Computational Methods in Plasma Physics ppt

Mattson, and Craig E Rasmussen INTRODUCTION TO SCHEDULING Yves Robert and Frédéric Vivien SCIENTIFIC DATA MANAGEMENT: CHALLENGES, TECHNOLOGY, AND DEPLOYMENT Edited by Arie Shoshani and D

Computational Methods in

Plasma Physics

Chapman & Hall/CRC Computational Science Series

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PETASCALE COMPUTING: Algorithms and Applications

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INTRODUCTION TO CONCURRENCY IN PROGRAMMING LANGUAGES

Matthew J Sottile, Timothy G Mattson, and Craig E Rasmussen

INTRODUCTION TO SCHEDULING

Yves Robert and Frédéric Vivien

SCIENTIFIC DATA MANAGEMENT: CHALLENGES, TECHNOLOGY, AND DEPLOYMENT

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COMPUTATIONAL METHODS IN PLASMA PHYSICS

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to Marilyn

List of Figures xiii

1.1 Introduction 1

1.2 Magnetohydrodynamic (MHD) Equations 4

1.2.1 Two-Fluid MHD 6

1.2.2 Resistive MHD 8

1.2.3 Ideal MHD 9

1.2.4 Other Equation Sets for MHD 10

1.2.5 Conservation Form 10

1.2.6 Boundary Conditions 12

1.3 Characteristics 14

1.3.1 Characteristics in Ideal MHD 16

1.3.2 Wave Dispersion Relation in Two-Fluid MHD 23

1.4 Summary 24

2 Introduction to Finite Difference Equations 27 2.1 Introduction 27

2.2 Implicit and Explicit Methods 29

2.3 Errors 30

2.4 Consistency, Convergence, and Stability 31

2.5 Von Neumann Stability Analysis 32

2.5.1 Relation to Truncation Error 36

2.5.2 Higher-Order Equations 37

2.5.3 Multiple Space Dimensions 39

2.6 Accuracy and Conservative Differencing 39

2.7 Summary 42

vii

viii Table of Contents

3.1 Introduction 45

3.2 One-Dimensional Poisson’s Equation 46

3.2.1 Boundary Value Problems in One Dimension 46

3.2.2 Tridiagonal Algorithm 47

3.3 Two-Dimensional Poisson’s Equation 48

3.3.1 Neumann Boundary Conditions 50

3.3.2 Gauss Elimination 53

3.3.3 Block-Tridiagonal Method 56

3.3.4 General Direct Solvers for Sparse Matrices 57

3.4 Matrix Iterative Approach 57

3.4.1 Convergence 59

3.4.2 Jacobi’s Method 60

3.4.3 Gauss–Seidel Method 60

3.4.4 Successive Over-Relaxation Method (SOR) 61

3.4.5 Convergence Rate of Jacobi’s Method 61

3.5 Physical Approach to Deriving Iterative Methods 62

3.5.1 First-Order Methods 63

3.5.2 Accelerated Approach: Dynamic Relaxation 65

3.6 Multigrid Methods 66

3.7 Krylov Space Methods 70

3.7.1 Steepest Descent and Conjugate Gradient 72

3.7.2 Generalized Minimum Residual (GMRES) 76

3.7.3 Preconditioning 80

3.8 Finite Fourier Transform 82

3.8.1 Fast Fourier Transform 83

3.8.2 Application to 2D Elliptic Equations 86

3.9 Summary 88

4 Plasma Equilibrium 93 4.1 Introduction 93

4.2 Derivation of the Grad–Shafranov Equation 93

4.2.1 Equilibrium with Toroidal Flow 95

4.2.2 Tensor Pressure Equilibrium 97

4.3 The Meaning of Ψ 99

4.4 Exact Solutions 102

4.4.1 Vacuum Solution 102

4.4.2 Shafranov–Solov´ev Solution 104

4.5 Variational Forms of the Equilibrium Equation 105

4.6 Free Boundary Grad–Shafranov Equation 106

4.6.1 Inverting the Elliptic Operator 107

4.6.2 Iterating on Jφ(R, Ψ) 107

4.6.3 Determining Ψ on the Boundary 109

4.6.4 Von Hagenow’s Method 111

4.6.5 Calculation of the Critical Points 113

