Synergism of Radio-Frequency Current Drive
with the Bootstrap Current
by
BARKF'
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
Joan Decker
JUL 3 1 2002
Ingenieur Dipl6m6 de l'Ecole Polytechnique
Ecole Polytechnique, Palaiseau, France (2000)
LIBRARIES
Submitted to the Department of Electrical Engineering and Computer Science
in partial fulfillment of the requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
June 2002
©
Massachusetts Institute of Technology 2002. All rights reserved.
...............................................
Author .......
Department of Electrical Engineering and Computer Science
May 10, 2002
7;1
Certified by........
. .. .. . .
. . . . . . . . . . . . . .
Abraham Bers
Professor of Electrical Engineering & Computer Science
Thesis Supervisor
Accepted by ...........
Arthur C. Smith
Chairman, Department Committee on Graduate Students
Synergism of Radio-Frequency Current Drive
with the Bootstrap Current
by
Joan Decker
Submitted to the Department of Electrical Engineering and Computer Science
on May 10, 2002, in partial fulfillment of the
requirements for the degree of
Master of Science in Electrical Engineering and Computer Science
Abstract
The presence of a large, non inductive toroidal current is necessary for confining the plasma
during steady-state tokamak operation. A substantial fraction of this current is expected to be
provided by the bootstrap current, which is driven by temperature and density gradients inside
the plasma. The supplementary current would be generated by radio-frequency (RF) waves that
are coupled to the plasma [9]. Therefore, a self-consistent kinetic calculation of the bootstrap
current and the RF-driven current is of interest so as to measure and understand the effects of the
bootstrap current on RF current drive (RFCD), and vice versa. A synergistic effect between the
bootstrap current and the current driven by lower-hybrid (LH) waves and by electron-cyclotron
(EC) waves has been demonstrated earlier [27].
This work extends these studies on the synergism, in order to understand the physics of
this effect. For this purpose, A 2-D quasilinear relativistic Fokker-Planck code was developed,
which solves the drift-kinetic equation and gives the steady-state electron distribution function.
In contrast to the code used in [27], this code uses an implicit method for the particle flux
conservation at the trapped-passing boundary, so that convergence to a steady-state is much
faster. As a consequence, a more extensive numerical study of the synergism is made possible.
Numerical simulations relevant to future experiments in C-Mod [9] are carried out, and find
that the synergistic current can be up to 10% of the LH driven current. However, with EC current
drive (ECCD) for the same scenarios, a larger synergistic current is found, which can reach up
to 25% of the EC driven current. In all simulations, the synergistic current is proportional to
the bootstrap current, which could be interesting for high bootstrap current configurations.
The physics of the synergism between the bootstrap current and RFCD is investigated using
a kinetic model. It is found that the synergistic current can be explained by considering the
effect of RF quasilinear diffusion on the perturbed distribution function due to radial drifts,
which leads to an enhanced bootstrap current. Conversely, this perturbed distribution function,
which drives the bootstrap current, leads to a larger number of particles in the RF diffusion
region, thereby increasing the RF driven current. The synergism is found to be the largest when
the RF diffusion domain, in momentum space, coincides with the maximum of the perturbed
distribution due to radial drifts, where most of the bootstrap current is generated.
Thesis Supervisor: Abraham Bers
Title: Professor of Electrical Engineering & Computer Science
3
4
Acknowledgments
I would like to thank my supervisors Pr. Abraham Bers and Pr. Ronald Parker for their advice,
support and interest in this research.
I am also very grateful to Dr. Yves Peysson for his large contribution to the present work,
particularly concerning the development of the code. I would also like to thank him for the warm
welcome I received in Cadarache last summer, and for our pleasant and fruitful collaboration on
this research.
5
6
Contents
List of Figures.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
1
Introduction
19
2
Theoretical Model
21
2.1
2.2
3
. . . . . . . . . . . . . . . . . . . . . . . .
21
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
Single Particle Motion in a Tokamak
2.1.1
Toroidal geometry
2.1.2
Particle motion in a tokamak
. . . . . . . . . . . . . . . . . . . . . . . .
23
Kinetic Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.2.1
Drift Kinetic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.2.2
Time scales and ordering . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
2.2.3
Expansion of the DKE . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
2.2.4
Set of Equations to be Solved . . . . . . . . . . . . . . . . . . . . . . . .
34
2.2.5
Moments of the Distribution Function
35
. . . . . . . . . . . . . . . . . . .
Computational Method
39
3.1
Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
3.2
Fokker-Planck Equation
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.2.1
Flux Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.2.2
Finite-Difference Method . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.2.3
Trapped-Passing Boundary . . . . . . . . . . . . . . . . . . . . . . . . . .
44
First-Order Neoclassical Calculations . . . . . . . . . . . . . . . . . . . . . . . .
45
3.3
f)
3.3.1
Radial Derivatives (calculation of
. . . . . . . . . . . . . . . . . . . .
45
3.3.2
Calculation of g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
3.3.3
Treatment of Trapped Particles in g . . . . . . . . . . . . . . . . . . . . .
47
7
4
4.1
General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
4.2
Lower-Hybrid Current Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.2.1
Lower-Hybrid Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
50
4.2.2
Simplified 1-D Analytical Model for LHCD . . . . . . . . . . . . . . . . .
52
4.2.3
Numerical Calculation of LHCD . . . . . . . . . . . . . . . . . . . . . . .
54
Electron Cyclotron Current Drive . . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.3
5
Electron Cyclotron Waves
. . . . . . . . . . . . . . . . . . . . . . . . . .
55
4.3.2
Mechanisms for Current Generation . . . . . . . . . . . . . . . . . . . . .
61
4.3.3
Numerical Calculation of ECCD and Benchmarking . . . . . . . . . . . .
66
70
5.1
Simplified Picture of the Bootstrap Current
. . . . . . . . . . . . . . . . . . . .
70
5.2
Numerical Calculation of the Bootstrap Current . . . . . . . . . . . . . . . . . .
72
5.2.1
Characteristics of the Distribution Function
. . . . . . . . . . . . . . . .
72
5.2.2
Benchmarking Calculation of the Bootstrap Current . . . . . . . . . . . .
74
. . . . . . . . . . . . . . . . . . . . . . . . .
75
Analysis in the Lorentz Model [13]
Self-Consistent Calculation of RFCD and the Bootstrap current
6.1
6.2
7
4.3.1
Bootstrap Current in a Maxwellian Plasma
5.3
6
48
Radio-Frequency Current Drive
Bootstrap Current in LHCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
77
. . . . . . . . . . . .
77
. . . . . . . . . . . . . . . . . . . . . .
78
6.1.1
Estimation of the synergism for a realistic scenario
6.1.2
Physical picture of the synergism
6.1.3
Prediction of the synergism in the Lorentz model
. . . . . . . . . . . . .
81
6.1.4
Test of the Predictive Model . . . . . . . . . . . . . . . . . . . . . . . . .
81
6.1.5
D iscussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
Bootstrap Current in ECCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
87
6.2.1
ECCD with Bootstrap Current in DIII-D . . . . . . . . . . . . . . . . . .
89
6.2.2
ECCD with Bootstrap Current in C-Mod . . . . . . . . . . . . . . . . . .
92
96
Conclusion
99
A Profiles for Tokamak parameters
A .1
77
Alcator C-M od
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
A .2 D III-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
100
8
B Collision Operator
B.1
102
Differential term Cee(f, fM) + Ce(f,f) . . . . . . . . . . . . . . . . . . . . . . .
102
B.1.1
Detailed Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
102
B.1.2
Expression in the Limit of a Lorentz Gaz . . . . . . . . . . . . . . . . . .
104
B.1.3
Expression as the Divergence of a Flux and Bounce-Averaged Operator .
104
B.2 Integral term Cee(fM, f) . . . .
. . . . . . . . . . . . . . . . .
. . .. .
. . .
105
B.2.1
Detailed Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
105
B.2.2
Bounce-Averaged Operator
106
. . . . . . . . . . . . . . . . . . . . . . . . .
107
C Quasilinear Operators
C.1
General Expression as the Divergence of a Flux
. . . . . . . . . . . . . . . . . .
107
C.2 Bounce Averaging of the Quasilinear Operator . . . . . . . . . . . . . . . . . . .
108
C.3 Lower-Hybrid Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
C.3.1
General Expression for the LH Diffusion Coefficient . . . . . . . . . . . .
109
C.3.2
Flat E 12 (kii) Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . .
109
C.3.3
Expression for the Fluxes due to LH Diffusion . . . . . . . . . . . . . . .
110
C.4 Electron-Cyclotron Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
112
C.4.1
General Expression for the EC-X2 Diffusion Coefficient . . . . . . . . . .
112
C.4.2
Gaussian E12 (kii) Spectrum . . . . . . . . . . . . . . . . . . . . . . . . .
112
C.4.3
Expression for the Fluxes due to EC Diffusion . . . . . . . . . . . . . . .
113
D Flux-Surface Averaging
D .1
D efinition
116
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
116
D.2 Electron Distribution f(p, ,0) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
117
D.2.1
Implicit 9-Dependence: case of fo and g.
D.2.2
Explicit 0-Dependence:f
. . . . . . . . . . . . . . . . . .
117
. . . . . . . . . . . . . . . . . . . . . . . . . . .
117
D.3 Flux-Surface Averaged Moment of f(p, ,0)
. . . . . . . . . . . . . . . . . . . .
118
D.3.1
Flux-Surface Averaged Density
. . . . . . . . . . . . . . . . . . . . . . .
120
D.3.2
Flux-Surface Averaged Current Density . . . . . . . . . . . . . . . . . . .
120
D.3.3
Flux-Surface Averaged Density of Power Absorbed
. . . . . . . . . . . .
121
D.3.4
Expression as a Moment of Particle Fluxes . . . . . . . . . . . . . . . . .
121
D.3.5
Flux-Surface Averaging . . . . . . . . . . . . . . . . . . . . . . . . . . . .
122
E Calculation of Bounce Average Coefficients
E.1
Calculation of A(o) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
123
123
E.1.1
Series Expansion
E.1.2
Calculation of the Integrals J 2m
124
E.1.3
Truncated Expression . . . . . .
126
. . . . . . . .
E.2
Calculation of s*
E.3
Calculation of *. . . . . . . . . . . ..
E.4
Calculation of A
E.5 Calculation of
. . . . . . . . . . . .
126
127
. . . . . . . . . . . .
Z(o)
123
. . . . . . . . . . .
F Listing of testfp2d9yp and fp2d9yp
128
130
132
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
132
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
158
F.1
testfp2d9yp
F.2
fp2d9yp
10
11
List of Figures
2-1
Toroidal coordinates system (r,0, ).
2-2
Velocity space coordinates (v, ,<b).
. . . . . . . . . . . . . . . . . . . . . . . .
22
. . . . . . . . . . . . . . . . . . . . . . . . .
23
2-3
Banana orbit projected onto the poloidal plane. . . . . . . . . . . . . . . . . . .
25
2-4
Trapping domain
3-1
Positioning of the discrete particle distribution and fluxes on the (pl, pi) grid.
.
41
3-2
Nine-points differentiation scheme . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3-3
Structure of the 9-diagonals differential operator matrix M.
43
3-4
Structure of the 15-diagonals differential operator matrix M. The index
I(o| <cot.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . .
26
_T_1
refers to the pitch-angle position in the counter-passing region which is closest
to the trapped-counterpassing boundary. The index
T
refers to the pitch-angle
position in the trapping region which is closest to the trapped-copassing boundary. 45
3-5
Particle fluxes conservation at the trapped-passing boundary. . . . . . . . . . . .
4-1
Distribution function calculated for C-Mod (vl = 3.5, v2 = 6, Do = 1). 2D contour
plot of fo(above) and logarithmic plot of the parallel distribution FO (below). . .
46
56
4-2
Launching angles and propagation for ECW. (a) top view, (b) poloidal cross-section 58
4-3
Radial frequency profiles in the plane 0 = 0 for ECW propagation in C-Mod
(Appendix A).
ECW propagation is represented by a horizontal arrow
(fEcw
is fixed by launching) from either the low-field side (top graphs (a),(b)) or the
high-field side (bottom graphs (c),(d)). The propagation of both the X-mode (left
graphs (a),(c)) and the 0-mode (right graphs (b),(d)) is considered. The two first
cyclotron harmonics fce and
2
fce are displayed. The shaded areas represent the
regions of frequencies unaccessible to the given mode, due to cut-offs (right-hand
cut-off f, for the X-mode, plasma frequency fpe for the O-mode) and resonances
(upper-hybrid resonance
fuh,
affecting the X-mode). . . . . . . . . . . . . . . . .
12
60
4-4
Resonance curves for C-mod parameters and varying Nil at fixed 2Q/w (above)
and for varying 2Q/w at fixed N (below).
4-5
. . . . . . . . . . . . . . . . . . . . .
63
Normalized particle fluxes due to EC quasilinear diffusion, as calculated in (C.49).
The length of the arrows is proportional to S/fo. The solid lines are a contour
plot of the steady-state distribution function and the dashed lines are a contour
plot of the diffusion coefficient DEC exp [-(kI,res - k1 ,o) 2 /Ak].
4-6
. . . . . . . . . .
65
Steady-state normalized particle fluxes due to EC quasilinear diffusion and collisions, as calculated in (C.49) and (B.18). The length of the arrows is proportional to S/fo. The solid lines are a contour plot of the steady-state distribution function and the dashed lines are a contour plot of the diffusion coefficient
DEC exp[-(i,res
4-7
-
kll, 0) 2 /Ak 1 ]..........
..
Resonance curves with C-mod parameters versus
...
6
b,
...
..
...
...
..
66
which is the poloidal angle
at the position where the EC beam crosses the flux-surface under consideration
(C .47). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4-8
67
Graph above: Steady-state ECCD distribution function (solid lines) and Maxwellian
distribution at the same temperature (dotted lines). The dashed lines are a contour plot of the magnitude of the diffusion coefficient DEC. Graph below: Anti. . . . . . . . . . . . . . . . . . .
69
5-1
Characteristics of trapped electron motion in the poloidal plane. . . . . . . . . .
71
5-2
Electron distribution function to first order in the neoclassical corrections. Top
symmetric (part in p1l) of the distribution FO.
graph: Maxwellian foM (solid) and corrected foM + flM (dashed) distributions.
Bottom graph: parallel first-order distribution FM, calculated from fiM using (5.7). 73
5-3
Bootstrap current calculated from the transport coefficients by [30] (dotted line),
[18] (dashed line) and the code (solid line) for C-Mod parameters (Appendix A).
5-4
74
Bootstrap current calculated from the transport coefficients by [30] (dotted line),
[18] (dashed line) and the result in the Lorenz limit (2.67) (solid line) for C-Mod
parameters (Appendix A).
6-1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
76
parallel projection of the first order distribution function, with (dashed line) and
without LHCD (solid line). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
79
6-2
The bottom graph is a zoom of the top graph in the region of the quasilinear
plateau. The dashed line represents the parallel electron distribution function for
LHCD only F0 (2.76). The dotted-dashed line is the first order, "bootstrap only"
distribution function FiM. The dotted line is associated with the total distribution
calculated non self-consistently Fnsc = Fo +FiM. The solid line represents the total
distribution function calculated self-consistently F = Fo + F1 (2.71). The "steps"
in the line contain no physics and are simply due to the numerical change of
variable (p, ) -- (pgp ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
6-3
Current densities in LHCD (top) and synergistic fraction (bottom) versus V1min
.
83
6-4
Current densities in LHCD (top) and synergistic fraction (bottom) versus vimaC
.
84
6-5
Current densities in LHCD (top) and synergistic fraction (bottom) versus the
bootstrap current in a case of high LH current . . . . . . . . . . . . . . . . . . .
6-6
Current densities in LHCD (top) and synergistic fraction (bottom) versus the
bootstrap current in a case of low LH current . . . . . . . . . . . . . . . . . . . .
6-7
85
86
ECCD in C-Mod at r = 0.15 m versus NI, > 0 and 2Q/w. Where N11 and 2Q/w
are too small (lower left corner), the resonance curve (4.48) is too far away in
the tail and very few electrons are resonant with the ECW so that no current is
generated. On the other hand, where NiI and 2Q/w are too large (upper right
corner), the resonance curve is too close to the trapped-passing boundary and the
Ohkawa effect reduces the current generated and eventually becomes predominant.
Between those limits, ECCD can be optimized, including when one parameter (NiI
or 2Q/w) is fixed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
6-8
Current Densities for ECCD in DIII-D at r = 0.3 m.
89
6-9
Synergistic current for ECCD with bootstrap in DIII-D at r = 0.3 m. . . . . . . .
. . . . . . . . . . . . . . .
90
6-10 Parallel first order distribution without (FIM, solid line), and with ECCD (F 1 ,
dashed line), and also for the case without ECCD but with an enhanced temperature (6.11) derived from
fo in ECCD
. . . . . . . . . . .
91
0.15 m. . . . . . . . . . . . . . . .
92
(FM,AT,
6-11 Current Densities for ECCD in C-Mod at r
=
dotted line).
6-12 Synergistic current for ECCD with bootstrap in C-Mod at r = 0.15 m.
. . . . .
93
6-13 Parallel first order distribution without (F1M, solid line), and with ECCD (F 1 ,
dashed line), and also for the case without ECCD but with an enhanced temperature (6.11) derived from fo in ECCD (FIM,AT, dotted line).
14
. . . . . . . . . . .
94
6-14 2-D contour plot of the first order distribution fi. The dotted line (flM) is the
case of a maxwellian plasma (no ECCD); the solid line (fi) is the case with ECCD.
The dashed line is a contour plot of the EC diffusion coefficient.
. . . . . . . . .
95
Temperature, density and toroidal B field profiles in C-Mod [23]
. . . . . . . . .
100
A-2 Temperature, density and toroidal B field profiles in DIII-D [19]
. . . . . . . . .
101
E-i
Bounce averaging coefficient A . . . . . . . . . . . . . . . . . . . . . . . . . . . .
126
E-2
Bounce averaging coefficient
. . . . . . . . . . . . . . . . . . . . . . . . . . .
129
E-3
Bounce averaging coefficient A . . . . . . . . . . . . . . . . . . . . . . . . . . . .
130
E-4 Flux-surface averaging coefficient \ . . . . . . . . . . . . . . . . . . . . . . . . .
131
A-i
*
15
16
List of Tables
4.1
LHCD calculations (current density j and density of power absorbed Pabs normalized respectively to enevTe and nemeV4eve) by M. Shoucri and I Shkarofsky [28]
(1) and from the present code (2). . . . . . . . . . . . . . . . . . . . . . . . . . .
57
A.1
On-axis parameters in C-Mod [23] . . . . . . . . . . . . . . . . . . . . . . . . . .
99
A.2
On-axis parameters in C-Mod [19] . . . . . . . . . . . . . . . . . . . . . . . . . .
101
17
18
Chapter 1
Introduction
The plasma confinement in a Tokamak requires the presence of a toroidal current in order to
create a poloidal magnetic field. The non-inductive generation of the toroidal current is of great
importance in the prospect of a steady-state reactor, since inductive methods, such as varying
magnetic fields, are by definition transient. Launching radio-frequency waves into the plasma has
been suggested as a possible means to transfer a parallel momentum to particles, thus driving
a current. This method, called Radio-frequency Current Drive (RFCD), has been successfully
used to generate a toroidal current, and steady-state operation with total non-inductive current
drive has been achieved experimentally with lower hybrid waves (Tore-Supra, [22]) and, more
recently, with electron cyclotron waves (TCV, [7],[8]).
Unfortunately, a significant amount of toroidal current is required to run a reactor-scale
tokamak (10 - 15 MA) such that a large fraction (100 to 500 MW) of the power generated by
fusion (1 GW) would have to be recycled in RFCD, critically reducing the power effectively
available. Fortunately, in hot, dense plasmas with relatively high inverse aspect ratio, a large
toroidal current, called bootstrap current, is driven by the pressure gradient, and requires no
additional supply of energy. This current can eventually represent a large fraction of the required
current. However, the bootstrap current can not supply the entire amount of current necessary to
confine the plasma. In particular, the bootstrap current vanishes at the plasma center. Moreover,
a control of the current profile is needed for stability, and this may not be possible with bootstrap
current only.
Many recent steady-state scenarios rely on a large bootstrap current with the additional
current being driven by RFCD. In C-Mod, where the bootstrap current is particularly large,
advanced scenarios relate on a bootstrap fraction
fBs
up to 70% of the total current ([9]).
Lower-hybrid current drive (LHCD) would supply the remaining 30%.
19
Although the mechanisms of both the bootstrap current and RFCD have been extensively
studied and are well understood, no self-consistent kinetic calculation of the bootstrap current
in the presence of RFCD had been carried until recently [27], where a synergism was found
between the bootstrap current and both Lower-Hybrid and Electron Cyclotron current drive.
Given that bootstrap current and RFCD are the two most often considered method to generate
non-inductive current, the study of this synergism is of great interest. This is the aim of the
present work, which extends the work done in [27].
A new code has been developed, which solves for the bootstrap current and the RF driven
current self-consistently in a much shorter time than the code used in [27], with a more accurate
treatment of the interaction between passing and trapped particles. This code allows a more
extensive investigation of the synergism.
In chapter 2, general considerations about particle motion in a tokamak are given and the
kinetic model used in this study is developed. Then, the new features of the code are described
to highlight the differences with the code used and described in [27]. In the next two chapters
(4 and 5), the problems of RFCD and bootstrap current are investigated separately to give an
understanding of their mechanism. The last chapter presents the study of the synergism between
the bootstrap current and both lower hybrid current drive (section 6.1) and electron cyclotron
current drive (section 6.2).
20
Chapter 2
Theoretical Model
Single Particle Motion in a Tokamak
2.1
2.1.1
Toroidal geometry
Magnetic Confinement
The confinement of the very energetic particles constituting the plasma is a major problem in
the attempt of achieving nuclear fusion. Particles in the plasma carry an electronic charge and
are subject to a gyromotion in a magnetic field due to the Lorentz force. The motion of a particle
with mass m and charge q in a plane perpendicular to a uniform magnetic field of magnitude
B
=
IBI is then forced to be circular, with a radius
(2.1)
PL _= Awhere v = JiY is the velocity of the particle and Q
qB/-ym is the relativistic gyro-frequency; -y
=
is the relativistic factor
1
1 -v
2
/c 2
=(2.2)
In a C-Mod like tokamak (description in Appendix A) with a magnetic field B = 4 T and a
temperature T = 7.5 keV, an electron with thermal velocity
frequency
VTe
=
/TIme would have a gyro-
e = 7 x 10" rad/s and a cyclotron radius, called Larmor radius PLe = 50 pm. For
protons at the same temperature, the respective quantities are Qi
=
3.8 108 rad/s and PLi
=
2.2
mm.
Therefore, ions and electrons can be well confined in the two dimensions perpendicular to a
magnetic field. The particle motion in the parallel direction is made quasi-periodic by closing
the magnetic field lines on themselves. Thus the particle motion can be decomposed into the
21
R
Ro
a
Figure 2-1: Toroidal coordinates system (r,0,).
fast rotation around the field line, with a cyclotron frequency Q and the slower motion of the
guiding center around the axis of the torus.
In the toroidal geometry (Fig. 2-1), positions are defined in the toroidal coordinate system
(r,6, ().
The distance from the torus axis is given by R
=
Ro + r cos 0 = Ro(1 + e cos 6)
where E = r/Ro is the local inverse aspect ratio. In the entire study, axisymmetry is assumed
(effects resulting from the finite number of toroidal magnetic coils are neglected) and therefore
(-dependencies will always be omitted.
Velocities (v, , 0) are calculated in spherical coordinates with respect to the local magnetic
field (Fig. 2-2) with
V
=
V
Vii
V
=
coslV±X
Vj
(2.3)
Guiding Center Drifts
Unfortunately, the bending of the magnetic field lines is responsible for a alteration of the
cyclotron motion resulting in a drift of the guiding center.
VD -
(v+vi/2)
1Q
I
22
B x V(B
2
B
(2.4)
z // B
V/
Cos E
y
x
Figure 2-2: Velocity space coordinates (v,
,
D).
In this expression, there is a contribution due to the parallel motion (vii) of the particle, which
experiences a centrifugal force due to the curvature of the field lines. The drift resulting from the
association of this force and the Lorentz force is called curvature drift. Moreover, the external
generation of a curved magnetic field creates a perpendicular variation of the magnitude of the
field. The cyclotron motion (v±) in this inhomogeneous magnetic field generates a drift called
VB drift which adds to the curvature drift in (2.4).
The magnitude of the guiding center drift (2.4) of thermal particles is of the order of
pLIV In BIvT. The spatial scale for the curvature of the magnetic field is the torus major radius
Ro; thus the guiding center drift is smaller than the particle velocity by a factor
6D - PLIRo
2.1.2
(2.5)
Particle motion in a tokamak
Poloidal Motion
The guiding center drift (2.4) is charge sensitive, so that electrons and ions are drifting in opposite
directions in the poloidal cross-section . This charge separation generates an electric field, which
creates a E x B drift. This drift is oriented in the same direction for both electrons and ions,
leading to a general loss of confinement.
In a tokamak, a poloidal component B9 is added to the magnetic field by the generation
of a toroidal current, so that the motion of the guiding center along the field line is now a
23
superposition of a toroidal rotation around the torus vertical axis and a poloidal rotation around
the toroidal axis. The motion of the particle along the field line is then confined in a given flux
surface. The effect of this poloidal motion is that the guiding center drift is directed alternately
outside and inside the flux surface. Therefore, the drift averaged over a poloidal (or bounce)
period vanishes, leading to an enhanced confinement of the particles.
Magnetic Trapping
Another important consequence of the poloidal motion of the particles is that the magnetic field
strength varies along the field line.
It is maximum when the particle is closest to the torus
vertical axis (high field side, 0 = 7r) and minimum when the particle is farthest away from this
axis (low field side, 0 = 0). Besides, the energy of the particle is a constant of the motion, since
there is no work produced by a constant magnetic field, and the periodicity of the cyclotron
motion results in the conservation of the magnetic moment
2
p
(2.6)
=
2B/-y
which is an adiabatic invariant, in the case of a slowly varying magnetic field. According to (2.6),
a particle moving along a magnetic field with increasing intensity will experience an increase in
its perpendicular velocity in order to keep p constant. As a consequence, the parallel velocity of
the particle will be reduced because of the conservation of energy, by what is called an effective
mirrorforce. Eventually, a particle with insufficient parallel energy will be reflected back toward
the low field region before completing a poloidal rotation. Such a particle is said to be trapped
in the magnetic field. Its trajectory projected onto the poloidal or the toroidal plane is called a
banana orbit (Fig. 2-3).
Using the pitch-angle coordinate system defined in (2.3) and adding a subscript 0 to the
quantities evaluated at 0 = 0 (low field side),
v2(1 _
2mB/
and the pitch-angle cosine
=
_
V2(1
_
g)
(2.7)
2mBo/y
can be evaluated according to its value at the midplane 0
= a [
with o-
2)
- T (r, 0) (1 -
02)
=
0:
(2.8)
sgn( o) and I(r, 0) = B(r,9)/Bo(r).
In the case of circular flux surfaces, which will always be considered in this study, the flux
surface can be parameterized by its radial position r. Moreover, the magnetic field at a given
24
particle
flux surface
grad B
Figure 2-3: Banana orbit projected onto the poloidal plane.
position can be easily evaluated from its central value BC = B(R = Ro):
B(r,6) =
R/Ro
=
(2.9)
1+ecos(
The magnetic field ratio T now takes the simple expression
T (r, 0) =
c
(2.10)
1 + ECos 0
(
Using (2.10) and (2.8), the condition for particle trapping
=
0 at 0 = 7r) becomes:
V2_6
6O
<0OT
(2.11)
+6
=
The trapping domain in the (vII, v±) plane is shown on fig. 2-4
Poloidal motion
* Passing particles
The circulation of passing (untrapped) particles in the poloidal cross-section defines a
transit time
27rr
27rr
27rq(r)Ro
vO
vIlBo/B
VII
(2.12)
where
q(r)
r B
RO Bo
25
r B
Ro Bo
(2.13)
Trapped
Particles
V
O
OT
7
OT
V
0
Figure 2-4: Trapping domain 16o <tot.
is the safety factor. Since the parallel velocity is of the order of the thermal velocity for
passing particles, the transit time can be estimated as
27q(r)Ro
(2.14)
VT
Now, it is possible to estimate the maximum drift off a flux surface along a passing particle
orbit. It is of the order of
Art
~
VD
VD- , 7rq(r)PL -
2
PL
(2.15)
since the safety factor q(r) is usually of the order of unity. Therefore, the excursion off a
flux surface is of the order of a Larmor radius for a passing particle.
Trapped particles
A characteristic time for the poloidal motion of trapped particle can be defined as the time
between two reflections (at 0 > 0 and 0 < 0), called bounce time The bounce time is also
of the order of (2.12)
2-rqRo
Tb -
(2.16)
Vil
According to (2.11),the typical parallel velocity of trapped particles is smaller than the
thermal velocity by a factor fE. Thus the bounce time is
Trb
26
~
Tt
(2.17)
and the maximum drift off a flux surface along a trapped particle orbit is of the order of
Arb
~
(2.18)
PL
Tb
2
-,e
which is no more than a few Larmor radii.
For electrons, the excursion off the flux surface is therefore rather small (Ar < (V In B)- 1 ,
(V In T)').
Electrons remains on the same flux surface to the leading order in 6D (2.5), thus the
along the trajectory (2.8). Changes
dependence upon r can be neglected in the evolution of
in
are mainly due to the poloidal motion. The poloidal angle at which a trapped electron is
reflected is then given by:
cos- 1 (1 - 2
O7
where the value of r in
OT,
2
(2.19)
2T)
given in (2.11), is considered as a fixed parameter for a given electron.
The electron transit or bounce time can be calculated more accurately:
f Oc
fOc
Tbt
where 0, =
7r
=
-rd9
JoI vo |
B
B
r
r
=
lvilI B0
J-oC
(2.20)
for passing electrons
for trapped electrons
OT
Since the excursion of an electron off its flux surface is small, r can be considered as a
parameter in (2.20), independent of 0. Using B/Bo ~ Roq(r)/r, the bounce or transit time
becomes:
Tb,-
2irRoq(r) fC
dO0
2wrRoq(r)A
~
27r
_0,v
| 1
(2.21)
0|v
where A is a normalized quantity defined as
A=
c!
dO o
(2.22)
-0c 27r
Bounce averaging
Finally, dependencies upon the poloidal angle can be removed by integration along the electron
orbit over a bounce or circulation period. This process is called bounce averaging
{A}=
In this expression, the symbol [1/2
-
Tb,t
I-
Za=±1ib
-
d
I
2=±1JT
A
lviii Bo
e
(2.23)
applies to trapped electrons only and means that
the integration must be made over both the forward and the backward motion. According to
(2.21) and (2.22), the bounce averaging (2.23) is equivalent to
A
=O
2
.
0-=il- T
27
rA
(2.24)
2.2
2.2.1
Kinetic Description
Drift Kinetic Equation
Boltzmann Equation
In the kinetic description, particles are described by a distribution function
the probability for a particle of the species a to be at the position
±'
f,(CF, V',
t) which gives
and with a velocity V' at
time t. The particle conservation in the phase space is expressed by the Boltzmann equation:
af
at
+
-
afa
ax-
af
-%-Q(fl(A, t) + v- x B(Y, t)) --
afI
apt 1C
(2.25)
where af /Ot~c describes collisions between particles and
(2.26)
S= 'ymav
is the momentum. The relativistic factor y is given in (2.2)
In the range of wave frequencies considered in this study: lower hybrid waves
GHz) and electron cyclotron waves (fEC
-
(fLH
1
100 GHz), only the electrons are subject to resonant
interaction with the waves. Therefore, ions remain mainly unaffected and their distribution can
be taken as a maxwellian. Unless explicitly specified, the distribution functions in the following
study will always refer to electrons.
Drift Kinetic Equation
Under the assumption of axisymmetry, and after averaging over the shortest periodic fluctuations,
namely the gyromotion and the RF wave oscillation, the Boltzmann equation (2.25) for electrons
reduces to the drift kinetic equation [27]
Of
+ &9c - Vf
where f
=
=
C(f) + Q(f)
(2.27)
(fe),w is the gyro-averaged, wave period-averaged electron distribution function.
The guiding center motion can be decomposed into the fast parallel motion and the slower
drift off the field lines given by 2.4
-+VD
S
(2.28)
|BI
Any radial component
VDO
can be neglected compared to the fast poloidal motion since
VDO
v1 Bo/B
~Dq(r)
PLq(r) <1
r
c
28
(2.29)
where the estimation (2.5) has been used to estimate the magnitude of the guiding center drift.
Under the assumption of axisymmetry, the variations of
f
along the field line depends upon
the poloidal angle only:
B
1 Bo 1
.V =
0'
r B
|B|
,
(2.30)
The radial drift can be evaluated by conservation of the canonical momentum, which gives
[10]
v11 0
V
(2.31)
V
r 0(!1
Finally, the steady-state form of the drift kinetic equation is
vrj Bo af
vjj o9
Of =OC(f) +
Q(f)
(2.32)
where the electron distribution f is function of the two real space variables (r, 9) and the two
velocity-space variables (p, ().
Collision Operator
The collisions are described by a Fokker-Planck operator, under the assumption that only elastic
binary collisions matter and that small angle collisions prevail. The electron distribution function
can be divided into its Maxwellian part fm at the electron temperature T and density ne, and
the contribution
fp
describing the perturbation from the Maxwellian:
f
= fM + fp. Assuming
that the perturbation is small, the non-linear term Cee (f,, f,) describing the collisions of the
perturbed part with itself can be neglected and a linearized collision operator can be used.
Using Cee(fM, fm)
=
0, the collision operator is reduced to
C(f) = Cee(f, fM) + Cee(fM, f) + Cei(f, fi)
(2.33)
where Cei describes collisions with ions.
Such a linearized, relativistic collision operator has been derived by Braams and Karney [15]
and is given by:
( f)
[2A(p)-'+
=
+
B(p) 1 F
(1-
F(p)f
f1
)--I + I1(fM, fl=1)
(2.34)
with the collision frequency
Ve =
4
rie In A
47re2m2ve
29
(2.35)
In (2.34), 1 is an integral term describing the evolution of the bulk distribution due to collisions
with the perturbed distribution
fp.
In this operator, only the first order term
f1=1 = 31
-_1
of the expansion in Legendre harmonics of Cee(fM,
f)
fd
(2.36)
has been kept (the zero-order term fi=o ~
fM(p) gives no contribution). This choice for the collision operator (called truncated operator [16]
[28]) ensures momentum conservation but not energy conservation. Therefore, it is not necessary
to add a energy loss term (radiation or radial diffusion) to reach a steady-state.
Detailed expressions for the integral 11, for the drag term F(p) and for the diffusion coefficient
A(p) and Bt(p) are given in Appendix B. For a more general description of the different collision
operators and their properties, see [16].
Quasilinear Operator
Since the model is intended for studying current drive, RF fields should be considered that can
modify significantly the distribution function. Then, a quasi-linear operator is necessary to take
into account the deviations from an equilibrium Maxwellian distribution engendered by the RF
fields. A quasilinear operator has been derived by Kennel and Engelmann [5] for arbitrary field
polarizations and propagation, and extended by Lerche [17] for relativistic plasmas:
lim
Qf) = V-+oo
1 f
V
d3 k
(27) 3
E-
p2(1 _ g2) Dk 6(w - k og - nQ)L(f)]
L
(2.37)
L...
where
6(w - k 1v11 - nQ)
(2.38)
is the resonance condition with vi = p11/mY
L(A) =1
)
+-
(2.39)
IA
is a momentum space differential operator and
2
Dk = qe Ek,+e-ln-
+ Ek,-eaJn+l + PEkiiJn
(2.40)
is the quasi-linear diffusion coefficient.
The electric field is decomposed into its Fourier components
E(Y, t) = Re
[J (2)3 e
30
-)Ek]
(2.41)
which are expressed in the rotating fields coordinates system defined by
E+,
1
Ek,I1 = Ek,z
iEk,y),
2
(2.42)
The transverse component of k is projected onto the x and y axes according to kx = k_ cos a
and k. = k1 sin a. The argument of the Bessel function is the cyclotron radius normalized to the
perpendicular wave number kv_/Q and gives the scaling of the interaction between electrons
and the wave.
Time scales and ordering
2.2.2
Time scales
" The operator terms in the drift kinetic equation (2.32) represent various time scales. A
quantitative evaluation of these time scales is given for a C-Mod plasma, with typical
parameters given in Appendix A.
The first term
v11 Bo 0
r B 90
is associated with the poloidal motion of the particles and its characteristic times are clearly
the bounce rb (2.17) and transit -r (2.14) times. In C-Mod at r = 0.15 m, their values are
typically of the order of:
'~~
2irq(r)Ro
Tt
Trb~ -
0.6 ps,
-
1.2 ps
(2.43)
VTe
"
The second term
V 11
(V9V11
r& ( Q
Or
describes the radial drift of particles. It applies mainly to particles whose parallel velocity
varies significantly in the poloidal motion, namely the trapped and barely passing particles.
Those particles are responsible for the so-called neoclassicaleffects, including the bootstrap
current. The characteristic time of the radial drift is given by rD= a/vD; this is the time
it takes for a particle with a drift velocity VD to travel a distance equivalent of the torus
minor radius a. Using (2.5), the characteristic drift time can be estimated for C-Mod at
r = 0.15 m:
TD
~
a
aR0
SDVTe
PLVTe
31
R ~ 0.3 ms
(2.44)
e The characteristic time associated with the collision operator is simply the thermal electron-
electron collision time obtained from (2.35), given by:
7e
I
472
Ve
e
M 2eT
n-
(2.45)
lnIA
In C-Mod at r = 0.15 m, the e/e collision time is
Te
= 34 Ms.
Trapped particles are also concerned with the characteristic detrapping time which is the
time for a trapped particle to be scattered out the trapped region. Since the angle of the
trapping region in the (pil,j
p)
scattering by an angle adt
-
v ',
plan is
sin- 1 OT
Detrapping requires pitch-angle collision
V . Since the collision results from a random
walk process, the time requires for a scattering by an angle a evolves like
1a2
and therefore
the detrapping time is shorter than the transverse collision diffusion time r- by a factor:
'Tdt
S
T1
adt
~.,'
7r/2
2
~
(2.46)
The transverse diffusion time is of the order of the electron collision time (2.45) [14].
Therefore, the detrapping time is shorter than the electron collision time by a factor
h
~t
(2.47)
Te
and its value in C-Mod at r = 0.15 m is typically -rdt~ 7.5 Ms.
* The quasi-linear diffusion due to RF fields can be characterized by a diffusion coefficient
DRF = Dovepre where ve (2.35) is the collision frequency and PTe
= mvTe is the ther-
mal momentum. With the RF fields under interest for experiments (with powers in the
MegaWatt range), the normalized diffusion coefficient Do is of the order of unity, as calculated for LHCD and for ECCD in Appendix C. Therefore, the quasi- linear time 7q, = 7e/Do
can be taken to be of the same order as the collision time.
Small Drift Expansion
Ordering the time scales allows solving the drift kinetic equation by using an expansion
f
=
fo + 6fi + -- - (following [27]). The small parameter in this expansion is the ration of the drift
time to the bounce time
6t=
t/TD
27rq(r)pL/a
for passing electrons
(2.48)
6b = Trb/TD - 27rq(r)PL/fEa for trapped electrons
This expansion parameter would be 6 t ~_2.1 x 10- 3 , 6b ~ 4.5 x 10-3 in C-Mod at r = 0.15 m.
32
Banana Regime Ordering
Moreover, the domain of collisionality in which the drift kinetic equation is expanded has to be
specified. This study aims to consider situations in which the neoclassical effects - including the
bootstrap current - are important. Those effects are intimately related to the existence of trapped
and barely passing particles. The presence of banana orbits is therefore required. Those orbits
are defined only if a given particle can bounce many times before being subject to detrapping
by collisions. The domain to be considered is therefore the low collisionality regime - or banana
regime - in which the detrapping time is much longer than the trapped particle bounce time.
This condition is expressed by
b
Tdt
where
lmrfp
rt/Te
~
=
1.6
VTere
27rq(r)- 3 /2 R o
~ 6- 3 / 2 T t
Te
<
(2.49)
lmfp
is the electron mean-free path length. In C-Mod at r = 0.15 m, we find
x 10-2 and
Tb/rdt -
0.15. Then, except at the edge of the plasma, where the
temperature usually drops faster than the density and the collisionality increases, and in the
center of the plasma, where c --> 0, the banana regime holds.
The ordering used to expand the drift kinetic equation, called banana regime ordering is then
given by [27]
2.2.3
Ttb <
<
T
6PT
-(2.50)
TD
Te
<
1(
<
Tdt
Expansion of the DKE
The electron distribution function is expanded in 6 ~ p
/a:
fo + 6fi +.-
f
(2.51)
Zero-order equation
The drift term is first order in the expansion and does not appear in the zero- order equation:
=
C(fo) + Q(fo)
r B 00
(2.52)
A sub-ordering can be performed by applying the banana regime ordering re,dt < rt,b (2.50).
The only term in (2.52) remaining on the bounce time scale is
B0 ~fo0
r B O9
v-
33
(2.53)
Therefore, the zero-order distribution fo is independent of 9. Then, fo can be evaluated by
performing the bounce averaging of (2.52) defined in (2.23), which removes the term related to
the bounce motion.
(2.54)
{C(fO)} + {Q(fo)} = 0
This equation is the bounce averaged Fokker-Planck equation with quasilineardiffusion solved
in so-called bounce averaged Fokker-Planck codes [28] [6], which include some neoclassical effects
(trapping of particles) but neglect the effects of radial drift, responsible for the bootstrap current.
First order equation
v1 B 0 96f 1 e0
rjBo
+
r B 00
roi9
V1
GQ
Ogfo_
r -=11(a
C(fi) + Q(fi)
Or
(2.55)
In this equation, the two first terms are of the same order J. Again, the contributions from
collisions and quasilinear diffusion are smaller by a ratio
Te,dt/rt,b
< 1. Using the banana regime
ordering therefore gives
v- B
-f
r B
90
--
-
-(VK'-fo
r Ok Qi Or
(2.56)
This equation can be integrated over 9 which gives
6 fi =
f+
g
(2.57)
with
f
~1
(9fo
and g constant (in 9)
Again, performing a bounce averaging of (2.55) removes the 9 derivatives in (2.55).
(2.58)
Then,
using the linearity of the quasilinear operator (2.37), and taking a linearized form of the collision
operator (2.33), an equation for the 9-independent function g is obtained:
{C(g)} + {Q(g)} = -{C(f)} - {Q(f)}
2.2.4
(2.59)
Set of Equations to be Solved
The problem that will be solved by the code can be condensed in a set of three equations that
must be solved in sequence:
* Zero-order bounce averaged radio-frequency current drive problem:
{C(fo)} + {Q(fo)} = 0
34
(2.60)
* First order radial dependence of the RFCD problem solution:
~vil (9fo
(2.61)
f = -fQ0 ar
* First order neoclassical corrections to the distribution function
{C(g)} + {Q(g)} = -{C(f)} - {Q(f)}
(2.62)
The total distribution is then obtained as
f = fo + f +g
(2.63)
Since fo is the homogeneous solution of the equation (2.62), g, = g + cfo is another
solution for (2.62). The particular solution for g, is found by using an equation imposing
the conservation of the density:
(2.64)
J(hf+ j + ge)d'p = ne
2.2.5
Moments of the Distribution Function
Current Densities
The total parallel electron current density at the position (r,9) is calculated from the
distribution function by integration in velocity space:
j(E)
qed3vIjf(p-) = 27rqej
1f
JO
p 2dP
f-1 'Ye
P Ofd
(2.65)
This current density can be separated into different contributions according to (2.63).
- RF driven current density only (RFCD problem, neglecting radial drift corrections)
j(RF)
=
11
22rqe
j/fo p j --1 'Y e f
2
dp
e
0d<
(2.66)
- Bootstrap current density in absence of RFCD
J(BS)
_ j(BSe)
+
(2.67)
j(BSi)
where
J BSe)
-2
n
+-d gdf
is the electron contribution and JS)
(2-6)
is the ion contribution, calculated from the
fluid model by Hirshman f30], and recalled in [27] p.91. The subscripts M denote that
f and g have been evaluated by taking fo = fM, maxwellian at temperature Te.
35
- Total current density, which includes the ion bootstrap contribution.
J(TOT)
j(E)
+
j(RF)
-
j(BSi)
(2.69)
- Synergistic current density
J(S)
-
(TOT)
_
j(BS)
-
j(E)
-
j(BSe)
-
j(RF)
(2.70)
Parallel distribution functions
The gyro-averaged relativistic distribution function f is a function of the momentum space
variables (pi , p 1 ). A 1-D representation is obtained by integrating over the perpendicular
momentum p1:
F(pi)
27r J
(2.71)
f(pii ,p)p± dpi
When the important features of the distribution functions are dependent mostly upon the
parallel momentum pil (which is the case for LHCD), the representation of the parallel
distribution function F(p1 ) is useful and often more physically understandable than the
2-D plot of
f.
The density can be calculated using F(p1i):
IMC
0
(2.72)
F(pjj)dpgj = ne
In the non-relativistic limit (-y -- + 1), the asymmetry of F in voj
=
p 1l/me gives the parallel
electron current density (2.65):
j(E)nr
11
qe
j
dpvlf (p
=
qe
JLF(pjj)dpij
fJ 0 0 me,
(2.73)
where the superscript nr stands for non-relativistic
The expansion of
f
in the drift parameter 6 (2.63)
f =fo+fi withfi=f+g
(2.74)
F = Fo + F
(2.75)
holds for F as well:
with
Fo(p) = 27r j
36
fo(pi1 , p 1 )pidp±
(2.76)
F1 (pjj) = 27r
fi(p 1 , pi)pidpi
(2.77)
Then, the electron current density, calculated from the parallel distribution function in the
non-relativistic limit (2.73) , can be divided into the contributions from RFCD, with (2.66)
becoming
j(RF)nr
= qe
Fo (p j)dpjj
f
(2.78)
and the electron bootstrap current, with (2.68) becoming
BSe)nr =qe
j
"Fi(p)dpj
(2.79)
The synergistic current is then given by (2.70)
j(S)nr
j(E)nr _
(BSe)nr
L(F 1(p1 )
_ j(RF)nr = ge
-
FM(p||))dp|\
(2.80)
Densities of Power Absorbed and Dissipated
The power densities can also be calculated by taking a moment of the distribution function.
If a differential operator can be expressed under the form of a divergence,
S= --V . S
(2.81)
which is the case for the collision and the quasilinear operators, as shown in Appendices B
and C, then the density of power absorbed by the plasma due to the action of this operator
0is given by (D.47)
P ymS
/o2rjcdg
(2.82)
Note that the power depends only upon fluxes in the p direction, since pitch-angle scattering
is a process without energy transfer and does not contribute to the power dissipated. The
expression for the fluxes associated with collisions and quasilinear diffusions are given
respectively by (B.16) and (C.13), leading to explicit formulations of the power densities:
-
Density of power dissipated by collisions
The dissipation of energy occurs through diffusion in the momentum magnitude p,
and slowing down.
PCOn = 2,T
d
-- 1
j
f0
37
dp pm (A(p)-'f + F(p)f
Iye
ap
(2.83)
- Density of power absorbed from RF waves
Pabs
= -27r
-1
1 dK-f(11d2\p3Fp'IcJ2(
-(
dp
dj
Y e
0
RF
0F
(2.84)
where DRF(p, ) is the RF diffusion coefficient given by (C.11) and L (f) is the differential operator (C.10).
In steady-state, the density of power dissipated should equal the density of power absorbed.
Current Drive Internal Figure of Merit
The current drive internal figure of merit on a given flux-surface can be defined as the
ratio of the RF current density plus the synergistic current density, divided by the total
absorbed power density:
j(RF) + j(S)
1(TOT)_ -
(TOT
(2.85)
p(TOT)
abs
which must be compared to the internal figure of merit for the RFCD problem only
j(RF)
77 (RF)
-
(2.86)
F
p (RF)
abs
where p(TOT) is calculated from (2.84) with
(2.84) with
f
f
= fo + f + g, while p(RF) is calculated from
= fo only.
Flux-Surface Averaging
It is common to present current densities and absorbed power as averaged over the flux
surface, so that they are a function of r only. The flux-surface element is dS
that, using R
=
=
Rd( rd9 so
Ro(1 + e cos 0), the flux surface average of a quantity A is becomes
f7
_
rdO f
(A(r))
Rd(A(r, 0)
rdO f
Rd
(1 +
e cos
O)A(r, 0)
=
(2.87)
The flux-surface averaged expressions for current and power densities are derived in Appendix
D.
38
Chapter 3
Computational Method
3.1
Background
A code (Listed in Appendix F) has been developed to solve the the drift kinetic equation for
the total distribution function
f
as formulated in Section 2.2.4. It is based on the neoclassical
Fokker-Planck FORTRAN code fastfp-nc written by S. Schultz [27], and on the Fokker-Planck
MATLAB code fp2dyp written by Y. Peysson [26]. Both codes contain initially the Fokker-Planck
solver FP2D created by M. Shoucri and I. Shkarovsky [28]. The further development of fp2dyp,
which is the major contribution of the present study, consists of two major extensions. First,
the resolution of the Fokker-Planck equation (Section 3.2) has been improved with a proper
treatment of boundary conditions on particle fluxes (Section 3.2.2) including at the trappedpassing boundary (Section 3.2.3). Then, the neoclassical treatment developed in fastfp-nc has
been carried to fp2dyp allowing this code to solve the drift kinetic equation and calculate the
bootstrap current consistently with radio-frequency current drive. Modifications have also been
made on the neoclassical development that allow a more consistent numerical evaluation of the
radial derivatives (Section 3.3). Furthermore, the new code is implemented in MATLAB which
is a matrix form code well-suited for the inversion of sparse multi-dimensional matrices. Another
advantage of MATLAB is that new modules or modifications are easily added.
39
3.2
3.2.1
Fokker-Planck Equation
Flux Formulation
The first step in solving the drift kinetic equation (Section 2.2.4) is to solve the equation for fo,
solution of the RFCD problem without first order corrections (2.54).
(3.1)
{C(fo)} + {Q(fo)} = 0
This Fokker-Planck equation is bounce averaged and therefore solved in the (p, o) space. It
is not fully implicit, because of the presence of an integral term in (2.34). Several iterations
are therefore needed to obtain the steady-state solution, by demanding that the time-dependent
Fokker-Planck equation
Of
- {C(fo)} - {Q(fo)} = 0
(3.2)
converges to a steady-state.
