to the restriction that wecannot cross those critical angles in subsequent iteration steps. This is taken careof by our choice of path variables.

In appendix E 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 [68] and Wheatley et al. [88]. For this case we havethat

u1 = (1.4000,−2.0000,−2.0000, 1.0000, 1.0000), (3.31)

u2 = (3.7333,−0.7500,−2.0000, 4.5000, 1.0000), (3.32)

u7 = (4.2000,−2.0000,−2.0000, 1.0000, 1.0000). (3.33)

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 3.5.

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;

SUPPRESSION OF THE RICHTMYER-MESHKOV INSTABILITY 71

• 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;

• 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 (B.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.


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 3.5: The cubic for respectively FR, SR, ST, FT.

Therefore, they do not allow for vorticity deposition on the contact, and theinterface must remain stable. Figure (3.6) shows snapshots of a numericalsimulation of the parameter regime (α, β, γ,M, η) =

(

π4 , 2,

75 , 2, 3

)

performed bythe Adaptive Mesh Refinement code AMRVAC. The first frame shows the density,ρ, during the shock refraction. Note that the CD remains stable, since the jumpin tangential velocity, [[vt]], vanishes. The second frame shows the tangentialmagnetic field component, Bt. The density is discontinuous in every new-formedsignal, while the CD is not visible in the Bt frame. SR is a slow shock, thereforeBt does not change sign across this signal, while ST on the other hand is anintermediate shock and Bt changes sign across it.

3.7 Conclusion and Future Work

We developed an exact Riemann solver based solution for regular shock refractionin planar ideal MHD. The Richtmyer-Meshkov instability is suppressed in idealMHD. We are able to reproduce results from the literature and results by numericalsimulations performed by AMRVAC.

CONCLUSION AND FUTURE WORK 73

Figure 3.6: 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. Down: 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.


Chapter 4

Parameter ranges for

intermediate shocks

We investigate under which parameter regimes the MHD Rankine-Hugoniotconditions, which describe discontinuous solutions to the MHD equations, allow forslow, intermediate and fast shocks. We derive limiting values for the upstream anddownstream shock parameters for which shocks of a given shock type can occur.We revisit this classical topic in nonlinear MHD dynamics, augmenting the recenttime reversal duality finding by [28] in the usual shock frame parametrization.

This work is published in Delmont & Keppens [17].

4.1 Introduction

4.1.1 Intermediate shocks in MHD

The dynamic behaviour 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 [25]; Anderson [5];Jeffrey & Taniuti [40]; Liberman & Velikhovich [48]; Sturtevant [70]; Gombosi [30];Goedbloed [28]) 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 perturbationswith MHD (switch-on and switch-off) shocks was studied both analytically by

75

76 PARAMETER RANGES FOR INTERMEDIATE SHOCKS

Todd [74] and numerically by Chu & Taussig [13]. Later on, the evolutionarityof intermediate shocks, which cross the Alfvén speed, has been adressed. 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 [89], De Sterck et al.[18] and others found intermediate shocks in one andtwo dimensional numerical simulations respectively. De Sterck & Poedts [19] werethe first to find intermediate shocks in three dimensional numerical simulations.The evolutionary condition became controversial and Myong & Roe (1997(a,b)),amongst others, argue that the evolutionary condition is not relevant in dissipativeMHD. Wu (1988, 1990) also argued that intermediate shocks are admissible. Shockobservations in the heliosphere can be found in favor of their existence as well:Chao et al.[11] reported a 2 → 4 intermediate shock observed by Voyager 1 in1980 and Feng & Wang [24] recognised 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[10] 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.[87]). The importance of compound waves is that they can bean alternative to intermediate shocks to cross the Alfvén speed. Barmin et al.[7]argue that if the full set of MHD equations is used to solve planar MHD, a smalltangential perturbation on the magnetic field vector splits the rotational jump fromthe non-evolutionary compound wave. This transforms the latter one into a slowshock, such that the compound wave is non-evolutionary. They investigate thereconstruction process of the non-evolutionary compound wave into evolutionaryshocks. Falle & Komissarov (1997, 2001) also reject intermediate shocks onevolutionary grounds, and suggest a shock capturing scheme based on Glimm’smethod (Glimm [26]) 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 argumentations.