4.6.6 Magnetic Feedback Systems 114

4.6.7 Summary of Numerical Solution 116

4.7 Experimental Equilibrium Reconstruction 116

4.8 Summary 117

5 Magnetic Flux Coordinates in a Torus 121 5.1 Introduction 121

5.2 Preliminaries 121

5.2.1 Jacobian 122

5.2.2 Basis Vectors 124

5.2.3 Grad, Div, Curl 125

5.2.4 Metric Tensor 127

5.2.5 Metric Elements 127

5.3 Magnetic Field, Current, and Surface Functions 129

5.4 Constructing Flux Coordinates from Ψ(R, Z) 131

5.4.1 Axisymmetric Straight Field Line Coordinates 133

5.4.2 Generalized Straight Field Line Coordinates 135

5.5 Inverse Equilibrium Equation 136

5.5.1 q-Solver 137

5.5.2 J-Solver 138

5.5.3 Expansion Solution 139

5.5.4 Grad–Hirshman Variational Equilibrium 140

5.5.5 Steepest Descent Method 144

5.6 Summary 147

6 Diffusion and Transport in Axisymmetric Geometry 149 6.1 Introduction 149

6.2 Basic Equations and Orderings 149

6.2.1 Time-Dependent Coordinate Transformation 151

6.2.2 Evolution Equations in a Moving Frame 153

6.2.3 Evolution in Toroidal Flux Coordinates 155

6.2.4 Specifying a Transport Model 158

6.3 Equilibrium Constraint 162

6.3.1 Circuit Equations 163

6.3.2 Grad–Hogan Method 163

6.3.3 Taylor Method (Accelerated) 164

6.4 Time Scales 165

6.5 Summary 168

7 Numerical Methods for Parabolic Equations 171 7.1 Introduction 171

7.2 One-Dimensional Diffusion Equations 171

7.2.1 Scalar Methods 172

7.2.2 Non-Linear Implicit Methods 175

7.2.3 Boundary Conditions in One Dimension 179

x Table of Contents

7.2.4 Vector Forms 180

7.3 Multiple Dimensions 183

7.3.1 Explicit Methods 183

7.3.2 Fully Implicit Methods 184

7.3.3 Semi-Implicit Method 184

7.3.4 Fractional Steps or Splitting 185

7.3.5 Alternating Direction Implicit (ADI) 186

7.3.6 Douglas–Gunn Method 187

7.3.7 Anisotropic Diffusion 188

7.3.8 Hybrid DuFort–Frankel/Implicit Method 190

7.4 Summary 192

8 Methods of Ideal MHD Stability Analysis 195 8.1 Introduction 195

8.2 Basic Equations 195

8.2.1 Linearized Equations about Static Equilibrium 195

8.2.2 Methods of Stability Analysis 199

8.2.3 Self-Adjointness of F 200

8.2.4 Spectral Properties of F 201

8.2.5 Linearized Equations with Equilibrium Flow 203

8.3 Variational Forms 204

8.3.1 Rayleigh Variational Principle 204

8.3.2 Energy Principle 205

8.3.3 Proof of the Energy Principle 206

8.3.4 Extended Energy Principle 207

8.3.5 Useful Identities 208

8.3.6 Physical Significance of Terms in δWf 210

8.3.7 Comparison Theorem 211

8.4 Cylindrical Geometry 213

8.4.1 Eigenmode Equations and Continuous Spectra 214

8.4.2 Vacuum Solution 215

8.4.3 Reduction of δWf 216

8.5 Toroidal Geometry 219

8.5.1 Eigenmode Equations and Continuous Spectra 220

8.5.2 Vacuum Solution 223

8.5.3 Global Mode Reduction in Toroidal Geometry 225

8.5.4 Ballooning Modes 226

8.6 Summary 232

9 Numerical Methods for Hyperbolic Equations 235 9.1 Introduction 235

9.2 Explicit Centered-Space Methods 235

9.2.1 Lax–Friedrichs Method 236

9.2.2 Lax–Wendroff Methods 237

9.2.3 MacCormack Differencing 238

9.2.4 Leapfrog Method 239

9.2.5 Trapezoidal Leapfrog 241

9.3 Explicit Upwind Differencing 242

9.3.1 Beam–Warming Upwind Method 244

9.3.2 Upwind Methods for Systems of Equations 245

9.4 Limiter Methods 247

9.5 Implicit Methods 249