The bounce averaged collision and quasilinear diffusion operators can be expressed as the
divergence of a flux in the momentum space (see Appendix B and C) , so that (3.2) becomes
+
.(AS()(f
0 )) = {I(fMf0,=1)0
(3-3)
where A is the normalized bounce time (2.22). The first-order term I, of the Legendre series, kept
to take into account the evolution of the bulk due to collisions with the perturbed distribution
(See Section 2.2.1), is an integral term that therefore cannot be expressed under the form of a
divergence and must be kept explicitly. In spherical momentum coordinates (p, o) this equation
becomes
+
(p2 O)(o)) _ 1
_
o
(fo))
=
(3.4)
{II(fUM, fo,1=)o}
where collisions and quasilinear diffusion contribute to particle fluxes:
S(0)(yo)
=SCO
(yo) + SF,(O) (fo
S (f)
=
(fo)+
The expression for the fluxes §C,(0 ) and
3.2.2
S
§RF,(o)
SRF(o)(fo)
(3.5)
are respectively given by (B.18) and (C.16).
Finite-Difference Method
Differential Operator in Momentum Space
The Fokker-Planck equation (3.4) contains second order derivatives in momentum space. It is a
boundary value problem that can be solved by finite difference methods. The momentum space
40
p 1 4+
l+1/2, j-1/2
Ei+11
A
M
S
10
S~~
+J/,1j1/
/2
A-1/+1/2
++
j+1
jj+3/2
P/
Figure 3-1: Positioning of the discrete particle distribution and fluxes on the (ps,p1 ) grid.
is discretized on a (p, o) grid of size (N x M) with
p=(i-)1)AP
with
AP=
"
+(j-1)A o
with
A~o-
M
o,=-1
andi=1,...,N
2
and j=1,...,M
(3.6)
Unlike the scheme used in previous codes [27] [28], the distribution function f is calculated on
the half grid
(Pi+1/2,
Oj+1/2) so that the fluxes, which are first order derivatives of the distribution
function, are defined on the (pi, o j) grid (Fig. 3-1).
This scheme is advantageous since the
boundary conditions are then automatically satisfied at p = 0 and
o = +1, since there is no
particle flux through those boundaries according to (3.4). In its discrete form, the expression for
the differential operator V - S in spherical coordinates, explicitly written in (3.4), becomes:
{V
- S}i+1/2,j+1/2 =
2
(P+±Zi+,j+1/2
(P
pi+1/2
A+1/2Pi+1/2
7 ji+1/2,j+1
O
(A
j
. i 1/2,j
(3.7)
The particle fluxes, calculated in Appendices B and C.13, are function of fo and its derivatives
with respect to (p, to). The discrete form of these fluxes is obtained by using the following rules
41
p1-112
pi -1/2
j - 1 /2
Pi - 1/ 2
j+1/2
i -1/2
j+3/2
1/2
i+1/2
j+3/2
I
Pi+1/2
j -1/2
R+2
i+ 3 / 2
j -1/2
Pi+3/
+ ri+3/2 40+/2
1j+1/2
Figure 3-2: Nine-points differentiation scheme
for the spatial derivatives:
Of
fi+1/2,j+1/2 -
fi-1/2j+1/2
AP
ij+1/2
af
fi+1/2, 3 -1/2
+1/2,+1/2 -
A O
'9~+1/2j
Of
fi-1/2,j
2Ap
f+3/2,j i+1/2,j
Of
fij+3/2
-
fij-1/2
2A~o
ij+1/2
and for the distribution function:
(1
fij+1/2=
+1/2
(1
=
-
6
ij+1/2)fi+1/2,j+1/2
-i+1/2,j)fi+1/2,j+1/2
+
6
+
6
ij+1/2fi-1/2,+1/2
i+1/2,jfi+1/2,j-1/2
(3.8)
where the coefficients 6ij are introduced in order to account for the exponential decrease of the
distribution function. They have been calculated by Chang and Cooper in [4].
With this discretization scheme, the bounce averaged collision and quasilinear operators (3.7)
are represented by a 9-diagonal square (NM x NM) matrix M connecting each point
(fi+1/2,j+1/2)
to its 8 neighboring points in velocity space, represented on Fig.3-2.
{V
.Si+1/2,j+1/2
=
(M.
The structure of the matrix M is shown on Fig.3-3.
42
f)i+1/2,j+1/2
(3.9)
(PI 1 ,Fi)
(pi,4
Figure 3-3: Structure of the 9-diagonals differential operator matrix M.
Convergence
The time dependence is discretized using an implicit (backward Euler) form
Of (f
t
(t + At) - f(t)
(3.10)
At
so that, using (3.9), the discrete (matrix) form of the equation (3.3) becomes
M+
where S(t)
=
f(t+ At)
=
hf(t) +S(t)
(3.11)
{I,(fm, f=1(t)) o} acts as a source term, and I is the identity tensor.
Without an integral term {Ii (fM, fl=1) o} in the equation above (for example, when this term
is omitted [161), a solution could be found in only one iteration with a time step At
>> 1 sufficient
to reach steady-state. Fortunately, the steady-state solution of the full equation, including the
first-order term I, in the Legendre expansion, can still be found within a few iterations (typically
5 to 30 iterations) with a time step much larger than the collision time (At
> 1).
The convergence to a steady-state is checked by calculating the norm of the residue Of/Ot
from (3.10), as done in [25] and [16]
f (Of /at) 2 fdp
c
f fd3p
1/2
<Rf
A good convergence was obtained with a convergence parameter Rfo = 10-10.
43
(3.12)
3.2.3
Trapped-Passing Boundary
The accurate treatment of the trapped particles is particularly important in the resolution of
the drift kinetic equation since those particles are responsible for most of the neoclassical effects
including the bootstrap current. Furthermore, the determination of the driven current (in the
abscence of bootstrap current) by solving the Fokker-Planck equation (3.2) requires a proper
treatment of the trapped particles because their fast bounce motion generates an enhanced
pitch-angle scattering that can affect the current drive efficiency (See Section 4.2.3).
Indeed, since the bounce frequency of the trapped particles is much larger than their collision
or detrapping frequency (rbT
< -rt,
Te),
any asymmetry in the trapped region is immediately
flattened by the bounce motion, and within the trapped domain (16oI < 6oT), the position of the
same electron is equally split between (p, -60)
and (p, 6o). Therefore, the distribution function
fo is intrinsically symmetric in the trapped region. Nevertheless, it may be useful to note that
neoclassical corrections generate asymmetry in the trapped region due to the finite radial drift
of trapped particles off the flux-surface associated with a radial non-uniformity. These effects
enter the first order distribution
f.
In most Fokker-Planck codes ([26], [27], [6]), the symmetrization of the distribution function,
due to the fast bounce motion, is forced explicitly at each time step in (3.11). However, because
the distribution function is not symmetric in 6o outside the trapped region, a large time step in
(3.11) would lead to a discontinuity in the distribution function, which is not satisfactory.
A different method, following [20], has been successfully implemented in the code.
The
symmetry in the trapped region is obtained by shrinking half of the trapped domain (-6oT < 6O <
0) for the calculation. Then, the particle fluxes between the counter-passing region (6o < -oT)
and the trapped region are treated by connecting the first point in the counter-passing region
( o = -6-T)
to the last point in the trapped region ( o = oT). This connection has a physical
sense since the particle at (6o = -JT)
and ( o =
-- ) are intrinsically the same. the flux
conservation is therefore implicitly included in the discrete scheme by inserting 6 extra non-zero
diagonals for the additional flux connections. The structure of the Matrix M is displayed on
Fig.3-4.
Beyond its consistency in the fluxes conservation, the main advantage of this fully implicit
method is that the trapped region is properly treated in an implicit manner, allowing large
time step At > 1000 while the former explicit method [27] required a much smaller time step
(At < 0.1). A considerable reduction of the calculation time, by orders of magnitude, is thus
obtained. Details of the fluxes exchanges at the trapped-passing boundary are shown on Fig.
44
(PN-T-1)
(PN'
T)
9 Diagonals
Figure 3-4: Structure of the 15-diagonals differential operator matrix M.
The index (_-1
refers to the pitch-angle position in the counter-passing region which is closest to the trappedcounterpassing boundary. The index
CT
refers to the pitch-angle position in the trapping region
which is closest to the trapped-copassing boundary.
3-5.
3.3
3.3.1
First-Order Neoclassical Calculations
Radial Derivatives (calculation of
f)
The zero-order distribution fo is calculated from the Fokker-Planck equation solved on a given
flux-surface, at a specific radial position r. However, the first-order neoclassical corrections to fo
are related to the radial variations of the distribution function. This radial dependence appears
in the calculation of
f
(2.61)
ffo
(3.13)
The radial derivative is performed numerically by taking the difference between the distribution function calculated on three neighboring flux-surfaces. However, the consistency of such a
process must be checked, considering that the radial variations of the distribution function can
have at least four different origins:
* The effects of the RF field on the distribution function fo depend upon the local parameters
of the RF field (magnitude, spectrum and poloidal localization of the power) that can vary
45
0O
ad., -o
-OT
O
OT
%
S2
in
In %
s3out %\
%%
'1
I
out
I
X
p//
Figure 3-5: Particle fluxes conservation at the trapped-passing boundary.
along the propagation and depend upon r. The accurate determination of these variations
requires ray-tracing techniques and are omitted in this work. Taking the three flux-surfaces
very close to each other reduces the effect of these variations.
" The effects of RF field on the distribution function fo also depends upon the local temperature and density (absorption, normalization of the diffusion coefficient). The RF parameters must therefore be normalized to the density and temperature specific to each
neighboring flux-surface.
" The radial variation of the electron density must be taken into account in the calculation
of f with a consistent normalization of the distribution functions fo.
" The radial variation of the temperature has to be handled carefully. Indeed, the three
distribution functions required for the calculation of f (3.13) must be defined on the same
momentum grid for consistency. Since the temperature is different at each flux surface, it
is necessary that the Fokker-Planck code be able to solve (3.11) on a given reference grid
independently from the electron and ion temperatures. In particular, the calculation of the
collision integrals must take into account the temperature dependence of the Maxwellian
distribution. This calculation has been carried out in [29 and has been used in our code.
46
3.3.2
Calculation of g
Equation Formulation
It is interesting to note that the equation for g (2.62) has the same structure as (3.2), from which
it differs only by the presence of a source term.
{C(g)} + {Q(g)}
=
-{C(f)}
- {Q(f)}
(3.14)
Consequently, the flux formulation in (3.2.1) can be reused for solving (3.14). A discrete form
analogous to (3.11) can be implemented, with a modified right-hand side source term
S = {I1(fM, fl=1)o} - {Cf)
-
{Q()}1
(3-15)
However, the treatment of trapped particle for g and fo are different.
3.3.3
Treatment of Trapped Particles in g
fo being symmetric in the trapped region,
depends upon
particles
(Tb
<
-, according to (2.8)).
Te, TQL),
f
(defined in 3.13) is antisymmetric since vij = v
However, because of the fast bounce motion of trapped
the collisions and quasilinear diffusion operators are symmetric in the
trapped region. Averaged over a bounce period (2.23), the collisions and quasilinear diffusion
operators applied to the antisymmetric distribution
f vanish and the source term S (3.15) is
zero in the trapped region. The solution for g in (3.14) can then be fixed to zero in the trapped
region.
47
Chapter 4
Radio-Frequency Current Drive
4.1
General Considerations
Among the most widely used techniques to generate a non-inductive toroidal current in a Tokamak is the Radio-frequency current drive (RFCD), which refers to a variety of scenarios for
driving a current through the damping and momentum deposition of externally launched waves .
For now, lower hybrid waves (LHW) and electron cyclotron waves (ECW) are both serious candidates for large-scale current drive in the future scenarios for steady-state tokamak operations;
therefore, both are considered in this study.
The radio-frequency current drive problem consists of determining the flux-surface averaged
current density
(j(RF))
(r) and power dissipated (Pabs)(r) resulting from the presence of given
fields E(, t) on the flux surface at radius r. These quantities can be evaluated by taking specific
moments of the electron distribution function.
When the effects of radial drifts are not taken into account, the distribution function to be
considered in fo, solution of the bounce averaged Fokker-Planck equation (3.2)
(4.1)
0
{C(f 0 )} + {Q(fo)}
Then, the flux-averaged moments are given by (See Appendix D):
(J(RF))
_
27rq,
I
(Pabs)
j
P
TMe
fo
27r
0
dp P
7
e
jf
J
where A is the normalized bounce time (2.22) and
do0o(1 + e)fo(p, o)
(4.2)
-1i
_1
df 0(1
+E)A{S
1
F}
(43)
{ } denotes the bounce averaging operation
(2.23). Expressions for the bounce averaged RF fluxes are given by (C.31) for LHCD and (C.49)
for ECCD.
48
Then, an internal current drive figure-of merit, for the flux-surface under consideration, can
be calculated from (4.2) and (4.3):
(J(RF))
11
(4.4)
(Pabs)
The quasilinear operator Q(f),which describes the kinetic resonance between waves and particles, is given in (2.37). The poloidal field Bp is neglected in the study of RFCD (Bo ~ BT).
For electrons, the resonance condition (2.38) can then be rewritten as
w - kjjvjj +
=0
(4.5)
~jq~IBTT
(4.6)
with
Wce
where
QT
Me
= qeBT/yme is the relativistic gyrofrequency associated to BT.
The case n
=
0 is called Landau resonance. A large absorption of the wave can be expected
when a the significant number of electrons have a parallel velocity v 1 of the order of the wave
parallel phase velocity w/kjj
=
c/N 1 . Therefore, a wave with a sufficiently large N is required,
which is the case of lower hybrid waves. The damping at n
=
0 of a unidirectional NgI spectrum
is indeed the mechanism of absorption required for LHCD (Section 4.2).
The cases n =
±1,
±2, ...
correspond to the electron cyclotron resonance and its harmonics.
Absorption now occurs when a the significant number of electrons have a parallel velocity v 1
of the order of the wave parallel doppler-shifted phase velocity (w - nQ)/kll. This condition is
fulfilled in the neighborhood of the electron cyclotron frequency (and its harmonics). Such waves
are called electron cyclotron waves and this resonance is the mechanism of absorption required
for ECCD (Section 4.3).
The study of RFCD will be illustrated using the advanced operation scenarios for C-Mod
([23],
[9]), which require a narrow RF driven current profile peaked around r
=
0.15 m or
r/a = 0.7 . Since the present code solves the RFCD problem only on a single flux surface, the
specific radius r = 0.15 m will be considered primarily. The C-Mod plasma parameters at this
radius are given in (A.1).
49
4.2
4.2.1
Lower-Hybrid Current Drive
Lower-Hybrid Waves
Cold plasma dispersion relation
We consider modes propagating mainly across the magnetic field between the ion and electron
cyclotron frequencies (Qi
< w <
Qce). They are given by the cold plasma dispersion relation
[32]
Do(x, k, w) = P4 nI + P2 ri + Po = 0
(4.7)
where the coefficients
(4.8)
P 4 = 61
P2 = (Ell + Ec)(n
Po = El [(n
4.9)
e1 ) + e
-
e2_]
- 6)-
(4.10)
are function of the parallel refractive index nil and the elements of the cold plasma dielectric
tensor which in this range of frequencies are
2
U2
e
~
1±
+
,
-
~
EXY -
(4.11)
The dispersion relation (4.7) gives two modes in terms of nI:
n
(-P2
k
VP 2 2
_ 4P 4 Po)
(4.12)
which are called slow wave and fast wave, respectively for the + and - signs, according to their
perpendicular phase velocity c/n
1
.
The slow wave is usually referred as the lower hybrid wave since it presents a resonance in
propagation (ni -
oo) at the lower-hybrid frequency EI -- 0
WLH
(4.13)
~1 + W21Qc
Accessibility
The value of nil for which the square root in (4.12) vanishes defines the accessibility condition.
In the case w > Wh = VQceQci, which holds for C-mod (A.2) (Wh,max/27r = 3.8 GHz, w/27r = 4.6
GHz), the accessibility condition derived from (4.12) is given by
ng =
wp
+
I+
ce
+
ce
50
(4.14)
W
At a given position (ne, ni, B fixed), modes with nil < n
are evanescent (n 1 imaginary).
Launched slow waves are then reflected back as fast waves. Consequently, lower hybrid waves
must be launched with a refractive index nl larger than the maximum value of na along its
propagation path.
plasma is nax
=
In C-Mod, the maximum value of the accessibility limit (4.14) over the
3.0
Launching
In the range of frequencies under consideration (wavelength of the order of a few centimeters),
the antenna structure is an array of waveguides. With an appropriate relative phasing of the
waveguides, it is possible to choose the refractive index nil so that the launched waves are
unidirectional and with nil > njj. [24]
However, close to the plasma edge, the density vanishes and the limiting form of ni1 for the
LHW is
ni ~- (1 - ni) + In
-
(4.15)
showing that there is a cut-off (n 1 -* 0) at the edge for nl ;> 1, which occurs at the plasma
frequency (wp = w) LHW must therefore be launched very close to the plasma edge so that
a significant fraction of the power can be transmitted through the cut-off by tunnel effect.
Consequently, the launching of LHW requires that the antenna be close to the plasma, which is
an issue for reactor-scale tokamaks (antenna deterioration).
LH resonance
Lower-hybrid waves are subject to the lower-hybrid resonance where w -*
WLH.
However, in a
hot plasma, LHW would not reach the resonance layer but be rather mode converted to a kinetic
ion plasma wave (IPW) and eventually a ion Bernstein wave (IBW) [31]. These waves would in
fact be damped on ions, which would be a process for ion heating [3]. In order to avoid such a
mode conversion, LHW in LHCD are launched with a frequency sensibly greater than the lower
hybrid frequency everywhere in the plasma (in C-mod,
WLH,max/7
=
2.2 GHz so that w >
2
WLH
everywhere in the plasma).
Landau damping and spectral gap problem
LHW can be subject to a significant wave-particle interaction if the Landau resonance condition w
=
kllv 11 is satisfied for a large number of electrons.
51
Indeed, a condition for significant
linear electron Landau damping (ELD) can be derived from the imaginary part of the dielectric
dispersion relation (see [1]) and leads to an energy damping condition
2/
where
AXD
minjj
iX3ee
me c
2e
eAxD
1
is the radial extent of the expected deposition profile and XOe
C-Mod at r = 0.15 m (see Appendix A), (4.16) with AXD
(4.16)
= C/(V"fnlIVTe).
1 mm gives nii
In
3.35.
However, the experimental refractive index at launching, nr1 = 2.75 (A.2), is below the
spectrum peak as calculated above and even below the accessibility condition at r = 0.15 m
(na = 2.9).
Therefore, an unexplained upshift of the refractive index must occur along the
propagation, referred to as the spectral gap problem.
Quasilinear operator for LHCD
With frequencies w
< Q, LHW are only affected by the Landau resonance n
=
0 in (2.37). The
LHCD power spectrum presents complicated features that are determined by the launching structures and the wave propagation. For instance, there is always a fraction of the power launched
in the opposite direction from the desired one, because the phase shifting of the waveguides is
unable to produce a purely unidirectional spectrum. To simplify the problem, it is assumed
that the spectrum can be considered as being flat between two limits, with a lower bound ngI,min
of the n11 spectrum is given by the accessibility condition (4.14), and the upper bound njma
is determined by the damping condition (4.16). Another approximation made to simplify the
problem is to assume that the LH power has a uniform poloidal repartition. In fact, a realistic
poloidal repartition of the power requires ray-tracing techniques not available for this study.
These two assumptions are clearly questionable in the prospect of considering the global
current drive problem. However, there are acceptable for the study of the LHCD mechanisms
at a given flux-surface. The derivation of the quasilinear operator for LHCD is detailed in the
Section C.3.
4.2.2
Simplified 1-D Analytical Model for LHCD
In the case of LHCD and in the non-relativistic limit, the quasi-linear diffusion operator 2.37
can be rewritten as
Q(f)
=
( Di (nr)
09vI
(V
52
09f
(4.17)
where the dependence upon vjj in the non-relativistic diffusion coefficient D (jnr is kept to treat
accurately the case of weak diffusion Do < 1:
(4.18)
DLH(vII) = m2D H VTe
Vil
with DLH given by (C.21).
We introduce the assumption that the dependence of the electron distribution upon the perpendicular velocity is a maxwellian at temperature T. This approximation is very questionable
because it neglects the increase in perpendicular temperature due to pitch-angle scattering, which
can be a dominant process, especially at high Zi.
fe
=
(4.19)
fN(v±-) f(vII)
the collision operator is greatly simplified and reads [33]
C(f) =
v(vI1)
Ef +V2
(4.20)
e f
where
(4.21)
v(vii) = ve(2+ Zi)
V'3
Introducing the dimensionless velocity
=V
= V(1ma
V2
=min
V,1
4.22)
VTe
VTe
VTe
the Fokker-Planck Equation (3.2) for the electron distribution function becomes
vfD(jdo
+
_f
(2 + Zi
+ 1
-f = 0
dv
(4.23)
and can be integrated:
j V
f(v) = f(0) exp
Idv'
[01 + Do/(2 + Z,)V'2
(4.24)
For v < v, Do = 0 (C.20) and the integration reduces to
v'dv' = v 2 /2
(4.25)
For v, < v < v 2 , Do / 0 so that
J
(2 + Zj) + Dov2
dv'_
V11min
1 + Do/(2 + Zi)v' 2
(2+Z )/2DO
(2 + Z2 ) + Dov2
For v > v 2 , Do = 0 and the integration reduces again to
v'dv' = v 2 /2
oVImax
53
-
Vm
/2
(4.27)
Using (4.25)-(4.27), the electron distribution function becomes
Sf(0) exp(-v
2
for v < vi
/2)
Dov 2 ])(2 +zi)/2 Do
f(0) exp(-v /2)([(2 + Zj) + Dov2]/[(2 + Zj) +
f
f(0) exp[(vi
for v, i v
v1)/2]([(2 + Z) + Dovf1/[(2 + Zi) + Dov ])( 2 +zi)/2 Do exp(-v
-
which in the case of strong RF fields Do
f =
for v < v,
f(0) exp(-v'/2)
for v,
-
/2)
for v2 ! V
(4.28)
> (2 + Zi)/2 reduces to
f(0) exp(-v 2 /2)
f(0) exp[(v2
2
2
v2)/2] exp(-v
/2)
for V2
(4.29)
<
5 v2
<
V
In the case of strong RF fields (f given by (4.29)), the normalized current density and the
normalized density of power dissipated by collisions (which in steady-state equals the density of
power absorbed from the waves) can be estimated:
fM
fV V2
eneVTe
e-
P 2V2
neMeVreVe
(2 ±zi)Ji 1
V2
- CQf )
fMv)~
V 2 _2
dv
(.0
(2±Zi)fm(v)In(V2
V1
dv
+(dv
(430
V-
2
V1
J
2ve
vdv
fM(Vl) f
fIVJ(v~dv
(4.31)
An analytical formula for the LHCD internal figure of merit can therefore be derived from
this ID model:
g(1D)
_
e
ITI/D
2
mevTeve
V
)
2(2 + Z )ln(v2 /v1)
A more accurate calculation, using the adjoint technique [11], gives a similar expression for
77 with a different multiplicative factor:
Ad
mevieve/e
2
-
(4.33)
5 + Zi ln(v 2 /v1)
This expression for the internal figure of merit, larger than the 1D result by a factor 4(2 +
Zi)/(5 + Zi), increases with Zi from 2 for Zi = 1 to 4 for Zi ->
oo. This difference confirms
the importance of the T 1 increase due to pitch angle scattering, which is neglected in the ID
model. The higher efficiency is explained by the fact that the resonant electrons with higher TL
are more energetic and therefore less collisional. The pitch-angle scattering of electrons increase
with Zi, which explains the dependence upon Zi in the factor q Adj /
4.2.3
7
(1D)
Numerical Calculation of LHCD
LHCD distribution function
The distribution fo has been calculated for the case of LHCD in C-Mod at r
=
15 cm (Appendix
A). The conditions for accessibility (4.14) and strong damping (4.16) give the spectrum limits
54
2
nlfmin= 2.8 and nfjm.a
= 4, which correspond in velocity space to
Volma =
6
VTe
and Vomi
=
3
.5VTe
according to (C.22). The normalized diffusion coefficient (C.21) is taken to be DLH = 1. The
distribution is plotted on Fig 4-1. The graph above is a 2D contour plot of the distribution.
The contour lines represent the values of a relativistic maxwellian at p = 1/4, 1/2, 3/4, -...
The
graph below shows the parallel distribution function FO, defined in (2.76).
The features of LHCD seem to be well described by the consideration of the LHCD plateau
between pi = vi/'y(vi) and
P2 = v 2 /y(v
in the parallel distribution Fo. This plateau creates
2 ),
an asymmetry in parallel direction and drives the current. The flux surface averaged LH driven
current calculated by the code is
(j
RF)) = 10.9 MA/M 2 . The normalized internal current drive
figure of merit calculated from the code (4.4) is
/ e
= 17.5
e/ (MeV2ve )
(4.34)
while the normalized current drive efficiency calculated from the adjoint technique (4.33) applied
to the present case gives 7Adj = 14.6. The adjoint technique, although it is closer to the numerical
result than the simpler ID model, remains somewhat smaller than the 2D numerical result. The
difference can be explained by the fact that the adjoint technique assumes a small perturbation
of the distribution, with respect to the maxwellian, which is not true for high-power LHCD.
Benchmarking of LHCD calculations
LHCD calculations from the present code have been benchmarked using the results presented
by M. Shoucri and I Shkarofsky in [28] and obtained with a 2-D code based on the same model.
Using the LH parameters
VI1min =
4
VTe,
V11ma =
7
VTe
(4.35)
the results for the normalized current densities and the density of power absorbed are in good
agreement (Table 4.1)
4.3
4.3.1
Electron Cyclotron Current Drive
Electron Cyclotron Waves
Launching
The terminology Electron Cyclotron Waves (ECW) refers to electromagnetic waves with frequencies in the range of the electron cyclotron frequency and its harmonics. In C-Mod at R = RO,
55
Electron Distribution Function in LHCD
12
8
..............
..................................................
CD
CL
CL 4
.................................. .. .................
..............
0
-12
-8
-4
0
......
......... .............
..
..
.
12
8
4
Parallel Electron Distribution
..................
...
..
..
....
...
..
.....
..
....
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..
.
.I.
..
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.-.
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I.......................................
.....................
10-1
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......................................
........ ..............I......
............................
...............
LL 10-3
..
..............
..
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..
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..
..
.............................. ........
-I.
..
.......
..
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I.
..
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10-4
.
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5...........
.........................
10
.
.
.
.
............
.......
..
...
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..........
......................................................
.........................................
...........
.........
10 0
10-6
-10
0
-5
5
10
PI/pTe
Figure 4-1: Distribution function calculated for C-Mod (vI = 3.5, v2 - 6, Do = 1). 2D contour
plot of fo(above) and logarithmic plot of the parallel distribution FO (below).
56
DLH
T'
-(2)
J\
3
x10-
DLH
Te
Ve2e
keV
x10-
1
1.5
5.0
0.3
1.5
1
0.3
(1)
(2)
Pabs
Pabs
x10- 4
x10- 4
4.7
1.4
1.8
3.2
3.0
1.2
1.2
5.1
3.3
3.2
1.3
1.3
5.1
1.8
1.8
0.8
0.8
3
Table 4.1: LHCD calculations (current density j and density of power absorbed Pabs normalized
respectively to enevTe and
nemevreve)
by M. Shoucri and I Shkarofsky [28 (1) and from the
present code (2).
where B
=
BC = 4 T, the electron cyclotron frequency (4.6) is
fce = Wce/27r
= 112 GHz. Elec-
tromagnetic waves in this range of frequency have a wavelength on the millimetric scale and can
be directed and focused with mirrors like optic beams.
ECW are right-circularly polarized electromagnetic waves, and as long as they remain far
from cut-offs and resonances, they propagate in the plasma essentially unchanged from their
launching in free space (k ~-
klaunch.,
N = kc/w ~ 1). As a consequence, the launching of ECW
does not require any specific coupling structure (no waveguide array as for LHW) and their
propagation is easily predicted, and well controllable with mirrors.
Even with a fixed frequency w, the launching of ECW has many degrees of freedom (see figure
4-2) allowing simultaneously the control of the damping radial position (adjusting the poloidal
launching angle
#/T).
#p)
and the parallel refraction index N (adjusting the toroidal launching angle
Mirrors can be reoriented in a short time (less than a millisecond) for a dynamic profile
control. Under appropriate conditions (see Section 4.3.3), the wave power can be absorbed in
a single path, over a short distance. The easy control of the wave propagation, along with a
narrow deposition profile, makes ECW well suited for current profile control [12].
the generation of high-power ECW at these high frequencies (e.g.
f
-
However,
200 GHz for second
harmonic in C-Mod) poses challenging technical problems that have not yet been solved. Hence,
the calculations for ECCD in advanced scenarios of C-Mod are not presently realizable.
Polarization
Electromagnetic waves propagating perpendicular to the magnetic field are in two different modes
according to their polarizations [32].
57
R
OT
,
/0
(a)
(b)
Figure 4-2: Launching angles and propagation for ECW. (a) top view, (b) poloidal cross-section
" The ordinary (0-)mode is linearly polarized, with the electric field in the direction of the
constant magnetic field (E(s, t) 11So(i)). The 0-mode must propagate at a frequency
higher that the electron plasma frequency
Wpe
=
2
ne
rome
(4.36)
which is a cut-off for this mode.
" The extraordinary (X-)mode is in general elliptically polarized in the plane perpendicular
to the constant magnetic field (E(5, t) 1 Bo(X)). The X-mode is subject to the right-hand
cut-off
W,-
1
+
J+
(2J)2(4.37)
and encounters a mode resonance (k1 -+ oc in a cold plasma) at the upper-hybrid (UH)
frequency
2
Wuh
e
W2e
(4.38)
where it is partly converted to electrostatic Electron Bernstein Waves (EBW).
In oblique propagation (N11 4 0), polarizations are mixed and 0- and X-modes cannot be
separated. However, modes showing a predominant 0 or X type of polarization can be approxi58
mately called 0 or X, respectively.
Propagation in a tokamak
On Fig. 4-3 are displayed the radial frequency profiles in the plane 0 = 0 for ECW propagation
in C-Mod (ne and BT profiles in Appendix A).
Both the X-mode (left graphs (a),(c)) and
the O-mode (right graphs (b),(d)) are considered, with launching from either the low-field side
(LFS)(top graphs (a),(b)) or the high-field side (HFS)(bottom graphs (c),(d)). Since ECCD
occurs near the cyclotron harmonics (w ~~nwce), ECW propagation must be considered from the
edge of the plasma to the location of these harmonics.
The electron cyclotron resonance
f
= fce cannot be reached from the LFS by the X-mode in
C-Mod (Fig. 4-3, graph (a)), because of the right-hand cut-off. With the O-mode (graph (b)),
only two narrow low-density layers at the edges (where fpe <
fe or fe, > max,(fpe) is satisfied)
are accessible from the LFS. On the other hand, from the HFS, all radii are accessible with the
X-mode (graph (c)), and a large layer on the HFS (where fce >
f,e)
can be reached with the
O-mode (graph (d)). However, launching from the HFS can be technically problematic because
of the lack of room on this side.
The first harmonic (fEcw ~ 2fce) and higher harmonics are accessible for all radii from the
low field side with both the X-mode and the O-mode.
Absorption and choice of EM mode
The resonance condition (4.5) being satisfied, the diffusion coefficient (2.40), which determines
the absorbtion magnitude (through (2.84)), can be rewritten as
2
1
Jjnj+ 1 + Ek,_ ei(--)n+1Jan_1 + ELEk,II(-_1)nJni
Dk = q2 Ek,+e-a(-)n-
e
~P1
(4.39)
since the condition for having a significant number of resonant electrons (4.5) requires n < 0.
The Bessel function property J-n = (-1)nJn was used in (4.39).
It has been shown above that only the harmonics
InI > 2 are accessible from the LFS, which
for a compact tokamak like C-Mod is the only technically acceptable launching option.
The argument of the Bessel Function in (4.39) is
kjv 1 = -N_
(4.40)
MC Wce
where (2.26) and (4.5) have been used. Since ECW remain mostly unchanged from their characteristics in free-space, N ~ 1, then N 1 ,< 1. In C-Mod at r
59
=
0.15 m (A.1), resonant electrons
Xo ic 0-mode in C-Mod. Launching from LFS
X-mode in C-Mod. Launching from LFS
350
.2fce
300
2t-
300
250
250
r
ECW (X-mode)
200
a 200
150
150
100
100
50
50
0-.
-0.5
0.5
ra
ECW. (0-mode)
ffuh
0-_
1
0.5
(b)
(a)
X-mode in C-Mod. Launching from HFS
300
N
E
300
2t
250
250
200
N 200
ECW (0-mode
1:.
150
150
foe
100 ECW (X-mode)
100
-ce-
*
pe.--
50
50
0.. I--
0-mode in C-Mod. Launching from HFS
350
-0.5
r9a
0_.
0.5
-0.5
ra
0.5
1
(d)
(C)
Figure 4-3: Radial frequency profiles in the plane 6 = 0 for ECW propagation in C-Mod (Appendix A). ECW propagation is represented by a horizontal arrow (fEcw is fixed by launching)
from either the low-field side (top graphs (a),(b)) or the high-field side (bottom graphs (c),(d)).
The propagation of both the X-mode (left graphs (a),(c)) and the 0-mode (right graphs (b),(d))
is considered. The two first cyclotron harmonics fce and
2
fce are displayed. The shaded areas
represent the regions of frequencies unaccessible to the given mode, due to cut-offs (right-hand
cut-off f, for the X-mode, plasma frequency fpe for the O-mode) and resonances (upper-hybrid
resonance fuh, affecting the X-mode).
60
have pi/mc < 0.25 so that up to the fourth harmonic Inj = w/wee = 4, the argument of the Bessel
functions (4.40) is kiv±/LQI < 1 and therefore the Bessel functions are accurately approximated
by
(4.41)
Jn(z) ~ (z/2)"/n!
The 0-mode is characterized by an E-field polarization parallel to the magnetic field so that
the dominant term in (4.39) is Ek,1I. The 0-mode diffusion coefficient therefore reads
D(On) =
-k
21
P
/
2
21
qe2e
qe
j 1Jn~iiOE~i2
(|n|!)2 p2
LL)
N
21nI
W~n~_v_/)
,
|11i
Ek,11 1
(4.42)
mc2wce
The X-mode has an F-field polarization perpendicular to the magnetic field. The electrons
are resonant with the X-mode is the wave is circularly polarized in the same direction as the
electron gyro-motion, which is right-hand polarized. This condition appears in the expression for
the diffusion coefficient (4.39), where the dominant term, for a given harmonic n, is Ek,_ Jin 1because of the fast decrease of the Bessel function with Inj for small arguments (4.41). With
such a polarization, the X-mode diffusion coefficient is
/2(InI-1)
2
D(Xn) = q,|Jinll(kiv/E
2
[(Inj
- e_
1)!]2
w(4.43)
N mc 24c3
)
(4.
Comparing (4.42) and (4.43) with kIvi/Wcel < 1, the X-mode diffusion coefficient is larger
the one for the 0-mode at a given harmonic. In conclusion, the X-mode at the first harmonic
(X2-mode) seems to be the most appropriate choice in the case of C-mod.
Quasilinear operator for ECCD
Assuming a gaussian transverse extent for the magnitude of the ECW beam E-field is convenient,
since it implies a gaussian spectrum in the Fourier space. However, ray-tracing techniques would
be required for a more realistic spectrum, for the prospect of solving the global ECCD problem.
Such a study is beyond the aim of the present work, which focuses on the ECCD mechanisms at
a kinetic level. The derivation and bounce averaging of the quasilinear operator for ECCD are
detailed in Section C.4.
4.3.2
Mechanisms for Current Generation
EC resonance curves
The resonance curves, that give the region in (pu, pi) space where the resonance condition (4.5)
is fulfilled, can be studied as a first step in the analysis of quasilinear diffusion due to ECW. The
61
resonance condition is recalled:
w - kvll +
(4.44)
0
Wce
-y
with
PH =Y
+
=
2
(4.45)
Defining the parallel refractive index and the ratio of the harmonic frequency to the wave
frequency as
kgc
Ni =
nQee
y=
(4.46)
leads to the following equation for the resonance condition in (pl1, p±) space:
2
= (NI, '+
y)
(4.47)
which can be rewritten as
(1 - N-1)
MC)
N1 y
PH
y2
___+
mc
1
-1N
1
-
(4.48)
11
This equation gives an ellipse centered in pil/mc = Ny/(1 - N ) and p'
=
0. A significant
wave absorbtion can be expected when a large enough number of electrons are positioned on this
resonance curve, which occurs when the resonance curve is sufficiently close to the thermal bulk.
As seen on Fig. 4-4, the absorbtion should be enhanced by larger N11 (fixed by the launching
angle
OT)
and should increase as the wave approaches a cyclotron harmonics (y
= nWce/w -
1.
Particle fluxes and current generation
The differential operator (2.39)
+
m)
1ggymew
k 11p0pi1
+ --
-0
-
(4.49)
P
can be rewritten in (pll, p±) variables as
I(f)
=
P1
ymew
- 1
)
(4.If
(.50
The direction of diffusion is derived from (4.50) using (4.5):
Ap=_ p'
Api-
klp 1 /(-ymeW)
pii kl 11
/(ymeW) - 1
_
p_
N11
mc nwce/w
(451)
In the case of ECCD in C-Mod at a radius r = 0.15 m, resonant electrons have pi/mc < 0.25;
N11
-
0.3 is an optimal choice for ECCD (see Fig.
6-7) and w/2wce
~1 for the X2-mode.
Therefore, less than 10% of the momentum increase of an electron due to EC diffusion is in
62
EC Resonance curve (C-Mod, 2WJw = 0.9 7 , r=0.15rr
1-
I
=0.26
=0.28= 0.30
= 0.32
H= 0.34
=0.36
10
0.38
5
0
P Pte
10
15
EC Resonance curve (C-Mod, r=0.15m, N1 =0.34)
15
2..= 0.965
2Uo= 0.970
2Qo= 0.975
2W= 0.98)
2'w= 0.985
10k0
5k
SL.
-5
.
0
5
10
15
14/PTe
Figure 4-4: Resonance curves for C-mod parameters and varying NiI at fixed 2Q/w (above) and
for varying 2Q/w at fixed NII (below).
63
the parallel direction. The current drive by direct transfer of parallel momentum is therefore
negligible.
The particle fluxes generated by quasilinear diffusion (C.49) are displayed on Fig. 4-5. As
expected, the diffusion is mainly perpendicular. Moreover, a contour plot of the diffusion coefficient (dashed lines) shows the finite width of the resonant region, according to (C.42). Another
visible characteristic of the EC quasilinear diffusion is that its magnitude increases with p1, as
it is also seen in (4.43). In particular, there is no diffusion at p± = 0.
While the distribution function is affected by the EC quasilinear diffusion, collisions tend to
bring it back to a Maxwellian . The competition between steady Quasilinear diffusion, collisions,
and an energy loss process (radiation, radial diffusion) eventually defines a steady-state. Note
that in the present code, energy is not conserved as a truncated, linearized collision operator is
used (section 2.2.1). Therefore, no energy loss term is necessary to reach a steady-state. The
bounce averaged particle fluxes in steady-state, due to EC quasilinear diffusion and collisions,
are given respectively by (C.49) and (B.18). Adding them gives the total steady-state fluxes,
displayed on Fig.
4-6.
It is interesting to see that the quasilinear diffusion itself does not
generate any current, since it is essentially in the perpendicular direction. However, it creates an
anisotropy in the distribution function that generates an asymmetry in collisional fluxes. Those
asymmetric collisional fluxes are eventually driving a current.
Then, two processes affecting
particle fluxes are competing in the generation of current:
" The first effect is that the perpendicular quasilinear diffusion transport resonant particles
to a region in the (p11,pi) plane with higher pi. There is therefore a depletion region
under the resonant curve, which is quickly filled collisionally by surrounding electrons
(with lower pi)) since lower energy particle are rather collisional.
On the other hand,
there is an accumulation region above the resonant curve from which particles are only
slowly diffusing toward lower (p1j, pI) region because they are more energetic and therefore
less collisional. In steady state, there is a resulting accumulation of particles at higher pi,
which generates a current due to a change in resistivity. This effect is called Fisch-Boozer
effect [2].
" A second effect opposite to the Fisch-Boozer effect is due to the presence of trapped particles. Because of their fast bounce motion, trapped particle can transfer almost instantaneously (on the time scale rb <
0 = -(-.
Te,QL)
particles from the region (o = (fr to the region
As a result, there is a sort of short-circuit across the trapped region, which
64
RF driven normalized fluxes in ECCD
4
-5
3
2
a)
1
0
-1
tokamak = C-Mod
r=0.68
2n /w =O97
-
N ,,=0.34,
-2 -4
-3
-2
0
-1
1
2
3
4
P//PTe
Figure 4-5: Normalized particle fluxes due to EC quasilinear diffusion, as calculated in (C.49).
The length of the arrows is proportional to S/fo. The solid lines are a contour plot of the steadystate distribution function and the dashed lines are a contour plot of the diffusion coefficient
DEC exp [- (kji,res
-
022
65
Total normalized fluxes in ECCD
5
4
3
.. .. . .. .. .
. .~.
0)
I0~
H
0~
.. .... ..-
2
1
....
0
'..
..
-1
..
tokamak = C-Mod
rla = 0.68
2Q o 0.97
21
0 =0
B
-2
-4
NJ/I
-3
-2
-1
0
P//PTe
0.34
1
2
3
4
Figure 4-6: Steady-state normalized particle fluxes due to EC quasilinear diffusion and collisions,
as calculated in (C.49) and (B.18). The length of the arrows is proportional to S/fo. The solid
lines are a contour plot of the steady-state distribution function and the dashed lines are a
contour plot of the diffusion coefficient DEC exp[-(kli,res - k-,o)2
accelerates the diffusion of particles out of the accumulating region described above, therefore reducing the current. When particles are diffused to become trapped in only one side
of v11, the current thus generated is called Ohkawa current[21]. It can be larger than the
Fisch-Boozer effect and drives a current in the opposite direction.
4.3.3
Numerical Calculation of ECCD and Benchmarking
ECCD calculation parameters
For comparison, the tokamak parameters (density, temperature and magnetic field profiles) are
chosen to be close to the experimental conditions for LHCD in C-Mod (Appendix (A).
Without the help of ray-tracing, local wave parameters have to be assumed from launching
parameters. Far away from cut-off and resonances, the wave vector k can be assumed to be
roughly conserved along the propagation. Then, the local parallel index N11 can be geometrically
66
EC Resonance curve at midplane
(C-Mod, r=0.15m , [4= 0.34, 2UO= 0.97)
10
=0
= 45
8 -
90
6
180
.b
-5
0
10
Figure 4-7: Resonance curves with C-mod parameters versus
6
b,
15
which is the poloidal angle at
the position where the EC beam crosses the flux-surface under consideration (C.47).
calculated from rT,Op and R. The ratio nQ/w is directly related to the local major radius R
through the profile of Bo (2.9).
In order to simplify their interpretation, all ECCD calculations presented in this study are
considering that the wave is absorbed on the LFS midplane (Ob
=
0, See Fig. 4-2). However,
ECCD in experiments occurs at various poloidal angles (this is necessary for reaching different
radial positions with the same wave frequency). The poloidal deposition angle does not influence
the absorbtion from well-passing particles since their parallel velocity is almost constant in 0.
However, it can affect significantly the absorbtion by barely passing particles, since their parallel
velocity - which appears in the resonance condition - largely varies with the poloidal angle.
Therefore, it is possible to account for poloidal angles
9
b
$
0 by considering a deformation of
the resonance curve, as seen on Fig. (4-7).
ECCD distribution function
Figure (4-8) shows a contour plot (solid lines) of the steady state distribution obtained by
numerical simulation of ECCD in C-Mod, as well as the antisymmetric part of the parallel
distribution FO, which is given by (2.76). The C-Mod tokamak parameters are given in Appendix
(A).
The wave parameters are N11 = 0.28 and 2wce/w = 0.97. They have been taken so that
the driven current is maximum (with a positive N11). The dotted line represents the relativistic
67
maxwellian at temperature T, (Appendix A).
Each contour lines represents the value of the
maxwellian at a half-integer values of p. The accumulation of particles above and on the right
of the resonance curve appears clearly as well as the depletion below the curve. The enhanced
pitch-angle scattering due to the presence of trapped particles is also visible.
68
Effect of ECCD on fo
7
6
5
4
CL72
1
0
-6
-4
6 x 1U
0
.. . . .. . . .
. . . . .. . . . .. .
-2
........
-4
-6
.. .
.. . . . .. . . . ..
.
. .. . . . .. . . .
. .. . . . .. .
.
.. .
........
....
6
4
2
6
Antisymmetric Part of the Distribution
4
2
6
-2
-4
4
:
. .. . . . . . . . .. . . .
.
. . .. . . .
.... . . ..
. .. .
...
. .. .
:
. .. . . . ..
..
. . .. . . . .
..
.....
....
. .. . . . . .. . . . ..
...
. .. . . . . .. . . . ..
.
. . . . . .. . . .
I. ............
.... .........
.........
......
.
... ...........................
2
-2
4
6
// Pe
Figure 4-8: Graph above: Steady-state ECCD distribution function (solid lines) and Maxwellian
distribution at the same temperature (dotted lines). The dashed lines are a contour plot of the
magnitude of the diffusion coefficient DEC.
Graph below: Antisymmetric (part in pgj) of the
distribution FO.
69
Chapter 5
Bootstrap Current in a Maxwellian
Plasma
5.1
Simplified Picture of the Bootstrap Current
A simple physical picture of the bootstrap current can be obtained by considering the motion of
particles in the vicinity of a given flux surface located at a radius r (See Figure 5-1).
At zero order in the expansion (2.51) (considering no radial drift of electrons), particles are
staying on the same flux surface along their orbit. This flux surface being characterized by a
given temperature T(r) and a density ne(r), the electron distribution on this flux surface is a
Maxwellian
fom(r)
=
exp
)7
/
ne(r) exp(-#- 2 ) 2
3
47r(me 2) ,3tK 2 (/3# )
47r(Me C2 )3 tK2(O
where £
= 'ymec 2
Te
2
))
(5.1)
meTe(r)(1 +
is the total energy of an electron, K 2 is the modified Bessel function of the
second kind, and with
3
th
(5.2)
e2
me c
=
To first order in the expansion, trapped particles and barely passing particles are drifting
out of their (mean) flux-surface. As shown on Fig. 5-1, particles that belong to the neighboring
flux-surfaces r- and r+can be found now on the flux-surface of radius r.
Let us assume that the toroidal magnetic field
BT
is directed clockwise around the torus
vertical axis, and that the toroidal current I (generated by RFCD, neutral beams or inductive
methods) is directed in the same direction(Fig. 5-1). Therefore, following Ampere's law, the
70
--
BP
VD
r-
r+
V// < 0
'
BT,1
V//> 0
Figure 5-1: Characteristics of trapped electron motion in the poloidal plane.
poloidal field Bp is directed counter-clockwise in the poloidal plane (direction 9). Moreover, the
direction of particle drift for electrons is given by (sign(qe)B x VB) and therefore is directed
upward on Fig. 5-1.
Hence, on the low field side (0 = 0), poloidal field and electron drift are in the same direction.
This property is true as well if the direction of BT and I are both inverted. Therefore, to perform
a banana orbit, an electron on the inside of its banana orbit must be going in the direction of
Bp and then has
vj1
> 0, while an electron on the outside of its banana orbit must be going in
the opposite direction and has vil < 0 (as shown on Fig. 5-1).
As a consequence, on the given flux surface at radius r, electrons having drifted from the
more dense and hot inbound flux-surface r- have vjj < 0 while those having drifted from the
less dense and hot flux surface r+ have vj1 > 0. These electrons collide with passing electrons
at radius r and because of the asymmetry between the pressures at r- and r+, there is a net
transfer of momentum to passing electrons at radius r in the negative direction vjj < 0. Carrying
a negative charge, those electrons drive a current in the same direction than BT and I, called
the bootstrap current.
If the toroidal current I is in the direction opposite to the toroidal field BT, the direction of
the poloidal field Bp is inverted and so are the signs of vj1 in the banana orbits. The bootstrap
current is then in the direction opposite to BT and therefore also adds to I. in conclusion, the
bootstrap current is always a positive contribution to the toroidal current I.
According to this simple physical picture, the bootstrap current can be expected to increase
71
with the plasma pressure gradient. A larger drift excursion of the particles off their flux surface
should also enhance the bootstrap current. According to (2.15), this drift is proportional to the
bounce time and therefore increases if the poloidal field Bp is reduced. It is also clear that the
bootstrap current should be proportional to the number of trapped particles, which fraction is
of the order of
E (2.11). This simple physical picture predicts a bootstrap current
(5.3)
dc
1 dr
~"Bp
5.2
5.2.1
Numerical Calculation of the Bootstrap Current
Characteristics of the Distribution Function
The bootstrap current in the absence of RF fields can be calculated using the simplified set of
equations (2.2.4):
{C(gM)}
=
(5.4)
-{C(fM)}
where f is derived from the zero-order, maxwellian distribution function fo = foM (5.1).
fM --
Vil
(5.5)
rfom
The total distribution is
fm = foM + fim = foM
fm + gm
(5.6)
and the electron bootstrap current density is given by (2.68).
A contour plot of the total distribution function fM including effects from radial drifts is
shown of Fig. 5-2 (top). The solid line is simply the maxwellian distribution foM and the dashed
line is the total distribution fM including the first order correction flM due to radial drifts. As
described by the physical picture in section 5.1, there are more particles with a velocity vil > 0
than with a velocity vil < 0, and this asymmetry concerns mostly the barely trapped and barely
passing particles, which experience the largest drift off their flux-surface.
The parallel first order distribution
F1M = 27r
j
flM(pjipI)p± dp 1
(5.7)
is shown on Fig. 5-2 as well (bottom). It is antisymmetric in pl1 as expected, and this asymmetry
generates the bootstrap current, according to (2.79). The case shown on Fig. 5-2 refers to the
distribution in C-Mod at radius r = 15 cm (r/a = 0.68).
The flux-surface averaged electron
bootstrap current density calculated from this distribution is 3.7 MA/m
72
2
tokamak = C-Mod
r/a = 0.68
Neoclassical Correction to the Maxwellian
........
4
2)3
. . .. . .. ..
..
...
..
1
. .. ..
.... .... ..
0-
-4
x 10-3
1
0.5
L:_
.. . .. ..
