PARAXIAL WKB SOLUTION OF A SCALAR WAVE EQUATION
G.V. Pereverzev
Introduction .................................................... 1
1 Eikonal approximation of the ray method ...................... 4
1.1 Wave equation and short-wavelength ordering ............. 4
1.2 Debye asymptotic expansion, eikonal approximation ....... 5
1.3 Ray tracing ............................................. 5
2 Paraxial WKB approach ........................................ 6
2.1 Short-wavelength asymptotic expansion ................... 6
2.2 Reference ray .......................................... 10
2.3 Wave-packet description ................................ 11
2.4 Paraxial expansion ..................................... 13
3 Beam tracing ................................................ 15
3.1 Ray coordinates ........................................ 15
3.2 First form of the beam-tracing equations ............... 16
3.3 Second form of the beam-tracing equations .............. 18
3.4 Initial conditions for the beam-tracing equations ...... 19
3.5 Discussion of the beam-tracing equations ............... 21
4 Equation for the wave amplitude ............................. 23
4.1 Solving the transport equation ......................... 23
4.2 Accounting for dissipation ............................. 25
5 Solution of the wave equation ............................... 26
5.1 Partial solution ....................................... 26
5.2 General solution ....................................... 28
5.3 Applicability of the paraxial WKB approach ............. 29
5.4 Example of a pWKB solution ............................. 31
6 Conclusions ................................................. 36
Appendices ..................................................... 37
A Geometric properties of ray trajectories .................... 37
A.l The Fermat principle for Eq. (1.1) ..................... 37
A.2 Rays as geodesies in a Riemannian space ................ 40
B Tensor form of the beam-tracing equations ................... 43
B.l Ray coordinates ........................................ 44
B.2 Derivation of the beam-tracing equations ............... 45
B.3 Initial conditions ..................................... 47
B.4 Metric properties of the ray coordinates ............... 48
MULTIPLE-MIRROR PLASMA CONFINEMENT
V.V. Mirnov and A.J. Lichtenberg
Introduction ................................................... 53
1 Qualitative consideration of multiple-mirror effects ........ 59
2 Plasma flow in a magnetic field with small-scale
corrugation ................................................. 68
2.1 Derivation of the macroscopic equation ................. 69
2.2 Calculation of the distribution function correction .... 75
2.3 Analysis of the macroscopic equations .................. 79
2.4 Plasma diffusion along a weakly corrugated magnetic
field with small-scale corrugation ..................... 84
3 Plasma dynamics with large-scale corrugation ................ 89
3.1 The description of plasma motion with two-fluid gas
dynamic equations ...................................... 90
3.2 The intermediate regime in a multiple-mirror field
with "point" mirrors ................................... 94
3.3 The "plateau" regime of plasma motion in a weakly
corrugated magnetic field ............................. 101
3.4 The effect of heavy impurities on plasma multiple-
mirror confinement .................................... 105
4 Multiple-mirror reactor concepts ........................... 108
4.1 Optimization of the axial confinement of a pulsed
reactor ............................................... 109
4.2 The complete pulsed reactor concept ................... 113
4.3 Optimization of the axial confinement of a steady-
state reactor ......................................... 117
4.4 Steady-state multiple-mirror reactor .................. 122
5 Experimental evidence of the multiple-mirror confinement ... 127
6 Summary and discussion ..................................... 133
References ................................................. 140
PLASMA ROTATION IN TOKAMAKS
V. Rozhansky and M. Tendler
Introduction .................................................. 147
1 Momentum balance ........................................... 151
1.1 Plasma flows within a magnetic surface. The
interrelation between the poloidal and the toroidal
rotations ............................................. 151
1.2 Flux surface average momentum balance ................. 158
2 Drift kinetic equation ..................................... 164
2.1 Distribution function in the plateau regime ........... 164
2.2 Regimes with fast poloidal rotation (|θ| ≥ θCS) ....... 168
2.3 Regimes with fast toroidal rotation ................... 174
3 The ontogeny of the poloidal and the toroidal rotations .... 179
3.1 Linear relaxation ..................................... 179
3.2 Distinctive features of the theory in the banana
regime ................................................ 181
3.3 Nonlinear effects resulting from fast rotations ....... 184
4 Fast poloidal rotation and L-H transitions ................. 187
4.1 Suppression of turbulence by a shear of the poloidal
rotation .............................................. 187
4.2 Anomalous transport and steep radial profiles of the
poloidal rotation velocity in edge plasmas ............ 193
4.3 The electric field at the separatrix .................. 197
4.4 Radial current in experiments with a biased
electrode ............................................. 200
4.5 Comparison with experiments ........................... 206
5 The effect of rotation on impurity transport ............... 211
5.1 Poloidal perturbation of the impurity densities and
their fluxes within a magnetic surface ................ 211
5.2 Radial transport of impurities ........................ 216
6 Plasma rotation and flows within the scrape-off
layer ...................................................... 218
6.1 Convection within the SOL in a tokamak with
a poloidal limiter .................................... 218
6.2 Flows within the SOL in a tokamak with a divertor ..... 224
6.3 The impact of the biasing radial electric field on
parameters of the SOL ................................. 231
Conclusions ................................................... 240
Appendices .................................................... 245
A.1 ........................................................ 245
A.2 ........................................................ 246
References .................................................... 249
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