9.5.1 θ-Implicit Method 251

9.5.2 Alternating Direction Implicit (ADI) 252

9.5.3 Partially Implicit 2D MHD 253

9.5.4 Reduced MHD 256

9.5.5 Method of Differential Approximation 257

9.5.6 Semi-Implicit Method 260

9.5.7 Jacobian-Free Newton–Krylov Method 262

9.6 Summary 264

10 Spectral Methods for Initial Value Problems 267 10.1 Introduction 267

10.1.1 Evolution Equation Example 268

10.1.2 Classification 269

10.2 Orthogonal Expansion Functions 269

10.2.1 Continuous Fourier Expansion 270

10.2.2 Discrete Fourier Expansion 272

10.2.3 Chebyshev Polynomials in (−1, 1) 273

10.2.4 Discrete Chebyshev Series 277

10.3 Non-Linear Problems 278

10.3.1 Fourier Galerkin 278

10.3.2 Fourier Collocation 279

10.3.3 Chebyshev Tau 280

10.4 Time Discretization 281

10.5 Implicit Example: Gyrofluid Magnetic Reconnection 283

10.6 Summary 286

11 The Finite Element Method 289 11.1 Introduction 289

11.2 Ritz Method in One Dimension 289

11.2.1 An Example 290

11.2.2 Linear Elements 290

11.2.3 Some Definitions 293

11.2.4 Error with Ritz Method 294

11.2.5 Hermite Cubic Elements 295

11.2.6 Cubic B-Splines 298

11.3 Galerkin Method in One Dimension 301

11.4 Finite Elements in Two Dimensions 304

11.4.1 High-Order Nodal Elements in a Quadrilateral 305

xii Table of Contents

11.4.2 Spectral Elements 309

11.4.3 Triangular Elements with C1 Continuity 310

11.5 Eigenvalue Problems 316

11.5.1 Spectral Pollution 318

11.5.2 Ideal MHD Stability of a Plasma Column 320

11.5.3 Accuracy of Eigenvalue Solution 322

11.5.4 Matrix Eigenvalue Problem 323

11.6 Summary 323

1.1 Gaussian pill box is used to derive jump conditions between

two regions 13

1.2 Characteristic curves dx/dt = u All information is propagated along these lines 14

1.3 Space is divided into two regions by characteristic curves 15

1.4 Domain of dependence (l) and domain of influence (r) 16

1.5 Characteristics in two spatial dimensions 17

1.6 Reciprocal normal surface diagram in low-β limit 20

1.7 Ray surface diagram in low-β limit 21

1.8 Typical dispersion relation for low-β two-fluid MHD for differ-ent angles of propagation relative to the background magnetic field 24

2.1 Space time discrete points 28

2.2 Domain of dependence of the point (xj, tn) for an (a) explicit and (b) implicit finite difference method 30

2.3 Amplification factor, r (solid), must lie within the unit circle (dashed) in the complex plane for stability 35

2.4 Density and velocity variables are defined at staggered locations 40

2.5 Finite volume method 41

3.1 Computational grid for Neumann boundary conditions 52

3.2 Multigrid method sequence of grids 67

3.3 Basic coarse grid correction 68

3.4 Full Multigrid V-cycle (FMV) 69

4.1 Cylindrical coordinates (R, φ, Z) 94

4.2 Calculate magnetic flux associated with disk in the z = 0 plane as shown 100

4.3 Poloidal magnetic flux contours for a typical tokamak dis-charge at three times 101

4.4 For an axisymmetric system, the vacuum toroidal field con-stant g0 is proportional to the total current in the toroidal field coils, IT F 104

xiii

xiv List of Figures

is obtained from a Green’s function External coils are sented as discrete circular current loops 1104.6 Singularity due to self-field term is resolved by taking the limit

