 | Goedbloed J.P. (Hans). Advanced magnetohydrodynamics: with applications to laboratory and astrophysical plasmas / J.P. (Hans) Goedbloed, R.Keppens, S.Poedts. - New York: Cambridge University Press, 2010. - xvi, 634 p.: ill. - Ref.: p.604-628. - Ind.: p.629-634. - ISBN 978-0-521-87957-6
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Preface page ................................................. xiii
Part III Flow and dissipation ................................. 1
12 Waves and instabilities of stationary plasmas ............... 3
12.1 Laboratory and astrophysical plasmas .................. 3
12.1.1 Grand vision: magnetized plasma on all
scales ........................................ 3
12.1.2 Differences between laboratory and
astrophysical plasmas ......................... 6
12.1.3 Plasmas with background flow ................. 12
12.2 Spectral theory of stationary plasmas ................ 13
12.2.1 Basic equations .............................. 13
12.2.2 Frieman-Rotenberg formulation ................ 16
12.2.3 Self-adjointness of the generalized force
operator ..................................... 22
12.2.4 Energy conservation and stability ............ 27
12.3 Solution paths in the complex ω plane ................ 35
12.3.1 Opening up the boundaries .................... 35
12.3.2 Approach to eigenvalues ...................... 40
12.4 Literature and exercises ............................. 47
13 Shear flow and rotation .................................... 49
13.1 Spectral theory of plane plasmas with shear flow ..... 49
13.1.1 Gravito-MHD wave equation for plane plasma
flow ......................................... 49
13.1.2 Kelvin-Helmholtz instabilities in interface
plasmas ...................................... 55
13.1.3 Continua and oscillation theorem  for
real eigenvalues ............................. 59
13.1.4 Complex eigenvalues and the alternator ....... 65
13.2 Case study: flow-driven instabilities in diffuse
plasmas .............................................. 71
13.2.1 Rayleigh-Taylor instabilities of magnetized
plasmas ...................................... 73
13.2.2 Kelvin-Helmholtz instabilities of ordinary
fluids ....................................... 76
13.2.3 Gravito-MHD instabilities of stationary
plasmas ...................................... 85
13.2.4 Oscillation theorem С for complex
eigenvalues .................................. 91
13.3 Spectral theory of rotating plasmas .................. 93
13.3.1 MHD wave equation for cylindrical flow ....... 93
13.3.2 Local stability .............................. 98
13.3.3 WKB approximation ........................... 102
13.4 Rotational instabilities ............................ 104
13.4.1 Rigid rotation of incompressible plasmas .... 104
13.4.2 Magneto-rotational instability: local
analysis .................................... 112
13.4.3 Magneto-rotational instability: numerical
solutions ................................... 118
13.5 Literature and exercises ............................ 123
14 Resistive plasma dynamics ................................. 127
14.1 Plasmas with dissipation ............................ 127
14.1.1 Conservative versus dissipative dynamical
systems ..................................... 127
14.1.2 Stability of force-free magnetic fields:
a trap ...................................... 128
14.2 Resistive instabilities ............................. 135
14.2.1 Basic equations ............................. 135
14.2.2 Tearing modes ............................... 138
14.2.3 Resistive interchange modes ................. 149
14.3 Resistive spectrum .................................. 150
14.3.1 Resistive wall mode ......................... 150
14.3.2 Spectrum of homogeneous plasma .............. 155
14.3.3 Spectrum of inhomogeneous plasma ............ 158
14.4 Reconnection ........................................ 162
14.4.1 Reconnection in 2D Harris sheet ............. 162
14.4.2 Petschek reconnection ....................... 168
14.4.3 Kelvin-Helmholtz induced tearing
instabilities ............................... 169
14.4.4 Extended MHD and reconnection ............... 171
14.5 Literature and exercises ............................ 175
15 Computational linear MHD .................................. 177
15.1 Spatial discretization techniques ................... 178
15.1.1 Basic concepts for discrete
representations ............................. 180
15.1.2 Finite difference methods ................... 182
15.1.3 Finite element method ....................... 186
15.1.4 Spectral methods ............................ 196
15.1.5 Mixed representations ....................... 201