.. .
. . .. .. .. . .. .
... ... ... ..
....
....
.... ..
.. .
...
...
...... ..
...... ..... ..... ....
.... ... ..... .
... ....
.. ....
4
2
0
P//re
Parallel First Order Distribution
..............................
.......-2
. .. . .. .. . .. ..
. . .. .. . .....
......
....
..
......... .... ...
. . ..
a
.... ........
0
. . .. .. . .. .. . .. .. . . ... . . . . .. .
-0.5
-1
.........
...............
..... . .. ..
..........
-4
.. .. ..
-2
. . .. . . .
. .. . . .. .. .
.. . .. . . .. . .. .. .
. ...... . . .. . .. .. . .. .. . .. . .. .. . .. .. . .. . .. .. .... . . .. ..
.. . .. ..
..... ....
.............
Pire
2
4
Figure 5-2: Electron distribution function to first order in the neoclassical corrections. Top
graph: Maxwellian foM (solid) and corrected f0M + f1M (dashed) distributions. Bottom graph:
parallel first-order distribution F1M, calculated from flM using (5.7).
73
Bootstrap current in CMod without RFCD
7
6 -
-
-
- -
.. .
-
DKE code
-
Sauter
Hirshman
E
5
0
0.2
0.4
0.6
0.81
Normalized radius: r/a
Figure 5-3: Bootstrap current calculated from the transport coefficients by [30] (dotted line),
[181 (dashed line) and the code (solid line) for C-Mod parameters (Appendix A).
It is interesting to note that the neoclassical corrections to the maxwellian is not described entirely by their projection on the parallel direction. Indeed, the correction affects mainly particles
with a significant perpendicular velocity, in the vicinity of the trapped region. The perpendicular velocity affects only slightly the value of bootstrap current (reduced because of relativistic
effects) but can be of great importance in the study of the synergism with RE fields, particularly
in the case of ECCD where a 2D description is necessary.
5.2.2
Benchmarking Calculation of the Bootstrap Current
The numerical calculation of the bootstrap current can be compared to the analytic result from
the transport calculations by Hirshman and Sigmar [30] and Sauter et. al [18]. Using those
models, the bootstrap can be derived from the temperature and density profiles assumed for
C-Mod (See Appendix A). The same profiles have been used with the code and the results are
shown on Fig. 5-3.
The current calculated from the code agrees well with the predictions of the fluid models. Note
that the contribution of ions to the bootstrap current, coming from the transport calculations
[30], is added (2.67). According to the analytical result derived in the Lorentz limit (Section 5.3),
74
the bootstrap current is maximum around r/a ~ 0.3 - 0.4, where the pressure is high and the
profiles are steep (since the bootstrap current is driven by temperature and density gradients).
Analysis in the Lorentz Model [13]
5.3
oo), pitch-angle scattering on ions is the dominant collision
In the Lorentz gaz model (limit Zi -*
process and electron/electron collisions are neglected. The collision operator is then simply given
by (B.12), which in the non-relativistic limit reduces to:
C(f)= 2 aa(1
[ -
-
(5.8)
Zive
(5.9)
2)
with the electon/ion collision frequency
Ze 4 ni ln A
3
4rm2v
Vei
=
being much larger than the electron/electron collision frequency ve, given in (2.35).
This simplification allows the derivation of an analytical expression for the bootstrap current
even in the kinetic description.
Starting from the zero-order, non-relativistic maxwellian, which is the limit of (5.1) for 3
Ith
-+
1,
-+
0:
n
oM
3/2
(5.10)
7)
exp(
we can calculate f:
fM
Vr dInn,
~ &fo
ag
Q
-
r
o
-
dr
+ (
MeV22
v
2Te
3 dInT~
_)_
2
dr
AM
(5.11)
I
The bounce averaged Lorentz collision operator (5.8) applied to fm and g have been derived
in [27] and reads
{C(fM)
{C(g
veiH(o I=
-
O)
1 '9 [(1 -
+
(V,
OfoM)
(5.12)
02)A (1
where the bounce average coefficients A and A are given respectively by (E.18) and (E.38).
As explained in Section 3.3.3, gM
to be finite at
=
0 for trapped particles. If we require in addition
O = ±1, the solution of (5.4) becomes
0
'o
(27]
91 = -- HV(|0| - GT)(G - GT)(1 + IfM
4 OT
75
'9M/
QO,min
or
(5.13)
Bootstrap Current in C-Mod
Lorentz
-
-- - Sauter
Hirshman
6
--
E5
-...
0
0.2
-..
0.4
0.6
Normalized Radius r/a
0.8
1
Figure 5-4: Bootstrap current calculated from the transport coefficients by [30] (dotted line),
[18] (dashed line) and the result in the Lorenz limit (2.67) (solid line) for C-Mod parameters
(Appendix A).
The electron contribution to the bootstrap current can then be evaluated from (2.67):
_
1.46.
J..s
nfeTe (dln ne
dr
or.By
+
dln Te
(5.14)
dr/
Confirming the simple physical picture (5.1), the bootstrap current increases with the fraction
of trapped particles
fi and
the pressure gradient dpe/dr. It is reduced if the poloidal field is
made stronger. A comparison of the bootstrap current calculations for C-Mod (at Ze = 1),
between the results from Hirshman, Sauter and the analytical result (2.67), is presented on Fig.
5-4. It shows that in spite of the limit Z,>> 1 and the approximations made in the derivation of
(2.67), the profiles for the bootstrap current are similar. However, the current predicted in the
Lorenz limit is about 15% too large.
76
Chapter 6
Self-Consistent Calculation of RFCD
and the Bootstrap current
In this section, RFCD and bootstrap current are calculated self-consistently, in the sense that
the modification of the radial drift corrections
fi
due to RF fields is taken into account, by
solving the full set of equations (Section 2.2.4). However, there is no self-consistent calculation
of the global currents and power dissipated, which requires ray-tracing and absorption codes
to calculate the evolution of RF fields. In addition, a self consistent calculation of the plasma
equilibrium is not carried out, since the poloidal field is determined by a toroidal current that is
not calculated by the code but taken from tokamak operation scenarios (e.g. [9] for C-Mod).
Bootstrap Current in LHCD
6.1
6.1.1
Estimation of the synergism for a realistic scenario
The calculations presented in this section describe LHCD in C-Mod at r
=
0.15 m (r/a = 0.68),
as proposed by an advanced operation scenarios [9]. As explained in Section 4.2.3, the LHCD parameters are chosen to match the conditions for accessibility and strong Landau damping (4.16).
They are also consistent with the launched power spectrum (A.2), when an upshift in nil, found
by ray-tracing techniques, is considered. These parameters are Vilmin
=
3
.5VTe,
Vlmax
=
6
VTe
and
Do = 1. All values displayed for the current densities are flux-surface averaged (Appendix D.2).
The bootstrap current calculated without LHCD (Section 5.2.1) is (J,(s)) = 3.7 MA/M 2 .
The current calculated for LHCD without first order corrections (Section 4.2.3) is (J(LH)) = 10.9
MA/m
2
. These values agree with the results obtained by S. Schultz [27] ((JLH)) = 10.6 MA/M
77
2
and (J 1"e))
-
2.9 MA/m 2 ). Note that the results presented by [27] for the bootstrap current
do not include the ion contribution
(J BSi)) -
0.8/ MA/M 2 .
The self consistent calculation of LHCD with the bootstrap current gives a total current
(J,(O')) = 15.2 MA/M 2 . We can thus evaluate the synergistic current
(J(S))
_
(BS))
(LH)
J(TOT)
2
0.6 MA/m
=
(6.1)
which is slightly larger than the result found in [27] ((J s)) = 0.44 MA/M 2 ) The fraction of
LHCD current supplied by the synergistic current remain small in this case
(J(s))
(6.2)
5.3%
(OT _(BS)
More relevant for the LHCD problem is the synergistic effect on the internal figure of merit,
given by the difference between (2.85) and (2.86). The densities of power absorbed found in the
numerical simulations are respectively
(p(LH)) = 6.2 MW/M 3
(p(OT)) =
6.9 MW/m 3
(6.3)
for the LHCD problem only, and for the self-consistent LHCD+BC calculation. It is found that
the density of power absorbed is larger when the bootstrap current is present, so that the internal
figure of merit is unchanged:
7(TO)
= 1.7 Am/W
((LH))
Therefore, in the present case, relevant to future experiments in Alcator C-Mod, there is no
synergistic effect on the internal figure of merit.
6.1.2
Physical picture of the synergism
On Fig. 6-1 is shown the parallel first order distribution (2.77)
F1 = 22r
j
f(pl, pi)pidpi
(6.5)
in the case of a Maxwellian distribution (fiM, solid line) and in case of LHCD (dashed line).
The asymmetry in F determines the bootstrap current (2.79) in the non relativistic limit.
With LHCD, a plateau is raised in F in the region of quasilinear diffusion. This observation
leads to a first interpretation of the synergism: From the 'bootstrap current' point of view, the
synergism is due to the LH quasilinear diffusion affecting the first order distribution function
which therefore drives an enhanced bootstrap current.
78
fi
1.5
63
1 -
First order parallel distribution
Bootstrap only
Bootstrap with LHCD
0.5-
LC'
-0.5
-1
-1.5
-10
5
0
-5
10
T//e/Pe
Figure 6-1: parallel projection of the first order distribution function, with (dashed line) and
without LHCD (solid line).
On the other hand, it can be interesting to consider the synergism as the effect of fi (first
order distribution in the drift expansion, which generates the bootstrap current) on LHCD. In
Fig.6-2, the dashed line represents the parallel electron distribution function for LHCD only F
(2.76). The dotted line is associated with the total distribution calculated non self-consistently.
It is the sum F + FM of the distribution for LHCD only F and the first order distribution
"bootstrap only" F1M (2.77).
At last, the solid line represents the total distribution function
F = Fo + F (2.71) calculated self-consistently.
According to (2.80), the synergism is then given by the shaded area between the solid and
the dotted line. This leads to another interpretation for the synergism: The asymmetric first
order distribution function fi brings more electrons to the maxwellian at velocity v 1 . Since the
height of the quasilinear plateau is fixed by this value of the distribution fm at velocity vi, there
is an enhancement in the height of the plateau, which corresponds exactly to the amount of
additional electrons brought by flM, as seen on Fig. 6-2.
In fact, the shaded area of Fig.6-2 is exactly the same as the area between the solid and the
dashed lines on Fig. 6-1, which confirms that the two interpretations for the synergism (from
the bootstrap or from the LHCD point of view) are equivalent.
79
Parallel Electron Distribution
5 x 10.4.5-
ElM
4-
a,
3.53-
N
2.50
-I
~~~---
2-
1M
-
-
-
1.51 0.5%
U
9
3
4
x 10 -3
5
Parallel
7
/P6
8
P///pTe
lectron Distribution
F=+
1.9
---- F=
nsc 0 +
1M
1 .8
.7
NO
.6
.5
011 .4
.
.....
C
01 .3
1 .2
-.
3
3.5
4
45
p
5.5
6
6.5
7
Figure 6-2: The bottom graph is a zoom of the top graph in the region of the quasilinear plateau.
The dashed line represents the parallel electron distribution function for LHCD only Fo (2.76).
The dotted-dashed line is the first order, "bootstrap only" distribution function FM. The dotted
line is associated with the total distribution calculated non self-consistently Fnsc = Fo + F1M.
The solid line represents the total distribution function calculated self-consistently F = FO + F
(2.71). The "steps" in the line contain no physics and are simply due to the numerical change
of variable (p,)
-*
(p, pi).
80
This ID picture of the synergism is consistent with the fact that no enhancement of the
internal figure of merit is observed, because the synergistic current, associated with an increase
in the height of the plateau (Fig.6-2), is generated at the same localization in p1l as the LH
current.
6.1.3
Prediction of the synergism in the Lorentz model
As seen on Fig. 6-2, the synergistic fraction of current seems to be given by the extra elevation of
the plateau due to the additional electrons brought at v
=
vi by fiM. Therefore, S. Schultz sug-
gested in his thesis [27] that the synergistic contribution to the LHCD current can be estimated
by the fraction
j(S)
11
-
F1M(Plmin)
- PImn
(6.6)
FOM(PlImin)
j(LH)
using the "bootstrap only" parallel distribution function FM given by (2.77)
pi(fm -- gM)dpL
Fim(pgj) = 27r
(6.7)
where fm and gm have been evaluated in the Lorentz model (5.11) (5.13), so that (6.6) becomes
F(plimin)
FM(||min)
dlnTe
1
~- VCpo
2
P11min 3
dr
If the temperature variations dominate (d in Te/dr
(6.8)
PTe )
> d In ne/dr), then the synergistic frac-
tion predicted takes the following form:
6.1.4
(S)
j(B)
J(LH)
en,3VTe
1
3
VT,
Test of the Predictive Model
According to the approximate model described above, the synergistic current is expected to
increase proportionally to the bootstrap current, and with a factor (vllmin/vTe) 3 having two
origins according to the expression (5.11) of fM: a contribution
(VII VTe) 2
derivative of a Maxwellian (term mev 2 /2T in (5.11)), and another
(VII
comes from the radial
/VTe) is present in the
3
expression of the drift (2.2.2) and thus in im (2.58). The resulting factor
(vjjmin/vTe)
at v11 = vI1min, since the elevation of the plateau is fixed by
as seen on Fig. 6-2. In
FM(VIjmin),
is evaluated
other words, the first-order distribution flM decreases more slowly than the maxwellian, so that
the synergism is larger when LH current is driven further in the tail. However, if the resonant
electrons are far in the tail, only a little current is driven.
81
In this section, a test of this predictive model is performed in various situations by modifying
the LHCD parameters. All situations are related to LHCD with C-Mod plasma parameters at
radius r = 0.15 m. (Appendix A)
First is considered the dependence upon
Keeping all other parameters fixed
(VIjmin/VTe).
according to the operation scenarios for LHCD in C-Mod (vmax = 6, Do = 1, 1p = 0.7 MA),
Vj1min
is let free to vary. Results are reproduced on Fig. 6-3. Currents are displayed on the top graph
and the dependence of the synergistic current upon
(VIlmin/VTe)
is shown on the bottom graph.
It is not surprising to see that the LH driven current decreases rapidly with higher
(VIImin/VTe)
since the number of resonant electrons drops according to the maxwellian distribution.
The synergistic fraction of the LH current is shown on the bottom graph of Fig. 6-3. It
increases with
and fits nicely an increase rate
(Vjjmin/VTe),
(VIjmin/VTe)
3
,
as predicted. However,
the actual value of the synergism predicted by our simple model (6.9), not shown on this graph
for clarity, is larger than the numerical result by a factor 2.
The influence of
(VIlmax/VTe)
(VIjmax/VTe)
is considered in Fig.
6-4.
with a parabolic evolution as expected (4.30).
current is independent of
(V11max/VTe)
The LH current increases with
The synergistic fraction of the LH
as expected from (6.9), but is lower than the predicted
value (6.9) by a factor of 2.
According to the predictions, the synergistic current should increase linearly with the bootstrap current. This dependence can be tested numerically by modifying the bootstrap current
independently from LHCD. This variations can be obtained by changing the total toroidal current Ip which affects the poloidal field Bp and thus directly the bootstrap current but not the
LH driven current. The currents and synergism are calculated both in the cases of high (Fig.
6-5) and low (Fig. 6-6) LH current compared to the bootstrap current.
In all cases, the predictions from (6.9) concerning the dependencies upon the LHCD parameters hold, whatever the relative scaling of the LH driven and the bootstrap current. However,there
is always a factor of 2 to 3 between the analytical and the numerical results, present in all the
simulations carried in this study. Possible reasons for such a difference are discussed below.
6.1.5
Discussion
It has been shown in the previous section that the parametric dependencies of the synergistic
current predicted analytically are well confirmed by the numerical calculations. In particular,
the synergistic fraction of the LH driven current increase linearly with the bootstrap current
and with
(VIjmin/VTe)
3
. This is true also when the bootstrap current is larger than the LH driven
82
r/a = 0.7
V//max = 6
Current Densities vs V//i /VT
lp = A0.7 MA
7
< (RF)
60
. . -. . .. .
. .. .-..
-
-<Jr >
-. <J IO)>
-.....-... .
-
50
. . . .. . .. .
- .-.
C~.J
E
. . .. .
-.
.. ..-
...
... ..
.-.. -.- --.
40
A30
-
.-
-
.-..
.. . . .- . . .
...
.. .-..
. . . . . . .. . .-.
V
20
-.- .
-
. . -.. -..
. . ..
.
. -.
10
..~ ~~~~~
~~
~
.
-- -
0,
4
3.5
V
/min
/V
4.5
5
Te
Synergistic fraction of LH current vs v.min /vTe
0.5
S>/jRF)
-
0.45
0.4
0.35
(H
-. -.
0.3
0.25
/
-.. -.............
.-.-.-
....
.....-........- ........-........-...
0.2
- . . .. . ... .. -. ..- -. .. . .. . -.. . .. . .. .-..
--
0.15
0.1
0.05
ffII
3
3.5
4.5
4
V//min
5
Te
Figure 6-3: Current densities in LHCD (top) and synergistic fraction (bottom) versus vin .
83
r/a = 0.7
3.5
0.7 MA
V//min =
lp
=
Current Densities vs V/I/vT
9R;,
<j (RF)>
20
-<jterOT)>
HI
C~j
E
15
.. ..ma
==
- -.
-
-
-
-
-
-
-
.'..
..
....... .
-
-
-
-
-
....
..
.... ... ......... ....
...
-
- .. - ..
- .
-
..-
..-
..-
...
-
- .. - .. - ..
V
5
4
4.5
5
5.5
6
6.5
7
Te
//smax t
Synergistic fraction of LH current vs v Hmx/V
0.:
2
<JS >/<j(RF)>
0.18--
-..
,..,,,.
)..
J B)>/(3 e n v ) (V/. ....
H
e VTe) (//min/VTe)
...
0.16- 0.14 -
-.. . . .. . . .. . .
...
. ..
....
.. .... .. . ...-
0.12
- -.
-
-.........
0.1
-...
.......
-.
.. .............
-
0.08
. . . . -... . . .- . . . . .
0.06
..
~~~~~~~~~
~~ ~
.
0.04
-
-.. ..- . .
0.02
4
4.5
5
5.5
/max
6
6.5
7
Te
Figure 6-4: Current densities in LHCD (top) and synergistic fraction (bottom) versus vlmax .
84
r/a = 0.7
V//mi
n =3.5
x= 6
Current Densities vs I
p
20
J(RF)
18
- J
16
.
.
.
.
.
.-.
.
>
<I
-.
T)
14
-
-~~< ->-
E12
)
-(
.......... : ......... e................
210
-
...
V
6
4
2
0.
0. 3
0.5
0.4
0.7
0.6
1
0.9
0.8
P
Synergistic fraction of LH current vs <J HI
0.5
.-- (S).><(RF)>
0.45
- -- - -
0.4
,,,, >/(3
~
fl
en v )V
~ ~
/
- -~e-
-
-mn/~
..
....
....
... ... ... ...
. ... ... .... ...... .... ... .....
. ...
0.35
. .- -..-..
..
-..
.. .- . .
... .
. .-..
. .. . . . .. .
0.3
..
. .-.
.-..
.. . . .
...
.
..
.
-..
.
-..
-..
-..
..
0.25
.....
I .........
. ... ....
. ... .... ... ...... ....I
0.2
...........
... ... .... .
. .... ... ..-
0.15
0.1
..-
-..-
........
-...
- -.
-.-.
..... . .
. ....
- -. -
0.05
0
3.5
4
4.5
5.5 (
5
6
6.5
7
7.5
//
Figure 6-5: Current densities in LHCD (top) and synergistic fraction (bottom) versus the bootstrap current in a case of high LH current .
85
r/a. = 0.7
V// min = 4.0
V// nax = 6
Current Densities vs I
P
10
(RF),
Jes
9
>
-J
8
-.
7
E
- --..-
. . ...-.-.
..
-.-.
-- --
--
-.
-s
"
-- ---.......-...--...-...-.--........--rT
-~~~< - - -.
-
6
5
:4
- -
-->
>
V
3
2
1
3
0.5
0.4
0.6
0.7
0.8
0.9
1
P
Synergistic fraction of LH current vs J(B)
0.5
<j(S)>/<j(RF)>
0.45
-- -
<J$)
<J
>/(3
(RF
e n VT)
//miV
)3
0.4
0.35
-
0.3
-~e
- - -....- .. .. . - .
-.
0.25
-.
-.. .. .
. .-
0.2
0.15
55
0.1
0.05
3.5
4
4.5
5
5.5
6
6.5
7
7.5
J(B)
//
Figure 6-6: Current densities in LHCD (top) and synergistic fraction (bottom) versus the bootstrap current in a case of low LH current .
86
current.
In fact, in the case of C-Mod with LHCD as described by operation scenarios [27] [9] (r = 0.15
m,
VIjmin/VTe
= 3.5, Vjmax/VTe = 6), a synergism of 6%, calculated numerically, is expected. The
analytical model predicts up to 12%, higher with a the factor 2 that is present in all simulations.
The origin for this discrepancy is due to either the ID interpretation of the synergism (6.6) or
the analytical approximations (5.11), (5.13) for the calculation of flM
=
M
-- gm in (6.7).
The ID approximation consists of saying that the synergistic fraction of the LH driven current
is given by
j(S)
______
J(LH)
Fim
-
lmin)
(6.10)
FOM(PIfmin)'
The error made then can be evaluated numerically. In the case of C-Mod described above, the
numerical calculation gives F1M(Pjjmin)/FM(PIImin) = 8%. In this particular case, one third of the
discrepancy can then be related to the approximation of a 1-D interpretation of the synergism.
Indeed, the mechanisms of both LHCD (high Ti) and the bootstrap current (see Section 5.2.1)
are affected by 2-D effects not accounted for in this simplified picture of the synergism.
The remaining factor 5/3 is related to the analytical evaluation of the bootstrap current and
of FM(plImin). A first correction can be brought in by adding the contribution of the logarithmic
density gradient in (5.14). In the case of C-Mod Te and ne profiles, this leads to a factor 4/15
instead of 1/3 in (6.9) and therefore the error factor 5/3 is reduced to 4/3.
The the bootstrap current calculated from the code for C-Mod at r = 0.15 m is (J BS)) -
3.7 MA/M 2 , close to the predictions by Sauter, et. al [18] (J sSaut)) = 3.8 MA/M 2 . The
analytical prediction in the Lorentz limit (5.14) gives
(JBSLor.)) =
4.6 MA/M 2 . (See Fig. 5-4).
The calculations in the Lorentz limit therefore give a bootstrap current too high by a factor
4.6/3.7 ~ 4/3 which explains the remaining difference factor.
6.2
Bootstrap Current in ECCD
The simulations presented here assume that EC driven current is expected to occur at a specific,
given radial position r (for example, for profile control). The poloidal deposition angle
9
b
clearly
has a strong effect on ECCD, since the fraction of trapped electrons strongly depends upon 6,
and vanishes on the hight-field side. We choose a deposition on the mid-plane (Ob = 0), which
is where the fraction of trapped particles is maximum, and therefore where the most significant
interaction between ECCD and bootstrap is expected. Moreover, only the X2-mode is considered
(see discussion Section 4.3.1).
87
Density of EC Driven Current
MA/m 2
1
2
0.995
1.8
1.6
J< 0
0.99
(Negative Ohkawa current dominant)
1.4
0.985
11.2
3
CS
6 0.98
0.8
0.975
J ~0
.
(No absorption)
0.97
-0.4
0.965
0.9G ,
0.2
1
0.3
0.2
0.4
0
N/
Figure 6-7: ECCD in C-Mod at r
=
0.15 m versus N11 > 0 and 2Q/w. Where N11 and 22/w are
too small (lower left corner), the resonance curve (4.48) is too far away in the tail and very few
electrons are resonant with the ECW so that no current is generated. On the other hand, where
Nil and 2Q/w are too large (upper right corner), the resonance curve is too close to the trappedpassing boundary and the Ohkawa effect reduces the current generated and eventually becomes
predominant. Between those limits, ECCD can be optimized, including when one parameter (NiI
or 2Q/w) is fixed.
Then, only two launching parameters
(OT
and w) can be adjusted to deposit the current
at the expected radius r. They are fixing respectively the parallel refractive index N11 and the
ratio y
=
2Q/w. By defining the resonance region in momentum space (Section 4.3.2), these
two parameters are affecting both the absorption magnitude and the competition between the
Fisch-Boozer and the Ohkawa effects. Since those parameters have comparable effects on ECCD,
one can be fixed and the other varied without loss of generality. Indeed, this statement can be
verified on Fig. 6-7 where it can be seen that current can still be optimized with one parameter
being fixed.
Then, the choice of fixing w is natural since it is experimentally much easier to modify the
launching angle (by rotating mirrors) than changing the frequency. Simulations are therefore
done with N11 as the only free parameter. Simulations have been carried for the cases of C-Mod
and DIII-D (See Appendix A).
88
Current Densities for ECCD in DIII-D
1.4
1 .2
RF
1.2
-
--
....
....
E
... .....
0 .8 -.....
A
V
0 .6 -.....
BS
-- --------------
0.4L 0 .4---
BS>
4#<
0B2
0.2 -
S
.-.
-.-- ..
r/a= 0.44
ni0=0
00
0.3
-
0-.96
B
0.32
0.34
N,
0.36
0.38
Figure 6-8: Current Densities for ECCD in DIII-D at r = 0.3 m.
6.2.1
ECCD with Bootstrap Current in DIII-D
On Fig. 6-8 are displayed the EC driven current (J 1 RF)) (dashed line), the (Nil independent)
bootstrap current without ECCD ((J
ergistic current ((J
BS))
BS))
solid line), and the bootstrap current with the syn-
+ (AS)), dotted line). The total current is then the sum of the dotted
and the dashed lines, and the synergistic current is the small difference between the dotted and
the solid line.
Concerning the EC driven current, it can be observed that it increases with N because the
resonance curve moves towards more populated regions (See Fig. 4-4), but only until a certain
point where the Ohkawa effect becomes important (as the diffusion region gets closer to the
trapped region). Besides, the synergistic current increases with N 1 , but remains a very small
fraction of the EC driven current for these experimental DIII-D parameters.
The synergistic current is plotted again on Fig. 6-9 (dashed line) where the solid line represents the increase AT in the bootstrap current due to the enhancement of temperature generated
by ECCD, and calculated by taking the second moment of the distribution function, as defined
in [16]:
AT=
f
3n b
2
1mev
e
2
2 (fo--fm)d 3p
(6.11)
Where both curves superpose could suggest that the synergism between ECCD and the
89
Synergistic current for ECCD in DIII-D
0.035
0.03
-
0.025
-
-
0.02---
rla =0.44
?0.015 nQ 0 /o
0.96
4
0 =0
0.015-B
0
0.3
0.32
0.34
N/1
0.36
0.38
Figure 6-9: Synergistic current for ECCD with bootstrap in DIII-D at r
=
0.3 m.
Bootstrap current is simply due to an enhancement of the bootstrap current due to electron
heating, at least for low N 11's. The difference between (J(S)) and (J
S)
increases with N11, as
the resonance curve gets closer to the trapped region. This suggests that the synergistic current
would be also subject to the Ohkawa effect.
On Fig. 6-10 is displayed the parallel first order distribution (2.77) without (FiM, solid line),
and with ECCD (F1 , dashed line). The dotted line represents FM,AT, first order distribution
without ECCD, but with an enhanced temperature derived from (6.11). It becomes clear that
the synergism is not only due to an increase in the temperature, which would lead to a uniform
increase (solid to dotted line) of F and therefore of the bootstrap current. On the contrary, the
synergism (solid to dashed line), is generated mostly by electrons in the tail. In the case of DIII-D
studied above, for which the bootstrap current is small compared to the ECCD, the synergistic
current is not a significant fraction of the EC driven current (less than 1% at the ECCD peak
N11 = 0.35). A larger synergism can be expected for a high-bootstrap current scenario, like in
Alcator C-Mod, which is the case considered next.
90
Parallel First Order Distribution
x 104
--
I
3.5 -
...
....
.
N6 RF
withRF
with A T =0.62 keV
3
2.5
LL-
2
.. . . . . . . .
1.5
1
0.5
. . . .
.-.
.
.
.....
. . . . . .
tokamak= DIII-D
rla =O0.44nni/o =n:0.96
,
-
I. .
. . . . ..-.
00
-
0
/B-
0.5
N
. ..
1
= 0.35
1.5
2
p/p
II Te
2.5
3
3.5
4
Figure 6-10: Parallel first order distribution without (F1M, solid line), and with ECCD (F,
dashed line), and also for the case without ECCD but with an enhanced temperature (6.11)
derived from fo in ECCD
(FM,AT,
dotted line).
91
Current Densities for ECCD in C-Mod
5
45
-
RE
__
--
Is
BS
4 - . !. < jB >+ JS
3 5. .
....
...... ....
.......... ...
...
...............
......
......
A
-,
V
-
-
B
0.5
%.25
.......
......
a2.5
.r=
0.68
2 n9 0/w=O.97
0=
1.5-
.................
.........
0.26
0.27
0.28 0.29
N//
d.3
0.31
0.3
Figure 6-11: Current Densities for ECCD in C-Mod at r = 0.15 m.
6.2.2
ECCD with Bootstrap Current in C-Mod
New scenarios for C-Mod [9] entail a large bootstrap fraction, up to 70% of the total current,
the remaining 30% being driven by RFCD. It would be interesting to determine if the synergism
between RFCD and the bootstrap can become more significant for these high-bootstrap scenarios,
eventually reducing the necessary amount of current to be driven. The simulations are carried
at r = 0.15 m, which is the location for RFCD in this scenario. On Fig. 6-11 are displayed the
EC driven current
(J RF))
(dashed line), the (NI, independent) bootstrap current without ECCD
((J(BS)), solid line), and the bootstrap current with the synergistic current ((JBS))
(S)
dotted line). Again, the total current is the sum of the dotted and the dashed lines, and the
synergistic current is the difference between the dotted and the solid line.
Unlike the previous case on DIII-D, the bootstrap current is larger than the RF driven current.
For the EC wave parameters leading to the highest EC current (NI = 0.28, 2Wce/W = 0.97,
according to Fig.6-11) and a hypothetical EC power on the flux surface of 10 MW, we find
JEC) = 1.7 MA/M2
BS)
= 3.4 MA/M
_
(EC)
2
jr(O)
= 5.5 MA/M2
(6.12)
which gives a synergistic current
(J(S)
((TOT))
92
BS)) =
0.4 MA/M
2
(6.13)
Synergistic current for ECCD in C-Mod
1.4
-- jJ
1.2 -
r = 0.15m
S
nQ/o = 0.97
A0=0
B
E
S0.8
0.4
-
0.2
.25
------------0.26
----
.27
028 0.29
N,,
0.
0.31
0.3:
Figure 6-12: Synergistic current for ECCD with bootstrap in C-Mod at r = 0.15 m.
The synergistic fraction of the driven current
(J(s) )
20%
(6.14)
( S)
(J EC )
is much larger than for the case of DIII-D.
The density of power absorbed is large in the present case, because of the strong Ohkawa
effect, but does not vary much when the effects of radial drifts are included:
(P(LH)) = 14.0 MW/M
3
(pTOT)) = 14.2 MW/m
3
(6.15)
so that there is a net synergism on the ECCD internal figure of merit, given by the difference
between (2.85) and (2.86):
77TOT) = 0.15 Am/W
7(LH) = 0.12 Am/W
(6.16)
On Fig. 6-12 are plotted the synergistic current (solid line) and the increase in the bootstrap
current due to the enhancement of temperature generated by ECCD (dashed line), calculated
by taking the second moment of the distribution function (6.11).
The argument of a synergism being due only to a temperature increase due to ECCD, which
seemed to hold in DIII-D, fails completely in the case of C-Mod where the increase in T can
explain only 10% of the synergism.
93
1.5 ,x 10-3
-No
Parallel First Order Distribution
RF
with RF
with A T= 0.10 keV
1
.. .....
tokamak =C-Mod
r/a - 0.68 -- nQ ho = 0.97
0
0=0
B
N =0.28
--
u--
0.5 k
.
.-
4--
0C
-
1
2
p/p
HI Te
3
4
5
Figure 6-13: Parallel first order distribution without (FM, solid line), and with ECCD (F1 ,
dashed line), and also for the case without ECCD but with an enhanced temperature (6.11)
derived from fo in ECCD
(FM,AT,
dotted line).
94
Effect of ECCD on First Order Distribution
A
3.5
3
tokamak =:C-Mod
r/a = 0.68:
ngoco =0:97
N
ttrapped passing
.28.
2.5
0
2
=
0
1.5
1
0.5
04
)
1
3
I
4
// Te
Figure 6-14: 2-D contour plot of the first order distribution fi. The dotted line (flM) is the case
of a maxwellian plasma (no ECCD); the solid line (fi) is the case with ECCD. The dashed line
is a contour plot of the EC diffusion coefficient.
Considering the parallel first order distribution on Fig. 6-13, it can be seen that unlike the
case of a simple increase in temperature (dotted line), the extra current due to the presence
of ECCD (dashed line) is driven mainly in the tail of the distribution fi. Indeed, the origin
of the synergism can be understood better by considering the 2D plot of fi (Fig. 6-14). The
dotted line (fiM) represents the first-order distribution, due to radial drifts, for a maxwellian
plasma (no ECCD), while the solid line (fi) is the case when ECCD is on. It appears that the
electrons described by this first order distribution are subject to EC quasilinear diffusion, in
a way similar to the Fisch-Boozer effect on the zero-order maxwellian (Fig.4-8). The resonant
region in momentum space (dashed contours) almost coincides with the maximum of the first
order distribution fi, leading to a large synergism. The synergism clearly results from a 2-D
interaction between ECCD and the radial-drift perturbed distribution fi.
95
Chapter 7
Conclusion
* General Considerations
A new two-dimensional code allows fast calculations of the steady-state solution of the drift
kinetic equation with quasi-linear diffusion and Fokker-Planck collisions. The trapped/passing
boundary conditions and the symmetrization in the trapped region, which are important
for neoclassical effects, are treated in a new way.
It has been shown in this study that the synergism between the bootstrap current and
both LHCD and ECCD could be a large fraction of the RF driven current, under the
condition of a large bootstrap current. In the case of off axis current drive in C-Mod [9],
which has been chosen for most scenarios used in this study because of the large bootstrap
current associated, a synergism of 6% has been found with LHCD, and up to 25% with
some hypothetic ECCD, which so far remains technically impossible because of the lack of
high power generators for 200GHz EC waves.
The mechanisms for the synergism have been investigated. In both the case of LHCD and
ECCD, the synergism can be understood as the effect of RF quasilinear diffusion on the
asymmetrically drifting particles (first order distribution
fi).
* LHCD and Bootstrap Current
In the case of LHCD, a simpler, 1D interpretation has been found to explain the synergism.
Indeed, the synergism can be related to the enhancement of the LH quasilinear plateau
thanks to the additional electrons with velocity v1 ..in, which are brought by the asymmetrical radial drift. A estimation of the synergism, based on this 1-D interpretation and on
the bootstrap current evaluated in the limit of a Lorentz gas, leads to a synergistic fraction
96
of the RF driven current given by
J(S) 111 j(BS) i ( V11min
J(RF)
enevTe
3
3
(7.1)
VTe
This estimation has been found to predict the numerical calculation with good accuracy.
The dependence upon the bootstrap current and upon
V1lmin
have been tested with an
excellent agreement. However, the prediction is constantly larger than the numerical calculation by a factor of 2. Several approaches have be considered to explain this difference,
which seems to be the accumulation of approximations due to both the 1D model for the
synergism and the Lorentz model for the distributions.
It is interesting to note that the synergism between LHCD and the bootstrap current
concerns the driven current and not the CD efficiency: the increase in the driven current
goes along with an increase in the power dissipated. The physical picture of the synergism
suggests an interpretation for this observation: if the synergism is due to an increase in
the number of electrons at v
= Vlmin,
it is equivalent to an overall increase in the density.
Since the CD efficiency does not depend upon the density, it remains constant with the
synergism.
" ECCD and Bootstrap Current
Although most of the features of LHCD and its synergism with the bootstrap current can
be well described considering the ID representation F(pj) only, the mechanism of ECCD
is essentially two-dimensional since the diffusion coefficient depends upon pi.
However,
since the corrections fi due to particle drifts have a strong component in the perpendicular
direction as well, there can be a strong synergism between ECCD and the bootstrap current.
Contrary to the case of LHCD, the synergism in EC driven current is not associated with
a significant increase in the power absorbed.
In ECCD, the current is not driven by a
specific portion of the distribution function as in LHCD (there is no plateau). However,
the synergism occurs well in the tail of the distribution (See Fig. 6-12) where electrons are
less collisional. The efficiency of the synergistic current is therefore higher than the overall
efficiency of ECCD.
* Suggestions for Future Work
This study can be extended to other tokamaks and scenarios. It would also be interesting
to use this code with results from ray-tracing calculations, in order to consider cases that
97
are closer to the experimental situation. Moreover, coupling to ray-tracing would allow a
global study of the synergism in the figure of merit, and therefore an estimation of the actual
additional synergistic current that could be driven for a given amount of input power. The
last developments of the code allow the self-consistent study of RFCD, bootstrap current
and radial transport, which also suggests to extend this work.
Moreover, it appears in the latest simulations that a strong positive current can be driven by
the Ohkawa method (OKCD), described in Section 4.3.2. It would be interesting to compare
the synergism with the bootstrap current for this case. Concerning the off-axis RFCD
problem, a comparison between OKCD on the low-field side (large fraction of trapped
particles) and ECCD on the high-field side (no trapped particles) can also be suggested.
98
Appendix A
Profiles for Tokamak parameters
A.1
Alcator C-Mod
The parameters chosen to simulate experimental conditions in C-Mod [23] are given by Table
A.1.
The radial profiles for the temperature, the density and the toroidal magnetic field on the
midplane are shown on Fig. A-1
The most recent advanced tokamak operation scenarios for C-Mod [9] rely on heating by
ICRF, and current drive with high bootstrap fraction, supplemented by LHCD peaked far from
the axis, around r = 0.15 m. At this location (r/a= 0.68), the plasma parameters are
Te(r) = 2.1 keV,
ne(r) = 1.8 x 1020 m- 3,
q(r) = 2.6
(A.1)
The parameters for LHCD are
fLHW = 4.6 GHz,
PLH = 2.4 MW,
no = 2.75,
C-MOD
Ro
0.68 m
a
0.22 m
Te0
7.5 keV
Tio
4.0 keV
neo
2.5 1020 m-3
BT
4T
Table A.1: On-axis parameters in C-Mod [23]
99
An= 0.25
(A.2)
Profiles in C-Mod (on midplane 0 = 0,n)
8
T (keV)
7-
6
B. (T)
5
3 -.--...
0
-1
(......-.
-0.5
0
0.5
Normalized Radius r/a
1
Figure A-1: Temperature, density and toroidal B field profiles in C-Mod [23]
A.2
DIII-D
The parameters chosen to simulate experimental conditions in DIII-D [19] are given by Table
A.2.
The radial profiles for the temperature, the density and the toroidal magnetic field on the
midplane are shown on Fig. A-2
We consider off-axis RFCD at r = 0.3 m. At this location (r/a = 0.44), the plasma parameters are
Te(r) = 6.1 keV,
ne(r) = 3.5 x 1019 m-,
100
q(r) = 1.5
(A.3)
DIII-D
1.67 m
a
0.67 m
Teo
7.8 keV
To
6.2 keV
neo
4.3 1019 m-3
Bt
2 T
Table A.2: On-axis parameters in C-Mod [19]
Profiles in DIll--D (on midplane 0 = 0,n)
8
7 -..... T '(kWV
-....
6
T (keV)
5
4
e-
3
2
BT(T)-
1
0
-1
-0.5
0.5
0
Normalized Radius r/a
1
Figure A-2: Temperature, density and toroidal B field profiles in DIII-D [19]
101
Appendix B
Collision Operator
The collision operator
C(f)
=
Cee(f, f)
(B.1)
+ Cee(fM, f) + Cei(f, fi)
includes a differential term that describes the collisions with the Maxwellian bulk and with ions
(Section B.1), and a integral term describing the effect on the bulk resulting of collisions with
the perturbed distribution function (Section B.2).
Differential term Cee(f, fM) + Cei(f, fi)
B.1
The collision operator describing collisions of the electron distribution f with the maxwellian
bulk electron distribution fM and with ions fi is given by (2.34)
Cee(f, fM) + Cei(f, fi)
B.1.1
=
2
(A(p) '
+ F(p)f)1 +
[(1
-
2)
(B.2)
Detailed Expression
The expression (B.2) shows that collisions affect the electron distribution
f
through three dif-
ferent physical processes:
9 Slowing Down, described by the force term F(p) = F(e)(p) + F()(p), which includes
effects of collisions with electrons and with ions. The expression for this force, normalized
to VePTe, is given by
-
Collisions with electrons
F(e)(p) = F1 + F 2
102
(B.3)
with
F1 =
47rf fm(p')%-
F2
4
I -Y
z
v (
=
dp'
[
(B.4)
) f p"p fM(p')dp'
and where p is normalized to the thermal momentum
PTe,
and where the following
definitions have been introduced
PTe
p
-y
- = sinh- 1 z
V
F(')(p)
v-2 Zi-'M
=
mec
2
V1+ z
I=
(B.5)
- Collisions with ions
(B.6)
* Diffusion in Momentum p, described by the diffusion coefficient A(p)
-
A(e)(p)+A()(p),
which also includes a contribution from collisions with electrons and with ions. A(p) is
normalized to the thermal diffusion coefficient vep
-
Collisions with electrons
(B.7)
F1 + F2
A(e)(p)
-
e
Collisions with ions
Ti
A(')(p) = 1Zime
3
V
(B.8)
mi Te
* Pitch-angle Scattering (Diffusion in c), described by the diffusion coefficient Bt(p) =
Bte)(p) + Btk )(p), normalized to
-
with
ep2
Collisions with electrons
(p) =
Bt
B1 +
(B.9)
B2
with
B1
= 4,7r
pfm (P') p21
v
10
('
2
4z 'y
B2
=
47r
p'fM(p
-
-
2
1
8y'y2 z2
I'z/2
Z* -
-'72
+ 8'y''z
I
-r
4z 2
2
pf2
6p 2
z
_
-z2)
3
V'
4
+
(-2
[
'
(4 + 2z')1
01 dp'
3-y2+ a
3
-Z+ 2z)
-...
2
-yp
2
z
dp'
(B.10)
Collisions with ions
Bt) (p) = -Z
2v
103
1-
1 me T
V
2
mi
Te
(B.11)
It is interesting to note that collisions with ions bring only a very small contribution (ratio
me/mi) to the slowing down and diffusion in p processes. However, the contribution of ions for
pitch-angle scattering is of the same order as the contribution from electrons.
B.1.2
Expression in the Limit of a Lorentz Gaz
> 1), pitch-angle scattering over ions becomes the dominant
In the limit of a Lorentz gas (Zi
collision process and the collision operator then reduces from (B.2) to:
C(
(f) =
Z 12 a ~_2)
2v
Va
[
p
_
-y2
(B.12)
2)!
q2
with
e 4 Zi 2 ni In A
1
'ei
B.1.3
4
23(B.13)
47re 2m2V3
Expression as the Divergence of a Flux and Bounce-Averaged
Operator
The differential part of the collision operator can also be expressed as the divergence of a flux:
(B.14)
Cee(f, fM) + Cei(f, fi) = -V ' §C
where the expression of the divergence operator in spherical coordinates is
V-. S
1_(p2
p2 ap
1P
_
(B.15)
(V
--
Therefore, the expressions for the collisions-related fluxes are
S
=
SC
-
p
(p)
p
+ F(p)fl
Bt p)
of
(3 6
A bounce-averaging of the collision operator is required in order to remove the dependence
upon the poloidal angle 6 and reduce the Fokker-Planck equation to a 2-D momentum space
equation. This operation is described in (2.23), and has been derived for the collision operator
in [27], for the case of fo, g and
f, which gives:
e For fo or g
{V
-
C(
=
o())
2S
104
-
1-
d2,A0O))
(B.17)
where
+ F(p)f(0)
(p,o)
(p)
A
SCO )
(B.18)
1
1 -
9 For f
1025C'(
)
I
(
( -
2A5 C'(4))
(B.19)
where
5f' (0)
c,(0)
>
A(1±E)
H( oj
A(H+±E)
A (p)
G)
'
±
F(p)f( 0 )(pjo)]
1
O) (p,
> OT) B
)
0 1()(p,_o)
(1+ E) o (o
o
2/2)
2
0)
(B.20)
The normalized bounce time A is defined in (2.22) and the bounce-average coefficient A is defined
as:
(B.21)
Both coefficients are described in Appendix E.
B.2
B.2.1
Integral term Cee(fM, f)
Detailed Expression
This term is included to ensure momentum conservation in the collision operator.
For this
purpose, only collisions with the first, dominant term in the Legendre polynomials expansion
need to be considered.
Cee(fMf)
=
U1(fM, fl)
(B.22)
with
I1(fM, f)
Ve fM(P) 14 7
fi =
j
105
) + I1,1(fi) + I1,2(f1)]
d
(B.23)
(B.24)
and
Ii,i(fi)
[
P3
47 o f
T(P)Y P2
2
+ Im'2
1'
+
5
3~Yy~y
+2
P2
0fi(P')
1
z -
+2z
2
B.2.2
-I
15
+
+
-
- 1iz']
7'P2 +
-3
5)
2z )
+3
7F 15 +6)
dp'
(B.25)
]
z
I..
3+ 2z
[-3+-
+22
}
15
z
..
-.-.
7 -0
z
12zot2
+ 2z' ) ]
Z/
_2
+ 3+ z
-
+ z') ...
+ o'l (
2z' 2 I-3
h 1'
3'z
(
+
Z/
15 +6
12z
--
3
6z'
+
- 1)-
yU
47r
I1,2(fl)
+
-
3y-''
z- 5)
+
-
Py/2
.z]
...
dp'
(B.26)
Bounce-Averaged Operator
The bounce-averaging of the integral term is derived in [27] for fo, g and
I, which gives:
" For fo or g
{Cee(fM, f)}fo,g
{U1(fM, fi)}og -
oI1(fI
, fI))
(B.27)
where
f "o = 3 j
2
_
(of (0)d~o
(B.28)
{i
(B.29)
s* is defined as
s* = H(J(oj >
'T)
This bounce-averaging coefficient is described in Appendix E.
" For f
{Cee(fM,
f)}y
=
{U1(fM, fi)}j
=
H(oI > GoT)oI1(fM, f1(o))
(B.30)
where
fIo) 1=
3()
2
o9_1~ )~
(B.31)
* is defined as
s* = H(IoI> o)
and is described in Appendix E.
106
{}
(B.32)
Appendix C
Quasilinear
C.1
Operators
General Expression as the Divergence of a Flux
The quasilinear operator is defined as (2.37):
Q(f)
=
lim I
I
d3 k +0
(27r) 3 ~
2
2[
1-~)
Dk 6(w-k
1
v
-Q)f)
(C.1)
where £ is a momentum-space differential operator:
L(A)
(k9i
1&A
-09 + ±
12
=
=
0 (pA)
P2
YMeU
mA
-( -
-
(C.2)
A)
+
(C.3)
1(p
Let us consider a particular cyclotron frequency n. Using the second formulation (C.3) of
the diffusion operator, (C.1) becomes
Q(f)
V[lim
Sp
+
1
lim
d k 2(1
d3 k
(27r)3
(
k11p
2)
_
-(
Dk
6(w - kjvil - nQ)12(f) ...
p2(1 _
2) Dk 6(W
- kjjvjj - nQ) L(f)])
(C.4)
The resonance condition is expressed by the delta function
6(w - kiivil - nQ)
(C.5)
Using
'yme
Yme
(C.6)
the resonance condition (C.5) gives
klw
- nQ
VII
wyrne (1
Wpy(
107
(C.7)
Then, the quasilinear operator can be rewritten as
Q(f)
=
L'
[p2 (I _
2
)DRF(p,
(C-8)
)(f)]
with the differential operators
L'(A) =
L(A)
=
nQ0A
I
a(pA) +
(I 1
l18A
1 (
-2
+ -
- 2
(C.9)
_]
(C.10)
and the RF quasilinear diffusion coefficient
DRF (P
=
V--+oo
1
V
3
(2)
(C.11)
J(w - ki vj - nQ)
D
Clearly, the operator L' (C.9) has the form of the divergence of a flux (B.15), so that the
quasilinear operator can be expressed as:
V.
SF
(9
=
F
_1~~ V/
2(2RF)
(C. 12)
with
SRF
=
_p(l
_
SCRF
C.2
2
)DRF(p,
2
1
2
C(f)
DRF(p,
-
W
)L(f)
(C -13)
Bounce Averaging of the Quasilinear Operator
The quasilinear operator must be bounce-averaged in order to reduce the equation to a 2-D
momentum space equation. The bounce averaging operation, described in (2.23), removes the
dependence upon the poloidal coordinate. However, the effects resulting from the poloidal localization of the waves on the flux-surface under consideration must eventually be taken into
account during this operation.
First, a transformation is required for commuting the bounce-averaging operation with the
divergence operator. From (B.15) one finds
2
P~
ji(PtSPj)
1
1P
2
a(p2SP) a
Ap0I
2 p2
-
~(p2{SI) - Ap8_o
(
1-
V1 -T2
108
[
o
dO
aG
V1
_T2S
))
J
dO ar S
_0C27
/-6SC
(C.14)
The bounce-averaged quasilinear operator can therefore be rewritten as
{V. SF}
1
a(p2RF,(O)\
-2RF,
(0))
0A o
19SR~O
Ap aIol
P
P2 ap
(C.15)
with
SRF,(o)
isF
=
SRF,(O)
C.3
C.3.1
{S1 F
(C-16)
Nq SRF
Lower-Hybrid Diffusion
General Expression for the LH Diffusion Coefficient
In the case of LHCD, the wave power is absorbed through Landau damping n = 0. The perpendicular wavelength 27r/k± is much larger than the Larmor radius of electrons, so that
1
k v)
JO
(C.17)
and (2.40) becomes
DL DP
(C.18)
2I2~~j~
Therefore, the diffusion coefficient becomes
2
DpLH
p
C.3.2
1
3
V-+00 V
(27)3
V-o
V
q21Ek, 1I
2
6(w - kiloj - nrQ)
(C.19)
Flat E12 (kii) Spectrum
It is common to approximate the k11 LH power spectrum as being flat between two limits ki,min
and k 1l,ma, respectively determined by the accessibility condition and the linear Landau damping
of the waves (See Section 4.2).