4.7 Poloidal flux at magnetic axis, Ψ0, is a local minimum Limitervalue of flux, Ψ`, is the minimum of the value of the flux atthe limiter points and at the saddle points 113

and boundary values 116

possible 123

θ coordinate 124

direction orthogonal to the other remaining coordinates, andnot in the ∇ψ direction 126

techniques 132

lines on a surface ψ=const appear as straight lines when ted in (θ, φ) space 134

relative to a fixed Cartesian frame 152

Euler method (BTCS) and the non-linear implicit Newton erative method 179

from the origin so that the condition of zero flux can be posed there 181

or a pressureless plasma, which is in turn surrounded by aconducting wall 196

8.3 The physical boundary conditions are applied at the turbed boundary, which is related to the unperturbed bound-

2πR 214

con-structed by taking a linear superposition of an infinite number

of offset aperiodic solutions 230

space-time points at odd and even values of n + j are completelydecoupled 240

decou-pled grid problem from leapfrog 241

the point where the characteristic curve intersects time level

for fully explicit and partially implicit methods 250

with N = 8 The two modes take on the same values at thegrid points 273

and are linear in between A half element is associated with

derivatives (b) for linear elements 292

v0(x) A linear combination of these two functions is associatedwith node j 296

value, first, and second derivatives 299

over-laps with itself and six neighbors 300

φ0 (Neumann) 301

method for a second-order differential equation is implementedusing cubic B-spline elements 304

xvi List of Figures

in-troducing hanging nodes Triangular elements can be locallyrefined, but require unstructured mesh data structures in pro-gramming 305

(x, y) is mapped into the unit square in the logical space forthat element, (ξ, η) 30611.10 Relation between global numbering and local numbering inone dimension 307

5.1 The poloidal angle θ is determined by the function h(ψ, θ) inthe Jacobian as defined in Eq (5.40) 133

determined by the function f (ψ) in the Jacobian definition,

What is computational physics? Here, we take it to mean techniques for lating continuous physical systems on computers Since mathematical physicsexpresses these systems as partial differential equations, an equivalent state-ment is that computational physics involves solving systems of partial differ-ential equations on a computer.

simu-This book is meant to provide an introduction to computational physics

to students in plasma physics and related disciplines We present most of thebasic concepts needed for numerical solution of partial differential equations.Besides numerical stability and accuracy, we go into many of the algorithmsused today in enough depth to be able to analyze their stability, efficiency,and scaling properties We attempt to thereby provide an introduction to andworking knowledge of most of the algorithms presently in use by the plasmaphysics community, and hope that this and the references can point the way

to more advanced study for those interested in pursuing such endeavors.The title of the book starts with Computational Methods , not All Com-putational Methods Perhaps it should be Some Computational Methods because it admittedly does not cover all computational methods being used

in the field The material emphasizes mathematical models where the plasma

is treated as a conducting fluid and not as a kinetic gas This is the most ture plasma model and also arguably the one most applicable to experiments.Many of the basic numerical techniques covered here are also appropriatefor the equations one encounters when working in a higher-dimensional phasespace The book also emphasizes toroidal confinement geometries, particularlythe tokamak, as this is the most mature and most successful configuration forconfining a high-temperature plasma

ma-There is not a clear dividing line between computational and theoreticalplasma physics It is not possible to perform meaningful numerical simulations

if one does not start from the right form of the equations for the questionsbeing asked, and it is not possible to develop new advanced algorithms unlessone has some understanding of the underlying mathematical and physicalproperties of the equation systems being solved Therefore, we include in thisbook many topics that are not always considered “computational physics,”but which are essential for a computational plasma physicist to understand.This more theoretical material, such as occurs in Chapters 1, 6, and 8 as well

as parts of Chapters 4 and 5, can be skipped if students are exposed to it inother courses, but they may still find it useful for reference and context