15.2 Linear MHD: boundary value problems ................. 204
15.2.1 Linearized MHD equations .................... 204
15.2.2 Steady solutions to linearly driven
problems .................................... 206
15.2.3 MHD eigenvalue problems ..................... 209
15.2.4 Extended MHD examples ........................ 211
15.3 Linear algebraic methods ............................ 217
15.3.1 Direct and iterative linear system
solvers ..................................... 217
15.3.2 Eigenvalue solvers: the QR algorithm ........ 220
15.3.3 Inverse iteration for eigenvalues and
eigenvectors ................................ 221
15.3.4 Jacobi-Davidson method ...................... 222
15.4 Linear MHD: initial value problems .................. 225
15.4.1 Temporal discretizations: explicit
methods ..................................... 225
15.4.2 Disparateness of MHD time scales ............ 233
15.4.3 Temporal discretizations: implicit
methods ..................................... 234
15.4.4 Applications: linear MHD evolutions ......... 236
15.5 Concluding remarks .................................. 240
15.6 Literature and exercises ............................ 241
Part IV Toroidal plasmas ...................................... 245
16 Static equilibrium of toroidal plasmas .................... 247
16.1 Axi-symmetric equilibrium ........................... 247
16.1.1 Equilibrium in tokamaks ..................... 247
16.1.2 Magnetic field geometry ..................... 252
16.1.3 Cylindrical limits .......................... 256
16.1.4 Global confinement and parameters ........... 260
16.2 Grad-Shafranov equation ............................. 269
16.2.1 Derivation of the Grad-Shafranov equation ... 269
16.2.2 Large aspect ratio expansion: internal
solution .................................... 271
16.2.3 Large aspect ratio expansion: external
solution .................................... 277
16.3 Exact equilibrium solutions ......................... 284
16.3.1 Poloidal flux scaling ....................... 284
16.3.2 Soloviev equilibrium ........................ 289
16.3.3 Numerical equilibria ........................ 293
16.4 Extensions .......................................... 299
16.4.1 Toroidal rotation ........................... 299
16.4.2 Gravitating plasma equilibria ............... 301
16.4.3 Challenges .................................. 302
16.5 Literature and exercises ............................ 304
17 Linear dynamics of static toroidal plasmas ................ 307
17.1 "Ad more geometrico" ................................ 307
17.1.1 Alfvén wave dynamics in toroidal geometry ... 307
17.1.2 Coordinates and mapping ..................... 308
17.1.3 Geometrical-physical characteristics ........ 309
17.2 Analysis of waves and instabilities in toroidal
geometry ............................................ 315
17.2.1 Spectral wave equation ...................... 315
17.2.2 Spectral variational principle .............. 318
17.2.3 Alfvén and slow continuum modes ............. 319
17.2.4 Poloidal mode coupling ...................... 322
17.2.5 Alfvén and slow ballooning modes ............ 326
17.3 Computation of waves and instabilities in
tokamaks ............................................ 334
17.3.1 Ideal MHD versus resistive MHD in
computations ................................ 334
17.3.2 Edge localized modes ........................ 340
17.3.3 Internal modes .............................. 344
17.3.4 Toroidal Alfvén eigenmodes and MHD
spectroscopy ................................ 347
17.4 Literature and exercises ............................ 352
18 Linear dynamics of stationary toroidal plasmas ............ 355
18.1 Transonic toroidal plasmas .......................... 355
18.2 Axi-symmetric equilibrium of transonic stationary
states .............................................. 357
18.2.1 General equations and toroidal rescalings ... 357
18.2.2 Elliptic and hyperbolic flow regimes ........ 365
18.2.3 Expansion of the equilibrium in small
toroidicity ................................. 366
18.3 Equations for the continuous spectrum ............... 374
18.3.1 Reduction for straight-field-line
coordinates ................................. 374
18.3.2 Continua of poloidally and toroidally
rotating plasmas ............................ 378
18.3.3 Analysis of trans-slow continua for small
toroidicity ................................. 385
18.4 Trans-slow continua in tokamaks and accretion
disks ............................................... 392
18.4.1 Tokamaks and magnetically dominated
accretion disks ............................. 393
18.4.2 Gravity dominated accretion disks ........... 396
18.4.3 A new class of transonic instabilities ...... 397
