Advanced Magnetohydrodynamics: With Applications to Laboratory and Astrophysical Plasmas

Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 9
Preface......Page 15
Part III Flow and dissipation......Page 19
12.1.1 Grand vision: magnetized plasma on all scales......Page 21
12.1.2 Differences between laboratory and astrophysical plasmas......Page 24
Interchanges in tokamaks and Parker instability in galaxies......Page 25
12.1.3 Plasmas with background flow......Page 30
12.2.1 Basic equations......Page 31
12.2.2 Frieman–Rotenberg formulation......Page 34
12.2.3 Self-adjointness of the generalized force operator......Page 40
12.2.4 Energy conservation and stability......Page 45
12.3.1 Opening up the boundaries......Page 53
12.3.2 Approach to eigenvalues......Page 58
Exercises......Page 65
13.1.1 Gravito-MHD wave equation for plane plasma flow......Page 67
13.1.2 Kelvin–Helmholtz instabilities in interface plasmas......Page 73
13.1.3 Continua and oscillation theorem R for real eigenvalues......Page 77
13.1.4 Complex eigenvalues and the alternator......Page 83
13.2 Case study: flow-driven instabilities in diffuse plasmas......Page 89
13.2.1 Rayleigh–Taylor instabilities of magnetized plasmas......Page 91
13.2.2 Kelvin–Helmholtz instabilities of ordinary fluids......Page 94
13.2.3 Gravito-MHD instabilities of stationary plasmas......Page 103
13.2.4 Oscillation theorem C for complex eigenvalues......Page 109
13.3.1 MHD wave equation for cylindrical flow......Page 111
13.3.2 Local stability......Page 116
13.3.3 WKB approximation......Page 120
13.4.1 Rigid rotation of incompressible plasmas......Page 122
13.4.2 Magneto-rotational instability: local analysis......Page 130
13.4.3 Magneto-rotational instability: numerical solutions......Page 136
Notes on literature......Page 141
Exercises......Page 142
14.1.1 Conservative versus dissipative dynamical systems......Page 145
14.1.2 Stability of force-free magnetic fields: a trap......Page 146
14.2.1 Basic equations......Page 153
14.2.2 Tearing modes......Page 156
(a) Outer ideal MHD regions......Page 158
(b) Inner resistive layer......Page 161
(c) Scaling and matching......Page 162
(d) Approximate solution......Page 165
14.2.3 Resistive interchange modes......Page 167
14.3.1 Resistive wall mode......Page 168
14.3.2 Spectrum of homogeneous plasma......Page 173
14.3.3 Spectrum of inhomogeneous plasma......Page 176
14.4.1 Reconnection in 2D Harris sheet......Page 180
14.4.2 Petschek reconnection......Page 186
14.4.3 Kelvin–Helmholtz induced tearing instabilities......Page 187
14.4.4 Extended MHD and reconnection......Page 189
Exercises......Page 193
15 Computational linear MHD......Page 195
15.1 Spatial discretization techniques......Page 196
15.1.1 Basic concepts for discrete representations......Page 198
15.1.2 Finite difference methods......Page 200
15.1.3 Finite element method......Page 204
15.1.4 Spectral methods......Page 214
15.1.5 Mixed representations......Page 219
15.2.1 Linearized MHD equations......Page 222
15.2.2 Steady solutions to linearly driven problems......Page 224
15.2.3 MHD eigenvalue problems......Page 227
15.2.4 Extended MHD examples......Page 229
15.3.1 Direct and iterative linear system solvers......Page 235
15.3.2 Eigenvalue solvers: the QR algorithm......Page 238
15.3.3 Inverse iteration for eigenvalues and eigenvectors......Page 239
15.3.4 Jacobi–Davidson method......Page 240
15.4.1 Temporal discretizations: explicit methods......Page 243
15.4.2 Disparateness of MHD time scales......Page 251
15.4.3 Temporal discretizations: implicit methods......Page 252
15.4.4 Applications: linear MHD evolutions......Page 254
15.5 Concluding remarks......Page 258
Notes on literature......Page 259
Exercises......Page 260
Part IV Toroidal plasmas......Page 263
16.1.1 Equilibrium in tokamaks......Page 265
16.1.2 Magnetic field geometry......Page 270
16.1.3 Cylindrical limits......Page 274
16.1.4 Global confinement and parameters......Page 278
16.2.1 Derivation of the Grad–Shafranov equation......Page 287
16.2.2 Large aspect ratio expansion: internal solution......Page 289
16.2.3 Large aspect ratio expansion: external solution......Page 295
16.3.1 Poloidal flux scaling......Page 302
16.3.2 Soloviev equilibrium......Page 307
Conformal mapping......Page 311
Two-dimensional finite elements......Page 313
16.4.1 Toroidal rotation......Page 317
16.4.2 Gravitating plasma equilibria......Page 319
16.4.3 Challenges......Page 320
Exercises......Page 322
17.1.1 Alfven wave dynamics in toroidal geometry......Page 325
17.1.2 Coordinates and mapping......Page 326
17.1.3 Geometrical–physical characteristics......Page 327
17.2.1 Spectral wave equation......Page 333