Recently, Goedbloed [28] classified the MHD shocks by rewriting the RH equationsin the de Hoffmann-Teller frame (De Hoffmann & Teller [34]) introducing theexistence of a distinct time reversal duality between entropy-allowed and entropy-forbidden solutions. This work encouraged us to revisit the classical RH conditionsand augment these results in terms of the commonly exploited shock parameters

INTRODUCTION 77

in 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, ρis the mass density, v is the velocity, p the thermal pressure and B the magneticfield. The ratio of specific heats, γ, is considered a constant parameter, as wewill assume an ideal gas equation of state. For a derivation of these well-knownexpressions, we refer to De Hoffmann & Teller[34]; Jeffrey & Taniuti [40]; Liberman& Velikhovich [48], Goedbloed & Poedts [27]. The eight governing MHD equationshave been reduced to six jump conditions in equation (B.1). Three equations havebeen dropped from the fact that tangential magnetic field components are forcedto lie 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 (B.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 otherunknown state, uu, in function of this known state. In order to do so, we introduce


three 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|an

. (4.4)

We introduced the normal Alfvén speed an ≡√

B2n

ρ . Analogously, we define the

tangential Alfvén speed at ≡√

B2t

ρ and the total Alfvén speed a ≡√

a2n + a2

t .

The full set of characteristic speeds of the full set of MHD equations is given byvn ± vf , vn ± an, vn ± vs, vn, where the fast speed vf and the slow speed vs aredefined as

v2f ≡ 1

2

(

c2 + a2 +

√

(c2 + a2)2 − 4c2a2

n

)

, (4.5)

v2s ≡ 1

2

(

c2 + a2 −√

(c2 + a2)2 − 4c2a2

n

)

, (4.6)

and where we introduced the sound speed c ≡√

γpρ . On the other hand, the

restricted set of planar MHD equations, describing MHD dynamics with all vectorquantities restricted to lie in the same plane, has only five characteristic speedsvn± vf , vn± vs, vn. Ignoring cold MHD, where the thermal pressure p vanishes,and assuming that the normal component of the magnetic field Bn (which can beseen as an equation parameter now) does not vanish, we conclude that the planarsystem is strictly hyperbolic, since vf = vs would imply that both a and c wouldvanish. Obviously c cannot vanish, and for the same reason also vs 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 characterisation: the shock type i → j. It is well-known ([48]) that once the shock type and one state is given, if a solution exists,

SOLUTION TO THE RANKINE-HUGONIOT CONDITIONS 79

it must be unique. Since 0 ≤ vs ≤ an ≤ vf , the full set of MHD equations ishyperbolic, but non-strictly hyperbolic. Those characteristic speeds categorize theup- and downstream states into the classical 1−2−3−4-classification as explainedin 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 dualityprinciple from Goedbloed [28]. 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 (B.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 [48]. Recently, a contribution to thisclassical theory was described by Goedbloed [28], 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 [34]). 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 (B.1)). Since nodistances or times are involved, the problem is now uniquely scaled. Note thatvelocities are scaled to the known normal Alfvén speed an. The known state isnow given by

uk ≡ (ρ, vn, p, Bt) =

(

1, σM,β(1 + θ2)

2, θ

)

, (4.7)

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, (4.8)

vn,u = σθ(M2 − 1) + ψ

Mψ, (4.9)

pu =

(

2

γ + 1+ψ2 + θψ + 2

M2 − 1

)

M2 − γ − 1

γ + 1

β(1 + θ2) + (ψ − θ)2

2, (4.10)

Bt,u = ψ, (4.11)

where ψ satisfies the cubic equation

C(ψ) ≡ ψ3 + τ2ψ2 + τ1ψ + τ0 = 0, (4.12)

and its coefficients are given by

τ2 = −θ(

(γ − 1)(M2 − 1) −M2)

, (4.13)

τ1 = (M2 − 1)(

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

, (4.14)

τ0 = −(γ + 1)θ(M2 − 1)2. (4.15)

The tangential velocity is then found from

[[vt]] =θ − ψ

M. (4.16)

We can now define the dimensionless parameters refering 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 (4.12) 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 81

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

Mu =

s

(M2 − 1)θ + ψ

ψ, (4.17)

θu = ψ, (4.18)

βu =

`

(γ − 1)((θ − ψ)2 + (1 + θ2)β) − 4M2´

(M2− 1) + 2M2(ψθ + ψ2)

(M2 − 1) (γ + 1) (1 + ψ2).(4.19)