xix

xx Computational Methods in Plasma Physics

The author has taught a semester class with the title of this book tograduate students at Princeton University for over 20 years The studentsare mostly from the Plasma Physics, Physics, Astrophysics, and Mechanicaland Aerospace Engineering departments There are no prerequisites, and moststudents have very little prior exposure to numerical methods, especially forpartial differential equations The material in the book has grown considerablyduring the 20-plus years, and there is now too much material to cover in aone-semester class Chapters 2, 3, 7, 9, and perhaps 10 and 11 form the core

of the material, and an instructor can choose which material from the otherchapters would be suitable for his students and their needs and interests Atwo-semester course covering most of the material is also a possibility.Computers today are incredibly powerful, and the types of equations weneed to solve to model the dynamics of a fusion plasma are quite complex.The challenge is to be able to develop suitable algorithms that lead to stableand accurate solutions that can span the relevant time and space scales This

is one of the most challenging research topics in modern-day science, and thepayoffs are enormous It is hoped that this book will help students and youngresearchers embark on productive careers in this area

The author is indebted to his many colleagues and associates for merable suggestions and other contributions He acknowledges in particular

innu-M Adams, R Andre, J Breslau, J Callen, innu-M Chance, J Chen, C Cheng,

C Chu, J DeLucia, N Ferraro, G Fu, A Glasser, J Greene, R Grimm, G.Hammett, T Harley, S Hirshman, R Hofmann, J Johnson, C Kessel, D.Keyes, K Ling, S Lukin, D McCune, D Monticello, W Park, N Pomphrey,

J Ramos, J Richter, K Sakaran, R Samtaney, D Schnack, M Sekora, S.Smith, C Sovinec, H Strauss, L Sugiyama, D Ward, and R White

Stephen C JardinPrinceton, NJ

x position coordinate

function for species j

species j

collisions betwen particles

of species j and j0

momen-tum, or energy for species j

density for species j

ˆ

momen-tum density for species j

density for species j

pressures

ran-dom heat fluxes

πigyr ion gyroviscous stress tensor

heat flux for species j

xxi

xxii Computational Methods in Plasma Physics

of time and space scales, but are more amenable to numerical solution.The most basic set of equations describing the six dimension plus time(x, v, t) phase space probability distribution function fj(x, v, t) for species j

of indistinguishable charged particles (electrons or a particular species of ions)

is a system of Boltzmann equations for each species:

∂fj(x, v, t)

∂t + ∇ · (vfj(x, v, t)) + ∇v· qj

mj(E + v × B)fj(x, v, t)

1

2 Computational Methods in Plasma Physics

These equations form the starting point for some studies of fine-scaleplasma turbulence and for studies of the interaction of imposed radio-frequency (RF) electromagnetic waves with plasma However, the focus herewill be to take velocity moments of Eq (1.1), and appropriate sums overspecies, so as to obtain fluid-like equations that describe the macroscopic dy-namics of magnetized high-temperature plasma, and to discuss techniques fortheir numerical solution

The first three velocity moments correspond to conservation of particlenumber, momentum, and energy Operating on Eq (1.1) with the velocityspace integrals R d3v, R d3vmjv, and R d3vmjv2/2 yields the following mo-ment equations:

Here, we have introduced for each species the number density, the fluid ity, the scalar pressure, the stress tensor, the heat flux density, and the externalsources of particles, momentum, and energy density which are defined as:

4 Computational Methods in Plasma Physics

The fluid equations given in the last section are very general; however,they are incomplete and still contain too wide a range of time and spacescales to be useful for many purposes To proceed, we make a number ofsimplifying approximations that allow the omission of the terms in Maxwell’sequations that lead to light waves and high-frequency plasma oscillations.These approximations are valid provided the velocities (v), frequencies (ω),and length scales of interest (L) satisfy the inequalities:

v  c,

ω  ωpe,

L  λD.Here ωpeis the electron plasma frequency, and λDis the Debye length Theseorderings are normally well satisfied for macroscopic phenomena in modernfusion experiments We also find it convenient to change velocity variablesfrom the electron and ion velocities ueand uito the fluid (mass) velocity andcurrent density, defined as:

we consider here only a single species of ions with unit charge The field tions in MHD are then (SI units):

elec-is referred to as quasineutrality and similarly replaces Eq (1.6) Note thatthere is no equation for the divergence of the electric field, ∇ · E, in MHD It

is effectively replaced by the quasineutrality condition, Eq (1.27)