17.2.2 Spectral variational principle......Page 336
17.2.3 Alfvén and slow continuum modes......Page 337
17.2.4 Poloidal mode coupling......Page 340
17.2.5 Alfvén and slow ballooning modes......Page 344
17.3.1 Ideal MHD versus resistive MHD in computations......Page 352
17.3.2 Edge localized modes......Page 358
17.3.3 Internal modes......Page 362
17.3.4 Toroidal Alfvén eigenmodes and MHD spectroscopy......Page 365
Exercises......Page 370
18.1 Transonic toroidal plasmas......Page 373
18.2.1 General equations and toroidal rescalings......Page 375
18.2.2 Elliptic and hyperbolic flow regimes......Page 383
18.2.3 Expansion of the equilibrium in small toroidicity......Page 384
18.3.1 Reduction for straight-field-line coordinates......Page 392
18.3.2 Continua of poloidally and toroidally rotating plasmas......Page 396
18.3.3 Analysis of trans-slow continua for small toroidicity......Page 403
18.4 Trans-slow continua in tokamaks and accretion disks......Page 410
18.4.1 Tokamaks and magnetically dominated accretion disks......Page 411
18.4.2 Gravity dominated accretion disks......Page 414
18.4.3 A new class of transonic instabilities......Page 415
Exercises......Page 420
Part V Nonlinear dynamics......Page 423
19 Computational nonlinear MHD......Page 425
19.1.1 Conservative versus primitive variable formulations......Page 426
19.1.2 Scalar conservation law and the Riemann problem......Page 433
19.1.3 Numerical discretizations for a scalar conservation law......Page 438
19.1.4 Finite volume treatments......Page 448
19.2 Upwind-like finite volume treatments for 1D MHD......Page 451
19.2.1 The Godunov method......Page 452
19.2.2 A robust shock-capturing method: TVDLF......Page 458
19.2.3 Approximate Riemann solver type schemes......Page 464
19.2.4 Simulating 1D MHD Riemann problems......Page 469
19.3 Multi-dimensional MHD computations......Page 472
19.3.1 delta · B = 0 condition for shock-capturing schemes......Page 473
19.3.2 Example nonlinear MHD scenarios......Page 479
19.3.3 Alternative numerical methods......Page 484
19.4 Implicit approaches for extended MHD simulations......Page 491
19.4.1 Alternating direction implicit strategies......Page 492
19.4.2 Semi-implicit methods......Page 493
19.4.3 Simulating ideal and resistive instability developments......Page 499
19.4.4 Global simulations for tokamak plasmas......Page 500
Notes on literature......Page 502
Exercises......Page 503
20.1.1 Flow in laboratory and astrophysical plasmas......Page 505
20.1.2 Characteristics in space and time......Page 506
20.2 Shock conditions......Page 508
20.2.1 Special case: gas dynamic shocks......Page 510
20.2.2 MHD discontinuities without mass flow......Page 516
20.2.3 MHD discontinuities with mass flow......Page 518
20.2.4 Slow, intermediate and fast shocks......Page 523
20.3.1 Distilled shock conditions......Page 525
20.3.2 Time reversal duality......Page 531
20.3.3 Angular dependence of MHD shocks......Page 538
20.3.4 Observational considerations of MHD shocks......Page 545
20.4 Stationary transonic flows......Page 547
20.4.1 Modeling the solar wind–magnetosphere boundary......Page 548
20.4.2 Modeling the solar wind by itself......Page 549
20.4.3 Example astrophysical transonic flows......Page 552
Exercises......Page 558
21 Ideal MHD in special relativity......Page 561
21.1.1 Space-time coordinates and Lorentz transformations......Page 562
21.1.2 Four-vectors in flat space-time and invariants......Page 565
21.1.3 Relativistic gas dynamics and stress–energy tensor......Page 569
21.1.4 Sound waves and shock relations in relativistic gases......Page 574
21.2.1 Electromagnetic field tensor and Maxwell’s equations......Page 582
21.2.2 Stress–energy tensor for electromagnetic fields......Page 587
21.2.3 Ideal MHD in special relativity......Page 588
21.2.4 Wave dynamics in a homogeneous plasma......Page 590
21.2.5 Shock conditions in relativistic MHD......Page 595
21.3 Computing relativistic magnetized plasma dynamics......Page 598
21.3.1 Numerical challenges from relativistic MHD......Page 601
21.3.2 Example astrophysical applications......Page 602
Exercises......Page 606
A.1 Vector identities......Page 609
A.2 Vector expressions in orthogonal coordinates......Page 610
A.2.1 Cartesian coordinates (x, y, z )......Page 612
A.2.2 Cylindrical coordinates…......Page 613
A.2.3 Spherical coordinates…......Page 614
A.2.4 Cylindrical coordinates for toroidal problems…......Page 615
A.2.7 Orthogonal flux coordinates…......Page 617
A.3 Vector expressions in non-orthogonal coordinates......Page 618
A.3.1 Non-orthogonal flux coordinates…......Page 620
Notes on literature......Page 621
References......Page 622
Index......Page 647