Ideal MHD:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York
Cambridge University Press
2014
|
| Ausgabe: | 1. publ. |
| Schlagworte: | |
| Online-Zugang: | Cover Inhaltsverzeichnis Klappentext |
| Beschreibung: | Updated version of: Ideal magnetohydrodynamics. 1987 |
| Beschreibung: | XX, 722 S. |
| ISBN: | 9781107006256 |
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| 020 | |a 9781107006256 |c hardback |9 978-1-107-00625-6 | ||
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| 100 | 1 | |a Freidberg, Jeffrey P. |e Verfasser |0 (DE-588)1053239998 |4 aut | |
| 245 | 1 | 0 | |a Ideal MHD |c Jeffrey Freidberg |
| 246 | 1 | 3 | |a Ideal magnetohydrodynamics |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a New York |b Cambridge University Press |c 2014 | |
| 300 | |a XX, 722 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Updated version of: Ideal magnetohydrodynamics. 1987 | ||
| 650 | 0 | 7 | |a Magnetohydrodynamik |0 (DE-588)4130803-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Magnetohydrodynamik |0 (DE-588)4130803-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 856 | 4 | |u http://assets.cambridge.org/97811070/06256/cover/9781107006256.jpg |3 Cover | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027498900&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027498900&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |3 Klappentext |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-027498900 | ||
Datensatz im Suchindex
| _version_ | 1804152494882816000 |
|---|---|
| adam_text | Ideal
MHD
Comprehensive, self-contained, and clearly written, this successor to
Ideal Maqnetohydrodynamics (Plenum Press,
1987)
describes the macroscopic
equilibrium and stability of high-temperature plasmas
-
the basic fuel for
the development of fusion power.
Now fully updated, this book discusses the underlying physical
assumptions for three basic
MHD
models: ideal, kinetic, and double-
adiabatic
MHD.
Included are detailed analyses of
MHD
equilibrium and
stability, with a particular focus on three key configurations at the cutting-
edge of fusion research: the tokamak, stellarator, and reversed field pinch.
Other new topics include continuum damping,
MHD
stability comparison
theorems, neoclassical transport in stellarators, and how quasi-
omnigeneity, quasi-symmetry, and quasi-isodynamic constraints impact
the design of optimized stellarators.
including full derivations of almost every important result, in-depth
physical explanations throughout, and a large number of problem sets
to help master the material, this is an exceptional resource for graduate
students and researchers in plasma and fusion physics.
Jeffrey Freidberg is Korea Electric Power Professor of Nuclear Science and
Engineering, and former Head of Nuclear Science and Engineering, at MIT,
and a former Associate Director of MIT s Plasma Science and Fusion Center.
He is a Fellow of the APS and the AAAS, and the author of Plasma Physics and
Fusion Energy
(2007).
Contents
и
Preface
page
xvii
Acknowledgements
xix
1
Introduction
1
1.1
The role of
MHD
in fusion energy
1
1.1.1
The plasma pressure in a fusion reactor
1
1.1.2
The dimensionless pressure,
β
4
1
.
1
.3
A variety of fusion concepts
4
1.
1
.4
Structure of the textbook
5
Í.2
Units
5
References
6
Further reading
6
2
The ideal
MHD
model
7
2.