Then, the LH diffusion coefficient (C.19) takes the simple form:
p2
DLH(p,)
=
DLH(
2 IVH(v, 1 - Vmin)H(Vma - V11)
pi lvii|
(C.20)
where
D LH(0)
0
=
rn
m -V-+oo
V
(27)3
q|Ek'1,0|2
e
(C.21)
VTe
and
wLU
Vmin = k
,Vm=
kl,max
109
max
(C.22)
k,min
Note that the
dependence in (C.20) is often omitted using the argument that D
VTe/vpI
> 1
so that the distribution is flattened anyway [28], [27]. This argument is not satisfactory in general
and this dependence should be kept in a numerical 2-D code.
A dependence of DLH(0) upon 0 is also necessary in order to take into account the poloidal
repartition of the LH power.
C.3.3
Expression for the Fluxes due to LH Diffusion
General Expression
Using (1
-
2)
= XIJ(1
), inserting (C.13) and (C.20) in (C.16) gives
-
2
_ {p(1
_
SLH,(O)
=
2
-
12
_2
p
2
q
e D(L)()H
{|D(LH)(O)H
=-pTe
SLH,(0)
)DLH(p, )Lf)}
_I
1-02
61
) DLH
2
1-
0{D
LH)(6)H
YPTe
Vminl H
-
16
[
Vin] H
Evmax
L(f)
(C.23)
I L(f)
-
t\ L
-D in] H
D LH)(0)H [
0
'Yme
'm
_
2
-
p|'{
vminl H
e -
7YMe
[vmax
-
7e
V
Lm
f)}
me /
L(f)
(C.24)
/Me
Approximation of Well-Passing Resonant Electrons
If the LH resonant region and the trapped region are far from each other in the momentum space,
which is often the case, especially for highly relativistic plasmas, the LH quasilinear diffusion
concerns only well passing particles. If those particles verify
to= -
6
< I
(C.25)
then
= Vr
(0) (1 -
02)
~
&p-6
(C.26)
so that the ratio
1- 6(' - 1)
can be taken as 1 at zero-order in 6.
110
J
(C.27)
However, the ratios
1
-
(C.28)
20-
and
1-
d
(C.29)
_
are not equal to 1 at zeros order and must be transformed according to (C.28) and (C.29).
Bounce-Averaged LH Quasilinear Operator
With this approximation, the bounce-averaging operation reduces to an average over 0 and the
bounce-averaged operator becomes:
e For fo or g
-10a
H)=
2
10
{gVLH
P(p2SLH,(O))
VI-72LH,(O)
a(\
(C-30)
where
_
SLH,(O)
SLH,(O)
6
af(0)(P, G) +
YPTe ID (LH)(O)(p,0)
AP
_
\0Op
}PTe sgn(Go)
1-
p
Do(p,
f 0)(AG
-(2
P0
f () (p,
+
(0)
Ap
))
(.1
0
(C.31)
The LH diffusion coefficient is simply averaged over 9:
D(LH),)(0)
0
LH)H
-- Vmin]
H
[max
7me
(C.32)
]
-
-Ymej
with
D(LH) =j
OD
(C.33)
LH)
* For f
{V.
LH}
2
10LH,()
LH,(O)
(C.34)
where
5vLH,(0)
_ _ YPTe
Ap
0 ;(LH),(0)(P
0
YPTesgn(O)
SLH,(O)
0)
op
X0
LH),()
Ap0
op
2
G
2
0
((C
c35
(C.35)
111
The 6 dependence of f in (D.9):
f(0)(p,
Pp,(, )
1 f(O)(P, o)
0)
(C.36)
now results in a weighting factor 1/Q as the LH diffusion is averaged over 0:
0)(p,
) = f (LH) H [
0
0
m
H [umax
-min]
yme
-
6 ]
YMe
(C.37)
with
0
b(LH)
j
7r
_7r
(C.38)
) D(LH)(O)
0
Note that with the assumption of a uniform poloidal repartition of the power, this flux-surface
averaged diffusion coefficient reduces to D0LH)
C.4
C.4.1
Electron-Cyclotron Diffusion
General Expression for the EC-X2 Diffusion Coefficient
We consider here the EC resonance of the X-Mode at second harmonic (X2-Mode); this mode is
chosen in many experimental situation because it allows the highest absorption with launching
from the low field side, when the 01-Mode is inaccessible because of a too high density wpe > w,,
. For the X2-Mode, (4.43) is
DEC-X 2
q2
Pe 2w
mec Wee
_ (N 1
Pe \
2
(C.39)
IEk,_ 12
Therefore, the diffusion coefficient (C.11) becomes
DEC-X2
C.4.2
P2
r
P2' V--*ac V
I
d 3k
(27r)3
q2
)2
(N± PTe 2w
IEk,-_1 2 6(w - k1vjj - nQ)
mec Wce
(C.40)
Gaussian JE12 (k 1 ) Spectrum
If the EC beam has a gaussian transverse extent, its power spectrum in k11 space is gaussian as
well, and can be expressed as
IEk,_1 2 = IEk,_,ol2 exp
(k-
kilo)2)
(C.41)
Then, the EC diffusion coefficient (C.40) takes the simple form:
DEC-X2(P,
D EC-X2
VTe
PDe VI I
112
(k,res-
k
Ak2
2
(C.42)
where
D EC-X2(0) =(Ni PTe2w
and
kil,res
d 2 k 1 q2Ek,_,O12
(27r)3
Ve
im 1-
V-OO V
MeC ce
(C.43)
is given by (C.7):
_
1
km,res =
(C.44)
Note that the 0 dependence in (C.43) takes into account the poloidal localization of the EC
beam.
C.4.3
Expression for the Fluxes due to EC Diffusion
General Expression
Using (1 SEC,(0)
(
= T(1 -
2)
2
{_p(1
=
), inserting
)DEC(p,
(C.13) and (C.42) in (C.16) gives
)I2(f)I
(-
-)2D("EC)(0) exp
EC, (0)
=
-
PTe
kol1
,resk
2
r
j~
(1 -
2)2
p21 _
PTe
f
k1 ,0
kres-
2
(C.45)
L(f)
2
k1
2)EC
ko
(-
0
PTe
1-
(
D(EC)(0)eXp
)
-p(1 -
(
2
)DPTeD(EC)(0)
exp
0I
PO~
! D(EC) (0) exp
J1
I]
(kll,res
(
kil 0 )2
Ak
~~
00
i0
k,)res-
-
ICf)j
(4)
Approximation of Well-Collimated EC Beam
The EC beams are in general well collimated. It is then justified to assume that the interaction
between electrons and the beam occur at a single, given poloidal position
Ob
on the flux-surface
considered. The EC diffusion coefficient (C.43) is then rewritten as
D0EC-X2(9)
=
A06(0
_
f)DC-X
2
(C.47)
where AO is the poloidal extent of the beam on the flux-surface under consideration.
Note also that the beam has a finite extent in the toroidal direction, over which the bounce
averaging process implicitly averages under the assumption of axisymmetry. In fact, this equivalence is valid only for passing particles, which follow circular orbits in the toroidal direction, but
113
not for trapped particles. Strictly speaking, with the present model, we should restrict ourselves
to situations where EC diffusion concerns only passing particles.
Bounce-Averaged EC Quasilinear Operator
With this approximation, the bounce-averaging operation becomes trivial and the bounceaveraged operator becomes:
* For fo or g
V- S.EC
10a
a
p26p
=
V1
2EC,(0)
0)
-7,ASC,M(
P
(C.48)
with
_y P (1
SEC,(O)
~
02
b| b 27
A PTe
(k-Ires,b
0 AO(EC) eXp
1
|,o)2
-
Ak2
0
Of(0)(p, a)
1( f( 0)(p, 0)
a6
0
+
ap
p
_
SEC,(0)
1 _ (1
1 0 O
'1b1b 27r
22
60
APTe
O(0)(p, w
(EC)
exp
0
- ko)
2
Ak2
I
X0
OP
-(kjires,b
(C.49)
where the following definitions have been introduced:
1 + Ecos(b)
Xb=
I +-
CO(S
(C.50)
()
(C.51)
kii,res,b = W7me
2Wce)
I
(C.52)
e For f
{V . SEC
a
1
2
Vj_
25EC,( ))
- EC,(O)
(C.53)
0 As o
Ap
with
-
A
PTe (
1
afb)(p,
ap
5EC,(0)
_YP
I2
I
OP
0)
1
-
27r
2 j(b)p,
po
114
kll, 0) 2
Ak 2
)1
0 AOD(EC) exp
0b b
0
(kI,res,b -
I
O
2)
of(b)(p,
(p,
-a(b)
Po
__--_
APTe
0 Ao(EC)eXp,
02)2*2 1
ako
0
]
(
k, 0) 2
(kI,res,b -
Ak 2
)
(C.54)
where
4)p
-(o)
115
(C.55)
Appendix D
Flux-Surface Averaging
We consider a single flux surface, which is taken to be at a constant normalized radius E = r/RO,
under the assumption of a circular cross-section.
Moreover, it is assumed that the electrons
mostly remain on this flux surface along their trajectory, which is an acceptable approximation
when the radial drift of electrons along their trajectories is much smaller that the characteristic
lengths of the radial dependence of macroscopic quantities like temperature, density, etc. This
condition is satisfied when (2.18)
< (V In B)-1 , (V In T)-
1
(D.1)
which holds in general, except at the plasma center where E -+ 0. Under the assumption (D.1),
the dependence upon the radial coordinate can be omitted in the following calculations.
D.1
Definition
The flux-surface averaging is the determination of the value of a quantity M(r, 0) averaged over
the volume between two adjacent flux surfaces of radius r and r + Ar, with Ar < r. The volume
element in toroidal geometry is dV = dr rdO Rd( so that the flux surface averaging operation is:
dr f7 rdO f *R 0 (1 +c cos(9))dCM(9)
fr+srd r
d6 f rR(1 + c cos(9))dC
fr+r
{M)
co(9))o(O)
rAr f_rrrd1
dR 70 (1 + --cos(9))M(6)
rAr f"
17
dRo(I + Ecos((0))
O(1 + 'E Cos (0)) M(0)
116
(D.2)
D.2
D.2.1
Electron Distribution f(p,
,
0)
Implicit 9-Dependence: case of fo and g.
The zero order distribution function fo and the first order distribution g have no explicit dependence upon the poloidal coordinate 9 (2.53), (2.58). However, the distribution function fo has
an implicit dependence upon 0 since an electron does not conserve the same pitch angle
along
its trajectory. Indeed, the relation between the pitch angle of an electron at a poloidal angle 0
and its pitch angle as it crosses the midplane 9 = 0 is given by the conservation of the magnetic
moment (See (2.6) and (2.8)) recalled here
(
1
=
(r, 0) (1 -
(D.3)
0)
with 'T(r,0) given by (2.10) and a being the sign of o.
Therefore, the electrons within the infinitesimal volume d 3r d 3p located at the midplane
position (r,9 = 0) and with a momentum (p, o) will be later within a volume d 3r' d'p' at a
position (r,0) with the momentum (p, ), with
given by (2.8). Since the number of particles
within this volume is conserved (f(r,6, p, )dar' d3 p' = f(r, 9 = 0, p, o)d'r d 3 p), and given the
incompressibility of phase space (d 3r' d3 p'
=
d 3r d 3 p), we get:
fo(p, , ) = fo(pdo,6 = 0) = fo (p,G0 )
(D.4)
and
g(p,,6) = g(po,
=
(D.5)
0) = g(O)(p, 0)
with
1
0-=o
(D.6)
2
and feo) (p, o) being the function calculated by the bounced-averaged Fokker-Planck code.
D.2.2
Explicit
-Dependence:f
The 9-dependent first-order correction to the distribution function is expressed as (2.58)
f(p,
)
(D.7)
fo(
V
Q
Or
0
The parallel velocity and the poloidal gyrofrequency are
_ p'
=
-
qeBo()
me
117
_qeIF(0)Bo(0
-
= 0)
(D.8)
In the expression of f(p,
, 0),
we can take fo(p,
given by (D.6). Then, we can express
, 0)
= fo(')(p, o) according to (D.4) with o
f as the product of a explicitly 0 dependent term and the
value of f at 0 = 0:
f (p,(, 0) =1(0)
(D.9)
(P, )
where
f(O) (P o) =
f(0 = 0)
po
_
qeBo(0 = 0)
af O(p,fo)
Or
(D.10)
is the function calculated by the code.
D.3
Flux-Surface Averaged Moment of f(p, , 0)
We consider a moment M(0) of the distribution function.
p2 dp f 1 d h(p, , 0)f(p, ,0)
Mf(0) = 27r
Applying the flux surface averaging (D.2) to (D.11) gives
cos(0))27r j
(Mf)
=
27r
I
(1
+E
P2dP
,
d h(p, , O)f(p, , )
P2dP
cos(0))
r27r
(D.11)
(D.12)
f
We have from (D.3)
(D.13)
d = 'F(0) od~o
and the integration domain transform as
S: -1 -0
4->
: 0 -1
<->
o :-l-+-
o:
1-
1-
(D.14)
so that (D.12) can be rewritten as:
(Mf)= 27r
P2dP
I~d(
1+
d~o H (02 > 1 S
ECos(0))f
1)
Lh(p, , )f(p, , 0)
(D.15)
where H is the Heaviside unit step function. Note that the integration domain can be rewritten
as
02 >
-
cos(0) > 1 - 21.
=
<OT
(D.16)
118
where 0, is defined in (2.20).
Since o is independent of 0, (D.12) can be transformed as follows:
(Mf)
=
27r
j
p 2 dp jdko
=
2w
j
p 2 dp jdo
h(p,
( + Ecos())H(II < 0c)
(1 +)
j
O)f(p, ,)
Gh(p, 0)f(p,
,
, 0)
(D.17)
where the expression (2.10) for T has been used.
Now, recalling the definition of the bounce averaging (2.23):
{A}
2b,t
2-
- T
dO r
B A
(D.18)
lI B9
|V
we note that if hf is symmetric in the trapped region, the sum over a can be removed. In the
cases where hf is antisymmetric in the trapped region, the sum over 0- would give zero. Nicely,
the integration over
o in (D.17) would also give zero if hf is antisymmetric in the trapped
region. As a consequence, under the assumption that hf is either symmetric or antisymmetric
in the trapped region, we can rewrite (D.17) as
p 2 dp
(Mf) = 27r
]
The total distribution function is (2.63)
d~o (1 + E)A{h(p,
f
=
, 6)f(p, , 0)}
(D.19)
fo + f + g so that we have
(D.20)
(M) = (MfO) + (M1 ) + (Mg)
" For fo or g
fo and g are symmetric in the trapped region.
Assuming that the moment h is either
symmetric or antisymmetric in the trapped region, which is always the case in this study
(quasilinear diffusion in the trapped region is nor allowed in our model, as discussed in
Section C.4.3), and using (D.4) and (D.5), we get:
(Mfo)
=
27r j
P2 dp
do fo0 (p, o) (1 + E)A{h(p, ,0)}
(D.21)
(Mg)
=
27r
p2 dP
d o g(O)(p, o) (1 + E)A{h(p, ,09)}
(D.22)
" For f
f is antisymmetric in the trapped region. Again, assuming that the moment h is either
symmetric or antisymmetric in the trapped region, and using (D.9), we get:
(M1 ) = 27r
P 2 dp
j
d~o f1()(p, o) (1 + E)A
119
h(p, , 0)
(D.23)
D.3.1
Flux-Surface Averaged Density
The electron density is the zero order moment of the distribution
ne = 27r
j
p 2 dp
d
f(p,
(D.24)
, )
so that the weighting function is simply
hn(p, ,O) = 1
(D.25)
{1} = 1
(D.26)
We have by definition
and it is easy to show that
{
which can be taken as 1/[A (1
1+ f sin(0c)/7r
'(O)
OJ
(D.27)
A (1 + c)
+ E)] since the electrons in the trapped region give no contribution,
because of the antisymmetry of
fO)
in the trapped region.
Thus, using (D.21), (D.22) and (D.23), we get:
=
27r jP2dP
do f ) (p, o) (1 + E)A
(ne)1 9 =
27r jp2dp
do 9 0)(p, o) (I1+
(n.e)y
27r
(ne)Ifo
=
j
p 2 dp
j
e) A
d<o f(O)(p, o)
(D.28)
(D.29)
(D.30)
(D.31)
D.3.2
Flux-Surface Averaged Current Density
The electron current density is a first order moment of the distribution
J1 = 27r qef
p 2 dp jd
v
ff(p, , 0)
(D.32)
with the weighting function given by
hn (A, , 0) = %eV1= q, X
IyMe
(D.33)
It is straightforward to show
(D.34)
for passing electrons, and
{{}
=
0 for trapped electrons.
120
Moreover, we define
2
6
=
T (0) 02
(D.35)
G-
A
where A is given by (E.40).
Then, using (D.21), (D.22) and (D.23):
(ne) Io
=
=27 q, 00
27r q, ej
1
3 d
3dp
1
27r q
P
0
(D.36)
do og(o)(p, o) (1 + c)H(I|oI > COT)
(D.37)
do of(0)(p, o)(1 + e)1
(D.38)
3
00
(ne) g
(ne) I
do ofo")(p, o) (1 + e)H(I oI > 'OT)
LI
dp
e
-
(D.39)
D.3.3
Flux-Surface Averaged Density of Power Absorbed
D.3.4
Expression as a Moment of Particle Fluxes
The kinetic energy associated with a relativistic electron of momentum p is
E = mec2 (7- 1)
(D.40)
Then, the local energy density of electrons is
(D.41)
S= Jd3p mec2 (Y - 1)f(p
The density of power absorbed from RF waves is defined as
Pabs -
aRF
d 3 p mec 2 (y-1)
-
at
f
(D.42)
RF
The effect of RF waves is described by a quasilinear operator, which can be expressed as the
divergence of a flux according to (C.12):
Of
at
1
P2
RF
p
\
P
2RF)
OR)
_
(D.43)
so that the density of power absorbed (D.42) becomes
Pabs
-mec 22r
The integration of the
dp p 2 (y
SRF
-
dIdi
1)
0- (p
P10
F
1_
2SRF)]
(D.44)
term gives no contribution, since the particle energy is function
of p only:
da
( V/1 -
[
(g,
121
_SR
RF
(D.45)
and (D.44) reduces to
Pabs =
-mec22
-
-mec221r
=
mec227rf
/
dp (-y - 1)
d
d
1
-1
RF
dp
j o2
/MP2
2
o
F
2dp XP2SRF
( [P2
d
(P2
SRF
7me
(D.46)
where (2.2) was used.
Finally, the density of power absorbed is
= 27r fd
Pabs
o
-1
D.3.5
dp
P3 SRF
7e
(D.47)
Flux-Surface Averaging
Using (D.19), (D.47) becomes
(Pabs) = 27r
dp
I
-l l
do (1 + E)A{SRF}
(D.48)
The flux-surface averaged particle fluxes due to RF diffusion are given in Appendix C.
122
Appendix E
Calculation of Bounce Average
Coefficients
E.1
Calculation of A( o)
Series Expansion
E.1.1
The bounce-averaging normalization coefficient is defined in (2.22)
c(E.1)
A(Go) =
with
for circulating particles
fir
OT =
We use the expression of
cos- 1 (1 - 2
for trapped particles
2T)
2/
from (2.8):
(E.2)
-2)
-W(
(=o
where T has a simple expression for the case of circular flux-surface, given in (2.10):
P (0) =
1 + + e(E.3)
1 + 6Cos (9)
Defining a = 0/2, we have cos(9) = 1 - 2 sin 2 (a) and then (E.3) becomes
, (0)
1
=
e
1 + e - 2esin 2 (a)
1 2(E.4)
1
where
GOT
-
2T sin(a)
is the pitch-angle at the trapped passing boundary, defined in (2.11).
123
Then, with a,c
0,/2, (E.1) becomes
t
'*C d6
A( o)
-c27r o-/1-
(o)(1 -
dO
afc~ /1-
fr o
21
_
7r
da
Q)(1
2 VolI
/
-
1
1
0o
(E.5)
02 sin2(a
02T/
We have the following series expansion, valid for
sin2(a))
02
sTin2(a)
2r
-
V/
02)
lxi
< 1:
00
V1-
XmX M
X =E
(E.6)
m=O
with
Xo
Applying (E.6) to V/1 -
1,
=
Xm =
2m - 3
2m Xm-i
(E.7)
o'T sin2(a), (E.5) becomes:
A( o)
=
(E.8)
J2m
EZXm
m=O
with
j0
J 2m =
E.1.2
da
sin 2 m (a)
Calculation of the Integrals
(E.9)
sin 2 (a)
V/1 - gT/
J 2m
The integrals J2m can be expressed in terms of complete ellipstic integrals of the first and second
kind [20]:
K(k 2 )
=
da
r/2
Jo
E(k2 )
2)
=
dx
_
/(1 - x 2 )(1
k 2 X2 )
V~~i2()jx
(1 -k 2 X22 )
(1
x )
1- k2 sin2(a)Jo
7r/2
da
- k2 sin2(a)1
O
0/(
-
(E.10)
-X2)
* For Trapped particles (Io1 < oT), we have from (E.2):
sin(ac)
=
sin(OT/2)
cos()
=
(E.11)
V2
Gor
so that
JO=
da
f("
o /1 - OT/
/sin(ac)
1
sin 2 (a)
x
o1
-T/o
dx
OTJo
1
K
-O/
OT
2
1x2
(E.12)
124
and
da
jo
J
sin 2 (a)
dac
S
01 -
(T/
2
sin (a)
/sin(ac) dx
1 -~ x2
VTT
I---
I
-
O X2
/
dx
G0T Jo
IoI
_
1
(f'd
-L
dx
dx
6T
(2
K
6
)
GRT I
2
2T
0/
1 _
1_
)]
-E'
)
2)
(E.13)
E 2T
'OT
* For Circulating particles (1ol >
To), we have
ac =
(E.14)
=
so that
r/ 2
da
Jo
2
1 OT4 O
=
sin (ai)
dx
2
2
V1-X2
(TIOT'X
Ko
=K (
(E.15)
and
/2
=2
L0 1
d
g02
sin 2 (a)
/ O sin (a)
da
f
2
Vo -
1
dx
1--
dx
2
2/
1T-
x
/0
S
02
-
1dx
d
K
(02T"
OT
O
1
VTT~
E
(
j
-
OX2
1-OT4
0o
dx
41
2TO2
X2
X
2T
EOT.6O
(E.16)
20
* For all particles, the integrals J2m, m > 1 can be derived from JO and J 2 from the
following recursive relation [20]:
(2m - 2)
(2m - 1)
(1+
2 f
125
J2m-2
-
(2m
-
3)
0
(2m - 1) 'COT
J2m-4
(E.17)
Bounce Averaging Coefficient X
-
1.8
-- M = 100
1.6
- - M = 2.
--
.
-..
-2
1.4
.~~
... ..
1.2
..... ............
-..
.....
0.6
-
...-..
. . ...
..
1
0.8
-..
.
-..
-.....
--......
....
-...............
..............-.
.......................
-.
-..
.
-..
. .........
............
-
0.4
0.2
-0.5
-1
0
0.5
1
Figure B-i: Bounce averaging coefficient A
E.1.3
Truncated Expression
It can be shown (See Figure E-1) that keeping only the two first terms in the expansion is
sufficient. Therefore A( o) can be evaluated with an excellent accuracy using:
A( o) =-
2
1 2j
JO - -dr2
7r (
2
E 8
(E.18
with JO and J 2 calculated above.
E.2
Calculation of s*
The coefficient s* is defined in (B.29). It is zero for trapped particles. For passing particles, it is
expressed by
s*
d
*
f
o1 + e
_, 27r I + c cos(O)
(E.19)
Defining
(E.20)
z = eiO
we get
d6 =
1
dz,
ZZ
cos(0) =
126
2
(Z + 1/z)
(E.21)
and (E.19) becomes
s**
2(1
+e)
E 27ri
dz
f
(E.22)
z2 + 2z/E + 1
where the contour -y is the unity circle in the complex plane.
The two roots of the denominator in (E.22) are
1
z
-- (1 ~
1-
(E.23)
2)
Only the greater root z+ lies inside the contour y, so that (E.22) becomes
1
)(27ri)Res
s* = 2 (1
e 2rz
Z2 + 2z/e +1V
=
)
Z+
(E.24)
where
1
Res (
=+ z+
2z/E + 1'
Z2 +
2/c /1 -
2
(E.25)
is the residue of the integral in (E.22) at z = z+.
Finally, inserting (E.25) in (E.24), we find
(E.26)
s*-:=
E.3
Calculation of 9*
The bounce averaging coefficient
*('o) is defined in (B.32). It is zero for trapped particles. Its
expression for passing particles can be rearranged as:
5*(60
,-
=
(2
rd6
-*7
f
2 76
6 20
-62)
rd 6 o 1 - T (1
A(~ 0
0
21r
J- '
(o
=
06-
(I
20
(2
(E.27)
- 02){,})
where the definition of A in (E.1)was used.
The expression for T is given by (E.4). Then, following the same derivation as for A from
(E.1) to (E.8), we get
f 'I
f_,
27r
2
7r/2
12
7,/2
A
1
=
(
da
da
sin (a)
s1
/1-T
1-
9/1 -_
2T/c2
127
sin2(a)/1 -%T
sin2()
(E.28)
We have the following series expansion, valid for
lxi < 1:
CY0
1
Y"m" M
=
-
/1
(E.29)
=
with
XO = 1,
Applying (E.29) to l//1
Xm
2m 2m
=
1
(E.30)
Xm-1
- Q sin(a), (E.28) becomes:
00
{O}(0 =
A12
7
~momJ
(E.31)
2m
m=O
where J 2 m is defined in (E.9).
Then, combining the results from (E.8) and (E.31), we get
*
[Xm -
()
%m-1(1 -
O)]
'
(E.32)
jJ2m
m=O
The series expansion (E.29) converging less rapidly than (E.6), the three first terms from the
two series are now required for a good accuracy (see Fig. E-2). The truncation to third order
gives:
s*()=-
2 ~
7r I
E.4
JO +
2 (
I)
+
O j2 O
(8
04TJ 4
2)
(E.33)
Calculation of A
The bounce-averaging coefficient A appears in the bounce averaging of the pitch-angle scattering
operator(B.18). It is defined as (B.21)
A =1
(E.34)
XFJ(0)}
Using
1
1 - cos(0)
=
1+
T2
2
(E.35)
and the bounce averaging expression from (2.24), (E.34) becomes
A=1
A
1
2 U±
]
I
2
J o(o2OT 1-- cos(0)
2dO
(E.36)
The sum over o-, which holds for trapped particles, can be removed since the function to be
integrated in independent of o-.
128
Bounce Averaging Coefficient s*
1.4
.
1.2 ...-
--
m = 100
m=2
- - m =3
.
-
1
0.8
-'
0.6
0.4- -...
.
.....
-.
.....
0.2OL
-1
-
0.5
0
0.5
1
0
Figure E-2: Bounce averaging coefficient
s*
This integral is very similar to the integral carried for A. Noting that 1 - cos(9) = 2 sin 2 (a)
with a = 6/2, and following the same derivation as for A from (E.1) to (E.8), we get
1 fOe d
o 2 1 - cos()
A f-oc 2-7r ( 72
1 2 f acl
Tsin2 (a)/I -
A ir J0
E
41
Xmd+
2
-
COT sinT(a)
sin(a)
(E.37)
J2m+2
m=O
Once again, we can see (Fig. E-3) that only the two first terms of the expansion need to be
kept. The truncated form of A therefore becomes:
A =j
1
(J02T2 + 1
129
4\
/QTJ4
(E.38)
Bounce Averaging Coefficient A
0.25
M =
100
m=2
--
0.2
-
-. -
.......
........
-a
-...
0.15
- --..
..
.-.
....
. ...... -.. -..
-
0.1 -.-.-.-.-
0.05|
1~
-i
'
-0.5
0
0.5
1
O
Figure E-3: Bounce averaging coefficient A
E.5
Calculation of
1(
o)
The flux-surface averaging coefficient A(
0)
is defined in (D.35). It can be evaluated using the
calculations of the previous coefficients. A can be rearranged according to:
J oce
dO
21r T(0) 0
o
SdO
X@(0)
Sc27r
f
2
dO o I - T(I IF (0)
J-0C 27r
dO G
27r
-ec
= A( 0 )
(1
02)
0
-10 '- ,1fI-
1
(E.39)
where the definitions of A in (E.1) and A in (E.34) were used.
Then, combining the results from (E.8) and (E.37), we get
00
2 JO + I: (AM
7r I
M=1
Xmn-i)
02TJ2m
(E.40)
We can take only the two first order terms from the two series, since higher order corrections
130
FluxSurface Averaging Coefficient X
0.9
-
0.8
-- m = 100
--.m. =....
. . . . . . ..
...-. .. ..
-m-2
- 0.7
.....
-. ..... ....
- -. ....
0.6 -.......
-....
0.5 - -..
0.4 -.....
- 0.3
-
-a
-..
..-...-.
..... ..-.
...........
-
........... ..........
- -..
-m
...........................
-.....
....
...
..... -.....
.....
0.2
0.1
0
0.5
0.5
1
Figure E-4: Flux-surface averaging coefficient A
are negligible (see Fig. E-4), which gives:
A(60)=
£r
-
(2
+
)
0)
131
0'2 J2 +
260
(oTJ
4
(E.41)
Appendix F
Listing of testfp2d9yp and fp2d9yp
F.1
testfp2d9yp
%
Test program of the 2-D relativistic bounce-averaged neoclassical Fokker-Planck solver
%
for the LH and EC absorption problem in presence of an ohmic electric field and toroidal
%
%
magnetic ripple.
Circular plasma cross-section.
See fp2d9yp.m, ptok9yp.m
%Last update: 04/18/2002
%by Y. PEYSSON CEA-DRFC 30/07/1997 <peysson~drfc. cad. cea.fr>
%Update for the drift kinetic calculations done by J. Decker and Y. Peysson 06-10/2001
clear all;pause(%.5);
clear mex;pause(0.5);
clear functions;pause(O.5);
warning off;
[s,dossier,machine]
= computyp;
%Physical constants
qe = 1.6e-19;%Electron charge (C)
me = 9.ie-31;/Electron rest mass (Kg)
re = 2.82e-15;%Classical electron radius (m)
mc2 = 511;%Electron energy rest mass (keV)
clum = 3e8;%Speed of light (m/s)
%Calculation mesh size
nmhu = 200;ANumber of angular values (WARNING: even number) (100 recommended for accurate run)
np = 200;/Number of momentum values (WARNING:
even number)
pnmax = 20;Maximum value of the momentum grid (pth-ref)
%Global plasma parameters
132
(200 recommended for accurate run)
[ap,Rp,Ip,Bt,Ncoils,Zi,mi,fi,BeLi,qmin,eq,TeO,Tea,eTe,neO,nea,ene,TiO,Tia,eTi,ZeffO,Zeffa,eZeff]
= ptok9yp('CMOD');
%TeO
=
ie-3;TiO
=
ie-3;
%.Normalizedradius values at which the distribution is calculated
X
drho = ie-2;/Xmaximum distance between two flux surfaces for radial derivatives.
%rho = [drho:drho:1-drho];%whole radial grid for dke calculations
rho = 15/22;
rhodisplay = rho(i);
ref-mode = 1;XReference values: (0) from core Te and ne,
%(M) from Te and ne at the normalized radius (paramater only active for a single radial
value, otherwise ref-mode = 0)
normref =
0;%Normalization
procedure for the ohmic electric field, LH and EC quasilinear
diffusion:
%(0) from local values Te and ne,
(1)
from reference values Te-ref and ne-ref
epsiX = O.OO;%Core Ohmic electric field (pthref*nhuthref/e or pth*nhuth/e)
epsia = 0.00;.Edge Ohmic electric field (pth-ref*nhuthref/e or pth*nhuth/e)
eepsi = 0;XExponent for Ohmic electric field
vparminh = 3.5;LH wave : Lower limit of the plateau (vth-ref or vth)
vparmaxlh = 6;%LH wave : Upper limit of the plateau (vthref or vth)
DOlh = 1;%LH quasi-linear diffusion coefficient (nhuth-ref*pth-ref^2 or nhuth*pth~2)
sm-lh =
0;ALH
quasi-linear diffusion coefficient smoothing parameter (no smoothing = 0)
DO-trap = 1;%LH quasi-linear diffusion coefficient in trapped region:
DO-vdep =
1;%LH
(0)
cut to 0,
or (1)
no cut
quasi-linear diffusion coefficient dependence over vpar: (0) no dep, or (1) 1/v
dep
Dperp-cy =
DOtrap-cy
0.0;%EC
= 0;%LH
quasi-linear diffusion coefficient (nhuth-ref*pth-ref~2 or nhuth*pth^2)
quasi-linear diffusion coefficient in trapped region:
(0) cut to 0, or (1) no
cut
-0.30;%EC wave parallel refractive index value
dnpar-cy = 0.02;%EC wave parallel refractive index width
omega-cy = 220;AEC frequency (GHz)
xomega-cyn0 = 0.98;
n-cy = 2;AHarmonic number;
npar-cy
=
polar-cy = 1;%EC wave polarization,
(0)
for the ordinary mode,
(1)
for the extraordinary mode
thetapoloidal-cy = 0;%Poloidal angle at which the EC wave is absorbed (deg)
sm-cy =
0;%EC
quasi-linear diffusion coefficient smoothing parameter (no smoothing = 0)
trunc = 0;%Collision operator:
(0) maxwellian, (1) truncated
bouncemode = 2;XBounce averaging option: (0) no bounce averaging, (1) explicit symmetrization
enforced at each time step,
133
%(2)
dke-mode =
DrO =
full implicit method with 15 diagonals
0;XFokker-Planck
equation (0), drift kinetic equation (1)
0.0;%Core radial diffusion coefficient (m^2/s)
pDr = 1;%Coefficient for the parabolic Dr radial profile
Dr-model =
0;%Radial
diffusion type:
(1)
magnetic turbulence model (vpar/vth),
(2)
no vpar
dependence
VrO =
0;XCore radial pinch coefficient (m/s)
pVr = 1;XCoefficient for the parabolic Vr radial profile
Vr-model = 0;%Radial pinch type:
(1)
magnetic turbulence model (vpar/vth),
(2)
no vpar dependence
vparminr = 3.5;%Lower limit of the parallel velocity dependence of the radial diffusion and
pinch (vth-ref or vth)
nhuloss = NaN;%Magnetic ripple loss frequency, nhuloss > 1000 (or NaN if no ripple losses)
(nhu-th) (WARNING: only for TORE SUPRA)
precOf = le-10;/Convergence level of the residu
nit-f = 30;XMaximum number of iterations for f
dtn = 1000;%integration time step (1/nhuth-ref)
(dtn > 0: fully implicit time differencing scheme
%dtn < 0: Crank-Nicholson time differencing scheme)
prec0_g = le-10;XConvergence level of the residu (if NaN, iterations stop with nit-g)
nit.g = 30;XMaximum number of iterations for f
initguess = 0;%Initial guess for the distribution function:
(0) Maxwellian,
(1) Fuchs model with Tperp for LH current drive only, (2) from a file
invproc = 1;%Matrix inversion method: (1) Full LU factorization, (2) qmr iterative method,
%(3) bicgstab iterative method,
(4)
cconv = 0;Coordinate conversion (p,mhu) ->
cgs iterative method,
(5)
pcg iterative method
(ppar,pperp) method: (0) Matlab built-in functions,
(1) Mex file
display-mode = 1;XDisplay mode:
(0): no, (1): partial, (2): full
nlevel = 30;%Number of level for contour plots
storemode = 0;Temporary storing option for sparse matrices calculations
(in RAM:
0,
file 1)
w2f_mode = 0;XWrite the final electron distribution in a file:
*** *******
*******
*** * ** ****
***
if display-mode > 1,
figure(i);clf;
figure(2);clf;
figure(i);
end
if bounce-mode ==
0,dke-mode
=
0;end
if dke-mode == 1,bounce_mode = 2;end
precO = [prec0_f,prec0_g];
134
(0)
no save,
(1)
save
in disk
nit = [nit-f,nit-g];
mhumin = -1;
mhumax = 1;
dmhu = (mhumax - mhumin)/nmhu;
mhu = [mhumin + dmhu/2: dmhu:mhumax - dmhu/2];
mhup = mhu + dmhu/2;
mhu2p = mhup.-2;
Xmhu = ones(np,1)*mhu;
Rp-norm = Rp/ap;%Normalized major radius
rdke
=
% list
[%;
of points for DKE calculations
if dke_mode == 1,
rho = sort(rho);
% list
in ascending order
rho = rho(rho>=drho & rho<=(1-drho)); % avoid dke calculations at the center or at the edge
xrho = [rho(1) - drho, rho(1)]; % r- point for radial derivative
rdke = 2;
if (length(rho) == 1)
X
dke calc. for one point only
xrho = [xrho, rho(i) + drho];% r+ point for radial derivative
end
for i = 2:length(rho)
if rho(i-1) < rho(i)-drho-eps
X
fp points are added only if radial gap between two
consecutive
% dke points is too large
xrho = [xrho, rho(i) - drho];
end
xrho = [xrho, rho(i)];
rdke =
[rdke, length(xrho)];
if (length(rho) == i) I (rho(i+1)>rho(i)+drho+eps)
xrho = [xrho, rho(i) + drho];
end
end
else
of points for fp calculations
xrho = rho; %list
end
nr = length(xrho);
%number of points for fp calculations
nr-dke = length(rdke);%number of points for dke calculations
xia = xrho/Rp.norm;
ir-display = find(xrho == rho-display);
display = [display-mode,ir-display] ;%Display parameters
%Local plasma parameters
xTe = (TeO
-
Tea)*(1
-
xrho.-2).^eTe + Tea;%Electron temperature (keV)
xne = (neo
-
nea)*(1
-
xrho.~2).^ene + nea;%Electron density (m-3)
xTi = (TiO
-
Tia)*(i
-
xrho.^2).~eTi + Tia;ADominant ion and impurity temperatures
Te/2) (keV)
135
(usually, Ti =
xZeff = (ZeffO - Zeffa)*(1 - xrho.~2).^eZeff + Zeffa;%Effective
charge (a.u.)
X
%Local safety factor value
qmax = ap*ap*Bt/Rp/Ip/0.2;%Ampere's theorem at the plasma edge (circular plasma cross-section)
xq = (qmax - qmin)*xrho.^eq + qmin;
%Ion density calculation (m-3) TO BE COMPLETED
if (length(Zi) == 1),
xni = xne/Zi;
elseif (length(Zi) == 2) & (max(max(Zi)) > 1)
%
[ngaz,nimp] = lowzyp(ne,Te,Zeffmax(max(Zi)),min(min(Zi))) ;%One light impurity fully stripped
model
%
ni = [ngaz,nimp];
elseif (length(Zi) <= 3) & (max(max(Zi)) == 1)
%
if sum(fi) > 1,
error('Wrong fraction of hydrogen isotops');
%
%
else
ni = fi*ne;
% end
else
% infoyp(2,'Use an impurity transport or equilibrium model');
% return;
end
%
XXfinit-ref = zeros(np,nmhu,nr);
XXDlhref = zeros(np,nmhu,nr);
XXDcyref = zeros(np,nmhu,nr);
XXnhulossref = zeros(np,nmhu,nr);
XXDrref = zeros(np,nmhu,nr);
XXVr-ref = zeros(np,nmhu,nr);
%Reference plasma parameters (for the normalization: pth-ref, nhuth-ref)
if length(rho) > 1 1 refmode == 0,
Teref = TeO;XCore temperature
ne-ref = neO;%Core density
if display-mode >= 1
infoyp(2,'Core reference temperature and density');
disp(' ------------------------------------------------------------------------------------------------------------------);
end
ref_mode = 0;
else % rho == 1 and ref-mode == 1
if nr-dke
Te-ref
ne_ref
==
=
=
0,
xTe(i);%Local temperature
xne(i);%Local density
if display-mode >= 1
136
infoyp(2,'Local reference temperature and density');
disp('-----------------------------------------------------------------------------------------------------------------')
end
else
Te-ref = xTe(2);%Local temperature
ne-ref = xne(2);%Local density
if display-mode >= 1
infoyp(2,'Local reference temperature and density');
disp( -----------------------------------------------------------------------------------------------------------------')
end
end
end
Tenref = Te-ref/mc2;
betath-ref = sqrt(Ten-ref);%Normalized pth-ref/mc as prescribed by Karney
pth-ref = betath-ref;
pmax-ref
=
pnmax*pth-ref;
dp-ref = pmax-ref/np;
p-ref = [dp-ref/2:dp-ref:pmax-ref - dp-ref/2];
pn-ref = p.ref/pth-ref;
dpn-ref = dp-ref/pth-ref;
pn2_ref = pn-ref.*pn-ref;
Xpn-ref = pn-ref'*ones(1,nmhu);
Xpn2_ref = pn2_ref'*ones(1,nmhu);
Ec-ref = mc2*(sqrt(i+p-ref.*p-ref)
- 1);%Fast electron kinetic energy in keV
Xpnpar-ref = Xpn-ref.*Xmhu;
Xpnperp2_ref = Xpn2_ref - Xpnpar-ref.*Xpnpar-ref;
lnc-ref = 15.2-0.5*log(ne-ref/1e20)+log(Te-ref);%Reference
Coulomb logarithm (in "Tokamaks" by
Wesson)
nhuref = 2.91e-6*(ne-ref/1e6)*lnc-ref*(Te-ref*1000)^(-1.5);%Reference bulk electron collision
rate: NRL formula value
xftp-normref = (me/qe)*xq*clum*betath-ref/ap/Bt./xia;%Normalization
equation (ftilde)
z-ref = betath-ref*pnref;
z2-ref = zref.*zref;
gamma-ref = sqrt(1 + z2_ref);
Xgamma-ref = gamma-ref'*ones(1,nmhu);
v-ref = pn-ref./gamma-ref;
Xvpar-ref = (v-ref'*ones(1,nmhu)).*(ones(length(v-ref) ,1)*mhu);
xTenorm
xTi-norm
xnenorm
xni-norm
=
=
=
xTe/Teref;
xTi/Teref;
xne/neref;
xni/ne-ref;
137
factor for drift kinetic
xTen = xTe/mc2;
xTin = xTi/mc2;
for ir = 1:nr,
if norm_ref == 0,
betath = sqrt(xTen(ir));XNormalized pth/mc as prescribed by Karney
pth = betath;
pmax = pnmax*pth;
dp = pmax/np;
p = [dp/2:dp:pmax - dp/2];
pn = p/pth;
pn2 = pn.*pn;
Xpn
=
pn'*ones(1,nmhu);
Xpn2 = pn2'*ones(1,nmhu);
Ec = mc2*(sqrt(1+p.*p)
- 1);%Fast electron kinetic energy in keV
Xpnpar = Xpn.*Xmhu;
Xpnperp2 = Xpn2 - Xpnpar.*Xpnpar;
lnc
= 15.2-0.5*log(xne(ir)/ie2O)+log(xTe(ir));%Coulomb
logarithm (in Tokamaks by Wesson)
nhu = 2.9ie-6*(xne(ir)/1e6)*lnc*(xTe(ir)*1000)^(-1.5);XBulk electron collision rate: NRL
formula value
z = betath*pn;
z2
=z.*z;
sqrt(i + z2);
gamma
Xgamma
=
gamma'*ones(1,nmhu);
v = pn./gamma;
Xvpar = (v'*ones(i,nmhu)).*(ones(length(v),i)*mhu);
rbetath = betath/betathref;
rnhuth = nhu/nhu-ref;
else
rbetath = 1;
rnhuth = 1;
end
xepsiO-ref(ir) = epsiO*rbetath*rnhuth;
xepsia-ref(ir) = epsia*rbetath*rnhuth;
xDOlhref(ir) = DO-lh*rbetath*rbetath*rnhuth;
xvparmin-lh-ref(ir) = vparmin-lh*rbetath;
xvparmax-lh-ref(ir) = vparmax-lh*rbetath;
xDperp-cy-ref(ir)
= Dperp-cy*rbetath*rbetath*rnhuth;
xvparmin-r-ref(ir) = vparmin-r*rbetath;
xnhuloss-ref(ir) = nhuloss*rnhuth;
%
LH diffusion coefficient
138
xepsi-ref(ir) = (xepsiO-ref(ir) - xepsia-ref(ir))*xrho(ir) .eepsi
+ xepsia-ref(ir);
%Parabolic dependence of the Ohmic electric field
mhutrap2
= 2*xia(ir)/(1+xia(ir));
if DOvdep == 1
XDlh-ref = smoothyp(xDO-lh-ref(ir)*xvparmin-lhref(ir)./Xvparref.*...
(Xvparref >= xvparmin-lh-ref(ir) & Xvpar.ref <= xvparmax-lh-ref(ir)),sm-1h);XLH
quasi-linear diffusion coefficient matrix with v1/vpar dependence
else
XDlh-ref = smoothyp(xDO-lh-ref(ir)*(ones(size(Xvpar-ref)).*...
(Xvpar-ref >= xvparmin-lh-ref(ir) & Xvpar-ref <= xvparmax-lhref(ir))),sm-1h);%LH
quasi-linear diffusion coefficient matrix
end
if
DO-trap == 0
XDlh-ref = XDlh-ref.*(Xmhu.*Xmhu > mhutrap2);%Dlh = 0 in trapped region
end
XDlh-ref(np-2:np,:) = O*XDlh-ref(np-2:np,:);%Boundary condition (used in Baranov's code)
XXDlh-ref(:,:,ir) = XDlh-ref;
X
EC diffusion coefficient
xomegac(ir) = qe*Bt./(I+xia(ir))/me/2/pi/e9;XElectron cyclotron resonance frequency at
thetapol = 0 (GHz)
xomega-cyn(ir) = xomega-cynO;WARNING:
set for X2 mode only
xpsib(ir) = (1+xia(ir))/(1+xia(ir)*cos(theta-poloidal.cy*pi/180));
xparam-cy(1:2,ir) =
xpsib(ir);xomega-cyn(ir)];
Xmhu-psib = sign(Xmhu).*sqrt(U - xpsib(ir)*(1 - Xmhu.^2));
Xpnperp2_psib-ref = Xpn2_ref.*(i - Xmhu-psib.^2);
Xpnpar2_psib-ref = Xpn2_ref - Xpnperp2_psibref;
Xnparres-ref
= (Xgammaref
-
xomegacyn(ir)) ./sqrt (Xpnpar2_psib-ref) .*sign(Xmhu-psib) /betath-ref;
XWARNING: For the second harmonic only
XDcyref = real(xDperp-cy-ref(ir)*Xpnperp2_psib-ref.*Xgamma-ref.*exp(-(Xnparres.ref
npar-cy).^2/dnpar-cy^2)./sqrt(Xpnpar2_psibref));X/*EC
-...
wave quasilinear diffusion
coefficient
XDcyref = smoothyp(XDcy-ref,sm-cy);%EC quasi-linear diffusion coefficient matrix
XDcy-ref(np-2:np,:) =
0*XDcy-ref(np-2:np,:);%Boundary
condition (used in Baranov's code)
if DO.trap-cy == 0
XDcyref = XDcy.ref.*(Xmhu.*Xmhu >
mhutrap2);%Dcy
= 0 in trapped region
else
XDcyref = XDcyref + fliplr(XDcy-ref.*(Xmhu.*Xmhu < mhutrap2));%Dcy symmetric in trapped
region
end
XXDcy-ref(:,:,ir) = XDcyref;
%Radial diffusion and pinch coefficients
139
if Drmodel==O,
XDrv-ref = DrO*ones(size(np,nmhu))/nhu-ref/ap/ap;%Radial
diffusion:
standard model
Dr(r) = DrO(r)
elseif Drmodel==1,
XDrv-ref = DrO*abs(Xvpar-ref)/nhu-ref/ap/ap;XRadial diffusion: magnetic turbulence model:
Dr(r) = DrO(r)*vpar-ref/vth-ref
end
XDrv-ref = XDrvref.*(Xvpar-ref >= xvparminrref(ir));
XXDrref(:,:,ir)
= XDrv-ref*(1+pDr*xrho(ir)^2);X3-D radial diffusion matrix
if Vr-model==O,
XVrv-ref = VrO*ones(size(np,nmhu))/nhu/ap;XRadial pinch: standard model : Vr(r) = VrO(r)
elseif Vrmodel==,
XVrv-ref = VrO*abs(Xvpar)/nhu/ap;%Radial pinch: magnetic turbulence model: Vr(r) =
VrO(r)*vpar-ref/vth-ref
end
XVrv-ref = XVrv-ref.*(Xvpar-ref
XXVrref(:,:,ir)
>= xvparmin-r-ref(ir));
= XVrv-ref*(1+pVr*xrho(ir)Y2);%3-D radial pinch matrix
%Loss term for the electrons trapped in the toroidal magnetic ripple
if bouncemode >= 1 & isnan(xnhulossref(ir))
== 0,
xNq(ir) = Ncoils*xq(ir);
[xthetabanana(ir),xthetaloss(ir),xpndetrap(ir),xEseuil(ir),xdeltaripple(ir)]
=
detrapmj3(xZeff(ir),ap,Bt,Rp,BeLi,Ip,xne(ir),xrho(ir)); %WARNING: Only valid for TORE
SUPRA
xpndetrap-ref(ir) = xpndetrap(ir)/betath-ref;
xmhusupertrap(ir) = sin(pi*xthetaloss(ir)/180);
Xmaskrip-ref = (Xpn-ref > xpndetrap-ref(ir)).*(abs(Xmhu)
< xmhusupertrap(ir));
Xnhuloss-ref = Xmaskripref*xnhuloss-ref(ir);
else
xdeltaripple(ir) = 0;
xNq(ir) = Ncoils*xq(ir);
Xnhuloss-ref = zeros(length(pn-ref),length(mhu));
end
XXnhuloss.ref(:,:,ir) = Xnhulossref;
if initguess == 0,
fM-ref = exp(-pn-ref.*pn-ref./(1 +
gamma-ref)/xTe-norm(ir))/2/pi/xTe-norm(ir)/sqrt(2*pi*xTe-norm(ir));
%Maxwellian distribution normalized to normalized density ne-norm (1st order gamma
corrections)
XfMref = fM-ref'*ones(1,nmhu);
normfMref = 2*pi*trapzyp([pn-ref',pnref '.*pnref'.*trapzyp([mhu',XfM-ref'])']);
XfMref = xne-norm(ir)*XfM-ref/normfMref;
Xf0_ref = XfM-ref;
title-initguess = 'Relativistic Maxwellian';
140
elseif initguess == 1,
%
Initial guess for the distribution function from ref. V.Fuchs et al., Phys. Fluids 28 (1985)
3619.
alpha = 2
beta = 2
-
2/(2+Zi(i));
alpha;
Tpinr-ref = (xvparmax-lh-ref(ir)^(alpha+1) - xvparmin-lhref(ir)^(alpha+i))*(alpha1)/(alpha+i)
-
(alpha-1)*(xvparmax-lh-ref(ir) - xvparmin-lh-ref(ir))*xvparmin-lhref(ir)^alpha +
xvparmaxlh-ref (ir) (alpha-1)
-
xvparmin-lh-ref(ir)^(alpha-1);
Tp2nr-ref = 2*(alpha-1)*(xvparmax-lh-ref(ir)
xvparmin-lhref(ir))*xvparmin-lh-ref(ir)^(alpha-2)
-
(xvparmax-lhref(ir)^(alpha-1) - xvparmin-lh-ref(ir)-(alpha-1));
Tpnr-ref = Tpinr-ref/Tp2nr-ref;
1.0*Tpnr-ref;%Relativistic correction (if necessary)
Xpperp2_ref = Xpn-ref.*Xpn-ref.*(1 - Xmhu.*Xmhu);
Xppar-ref = Xpn-ref.*Xmhu;
Xppar2_ref = Xppar-ref.*Xppar-ref;
Xgamma-ref = gamma-ref'*ones(1,length(mhu));
Xpparmin-ref = xvparmin-lh-ref (ir) *sqrt (1 +Xpperp2_ref*betathref ^2) /(1betath-ref-2*xvparmin-lh-ref(ir)^2));
Xpparmax-ref = xvparmax.lh-ref (ir)*sqrt (1 +Xpperp2_ref *betath-ref ^2) /(1betath-ref^2*xvparmax-lhref(ir)^2));
Xmaskplat-ref = (Xppar-ref >= Xpparmin-ref).*(Xppar-ref <= Xpparmaxref);
XfsthOref = Xmaskplatref.*exp(-Xpperp2_ref/2/Tpr.ref)/2/pi/Tpr-ref;
Xmaskback-ref = (Xppar-ref <= Xpparmin-ref);
Xfsth-ref = Xmaskback-ref.*exp(Xpn-ref.*Xpn-ref/2/Tpr-ref).*exp(Xpparmin-ref.*Xpparmin-ref/2/Tpr-ref)/2/pi/Tpr-ref;
Xmaskup-ref = (Xppar-ref > Xpparmax-ref);
Xfsth2_ref = Xmaskupref.*exp(-(Xpn-ref.*Xpn-ref.*(Xppar-ref./Xpparmaxref).^(2/Zi(i))
Tpr.ref =
Xpparmax-ref . *Xpparmax-ref)/Tpr-ref/2)/2/pi/Tpr-ref ;%See thesis of R.Arslanbekov
Xfsthref = XfsthOref+Xfsthiref+Xfsth2_ref;
fM-ref = xne-norm(ir)*exp(pn-ref.*pn-ref./(i+gamma-ref)/xTe-norm(ir))/2/pi/xTe-norm(ir)/sqrt(2*pi*xTenorm(ir));
XfM-ref = fMref'*ones(1,nmhu);
normfMref = 2*pi*trapzyp([pn-ref(:),pn2-ref(:).*trapzyp([mhu',XfMref'])']);
normfsth-ref = 2*pi*trapzyp([pn-ref(:),pn2_ref(:).*trapzyp([mhu',Xfsth.ref])']);
XfMref = XfM-ref/normfMref;
Xfsthref = Xfsthref/normfsth-ref;
lambdaref = 1/(1 + Xfsthref(min(find(Xmaskplatref(:,nmhu-)==)),nmhu-)/...