The fluid equations are the continuity equation for the number density

In addition, from the momentum equation for the electrons, Eq (1.9) with

j = e, we have the generalized Ohm’s law equation, which in the limit of

obey time advancement equations The variables E, J, and the ion pressure

pi = p − peare auxiliary variables which we define only for convenience Wehave also introduced the total heat flux q = qe+qiand the total non-isotropicstress tensor π = πe+ πi

To proceed with the solution, one needs closure relations for the ing terms; the collisional friction term Re, the random heat flux vectors qj,the anisotropic part of the stress tensor πj, and the equipartition term Q∆ei

remain-It is the objective of transport theory to obtain closure expressions for these

in terms of the fundamental variables and their derivatives There is sive and evolving literature on deriving closures that are valid in differentparameter regimes We will discuss some of the more standard closures in thefollowing sections We note here that the external sources of particles, total

initial and boundary conditions

6 Computational Methods in Plasma Physics

1.2.1 Two-Fluid MHD

The set of MHD equations as written in Section 1.2 is not complete because

of the closure issue The most general closures that have been proposed arepresently too difficult to solve numerically However, there are some approxi-mations that are nearly always valid and are thus commonly applied Otherapproximations are valid over limited time scales and are useful for isolatingspecific phenomena

The two-fluid magnetohydrodynamic equations are obtained by taking anasymptotic limit of the extended MHD equations in which first order terms

in the ratio of the ion Larmor radius to the system size are retained This issometimes called the finite Larmor radius or FLR approximation The princi-pal effects are to include expressions for the parts of the ion stress tensor πiand ion and electron heat fluxes qi and qe that do not depend on collisions.This contribution to the ion stress tensor is known as the gyroviscous stress

or gyroviscosity Let b be a unit vector in the direction of the magnetic field.The general form of gyroviscosity, assuming isotropic pressure and negligibleheat flux, can be shown to be [1, 2, 3]:

pjkB

The remaining contributions to the ion stress tensor are dependent on thecollisionality regime and magnetic geometry in a complex way These are oftenapproximated by an isotropic part and a parallel part as follows:

∇ · πiso

We note the positivity constraints µ ≥ 0, µc≥ 2

3µ, µk≥ 0

The electron gyroviscosity can normally be neglected due to the smallelectron mass However, for many applications, an electron parallel viscositycan be important This is similar in form to the ion one in Eq (1.40) but with acoefficient µe

k In addition, it has been shown by several authors that in certainproblems involving two-fluid magnetic reconnection, an electron viscosity termknown as hyper-resistivity is required to avoid singularities from developing inthe solution [6, 7] It is also useful for modeling the effect of fundamentallythree-dimensional reconnection physics in a two-dimensional simulation [8, 9].Introducing the coefficient λH, we can take the hyper-resistivity to be of theform:

If h is the smallest dimension that can be resolved on a numerical grid, then

it has been proposed that λH ∼ h2 is required for the current singularity to

be resolvable, while at the same time vanishing in the continuum limit h → 0.The plasma friction force is generally taken to be of the form:

The standard two-fluid MHD model can now be summarized as follows

8 Computational Methods in Plasma Physics

Using the equations of Section 1.2, we use the following closure model:

k, and λH These can be either constants or functions

of the macroscopic quantities being evolved in time according to the transportmodel being utilized

1.2.2 Resistive MHD

The resistive MHD model treats the electrons and ions as a single fluid

a limiting case of the two-fluid equations The first limit is that of collisiondominance, which allows one to neglect the parallel viscosities compared to

∇p The second limit is that of zero Larmor radius This implies the neglect ofthe gyroviscous stress and the collision-independent perpendicular heat fluxes

It follows that the electron hyperviscosity is also not required With thesesimplifications, neglecting external sources, and introducing the mass density

ρ ≡ nmi, the equations of Section 1.2 become:

qi= −κk∇kT − κ⊥∇⊥T (1.58)

in Eq (1.54) is evaluated using Eq (1.39) This model requires only the fivetransport coefficients: η, κe

k , κ⊥, µ, and µc

1.2.3 Ideal MHD

A further approximation has to do with the smallness of the plasmaresistivity and the time scales over which resistive effects are important,