18.5 Literature and exercises ............................ 402
Part V Nonlinear dynamics .................................... 405
19 Computational nonlinear MHD ............................... 407
19.1 General considerations for nonlinear conservation
laws ................................................ 408
19.1.1 Conservative versus primitive variable
formulations ................................ 408
19.1.2 Scalar conservation law and the Riemann
problem ..................................... 415
19.1.3 Numerical discretizations for a scalar
conservation law ............................ 420
19.1.4 Finite volume treatments .................... 430
19.2 Upwind-like finite volume treatments for ID MHD ..... 433
19.2.1 The Godunov method .......................... 434
19.2.2 A robust shock-capturing method: TVDLF ...... 440
19.2.3 Approximate Riemann solver type schemes ..... 446
19.2.4 Simulating ID MHD Riemann problems ........... 451
19.3 Multi-dimensional MHD computations .................. 454
19.3.1  • В = 0 condition for shock-capturing
schemes ..................................... 455
19.3.2 Example nonlinear MHD scenarios ............. 461
19.3.3 Alternative numerical methods ............... 466
19.4 Implicit approaches for extended MHD simulations .... 473
19.4.1 Alternating direction implicit strategies ... 474
19.4.2 Semi-implicit methods ....................... 475
19.4.3 Simulating ideal and resistive instability
developments ................................ 481
19.4.4 Global simulations for tokamak plasmas ...... 482
19.5 Literature and exercises ............................ 484
20 Transonic MHD flows and shocks ............................ 487
20.1 Transonic MHD flows ................................. 487
20.1.1 Flow in laboratory and astrophysical
plasmas ..................................... 487
20.1.2 Characteristics in space and time ........... 488
20.2 Shock conditions .................................... 490
20.2.1 Special case: gas dynamic shocks ............ 492
20.2.2 MHD discontinuities without mass flow ....... 498
20.2.3 MHD discontinuities with mass flow .......... 500
20.2.4 Slow, intermediate and fast shocks .......... 505
20.3 Classification of MHD shocks ........................ 507
20.3.1 Distilled shock conditions .................. 507
20.3.2 Time reversal duality ....................... 513
20.3.3 Angular dependence of MHD shocks ............ 520
20.3.4 Observational considerations of MHD
shocks ...................................... 527
20.4 Stationary transonic flows .......................... 529
20.4.1 Modeling the solar wind-magnetosphere
boundary .................................... 530
20.4.2 Modeling the solar wind by itself ........... 531
20.4.3 Example astrophysical transonic flows ....... 534
20.5 Literature and exercises ............................ 540
21 Ideal MHD in special relativity ........................... 543
21.1 Four-dimensional space-time: special relativistic
concepts ............................................ 544
21.1.1 Space-time coordinates and Lorentz
transformations ............................. 544
21.1.2 Four-vectors in flat space-time and
invariants .................................. 547
21.1.3 Relativistic gas dynamics and stress-
energy tensor ............................... 551
21.1.4 Sound waves and shock relations in
relativistic gases .......................... 556
21.2 Electromagnetism and special relativistic MHD ....... 564
21.2.1 Electromagnetic field tensor and Maxwell's
equations ................................... 564
21.2.2 Stress-energy tensor for electromagnetic
fields ...................................... 569
21.2.3 Ideal MHD in special relativity ............. 570
21.2.4 Wave dynamics in a homogeneous plasma ....... 572
21.2.5 Shock conditions in relativistic MHD ........ 577
21.3 Computing relativistic magnetized plasma dynamics ... 580
21.3.1 Numerical challenges from relativistic
MHD ......................................... 583
21.3.2 Example astrophysical applications .......... 584
21.4 Literature and exercises ............................ 588
Appendices .................................................... 591
A Vectors and coordinates ................................ 591
A.1 Vector identities ................................. 591
A.2 Vector expressions in orthogonal coordinates ...... 592
A.3 Vector expressions in non-orthogonal
coordinates ....................................... 600
References .................................................... 604
Index ......................................................... 629
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