1 Introduction
7
2.2
Description of the model
8
2.3
Derivation of the ideal
MHD
model
12
2.3.1
Starting equations
12
2.3.2
Two-fluid equations
15
2.3.3
Low-frequency, long-wavelength, asymptotic expansions
18
2.3.4
The single-fluid equations
19
2.3.5
The ideal
MHD
limit
22
2.4
Region of validity
28
2.4.1
Overall criteria
28
2.4.2
Conservation of mass
30
2.4.3
Momentum equation
30
2.4.4
Energy equation
32
2.4.5
Ohm s Jaw
33
2.4.6
Summary of validity conditions
34
VII
viii Contents
2.5 Overall
summary
35
References
35
Further reading
35
Problems
36
3
General properties of ideal
MHD
39
3.1
Introduction
39
3.2
Boundary conditions
39
3.2.1
Perfectly conducting wall
40
3.2.2
Insulating vacuum region
40
3.2.3
Plasma surrounded by external coils
43
3.3
Local conservation relations
45
3.3.1
Conservation of mass
45
3.3.2
Conservation of momentum
45
3.3.3
Conservation of energy
46
3.4
Global conservation laws
47
3.4.1
Perfectly conducting wall
48
3.4.2
Insulating vacuum region
49
3.4.3
Plasma surrounded by external coils
51
3.5
Conservation of flux: the frozen-in field line concept
51
3.6
Summary
54
Further reading
54
Problems
55
4
MHD
equilibrium: general considerations
58
4.1
Introduction
58
4.2
Basic equilibrium equations
58
4.3
The virial theorem
59
4.4
The need for toroidicity
61
4.5
Flux surfaces
62
4.6
Surface quantities: basic plasma parameters and figures of merit
66
4.6.1
Fluxes and currents
66
4.6.2
Normalized plasma pressure,
β
68
4.6.3
Kink safety factor, q*
70
4.6.4
Rotational transform,
/,
and the
MHD
safety factor,
q
70
4.6.5
Summary
72
4.7
Equilibrium degrees of freedom
72
4.8
The basic problem of toroidal equilibrium
73
4.9
A single particle picture of toroidal equilibrium
78
4.10
Summary
80
Contents ix
References
81
Further reading
82
Problems
82
Equilibrium: one-dimensional configurations
85
5.1
Introduction
85
5.2
The 0-pinch
85
5.3
The Z-pinch
90
5.4
The
generai
screw pinch
94
5.4.1
General properties
94
5.4.2
The parallel pinch
98
5.4.3
The perpendicular pinch
100
5.5
Inherently 1-D fusion configurations
102
5.5.1
The reversed field pinch
102
5.5.2
The low
β
ohmic
tokárnak
108
5.6
Summary
116
References
117
Further reading
117
Problems
117
Equilibrium: two-dimensional configurations
123
6.1
Introduction
123
6.2
Derivation of the Grad-Shafranov equation
124
6.2.1
The V
-
В
= 0
equation
124
6.2.2
Ampere s law
126
6.2.3
Momentum equation
126
6.3
Plasma parameters and figures of merit
128
6.3.1
Simple flux coordinates
128
6.3.2
The volume of a flux surface
130
6.3.3
The plasma beta
130
6.3.4
The kink safety factor
131
6.3.5
Rotational transform and the
MHD
safety factor
131
6.3.6
The
MHD
safety factor on axis
132
6.3.7
Alternate choices for F(y/)
134
6.4
Analytic solution in the limit
ε<1
and
βρ ~
1 136
6.4.1
The coordinate transformation
137
6.4.2
The asymptotic expansion
138
6.4.3
The
ε°
equation: radial pressure balance
139
6.4.4
The
ε]
equation: toroidal force balance
140
6.4.5
Application to early reversed field pinches (RFP)
142
:
Contents
6.4.6
Application
to early ohmic
tokamaks
and modern
reversed field pinches
144
6.4.7
Summary
151
6.5
Analytic solution in the limit Kl and
ßp
~ XI
ε
(the high
β
tokamak)
152
6.5.1
The high
β
tokamak
expansion
153
6.5.2
The circular high
β
tokamak
154
6.5.3
The flux conserving tokamak
-
avoiding the
equilibrium
β
limit
164
6.5.4
The elliptic high
β
tokamak
171
6.6
Exact solutions to the Grad-Shafranov equation
(standard and spherical
tokamaks)
176
6.6.1
Mathematical formulation
177
6.6.2
Examples: TFTR and JET
182
6.6.3
Example: the spherical tokamak (ST)
183
6.6.4
The equilibrium
β
limit
187
6.6.5
Up-down asymmetric solutions
189
6.6.6
Example: the International Thermonuclear
Experimental Reactor
(ITER)
192
6.6.7
Example: the National Spherical Torus Experiment
(NSTX)
194
6.6.8
Summary
195
6.7
The helical
Grad—Shafranov
equation (the straight stellarator)
196
6.7.1
Overview
196
6.7.2
The helical Grad-Shafranov equation
197
6.7.3
Low
β
analytic solution
199
6.7.4
The rotational transform
207
6.7.5
Summary
209
6.8
Overall summary
210
References
213
Further reading
213
Problems
215
7
Equilibrium: three-dimensional configurations
223
7.1
Introduction
223
7.2
The high
β
stellarator expansion
226
7.2.1
Introduction
226
7.2.2
The basic equations
227
7.2.3
The high
β
stellarator expansion
228
7.2.4
Reduction of the equations
230
Contents xi
7.3 Relation
of the
high
β
stellarator expansion to other models
236
7.3.1
The high
β
tokamak
237
7.3.2
The straight stellarator
237
7.4
The Greene-Johnson limit
240
7.4.1
Comparison of expansions
240
7.4.2
The Greene-Johnson limit of the
HBS
model
241
7.4.3
The Greene-Johnson model
243
7.4.4
Summary
244
7.5
Vacuum flux surfaces
244
7.5.1
Single helicity
-
the limiting helical field amplitude
244
7.5.2
Multiple helicity stellarators
249
7.6
Effects of finite
β
251
7.6.1
Low
β
single helicity solutions
253
7.6.2
Toroidal force balance in a current-free stellarator
257
7.6.3
How does a vertical field shift a stellarator with no
net current?