XfMref (min(find(Xmaskplat-ref(: ,nmhu-1)==)),nmhu-1));
if (xDlhOref(ir) == 0),lambda-ref = O;end
XfOref = XfM-ref + Xfsth-ref*lambda-ref;
normfOref = 2*pi*trapzyp([pn.ref(:),pn2_ref(:).*trapzyp([mhu',XfOref'])']);
XfOref = xne-norm(ir)*XfQref/normfQref;
titleinitguess = 'Fuchs et al. LH model';
141
-
elseif initguess == 2,
eval(['load ',dossier,s,'XXfinit.mat'])
XfOref = squeeze(XXfinit(:,:,ir));
title-initguess = 'From data file';
end
XXfinitref(:,:,ir) = XfOref;
%Display initial distribution function
if display-mode == 2,
normfOref = 2*pi*trapzyp([pnref(:),pn2_ref(:).*trapzyp([mhu',Xf&_ref'])']);
[XfOcyl-ref,ppar-cyl-ref,pperpcyl-ref]
= s2cyp(full(XfOref),pnref,mhu,cconv);
Xppar-cyl-ref = ppar-cylref(:)*ones(i,length(pperp-cyl.ref));
Xpperp-cyl-ref = ones(length(ppar-cyl-ref),i)*pperp-cyl-ref(:)';
Xp-cyl-ref = sqrt(Xppar-cyl-ref.*Xppar-cyl-ref + Xpperp-cyl-ref.*Xpperp-cyl-ref);
Xmhu-cyl-ref = Xppar-cyl-ref./sqrt(Xppar-cyl-ref.~2+Xpperp-cyl-ref.^2);
Xgammacyl-ref = sqrt(i + Xp-cyl-ref.*Xp-cyl-ref*betath-ref*betath-ref);
Xgammapar-cyl-ref = sqrt(i +
Xppar-cyl-ref.*Xppar-cyl-ref*betath-ref*betath-ref);/??????????
Xvpar-cyl-ref = Xppar-cyl-ref./Xgamma-cyl-ref;
dpperp-cyl-ref = pperp-cyl-ref(2) - pperp-cyl-ref(1);
dppar-cyl-ref = ppar-cyl-ref(2) - ppar-cyl-ref(1);
if xDO-lh-ref(ir) ~= 0,
pparresmin-lh-cyl-ref = xvparminlh-ref(ir)*sqrt(i +
betath-ref^2.*pperp-cyl-ref.^2)/sqrt(1
-
xvparminjlh-ref(ir)^2*betath-ref^2);%LH resonance condition
pparresmax-lh-cyl-ref = xvparmaxlh-ref(ir)*sqrt(1 +
betath-ref^2.*pperp-cyl-ref.^2)/sqrt(i
-
xvparmax.lh-ref(ir)^2*betath-ref^2);XLH resonance condition
end
XfO-cyl-ref = XfOcyl-ref.*(Xf0_cyl-ref>O);%Remove negative values
XfOcyl-ref = XfO-cyl-ref.*(Xp-cyl-ref<max(ppar-cyl-ref));XRemove values above max(ppar)
XfOcyl-ref = Xf0_cyl-ref.*(abs(Xmhu-cyl-ref)<=1 - dmhu/2);%Remove values outside from
the domain of calculation
currOref
=
2*pi*trapzyp([pn-ref(:),pn2_ref(:).*(pn-ref(:)./gammaref(:)).*trapzyp([mhu',Xmhu'.*XfOref'])'])
currO-cyl-ref =
2*pi*trapzyp([pperp-cyl-ref(:),pperp-cyl-ref(:).*trapzyp([ppar-cyl-ref(:),Xvpar-cyl-ref.*Xf0_cyl_
ref])']);
if nrdke >= 1,
if ir == ir-display-1,
figure(i),subplot(131);zoom on
elseif ir == ir-display,
142
figure(i),subplot(132);zoom on
elseif ir == ir-display+1,
figure(1),subplot(133);zoom on
end
C = contour(ppar-cyl-ref,pperp-cyl-ref,log(XfOcyl_ref'),[20:20/nlevel:0]);axis('equal');
faxis = axis;dfaxis-x = (faxis(2) - faxis(1))/100;dfaxis_y
= (faxis(4) -
faxis(3))/100;
text(faxis(i) + dfaxis-x*5,faxis(4)
text(faxis(i) + dfaxis-x*5,faxis(4)
- dfaxis-y*5,['r/a = ',num2str(xrho(ir))]);
- dfaxis-y*10,['Te(norm)
=
',num2str(xTenorm(ir))]);
text(faxis(i) + dfaxis-x*5,faxis(4)
- dfaxis-y*15, ['ne(norm) =
',num2str(xne.norm(ir))]);
xlabel('ppar/pth-ref');ylabel('pperp/pth-ref');title(title-initguess);
pause(Q.1);
-= 0,
hold on,plot(pparresmin-lh-cyl-ref,pperp-cyl-ref,'k-'),plot (pparresmax-lh-cyl-ref ,pperp-cyl-ref, 'k--'),hold off
end
if xDO-lh-ref(ir)
if ir == ir-display-1,
figure(2),subplot(231);zoom on
elseif ir == irdisplay,
figure(2),subplot(232);zoom on
elseif ir == ir-display+1,
figure(2),subplot(233);zoom on
end
semilogy(-v-ref,XfM-ref(:,i),'g--',v_refXfM-ref(:,nmhu),'g--',v-ref,Xf0_ref(:,1),'c-',vref,XfO._ref(:,nmhu),'c-');
%Forward and backward directions only
axis([-max(v-ref),max(v-ref),ie-20,10]);grid on;zoom on
xlabel('vpar/vth-ref');ylabel('f(pperp = 0)');title(['Norm.(',num2str(xrho(ir)),')
',num2str(normfOref)]);
pause(O.1);
if ir == ir-display-1,
figure(2),subplot(234);zoom on
elseif ir == ir-display,
figure(2),subplot(235);zoom on
elseif ir == ir-display+1,
figure(2),subplot(236);zoom on
end
plot(pn-ref,pn-ref,'r-.',pn-ref,vref,'b:');grid on;zoom on
xlabel('p/pth-ref');ylabel('v/vth-ref');title(['j = ',num2str(curr0_ref)]);
pause(0.1);
elseif ir == ir-display
%if ir == irdisplay
figure(1),subplot(132);zoom on
143
=
%end
C = contour(ppar-cyl-ref ,pperp-cyl-ref ,log(Xf0_cyl-ref'),[20:20/nlevel:0]);axis('equal');
faxis = axis;dfaxis-x = (faxis(2) - faxis(1))/100;dfaxisy
faxis(3))/100;
text(faxis(i)
+ dfaxis-x*5,faxis(4)
text(faxis(1) + dfaxis-x*5,faxis(4)
= (faxis(4) -
- dfaxis-y*5,['r/a = ',num2str(xrho(ir))]);
- dfaxisy*10, ['Te(norm) =
,num2str(xTenorm(ir))]);
text(faxis(i) + dfaxis-x*5,faxis(4) - dfaxis-y*15, ['ne(norm) =
',num2str(xnenorm(ir))]);
xlabel('ppar/pth-ref');ylabel('pperp/pth-ref');title (title-initguess);
pause(O.1);
0,
hold on,plot(pparresmin-lh-cylref ,pperp-cyl-ref, 'k-'),plot(pparresmax-lhcyl-ref ,pperp-cyl-ref, 'k--'),hold off
end
if xDOlhref(ir)
-=
figure(2),subplot(232);zoom on
%end
semilogy(-vref ,XfM_ref(: ,1),'g--',v-ref,XfM-ref(: ,nmhu),'g--',-
v_ref,XfOref(:,),c-',v-ref,XfOref(:,nmihu),'c-');
/Forward and backward directions only
axis([-max(v-ref),max(vref),le-20,10]);grid on;zoom on
xlabel('vpar/vth-ref');ylabel('f(pperp
=
0)');title(['Norm.
(',num2str(xrho(ir)),')
=
',num2str(normfOref)]);
pause(O.1);
figure(2),subplot(235);zoom on;pause(O.1)
Xend
plot(pn-refpn-ref,'r-.',pn-ref v_ref ,'b:');grid on;zoom on
xlabel('p/pth-ref');ylabel('v/vth-ref');title(['j
= ',num2str(currOref)]);
pause(0.1);
end
end
end
X
2D Fokker-Planck calculations
timeO = clock;
[XXf0_ref ,XXf0tp-ref,XXf0_g-ref,XXSpOref,XXSmhuOref,XXSpO_g_ref,XXSmhuo_g_ref,XXSpotp-ref,...
XXSmhuOtpref ,xPcollref ,xPcoll-totref ,xPcollref_fsav,xPcolltotref_fsav, xPabs-ref _fsav,...
xPabs-tot-ref_fsav, xnormfref ,xnormf-ref_fsav,xcurrO_ref ,xcurrtotref ,xcurr0_ref_fsav,...
xcurr-tot-ref-fsav,xcurrB-ref_fsav,xRR-reffsavxTe_normoutXpn-ref,Xmhu-ref,xmhubounce2]
=
fp2d9yp-new2 (betath-ref ,Rp-norm, xTenorm,xnenorm, Zi,mi, xninorm,xTi-norm,pnref ,mhu,XXDlh_ref,.
144
XXDcy-ref,xparam-cy,XXnhuloss-ref,xepsi-refbounce-mode,rdke,xia,xftpnorm-refXXDr-ref,XXVrref,
xdeltaripple,xNq,trunc,dtn,precO,nit,invproc,display,storemodeXXfinitref);%
if display-mode >= 1,
if nr-dke == 0,
infoyp(2,['Full elapsed time for solving the Fokker-Planck equation:
',num2str(etime(clock,time0)),' Cs)'])
else
infoyp(2,['Full elapsed time for solving the drift kinetic equation:
',num2str(etime(clock,time0)),' Cs)'])
end
disp(' --------------------------------------------------------------------------------------;
-----------------------------
end
if
v2fmode
== 1,
XXfp-out = XXfOref;
XXfp-tp-out
XXfp-g-out
eval(['save
XXfOtp-ref;
= XXfOg-ref;
',dossier,s,'XXdke_out.mat
=
XXfp-out XXfp-tp-out XXfp-gout Xmhu Xpn-ref'])
end
P-ref = 9.le-31*(clum*betath-ref)^2*nhu-ref*ne-ref/1e6;%MKSA power unit
j-ref = 1.6e-19*clum*betath-ref*ne-ref/le6;.MKSA current unit
Edreicer-ref = me*clum*betath-ref*nhu-ref/qe;%MKSA Dreicer field unit
xcollisionnality =
0.012*(xne/le20).*xZeff.*xq*Rp./xia.^1.5./xTe.^2;XFrom 0. Sauter et al. Phys.
Plasma 6 (1999) 2834
[xcurr-hir-mksa-fsav,xcurr-saumksafsav,xcurr-i_mksa-fsav,xcurrjlormksafsav
hsneomodels9jd(xrho,ap,Rp,Zi,xq,Bt,TeO,Tea,eTe,neO,nea,ene,TiO,Tia,eTi);
=.
%Calculation of the flux averaged bootstrap current from Hirshman and Sauter models (bulk
contribution)
if display-mode >=I
=
XfOref = squeeze(XXf _ref(:,:,ir display));%Display results at the required radial
Xf0_tp~ref
value
=
value
squeeze(XXf0_tp.ref(:,:,irdisplay));.Display results at the required radial
XfOg-ref = squeeze(XXfOg-ref(:,:,irdisplay));%Display results at the required radial value
Xfinit-ref = squeeze(XXfinit-ref(:,:,ir-display));%Display
results at the required radial
value
XDlh-ref = squeeze(XXDlh-ref(:,:,ir-display));%Display
results at the required radial value
XDcyref = squeeze(XXDcy-ref(:,:,ir-display));%7Display results at the required radial value
XSpOref = squeeze(XXSpOref(:,:,ir.display));%Display results at the required radial value
XSmhu0_ref = squeeze(XXSmhuOref(:,:,ir-display));Y.Display results at the required radial
145
value
infoyp(2,['Reference electron temperature (keV): ',num2str(Ten-ref*mc2)])
infoyp(2,['Reference electron density (10+19 m-3): ',num2str(neref/1e19)])
infoyp(2,['MKSA power unit (P.ref): ',num2str(P-ref),' (MW/m3)'])
infoyp(2,['MKSA current unit (jref): ',num2str(jref),' (MA/m2)'])
infoyp(2,['MKSA Dreicer field unit (Edreicer-ref):
',num2str(Edreicer-ref),'
(V/m)'])
disp(' ------------------------------------------------------------------------------------------------------------------);
infoyp(2, ['Radial position at which results are displayed:
',num2str(xrho(irdisplay)),'
(ap)'])
disp(' -----------------------------------------------------------------------------------------------------------------'Y);
infoyp(2,['Electron temperature (keV): ',num2str(xTe(irdisplay))])
infoyp(2, ['Electron temperature increment
(keV):
',num2str ( (xTe-norm-out (ir-display) -
xTe-norm(ir-display))*Ten-ref*mc2)J)
infoyp(2,['Electron density (10+19
m-3):
',num2str(xne(ir-display)/iei9)])
disp('---------------------------------------------------------------------------------------------------------------');
infoyp(2,['Inverse aspect ratio:
',nuim2str(xia(irdisplay))])
infoyp(2,['Collisionality parameter:
',num2str(xcollisionnality(ir-display))])
disp(I --------------------------------------------------------------------------------------');
----------------------------
%
Magnetic ripple losses
(*********
to be cross-checked **********)
if bouncemode >= 1 & isnan(xnhuloss-ref(ir-display)) == 0,
Gripmhuref = 2*pi*trapzyp([pn-ref',Xmaskrip-ref.*XSmhu0_ref.*Xpn-ref])*sqrt(i-...
min(mhu(find(abs(mhu)<xmhusupertrap(ir-display))))-2);
Grippn-ref =
2*pi*pn(max(find(pn-ref<xpndetrap-ref(ir-display))))
^2*trapzyp([mhu',Xmaskrip-ref'
Gripflux-ref = Gripmhu-ref (max(f ind (abs (mhu) <xmhusupertrap(ir-display))))
Gripmhu-ref(min(find(abs(mhu)<xmhusupertrap(ir-display)))) +
.*XSpO-ref']);
-
Grippn-ref(max(find(pn-ref<xpndetrap-ref(ir-display)))+1)
GriptauOref
=
trapzyp( [pn-ref' ,Xnhuloss-ref .*Xf0_ref .*Xpn-ref .*Xpn-ref]);
Griptau-ref = 2*pi*trapzyp([mhu',Griptauref']),%Flux
of electrons leaving the plasma
through the loss cone
infoyp(2, ['Ripple loss (s-i.m-3):
',num2str(Griptau-ref*nhu-ref*xne(ir-display))]);
disp(' -----------------------------------------------------------------------------------------------------------------')
end
[Xfinit-cyl.ref,ppar-cyl-ref,pperp-cyl-ref]
= s2cyp(Xfinit-ref,pn.ref,mhu,cconv);
%Spherical to cylindrical coordinate transformation
[XfOcyl-ref,ppar-cyl-ref,pperp-cyl-ref]
s2cyp(Xf0_ref ,pnref,mhu,cconv);%Spherical
=
to
cylindrical coordinate transformation
[XDlh-cyl-ref,ppar-cyl-ref,pperp-cyl-ref]
=
s2cyp(XDlh-ref ,pn-ref ,mhu,cconv); %Spherical to
=
s2cyp(XDcyref ,pn-ref ,mhu, cconv);%Spherical to
cylindrical coordinate transformation
[XDcy.cyl-ref,ppar-cyl-ref,pperp-cyl-ref]
146
cylindrical coordinate transformation
Xppar-cyl-ref = ppar-cyl-ref(: )*ones(1,length(pperp-cyl-ref));
Xpperp-cyl-ref = ones(length(ppar-cyl-ref) , 1)*pperp-cyl-ref(:)';
Xmhu-cyl-ref = Xppar-cyl-ref./sqrt(Xppar-cyl-ref.^2+Xpperp-cyl-ref.~2);
Xp-cyl-ref = sqrt(Xppar-cyl-ref.*Xppar-cyl-ref + Xpperp.cyl-ref.*Xpperp-cyl-ref);
Xganmmacyl-ref = sqrt(1 + Xp-cyl-ref.*Xp-cyl-ref*betath-ref*betath-ref);
Xgammapar-cyl-ref = sqrt(1 + Xppar-cylref .*Xppar-cylref *betath-ref*betath-ref) ;??????????
Xvpar-cyl-ref = Xpparcyl_ref./Xgammacyl_ref;
dpperp-cyl-ref = pperp-cyl-ref(2) - pperp-cyl-ref(1);
dppar-cyl-ref = ppar-cyl-ref(2) - ppar-cyl-ref(1);
XfOcyl-ref = XfOcyl-ref.*(XfOcyl-ref>O);%Remove negative values
XfOcyl-ref = XfO-cyl-ref.*(Xp-cyl-ref<max(ppar-cyl-ref));%Remove values above max(ppar)
XfOcyl-ref = Xf0_cyl-ref.*(abs(Xmhu-cyl-ref)<=1-dmhu/2);%Remove values outside from the
domain of calculation
= XDlh-cyl-ref.*(XDlh-cyl-ref>O);%Remove negative values
XDlh-cyl-ref
XDlh-cyl-ref
XDlh-cyl.ref
XDlh-cyl-ref
= XDlh-cyl-ref.*(Xp-cyl-ref<max(ppar-cyl-ref));%Remove values above max(ppar)
= XDlh-cyl-ref. * (XDlh-cyl-ref<=max(max(XDlh-ref))) ;%Remove too large values
= XDlh-cyl-ref.*(abs(Xmhu-cyl-ref)<=-dmhu/2);Remove
values outside from the
domain of calculation
XDcy-cyl-ref = XDcy-cyl-ref.*(XDcy-cyl-ref>O);%Remove negative values
XDcy-cyl-ref = XDcy-cyl-ref.*(Xp-cyl-ref<max(ppar-cyl-ref));%Remove values above max(ppar)
XDcy-cyl-ref = XDcy-cylref.*(XDcy-cyl-ref<=max(max(XDcy-ref)));XRemove too large values
XDcy-cyl-ref = XDcy-cyl-ref.*(abs(Xmhu-cyl-ref)<=-dmhu/2);XRemove values outside from the
domain of calculation
%LH and EC quasi-linear domains
pparresmin-lh-cyl-ref = xvparmin-lh-ref (ir-display)*sqrt (1 +
betathref^2.*pperp-cyl-ref.^2)/sqrt(i xvparmin-lh-ref (ir-display) 2*betath-ref^2) ;%LH resonance condition
pparresmax-lh-cyl-ref = xvparmax-lh-ref(ir-display)*sqrt(1
betath-ref^2.*pperp-cyl-ref.^2)/sqrt(1
+
-.
xvparmaxlh-ref (ir-display) 2*betath-ref 2) ;%LH resonance condition
pperp2res.cy-cyl-ref
1)/betath-ref2
=
((xomega-cyn(ir-display)
+ npar-cy*betath-ref*ppar-cyl-ref).^2-
-
ppar-cyl-ref.^2;XEC resonance condition
mask-pperp2res-cy-cyl-ref = find(pperp2res.cy-cyl-ref >=O);
pperpres-cy-cyl-ref = sqrt(pperp2res-cy-cyl-ref(mask-pperp2res-cy-cyl-ref));
pparres-cy-cyl-ref = ppar-cyl-ref(mask-pperp2res-cy-cyl-ref);
pparres-cy-cy10_ref = (npar-cy*xomega-cyn(ir-display) - sign(npar-cy)*sqrt(npar-cy^2 xomega-cyn(ir-display)^2))/betath-ref/(1
1+
- npar-cy^2);%EC resonance condition at pperp =
0
%calculation in cylindrical coordinates
147
xnormf-cyl-ref(ir-display) =
2*pi*trapzyp([pperp-cyl-ref (:) ,pperp-cyl-ref
(:).*trapzyp([ppar-cyl-ref (:),XfQcyl-ref])']);
xcurrOcylref (ir-display) = 2*pi*trapzyp([pperp-cyl-ref(:),pperp-cyl-ref(:).*...
trapzyp([ppar-cyl-ref(:),Xvpar-cyl-ref.*XfOcyl-ref])']);
%Flux calculations
%
Calculation of F// and Tperp
if display-mode >=2
FparOcyl-ref = 2*pi*trapzyp( [pperp-cyl-ref(:),Xpperp-cyl-ref'
.*XfOcyl-ref ']);
Fparinit-cyl-ref = 2*pi*trapzyp([pperp-cyl-ref(:),Xpperp-cyl-ref'.*Xfinit-cyl-ref']);
zFarOcyl-ref = find(FparQcyl-ref==O);FparOcyl-ref(zFarO-cylref) =
FparOcyl-ref(zFarOcyl-ref)+le-60;%zero removal
Fpar3_cyl-ref =
2*pi*trapzyp([pperp-cyl-ref(:),Xpperp-cyl-ref'.*Xpperp-cyl-ref'.*Xpperp-cyl-ref'.*XfO-cyl-ref']);
Tperp-cyl-ref = Fpar3_cyl-ref./FparOcyl-ref/2;
Tperpn-cyl-ref = betath-ref*betath-ref*Tperp-cyl.ref*mc2/Te-ref;
figure(3);clf
subplot (121),semilogy(ppar-cylref ,FparOcyl-ref, 'r-' ,ppar-cyl-ref ,Fparinit-cyl-ref, 'b-');grid on;
axis([-max(pn-ref),max(pnref),le-20,10]);zoom on
ylabel('FO//');xlabel('ppar/pth-ref');title( ['r/a = ',num2str(xrho(ir.display)),')']);
subplot(122) ,plot(ppar-cyl-ref ,Tperpn-cyl-ref, 'b-') ;grid on;zoom on
ylabel('Tperp/Te-ref');xlabel('ppar/pth-ref');title(['(r/a =
',num2str(xrho(ir-display)),')']);
pause(O.1);
figure(4),clf
C = contour(ppar-cyl-ref ,pperp-cyl-ref ,log(Xf0_cylref'), [-20:20/nlevel:0]) ;grid on;zoom
on;
%Distribution function and wave quasilinear diffusion coefficient contour plot
axis('equal')
title('2-D steady state electron distribution function');
xlabel('ppar/pth-ref');
ylabel('pperp/pth-ref');
if bounce-mode >= 1,%Draw bounce domain boundaries
line ( [0, pnmax*sqrt (xmhubounce2(ir-display))] , [O,pnmax*sqrt (
-
xmhubounce2(ir-display))]);
line( [,-pnmax*sqrt(xmhubounce2(ir-display))],[O,pnmax*sqrt(i
-
xmhubounce2(ir-display))]);
if isnan(nhuloss) ==
0,XDraw
the loss cone
line( [O,pnmax*sqrt(icos(xthetaloss(ir-display)*pi/180)^2)],[0,pnmax*cos(xthetaloss(ir-display)*pi/180)],...
'LineStyle','-','color','r');
line([O,-pnmax*sqrt(1-
148
cos(xthetaloss(ir-display)*pi/180)^2)],
[O,pnmax*cos(xthetaloss(ir-display)*pi/180)],...
'LineStyle','-','color','r');
end
end
if xDO-lh-ref(ir-display)
-=
0,
hold on,plot(pparresmin-lhcyl-ref,pperp-cylref,'k--
'),plot (pparresmaxlh-cylref ,pperp-cyl-ref, 'k--'),hold off
end
if xDperp-cy-ref(ir-display)
-=
0,
hold on;plot(pparres-cy-cyl-ref,pperpres-cy-cyl-ref,
'k-. ');hold off
hold on,plot(pparres-cy-cylOref,0,'k*'),hold off;
end
faxis = axis;dfaxis-x = (faxis(2) - faxis(1))/100;dfaxis-y
text(faxis(1)
+ dfaxis-x*5,faxis(4)
= (faxis(4) - faxis(3))/100;
- dfaxis.y*5,['r/a = ',num2str(xrho(ir.display))]);
pause(0.1);
if xDO-lh-ref(ir-display)
-= 0,
figure(7);clf
C = contour(ppar-cyl-ref,pperp-cyl-ref,XDlh-cyl-ref',nlevel);grid on;zoom on;%LH
diffusion coefficient
axis('equal')
title('Lower Hybrid diffusion coefficient (nhu-ref*pth-ref^2)');
xlabel('ppar/pth-ref');
ylabel('pperp/pth-ref');
hold on,plot(pparresmin-lh-cyl-ref,pperp-cylref,'k-'),plot (pparresmax-lh-cyl-ref ,pperp-cyl-ref,'k--'),hold off
if bounce-mode >= 1,%Draw bounce domain boundaries
line( [O,pnmax*sqrt(xmhubounce2(ir-display))], [0,pnmax*sqrt(i
xmhubounce2(ir-display))]);
line( [O,-pnmax*sqrt(xmhubounce2(ir-display))] , [O,pnmax*sqrt(i
-
-
xmhubounce2(ir-display))]);
if isnan(nhuloss) == 0,YDraw the toroidal ripple loss cone
line ( [O,pnmax*sqrt(icos(xthetaloss(ir-display)*pi/180)-2)] ,[0,pnmax*cos(xthetaloss(ir-display)*pi/180)]
'LineStyle','-','color','r');
line([O,-pnmax*sqrt(icos(xthetaloss(ir-display)*pi/180)^2)]
,
[O,pnmax*cos(xthetaloss(irdisplay)*pi/180)],
'LineStyle','-','color','r');
end
end
faxis = axis;dfaxisx = (faxis(2) - faxis(1))/100;dfaxis-y
faxis(3))/100;
text(faxis(i) + dfaxis-x*5,faxis(4) - dfaxis-y*5,['r/a
',num2str(xrho(ir-display))]);
pause(O.1);
end
if xDperp-cy-ref(ir-display) -= 0,
figure(8);clf
149
(faxis(4)
-
C = contour(ppar-cyl-ref,pperp-cyl-ref,XDcy-cyl-ref',nlevel);grid on;zoom on;%EC
diffusion coefficient
axis('equal')
title('Electron cyclotron diffusion coefficient
(nhu-ref*pth-ref^2)');
xlabel('ppar/pth-ref');
ylabel('pperp/pth-ref');
hold on;plot(pparres-cy-cyl-ref,pperpres-cy-cyl-ref,
'k-. ');hold off
hold on,plot(pparres-cy-cylOref,0,'k*'),hold off;
if bounce-mode >= 1,%Draw bounce domain boundaries
line([O,pnmax*sqrt(xmhubounce2(ir-display))], [O,pnmax*sqrt(i
-
xmhubounce2(ir-display))]);
line( [0,-pnmax*sqrt(xmhubounce2(ir-display))],
[0,pnmax*sqrt(i
-
xmhubounce2(ir-display))]);
if isnan(nhuloss) == 0,%Draw the toroidal ripple loss cone
line ( [0,pnmax*sqrt(1cos(xthetaloss(irdisplay)*pi/180)^2)],
[0,pnmax*cos(xthetaloss(ir-display)*pi/180)]
'LineStyle','-','color','r');
line( [0,-pnmax*sqrt (1cos(xthetaloss(ir-display)*pi/180)^2)],
[0,pnmax*cos(xthetaloss(irdisplay)*pi/180)],...
'LineStyle','-','color','r');
end
end
faxis = axis;dfaxisx = (faxis(2) - faxis(1))/100;dfaxisy
=
(faxis(4)
-
faxis(3))/100;
text(faxis(1) + dfaxis-x*5,faxis(4)
- dfaxis-y*5,['r/a =
',num2str(xrho(ir_display))]);
pause(0.1);
end
end
if isempty(find(rdke==ir-display))%no
xcurr-tot-cyl-ref(ir-display)
display of dke calculations
= xcurrOcyl-ref(ir-display);
xnormf-tot-cyl-ref(ir-display)
=
xnormf-cyl-ref(ir-display);
% only the flux-surface averaged density of absorbed power is physical
disp(' -----------------------------------------------------------------------------------------------------------------'));
infoyp(2,'Global results (normalized to reference units) at Bmin [theta-pol = 0
(deg)]:');
- Density [spherical]:
disp(['
',nuim2str(xnormf-ref(ir-display)),',
[cylindrical]:
',num2str(xnormf-cyl-ref(ir-display))])
disp(['
[cylindrical]:
-
RF current density [spherical]:
',num2str(xcurrcref(ir-display)),',
',...
num2str(xcurrOcyl-ref(ir-display))])
disp(['
- RF power density lost on collisions (coll) [spherical]:
',num2str(xPcoll-ref(irdisplay))])
150
disp('-----------------------------------------------------------------------------------------------------------------');
infoyp(2,'Global results (normalized to MKSA units) at Bmin [theta-pol = 0 (deg)]:');
disp(['
- Density (10+19 m-3) [spherical]:
,num2str(xnormf-ref (ir-display)*ne-ref/e19),', [cylindrical]: ',...
num2str(xnormf-tot-cyl-ref(ir-display)*ne-ref/e9)])
disp(['
- RF current density (MA/m2) [spherical]:
[cylindrical]:
,num2str(j.ref*xcurrOref(irdisplay)),',
disp(['
',...
num2str(j-ref*xcurr.cyl-ref(irdisplay))])
- RF power density lost on collisions (MW/m3) [spherical]:
',num2str(P-ref*xPcoll-ref(ir-display))])
disp(' ------------------------------------------------------------------------------------------------------------------);
infoyp(2,'Global flux surface averaged results (normalized to reference units):');
',num2str(xnormf-ref.fsav(ir-display))])
disp(['
- Density:
disp(['
- RF current density:
disp(['
- RF power density lost on collisions (coll) [spherical]:
[spherical]:
[spherical]:
',num2str(xcurrOref_fsav(ir-display)))
',num2str(xPcoll-ref-fsav(irdisplay))])
disp(['
- RF absorbed power density (drag)
[spherical]:
',num2str(xPabs-ref-fsav(ir-display))])
if xPabs-ref-fsav(ir-display) ~= 0,
disp(['
- RF current drive efficiency [spherical]:
',...
num2str(xcurrOref_fsav(ir-display)/(xPabs-ref_fsav(ir-display)+eps))])
else
disp(['
- RF current drive efficiency
[spherical]:
Inf,
[cylindrical]:
Inf'])
end
disp(['
- Runaway rate [spherical]:
',num2str(xRR-ref-fsav(ir-display,np-1))])
disp(['
- Bulk current [spherical]:
',num2str(xcurrB-ref_fsav(ir-display))])
disp('---------------------------------------------------------------------------------------------------------------);
infoyp(2,'Global flux surface averaged results (normalized to MKSA units):');
disp(['
- Density (10+19 m-3) [spherical]:
,num2str(xnormf-ref_fsa(ir-display)*ne-ref/1e19)])
disp(['
- RF current density (MA/m2) [spherical]:
',num2str(j-ref*xcurraref-fsav(ir-display))])
disp(['
- RF power density lost on collisions (coll) [spherical]:
',num2str(P-ref*xPcoll-ref-fsav(ir-display))])
disp(['
- RF absorbed power density (MW/m3) [spherical]:
,num2str(P-ref*xPabs-ref-fsav(ir-display))])
if xPabs-ref-fsav(ir-display)
disp(['
-=
0,
- RF current drive efficiency (A.m/M) [spherical]:
,...
num2str(jref*xcurroref_fsav(ir-display)/P-ref/xPabs-ref_fsav(ir-display))])
else
disp(['
- RF current drive efficiency (A.m/M)
[spherical]:
Inf,
[cylindrical]:
Inf'])
end
disp(['
- Runaway rate
(Kg.m-3.s-1)
[spherical]:
151
',num2str(xRR-ref-fsav(ir-display,np-
1)*nhu-ref*ne-ref*me)])
- Bulk current (MA/m2) [spherical]:
disp(['
',num2str(j-ref*xcurrB-ref-fsav(ir-display))])
disp('------------------------------------------------------------------------------------------------------------------
)
else
irdke-display = find((rdke==ir-display));
[XfOg-cyl-ref,ppar-cyl-ref,pperp-cyl-ref] = s2cyp(XfO-g-ref,pnrefmhucconv);
[XfOtp-cyl-ref,ppar-cyl-ref,pperp-cyl-ref] = s2cyp(XfO-tp-ref,pn-ref,mhu,cconv);
XfO-tpg-cyl-ref = XfO-g-cyl1ref + XfOtp-cyl-ref;
XfO-tpg-cyl-ref = XfO-tpg-cyl-ref.*(Xp-cyl-ref<max(ppar-cyl-ref));%Remove
values above
max(ppar)
XfO-tpg-cyl-ref = Xf 0_tpg-cyl-ref.*(Xmhu-cyl-ref<1-dmhu/2);%Remove
values outside from
the domain of calculation
Flbootstrap-cyl-ref = 2*pi*trapzyp([pperp-cyl-ref(:),Xpperp-cylref'.*XfOtpg-cyl-ref']);
% non cylindric term calculated by fp2dyp
XfO-tot-cyl-ref = Xf0_cyl-ref + XfO-g-cyl-ref + XfOtp.cyl-ref;
xcurr-tot-cyl-ref(ir-display)
=
2*pi*trapzyp( [pperp-cyl-ref(:) ,pperp-cylref(:)
*trapzyp( [ppar-cyl-ref(:),...
Xvpar-cyl-ref.*Xf0_tot-cyl-ref])']);
if display-mode >= 2
figure(9) ,plot(ppar-cyl-ref ,Flbootstrap-cyl-ref),ylabel('F1(bootstrap)') ,xlabel('ppar') ;zoom on
pnplot-ref = min(find(pn-ref>=4));
figure(10),plot(mhu,XfOtp-ref(pnplot-ref,:)','b--',mhu,XfQg-ref(pnplotref,:)','rylabel('ftp (b--) and g (r-.)');xlabel('mhu')
title('pn-ref = 4')
pause(O.1);
end
disp('---------------------------------------------------------------------------------------------------------------');
infoyp(2,'Global results (normalized to reference units) at Bmin
[theta-pol = 0
(deg)]:');
- Density [spherical] : ',num2str(xnormf-ref (ir-display)),',
disp(['
[cylindrical]:
',num2str(xnormf-cyl-ref(ir-display))])
disp(['
[cylindrical]:
-
RF only current density [spherical]:
',num2str(xcurrOref(ir-display)),',
,...
num2str(xcurrOcyl-ref(ir-display))])
disp(['
- Electron bootstrap current density [spherical]:
',...
num2str(xcurr-tot-ref (ir-display) - xcurr0_ref (ir-display)),...
',
[cylindrical]: ',num2str(xcurr-tot-cyl-ref (ir-display) xcurrOcyl-ref(ir-display))])
disp(['
- RF + electron bootstrap current density [spherical]:
152
',...
[cylindrical]:
num2str(xcurrtotref(ir-display)),',
,num2str(xcurr-tot-cyl-ref(ir-display))])
- Power density dissipated on collisions (FP solution only)
disp(['
[spherical]:
,num2str(xPcoll-ref(ir-display))])
- Power density dissipated on collisions (DKE solution)
disp(['
[spherical]:
,num2str(xPcoll-tot-ref(ir-display))])
disp(' ------------------------------------------------------------------------------------------------------------------)
-;
infoyp(2,'Global results (normalized to MKSA units) at Bmin [theta-pol = 0 (deg)]:');
- Density (10+19 m-3)
disp(['
[spherical]:
[cylindrical]: ,,...
num2str(xnormf-cyl-ref(ir-display)*ne-ref/1e19)])
',num2str(xnormf-ref(irdisplay)*ne.ref/1e19),',
- RF only current density (MA/m2) [spherical]:
disp(['
',num2str(j-ref*xcurrOref(ir-display)),...
1,
disp(['
[cylindrical]:
',num2str(j-ref*xcurro-cyl-ref(ir-display))])
- Electron bootstrap current density (MA/m2) [spherical]:
',...
num2str(jref*(xcurr-tot-ref (ir.display) - xcurr0_ref (ir-display))),...
, [cylindrical]: ',num2str(jref*(xcurr-tot-cyl-ref(ir-display)
xcurrOcyl-ref(ir-display)))])
disp(['
- RF + electron bootstrap current density (MA/m2)
num2str(j-ref*xcurr-tot-ref(ir-display)),',
[spherical]:
',...
[cylindrical]:
',num2str(j-ref*xcurr-tot-cyl-ref(ir-display))])
disp(['
- RF power density lost on collisions (FP solution only)(MW/m3)
[spherical]:
num2str(Pref*xPcoll-ref(ir-display))])
disp(['
- RF power density lost on collisions (DKE solution)(MW/m3)
[spherical]:
num2str(P-ref*xPcoll-tot-ref(ir.display))])
disp( ---------------------------------------------------------------------------------------------------------------);
infoyp(2,'Global flux surface averaged results (normalized to reference units):');
[spherical]:
',num2str(xnormf.-ref.-fsa(ir-display))])
disp(['
- Density:
disp(['
- RF only current density:
[spherical]:
,num2str(xcurrOref-fsav(ir-display))])
disp(['
- Electron Bootstrap current density:
',num2str(xcurr-tot-ref-fsav(ir-display)
[spherical]:
-
xcurrOref-fsav(ir-display))])
disp(['
- RF + electron bootstrap current density:
[spherical]:
',num2str(xcurr-tot-reffsav(irdisplay))])
disp(['
- Ion bootstrap current density:
',num2str(xcurr-i-mksafsav(ir-display)/j-ref)])
disp(['
- Total bootstrap current density [spherical]:
,num2str(xcurr-tot-ref-fsav(ir-display)
-
xcurrOref -fsav(ir-display) + xcurr-i-mksa-fsav(ir-display)/j-ref)]);
disp(['
- Hirshman model for bootstrap:
,num2str(xcurr-hir-mksa-fsav(ir-display)/j-ref)])
disp(['
- Sauter model for bootstrap:
,num2str(xcurr-sau-mksafsav(irdisplay)/j-ref)])
153
,...
disp(['
- RF + total bootstrap current density [spherical]:
',...
num2str(xcurr-tot-reffsav(ir-display) + xcurr_i-mksafsav(ir-display)/j-ref)]);
disp(['
- Power density dissipated on collisions (FP solution only)
[spherical]:
',num2str(xPcoll-ref-fsav(irdisplay))])
disp(['
- Power density dissipated on collisions (DKE solution) [spherical]:
',num2str(xPcoll-tot-ref-fsay(ir-display))])
disp(['
- RF absorbed power density (FP solution only) [spherical]:
',num2str(xPabs-reffsav(ir-display))])
disp(['
- RF absorbed power density (DKE solution) [spherical]:
',num2str(xPabs-tot-ref-fsav(ir-display))])
if xPabsreffsav(ir-display)
-=
0,
- RF current drive efficiency (FP solution only)
disp(['
[spherical]:
,...
num2str(xcurrOref-fsav(ir-display)/(xPabs-ref-fsa(ir-display)+eps))])
else
disp(['
[cylindrical]:
- RF current drive efficiency (FP solution only)
[spherical]:
Inf,
Inf'])
end
disp(['
- Runaway rate [spherical]:
',num2str(xRR-ref-fsav(ir-display,np-1))])
disp(['
- Bulk current [spherical]:
',num2str(xcurrB-ref-fsav(ir-display))])
disp('---------------------------------------------------------------------------------------------------------------');
infoyp(2,'Global flux surface averaged results (normalized to MKSA units):');
disp(['
- Density (10+19 m-3) [spherical]:
,num2str(xnormf-ref-fsav(ir-display)*ne-ref/1e19)])
disp(['
- RF only current density (MA/m2) [spherical]:
,num2str(j-ref*xcurrOreffsav(irdisplay))])
disp(['
- Electron bootstrap current density (MA/m2) [spherical]:
num2str(j-ref*(xcurr-tot-ref-fsa(ir-display)
disp(['
',...
- xcurr0_ref-fsa(ir.display)))])
- RF + electron bootstrap current density (MA/m2) [spherical]:
,num2str(j-ref*xcurr-tot-ref-fsa(ir-display))])
disp(['
- Ion bootstrap current density (MA/m2):
,num2str(xcurr-i-mksa-fsa(ir-display))])
disp(['
- Total bootstrap current density (MA/m2) [spherical]:
num2str(j-ref*(xcurr-tot-ref-fsa(ir-display)
xcurr-i-mksa-fsav(irdisplay))])
disp(['
',...
- xcurr0_ref-fsav(ir-display))
+
- Hirshman model for bootstrap (MA/m2):
,num2str(xcurr-hir-mksa-fsav(ir-display))])
disp(['
- Sauter model for bootstrap (MA/m2):
,num2str(xcurr-saumksa-fsav(ir-display))])
disp(['
- RF + total bootstrap current density (MA/m2):
[spherical]:
',...
num2str(j-ref*xcurrtot-ref-fsav(ir-display) + xcurr_i_mksa-fsa(ir-display))])
disp(['
- Power density dissipated on collisions (FP solution only)
[spherical]:
,num2str(xPcoll-ref-fsav(ir-display))])
disp(['
- Power density dissipated on collisions (DKE solution) [spherical]:
,num2str(xPcoll-totref-fsav(ir-display))])
disp(['
- RF absorbed power density (FP solution only) [spherical]:
,num2str(P-ref*xPabs-ref-fsav(ir-display))])
disp(['
- RF absorbed power density (DKE solution) [spherical]:
154
',num2str(P-ref*xPabs-totrefjfsav(ir-display))])
if xPabs-ref-fsav(ir-display) ~= 0,
- RF current drive efficiency (FP solution only) [spherical]:
disp(['
',...
num2str(j-ref*xcurrO-ref-fsav(ir-display)/P-ref/(xPabs-ref-fsav(ir-display)+eps))])
else
- RF current drive efficiency (FP solution only) [spherical]: Inf,
disp(['
[cylindrical]: Inf'])
end
disp(['
- Runaway rate (Kg.m-3.s-1) [spherical]: ',num2str(xRR-ref-fsav(ir-display,np1)*nhu-ref*ne-ref*me)])
- Bulk current (MA/m2) [spherical]:
disp(['
',num2str(j-ref*xcurrB-ref-fsav(ir-display))])
disp(' -------------------------------------------------------------------------------------------------------------------
end
end
if display-mode == 2
if nr-dke == 0 & nr >
1
figure(200),subplot (221) ,plot(xrho,j-ref*xcurrOref_fsav,
set(gca,'xlim',[0,1])
zoom on;grid on;
xlabel('r/a');ylabel('<J> (MA/m^2)');
title('Flux Surface Averaged Current density');
figure(200),subplot(222) ,plot(xrho,P-ref*xPabs-ref,
set(gca,'xlim',[0,1])
zoom on;grid on;
xlabel('r/a');ylabel('P_{abs} (MW/m^3)');
title('Density of Absorbed Power');
'+');
'+');
figure(200),subplot(223) ,plot(xrho,xTe, 'ro' ,xrho,xTi, 'bx');
set(gca,'xlim',[0,1])
zoom on;grid on;
xlabel('r/a');ylabel('T (keV)');
title('Temperature')
legend('Electron','Ion');
figure(200),subplot(224) ,plot(xrho,xne/1e20, 'ro' ,xrho,xni/1e20, 'bx');
set(gca,'xlim',[0,1])
zoom on;grid on;
xlabel('r/a');ylabel('n (m-3)\times 10-20');
title('Densities')
legend('Electron','Ion');
155
elseif nrdke >1
figure(200),subplot(221),plot(xrho,jref*xcurrOreffsav,
set(gca,'xlim',[0,1])
zoom on;grid on;
'+');
xlabel('r/a');ylabel('<J> (MA/m^2)');
title('Flux Surface Averaged Current density');
figure(200) ,subplot(222) ,plot(xrho,P-ref*xPabs-ref,'+');
set(gca,'xlim',[0,1])
zoom on;grid on;
xlabel('r/a');ylabel('P_{abs} (MW/m^3)');
title('Density of Absorbed Power');
figure(200) ,subplot(223) ,plot(xrho,xTe, 'ro' ,xrho,xTi, 'bx');
set(gca,'xlim',[0,1])
zoom on;grid on;
xlabel('r/a');ylabel('T (keV)');
title('Temperature')
legend('Electron','Ion');
figure(200) ,subplot(224) ,plot(xrho,xne/1e20, 'ro' ,xrho,xni/1e20, 'bx');
set(gca,'xlim',[0,1])
zoom on;grid on;
xlabel('r/a');ylabel('n (m-3)\times 10-20');
title('Densities')
legend('Electron','Ion');
srho = 0:0.01:1;
sq = (qmax - qmin)*srho.-eq + qmin;
[scurr-hir-mksa-fsav,scurr-sau-mksafsavscurr-i-mksa-fsav,scurr_lor-mksa-fsav]
hsneomodels9jd(srho, ap,Rp,Zi, sq,Bt,TeO,Tea,eTe,neO,nea,ene,TiO,Tia,eTi);
=
%Calculation of the flux averaged bootstrap current from Hirshman and Sauter models (bulk
contribution)
figure(201),
plot(xrho(rdke),j-ref*xcurr-totref-fsav(rdke) + xcurr_i_mksafsav(rdke),'ro',...
xrho(rdke) ,j-ref*xcurrO-ref-fsav(rdke), 'bx',...
xrho(rdke),j-ref*(xcurr-totref-fsav(rdke) - xcurr0_reffsav(rdke)) +
xcurr_i_mksa-fsav(rdke),'k+',...
srho,scurr-hir-mksafsav,':',...
srho,scurr-saumksajfsav,'-');
set(gca,'xlim',[0,1])
legend('J^{tot}','J^{RF}','J^{BS}','J^{BS} (Hirshman),'J^{BS}
zoom on;grid on;
xlabel('r/a');ylabel('<J> (MA/m~2)');
title('Flux Surface Averaged Current densities');
end
156
(Sauter)');
end
157
F.2
fp2d9yp
function [XXf0,XXf _tp,XXf Og,XXSpO,XXSmhuO,XXSpOg,XXSmhuOg,XXSpOtp,...
XXSmhuOtp,PcollO,PcolltotPcollOfsavPcolltot_fsav,PabsO_fsav,...
Pabs_totfsav,normf,normfjfsav,curr_0,currtot,currOfsav,curr_tot_fsav,...
currB-fsav,RR-fsav,Te-out-norm,Xpn,Xmhu,mhubounce2 = ...
fp2d9yp-new_2(betath,Rp-norm,Te-norm,ne-norm,Zi,mi,ni-norm,Ti-norm,...
pn,mhu,XXDlh,XXDcy,param-cy,XXnhuloss,epsi,bouncemode,rdke,ia,ftp-norm,...
XXDr,XXVr,delta,Nq,trunc,dtn,precO,nit,invproc,display,storemode,XXfinit);
%
Self-consistent 2-D relativistic solver of the drift kinetic equation in momentum space
%
with Beliaev-Budker collision integrals and toroidal magnetic losses
%
Fully implicit or Crank-Nicholson time differencing schemes
%
LH and EC wave quasilinear diffusion calculations + ohmic electric field
%
See references:
%
-
C. Karney, Comput. Phys. Rep. 4 (1986) 183.
%
-
B. Braams, C. Karney, Phys.fluids B1 (1989) 1355
%
-
M. Shoucri and I. Shkarofsky, Comp. Phys. Comm. 82 (1994) 287
%-
I. Shkarofsky and M. Shoucri, Nuclear Fusion, 37 (1997) 539
%
-
J.P.S. Bizarro and P. Rodrigues, Nuclear Fusion, 37 (1997) 1509
Y. Peysson and M. Shoucri, Comp. Phys. Comm. 109 (1998) 55
%%
-
S.D. Schultz, A. Bers and A.B. Rahm, AIP Proc. of the 12th Conf.