µ0nMi, a, and the Alfv´en time τA= a/VA, we find that the plasmaresistivity becomes multiplied by the inverse magnetic Lundquist number S−1,where

In modern fusion experiments, this number is typically in the range S ∼

106− 1012

The other dissipative quantities qj and πiso are related to the resistivity

im-portant for the longer time dynamics of the plasma or to describe resistiveinstabilities that involve internal boundary layers [11], if we are only inter-ested in the fastest time scales present in the equations, we can neglect all

equations [14]

These equations have been extensively studied by both physicists andmathematicians They have a seemingly simple symmetrical structure withwell-defined mathematical properties, but at the same time can be exceedinglyrich in the solutions they admit Adding back the dissipative and dispersiveterms will enlarge the class of possible solutions by making the equationshigher order, but will not fundamentally change the subset of non-dissipativesolutions found here for macroscopic motions It is therefore important to un-derstand the types of solutions possible for these equations before studyingmore complex equation sets

The ideal MHD equations can be written (in SI units), as follows:

10 Computational Methods in Plasma Physics

∂p

where the current density is given by J ≡ µ−1o ∇ × B Here we have introduced

γ = 53, which is the ratio of specific heats, sometimes called the adiabaticindex It is sometimes useful to also define another variable

s ≡ p/ργ,the entropy per unit mass It then follows from Eqs (1.60) and (1.63) that sobeys the equation

∂s

Eq (1.64) can be used to replace Eq (1.63)

1.2.4 Other Equation Sets for MHD

The equation sets corresponding to the closures listed in the previous tions are by no means exhaustive Higher-order closures exist which involveintegrating the stress tensor and higher-order tensors in time [15] These be-come very complex, and generally require a subsidiary kinetic calculation tocomplete the closure of the highest-order tensor quantities Additional clo-sures of a more intermediate level of complexity exist in which the pressurestress tensor remains diagonal but is allowed to have a different form paralleland perpendicular to the magnetic field [16] These allow the description ofsome dynamical effects that arise from the fact that the underlying kineticdistribution function is not close to a Maxwellian

sec-There are also widely used sets of starting equations for computationalMHD that are in many ways easier to solve than the ones presented here Thereduced MHD equations follow from an additional expansion in the inverseaspect ratio of a toroidal confinement device [17, 18] These are discussed more

in Chapters 9 and 11

The surface-averaged MHD equations are discussed in Chapter 6 Theseare a very powerful system of equations that are valid for modeling the evo-lution of an MHD stable system with good magnetic flux surfaces over longtimes characterized by τR∼ S τA The equations are derived by an asymptoticexpansion in which the plasma inertia approaches zero This leads to consid-erable simplification, effectively removing the ideal MHD wave transit timescales from the problem At the same time, the flux surface averaging allowsthe implementation of low collisionality closures that would not otherwise bepossible

1.2.5 Conservation Form

A system of partial differential equations is said to be in conservation form

if all of the terms can be written as divergences of products of the dependent

variable, i.e., in the form

∂

∂t(· · · ) + ∇ · (· · · ) = 0.

This form is useful for several purposes It allows one to use Gauss’s theorem

to obtain global conservation laws and the appropriate boundary conditionsand jump conditions across shocks or material interfaces It can also be used

as a starting point to formulate non-linear numerical solutions which maintainthe exact global conservation properties of the original equations

By using elementary vector manipulations, we obtain the conservationform of the MHD equations of Section 1.1, also incorporating the commonapproximations discussed in Section 1.2 Assuming the external source terms

Sm, Se, See, etc are also zero, we have



22µ0



µ0BB

12 Computational Methods in Plasma Physics

As discussed above and further in Section 2.6, a numerical solution based

on the conservation form of the equations offers some advantages over otherformulations in that globally conserved quantities can be exactly maintained in

a computational solution However, in computational magnetohydrodynamics

of highly magnetized plasmas, the conservative formulation is not normallythe preferred starting point for the following reasons: While the mass andmagnetic flux conservation equations, Eqs (1.65) and (1.66) (in some form)offer clear advantages, the conservation form for the momentum and energyequations, Eqs (1.67) and (1.68), can lead to some substantial inaccuracies.The primary difficulty is due to the difference in magnitude of the kineticenergy, pressure, and magnetic energy terms in a strongly magnetized plasma