262
7.6.4
The equilibrium
β
limit in a stellarator
269
7.6.5
The flux conserving steliarator
273
7.6.6
Multiple helicity, finite
β
stellarators
275
7.7
Neoclassical transport in stellarators
277
7.7.1
Review of transport in a tokamak
278
7.7.2
The problem with neoclassical transport in a stellarator
282
7.7.3
One solution
-
the omnigenous stellarator
287
7.7.4
The isodynamic stellarator
292
7.7.5
The symmetric stellarator
297
7.7.6
Boozer coordinates
307
7.7.7
Summary of neoclassical transport in a
stellarator
318
7.8
Modern stellarators
319
7.8.1
The Large Helical Device (LHD)
319
7.8.2
The
Wendelstein 7-Х
(W7-X) stellarator
321
7.9
Overall summary
323
References
324
Further reading
325
Problems
325
8
MHD
stability
-
general considerations
327
8.1
Introduction
327
8.2
Definition of
MHD
stability
328
8.3
Waves in an infinite homogeneous plasma
330
xii Contents
8.3.1
The shear
Alfven
wave
332
8.3.2
The fast magnetosonic wave
332
8.3.3
The slow magnetosonic wave
333
8.3.4
Summary
334
8.4
General linearized stability equations
334
8.4.1
Initial value formulation
334
8.4.2
Normal mode formulation
336
8.5
Properties of the force operator
¥{ξ)
ЪЪ1
8.5.1
Self-adjointness of
¥(ζ)
337
8.5.2
The standard form of
ÔW
338
8.5.3
The intuitive form of
ÖW 340
8.5.4
The intuitive self-adjoint form of
ÖW 342
8.5.5
Real
ω1
344
8.5.6
Orthogonality of the normal modes
345
8.5.7
Spectrum of
¥(ξ)
345
8.6
Variational formulation
347
8.7
The Energy Principle
349
8.7.1
Statement of the Energy Principle
350
8.7.2
Proof of the Energy Principle
351
8.7.3
Advantages of the Energy Principle
354
8.8
The Extended Energy Principle
355
8.8.1
Statement of the problem
356
8.8.2
The boundary conditions
356
8.8.3
The natural boundary condition
358
8.8.4
The surface energy
359
8.8.5
The vacuum energy
360
8.8.6
Summary of the Extended Energy Principle
362
8.9
Incompressibility
363
8.9.1
The general minimizing condition
363
8.9.2
Ergodic systems
364
8.9.3
Closed line systems
365
8.9.4
Summary and discussion
366
8.10
Vacuum versus force-free plasma
367
8.10.1
The nature of the problem
367
8.10.2
Vacuum vs. force-free plasma: the same results
367
8.10.3
Vacuum vs. force-free plasma: different results
368
8.10.4
The real situation: a resistive region
368
8.11
Classification of
MHD
instabilities
369
8.11.1
Internal/fixed boundary modes
370
8.11.2
External/free boundary modes
370
Contents xiii
8.11.3
Pressure-driven modes
370
8
Л
1.4
Current-driven modes
373
8.12
Summary
375
References
376
Further reading
377
Problems
377
9
Alternate
MHD
models
381
9.1
Introduction
381
9.2
Transition from collision dominated to coUisionless regimes
382
9.3
General formulation
384
9.4
Ideal
MHD
closure
385
9.5
Kinetic
MHD
385
9.5.1
Basic assumptions
385
9.5.2
Derivation of the kinetic
MHD
model
386
9.6
The Chew, Goldberger, Low (CGL) double adiabatic model
391
9.6.1
Formulation of the problem
392
9.6.2
Derivation of the double adiabatic model
393
9.6.3
The modified double adiabatic model
395
9.7
Summary
396
9.7.1
Ideal
MHD
396
9.7.2
Kinetic
MHD
396
9.7.3
The double adiabatic model
397
References
398
Further reading
398
Problems
398
10
MHD
stability comparison theorems
400
10.1
Introduction
400
10.2
Ideal
MHD
equilibrium and stability
401
10.3
Double adiabatic
MHD
equilibrium and stability
402
10.3.1
Relation between the perturbed quantities and
ξ
403
10.3.2
The double adiabatic
MHD
Energy Principle
404
1033
Summary of CGL stability
406
10.4
Kinetic
MHD
equilibrium and stability
406
10.4.1
Equilibrium
407
10.4.2
Stability of a closed line cylindrical system
408
10.4.3
Stability of an ergodic cylindrical system