%
%
Radio Frequency Power in Plasmas, Savannah
(USA),
403 1997,327
(see lhfp2d9yp.m)
%Reste
faire:
%3) Diffusion radiale avec pas variable
%6) Calcul selfconsistent du champ electrique ohmique avec bootstrap
%7) Valider les integrales de bounce non-axisymetrique
%Lower Hybrid wave: D(p//,pper) = Dlh(v//min,v//max)
%Electron cyclotron wave: D(p//,pper) =
%
%
Dcy*gamma*pperp^2*exp(-((gamma-omega)/p//betath
- n//)^2/dn//^2)/abs(p//)
Input:
%
- betath: Normalized reference thermal velocity (vth/c) [1,1]
%
- Rp-norm: Normaized plasma major radius (ap) [1,1]
%
- Tenorm: normalized electron temperature (Te/Te-ref) [1,nr]
'
- ne-norm: normalized electron density (ne/ne-ref) [1,nr]
I
- Zi: ion charges [1,m]
%
- mi:
'
- ni-norm: normalized ion density (ni/neref) [m,nr]
'
- pn: normalized electron momentum space grid (pth-ref) [1,np]
ion masses (u.m.a.)
[1,m]
158
- mhu: cos(theta) angular grid [1,nmhu]
7
7
- XXDlh: LH normalized quasi-linear diffusion coefficient
(nhuthref*pthref^2)
[np,nmhu,nr]
- XXDcy: EC normalized quasi-linear diffusion coefficient (nhuth-ref*pth-ref-2)
%
[np,nmhu,nr]
%
- param-cy: *******************normalized EC frequency [1,nr]
%
%
- XXnhuloss: normalized loss frequency (nhuth-ref) [np,nmhu,nr]
- epsi: normalized ohmic electric field (pth-ref*nhuth-ref/e) [1,nr]
%
- bouncemode: bounce averaged option: (0) no trapping, (1) explicit flux
and distribution symmetrization, (2) full implicit calculations
%
- rdke: list for positions for dke calculations [1,nr-dke]. if rdke==[],
fokker-planck calculations only
%
%
%
- ia: inverse aspect ratio [1,nr]
%
- ftp-norm: normalization factor for the drift kinetic equation
%
- XXDr: Fast electron radial diffusion (nhuth-ref*ap^2) [np,nmhu,nr]
%
- XXVr:
[1,nr]
[np,nmhu,nr]
%
Fast electron radial pinch (nhuth-ref*ap)
- delta: Toroidal magnetic ripple depth [1,nr]
%
- Nq: Number of toroidal magnetic coils * local safety factor [1,nr]
%
- trunc: 1 ->
truncated e-e collision operator, 0 ->
Maxwellian e-e collision operator
[1,1]
.
(default = 0)
- dtn: normalized integration time step (dtn > 0: fully implicit time differencing
%
scheme,
%
dtn < 0: Crank-Nicholson time differencing scheme) (1/nhu-th) [1,1]
%
%
(default = 1000)
- precO: required accuracy on the residu for f and for g (M) [1,2]
(default = [ie-10,le-10])
%
%
- nit: maximum number of iterations
%
%
- invproc: matrix inversion procedure (1:
- display: display mode parameter (full: 2,
2-5: Iterative method)
[1,1]
partial:
1,
no: = 0)
index of the radial
position where results are displayed
(default = [0,1])
%
%
LU decomposition,
(default = 1)
%
%
%
for f and g [1,2]
(default = [20,20])
- storemode: Temporary storing option for sparse matrices calculations (in RAM: 0, in
disk file 1)
%
%
%
%
%
(default = 0)
- XXfinit: initial
guess for the 2D electron momentum distribution function
Output:
- XXf0: 2D electron momentum distribution function normalized to the normalized density
[np,nmhiu,nr]
%%
-
XXfOg: Second neoclassical correction (g) [np,nmhu,nr]
%
-
%'
XXSpO: p-component of the 2D electron momentum flux for fO[np,nmhu,nr]
XXSmhuO: mhu-component of the 2D electron momentum flux for fO[np,nmhu,nr]
-
XXSpOg:
%-
[np,nmhu,nr]
(default = Maxwellian)
XXf0_tp: First neoclassical correction
(f-tp) [np,nmhu,nr]
p-component of the 2D electron momentum flux for g[np,nmhu,nr]
XXSmhuOg: mhu-component of the 2D electron momentum flux for g[np,nmhu,nr]
159
%
- XXSpOtp: p-component of the 2D electron momentum flux for ftp[np,nmhu,nr]
%
- XXSmhuOtp:
%
- PcollO: Power loss on collisions from fO(me*ne*nhuth*vth-2)[1,nr]
%
%
- Pcoll-tot: Power loss on collisions from f0+g+f-tp(me*ne*nhuth*vth~2)[1,nr]
%
- Pcoll-tot-fsav: FS averaged Power loss on collisions from
- PcollOfsav:
mhu-component of the 2D electron momentum flux for ftp[np,nmhu,nr]
FS averaged Power loss on collisions from f0(me*ne*nhuth*vth^2)[1,nr]
fO+g+f-tp(me*ne*nhuth*vth^2)[1,nr]
%
%
- PabsOfsav:
FS averaged
RF absorbed power from fO(me*ne*nhuth*vth~2) [1,nr]
RF absorbed power from f0+g+f-tp(me*ne*nhuth*vth^2)
%
- currO0: RF + ohmic current density [1,nr]
%
- curr-tot: Total current density (RF + ohmic + boostrap current)
%
- curr_0_fsav:
%
- currtotfsav:
- Pabstotfsav: FS averaged
FS averaged
RF + ohmic current density
FS averaged
[1,nr]
[1,nr]
[1,nr]
Total current density (RF + ohmic + boostrap current)
[1,nr]
%
- currB: Bulk current (RF + ohmic + boostrap - runway current)
%
- RR: Runaway rate through the surface of a sphere of radius p [nr,np]
%
- Tenout: New temperature of the bulk (mec2)
%
- normf-tot-fsav:
%
- curr0-fsav: Flux surface averaged RF + ohmic current density [1,nr]
%
- currtotfsav: Total flux surface averaged current density (RF + ohmic + boostrap
[1,nr]
[1,nr]
Flux surface averaged norm of the distribution function
[1,nr]
current)
%
- Xpn: Momentum grid normalized to the reference momentum value [np,nmhu]
%
- Xmhu:
%
- mhubounce2:
%
Angular grid [np,nmhu]
Square of the bounce averaged cosine angle
[1,1]
Warning: none
%Last Update: 05/05/2002
%by Y. Peysson CEA-DRFC 21/07/1997 <peysson~drfc.cad.cea.fr>
%Update for the drift kinetic calculations done by J. Decker and Y. Peysson 06-10/2001
if nargin < 23,
error('Wrong number of input arguments in fp2d9yp')
elseif nargin == 23,
trunc = 0;
dtn = 1000;
precO
[le-10,le-10];
nit = 50;
invproc =
1;
display = [0,1];
store_mode = 0;
elseif nargin == 24,
dtn = 1000;
precO
=
[le-10,1e-10];
nit = [20,20];
invproc = 1;
display = [0,1];
160
store-mode = 0;
elseif nargin == 25,
precO = [le-10,le-10];
nit = [20,20];
invproc = 1;
display = [0,1];
store-mode
elseif nargin
=
0;
== 26,
nit = [20,20];
invproc = 1;
display =
[0,1];
store-mode = 0;
elseif nargin
== 27,
invproc = 1;
display = 1;
store-mode
=
0;
elseif nargin == 28,
display = [0,1];
store-mode = 0;
elseif nargin == 29,
store-mode = 0;
end
narginmax = 31;
timeCPUO = cputime;
pfactor = 20;%Supergrid for very accurate collison integral calculation
me = 1/1860;%Electron mass (u.m.a.)
if dtn > 0,
scheme = 1;%Implicit differencing scheme
elseif dtn < 0,
dtn
= abs(dtn);
scheme = 2;%Crank-Nicholson differencing scheme
end
spparms('default');XSet parameters for sparse matrix ordering algorithms and direct linear
equation solver
spparms('supernd',4);
spparms('rreduce',4);
precOf = precO(1);
precOg = precO(2);
display-mode = display(1);
ir-display = display(2);
nit_f = nit(1);
nitg = nit(2);
nr-dke = length(rdke);
161
xpsib = paramncy(1,:);
xomega-cy = param-cy(2,:);
if bouncemode == O,nr-dke
==
O;end
if nr-dke >= 1,bouncemode = 2;end
nmhu = length(mhu);%Size of angular grid
npn = length(pn);%Size of momentum grid
nr = length(ia);%Size of radial grid
if store-mode == 1,
delete MXPmat.mat;
save MXPmat.mat nr;
end
%Masks for collision integrals
Xa
masku = triu(ones(npn,npn));
maskl = tril(ones(npn,npn));
%Momentum grid
pth = betath;
dpn = pn(3) - pn(2);
pn2 = pn.*pn;
pn3 = pn2.*pn;
xpn = pn'*ones(i,npn);
xpn2 = xpn.*xpn;
xpn3 = xpn2.*xpn;
Xpn
=
pn'*ones(i,nmhu);
Xpn2 = Xpn.*Xpn;
Xpn3 = Xpn2.*Xpn;
pnm = pn - dpn/2;pnm(1) = 0;
pn2m = pnm.*pnm;
Xpnm = pnm'*ones(i,nmhu);
Xpn2m = Xpnm.*Xpnm;
pnp = pn + dpn/2;
pn2p = pnp.*pnp;
Xpnp = pnp'*ones(i,nmhu);
Xpn2p = Xpnp.*Xpnp;
z = betath*pn;
z2 =z.*z;
z3 = z2.*z;
zm = betath*pnm;
z2m = zm.*zm;
zp = betath*pnp;
z2p = zp.*zp;
162
xz = z'*ones(1,length(z));
xz2 = xz.*xz;
xz3
= xz2.*xz;
gamma = sqrt(1 + z2);
gamma2
=
gamma.*gamma;
gammam
=
sqrt(1 + z2m);
gamma2m
=
gammam. *gammam;
gammap = sqrt(1 + z2p);
gamma2p = gammap.*gammap;
xgamma = gamma'*ones(1,length(gamma));
xgamma2 = xgamma.*xgamma;
Xgamma = gamma'*ones(1,nmhu);
Xgammam = gammam'*ones(1,nmhu);
Xgammap = gammap'*ones(1,nmhu);
sigma
asinh(z);
sigmam = asinh(zm);
sigmap = asinh(zp);
xsigma
=
sigma'*ones(i,length(sigma));
J1 = -3*gamma
+ sigma.*(3.0./z + 2*z);
J2 = gamma - sigma./z -2*gamma.*z2/3;
J3 = -3*gamma.*sigma./z2 + 3.0./z + z
-
2*z2.*z/5;
J4 = gamma.*sigma.*(15.0./z2 + 6) - 15.0./z - 11*z;
xJ1 = -3*xgamma + xsigma.*(3.0./xz + 2*xz);
xJ2 = xgamma - xsigma./xz - 2*xgamma.*xz2/3;
xJ3 = -3*xgamma.*xsigma./xz2 + 3.0./xz + xz - 2*xz2.*xz/5;
xJ4 = xgamma.*xsigma.*(15.0./xz2 + 6)
-
15.0./xz
-
11*xz;
v = pn./gamma;
v.*v;
vm = pnm./gammam;
v2
v2m = vm.*vm;
vp = pnp./gammap;
v2p = vp.*vp;
xv =
v'*ones(1,length(v));
xv2 = xv.*xv;
Ungular grid
dmhu = mhu(3)
- mhu(2);
mhum = mhu - dmhu/2;mhum(i)
=
-1;
mhup = mhu + dmhu/2;mhup(nmhu) = 1;
Xmhu = ones(npn,i)*mhu;
Xmhu2 = Xmhu.*Xmhu;
X1mmhu2 = 1 - Xmhu2;
Xmhum = ones(npn,1)*mhum;
Xmhum(find(Xmhum==O))
= eps;
163
Xmhu2m = Xmhum.*Xmhum;
Xmhu2m(find(Xmhu2m==O)) = eps;
X1mmhu2m = 1 - Xmhu2m;
Xmhup = ones(npn,1)*mhup;
Xmhup(find(Xmhup==O))= eps;
Xmhu2p = Xmhup.*Xmhup;
X1mmhu2p = 1 - Xmhu2p;
mp = [2:npn-1];
mpm = mp - 1;
mpp = mp + 1;
mm = [2:nmhu-1];
mmm = mm - 1;
mmp = mm + 1;
%Super momentum grid definitions for collision integrals
spn = [eps:dpn/pfactor:max(pn)];
nspn = length(spn);
spn2 = spn.*spn;
spn3
sz
= spn2.*spn;
= betath*spn;
sz2 = sz.*sz;
sz3 = sz2.*sz;
sgamma = sqrt(1 + sz2);
sgamma2 = sgamma.*sgamma;
ssigma = asinh(sz);
sv = spn./sgamma;
sv2
=
sJ1
= -3*sgamma
sv.*sv;
+ ssigma.*(3.0./sz + 2*sz);
sJ2 = sgamma - ssigma./sz - 2*sgamma.*sz2/3;
sJ3 = -3*sgamma.*ssigma./sz2 + 3.0./sz + sz
sJ4
= sgamma.*ssigma.*(15.0./sz2 + 6) -
-
2*sz2.*sz/5;
15.0./sz -
11*sz;
if nargin < narginmax,XXfinit = zeros(np,nmhu,nr);end;%For
normf0 = zeros(nr,nit-f);
normfl = zeros(nr,nit-f);
currO = zeros(nr,nit-f);
curri = zeros(nr,nit-f);
prec-curr = zeros(nr,nit-f);
residu-f
=
ones(nr,nit-f);
residu-g = ones(nr,nit-g);
ns = zeros(i,nr);
164
the initial
guess
nstp = zeros(1,nr);
ns-g = zeros(i,nr);
mhubounce2 = zeros(1,nr);
rho = zeros(1,nr);
Pabs = zeros(1,nr);
Pcoll = zeros(1,nr);
Tenormout = zeros(1,nr);
RR = zeros(nr,npn);
mhubounce2 = zeros(1,nr);
XXI
XXf
zeros(npnnmlunr);
=
zeros(npn,nmhu,nr);
XXfOtp= zeros(npn,nmu,nr);
XXfO._g = zeros(npn,nmhu,nr);
XXSp = zeros(npn,nmhu,nr);
XXSmhu = zeros(npn,nmhu,nr);
XXfM = zeros(npn,nmu,nr);
XXfMgth = zeros(npn,nmhu,nr);
XXF = zeros(npnnnhu,nr);
XXFm = zeros(npn,nmhu,nr);
XXFp = zeros(npn,nmhu,nr);
XXA = zeros(npn,nmu,nr);
XXAm = zeros(npn,nmhu,nr);
XXAp = zeros(npn,nmhu,nr);
XXBt = zeros(npn,nmhu,nr);
XXDlhim = zeros(npn,nhu,nr);
XXDlhip = zeros(npnnmhunr);
XXDlhjm = zeros(npn,nmhu,nr);
XXDlhjp = zeros(npn,nmhu,nr);
XXDcyim = zeros(npn,nmbu,nr);
XXDcyip = zeros(npn,nmhu,nr);
XXDcyjm
=
zeros(npnnmhunr);
XXDcyjp = zeros(npn,nmhu,nr);
XXsssl = zeros(npn,nmu,nr);
XXlambda = zeros(npn,nmu,nr);
XXmsomOs = zeros(npn,nmuli,nr);
XXdphib = zeros(npn,nmhu,nr);
if nrdke >= 12
I
(nr-dke == 0 & nr >= 12)
radialmode = 1;
else
radialmode = 0;
if display-mode >= 1
infoyp(2,['WARNING:
radial mesh too coarse,
for radial diffusion or pinch calculations'])
disp('---------------------------------------------------------------------------------------------------------------);
165
end
end
if display-mode >=I,
disp('Calculation parameters:');
'I
if (storemode == 0), disp('
- Matrices storage in RAM');
else
- Matrices storage in disk file');end
disp('
if (invproc == 1),
disp('
- Full LU factorization method');
elseif (invproc
elseif (invproc
elseif (invproc
elseif (invproc
== 2),
disp('
- QMR iterative method');
== 3),
disp('
- BICGSTAB iterative method');
== 4),
disp('
- CGS iterative method');
== 5),
disp('
- PCG iterative method');end
if (scheme == 1),
disp(['
- Implicit time differencing scheme']);
elseif (scheme == 2),disp(['
- Crank-Nicholson time differencing scheme']);end
if (trunc == 0),
disp(['
- Maxwellian e-e collision operator']);
elseif (trunc == 1),
disp(['
- Truncated e-e collision operator']);end
if (bounce-mode == 0),
- No bounce averaging']);
disp(['
elseif (bounce-mode == 1),
- Bounce averaging + explicit symmetrization in the bounce domain (9 diagonals
disp(['
operator, dtn <<
1)']);
elseif (bounce-mode == 2),
disp(['
- Bounce averaging + implicit symmetrization in the bounce domain (15 diagonals
operator)']);end
if (nr-dke == 0),
disp(['
- Resolution of the Fokker-Planck equation']);
elseif (nrdke >= 1),
disp(['
- Resolution of the drift kinetic equation']);end
disp(['
- Momentum grid: ',int2str(npn)]);
disp(['
- Angular grid: ',int2str(nmhu)]);
disp(['
- Total radial grid: ',int2str(nr)]);
if nr-dke == 0
disp(['
- Effective radial grid:
',int2str(nr)]);
disp(['
- Effective radial grid:
',int2str(nrdke)]);
else
end
disp(I -----------------------------------------------------------------------------------------------------------------'Y);
end
timeO = clock;
for ir = 1:nr,
rho(ir) = ia(ir)*Rp-norm;
if (bouncemode >= 1) & (ia(ir) > 0),XBounce-averaged factors
mhubounce2(ir) = 2*ia(ir)/(1+ia(ir));
166
XY = zeros(size(Xmhu));
XY(Xmhu2 > mhubounce2(ir)) = mhubounce2(ir)./Xmhu2(Xmhu2 > mhubounce2(ir));
XY(Xmhu2 <= mhubounce2(ir)) = Xmhu2(Xmhu2 <= mhubounce2(ir))/mhubounce2(ir);
[XK,XE] = ellipke(XY);
XJO = zeros(size(Xmhu));
XJO(Xmhu2 > mhubounce2(ir)) = XK(Xmhu2 > mhubounce2(ir));
XJO(Xmhu2 <= mhubounce2(ir)) = abs(Xmu(Xmhu2 <= mhubounce2(ir))).*XK(Xmhu2 <=
mhubounce2(ir) )/sqrt (mhubounce2(ir));
XJ2 = zeros(size(Xmhu));
XJ2(Xmhu2 > mhubounce2(ir)) =
Xmhu2(Xmhu2 > mhubounce2(ir)).*(XK(Xmhu2 > mhubounce2(ir)) - XE(Xmhu2 >
mhubounce2(ir)))/mhubounce2(ir);
XJ2(Xmhu2 <=
mhubounce2(ir))
=
abs(Xmhu(Xmhu2 <= mhubounce2(ir))).*(XK(Xmhu2 <=
mhubounce2(ir))
- XE(Xmhu2 <=
mhubounce2(ir))) /sqrt (mhubounce2(ir));
XJ4 = (2*(i + Xmhu2/mhubounce2(ir)).*XJ2
- Xmhu2.*XJO/mhubounce2(ir))/3;
XYm = zeros(size(Xmhum));
XYm(Xmhu2m > mhubounce2(ir)) = mhubounce2(ir)./Xmhu2m(Xmhu2m > mhubounce2(ir));
XYm(Xmhu2m <= mhubounce2(ir)) = Xmhu2m(Xmhu2m <= mhubounce2(ir))/mhubounce2(ir);
[XKm,XEm] = ellipke(XYm);
XJ~m = zeros(size(Xmhum));
XJOm(Xmhu2m > mhubounce2(ir)) = XKm(Xmhu2m > mhubounce2(ir));
XJ~m(Xmhu2m <= mhubounce2(ir)) = abs(Xmhum(Xmhu2m <= mhubounce2(ir))).*XKm(Xmhu2m
mhubounce2(ir) )/sqrt (mhubounce2(ir));
XJ2m = zeros(size(Xmhum));
XJ2m(Xmhu2m > mhubounce2(ir)) =
Xmhu2m(Xmhu2m > mhubounce2(ir)).*(XKm(Xmhu2m > mhubounce2(ir)) - XEm(Xmhu2m >
mhubounce2(ir)))/mhubounce2(ir);
XJ2m(Xmhu2m <= mhubounce2(ir)) =
abs(Xmhum(Xmhu2m <= mhubounce2(ir))).*(XKm(Xmhu2m <= mhubounce2(ir)) XEm(Xmhu2m <= mhubounce2(ir)))/sqrt(mhubounce2(ir));
XJ4m = (2*(i + Xmhu2m/mhubounce2(ir)).*XJ2m - Xmhu2m.*XJOm/mhubounce2(ir))/3;
XYp = zeros(size(Xmhup));
XYp(Xmhu2p > mhubounce2(ir))
XYp(Xmhu2p <= mhubounce2(ir))
mhubounce2(ir)./Xmhu2p(Xmhu2p
=
=
> mhubounce2(ir));
Xmhu2p(Xmhu2p <= mhubounce2(ir))/mhubounce2(ir);
[XKp,XEp] = ellipke(XYp);
XJOp = zeros(size(Xmhup));
XJ~p(Xmhu2p > mhubounce2(ir)) = XKp(Xmhu2p > mhubounce2(ir));
XJOp(Xmhu2p <= mhubounce2(ir)) = ...
abs(Xmhup(Xmhu2p <= mhubounce2(ir))).*XKp(Xmhu2p <=
mhubounce2(ir))/sqrt(mhubounce2(ir));
XJ2p = zeros(size(Xmhup));
XJ2p(Xmhu2p >
mhubounce2(ir))
=
Xmhu2p(Xmhu2p > mhubounce2(ir)).*(XKp(Xmhu2p
> mhubounce2(ir))
- XEp(Xmhu2p >
mhubounce2(ir)))/mhubounce2(ir);
XJ2p(Xmhu2p <= mhubounce2(ir)) =
abs(Xmhup(Xmhu2p <= mhubounce2(ir))).*(XKp(Xmhu2p <= mhubounce2(ir))
167
-
<=
XEp(Xmhu2p <=
mhubounce2(ir)))/sqrt(mhubounce2(ir));
XJ4p = (2*(1 + Xmhu2p/mhubounce2(ir)).*XJ2p
- Xmhu2p.*XJOp/mhubounce2(ir))/3;
%As done by M. Ju et al. in EUR-CEA-FC-1712 report
Xlambda = zeros(size(XY));Xlambdam = zeros(size(XYm));Xlambdap = zeros(size(XYp));Xlambda
Xsssl = zeros(size(XY));Xssslm = zeros(size(XYm));Xssslp = zeros(size(XYp));%s*/lambda
Xdphib
zeros(size(XY));Xdphibm = zeros(size(XYm));Xdphibp = zeros(size(XYp));%<Delta>
%Axisymmetric bounce-averaging terms
Xlambda(Xmhu2 > mhubounce2(ir)) =
2*(XJO(Xmhu2 > mhubounce2(ir))
0.5*mhubounce2(ir)*XJ2(Xmhu2
-
> mhubounce2(ir)))/pi;
%Parameter lambda for circulating electrons
Xlambdam(Xmhu2m > mhubounce2(ir)) = ...
2*(XJOm(Xmhu2m > mhubounce2(ir)) -
0.5*mhubounce2(ir)*XJ2m(Xmhu2m
>
0.5*mhubounce2(ir)*XJ2p(Xmhu2p
>
mhubounce2(ir)))/pi;
Xlambdap(Xmhu2p > mhubounce2(ir)) =
2*(XJOp(Xmhu2p > mhubounce2(ir))
-
mhubounce2(ir)))/pi;
Xlambda(Xmhu2 <= mhubounce2(ir)) =
2*(XJO(Xmhu2 <= mhubounce2(ir))
-
0.5*mhubounce2(ir)*XJ2(Xmhu2
<=
mhubounce2(ir)))/pi;
%Parameter lambda for bouncing electrons
Xlambdam(Xmhu2m <= mhubounce2(ir)) = ...
2*(XJOm(Xmhu2m <= mhubounce2(ir)) -
0.5*mhubounce2(ir)*XJ2m(Xmhu2m
<=
0.5*mhubounce2(ir)*XJ2p(Xmhu2p
<=
mhubounce2(ir)))/pi;
Xlambdap(Xmhu2p <= mhubounce2(ir)) =
2*(XJOp(Xmhu2p <= mhubounce2(ir))
-
mhubounce2(ir)))/pi;
0/=
Xlambdam(find(Xlambdam==O))
Xlambdap (find (Xlambdap==O) )
= eps;
=eps ;
Xsssl(Xmhu2 > mhubounce2(ir))
=
sqrt((1 + ia(ir))/(1 - ia(ir)))./Xlambda(Xmhu2 > mhubounce2(ir)); %Parameter s*/lambda
for circulating electrons
Xssslm(Xmhu2m > mhubounce2(ir)) = sqrt((i + ia(ir))/(i - ia(ir)))./Xlambdam(Xmhu2m >
mhubounce2(ir));
Xssslp(Xmhu2p > mhubounce2(ir)) = sqrt((i + ia(ir))/(1 - ia(ir)))./Xlambdap(Xmhu2p >
mhubounce2(ir));
Xdphib(Xmhu2 > mhubounce2(ir))
=
-2/pi*mhubounce2(ir)*(XJ2(Xmhu2 > mhubounce2(ir))
-
0.5*mhubounce2(ir)*XJ4(Xmhu2 > mhubounce2(ir)))./Xlambda(Xmhu2 > mhubounce2(ir));
%Parameter <Delta>0 for circulating electrons
Xdphibp(Xmhu2p > mhubounce2(ir)) = ...
-2/pi*mhubounce2(ir)*(XJ2p(Xmhu2p > mhubounce2(ir))
0.5*mhubounce2(ir)*XJ4p(Xmhu2p
-
> mhubounce2(ir))) ./Xlambdap(Xmhu2p > mhubounce2(ir));
Xdphibm(Xmhu2m > mhubounce2(ir)) = ...
168
-2/pi*mhubounce2(ir)*(XJ2m(Xmhu2m > mhubounce2(ir)) 0.5*mhubounce2(ir)*XJ4m(Xmhu2m > mhubounce2(ir)))./Xlambdam(Xmhu2m > mhubounce2(ir));
Xsssl(Xmhu2 <= mhubounce2(ir)) = 0;%Parameter s*/lambda for bouncing electrons
Xssslm(Xmhu2m <= mhubounce2(ir)) = 0;
Xssslp(Xmhu2p <= mhubounce2(ir)) = 0;
Xdphib(Xmhu2 <= mhubounce2(ir)) = ...
-2/pi*mhubounce2(ir)*(XJ2(Xmhu2 <= mhubounce2(ir)) 0.5*mhubounce2(ir)*XJ4(Xmhu2 <= mhubounce2(ir)))./Xlambda(Xmhu2 <= mhubounce2(ir));
%Parameter <Delta>0 for bouncing electrons
Xdphibp(Xmhu2p <= mhubounce2(ir)) = ...
-2/pi*mhubounce2(ir)*(XJ2p(Xmhu2p <= mhubounce2(ir))
-
0.5*mhubounce2(ir)*XJ4p(Xmhu2p <= mhubounce2(ir))) ./Xlambdap(Xmhu2p <=
mhubounce2(ir));
Xdphibm(Xmhu2m <= mhubounce2(ir)) =
-2/pi*mhubounce2(ir)*(XJ2m(Xmhu2m <= mhubounce2(ir)) 0.5*mhubounce2(ir)*XJ4m(Xmhu2m <= mhubounce2(ir)))./Xlambdam(Xmhu2m <=
mhubounce2(ir));
Xmhu-psib = sign(Xmhu).*sqrt(i - xpsib(ir)*(i - Xmhu.~2));
Xmhu-psib = Xmhu-psib + eps*(Xmhu-psib==0);
Xmhu-psibm = sign(Xmhum).*sqrt(i - xpsib(ir)*(i - Xmhum.-2));
Xmhu-psibm = Xmhupsibm + eps*(Xmhu-psibm==0);
Xmhu-psibp = sign(Xmhup).*sqrt(i - xpsib(ir)*(1 - Xmhup.^2));
Xmhu-psibp = Xmhu-psibp + eps*(Xmhu.psibp==O);
XAsAeffcy = real(Xmhu./Xlambda./Xmhupsib);XEffective area of a magnetic surface for
localized EC absorption
XAsAeffcym = real(Xmhum./Xlambdam./Xmhu-psibm);
XAsAeff-cyp = real(Xmhup./Xlambdap./Xmhu-psibp);
XRlambdam = Xlambdam./Xlambda;
XRlambdap = Xlambdap./Xlambda;
%Bounce average correction term for I(fm,ftildei)
Xmsom0s = zeros(size(XY)); % = {mhu^2/mhuO~2} = [1 - {psi}*[1-mhu0^2]]/mhuO^2
Xmsom0s(Xmhu2 <= mhubounce2(ir)) = 0;
Xmsom0s(Xmhu2 > mhubounce2(ir))
2/pi*(XJO(Xmhu2 > mhubounce2(ir)) + ...
(0.5 - 1./Xmhu2(Xmhu2 > mhubounce2(ir)))*mhubounce2(ir).*XJ2(Xmhu2 > mhubounce2(ir))
+ (3/8 - 0.5./Xmhu2(Xmhu2 > mhubounce2(ir)))* ...
mhubounce2(ir) .~2.*XJ4(Xmhu2 > mhubounce2(ir))) ./Xlambda(Xmhu2 > mhubounce2(ir));
XXmsom0s(:,:,ir) = Xmsom0s;
%Non-axisymmetric corrections
(********* to be cross-checked ***********)
if 0,
%
if delta(ir) > 0,
169
Xlambdacorr = zeros(size(XY));Xlambdacorrm = zeros(size(XYm));Xlambdacorrp
zeros(size(XYp));%lambdal
Xdphiblambdacorr = zeros(size(XY));%<Delta>i*(lambdaO+1ambda1)
Xdphiblambdacorrm = zeros(size(XYm));
=
Xdphiblambdacorrp = zeros(size(XYp));
thetaO = zeros(size(Xmhu));thetaom = zeros(size(Xmhum));theta0p
=
zeros(size(Xmhup));%pitch angle
thetaO(Xmhu2 > mhubounce2(ir))=acos(Xmhu(Xmhu2 > mhubounce2(ir)));
theta0m(Xmhu2m > mhubounce2(ir))=acos(Xmhum(Xmhu2m > mhubounce2(ir)));
theta0p(Xmhu2p > mhubounce2(ir))=acos(Xmhup(Xmhu2p > mhubounce2(ir)));
mhubounce2(ir))=acos(Xmhu(Xmhu2 <= mhubounce2(ir)));
theta0m(Xmhu2m <= mhubounce2(ir))=acos(Xmhum(Xmhu2m <= mhubounce2(ir)));
theta0p(Xmhu2p <= mhubounce2(ir))=acos(Xmhup(Xmhu2p <= mhubounce2(ir)));
thetaO(Xmhu2 <=
thetaB = zeros(size(Xmhu));thetaBm = zeros(size(Xmhum));thetaBp =
zeros(size(Xmhup)) ;Y.bounce angle
thetaB(Xmhu2 > mhubounce2(ir)) = pi;
thetaBm(Xmhu2m > mhubounce2(ir)) = pi;
thetaBp(Xmhu2p > mhubounce2(ir)) = pi;
thetaB(Xmhu2 <= mhubounce2(ir))
= acos(1-2./(mhubounce2(ir)./cos(thetaO(Xmhu2
<=
mhubounce2(ir))).^2));
thetaBm(Xmhu2m <= mhubounce2(ir))
mhubounce2(ir))).~2));
thetaBp(Xmhu2p <= mhubounce2(ir))
= acos(1-2./(mhubounce2(ir)./cos(thetaom(Xmhu2m
<=
= acos(1-2./(mhubounce2(ir)./cos(thetaop(Xmhu2p <=
mhubounce2(ir))).~2));
thetaBm(find(thetaBm==O))
=
thetaBp(find(thetaBp==O))
= eps;
eps;
%Circulating electrons
ssslIcorr = -pi/2*delta(ir)./sqrt(i-mhubounce2(ir)).*((1
-
sqrt(i-mhubounce2(ir)))./sqrt(mhubounce2(ir))).~(2.*N(ir)); /*si*
Xlambdacorr(Xmhu2 > mhubounce2(ir)) = ...
-pi/4*delta(ir) .*tan(thetaO(Xmhu2 >
mhubounce2(ir))).*((
mhubounce2(ir))).~2./sqrt(1-XY(Xmhu2
>
-.
sqrt(i-XY(Xmhu2 > mhubounce2(ir)))) ./sqrt(XY(Xmhu2 >
mhubounce2(ir)))).^(2.*Nq(ir));
Xlambdacorrm(Xmhu2m > mhubounce2(ir)) =
-pi/4*delta(ir) .*tan (theta0m(Xmhu2m > mhubounce2(ir))) .2./sqrt(i-XYm(Xmhu2m
>
mhubounce2(ir))).*((1-...
sqrt(i-XYm(Xmhu2m > mhubounce2(ir)))) ./sqrt(XYm(Xmhu2m >
mhubounce2(ir)))).~(2.*Ng(ir));
Xlambdacorrp(Xmhu2p > mhubounce2(ir)) =
-pi/4*delta(ir).*tan(theta0p(Xmhu2p >
mhubounce2(ir))).^2./sqrt(I-XYp(Xmhu2p
mhubounce2(ir))).*((1-...
sqrt(i-XYp(Xmhu2p >
mhubounce2(ir))))./sqrt(XYp(Xmhu2p
mhubounce2(ir)))).^(2.*Nq(ir));
170
>
>
Xdphiblambdacorr(Xmhu2 > mhubounce2(ir)) = ...
3/16*delta(ir)*pi*mhubounce2(ir).*tan(thetaO(Xmhu2 > mhubounce2(ir))).^2./sqrt(1
XY(Xmhu2 > mhubounce2(ir))).*((1-sqrt(1-XY(Xmhu2 >
mhubounce2(ir))))./sqrt(XY(Xmhu2
mhubounce2(ir)))).^(2.*Nq(ir));
>
Xdphiblambdacorrp(Xmhu2p > mhubounce2(ir)) = ...
3/16*delta(ir)*pi*mhubounce2(ir)*tan(theta0p(Xmhu2p > mhubounce2(ir))).^2./sqrt(1
XYp(Xmhu2p> mhubounce2(ir))).*((1-sqrt(1
-
XYp(Xmhu2p> mhubounce2(ir))))./sqrt(XYp(Xmhu2p> mhubounce2(ir)))).^(2.*Nq(ir));
Xdphiblambdacorrm(Xmhu2m > mhubounce2(ir)) = ...
3/16*delta(ir)*pi*mhubounce2(ir).*tan(theta0m(Xmhu2m >
mhubounce2(ir))).-2./sqrt(1
-
XYm(Xmhu2m> mhubounce2(ir))) .*((1-sqrt(1
XYm(Xmhu2m>
-
mhubounce2(ir)))) ./sqrt(XYm(Xmhu2m> mhubounce2(ir)))).^(2.*Nq(ir));
%bouncing electrons
Xlambdacorr(Xmhu2 <=
mhubounce2(ir))
=
sqrt(pi*Nq(ir))/2*delta(ir).*tan(thetaO(Xmhu2 <= mhubounce2(ir))).^2.*((1
-
cos(thetaB(Xmhu2 <= mhubounce2(ir))))./sin(thetaB(Xmhu2 <=
mhubounce2(ir)))).^(3/2).*sin(pi/4 - ...
Nq(ir).*thetaB(Xmhu2 <= mhubounce2(ir)));
Xlambdacorrm(Xmhu2m <= mhubounce2(ir)) = ...
sqrt(pi*Nq(ir))/2*delta(ir).*tan(theta0m(Xmhu2m <= mhubounce2(ir))).^2.*((1
-
cos(thetaBm(Xmhu2m <= mhubounce2(ir)))). /sin(thetaBm(Xmhu2m <=
mhubounce2(ir)))).-(3/2).*sin(pi/4 - ...
Nq(ir) .*thetaBm(Xmhu2m <= mhubounce2(ir)));
Xlambdacorrp(Xmhu2p <= mhubounce2(ir)) = ...
mhubounce2(ir))).^2.*((1
mhubounce2(ir)))) ./sin(thetaBp(Xmhu2p <=
sqrt(pi*Nq(ir))/2*delta(ir).*tan(theta0p(Xmhu2p <=
cos(thetaBp(Xmhu2p <=
mhubounce2(ir)))).^(3/2).*sin(pi/4
-
Nq(ir).*thetaBp(Xmhu2p <=
mhubounce2(ir)));
Xdphiblambdacorr(Xmhu2 <= mhubounce2(ir))
= ...
-3/8*delta(ir)*sqrt(pi*Nq(ir))*mhubounce2(ir).*tan(thetaO(Xmhu2 <=
mhubounce2(ir))).^2.*(1
-
cos(thetaB(Xmhu2 <= mhubounce2(ir)))).^(5/2)./sin(thetaB(Xmhu2 <=
mhubounce2(ir))).^(3/2).*sin(pi/4
-
Nq(ir).*thetaB(Xmhu2 <= mhubounce2(ir)));
Xdphiblambdacorrp(Xmhu2p <= mhubounce2(ir)) = ...
-3/8*delta(ir)*sqrt(pi*Nq(ir))*mhubounce2(ir).*tan(theta0p(Xmhu2p <=
mhubounce2(ir))).^2.*(1
-
cos(thetaBp(Xmhu2p <= mhubounce2(ir)))).^(5/2)./sin(thetaBp(Xmhu2p <=
mhubounce2(ir))).^(3/2).*sin(pi/4 -...
Nq(ir).*thetaBp(Xmhu2p <= mhubounce2(ir)));
Xdphiblambdacorrm(Xmhu2m <= mhubounce2(ir)) = ...
-3/8*delta(ir)*sqrt(pi*Nq(ir))*mhubounce2(ir).*tan(theta0m(Xmhu2m <=
mhubounce2(ir))).^2.*(1
-
cos(thetaBm(Xmhu2m <= mhubounce2(ir)))).^(5/2)./sin(thetaBm(Xmhu2m <=
171
-
mhubounce2(ir))). (3/2) .*sin(pi/4
-
Nq(ir).*thetaBm(Xmhu2m <= mhubounce2(ir)));
%New bounce-averaged coefficients due to non-axisymmetric geometry
Xsssl = Xsssl + ssslicorr;
Xlambda(Xmhu2 > mhubounce2(ir)) = Xlambda(Xmhu2 > mhubounce2(ir)) + Xlambdacorr(Xmhu2
> mhubounce2(ir));
Xlambdam(Xmhu2m > mhubounce2(ir)) = Xlambdam(Xmhu2m > mhubounce2(ir)) +
Xlambdacorrm(Xmhu2m > mhubounce2(ir));
Xlambdap(Xmhu2p > mhubounce2(ir)) = Xlambdap(Xmhu2p > mhubounce2(ir)) +
Xlambdacorrp(Xmhu2p > mhubounce2(ir));
Xlambda(Xmhu2 <= mhubounce2(ir)) = Xlambda(Xmhu2 <= mhubounce2(ir)) +
Xlambdacorr(Xmhu2 <= mhubounce2(ir));
Xlambdam(Xmhu2m <= mhubounce2(ir)) = Xlambdam(Xmhu2m <= mhubounce2(ir)) +
Xlambdacorrm(Xmhu2m <= mhubounce2(ir));
Xlambdap(Xmhu2p <= mhubounce2(ir)) = Xlambdap(Xmhu2p <= mhubounce2(ir))
Xlambdacorrp(Xmhu2p <= mhubounce2(ir));
XRlambdam = Xlambdam./Xlambda;
XRlambdap = Xlambdap./Xlambda;
+
>>
Xsssl(Xmhu2 > mhubounce2(ir))
= Xsssl./Xambda(Xmhu2 > mhubounce2(ir));
Xssslm(Xmhu2m > mhubounce2(ir)) = Xsssl./Xlambdam(Xmhu2m > mhubounce2(ir));
Xssslp(Xmhu2p > mhubounce2(ir)) = Xsssl./Xlambdap(Xmu2p > mhubounce2(ir));
Xsssl(Xmhu2 <= mhubounce2(ir)) = 0;
Xssslm(Xmhu2m <= mhubounce2(ir)) = 0;
Xssslp(Xmhu2p
<=
mhubounce2(ir))
=
0;
Xdphib(Xmhu2 > mhubounce2(ir)) =
Xdphib(Xmhu2 > mhubounce2(ir)) + Xdphiblambdacorr(Xmhu2 >
mhubounce2(ir)) ./Xlambda(Xmhu2 > mhubounce2(ir));
Xdphibp(Xmhu2p > mhubounce2(ir)) = ..
Xdphibp(Xmhu2p > mhubounce2(ir)) + Xdphiblambdacorrp(Xmhu2p >
mhubounce2(ir)) ./Xlambdap(Xmhu2p > mhubounce2(ir));
Xdphibm(Xmhu2m > mhubounce2(ir)) = ...
Xdphibm(Xmhu2m > mhubounce2(ir)) + Xdphiblambdacorrm(Xmhu2m >
mhubounce2(ir)) ./Xlambdam(Xmhu2m > mhubounce2(ir));
Xdphib(Xmhu2 <= mhubounce2(ir)) =
Xdphib(Xmhu2 <= mhubounce2(ir)) + Xdphiblambdacorr(Xmhu2 <=
mhubounce2(ir))./Xlambda(Xmhu2
<= mhubounce2(ir));
Xdphibp(Xmhu2p <= mhubounce2(ir)) = ...
Xdphibp(Xmhu2p <= mhubounce2(ir)) + Xdphiblambdacorrp(Xmhu2p <=
mhubounce2(ir)) ./Xlambdap(Xmhu2p <= mhubounce2(ir));
Xdphibm(Xmhu2m <= mhubounce2(ir)) = ...
Xdphibm(Xmhu2m <= mhubounce2(ir)) + Xdphiblambdacorrm(Xmhu2m <=
mhubounce2(ir)) ./Xlambdam(Xmhu2m <= mhubounce2(ir));
end
172
else
mhubounce2(ir) = 0;
Xsssl = ones(size(Xmhu));
Xssslm = Xsssl;
Xssslp
=
Xsssl;
Xdphib = zeros(size(Xmhu));
Xdphibm = Xdphib;
Xdphibp = Xdphib;
Xlambda = ones(size(Xmhu));
XRlambdam = ones(size(Xmhu));
XRlambdap = ones(size(Xmhu));
XAsAeff-cy = ones(size(Xmhu));
XAsAeffcym = ones(size(Xmhu));
XAsAeff-cyp = ones(size(Xmhu));
Xmsom~s = ones(size(Xmhu));
end
XXsssl(:,:,ir) = Xsssl;
XXlambda(:,:,ir) = Xlambda;
XXmsom0s(:,:,ir) = Xmsom0s;
%Relativistic Maxwellian distribution function for collision integrals
fM = exp(-pn2./(1 + gamma)/Te-norm(ir))/2/pi/Te-norm(ir)/sqrt(2*pi*Te-norm(ir));
normfM = 4*pi*trapzyp([pn',pn'.*pn'.*fM']);
fM = fM*nenorm(ir)/normfM;
xfM = fM'*ones(i,length(fM));
XXfM(:,:,ir) = fM'*ones(1,nmhu);
if nargin < narginmax,XXfinit(:,:,ir) = fM'*ones(i,nmhu);end;XInitial
ne-norm)
guess (normalized to
sfM = exp(-spn2./(i + sgamma)/Te-norm(ir))/2/pi/Te-norm(ir) /sqrt(2*pi*Te-norm(ir));
normsfM = 4*pi*trapzyp([spn',spn'.*spn'.*sfM']);
sfM = sfM*ne-norm(ir)/normsfM;
Si
fMm = exp(-pn2m./(i + gammam)/Te-norm(ir))/2/pi/Te-norm(ir) /sqrt(2*pi*Te-norm(ir));
normfMm = 4*pi*trapzyp([pnm',pnm' .*pnm'.*fMm']);
fMm = fMm*ne-norm(ir)/normfMm;
fMp = exp(-pn2p./(1 + gammap)/Te-norm(ir))/2/pi/Te-norm(ir) /sqrt(2*pi*Te-norm(ir));
normfMp = 4*pi*trapzyp([pnp',pnp'.*pnp'.*fMp']);
fMp = fMp*ne-norm(ir)/normfMp;
%Lower Hybrid quasilinear diffusion coefficient
(half-grid)
XDlh = squeeze(XXDlh(:,:,ir));
=
XDlhim = zeros(size(XDlh));XDlhim(2:npn,:)
XDlh(1,:)/2;
XDlhip = zeros(size(XDlh));XDlhip(i:npn-1,:)
= (XDlh(i:npn-1, :) + XDlh(2:npn,:))/2;XDlhim(i,:)
= (XDlh(1:npn-i,:)
173
+
XDlh(2:npn,:))/2;XDlhip(npn,:) = XDlh(npn,:)/2;
XDlhjm = zeros(size(XDlh));XDlhjm(:,2:nmhu)
(XDlh(:,l:nmhu-1)
=
+
XDlh(:,2:nmhu))/2;XDlhjm(:,l) = XDlh(:,i)/2;
XDlhjp = zeros(size(XDlh));XDhjp(:,1:nmhiiu-1)
XDlh(:,2:nmhu))/2;XDlhjp(:,nmhu)
=
(XDlh(:,1:nmhu-1)
+
= XDlh(:,nmhu)/2;
XXDlhim(:,:,ir) = XDlhim;
XXDlhip(:,:,ir) = XDlhip;
XXDlhjm(:,:,ir) = XDlhjm;
XXDlhjp(:,:,ir) = XDlhjp;
%Electron cyclotron quasilinear diffusion coefficient
(half-grid)
XDcy = squeeze(XXDcy(:,:,ir));
XDcyim = zeros(size(XDcy));XDcyim(2:npn,:)
= (XDcy(i:npn-i,:)
+ XDcy(2:npn,:))/2;XDcyim(l,:)
= XDcy(l,:)/2;
XDcyip = zeros(size(XDcy));XDcyip(l:npn-l,:)
=
(XDcy(i:npn-i,:)
+
(XDcy(:,l:nmhu-1)
+
XDcy(2:npn,:))/2;XDcyip(npn,:) = XDcy(npn,:)/2;
XDcyjm = zeros(size(XDcy));XDcyjm(:,2:nmhu)
=
XDcy(:,2:nmhu))/2;XDcyjm(:,l) = XDcy(:,l)/2;
XDcyjp = zeros(size(XDcy));XDcyjp(: ,1:nmhu-1)
XDcy(:,2:nmihu))/2;XDcyjp(:,nmhu)
= (XDcy(:,1:nmhu-1)
+
= XDcy(:,nmhu)/2;
XXDcyim(:,:,ir) = XDcyim;
XXDcyip(:,:,ir) = XDcyip;
XXDcyjm(:,:,ir) = XDcyjm;
XXDcyjp(:,:,ir) = XDcyjp;
XDcyim = XDcyim.*XAsAeff-cy;
XDcyip = XDcyip.*XAsAeff-cy;
XDcyjm = XDcyjm.*XAsAeff-cym;
XDcyjp
= XDcyjp.*XAsAeff-cyp;
%Collision integrals
F1_1 = zeros(l,npn);
F2_1 = zeros(1,npn);
F1_2 = zeros(1,npn);
B1_1 = zeros(l,npn);
B1_2 = zeros(l,npn);
B1_3 = zeros(l,npn);
B1_4 = zeros(l,npn);
B1_5 = zeros(1,npn);
B2_1 = zeros(i,npn);
B2_2 = zeros(1,npn);
B2_3 = zeros(l,npn);
for i = 1:npn,%Loop for very accurate collision integral calculation close to p =
especially
smasku = spn <= pn(i);
smaskl = spn > pn(i);
174
0
F1_1(1,i) = trapzyp([spn',sfM'.*spn'.*sv'.*smasku']);
F1_2(i,i) = trapzyp([spn',sfM'.*spn'.*sv'.*(1 - sgamma'.*ssigma'./sz').*smasku']);
F2_1(1,i) = trapzyp([spn',sfM'.*spn'.*smaskl']);
BX1(1,i) = trapzyp([spn',sfM'.*smasku.*spn2I);
B1_2(1,i) = trapzyp([spn',sfM'.*smasku'.*spn2'.*spn2]);
);
B1_3(1,i) = trapzyp([spn',sfM'.*smasku'.*spn2'.*sJl'./sgamma])
B1_4(1,i) = trapzyp([spn',sfM'.*smasku'.*spn2'.*sJ2'./sgamma']);
= trapzyp([spn',sfM'.*smasku.*spn2.*((sgmma
B1_5(1,i)
B2_1(1,i)
B2_2(1,i)
B2_3(1,i)
- ssigma'./sz')./sgMma')]);
= trapzyp([spn',sfM'.*smaskl'.*spn2'./sv']);
= trapzyp([spn',sfM'.*smaskl'.*spn2'./sv'./sgamma2']);
= trapzyp([spn',sfM'.*smaskl'.*spn2'.*sv']);
end
F1 = F1.1./v2 + F1-2./pn2;
F2 = (1 - sigma./z./gamma).*F2_1./v;
F = 4*pi*(F1 + F2) + sum(Zi.^2.*(me./mi).*ni-norm(ir))./v2;
XF = F'*ones(1,nmhu);
Fim = (F1_1 - 0.50*dpn*fMm.*pnm.*vm)./v2m + (Fl_2 - 0.50*dpn*fMm.*pnm.*vm.*(1
gammam.*sigmam. /zm)) . /pn2m;
F2m = (1 - sigmam./zm./gammam).*(F2_1 + 0.50*dpn*fMm.*pnm)./vm;
Fm = 4*pi*(Fim + F2m) + sum(Zi.-2.*(me./mi).*ninorm(ir))./v2m;
Fm(1) = 1;%Avoid singularity at p = 0, but the value is not used
Fip = (F11 + 0.50*dpn*fMp.*pnp.*vp)./v2p + (F1.2 + 0.50*dpn*fMp.*pnp.*vp.*(1
gammap.*sigmap./zp))./pn2p;
F2p = (1 - sigmap./zp./gammap).*(F2-1 - 0.50*dpn*fMp.*pnp)./vp;
Fp = 4*pi*(Flp + F2p) + sum(Zi.^2.*(me./mi).*ni-norm(ir))./v2p;
-
-
XFm = Fm'*ones(Inmhu);
XFp = Fp'*ones(1,nmhu);
XXF(:,:,ir)
sXF;
XXFm(:,:,ir) =
XXFp(:,:,ir)
=
XFm;
XFp;
A = 4*pi*Te-norm(ir)*(Fi + F2)./v + sum(Zi.^2.*(me./mi).*ninorm(ir).*Ti-norm(ir))./v./v2;
XA = A'*ones(1,nmhu);
Am = 4*pi*Te-norm(ir)*(Flm + F2m)./vm +
sum(Zi.^2.*(me./mi).*ni-norm(ir).*Ti-norm(ir))./vm./v2m;
Am(1) = 1;%Avoid singularity at p = 0, but the value is not used
Ap = 4*pi*Te-norm(ir)*(Flp + F2p)./vp +
sum(Zi.^2.*(me./mi).*ni-norm(ir).*Ti-norm(ir))./vp./v2p;
XAm = Am'*ones(1,nmhiu);
XAp = Ap'*ones(1,nmhu);
XXA(:,:,ir) = XA;
XXAm(:,:,ir) = XAm;
XXAp(:,:,ir)
=
XAp;
175
Bi = B1_1./v/2 - B1_2./v./pn2/6 + B1_3./v./gamma2./z2/8 - Bi_4./v./z2/4 - B1_5./v./gamma2/4;
B2 = B2_1/2 + B2_2.*(-gamma2/6 + J1./z2./gamma/8 - gamma.*J2./z2/4)
- B2_3.*(ganmma -
sigma./z)./pn2./gamma/4;
Bt = 4*pi*(Bi + B2) + sum(Zi.^2.*ni_norm(ir))./v/2 sum(Zi.^2.*ni-norm(ir) .*(me./mi) .*Ti-norm(ir))./v./v2/2;
XBt = Bt'*ones(i,nmhu);
XXBt(:,:,ir) = XBt;
%Chang & Cooper weights: link between full grid (distribution mesh) and half grid (flux mesh) in
the momentum direction
Xdeltaip = gyp(-dpn*(epsi(ir)*Xssslp.*Xmhu - XFp)./(XAp + XDlhip.*Xmhu2 +
XDcyip.*Ximmhu2*xpsib(ir)));
%WARNING: EC term to be added
Xdeltaim = gyp(-dpn*(epsi(ir)*Xssslm.*Xmhu - XFm)./(XAm + XDlhim.*Xmhu2 +
XDcyim. *Ximmhu2*xpsib(ir)));
%WARNING: EC term to be added
Xdeltajp =
Xdeltajm =
0.5*ones(size(Xdeltaip));
0.5*ones(size(Xdeltaim));
XXdeltaip(:,:,ir) = Xdeltaip;
XXdeltaim(:,:,ir) = Xdeltaim;
XXdeltajp(:,:,ir) = Xdeltajp;
XXdeltajm(:,:,ir) = Xdeltajm;
XLeft-hand side matrix terms: collisions, dc electric field, LH and EC wave, losses, etc...