In terms of the dimensionless parameters, these equations lead to the followingwell-known (see e.g. [28]) invariants across a shock:

[[(M2 − 1)θ]] = 0, (4.20)

[[

2M2 + β(1 + θ2) + θ2]]

= 0, (4.21)

[[

(γ

γ − 1β +M2)(1 + θ2)M2

]]

= 0. (4.22)

When we define the new quantity Ω, computed from the coefficients of thegoverning cubic equation (4.12), as

Ω ≡ 27τ20 + 4τ3

1 + 4τ22 τ0 − τ2

2 τ21 − 18τ2τ1τ0, (4.23)

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

Ω < 0. (4.24)

Note that this criterium is equivalent to equation .

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 [28] 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 , Alfvén: vn = an and slow: vn = vs. These curves separatethe (θ,M) plane in the classical 1 − 2 − 3 − 4 state regions. Right: Shown is thecurve Ω = 0. Note that states with M = 1 appear as a double zero of Ω. WhereΩ > 0, only one real solution to the RH conditions exist. On the other hand,where Ω < 0, there exist three real solutions to the RH conditons. On Ω = 0, theRH solutions allow for 2 distinct real solutions.

(M, θ, β). A related observation is that equation (4.24) 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 [48]).

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−an)(v2

n− v2f )(v

2n− v2

s) = 0. One can visualize these conceptsby drawing the corresponding curves in the (θ,M)-plane, for fixed β and γ. Thisis done in figure (4.1), where the plot at left concentrates on the 1−2−3−4 stateregions, 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 83

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 or a 3 = 4-state, where vn = vs, 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 . Straightforward algebra showsthat 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 seperated by a slow 3 → 4 shock. Both thesesolutions have our specific known upstream state uk, where vn = vf , as a doublesolution, and therefore satisfy Ωu,a = Ωu,b = 0.

Completely analogously, consider now a downstream state with vn = vs.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, as a double solution, and therefore satisfy Ωu,a = Ωu,b = 0.

Finally, consider a state where vn = an. 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 , vn = an, vn = vs)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 and vn = an have a common point at (θ,M) = (0, 1). When

RESULTS 85

the plasma is thermally dominated, β > 2γ , the curves vn = vs and vn = an 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 [28] 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 87

(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 [28].

4.4.2 Equivalence classes introduced by the RH conditions

The RH conditions are equivalent to equations (4.20-4.22), 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)(v2n − v2

f ) = 0. In this case, the only real solution tothe 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)(v2n − v2

f ) 6= 0. In this case the equivalence classcontains A itself, and the state introduced by the unique (non-trivial) realsolution to the cubic equation (4.12).

• If Ω = 0 and (v2n − v2

s)(v2n − v2

f ) = 0. Since Ω = 0, the solutions to the RHconditions crossing the Alfvén speed are coinciding. Since vn is exactly fastor 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)(v2n − v2

f ) 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 or vn = vs. 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 and vn 6= vs. 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 = an. The RH conditions now have a switch-on shock, a switch-off shockand a rotational wave as a solution. The considered state itself completesthe 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 89

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 analoguously 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 tothe V −D-region, and vice versa. Further, we conclude that for any state in theregions I −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 throughthe RH jump condition. The initial eight regions of parameter space have thusbeen divided into two groups of two mutually connectable regions and one groupof four mutually 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 subregions 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 subregions 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, vn = an and vn = vf ; (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 91

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, vn = an, vn = vf 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 93

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. (4.25)

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, (4.26)

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.[41]).

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 (4.27)

Mu = 1 (4.28)

θu = ±√

((γβ − 2) − (γ − 1)(M2 − 1))(M2 − 1). (4.29)

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

)

, (4.30)

which only has positive pressure for β < 6M2.

Figure (4.4) shows the plane given by θ = 0 in parameter space. The graphsplotted are (i) the curves given by Ω = 0; (ii) the curve vn = an; (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, demarkating 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.

(4.31)

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. (4.32)

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 95

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 ,Ω = 0.

(4.33)

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 isdecreasing for increasing β, and the limit value for β = 0, equals 0.65633, weconclude that −0.65633 < θ1,1→3 < 0.65633.

The downstream state.


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.

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

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 shocks

RESULTS 97

to 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.

(4.34)

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 thusbe found by solving equations (4.33). The left panel of figure (4.7) also shows aplot of 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.