411
10.4.4
Stability of a general toroidal configuration
414
10.5
Stability comparison theorems
422
10.5.1
Closed line cylindrical geometry
422
xiv
Contents
10.5.2
Ergodic cylindrical geometry
423
10.5.3
Closed line toroidal geometry
424
10.5.4
Ergodic toroidal geometry
425
10.6
Summary
426
References
427
Further reading
427
Problems
427
11
Stability: one-dimensional configurations
428
11.1
Introduction
428
11.2
The basic stability equations
430
11.2.1
The Energy Principle
430
11.2.2
The normal mode eigenvalue equations
430
11.2.3
Incompressible
MHD
431
11.3
Stability of the tf-pinch
431
11.3.1
Application of the Energy Principle to the
0-pinch
432
11.3.2
Minimizing SWF for
a Ö-pinch
433
11.3.3
Continuum damping in a slab 0-pinch
434
11.4
Stability of the Z-pinch
443
11.4.1
Energy Principle analysis of
m
^
0
modes
444
11.4.2
Energy Principle analysis of the
m
= 0
mode
447
11.4.3
Double adiabatic Energy Principle analysis of the
m
= 0
mode
450
11.4.4
The hard-core Z-pinch
452
11.5
General stability properties of the screw pinch
458
11.5.1
Evaluation of
ÔW
for a general screw pinch
459
11.5.2
Suydam
s
criterion
465
11.5.3
Newcomb
s
procedure
470
11.5.4
The normal mode eigenvalue equation
477
11.5.5
The oscillation theorem
482
11.5.6
The resistive wall mode
486
11.6
The straight
tokárnak
494
11.6.1
Reduction of
ôW
for the straight
tokárnak
495
11.6.2
Sawtooth oscillations
-
the internal
m
—
í
mode
497
11.6.3
Current-driven disruptions
-
external low
m
modes
500
11.6.4
Density-driven disruptions
-
external low
m
modes
512
11.6.5
Resistive wall modes
-
external low
m
modes
518
11.6.6
Edge localized modes (ELMs)
-
external high
m
modes
519
11.6.7
Summary of the straight tokamak
527
Contents xv
11.7
The reversed field pinch
(RFP) 529
11.7.1
Physical parameters describing an RFP
530
11.7.2
Instability of an RFP without a Bz reversal
532
11.7.3
The
m
= 0
instability
534
11.7.4
Suydam
s
criterion
535
11.7.5
Internal modes
537
11.7.6
Taylor s theory
538
11.7.7
Ideal external modes
552
11.7.8
Overview of the RFP
5 57
11.8
Summary
558
References
562
Further reading
562
Problems
563
12
Stability: multi-dimensional configurations
570
12.1
Introduction
570
12.2
Ballooning and interchange instabilities
572
12.2.1
Introduction
572
12.2.2
Shear, periodicity, and localization
574
12.2.3
General reduction of SWfor ballooning modes
578
12.3
The ballooning mode equations for
tokamaks
581
12.4
The ballooning mode equation for stellarators
585
12.5
Stability of
tokamaks
-
the
Mercier
criterion
588
12.5.1
Introduction
588
12.5.2
Cylindrical limit: the Suydam criterion
589
12.5.3
Toroidal geometry: the
Mercier
criterion
591
12.5.4
Analytic limits of the
Mercier
criterion
596
12.5.5
Summary
598
12.6
Stability of
tokamaks
-
ballooning modes
598
12.6.1
Introduction
598
12.6.2
The
s
-а
model for ballooning modes
599
12.6.3
β
limits due to ballooning modes
605
12.6.4
Summary
608
12.7
Stability of
tokamaks
-
low
η
internal modes
608
12.7.1
Introduction
608
12.7.2
Low
η
internal modes with finite shear
609
12.7.3
Low
η
internal modes with small shear
610
12.7.4
Summary
611
12.8
Stability of
tokamaks
-
low
η
external ballooning-kink modes
611
12.8.1
Introduction
611
xvi Contents
12.8.2
Simplification
of
ôWF
by the high
β
tokamak
expansion
613
12.8.3
High
β
stability of the surface current model
615
12.8.4
Numerical studies of ballooning-kink instabilities
627
12.9
Stability of
tokamaks
-
advanced tokamak (AT) operation
630
12.9.1
Introduction
630
12.9.2
Bootstrap current profile
-
the
βΝ
limit
632
12.9.3
Wall stabilization in an advanced tokamak