It can be seen that there is no explicit equation to advance the pressure inthe set of Eqs (1.65)–(1.68) The pressure evolution equation is effectivelyreplaced by Eq (1.68) to advance the total energy density:

1.2.6 Boundary Conditions

The global conservation laws Eqs (1.65)–(1.68) apply to any volume andcan be used to obtain boundary conditions and jump conditions across inter-faces by applying them appropriately For example, if we have two regions,

Region 2 Region 1

14 Computational Methods in Plasma Physics

x t

FIGURE 1.2: Characteristic curves dx/dt = u All information is propagatedalong these lines

The ideal MHD equations are a quasilinear symmetric hyperbolic system

of equations and thus have real characteristics Because they are real, thecharacteristic directions or characteristic manifolds have important physicalmeaning since all information is propagated along them In Section 1.3.1 wereview the nature of the ideal MHD characteristics

To establish the concepts, let us first consider the simplest possible bolic equation in one-dimension plus time,

of x, then s(x, t) = f (x − ut) is the solution at a later time The characteristiccurves, as shown in Figure 1.2, form a one-parameter family that fills all space

To find the solution at some space-time point (x, t) one need only trace backalong the characteristic curve until a boundary data point is intersected

We note that the boundary data that define the solution may be givenmany different ways The data may be given entirely along the t axis, inwhich case they are commonly referred to as boundary conditions, or theymay be given entirely along the x axis in which case they are referred to asinitial conditions However, more general cases are also allowable, the onlyrestriction is that all characteristic curves intersect the boundary data curvesonce and only once In particular, the boundary data curve is not allowed

t

timelike region

spacelike region

FIGURE 1.3: Space is divided into two regions by characteristic curves

to be tangent to the characteristic curves anywhere We will see in the nextsection that this restriction allows us to determine the characteristics for amore complicated system

Finally, let us review another simple concept that is of importance forhyperbolic equations It is that information always travels at a finite velocity.Consider the simple wave equation

16 Computational Methods in Plasma Physics

FIGURE 1.4: Domain of dependence (l) and domain of influence (r)

region of the boundary data for which that point is timelike This region isthe domain of dependence of the point Similarly, a given region of boundarydata can only affect points that are timelike with respect to at least part ofthat region This is the domain of influence of that region; see Figure 1.4

1.3.1 Characteristics in Ideal MHD

We saw that in one spatial dimension the characteristics were lines, as

in Figure 1.2 In two spatial dimensions, the characteristics will be dimensional surfaces in three-dimensional space time, as shown in Figure 1.5

two-In full three-dimensional space, the characteristics will be three-dimensionalmanifolds, in four-dimensional space time (r, t) They are generated by themotion of surfaces in ordinary three-dimensional space r Let us determinethese characteristics for the ideal MHD equations [19, 20]

We start with the ideal MHD equations, in the following form:

character-FIGURE 1.5: Characteristics in two spatial dimensions.

the three-dimensional surface φ(r, t) = φ0, and ask under what conditions isthis insufficient to determine the solution away from this surface

We perform a coordinate transformation to align the boundary data face with one of the coordinates We consider φ as a coordinate and introduceadditional coordinates χ, σ, τ within the three-dimensional boundary datamanifold Thus we transform

φ 0

∂χ

φ0

+ (σ − σ0)∂v

∂σ

φ0

+ (τ − τ0)∂v

∂τ

φ0+ · · · ,

and similarly for B, p, and s The problem is solvable if the normal derivatives

∂v/∂φ|φ0, ∂B/∂φ|φ0, ∂p/∂φ|φ0, and ∂s/∂φ|φ0 can be constructed since allsurface derivatives are known and higher-order derivatives can be constructed

by differentiating the original PDEs

by differentiating the original PDEs