(4.35)

A straighforward 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. Thevariation with β is shown in the right panel of figure (4.7).


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.


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 → 4shock, must be located in region (xvi).

We first find the minimal value of the Mach number M at which 1 → 4 can occur.

RESULTS 99

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.

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,

(4.36)

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.

(4.37)

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).


4.5 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 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 [28], 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

Dit proefschrift behaldelt schokbreking in hydro- en magnetohydrodynamica.Vooraleer dieper in te gaan op schokken en schokbreking, lichten we eerst debegrippen hydrodynamica (HD) en magnetohydrodynamica (MHD) toe.

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 in functie van de tĳd te beschrĳven, maar dit zou leiden totenorm stelsel van vergelĳkingen. Zelfs voor zeer kleine systemen is het oplossenvan zo’n stelsel onbegonnen werk. Er is dus nood aan verstandigere manierenom het gedrag van fluïda te beschrĳven. In plaats van de beweging van elkdeeltje afzonderlĳk te beschrĳven, wordt het fluïdum beschouwd als een continuüm.In deze continuümbeschrĳving spreken we over massadichtheid, momentum enenergiedichtheid, wat essentieel statistische begrippen zĳn. De aannames die HDdan mogelĳk maken zĳn het behoud van massa, het behoud van impuls en hetbehoud 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 mexican wave, een microgolf, een radiogolf of een geluidsgolf.

Elk gas heeft een karakteristieke snelheid. Deze karakteristieke snelheid wordt de

101

102 NEDERLANDSTALIGE SAMENVATTING

geluidssnelheid van het gas genoemd. Het karakteristieke aan deze snelheid is dater een schokgolf onstaat wanneer het medium lokaal de geluidssnelheid overschrĳdt.Dit is bĳvoorbeeld het geval wanneer een vliegtuig door de geluidsmuur vliegt.Exacter uitgedrukt: een stationaire HD schok scheidt een supersonische toestandvan een subsonische toestand. Deze eigenschap wordt de Prandtl-Meyer eigenschapgenoemd. Wiskundig valt dit te begrĳpen door op te merken dat de HDvergelĳkingen in dit geval enkel discontinue oplossingen hebben, die exact dezeschokgolven beschrĳven. Verder kan aangetoond worden dat de geluidssnelheidisotroop is: deze snelheid is namelĳk even groot in elke richting. Daarom verplaatsteen 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. Alsde temperatuur van het gas hoog genoeg is, overwint de centrifugale de elektronende aantrekkingskracht van de protonen, zodat het has bestaat uit protonen envrĳ bewegende elektronen. Wanneer een gas voldoende opgewarmt wordt, zodatde elektronen zo snel bewegen dat hun momentum de aantrekkingskracht van deprotonen overwint, komen de elektronen los van de ionen, zodat het gas bestaat uitionen 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 met met dedrie karakteristieke MHD snelheden nu vier verschillende toestanden definieren.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 de vergelĳkingen dieschokgolven beschrĳven, maar er is geen eensgezindheid over het bestaan van dezeschokken in de fysische wereld. Alleszins, in hoofdstuk 4 van deze thesis leiden weaf in welke omgevingen de wiskunde het bestaan van zo’n intermediaire schokkentoelaat.

NEDERLANDSTALIGE SAMENVATTING 103

Een contactdiscontinuïteit (CD) is het oppervlak dat twee gassen of plasmasscheidt. In schoktubes worden CD’s nagebootst met behulp van een heel lichtmembraan. Zo gebeuren fysische experimenten i.v.m. schokbreking. In deze thesisbestuderen we schokbreking vanuit een andere hoek.

In hoofdstuk 2 bestuderen we de interactie van een HD-schok met zo’n CD enin hoofdstuk 3 veralgemenen we deze studie naar de interactie tussen een MHD-schok en een CD. We tonen aan dat het magneetveld de natuur van deze interactiegrondig verandert. Onze aanpak is semi-analytisch. Dit betekent dat de oplossingexact is, maar er zĳn iteraties nodig om deze exacte oplossing te benaderen.We voeren ook computerexperimenten uit met de numerieke code AMRVAC envergelĳken de exacte oplossing met de numerieke oplossing. De oplossingen komenovereen.

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 het geschokte CD ertussenin. Wanneer deze nieuwe signalen reflecteren aan de wand van de schoktube,wordt in het HD-geval de CD instabiel en rolt ze op. Deze instabiliteit wordt deRichtmyer-Meshkov instabiliteit genoemd. In het MHD-geval blĳft de CD echterstabiel.