635
12.9.4
Infernal modes
636
12.10
Stability of
tokamaks
-
η
= 0
axisymmetric modes
637
12.10.1
Introduction
637
12.10.2
Vertical instabilities in a circular plasma
637
12.10.3
Horizontal instabilities in a circular plasma
640
12.10.4
Vertical instabilities in an elongated plasma
641
12.11
Overview of the tokamak
647
12.12
Stellarator stability
649
12.12.1
High
«
modes in a stellarator
650
12.12.2
The parallel current constraint
651
12.12.3
The relation between average curvature and
magnetic well
652
12.12.4
Average curvature of a straight helix
656
12.12.5
Shear stabilization of a straight helix
658
12.12.6
Stabilization of a toroidal stellarator
659
12.12.7
Numerical results
665
12.13
Summary
666
12.14
The final word
669
References
670
Further reading
671
Problems
672
Appendix A Heuristic derivation of the kinetic equation
678
Appendix
В
The Braginskii transport coefficients
688
Appendix
С
Time derivatives in moving plasmas
691
Appendix
D
The curvature vector
695
Appendix
E
Overlap limit of the high
β
and Greene—Johnson stellarator
models
697
Appendix
F
General form for q(y/)
706
Appendix
G
Natural boundary conditions
707
Appendix
H
Upper and lower bounds on bQKIN
711
Index
718
|
| any_adam_object | 1 |
| author | Freidberg, Jeffrey P. |
| author_GND | (DE-588)1053239998 |
| author_facet | Freidberg, Jeffrey P. |
| author_role | aut |
| author_sort | Freidberg, Jeffrey P. |
| author_variant | j p f jp jpf |
| building | Verbundindex |
| bvnumber | BV042057960 |
| classification_rvk | UR 7000 |
| ctrlnum | (OCoLC)900409422 (DE-599)GBV783071140 |
| discipline | Physik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV042057960 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:11:37Z |
| institution | BVB |
| isbn | 9781107006256 |
| language | English |
| lccn | 2014002053 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027498900 |
| oclc_num | 900409422 |
| open_access_boolean | |
| owner | DE-703 |
| owner_facet | DE-703 |
| physical | XX, 722 S. |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| publisher | Cambridge University Press |
| record_format | marc |
| spelling | Freidberg, Jeffrey P. Verfasser (DE-588)1053239998 aut Ideal MHD Jeffrey Freidberg Ideal magnetohydrodynamics 1. publ. New York Cambridge University Press 2014 XX, 722 S. txt rdacontent n rdamedia nc rdacarrier Updated version of: Ideal magnetohydrodynamics. 1987 Magnetohydrodynamik (DE-588)4130803-7 gnd rswk-swf Magnetohydrodynamik (DE-588)4130803-7 s DE-604 http://assets.cambridge.org/97811070/06256/cover/9781107006256.jpg Cover Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027498900&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027498900&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Freidberg, Jeffrey P. Ideal MHD Magnetohydrodynamik (DE-588)4130803-7 gnd |
| subject_GND | (DE-588)4130803-7 |
| title | Ideal MHD |
| title_alt | Ideal magnetohydrodynamics |
| title_auth | Ideal MHD |
| title_exact_search | Ideal MHD |
| title_full | Ideal MHD Jeffrey Freidberg |
| title_fullStr | Ideal MHD Jeffrey Freidberg |
| title_full_unstemmed | Ideal MHD Jeffrey Freidberg |
| title_short | Ideal MHD |
| title_sort | ideal mhd |
| topic | Magnetohydrodynamik (DE-588)4130803-7 gnd |
| topic_facet | Magnetohydrodynamik |
| url | http://assets.cambridge.org/97811070/06256/cover/9781107006256.jpg http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027498900&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027498900&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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