Xa = zeros(npnnmhu);%f(ij-i)
Xb = zeros(npn,nmhu);%f(i-i,j)
Xc = zeros(npn,nmhu);%f(i,j)
Xd = zeros(npn,nmhu);%f(i+l,j)
Xe = zeros(npn,nmhu);Xf(i,j+1)
Xf
Xg
Xh
Xi
= zeros(npn,nmhu);%f(i-1,j-1)
= zeros(npn,nmhu);%f(i+i,j+)
= zeros(npn,nmhu);%f(i-1,j+1)
= zeros(npn,nmhu);%f(i+1,j-1)
%Collision matrix coefficients
Xa
Xb
Xc
Xc
= - X1mmhu2m.*XRlambdam.*XBt.*(1
+ Xdphibm./Xmhu2m)/dmhu/dmhu;Xf (i,j-1)
= Xpn2m.*(XFm.*Xdeltaim - XAm/dpn)/dpn;%f(i-,j)
= Ximmhu2p.*XRlambdap.*(XBt .*(1 + Xdphibp. /Xmhu2p))/dmhu/dmhu; %f(i,j)
= Xc + Ximmhu2m.*XRlambdam.*(XBt.*(1
Xc = Xc + Xpn2m.*(XFm.*(i
+ Xdphibm./Xmhu2m))/dmhu/dmhu;%f (i,j)
- Xdeltaim) + XAm/dpn)/dpn;%f(i,j)
Xc = Xc - Xpn2p.*(XFp.*Xdeltaip - XAp/dpn)/dpn;%f(i,j)
Xd = - Xpn2p.*(XFp.*(i - Xdeltaip) + XAp/dpn)/dpn;%f(i+i,j)
Xe = -Ximmhu2p.*XRlambdap.*XBt.*(i
+ Xdphibp./Xmhu2p)/dmhu/dmhu;Xf(i,j+i)
%DC electric field matrix coefficients
176
Xa = Xa - epsi(ir)*Xmmhu2m.*Xssslm.*XRlambdam.*Xpn.*Xdeltajm/dmhu;X.f(i,j-1)
Xb = Xb - epsi(ir)*Xpn2m.*Xmhu.*Xsssl.*Xdeltaim/dpn;%f(i-1,j)
Xc = Xc + epsi(ir)*Xsssl.*Xmhu.*(Xpn2p.*Xdeltaip
- Xpn2m.*(1 -
Xdeltaim))/dpn;Xf(i,j)
Xc = Xc + epsi(ir)*Ximmhu2p.*XRlambdap.*Xssslp.*Xpn.*Xdeltajp/dmhu;%f(i,j)
Xc = Xc - epsi(ir)*Xlmmhu2m.*Xssslm.*XRlambdam.*Xpn.*(1
-
Xdeltajm)/dmhu;%f(i,j)
Xd = Xd + epsi(ir)*Xsssl.*Xpn2p.*Xmhu.*(l - Xdeltaip)/dpn;Xf(i+i,j)
Xe = Xe + epsi(ir)*Xssslp.*XRlambdap.*Xlmmhu2p.*Xpn.*(1
-
Xdeltajp)/dmhu;%f(i,j+1)
%LH wave matrix coefficients
X
%Discretization technique of Simonetto et al.
simonetto = 1;
if abs(simonetto) == 1,
Xa = Xa - Xlmmhu2m.
2.*XRlambdam.*XDlhjm/dmhu/dmhu; Xf(i,j-1)
Xa = Xa - Xmhu.*Ximmhu2.*Xpnm.*XDlhim/4/dpn/dmhu;
Xa = Xa + Xmhu.*Xlmmhu2.*Xpnp.*XDlhip/4/dpn/dmhu;
Xb = Xb - Xmhu2.*XDlhim.*Xpn2m/dpn/dpn;%f(i-1,j)
Xb = Xb + Xpn.*(XRlambdap.*XDlhjp.*Xmhup.*Ximmhu2p
-
XRlambdam.*XDlhjm.*Xmhum.*Ximmhu2m)/4/dmhu/dpn;
Xc = Xc + Xmhu2.*XDlhip.*Xpn2p/dpn/dpn;.f(i,j)
Xc = Xc + Xmhu2.*XDlhim.*Xpn2m/dpn/dpn;%f(i,j)
Xc = Xc + Ximmhiu2p. ^2.*XRlambdap.*XDlhjp/dmhu/dmhu;%f(i,j)
Xc = Xc + Xlmmhu2m. -2.*XRlambdam.*XDlhjm/dmhu/dmhu;%f (i, j)
Xd = Xd - Xmhu2.*XDlhip.*Xpn2p/dpn/dpn;%f(i+1,j)
Xd = Xd - Xpn.*(XRlambdap.*XDlhjp.*Xmhup.*Ximmhu2p
-
XRlambdam.*XDlhjm. *Xmhum.*Ximmhu2m)/4/dmhu/dpn;
Xe = Xe - Xlmmhiu2p.~2.*XRlambdap.*XDlhjp/dmhu/dmhu;%f(i,j+1)
Xe = Xe + Xmhu.*Xlmmhu2.*Xpnm.*XDlhim/4/dpn/dmhu;
Xe = Xe - Xmhu.*Ximmhu2.*Xpnp.*XDlhip/4/dpn/dmhu;
Xf = Xf - Xmhu. *Xmmhu2. *XDlhim. *Xpnm/dpn/dmhu/4; %f (i-1, j-1)
Xf = Xf - Xmhum.*XImmhi2m.*XRlambdam.*XDhjm.*Xpn/dpn/dmhu/4;
Xg = Xg - Xmhu.*Xlmmhu2.*XDlhip.*Xpnp/dpn/dmhu/4; %f(i+1,j +1)
Xg = Xg - Xmhup.*Xlmmhiu2p.*XRlambdap.*XDlhjp.*Xpn/dpn/dmhu/4;
Xh = Xh + Xmhu.*Xmmhu2.*XDlhim.*Xpnm/dpn/dmhu/4;%Xf(i-1,j+1)
Xh = Xh + Xmhup.*Xlmmhu2p.*XRlambdap.*XDlhjp.*Xpn/dpn/dmhu/4;
Xi = Xi + Xmhu.*Xlmmhu2.*XDlhip.*Xpnp/dpn/dmhu/4; %f(i+1,j-1)
Xi = Xi + Xmhum.*Xmmhu2m.*XRlambdam.*XDlhjm.*Xpn/dpn/dmhu/4;
%Discretization technique of Shoucri et al.
else
Xa = Xa - Ximmhu2m.
2.*XRlambdam.*XDlhjm/dmhu/dmhu; Xf(i,j-i)
Xa(mp,mm) = Xa(mp,mm)
+ Xmhu(mp,mm).*Xmmhu2(mp,nmm).*(XDlh(mp,mm)
Xpn(mp,mm).*(XDlh(mpp,mm)
+
- XDlh(mpm,mm))/2/dpn)/2/dmhu;
Xb = Xb - Xmhu2.*XDlhim.*Xpn2m/dpn/dpn;%f(i-1,j)
Xb(mp,mm) = Xb(mp,mm) +
Xpn(mp,mm)
.
*(XDlh(mp,mmp)
.
*Xmhu(mp,mmp)
.
*Xlmmhu2(mp,mmp)
177
. *Xlambda(mp,mmp)
-
./Xlambda(mp,mm)/dpn/dmhu/4;
XDlh(mp,mmm) .*Xmhu(mp,mmm) .*Ximmhu2(mp,mmm) .*Xlambda(mp,mmm))
% Xb = Xb + Xpn.*(XDlhjp.*Xmhup.*Xmmhu2p.*Xlambdap XDlhjm.*Xmhum.*Xlmmhu2m.*Xlambdam)./Xlambda/dpn/dmhu/2;
Xc = Xc + Xmhu2.*XDlhip.*Xpn2p/dpn/dpn;Xf(i,j)
Xc = Xc + Xmhu2.*XDlhim.*Xpn2m/dpn/dpn;
Xc = Xc + Ximmhu2p.^2.*XRlambdap.*XD1hjp/dmhu/dmhu;
Xc = Xc + Ximmhu2m.^2.*XRlambdam.*XD1hjm/dmhu/dmhu;
Xd = Xd - Xmhu2.*XDlhip.*Xpn2p/dpn/dpn;Xf(i+,j)
Xd(mp,mm)
=
Xd(mp,mm)
-
Xpn(mp,mm).*(XD1h(mp,mmp).*Xmhu(mp,mmp).*Ximmhu2(mp,mmp).*Xlambda(mp,mmp)
./Xlambda(mp,mm)/4/dpn/dmhu;
XDlh(mp,mmm) .*Xmhu(mp,mmm) .*Ximmhu2(mp,mmm) .*Xlambda(mp,mmm))
% Xd = Xd - Xpn.*(XDlhjp.*Xmhup.*Ximmhu2p.*Xlambdap
-
-
XDlhjm.*Xmhum. *Ximmhu2m. *Xlambdam) ./Xlambda/dpn/dmhu/2;
Xe = Xe - Ximmhu2p.^2.*XRlambdap.*XD1hjp/dmhu/dmhu;X/f(i,j+1)
Xe(mp,mm)
= Xe(mp,mm)
-
Xpn(mp,mm) .*(XD1h(mpp,mm)
Xmhu(mp,mm).*Ximmhu2(mp,mm).*(XDlh(mp,mm)
+
- XD1h(mpm,mm))/2/dpn)/2/dmhu;
Xf = Xf - Xmhu.*Ximmhu2.*XD1h.*Xpn/dpn/dmhu/2;Xf(i-i,j-1)
Xg = Xg - Xmhu.*Ximmhu2.*XD1h.*Xpn/dpn/dmhu/2;Xf(i+,j+)
Xh = Xh + Xmhu.*Ximmhu2.*XDlh.*Xpn/dpn/dmhu/2;Xf(i-,j+)
Xi = Xi + Xmhu.*Xlmmhu2.*XDlh.*Xpn/dpn/dmhu/2;Xf (i+i,j-1)
end
%EC wave matrix coefficients
Xcym = xpsib(ir)*Ximmhu2m - xomega-cy(ir)./Xgamma;
XcyOm = xpsib(ir)*Ximmhu2 - xomega_ cy(ir)./Xgammam;
Xcy0p = xpsib(ir)*Xlmmhu2 - xomega-cy(ir)./Xgammap;
Xcyp = xpsib(ir)*Xlmmhu2p - xomega.cy(ir)./Xgamma;
Xa = Xa + Xpnp.*XDcyip.*Ximmhu2.*Xcy0p./Xmhu/4/dmhu/dpn
-
Xpnm. *XDcyim. *Ximmhu2. *Xcy0m. /Xmhu/4/dmhu/dpn; %f (i,j-1)
Xa = Xa - XRlambdam.*Ximmhu2m. *XDcyjm.*Xcym. *Xcym./(Xmhu2m+eps) ./xpsib(ir)/dmhu/dmhu;
Xb = Xb - Xpn2m.*XDcyim.*Xmmhu2.*xpsib(ir)/dpn/dpn;
Xf(i-i,j)
Xb = Xb + Xpn.*XRlambdap.*Ximmhu2p.*XDcyjp.*Xcyp./(Xmhup+eps)/dpn/dmhu/4
-
Xpn.*XRlambdam.*Ximmiu2m.*XDcyjm.*Xcym./(Xmhum+eps)/dpn/dmhu/4;
Xc = Xc + Xpn2p.*XDcyip.*Xlmmhu2.*xpsib(ir)/dpn/dpn +
Xpn2m. *XDcyim. *Xlmmhu2. *xpsib(ir) /dpn/dpn ;*f(i,j)
Xc = Xc +
XRlambdap.*Xlmmhu2p.*XDcyjp.*Xcyp.*Xcyp. /(Xmhup+eps)
./(Xmhup+eps)/xpsib(ir)/dmhu/dmhu;
Xc = Xc +
XRlambdam.*Ximmhu2m.*XDcyjm.*Xcym.*Xcym. /(Xmhum+eps)
./(Xmhum+eps)/xpsib(ir)/dmhu/dmhu;
Xd = Xd - Xpn2p.*XDcyip.*Ximmhu2.*xpsib(ir)/dpn/dpn;Xf(i+1,j)
Xd = Xd - Xpn.*XRlambdap.*Ximmhu2p.*XDcyjp.*Xcyp./(Xmhup+eps)/dpn/dmhu/4 +
Xpn. *XRlambdam. *X1mmhu2m. *XDcyjm. *Xcym.
/
(Xmhum+eps) /dpn/dmhu/4;
Xe = Xe - Xpnp.*XDcyip.*Xmmhu2.*Xcy0p./Xmhu/4/dmhu/dpn +
Xpnm. *XDcyim. *Ximmhu2. *Xcy0m. /Xmhu/4/dmhu/dpn;Xf (i,j+i)
Xe = Xe - XRlambdap.*Xmmiu2p.*XDcyjp.*Xcyp.*Xcyp./(Xmhu2p+eps)./xpsib(ir)/dmhu/dmhu;
178
Xf = Xf - Xpnm.*XDcyim.*Xmmhu2.*Xcy0m./Xmhu/dmhu/dpn/4 Xpn. *XRlambdam. *Ximmhu2m.*XDcyjm. *Xcym. / (Xmhum+eps)/dpn/dmhu/4; Xf (i-1, j-1)
Xg = Xg - Xpnp.*XDcyip.*Xlmmhu2.*Xcy0p./Xmhu/dmhu/dpn/4 Xpn.*XRlambdap.*Xlmmhu2p.*XDcyjp.*Xcyp./(Xmhup+eps)/dpn/dmhu/4;*/f(i+l,j+1)
Xh = Xh + Xpnm.*XDcyim.*Xmmhu2.*Xcy0m./Xmhu/dmhu/dpn/4 + ...
Xpn. *XRlambdap. *Ximmhu2p. *XDcyjp. *Xcyp./(Xmhup+eps)/dpn/dmhu/4;%f (i-1,j+1)
Xi = Xi + Xpnp.*XDcyip.*X1mmhu2.*Xcy0p./Xmhu/dmhu/dpn/4 + ...
Xpn.*XRlambdam.*Ximmhu2m.*XDcyjm.*Xcym./(Xmhum+eps)/dpn/dmhu/4; Xf (i+i,j-1)
%Loss matrix coefficients
Xnhuloss = squeeze(XXnhuloss(: , :,ir));
Xc = Xc + Xnhuloss.*Xpn2;
%Time differencing scheme
Xa = Xa/scheme;
Xb = Xb/scheme;
Xc = Xc/scheme + Xpn2/dtn;
Xd = Xd/scheme;
Xe = Xe/scheme;
Xf = Xf/scheme;
Xg = Xg/scheme;
Xh = Xh/scheme;
Xi = Xi/scheme;
%Boundaries of the bounce domain for f
if bounce-mode == 0,%No bounce averaging
Xmask = [1: :nmhu] ;
ns(ir) = 0;XNumber of mhu values that are removed for implicit bounce averaged
calculations
Xa-t = Xa;
Xb-t = Xb;
Xc-t = Xc;
Xd_t
=
Xd;
Xe-t = Xe;
Xft = Xf;
Xg-t = Xg;
Xh-t = Xh;
Xi-t = Xi;
mhu_t
=
mhu;
elseif bounce-mode == 1,
%Explicit bounce-averaging calculations,
enforced on operators
five or nine diagonals modes: symetrization
Xmask = [1:1:nmhu];
ns(ir) =
0;%Number
of mhu values that are removed for implicit bounce averaged
calculations
179
Xat = Xa;
Xb-t = Xb;
Xct = Xc;
Xdt = Xd;
Xet = Xe;
Xf t
= Xf;
Xg-t
=
Xg;
Xh-t = Xh;
Xit = Xi;
mhut
=
mhu;
= min(find(mhu.*mhu<mhubounce2(ir)));
jmhutXm
jmhut-max = max(find(mhu.*mhu<mhubounce2(ir)));
djmhut = (jmhutmax - jmhutmin+i)/2;
if djmhut > 3,%At least three points in the bounce domain to enforce operator
symmetrization
Xa-tp = fliplr(Xa-t(: ,find((mhu>=0) & (abs(mhu) < sqrt(mhubounce2(ir))))));
Xatm = Xa-t(:,find((mhu<=O) & (abs(mhu) < sqrt(mhubounce2(ir)))));
Xa-ts = (Xa-tp + Xatm)/2;
Xa-t(:,find((mhu<=O) & (abs(mhu) < sqrt(mhubounce2(ir))))) = Xats;
Xa-t(:,find((mhu>=O) & (abs(mhu) < sqrt(mhubounce2(ir))))) = fliplr(Xa-ts);
Xbtp = fliplr(Xb-t(:,find((mhu>=Q) & (abs(mhu) < sqrt(mhubounce2(ir))))));
Xb_tm = Xbrt(:,find((mhu<=O) & (abs(mhu) < sqrt(mhubounce2(ir)))));
Xb-ts = (Xb-tp + Xb-tm)/2;
Xb-t(:,find((mhu<=Q)
Xb-t(:,find((mhu>=O)
& (abs(mhu) < sqrt(mhubounce2(ir)))))
& (abs(mhu) < sqrt(mhubounce2(ir)))))
= Xb-ts;
= fliplr(Xb-ts);
Xctp = fliplr(Xct(:,find((mhu>=O) & (abs(mhu) < sqrt(mhubounce2(ir))))));
Xc_tm = Xcit(:,find((mhu<=O) & (abs(mhu) < sqrt(mhubounce2(ir)))));
Xc-ts = (Xc-tp + Xc-tm)/2;
Xc-t(:,find((mhu<=O) & (abs(mhu) < sqrt(mhubounce2(ir))))) = Xcts;
Xc-t(:,find((mhu>=O) & (abs(mhu) < sqrt(mhubounce2(ir))))) = fliplr(Xcts);
XX-tp = fliplr(Xd-t(:,find((mhu>=O) & (abs(mhu) < sqrt(mhubounce2(ir))))));
& >=)
(abs(mhu) < sqrt(mhubounce2(ir)))));
Xd_tm = XLt(:,find((mhu<
Xd-ts = (Xd.tp + Xd-tm)/2;
Xd-t(:,find((mhu<=O)
Xd-t(:,find((mhu>=Q)
& (abs(mhu) < sqrt(mhubounce2(ir)))))
& (abs(mhu) < sqrt(mhubounce2(ir)))))
= Xdts;
= fliplr(Xd-ts);
=<
Xetp = fliplr(Xet(: ,find((hu>=0) & (abs(mhu) < sqrtnhubounce2(ir))))));
Xe-tm = Xet(:,find((mhu<=O) & (abs(mhu) < sqrt(mhubounce2(ir)))));
Xe-ts = (Xe-tp + Xetm)/2;
Xe-t(:,find((mhu<=O) & (abs(mhu) < sqrt(mhubounce2(ir)))))
Xe-t(:,find((mhu>=O) & (abs(mhu) < sqrt(mhubounce2(ir)))))
= Xe-ts;
= fliplr(Xe-ts);
Xf-tp = fliplr(Xf-t(:,find((mhu>=0) & (abs(mhu) < sqrt(mhubounce2(ir))))));
Xftm = Xf-t(:,find((mhu<=0) & (abs(mhu) < sqrt(mhubounce2(ir)))));
180
Xf-ts = (Xf-tp + Xf-tm)/2;
Xf-t(:,find((mhu<=O)
& (abs(mhu) < sqrt(mhubounce2(ir)))))
= Xf-ts;
Xft(:,find((mhu>=O)
& (abs(mhu) < sqrt(mhubounce2(ir)))))
= fliplr(Xf-ts);
Xg-tp = fliplr(Xg-t(:,find((mhu>=O) & (abs(mhu) < sqrt(mhubounce2(ir))))));
Xg-tm = Xg-t(:,find((mhu<=O) & (abs(mhu) < sqrt(mhubounce2(ir)))));
Xg-ts = (Xg-tp + Xg-tm)/2;
Xgt(:,find((mhu<=O) & (abs(mhu) < sqrt(mhubounce2(ir))))) = Xg-ts;
Xg-t(:,find((mhu>=)
& (abs(mhu)
< sqrt(mhubounce2(ir)))))
= fliplr(Xg-ts);
Xhtp = fliplr(Xh-t(:,find((mhu>=O) & (abs(mhu) < sqrt(mhubounce2(ir))))));
(mhu)
< sqrt(mhubounce2(ir)))));
Xh-tm = Xhrt(:,find((mhu<=O) & (abs
Xhts = (Xhtp + Xh-tm)/2;
Xh-t(:,find((mhu<=Q) & (abs(mhu) < sqrt(mhubounce2(ir))))) = Xhts;
Xh-t(:,find((mhu>=O) & (abs(mhu) < sqrt(mhubounce2(ir))))) = fliplr(Xh-ts);
Xi-tp
=
fliplr(Xit(:,find((mhu>=O) & (abs(mhu) < sqrt(mhubounce2(ir))))));
Xi-tm = Xit(:,find((mhu<=O) & (abs(mhu) < sqrt(mhubounce2(ir)))));
Xi-ts = (Xi.tp + Xi-tm)/2;
Xi-t(:,find((mhu<=O)
& (abs(mhu) < sqrt(mhubounce2(ir)))))
= Xits;
Xi-t(:,find((mhu>=O)
& (abs(mhu) < sqrt(mhubounce2(ir)))))
= fliplr(Xi-ts);
%
Xa-t(2:npn,jmhut-min-1)
= Xa-t(2:npn,jmhut-min-)/2;%f(i,jmhut-min-1)
%
%
%
%
%
Xet(2:npn,jmhut-max+i)
Xf-t(2:npn,jmhut-min-1)
Xg-t(2:npn,jmhut-max+i)
Xh-t(2:npn,jmhut-max+)
Xi-t(2:npn,jmhut-min-1)
= Xe-t(2:npn,jmhut-max+)/2;f(i,jmhut-max+)
= Xf-t(2:npn,jmhut-min-1)/2;%f(i-,jmhut-min-1)
= Xg-t(2:npn,jmhut-max+)/2;Xf(i+,jmhutmax+1)
= Xh-t(2:npn,jmhut-max+i)/2;%f(i-i,jmhut-max+)
= Xi-t(2:npn,jmhut-min-i)/2;%f(i+1,jmhut-min-1)
end
elseif bounce-mode == 2,%Full implicit bounce-averaging calculations, fifteen diagonals mode
jmhut-min = min(find(mhu. *mhu<mhubounce2(ir))) ;
jmhut-max = max(find(mhu.*mhu<mhubounce2(ir)));
djmhut = (jmhut-max - jmhut-min+i)/2;
if djmhut > 3,%At least three points in the bounce domain
=d
Xj = zeros(npn,nmhu);%Extra diagonals
Xk
= zeros(npn,nmhu);XExtra diagonals
Xl = zeros(npn,nmhu);%Extra diagonals
Xm = zeros(npn,nmhu);XExtra diagonals
Xn = zeros(npn,nmhu);%Extra diagonals
Xo = zeros(npn,nmhu);X.Extra diagonals
Xmask = (Xmhu2 < mhubounce2(ir)) & (Xmhu <
O);%Only
the half-bounce region is removed
Xmask = nonzeros(~(Xmask(2,:)==1).*cumsum(ones(i,nmhu)))';
ns(ir) = nmihu - length(Xmask);%number of mhu values that are removed
Xa-t = Xa(:,Xmask);%Matrix reduction along the angular direction
Xb-t = Xb(:,Xmask);
Xc-t = Xc(:,Xmask);
181
Xd_t
Xd(:,Xmask)
Xe_t
Xf_t
Xe(:,Xmask)
Xf(:,Xmask)
Xg-t
Xg(:,Xmask)
Xh_t
Xh(:,Xmask)
Xi(:,Xmask)
Xj(:,Xmask)
Xk(:,Xmask)
Xl(:,Xmask)
Xm(:,Xmask)
Xn(:,Xmask)
Xo(:,Xmask)
Xi_t
Xj -t
Xk_t
X1-t
Xm-t
Xn_t
Xo_t
X~
Xft(: ,jmhutmax
Xa-t(:,jmhutmax
Xi-t(:,jmhutmax
ns (ir))
ns(ir))
ns(ir))
Xft (:,jmhutmax - ns(ir))/2;
Xa-t(: ,jmhut-max - ns(ir))/2;
Xj-t(:,jmhut-max
Xk-t(:,jmhut-max
Xlt(:,jmhutmax
ns(ir))
ns(ir))
ns(ir))
Xf_t(: ,jmhut-max - ns(ir));Xf(i-i,j-djmhut)
Xat(:,jmhut-max - ns(ir));%f(i,j-djmhut)
Xi-t(: ,jmhut-max - ns(ir));Xf(i+,j-djmhut)
,jmhut-max - ns(ir))/2;
0X
Xm.t
C:,jmhut-min
Xn-t
,jmhut_min
Xo~t(: ,jmhut-min
1) Xh-t(:,jmhut-min
1) Xe-t(:,jmhut-min
1)
- 1);%f(i-1,j+djmhut)
- 1);%f(i,j+djmhut)
Xg-t(: ,jmhut-min - 1);%f(i+1,j+djmhut)
XJ
Xh-t(: ,jmhut_min
Xet(: ,jmhut_min
xg-t( ,jmhut-min
1) 0;
1)
0;
1) 0;
=0
ns(ir))
0;%df(pn,mhu=pi/2)
0;%df(pn,mhu=pi/2)
ns(ir))
1;%df(pn,mhu=pi/2)
=0
Xd-t(:,nmhu/2
ns(ir))
0;%df(pn,mhu=pi/2)
Xe-t(: ,nmhu/2
ns(ir))
-1;%df(pn,mhu=pi/2)
Xf.t(:,nmhu/2
ns(ir))
0;%df(pn,mhu=pi/2)
=0
Xg-t(:,nmhu/2
ns(ir))
=0
Xh-t(:,nmhu/2
ns(ir))
0;%df(pn,mhu=pi/2)
0;%df(pn,mhu=pi/2)
Xi-t(:,nmhu/2
ns(ir))
ns(ir))
0;Xdf(pn,mhu=pi/2)
0;%df(pn,mhu=pi/2)
=0
Xjt(:,nmhu/2
Xk-t(:,nmhu/2
ns(ir))
=0
Xlt(:,nmhu/2
ns(ir))
0;%df(pn,mhu=pi/2)
0;%df(pn,mhu=pi/2)
Xmt(:,nmhu/2
ns(ir))
ns(ir))
Xo.t(:,nmhu/2
ns(ir))
0;%df(pn,mhu=pi/2)
0;%df(pn,mhu=pi/2)
0;%df(pn,mhu=pi/2)
=0
Xn-t(:,nmhu/2
Xat(:,nmhu/2
ns(ir))
Xb-t(:,nmhu/2
Xct(:,nmhu/2
=0
=0
=0
=0
=0
=0
=0
=0
0;%df/dmhu(abs(mhu)
Xj -t(2: npn, 3)
Q;Xj-t(2:npn,nmhu - ns(ir)
1) =
1)
= 0
Xkt(2:npn,3)
0;Xk-t(2:npn,mihu
- ns(ir)
1) = 0;%df/dmhu(abs(mhu) == 1)
= 0
Xlt(2:npn,3)
Q;Xlt(2:npn,nmhu - ns(ir)
Q;Xm-t(2:npn,nmhu - ns(ir)
1) = 0;%df/dmhu(abs(mhu) == 1)
= 0
1)
= 0
Xmt(2:npn,3)
182
=
0;%df/dmhu(abs(mhu)
==
== 1)
Xn_t(2:npn,3) = O;Xn-t(2:npn,nmhu - ns(ir) - 1) = 0;%df/dmhu(abs(mhu)
Xot(2:npn,3) =
0;Xo-t(2:npn,nmhu
- ns(ir) - 1) =
0;%df/dmhu(abs(mhu)
== 1) =
==
0
1) = 0
mhu_t = mhu(Xmask);
else
Xmask = [1:1:nmhu];
ns(ir) = 0;%Number of mhu values that are removed for implicit bounce averaged
calculations
bounce_mode = 1;
if display-mode>=0,
infoyp(2,'WARNING: Not enough grid point in the bounce region for bounce-averaged
!')
calculations
end
Xa_t = Xa;
Xb-t = Xb;
Xc-t = Xc;
Xdt
Xe_t
Xd;
=
= Xe;
Xft = Xf;
Xgt
=
Xh_t
= Xh;
Xg;
Xit = Xi;
mhu-t
=
mhu;
end
end
%Boundary conditions at p = dp/2
Xa-t(i,:) = 0;
Xb.t(i,:) = 0;
Xc-t(1,:) = Xpn2(1,Xmask)/dtn;
Xd.t(1,:) = 0;
Xet(1,:) = 0;
Xf.t(1,:) = 0;
Xg.t(1,:) = 0;
Xh_t(1,:)
Xi-t(1,:)
=
0;
=0;
if bouncemode == 2,
Xj-t(i,:)
=
0;
Xkt(,:) = 0;
X1_t(1,:)
=0;
Xm.t(1,:) = 0;
Xn-t(i,:)
Xo-t(i,:)
=
=
0;
0;
end
if ~isempty(find(ir==rdke))
183
%Matrix coefficients calculations for ftp
Xatp = zeros(size(Xa));%ftp(i,j-1)
Xbtp = zeros(size(Xb));%ftp(i-i,j)
Xc-tp = zeros(size(Xc));%ftp(i,j)
Xdtp = zeros(size(Xd));%ftp(i+,j)
Xe.tp = zeros(size(Xe));%ftp(i,j+1)
Xftp = zeros(size(Xf));%ftp(i-,j-1)
Xg-tp = zeros(size(Xg));%ftp(i+i,j+1)
Xh-tp = zeros(size(Xh));%ftp(i-,j+1)
Xi.tp = zeros(size(Xi));%ftp(i+i,j-1)
%Collision matrix coefficients for ftp
Xsssl-tp = Xsssl*sqrt((i - ia(ir))/(i + ia(ir)))/(i + ia(ir));
alpha-tp = (ia(ir) - ia(ir)*ia(ir)/2)/(1 + ia(ir));
Xa_tp(:,1) = 0;
Xa-tp(: ,mm) = -Xsssl-tp(: ,mm).*XBt(: ,mm).*Xmmhu2m(: ,mm).*(1 alpha-tp. /Xmhum(: ,mm) ./Xmhu(: ,mmm) )/dmhu/dmhu;
Xa.tp(: ,nmhu) = -Xsssltp(: ,nmhu).*XBt(:,nmhu).*Xmmhu2m(: ,nmhu).*(1 alpha-tp./Xmhum(: ,nmhu)./Xmhu(: ,nmhu-1))/dmhu/dmhu;
Xb-tp = Xsssl-tp.*Xpn2m.*(XFm.*Xdeltaim - XAm/dpn)/dpn;
Xctp = Xsssl-tp.*Xpn2m.*(XFm.*(i - Xdeltaim) + XAm/dpn)/dpn;
Xc-tp = Xc-tp - Xsssltp.*Xpn2p.*(XFp.*Xdeltaip - XAp/dpn)/dpn;
Xc-tp = Xc.tp + Xsssltp.*XBt.*Xlmmhu2m.*(1 - alpha-tp./Xmhum./Xmhu)/dmhu/dmhu;
Xctp = Xc-tp + Xsssltp.*XBt.*Ximmhu2p.*(1 - alpha-tp./Xmhup./Xmhu)/dmhu/dmhu;
Xd-tp = -Xsssl-tp.*Xpn2p.*(XFp.*(i - Xdeltaip) + XAp/dpn)/dpn;
Xe-tp(:,i) = -Xsssl-tp(:,1).*XBt(:,i).*Xmmhu2p(:,i).*(1 alpha-tp./Xmhup(: ,i)./Xmhu(: ,2))/dmhu/dmhu;
Xe-tp(:,mm) = -Xsssl-tp(:,mm).*XBt(:,mm).*Xlmmhu2p(:,mm).*(1
-
alpha-tp./Xmhup(: ,mm)./Xmhu(: ,mmp))/dmhu/dmhu;
Xe-tp(:,nmhu) = 0;
%DC electric field matrix coefficients for ftp
%
Xa-tp = Xa-tp +;
%
Xbtp = Xb-tp +;
%
Xctp = Xc-tp +;
%
Xctp = Xc-tp +;
Xc-tp = Xc.tp +;
Xd-tp = Xd-tp +;
Xetp = Xe-tp +;
%
%
%
(******* NOT YET IMPLEMENTED ********)
XLH wave matrix coefficients for ftp
X
if simonetto == -1 %previous scheme
Xatp = Xa-tp - Ximmhu2m.~2.*XDlhjm/dmhu/dmhu;X/f(i,j-1)
184
Xa-tp = Xa-tp - Xmhu.*Xlmmhu2.*Xpnm.*XDlhim/4/dpn/dmhu;
Xa-tp = Xa-tp + Xmhu.*Xlmmhu2.*Xpnp.*XDlhip/4/dpn/dmhu;
Xb-tp = Xb-tp - Xmhu2.*XDlhim.*Xpn2m/dpn/dpn;%f(i-1,j)
Xb-tp = Xb-tp + Xpn.*(XDlhjp.*Xmhup.*Xlmmhu2p - XDlhjm.*Xmhum.*Xlmmhu2m)/4/dmhu/dpn;
Xc-tp = Xc-tp + Xmhu2.*XDlhip.*Xpn2p/dpn/dpn;%f(i,j)
Xc-tp = Xc-tp + Xmhu2.*XDlhim.*Xpn2m/dpn/dpn;%f(i,j)
Xc-tp = Xc-tp + Xlmmhu2p.^2.*XDlhjp/dmhu/dmhu;X.f(i,j)
Xc-tp = Xc-tp + Xlmmhu2m.-2.*XD1hjm/dmhu/dmhu;Xf(i,j)
Xd-tp = Xd-tp - Xmhu2.*XDlhip.*Xpn2p/dpn/dpn;
Xf(i+1,j)
Xd-tp = Xd-tp - Xpn.*(XDlhjp.*Xmhup.*Xlmmhu2p - XDlhjm.*Xmhum.*Xlmmhiu2m)/4/dmhu/dpn;
Xe-tp = Xe-tp - Xlmmhu2p.^2.*XDlhjp/dmhu/dmhu;%Xf(i,j+1)
Xe-tp = Xe-tp + Xmhu.*Xlmmhu2.*Xpnm.*XDlhim/4/dpn/dmhu;
Xe-tp = Xe-tp - Xmhu.*Xlmmhu2.*Xpnp.*XDlhip/4/dpn/dmhu;
Xftp = Xf-tp - Xmhu.*Xlznmhi2.*XDlhim.*Xpnm/dpn/dmhu/4; /f(i-1,j-1)
Xf-tp = Xf-tp - Xmhum.*Xlmmhu2m.*XDlhjm.*Xpn/dpn/dmhu/4;
Xg-tp = Xg-tp - Xmhu.*Xlmmhu2.*XDlhip.*Xpnp/dpn/dmhu/4; %f(i+1,j+1)
Xg-tp = Xg-tp - Xmhup.*Xlmmhu2p.*XDlhjp.*Xpn/dpn/dmhu/4;
Xh-tp
= Xh-tp + Xmhu.*Xlmmhu2.*XDlhim.*Xpnm/dpn/dmhu/4;X/f(i-1,j+1)
Xh-tp = Xh-tp + Xmhup.*Xlmmhu2p.*XDlhjp.*Xpn/dpn/dmhu/4;
Xi-tp = Xi-tp + Xmhu.*Xlmmhu2.*XDlhip.*Xpnp/dpn/dmhu/4;/f (i+1,j-1)
Xi-tp = Xi-tp + Xmhum.*Xlmmhu2m.*XDlhjm.*Xpn/dpn/dmhu/4;
elseif simonetto
== 1,
Xa-tp = Xa-tp - Ximmhu2m. ^2.*XRlambdam.*XDlhjm/dmhu/dmhu; Xf(i,j-1)
Xa-tp = Xa-tp - Xmhu.*Xlmmhu2.*Xpnm.*XDlhim/4/dpn/dmhu;
Xa-tp = Xa-tp + Xmhu.*Xlmmhu2.*Xpnp.*XDlhip/4/dpn/dmhu;
Xb-tp = Xb-tp - Xmhu2.*XDlhim.*Xpn2m/dpn/dpn; %f(i-i,j)
Xb-tp = Xb-tp + Xpn.*(XRlambdap.*XDlhjp.*Xmhup.*Xlmmhu2p XRlambdam.*XDlhjm.*Xmhum.*Xlmmhu2m)/4/dmhu/dpn;
Xc-tp = Xc-tp + Xmhu2.*XDlhip.*Xpn2p/dpn/dpn;%f(i,j)
Xc-tp = Xc-tp + Xmhu2.*XDlhim.*Xpn2m/dpn/dpn;%f(i,j)
Xc-tp = Xc-tp + Xlmmhu2p.^2.*XRlambdap.*XDlhjp/dmhu/dmhu;Xf(i,j)
Xc-tp = Xc-tp + Xlmmhu2m.^2.*XRlambdam.*XDlhjm/dmhu/dmhu;%f(i,j)
Xd-tp = Xd-tp - Xmhu2.*XDlhip.*Xpn2p/dpn/dpn; Xf (i+1, j)
Xd.tp = Xd-tp - Xpn.*(XRlambdap.*XDlhjp.*Xmhup.*Xlmmhu2p XRlambdam.*XDlhjm.*Xmhum.*Xlmmhiu2m)/4/dmhu/dpn;
Xe-tp = Xetp - Xlmmhu2p. ^2.*XRlambdap.*XD1hjp/dmhu/dmhu; Xf(i,j+1)
Xe-tp = Xe-tp + Xmhu.*Xlmmh12.*Xpnm.*XDlhim/4/dpn/dmhu;
Xe-tp = Xe-tp - Xmhu.*Xlmmhu2.*Xpnp.*XDlhip/4/dpn/dmhu;
Xftp = Xftp - Xmhu.*Xlmmhu2.*XDlhim.*Xpnm/dpn/dmhu/4;Xf(i-1,j-1)
Xftp = Xftp - Xmhum.*Xlmmhu2m.*XRlambdam.*XDlbjm.*Xpn/dpn/dmhu/4;
Xg-tp = Xg-tp - Xmhu.*Xlmmhu2.*XDlhip.*Xpnp/dpn/dmhu/4; Xf (i+1,j+1)
Xg-tp = Xg-tp - Xmhup.*Xlmmhu2p.*XRlambdap.*XDlhjp.*Xpn/dpn/dmhu/4;
Xh-tp
Xh-tp
= Xh-tp + Xmhu.*Ximmhu2.*XDlhim.*Xpnm/dpn/dmhu/4;.f (i-1,j+i)
= Xh-tp + Xmhup.*Xlmmihu2p.*XRlambdap.*XDlhjp.*Xpn/dpn/dmhu/4;
Xi-tp = Xi-tp + Xmhu.*Ximmhu2.*XDlhip.*Xpnp/dpn/dmhu/4;Xf (i+i,j-1)
Xi-tp = Xi-tp + Xmhum.*Xlmmhu2m.*XRlambdam.*XDlhjm.*Xpn/dpn/dmhu/4;
%Discretization technique of Shoucri et al.
185
else
Xa-tp = Xa-tp
Xa-tp(mp,mm)
Ximmhu2m.^2.*XRlambdam.*XDlhjm/dmhu/dmhu;/f (i,j-1)
-
Xa-tp(mp,mm) + Xmhu(mp,mm).*Ximmhu2(mp,mm).*(XDlh(mp,mm) +
=
Xpn(mp,mm) .*(XDlh(mpp,mm)
-
XDlh(mpm,mm)) /2/dpn)/2/dmhu;
Xb-tp = Xb-tp
Xb-tp(mp,mm)
-
Xmhu2. *XDlhim. *Xpn2m/dpn/dpn; X/f (i-i, j)
Xb-tp(mp,mm) +
=
Xpn(mp,mm).*(XDlh(mp,mmp).*Xmhu(mp,mmp).*Xlmmhu2(mp,mmp).*Xlambda(mp,mmp)
XDlh(mp,mmm). *Xmhu(mp,mmm) .*Xlmmhu2(mp,mmm).
-
*Xlambda(mp,mmm)). /Xlambda(mp,mm) /dpn/dmhu/4;
% Xb-tp = Xb-tp + Xpn.*(XDlhjp.*Xmhup.*Xlmmhiu2p.*Xlambdap XDlhjm. *Xmhum. *Xlmmhu2m.*Xlambdam) ./Xlambda/dpn/dmhu/2;
Xc-tp = Xc-tp + Xmhu2.*XDlhip.*Xpn2p/dpn/dpn;Xf(i,j)
Xc-tp = Xc-tp + Xmhu2.*XDlhim.*Xpn2m/dpn/dpn;
Xc-tp = Xc-tp + Ximmhu2p.^2.*XRlambdap.*XDlhjp/dmhu/dmhu;
Xc-tp = Xc-tp + Ximmhu2m.-2.*XRlambdam.*XDlhjm/dmhu/dmhu;
Xd-tp
= Xd-tp
Xd-tp(mp,mm)
-
Xmhu2.*XDlhip. *Xpn2p/dpn/dpn; .f(i+1, j)
Xd-tp(mp,mm) -
=
Xpn(mp,mm) .*(XDlh(mp,mmp) .*Xmhu(mp,rmmp) .*Xmmbihu2(mp,mmp) .*Xlambda(mp,mmp)
-...
XDlh(mp,mmm) .*Xmhu(mp,mmm) .*Xlmmhu2(mp,mmm) .*Xlambda(mp,mmm)) ./Xlambda(mp,mm)/4/dpn/dmhu;
% Xd-tp
= Xd-tp -
Xpn.*(XDlhjp.*Xmhup.*Xlmmhu2p.*Xlambdap -
XDlhjm.*Xmhum.*Xlmmhu2m.*Xlambdam) ./Xlambda/dpn/dmhu/2;
Xetp = Xe-tp
Xe-tp(mp,mm)
=
Ximmhu2p.
2.*XRlambdap.*XDlhjp/dmhu/dmhu; Xf(i,j+i)
Xe-tp(mp,mm) - Xmhu(mp,mm).*Ximmhu2(mp,mm).*(XDlh(mp,mm)
Xpn(mp,mm) .*(XDlh(mpp,mm)
+
...
XDlh(mpm,mm))/2/dpn)/2/dmhu;
Xf-tp = Xf-tp - Xmhu.*Ximmhiu2.*XD1h.*Xpn/dpn/dmhu/2;.f(i-i,j-1)
Xg-tp = Xg-tp - Xmhu.*Ximmhu2.*XDlh.*Xpn/dpn/dmhu/2;Xf(i+1,j+1)
Xh-tp = Xh-tp + Xmhu.*Xlmmhu2.*XDlh.*Xpn/dpn/dmhu/2;/f(i-1,j+)
Xi-tp = Xi.tp + Xmhu.*Xlmmhu2.*XDlh.*Xpn/dpn/dmhu/2;Xf(i+,j-1)
end
%EC wave matrix coefficients for ftp
Xcym = xpsib(ir)*Xlmmhu2m
-
xomegacy(ir).Xgama;.2
XcyOm = xpsib(ir)*Ximmhu2 - xomega-cy(ir)./Xgammm;%3
Xcy0p = xpsib(ir)*Xlmmhu2 - xomega-cy(ir)./Xgammap;4
Xcyp = xpsib(ir)*Ximmhu2p - xomega-cy(ir)./Xgamma;6
Xmhub = sign(Xmhu).*sqrt(i - xpsib(ir)*Ximmhu2);
Xmhubm = sign (Xmhum). *sqrt (1
- xpsib(ir)*Xmmhu2m);
Xmhubp = sign(Xmhup).*sqrt(1 - xpsib(ir)*Ximmhu2p);
Xatp = Xa-tp + Xpnp.*XDcyip.*Xlmmhu2.*Xcy0p.*Xmhub./Xmhu2/xpsib(ir)/4/dmhu/dpn;/f(i,j-1)
Xatp = Xa-tp - Xpnm.*XDcyim.*Xlmmhu2.*Xcy0m.*Xmhub./Xmhu2/xpsib(ir)/4/dmhu/dpn;
Xatp = Xa-tp XRlambdam. *X1mmhu2m. *XDcyjm. *Xcym. *Xcym. *Xmhubm. / (Xmhum+eps) ./ (Xmhu2m+eps) ./xpsib(ir) /xpsib(ir) /d
mhu/dmhu;
186
Xa-tp = Xa-tp + XRlambdam.*Ximmhu2m.*XDcyjm.*Xcym.*Xcym.*(xpsib(ir)./(Xmhu2m+eps) ./(Xmhubm+eps) . . .