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 [28], 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

105

106 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 Hols & Keppens [83]; Keppens et al.[43]) can be read inby visualization software as ParaView and VisIt.

Appendix A

USR-file

This appendix shows the USR file used for the simulation of the shock tube problempresented in van der Holst & Keppens [83]. It is written in LASY syntax, asintroduced in Tóth[78].

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

107

108 USR-FILE

eqpar (gamma_)=1.4 d0M=10.0d0eta =0.1d0tang=s q r t ( 3 . 0 d0 ) / 2 . 0 d0

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

vpost=M−(( eqpar (gamma_)−1)∗M∗∗2+2)/(( eqpar (gamma_)+1)∗M)rhopost=−1.0d0 ∗( vpost+M)∗ eqpar (gamma_)∗Mppost=(2∗eqpar (gamma_)∗M∗∗2−eqpar (gamma_)+1)/( eqpar (gamma_)+1)

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 B

Stationary planar

Rankine-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, (B.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[27]). From (B.1) we know that m ≡ ρvn and Bn are constant across a shock.

109

110 STATIONARY PLANAR RANKINE-HUGONIOT CONDITIONS

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

m2[[1

ρ]] + [[p]] +

1

2[[B2

t ]] = 0, (B.2)

m[[vt]] −Bn[[Bt]] = 0, (B.3)

m[[Btρ

]] −Bn[[vt]] = 0, (B.4)

γ

γ − 1[[p

ρ]] +

m2

2[[

1

ρ2]] + [[

B2t

ρ]] − B2

n

2m2[[B2

t ]] = 0, (B.5)

where we used equation (B.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, (B.6)

which is equivalent to equation (B.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 (B.2-B.4) drastically:

[[(vt, Bt, p)t]] = 0. (B.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 C

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 (B.4), we arriveat the following 3 × 3-system (see also Torrilhon [81]):

p− 1 + C(1

ρ− 1) +

1

2(B2

t −A2) = 0, (C.1)

C(Btρ

−A) −B2(Bt −A) = 0, (C.2)

1

γ − 1(p

ρ− 1) +

1

2(1

ρ− 1)(p+ 1) +

1

4(1

ρ− 1)(Bt −A)2 = 0, (C.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.

111

112 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. (C.4)

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

ρ =CBt

AC +B2(Bt − A), (C.5)

p =2C

γ + 1− γ − 1

γ + 1

(

(A− Bt)2

2+ 1

)

− BtC(A+ Bt)

(γ + 1)(C −B2), (C.6)

where Bt must satisfy a cubic relation

Σi=0,3τiBit = 0, (C.7)

with its coefficients given by

τ3 = B2, (C.8)

τ2 =(

(γ − 1)(B2 − C) + C)

A, (C.9)

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

τ0 = −(γ + 1)A(B2 − C)2. (C.11)

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

√pk. Finally equation (B.1) delivers vn,u and Bn,u, while from (B.4),

we know vt,u:

vn,u =m

ρρk, (C.12)

vt,u = vt,k +Bn,kρkvn,k

(Bt,n −Bt,u), (C.13)

Bn,u = Bn,k. (C.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 113

velocity, or equivalently when Bt,k does not vanish. In this special case the system(C.1-C.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(C.5-C.6)) and finally the following non-trivial switch-off shock:

Bt = 0, (C.15)

ρ =2(γ + 1)C

γ(A2 + 2C + 2) ±√

(γ(A2 + 2))2

+ 4C(C +A2 − 2γ), (C.16)

p =2 + 2C +A2 ∓

√

(γ(A2 + 2))2

+ 4C(C +A2 − 2γ)

2(γ + 1). (C.17)

Appendix D

Solving the cubic analytically

The cubic (C.7) is solved analytically by the following procedure. Defining thereal quantities

D = 4(τ22 − 3τ3τ1), (D.1)

N = 4(2τ32 − 9τ3τ2τ1 + 27τ2

3 τ0), (D.2)

one notes that D is the discriminant of the derivative function of the cubic. WhenD > 0, the inequality N2 − D3 < 0 gives limiting values for τ0, for which the 2extrema of the cubic have opposite signs. Since N2 −D3 < 0 implies D > 0, thecriterium on the coefficient of the cubic to have 3 different real solutions is

N2 −D3 < 0. (D.3)

From these real-valued quantities, we define the following, possibly complex,quantities:

H =(

√

N2 −D3 −N)

13

, (D.4)

J =D

H. (D.5)

114

SOLVING THE CUBIC ANALYTICALLY 115

In terms of these introduced quantities, the 3 complex solutions of the cubicequation (C.7) are given by

Bt,0 =J +H − 2τ2

6τ3, (D.6)

Bt,1 = −J +H + 4τ2 −√

3(H − J)i

12τ3, (D.7)

Bt,2 = −J +H + 4τ2 +√

3(H − J)i

12τ3. (D.8)

The evaluation of expressions (D.6 - D.8) requires the evaluation of√N2 −D3 and

(√N2 −D3 −N

)13 .

√N2 −D3 is discontinuous when N2 −D3 is a negative (and

thus real) number and(√N2 −D3 −N

)

13 is discontinuous when

√N2 −D3 −N

is a negative (and thus real) number. Since N2 − D3 is an (even) polynomial ofdegree 18 in M , it renders a mathematical classification rather difficult.

Before we explain how to permute the indices of the roots (D.6-D.8), let us firstrewrite these expressions as

Bι = Hι + Jι, (D.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 natural value of ι;

• Note that τ3 > 0. Therefore, equation (D.9) is well-defined. Also, when weknow that one of those expressions is real, then its sign equals the sign of itsnominator;

• 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, (D.10)

and Li, the lines separating those sextants, i.e.

Lj ≡ r(cosϕ+ i sinϕ)|ϕ = jπ

3, r > 0. (D.11)

These regions are shown in figure (D.1).

116 SOLVING THE CUBIC ANALYTICALLY

Figure D.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 .

SOLVING THE CUBIC ANALYTICALLY 117

• 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.

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. Infigure D.2 we show the real part of the unpermuted roots of the cubic (C.7),together with 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 (C.7) has only one real solution and three realsolutions for φ < φCr,T .

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.

118 SOLVING THE CUBIC ANALYTICALLY

-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 D.2: Left: the real parts of the roots of the cubic (C.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.

Appendix E

Integration across rarefaction

waves

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.[18]), 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

(E.1)

where the prime ′ denotes partial derivation in the normal direction. Also notethat Bt is eliminated from the system since it is completely determined by the

119

120 INTEGRATION ACROSS RAREFACTION WAVES

other variables. Only when (E.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, (E.2)

m

v⊥(v2⊥ − v2

f,⊥)(v2⊥ − v2

s,⊥) = 0, (E.3)

which is equivalent to characteristic equation (3.7). This relation should holdinside of the expansion fans, since u is partially differentiable there. The solutionto (E.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, (E.4)

p′(φ) = 2γpvnvt(2v

2n − a2 − c2) + c2anat

(4 + 2γ)v4n − ((2γ + 1)a2 + (3 + γ)c2)v2

n + γa2nc

2, (E.5)

B′n(φ) = −Bt, (E.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,(E.7)

v′t(φ) =Σ6i=0τiv

in

((4 + 2γ)v4n − ((2γ + 1)a2 + (3 + γ)c2)v2

n + γa2nc

2)vnat. (E.8)

Here we introduced the symbols an ≡ Bn√2p

and at ≡ Bt√2p

, and the coefficients τiare given by

τ0 = −γa4nc

3, (E.9)

τ1 = 0, (E.10)

τ2 = ((γ − 2)anat + (γ + 3)c2 + (2γ + 1)a2)anc2, (E.11)

τ3 = 2(a2 + c2)anvt, (E.12)

τ4 = −(2γ + 1)ata2 − ((γ + 3)at + (2γ + 4)an)c

2, (E.13)

τ5 = −4anvt, (E.14)

τ6 = (2γ + 4)at. (E.15)

INTEGRATION ACROSS RAREFACTION WAVES 121

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

• 2006-2010Doctor in the Sciences, Centre for Plasma Astrophysics, 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, accepted

• P. Delmont & R. Keppens, ’Parameter regimes for slow intermediate andfast MHD shocks’, 2010, J. Plasma Phys., doi:10.1017/S0022377810000115

129

130 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 131

• 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.



Documents

On regular shock refraction in hydro- and magnetohydrodynamics€¦ · Arenberg Doctoral School of Science, Engineering & Technology Faculty of Science Department of Mathematics Centre