./(Xmhu2m+eps)
1)
/xpsib(ir)/xpsib(ir)/dmhu;
Xb-tp = Xb-tp - Xpn2m. *XDcyim. *Xlmmhu2.*Xmhub. /Xmhu/dpn/dpn; %f (i-1,
Xb-tp = Xb-tp + Xpnm.*XDcyim.*Ximmhu2.*Xcy0m*(xpsib(ir)-
j)
1) ./Xmhu./Xmhu2./Xmhub/xpsib(ir)/dpn;
Xb-tp = Xb-tp +
Xpn. *XRlambdap. *Xlmmhu2p. *XDcyjp. *Xcyp. *Xmhubp. /Xmhu2p/xpsib(ir) /dpn/dmhu/4;
Xb-tp = Xb-tp -
Xpn.*XRlambdam. *Xlmmhu2m. *XDcyjm. *Xcym. *Xmhubm. /(Xmhu2m+eps)/xpsib(ir)/dpn/dmhu/4;
Xc-tp = Xc.tp + Xpn2p. *XDcyip. *Xlmmhu2.*Xmhub. /Xmhu/dpn/dpn;%f (i,j)
Xc-tp = Xc-tp + Xpn2m.*XDcyim.*Ximmhu2.*Xmhub./Xmhu/dpn/dpn;
Xc-tp = Xc-tp +
XRlambdap.*Xlmmhu2p.*XDcyjp.*Xcyp. *Xcyp. *Xmhubp. /Xmhu2p./Xmhup/xpsib(ir)/xpsib(ir)/dmhu/dmhu;
Xc.tp = Xc-tp +
XRlambdam.*Xlmmhu2m.*XDcyjm.*Xcym.*Xcym.*Xmhubm./(Xmhu2m+eps)./(Xmhum+eps)/xpsib(ir)/xpsib(ir)/dm
hu/dmhu;
Xd.tp = Xd-tp - Xpn2p.*XDcyip.*Xlmmhu2.*Xmhub./Xmhu/dpn/dpn;%f (i+,j)
Xd-tp
= Xd-tp - Xpnp.*XDcyip.*Ximmhu2.*Xcy0p.*(xpsib(ir)-
1). /Xmhu2./Xmhub. /Xmhu/xpsib(ir)/dpn;
Xd-tp = Xd-tp -
Xpn.*XRlambdap.*Xlmmhu2p.*XDcyjp.*Xcyp.*Xmhubp./Xmhu2p/xpsib(ir)/dpn/dmhu/4;
Xd-tp = Xd-tp +
Xpn.*XRlambdam.*Xlmmhu2m.*XDcyjm.*Xcym.*Xmhubm./(Xmhu2m+eps)/xpsib(ir)/dpn/dmhu/4;
Xe-tp = Xe-tp - Xpnp.*XDcyip.*Xlmmhu2.*Xcy0p.*Xmhub./Xmhu2/4/dmhu/dpn/xpsib(ir); f(ij+1)
Xe-tp = Xe.tp + Xpnm.*XDcyim.*Ximmhu2.*Xcy0m.*Xmhub./Xmhu2/4/dmhu/dpn/xpsib(ir);
Xe-tp = Xe-tp -
XRlambdap.*Ximmhu2p.*XDcyjp.*Xcyp.*Xcyp.*Xmhubp./Xmhup./Xmhu2p./xpsib(ir)/xpsib(ir)/dmhu/dmhu;
Xe-tp = Xe-tp - XRlambdap.*Xlmmhu2p.*XDcyjp.*Xcyp.*Xcyp.*(xpsib(ir)1) ./Xmhubp./Xmhu2p./Xmhu2p./xpsib(ir)/xpsib(ir)/dmhu;
Xf._tp = Xftp - Xpnm. *XDcyim.*Ximmhu2.*Xcy0m. *Xmhub. /Xmhu2/xpsib(ir) /dmhu/dpn/4;
Xfttp = Xfttp -
Xpn. *XRlambdam. *Xlmmhu2m. *XDcyjm. *Xcym.*Xmhubm./(Xmhu2m+eps)/dpn/dmhu/xpsib(ir)/4; Xf (i-1,j-1)
Xg-tp = Xg-tp Xpnp.*XDcyip.*Xlmmhu2.*Xcy0p.*Xmhub./Xmhu2/dmhu/dpn/xpsib(ir)/4; %f(i+1,j+1)
Xg-tp = Xg.tp -
Xpn. *XRlambdap. *Ximmhu2p. *XDcyjp. *Xcyp. *Xmhubp. /Xmhu2p/dpn/dmhu/xpsib (ir)/4;
Xh.tp = XhWtp + Xpnm.*XDcyim.*Ximmhu2.*Xcy0m.*Xmhub./Xmhu2/dmhu/dpn/xpsib(ir)/4;%f(i1,j+i)
Xh-tp = Xh-tp +
Xpn.*XRlambdap.*Xlmmhu2p.*XDcyjp.*Xcyp.*Xmhubp./Xmhu2p/dpn/dmhu/xpsib(ir)/4;
Xi-tp
= Xi-tp + Xpnp.*XDcyip.*Ximmhu2.*Xcy0p.*Xmhub./Xmhu2/xpsib(ir)/dmhu/dpn/4;Xf(i+1,j-
1)
Xitp = Xi-tp +
Xpn.*XRlambdam.*Ximmhu2m.*XDcyjm.*Xcym.*Xmhubm./(Xmhu2m+eps)/xpsib(ir)/dpn/dmhu/4;
%Boundaries of the bounce domain for ftp
Xmask-tp = Xmhu2 < mhubounce2(ir);
187
Xmask-tp
=
nstp(ir)
nonzeros(~(Xmask-tp(2, :)==1).*cumsum(ones(1,nmhu)))';
= nmhu - length(Xmask-tp);Xnumber of mhu values that are removed
Xa-tp-t = Xa-tp(:,Xmasktp);%Matrix
reduction along the angular direction
Xbtp-t = Xb.tp(:,Xmask-tp);
Xc-tp-t = Xc-tp(:,Xmasktp);
Xd-tp-t = Xdtp(:,Xmasktp);
Xetp-t = Xe-tp(:,Xmasktp);
Xf-tp-t
Xg-tp-t
Xh-tp-t
Xi-tp-t
= Xf-tp(:,Xmask-tp);
= Xg-tp(:,Xmasktp);
= Xh-tp(:,Xmasktp);
= Xi-tp(:,Xmasktp);
Xf-tp.t(:,(nmhu - ns-tp(ir))/2 + 1) =;%ftp(i-ij-i)
Xa-tp-t(:,(nmhu - ns-tp(ir))/2 + 1) = 0;%ftp(i,j-1)
Xatp.t(:,(nmhu - ns..tp(ir))/2 + i) = O;X.ftp(i,j-)
Xi-tp-t(:,(nmhu
- ns-tp(ir))/2 + 1)
=
0;%ftp(i+1,j-1)
Xh_tpt(:,(nmhu - ns-tp(ir))/2) =
0;Yftp(i-1,j+)
Xe_tp_t(:,(nmhu
0;Xftp(i,j+l)
- ns-tp(ir))/2) =
Xg-tp-t(:,(nmhu - ns-tp(ir))/2) =
0;%ftp(i+,j+i)
%Matrix coefficients calculations for g
Xag = Xa*scheme;%g(i,j-1)
Xb-g
Xc-g
Xdg
Xe.g
Xf-g
= Xb*scheme;%g(i-1,j)
= (Xc - Xpn2/dtn)*scheme;Xg(i,j)
= Xd*scheme;%g(i+i,j)
= Xe*scheme;%g(i,j+1)
= Xf*scheme;Xg(i-i,j-1)
Xg-g = Xg*scheme;%g(i+i,j+i)
Xh-g = Xh*scheme;Yg(i-i,j+i)
Xi-g = Xi*scheme;%g(i+i,j-i)
%Boundary conditions for g (the Lorentz model is forced at p = 0)
XIth = (abs(Xmhu) - sqrt(mhubounce2(ir))).*(1 +
0.25*sqrt(mhubounce2(ir)). /abs (Xmhu));%Large asoect ratio limit
XfOg-th = -XIth.*Xftp./abs(Xmhu);
XfOg-th(:,mhu.*mhu<mhubounce2(ir)) = 0;
XXfOg-th(:,:,ir) = XfOg_th;
Xag(i,:) =
0;%g(i,j-1)
Xb-g(i,:) =
0;Xg(i-i,j)
Xc.g(i,:)
= 1;%g(i,j)
Xd-g(,:) = 0;%g(i+1,j)
Xe.g(1,:) = 0;%g(i,j+i)
Xfg(i,:) = 0;%g(i-i,j-1)
Xg-g(,:) = 0;%g(i+i,j+i)
Xhg(1,:) = 0;%g(i-i,j+i)
Xi-g(,:) = 0;%g(i+1,j-1)
188
XBoundaries of the bounce domain for g
Xmaskg
=
Xmask-tp;
ns-g(ir) = ns-tp(ir);%number of mhu values that are removed
Xa-g-t = Xa-g(:,Xmask-g);XMatrix reduction along the angular direction
Xb-g-t = Xb.g(:,Xmask-g);
Xc-g-t = Xc-g(:,Xmask-g);
Xd-g-t = Xd-g(:,Xmask-g);
Xe-g-t = Xe-g(:,Xmask-g);
Xf-g-t = Xf-g(:,Xmask.g);
Xg-g-t = Xg-g(:,Xmask-g);
Xh-g-t =
Xh-g(:,Xmask-g);
Xi-g-t = Xi-g(:,Xmask-g);
Xe-gt(:,(nmhu - ns-g(ir))/2) = ;%g(pnmhu = -mhut) = 0
Xg-g-t(:,(nmhu - ns-g(ir))/2) = 0;%g(pn,mhu = -mhut) = 0
Xh-gt(:,(nmhiu - ns-g(ir))/2) = 0;%g(pn,mhu = -mhut) = 0
Xag-t(:,(nmhu - ns-g(ir))/2 + 1) = 0;%g(pn,mhu = mhut) = 0
Xf .g-t(:,(nmhu - nsg(ir))/2 + 1) = O;%g(pnmhu = mhut) = 0
Xig-t(:,(nmhu - ns-g(ir))/2 + 1) = 0;7g(pn,mhu = mhut) = 0
end
%Matrices calculation
if bounce-mode <= 1,
MXc = reshape(Xct' ,(nmh-ns(ir))*npn,i);
MXP-f-t = sparse([1:(nmhu-ns(ir))*npn], [1: (nmhu-ns(ir))*npn],MXc,(nmhu-ns(ir))*npn,(nmhuns(ir))*npn);
MXa = reshape(Xa-t',(nmhu-ns(ir))*npn,1);
MXPft = MXP_f_t + sparse([1+1:(nmlhu-ns(ir))*npn],[1:(nmhu-ns(ir))*npn1,MXa(1+1:length(MXa)),...
(nmhu-ns (ir) )*npn, (nmhu-ns (ir) )*npn);
MXe = reshape(Xet' ,(nmhu-ns(ir))*npn,i);
MXP_f_t = MXP_f_t + sparse([1:(nmhu-ns(ir))*npn-1],[1+1:(nmhu-
ns(ir))*npn],MXe(1:length(MXe)-1),...
(nmhu-ns (ir))*npn, (nmhu-ns (ir))*npn);
MXb = reshape(Xb-t',(nmhh-ns(ir))*npn,1);
MXP-f-t = MXP-f-t + sparse([1+(nmhu-ns(ir)):(nmhu-ns(ir))*npn],[1:(nmhu-ns(ir))*npn-
(nmhu-ns(ir))],...
MXb(1+(nmhu-ns(ir)):length(MXb)) ,(nmhu-ns(ir))*npn, (nmhu-ns(ir))*npn);
MXd = reshape(Xd-t',(nmihu-ns(ir))*npn,i);
MXP_f_t = MXP-f-t + sparse([i:(nmhu-ns(ir))*npn-(nmhu-ns(ir))],[i+(nmhu-ns(ir)):(nmhuns(ir))*npn],...
MXd(1:length(MXd)-(nmhu-ns(ir))),(nmhu-ns(ir))*npn, (nmhu-ns(ir))*npn);
MXf = reshape(Xft' ,(nmhu-ns(ir))*npn,1);
MXP_f_t = MXPf-t + sparse([i+(nmhu-ns(ir))+1:(nmhu-ns(ir))*npn],[:(nmhu-ns(ir))*npn-
189
(nmhu-ns(ir))-1],...
MXf(l+(nmhu-ns(ir))+l:length(MXf)),(nmhu-ns(ir))*npn,(nmhu-ns(ir))*npn);
MXh = reshape(Xht' , (nmhu-ns(ir))*npn,1);
MXP_f_t = MXP-f-t + sparse([1+(nmhu-ns(ir))-1: (nmhu-ns(ir))*npn]
,
[1: (nmhu-ns(ir))*npn-
(nmhu-ns(ir))+1],...
MXh(l+(nmhu-ns(ir))-l:length(MXh)),(nmhu-ns(ir))*npn,(nmhu-ns(ir))*npn);
MXi = reshape(Xi-t',(nmhu-ns(ir))*npn,1);
MXP_f_t = MXPf -t + sparse([1: (nmhu-ns(ir))*npn-(nmhu-ns(ir))+l],
[1+(nmhu-ns(ir))-
1:(nmhu-ns(ir))*npn],...
MXi(1:length(MXi)-(nmhu-ns(ir))+1)
,
(nmhu-ns(ir))*npn, (nhu-ns(ir))*npn);
MXg = reshape(Xgt' ,(nmhiu-ns(ir))*npn,1);
MXP-f-t = MXP-f-t + sparse( [1: (nmhu-ns(ir) )*npn-(nmhu-ns(ir) )-1] , [1+(nmhuns(ir))+l:(nmhu-ns(ir))*npn],...
MXg(1:length(MXg)-(nmhu-ns(ir))-1) , (nmhu-ns(ir))*npn, (nmhu-ns (ir))*npn);
elseif bounce-mode
== 2,
MXc = reshape(Xc-t',(nmhu-ns(ir))*npn,l);c(i,j)
MXP-f-t = sparse( [1: (nmhu-ns(ir))*npn]
,
[1: (nmliu-ns(ir))*npn],MXc, (nmhu-ns(ir))*npn, (nmhu-
ns(ir))*npn);
MXa = reshape(Xa.t',(nmhu-ns(ir))*npn,l);%a(i,j-1)
MXP-f-t = MXP-f-t + sparse([1+1: (nmhu-ns(ir))*npn]
,
[1: (nmhu-ns(ir))*npn-
1],MXa(1+1:length(MXa)),...
(nmhu-ns(ir) )*npn, (nmhu-ns (ir) )*npn);
MXe = reshape(Xe-t',(nmhu-ns(ir))*npn,l);Xe(i,j+l)
MXP_f_t = MXP-f-t + sparse([l:(nmhu-ns(ir))*npn-1],[1+1:(nmhuns(ir))*npn],MXe(l:length(MXe)-l),...
(nmhu-ns(ir))*npn,(nmhu-ns(ir))*npn);
MXj = reshape(Xj-t',(nmhu-ns(ir))*npn,1); %j(i-1,j-djmhut)
MXP-ft = MXPft + sparse ([l+(nmhu-ns (ir))+djmhut: (nmhu-ns(ir))*npn] , [1: (nmhuns(ir))*npn-(nmhu-ns(ir))-djmhut],...
MXj (1+(nmhu-ns(ir))+djmhut:length(MXj)), (nmhu-ns(ir))*npn, (nmhu-ns(ir))*npn);
MXk = reshape(Xk-t',(nmh-ns(ir))*npn,1);%k(i,j-djmhut)
MXP-f-t = MXP-f-t + sparse( [l+djmhut: (nmhu-ns(ir))*npn] , [1: (nmhu-ns(ir))*npndjmhut],MXk(l+djmhut:length(MXk)),...
(nmhu-ns(ir))*npn, (nmhu-ns(ir))*npn);
MX1 = reshape(X1_t',(nmhu-ns(ir))*npn,1); %1(i+1,j-djmhut)
MXP_f_t = MXP-f-t + sparse( [1: (nmhu-ns(ir))*npn-(nmhu-ns(ir))+djmhut]
,
[1+(nmhu-ns(ir))-
djmhut:(nmhu-ns(ir))*npn] ....
MX1(1:length(MX1)-(nmhu-ns(ir))+djmhut),
(nmhu-ns(ir))*npn, (nmhu-ns(ir))*npn);
MXf = reshape(Xf-t',(nmhu-ns(ir))*npn,l);Xf(i-l,j-1)
MXP-f-t = MXP-f-t + sparse([i+(nmhu-ns(ir))+l: (nmhu-ns(ir))*npn], [1: (nmhu-ns(ir))*npn(nmhu-ns(ir))-1],...
MXf(1+(nmhiu-ns(ir))+1:length(MXf))
,(nmhu-ns(ir))*npn,
(nmhu-ns(ir))*npn);
MXb = reshape(Xb-t',(nmhu-ns(ir))*npn,l);Xb(i-l,j)
MXP-f-t = MXP-f-t + sparse ([l+(nmhu-ns(ir)): (nmhu-ns(ir))*npn]
,
[1: (nmhu-ns(ir))*npn-
(nmhu-ns(ir))],...
MXb(l+(nmhu-ns(ir)):length(MXb)),(nmhu-ns(ir))*npn,(nmhu-ns(ir))*npn);
MXh = reshape(Xh-tt',(nmhu-ns(ir))*npn,1);XYh(i-1,j+1)
MXP_f_t = MXP-f-t + sparse( [1+(nmhu-ns(ir))-1:(nmhu-ns(ir))*npn],[1:(nmhu-ns(ir))*npn(nmhu-ns(ir))+1]....
190
MXh(1+(nmhu-ns(ir))-1:length(MXh)),
(nmhu-ns(ir))*npn,(nmhu-ns(ir))*npn);
MXi = reshape(Xi-t',(nmhu-ns(ir))*npn,1); %i(i+1,j-1)
MXP_f-t = MXP-f-t + sparse([1: (nmhu-ns(ir))*npn-(nmhu-ns(ir))+]
,
[1+(nmhu-ns(ir))-
1:(nmhu-ns(ir))*npn],...
MXi(1:length(MXi)-(nmhu-ns(ir))+1),
(nmhu-ns(ir))*npn, (nmhu-ns(ir))*npn);
MXd = reshape(Xd-t',(nmhiu-ns(ir))*npn,1);%d(i+i,j)
MXP-f-t = MXP-f-t + sparse([1:(nmhu-ns(ir))*npn-(nmhu-ns(ir))],[1+(nmhu-ns(ir)):(nmhuns(ir))*npn],...
MXd(1:length(MXd)-(nmhu-ns(ir))),
(nmhu-ns(ir))*npn, (nmhu-ns(ir))*npn);
MXg = reshape(Xg-t',(nmh-ns(ir))*npn,1); %g(i+1,j+1)
MXP_f_t = MXP-f-t + sparse([1:(nmhu-ns(ir))*npn-(nmhu-ns(ir))-1],[1+(nmhuns(ir))+1:(nmhu-ns(ir))*npn],...
MXg(1:length(MXg)-(nmhu-ns(ir))-1), (nmhu-ns(ir))*npn, (nmhu-ns(ir))*npn);
MXm = reshape (Xm-t' ,(nmhu-ns(ir))*npn,i) ;Ym(i-1, j+djmhut)
MXP-f-t = MXP-f-t + sparse ([i+(nmhu-ns(ir))-djmhut: (nmhu-ns(ir))*npn], [1: (nmhuns(ir))*npn-(nmihu-ns(ir))+djmhut],
...
MXm(1+(nmhu-ns(ir))-djmhut:length(MXm)),(nmhu-ns(ir))*npn,(nmhu-ns(ir))*npn);
MXn = reshape(Xn-t', (nmhu-ns(ir))*npn,1);%n(i, j+djmhut)
MXPf_t = MXP-f.t + sparse([[1: (nmhu-ns(ir))*npn-djmhut], [1+djmhut: (nmhuns(ir))*npn],MXn(1:length(MXn)-djmhut),...
(nmhu-ns (ir))*npn, (nmhiu-ns (ir))*npn);
MXo = reshape(Xo-t',(nmhiu-ns(ir))*npn,1);%o(i+i,j+djmhut)
MXPf_t = MXPft + sparse([1: (nmhu-ns(ir))*npn-(nmhu-ns(ir))-djmhut],
[1+(nmhu-
ns(ir))+djmhut:(nmhu-ns(ir))*npn],...
MXo(1: length(MXo)-(nmhilu-ns(ir))-djmhut)
,
(nmhu-ns(ir))*npn, (nmhu-ns(ir))*npn);
end
MXPDI-f-t = inv(diag(diag(MXP_f_t)));
MXPCft = MXPDI-f-t*MXP-f-t;%Main diagonal matrix preconditionning
MXPDI-f.t;'1);
eval(['MXPDIXint2str(ir),'ft
eval(['MXPC',int2str(ir),'_f_t
eval(['Xmask',int2str(ir),'
= MXPCf-t;']);
= Xmask;']);
if store-mode == 1,
eval(['save MXPmat.mat MXPDI',int2str(ir),'_f-t MXPC',int2str(ir),'_f_t
Xmask',int2str(ir),'
-append']);
eval(['clear MXPDI',int2str(ir),'_f-t MXPC',int2str(ir),'_f-t
Xmask',int2str(ir)]);
end
if ~isempty(find(ir==rdke))
%Matrix calculation for ftp
MXc-tp-t = reshape(Xc-tpt' ,(nmihu - ns-tp(ir))*npn,i);
MXP-tp-t = sparse([i:(nmhu
-
ns-tp(ir))*npn],[i:(nmhu - nstp(ir))*npn],MXctpt,(nmhu
ns-tp(ir))*npn,...
(nmhu - ns-tp(ir))*npn);
MXa-tp-t = reshape(Xa-tp-t',(nmhu - ns-tp(ir))*npn,1);
191
-
MXPtp.t = MXP-tp-t + sparse([1+1:(nmhu - ns-tp(ir))*npn],[1:(nmhu - nstp(ir))*npn1],MXa-tp-t(1+1:length(MXa-tp-t)),...
(nmhu
-
ns-tp(ir))*npn,(nmhu
ns-tp(ir))*npn);
-
MXetp-t = reshape(Xe-tp-t',(nmhu
-
ns-tp(ir))*npn,1);
MXP_tpt = MXP-tpt + sparse([1:(nmhu - nstp(ir))*npn-1],[1+1:(nmhu
-
ns-tp(ir))*npn],MXe-tp-t(1:length(MXe-tp-t)-),...
(nmhu
MXbtp-t
-
ns-tp(ir))*npn,(nmhu
ns-tp(ir))*npn);
-
= reshape(Xb-tp-t',(nmhu -
ns-tp(ir))*npn,1);
MXP-tp-t = MXP_tpt + sparse([1+(nmhu - nstp(ir)):(nmhu
nstp(ir))*npn],[1:(nmhu
-
-
ns-tp(ir))*npn-(nmhu - ns-tp(ir))],...
MXb-tp-t(1+(nmhu - ns-tp(ir)):length(MXb-tpt)),
(nmhu
-
nstp(ir))*npn,(nmhu -
ns-tp(ir))*npn);
MXdtp.t = reshape(Xd-tp-t',(nmhu - ns-tp(ir))*npn,1);
MXP_tpt = MXP_tpt + sparse([1:(nmiu - nstp(ir))*npn-(nmhu - ns-tp(ir))],[1+(nmhu
-
nstp(ir)):(nmihu - ns-tp(ir))*npn],...
MXd-tp-t(1: length(MXd-tp-t)-(nmhu - nstp(ir))),(nmhu - nstp(ir))*npn,(nmhu ns-tp(ir))*npn);
MXftp-t = reshape(Xftp-t',(nmhu
- ns-tp(ir))*npn,1);
MXPtpt = MXPtpt + sparse([1+(nmhiu - nstp(ir))+1:(nmhu - ns-tp(ir))*npn,...
[1:(nmhu - nstp(ir))*npn-(nmhu - nstp(ir))-1],MXftpt(1+(nmhu ns-tp(ir))+1:length(MXf-tpt)),...
(nmhu
MXh-tpt
MXPtp-t
-
ns-tp(ir))*npn,(nmhu
ns-tp(ir))*npn);
-
= reshape (Xh-tp-t' , (nmhu -
ns-tp(ir))*npn,i);
= MXPtpt + sparse([1+(nmhu
[1:(nmhu - nstp(ir))*npn-(nmhu
- ns_tp(ir))-1:(nmhu
- ns.tp(ir))*npn],...
- ns-tp(ir))+1],MXh-tpt(1+(nmlhu
- ns-tp(ir))-
1:length(MXh-tp-t)),...
(nmhu
-
ns-tp(ir))*npn,(nmhu
MXi-tp-t = reshape(Xi-tp-t'
,
ns-tp(ir))*npn);
-
(nmhu
-
ns.tp(ir))*npn,1);
MXPtpt + sparse([1:(nmhu - nstp(ir))*npn-(nmhu - ns-tp(ir))+1],...
MXP_tpt
[1+(nmhu - nstp(ir))-1:(nmhu
- ns-tp(ir))*npn],MXi-tp-t(1:length(MXi-tp-t)-(nmhu
-
nstp(ir))+1),...
(nmhu
MXg-tp-t
MXPtpt
-
ns-tp(ir))*npn,(nmhu - ns-tp(ir))*npn);
= reshape(Xg-tp-t',(nmhu
=
- ns-tp(ir))*npn,1);
MXPtpt + sparse([1:(nmhu - nstp(ir))*npn-(nmhu - nstp(ir))-1],...
[1+(nmhu - ns-tp(ir))+1:(nmhu
-
ns-tp(ir))*npn] ,MXg-tpt(1:length(MXg-tpt)-(nmhu
ns_tp(ir))-1),...
(nmhu - ns-tp(ir))*npn,(nmhu
-
ns-tp(ir))*npn);
eval(['MXP',int2str(ir),'_tpt = MXP.tp-t;']);
if store-mode == 1,
eval(['save MXPmat.mat MXP',int2str(ir),'_tp-t -append']);
eval(['clear MXP',int2str(ir),'_tpt']);
end
%Matrix calculation for g
MXc_g_t = reshape(Xcg-t' ,(nmhu
- ns-g(ir))*npn,1);
MXP-g-t = sparse([1:(nmhu - ns-g(ir))*npn],[i:(nmhu
192
- ns-g(ir))*npn],MXc-g-t,(nmhu
-
-
ns-g(ir))*npn,(nmhu - ns-g(ir))*npn);
MXa_g_t = reshape(Xag-t' ,(nmhu - ns-g(ir))*npn,1);
MXP_g-t = MXP_g_t + sparse([1+1:(nmhu
MXa_g_t(1+1:length(MXa-g-t))
ns-g(ir))*npn],[1:(nmhu
-
,(nmhu
-
- ns-g(ir))*npn-1],...
ns-g(ir))*npn,(nmhu - ns-g(ir))*npn);
MXe_gt = reshape(Xe-gt',(nmhu - ns-g(ir))*npn,1);
MXPgt = MXP_g_t + sparse([1:(nmhu - ns-g(ir))*npn-1],[1+1:(nmhu - ns-g(ir))*npn],...
(nmhu - ns-g(ir))*npn,(nmhu - ns-g(ir))*npn);
MXe_g_t(1:length(MXeg-t)-1),
MXb_g.t = reshape(Xbg-t' ,(nmhu
- ns-g(ir))*npn,1);
MXP_g-t = MXP_g-t + sparse([1+(nmhu - asg(ir)):(nmhu - ns-g(ir))*npn],...
[1:(nmhu - ns-g(ir))*npn-(nmhu
nsg(ir))],MXb-g-t(1+(nmhu -
-
nsg(ir)):length(MXb_g_t)),...
(nmhu - ns-g(ir))*npn,(nmhu - ns-g(ir))*npn);
MXd_g_t = reshape(Xd-g-t',(nmhu - ns-g(ir))*npn,1);
MXP_g_t = MXP_g_t + sparse([i:(nmhu - nsg(ir))*npn-(nmhu - ns.g(ir))],...
[1+(nmhu - nsg(ir)):(nmhu
ns-g(ir))*npn],MXd_g_t(1:length(MXd-g-t)-(nmhu
-
-
ns.g(ir))),...
(nmhu - ns-g(ir))*npn,(nmhu
ns-g(ir))*npn);
-
MXf_g_t = reshape(Xf-g-t',(nmhu - ns-g(ir))*npn,1);
MXP_g_t = MXP_gt + sparse([i+(nmhu - ns-g(ir))+1:(nmhu - ns-g(ir))*npn],...
[1:(nmhu - nsg(ir))*npn-(nmhu
nsg(ir))-1],MXf_g_t(1+(nmhu
-
-
nsg(ir))+1:length(MXf_g-t)),...
(nmhu
- ns-g(ir))*npn,(nmhu - nsg(ir))*npn);
MXh_gt = reshape (Xh-g-t'
,
(nmhu - ns-g(ir) ) *npn,1);
MXP_g-t = MXP_g_t + sparse([1+(nmhu
[1:(nmhu - nsg(ir))*npn-(nmhu
-
nsg(ir))-1:(nmhu
- ns-g(ir))*npn],...
nsg(ir))+I],MXh_g_t(1+(nmhu
- ns-g(ir))-
1:length(MXh-g-t)),...
(nmhu - nsg(ir))*npn,(nmhu - ns-g(ir))*npn);
MXi_gt = reshape(Xi-g-t',(nmhlu - ns-g(ir))*npn,1);
MXP_g_t = MXP_g_t + sparse([1:(nmhu - ns-g(ir))*npn-(nmhu
[1+(nmhu - nsg(ir))-1:(nmhu
-
- ns-g(ir))+1],...
ns-g(ir))*npn],MXi_g_t(1:length(MXi-g-t)-(nmhu
-
ns-g(ir))+1),...
(nmhu - ns-g(ir))*npn,(nmhu
-
ns-g(ir))*npn);
MXg_g-t = reshape(Xgg-t' ,(nmhu
-
ns-g(ir))*npn,1);
MXP_g_t = MXP-g-t + sparse([1:(nmhu - ns-g(ir))*npn-(nmhiu
[1+(nmhu - nsg(ir))+i:(nmhu
-
- ns-g(ir))-1],...
ns-g(ir))*npn],MXg_g_t (1:length(MXg-g-t)-(nmhu
nsg(ir))-1),...
(nmhu
MXPDI-g-t
-
ns-g(ir))*npn,(nmhu
-
ns-g(ir))*npn);
inv(diag(diag(MXP_g_t)));
MXPC-g-t = MXPDI-g-t*MXP-g-t;XMain diagonal matrix preconditionning
eval(['MXPDI',int2str(ir),'_g-t
eval(['MXPC',int2str(ir),'_g_t
= MXPDI-g.t;']);
= MXPC-gt;']);
eval(['Xmask',int2str(ir),'_g = Xmask-g;']);
if store-mode == 1,
eval(['save MXPmat.mat MXPDI',int2str(ir),'_g-t MXPC',int2str(ir),'
Xmask',int2str(ir),'_g -append']);
eval(['clear MXPDI',int2str(ir),'_g_t MXPC',int2str(ir),'_g_t
193
g-t
-
Xmask',int2str(ir),'_g']);
end
end
if display-mode >= 1,infoyp(2, ['Matrix calculations (',int2str(ir),'/',int2str(nr),')
done.']);end
end
clear ir ssslO XY XE XK XJO XJ2 XJ4 XYm XEm XKm XJOm XJ2m XJ4m XYp XEp XKp XJOp XJ2p XJ4p
clear alphatp Xsssl Xsssl-tp Xssslm Xssslp Xdphib Xdphibm Xdphibp Xlambda Xlambdam Xlambdap
clear XRlambdam XRlambdap XAsAeff-cy XAsAeff-cym XAsAeff-cyp
clear Xmhu-psib Xmhu-psibm Xmhu-psibp XAsAeff-cy XAsAeff-cym XAsAeff-cyp XmsomOs
clear Xdphiblambdacorr Xdphiblambdacorrm Xdphiblambdacorrp thetaO theta0m theta0p thetaB thetaBm
thetaBp
clear sssllcorr Xlambdacorr Xlambdacorrm Xlambdacorrp
clear Xa Xb Xc Xd Xe Xf Xg Xh Xi Xj Xk Xl Xm Xn Xo
clear Xat Xbt Xct Xdt Xet Xf-t Xg.t Xh.t Xi-t Xj-t Xk-t Xlt
Xm-t Xnt Xo_t
clear Xats Xbts Xcts Xdts Xe-ts Xf-ts Xg-ts Xh-ts Xi-ts
clear Xatm Xbtm Xctm Xdtm Xe-tm Xf-tm Xg-tm Xh-tm Xitm
clear Xa-tp Xbtp Xc-tp Xd-tp Xe.tp Xf-tp Xg-tp Xh-tp Xi-tp
clear Xa-tp-t Xb-tp-t Xc-tp-t Xd-tp-t Xe-tp-t Xf-tp-t Xg-tp-t Xh-tp-t Xi-tp-t
clear Xa-g Xb-g Xc-g Xd-g Xe-g Xf-g Xg-g
XIhg
Xi-g
clear Xa-g-t Xb-g-t Xc-g-t Xd-g-t Xe-g.t Xf-g.t Xg-g-t Xh-g-t Xi-g.t
clear MXa MXb MXc MXd MXe MXf MXg MXh MXi MXj MXk MX1 MXm MXn MXo
clear MXa-tp-t MXb-tp-t MXc-tp-t MXd-tp-t MXetp-t MXf-tp-t MXg-tp-t MXh-tp-t MXi-tp-t
clear MXa-g-t MXb_g_t MXc-g-t MXd.g-t MXe_g_t MXftg-t MXg_g_t MXhgt MXi-g-t
clear fM normfM xfM sfM normsfM fMm normfMm fMp normfMp
clear XDlh XDlhim XDlhip XDlhjm XDlhjp
clear XDcy XDcyim XDcyip XDcyjm XDcyjp
clear Xmhu-t Xmhub Xmhubm Xmhubp Xcym Xcyp XcyOm Xcy0p
clear Xdeltaip Xdeltaim Xdeltajp Xdeltajm
clear Fl_1 F2_1 F1_2 B1_1 B1_2 B1_3 B1_4 B1_5 B2_1 B2_2 B2_3
clear F1 F2 F XF Fim F2m Fm Fip F2p Fp XFm XFp A XA Am Ap XAm XAp B1 B2 Bt XBt
clear MXP_f_t MXPC_f_t MXPDI_f_t MXPtpt MXP_g_t MXPC_g_t MXPDI-g-t
clear Xmask Xmask-g Xmask-tp
clear Xnhuloss XI-th XfOg-th
clear jmhut.min jmhut-max djmhut
if display-mode >= l,infoyp(2,['Full elapsed time for matrix calculation:
',num2str(etime(clock,timeO)),'
(s)']);end
X**************************************************
Main Fokker-Planck loop
if display-mode >= 1
disp(' -----------------------------------------------------------------------------------------------------------------')
end
194
XXfO = XXfinit;%Initial guess (normalized to ne-norm)
while 1,
for ir =
1:nr,
XfM = squeeze(XXfM(:,:,ir));
fM = XfM(:,1)';
Xsssl = squeeze(XXsssl(:,:,ir));
XI = squeeze(XXI(:,:,ir));
if store-mode == 1,
eval(['load MXPmat.mat MXPDI',int2str(ir),'_f-t MXPC',int2str(ir),'_ft
Xmask',int2str(ir)]);
end
eval(['MXPDItft = MXPDI',int2str(ir) , 'ft; ') ;Load local value
eval(['MXPCIft = MXPC' ,int2str(ir),'_f-t;]);%Load local value
eval(['Xmask = Xmask',int2str(ir),';']);XLoad local value
eval( ['clear MXPDI' ,int2str(ir)
,_f
_t MXPC' ,int2str(ir)
,_f
_t Xmask' ,int2str(ir)]);
if invproc == 1,%MatLab built-in full LU factorization
[MXLftMXU_f-t] = lu(MXPC-f-t);
elseif invproc == 2,%MatLab built-in iterative inversion method
[MXLf_tMXU_f-t] = luinc(MXPC_f_t,1e-5);
end
for it
= 2:nit-f,
Xf0 = squeeze(XXfO(:,:,ir));%Load value only
if residu-f(ir,it-1) >= precOf,
Xq = (XI + scheme*XfO/dtn).*Xpn2;
Xqt = Xq(:,Xmask);
XfO_t
=
XfO(:,Xmask);
if scheme == 2,Xq-t = Xq-t + reshape(-MXPDI_f_t*MXPC-f_t*reshape(Xf0_t',(nmhu
-
ns(ir))*npn,1),...
(nmhu - ns(ir)),npn)';end;%Crank-Nicholson
if bounce-mode == 2,Xq-t(2:npn,nmhu/2 + 1 - ns(ir)) = O;end;%df/dmhu(mhu == pi/2)
=0
%A
XRt = reshape(MXPDI-f_t*reshape(Xq-t',(nmhu
- ns(ir))*npn,i),(nmhu -
ns(ir)),npn)' reshape(MXPC-ft*reshape(Xf0t'
,
(nmhu - ns(ir))*npn,i),(nmhu - ns(ir)),npn)';
XA
if invproc == 1,%Full LU factorization
XYt = reshape(MXL-f-t\reshape(XR-t',(nmhu - ns(ir))*npn,1),(nmhu -
195
ns(ir)),npn)';
%Solving the simultaneous linear equations
XS-t = reshape(MXU-f-t\reshape(XY-t',(nmhu
- ns(ir))*npn,1),(nmhu -
ns(ir)),npn)';
%Solving the simultaneous linear equations
elseif invproc == 2,%qmr iterative method
XS.t = reshape(qmr(MXPC-f-t,reshape(XRt'
,
(nmihu - ns(ir))*npn,1),1e-6,(nmhu
ns(ir))*npn,...
MXL_f_t,MXUf-t),(nmhu - ns(ir)),npn)';
elseif invproc == 3,Xbicgstab iterative method
XS.t = reshape(bicgstab(MXPC-f-t,reshape(XR_t', (nmhu - ns(ir))*npn,1),1e6,(nmhu - ns(ir))*npn,...
MXL-f-t,MXU-f-t),(nmhu - ns(ir)),npn)';
elseif invproc == 4,%cgs iterative method
XS-t = reshape(cgs(MXPC-f_treshape(XR-t',(nmhu - ns(ir))*npn,1),Ie-6,(nmhu
-
ns(ir))*npn,MXL-f-t,MXU-f-t),...
(nmhu - ns(ir)),npn)';
elseif invproc == 5,%pcg iterative method
XS-t = reshape(pcg(MXPC-f-t,reshape(XRt',(nmhiu - ns(ir))*npn,1),1e-6,(nmhu
ns(ir))*npn,MXL-f-t,MXU-f-t),...
(nmhu - ns(ir)),npn)';
end
Xfit = XfO-t + XS_t;
XMatrix expansion for bounce averaged calculations
if bouncemode == 0,
Xf1 = Xfit;
elseif bouncemode == 1,
Xfltp = fliplr(Xfit(:,find((mhu>=0) & (abs(mhu) <=
sqrt(mhubounce2(ir))))));
Xfitm
Xfits
=
Xfit(:,find((mhu<=0) & (abs(mhu) <= sqrt(mhubounce2(ir)))));
=
(Xfitp + Xfitm)/2;
Xflt(:,find((mhu<=O)
& (abs(mhu)
<= sqrt(mhubounce2(ir)))))
= Xf1_ts;
Xflt(:,find((mhu>=O)
& (abs(mhu)
<= sqrt(mhubounce2(ir)))))
=
fliplr(Xfl-ts);
Xfi
=
Xf1_t;
elseif bounce-mode == 2,
XfI = zeros(size(Xf0));
Xfl(:,1:nmhiu/2 - ns(ir)) = Xflt(:,1:nmhu/2 - ns(ir));
Xfi(:,nmhu/2 + 1:nmhu) = Xfi-t(:,nmhu/2 + 1 - ns(ir):nmhu - ns(ir));
Xfi(:,nmhu/2 + 1 - ns(ir):nmhu/2) = fliplr(Xfit(:,nmhu/2 + 1 ns(ir):nmhu/2));
end
=A
normf0(ir,it) = 2*pi*trapzyp([pn',pn2'.*trapzyp([mhu',XfO'])']);
normfi(ir,it) = 2*pi*trapzyp([pn',pn2'.*trapzyp([mhu',Xfl'])']);
currO(ir,it) =
2*pi*trapzyp([pn',pn2'.*(pn'./gamma').*trapzyp([mhu',Xmhu'.*Xf0'])'])*nenorm(ir)/normf0(ir,it);
196
-
curri(ir,it) =
2*pi*trapzyp([pn',pn2'.*(pn'./gamma').*trapzyp([mhu',Xmhu'.*Xf1'])'])*ne-norm(ir)/normf1(ir,it);
prec-curr(ir,it) = abs((currl(ir,it) - currO(ir,it))/currO(ir,it));
= sqrt(2*pi*trapzyp([pn',pn2'.*trapzyp([mhu',(Xfl'-
residu-f(ir,it)
XfO').~2.*XfO'])'])/normfl(ir,it)/dtn/dtn);
%As done by J.P. Bizarro et al., Nuclear Fusion, 37 (1997) 1509
if display-mode >= 1,
infoyp(2, ['Normalized minor radius:
ratio:
Inverse aspect
',num2str(rho(ir)),',
',num2str(ia(ir)),...
',
iteration : ',int2str(it),'
/
',int2str(nit-f),',
Residu =
,num2str(residuf(ir,it))]);
disp(['
- Density [t-1]:
',num2str(normf0(ir,it-i)),'
- Current [t-1]:
',num2str(curr0(ir,it-1)),'
-
[t]:
',num2str(normf1(ir,it))]);;
disp(['
-
[t]:
',num2str(curr1(ir,it))]);
disp(['
- Current relative accuracy [t]:
',num2str(prec-curr(ir,it))]);
end
if display-mode == 2,
if nr-dke >= 1,
if ir == ir-display-1,
figure(100),subplot (321),semilogy(-pn,XfM(: ,2), 'g--',pn,XfM(: ,nmhu1), 'g--',
..
-pn,Xf1(:,2),'r-',pn,Xf1(:,nmhu-i),'r-');%f(perp = 0)
axis('square');axis([-max(pn),max(pn),1e-20,10]),zoom on
title(['Iteration:
',int2str(it)]);xlabel('pn-
ref');ylabel('f(pperp=0)')
hold on;zoom on
figure(100),subplot(322),semilogy([1:it] ,residu-f(ir,1:it),'rIr+', [1:it],...
precOf*ones(size([1:it])),'b--');
axis('square');axis([0,it,precO-f/10,1]);
title('Convergence - f');xlabel('Iteration');ylabel('Residu')
hold on;zoom on
pause(0.1);
elseif ir == ir-display,
figure(100) ,subplot (323),semilogy(-pn,XfM(: ,2), 'g--' ,pn,XfM(: ,nmhu1), 'g--' ,..
-pn,Xfi(:,2),'r-',pn,Xf1(:,nmhu-1),'r-');%f(perp = 0)
axis('square');axis([-max(pn),max(pn),1e-20,10]),zoom on
title(['Iteration:
',int2str(it)]);xlabel('pn-
ref');ylabel('f(pperp=0)')
hold on;zoom on
figure(100),subplot(324),semilogy([i:it] ,residu-f(ir,1:it),'r+', [1:it] ,precOf*ones(size([1:it]))
,'b--');
axis('square');axis([0,it,precOf/10,1]);
197
title('Convergence - f');xlabel('Iteration') ;ylabel('Residu')
hold on;zoom on
pause(O.1);
elseif ir == ir-display+1,
figure(100) ,subplot(325),semilogy(-pn,XfM(: ,2), 'g--',pn,XfM(: ,nmhu1), 'g--',..
-pn,Xfi(:,2),'r-',pn,Xfi(:,nmhu-1),'r-');Xf(perp
= 0)
axis('square');axis([-max(pn),max(pn),ie-20,10]),zoom
title(['Iteration:
on
',int2str(it)]);xlabel('pn-
ref');ylabel('f(pperp=0)')
hold on;zoom on
figure(100),subplot(326),semilogy([i:it],residu-f(ir,i:it),'r+', [:it],...
precOf*ones(size([1:it])),'b--');
axis('square');axis([O,it,precOf/10,1]);
title('Convergence
- f');xlabel('Iteration');ylabel('Residu')
hold on;zoom on
pause(0.1);
end
else
if ir == ir-display,
figure(100) ,subplot(121),semilogy(-pn,XfM(: ,2), 'g--',pn,XfM(: ,nmhu1),...
'g--',-pn,Xfi(:,2),'r-',pn,Xfl(:,nmhu-1),'r-');%f(perp
axis('square');axis([-max(pn),max(pn),ie-20,10]),zoom on
title(['Iteration:
=
',int2str(it)]);xlabel('pn-
ref');ylabel('f(pperp=O)')
hold on;
figure(100),subplot(122),semilogy([1:it],residu-f(ir,i:it),'r+',[i:it],...
precOf*ones(size([1:it])),'b--');
axis('square');axis([O,it,precOf/10,1]);
title('Convergence - f');xlabel('Iteration') ;ylabel('Residu')
hold on;
pause(O.1);
end
end
end
%First order Legendre correction (2nd order has a negligible effect)
if trunc == 1,
fl = 1.5*trapzyp([mhu',Xfl'.*(mhu'*ones(1,npn))]);
xfl = f1'*ones(1,length(fi));
Il = trapzyp([pn',xf1.*masku.*xpn3./xgamma])/3/Te-norm(ir)
-
...
trapzyp([pn',xf1.*masku.*xpn3]).*gamma*2/3/Te-norm(ir)
trapzyp([pn',xf1.*masku.*xpn3.*xpn2./xgamma]).*gamma/5/Te-norm(ir)/Te-norm(ir)
198
...
...
0)
+ trapzyp([pn' ,xf1.*masku.*(xgamma - xsigma./xz) .*xpn./xgamma])
-
...
trapzyp( pn' ,xf1.*masku. *xpn3. *xJ2. /xgamma. /xz2]). *gamma/Te-norm(ir)
+ trapzyp([pn',xfl.*masku.*xpn3.*(1
3*xgamma.*xsigma./xz2./xz)./xgamma]).*(gamma.*pn2
+ 3.0./xz2 -
-
5*Te-norm(ir))/6/Te-norm(ir)/Te-norm(ir) ...
trapzyp([pn',xf1.*masku.*xpn3.*xJ3./xz./xganmma]).*gAnma/2/betath/betath/Te-norm(ir)/Te-norm(ir)
+ trapzyp([pn',xf1.*masku.*xpn3.*xJ1./xz2./xgamma]).*gamma/2/Te-norm(ir)
-
+ trapzyp([pn',xf1.*masku.*xpn.*(xgamma.*xsigma./xz
1)./xgamma]).*pn2/Tenorm(ir) ...
trapzyp([pn',xf1.*masku.*xpn3.*xJ4./xz./xgamma]).*gamma/12/betath/betath/Te-norm(ir)/Te.norm(ir);
12 = trapzyp([pn',xf1.*maskl./xgamma])/3/Te-norm(ir) ...
+ trapzyp([pn',xf1.*maskl]).*(-2*gamma/3/Te-norm(ir) +
pn2/5/Tenorm(ir)/Tenorm(ir)) ...
+ trapzyp([pn',xfl.*maskl./xgamma).*(gimma
- sigma./z)./pn2
...
- trapzyp([pn',xf1.*maskl]).*J2./z2/Te-norm(ir) ...
+ trapzyp([pn',xfl.*maskl.*(xgamma.*xpn2
3.0./z2
- 5*Tenorm(ir))./xgamma]).*(1
-
3*gamma.*sigma./z2./z)/6/Te-norm(ir)/Te-norm(ir)
trapzyp([pn',xf1.*maskl]).*(J3./z/2/betath/betath/Te-norm(ir)/Te-norm(ir)
J4./z/12/betath/betath/Te-norm(ir)/Te-norm(ir))
...
+ J1./z2/2/Tenorm(ir)
...
+ trapzyp([pn',xf1.*maskl.*xpn2./xgamma]).*(gamma.*sigma./z
1)./pn2/Tenorm(ir);
XI = 4*pi*((fM.*(fl + I1./pn2 + 12.*pn)./gamma)'*mhu).*Xsssl;
XI(1,:) =
0;%Boundary
conditions at p =
0
XXI(:,:,ir) = XI;
end
XXfO(:,:,ir) = Xfl;
else
normfl(ir,it) = normfl(ir,it-1);
residu-f(ir,it) = residu-f(ir,it-1);
end
end
XXfO(:,:,ir) = ne-norm(ir)*Xfl/normfl(ir,nit-f);
end
%
%
%
%
if radialmode == 1,%Radial diffusion or pinch
for imhu = 1:nmhu,
for ip = 1:np,
[Mrl,MrO,Mrr] =
transprad8yp(dtn,xrho(:),squeeze(XXDr(ip,im,:)),squeeze(XXVr(ip,im,
%
Ar = Mrl\MrO;
199
:)),1);
-
+
XXfi(ip,im,:) = Ar*squeeze(XXfO(ip,im,:));%Radial transport calculation (with p
0j
dependence)
%
end
%
end
%
end
%
else
break;
%
end
end
clear ir fM XfM Xsssl MXPC_f_t MXPDI-ftt Xmask MXL_f-t MXU-f-t
clear XfO XfOt Xq Xq-t Xf1 Xf1_t XfItp Xfitm Xfits XRt XSt XY_t
clear fl xfl Ii 12 XI
if display-mode == 2,
figure(100),subplot(121),hold off;
figure(100),subplot(122),hold off;
end
if nr-dke >= 1,
for ir
1:nrdke,
=
jri
rdke(ir)-1;
=
jr2 = rdke(ir);
jr3
rdke(ir)+i;
=
XfOm = squeeze(XXfO(:,:,jrl));
XfO
= squeeze(XXfO(:,:,jr2));
XfOp = squeeze(XXfO(:,:,jr3));
XfM
fM
=
squeeze(XXfM(:,:,jr2));
XfM(:,1)';
=
Xmsom0s
Xsssl
Xlambda
= squeeze(XXmsom0s(:,:,jr2));
= squeeze(XXsssl(:,:,jr2));
= squeeze(XXlambda(:,:,jr2));
XfOg-th = squeeze(XXfOgth(:,:,jr2));
if store-mode == 1,
eval( ['load MXPmat.mat MXPDI',int2str(jr2), '_g_t MXPC',int2str(jr2),
Xmask',int2str(jr2),'_g'];
eval(['load MXPmat.mat MXP',int2str(jr2),'_tpt']);
end
eval(['MXPltpt = MXP',int2str(jr2),'_tp-t;']);
eval(['MXPDI-g-t = MXPDI',int2str(jr2),'_g.t;']);
eval(['MXPC-g-t = MXPC',int2str(jr2),'_g-t;']);
eval(['Xmask-g = Xmask',int2str(jr2),'_g;']);
200
gt
Xmask-tp = Xmask-g;
eval(['clear MXPDI',int2str(jr2),'.g.t MXPC',int2str(jr2),'-g-t
Xmask',int2str(jr2),'_g']);
eval(['clear MXP',int2str(jr2),'_tpt']);
if invproc == 1,%MatLab built-in full LU factorization
[MXL-g-t,MXU-g-t] = lu(MXPC-g-t);
elseif invproc == 2,XMatLab built-in iterative inversion method
[MXL-g-t,MXU-g-t] = luinc(MXPC-g-t,ie-5);
end
dr = (ia(jr3)
ia(jrl))*Rp-norm;%radial step for gradient calculations (dke)
-
drm = (ia(jrl)
-
ia(jr2))*Rp-norm;
drp = (ia(jr3)
-
ia(jr2))*Rp-norm;
Xdf0drho = -drm*XfOp/dr/drp + drp*XfOm/dr/drm - (drp + drm)*XfO/drp/drm;%Calculation of
the radial gradient of f
XfOtp = - ftp-norm(jr2)*(Xpn.*Xmhu.*Xdf0drho);
XXfOtp(:,:,jr2) = XfOtp;
XFirst order Legendre correction for ftp
fitp
=
1.5*trapzyp([mhu',XfO._tp'.*XmsomOs'.*Xlambda' .*(mhu'*ones(i,npn))]);
xfltp = fLtp'*ones(1,length(f1_tp));
XI-tp = zeros(size(XfOtp));
if trunc == 1,
I1_tp = trapzyp([pn',xfltp.*masku.*xpn3./xgamma])/3/Te-norm(jr2)
-
...
trapzyp( [pn' ,xfi-tp.*masku.*xpn3]) .*gamma*2/3/Te-norm(jr2) ...
trapzyp( [pn' ,xf1_tp.*masku. *xpn3. *xpn2./xgAmma]) .*gamma/5/Te-norm(jr2)/Te-norm(jr2) ...
+ trapzyp([pn',xfitp.*masku.*(xgamma - xsigma./xz).*xpn./xgamma])
...
- trapzyp( [pn' ,xfl..tp. *masku.*xpn3.*xJ2./xgamma./xz2]) .*gamma/Te-norm(jr2)
...
+ trapzyp([pn',xfitp.*masku.*xpn3.*(I + 3.0./xz2 3*xgamma.*xsigma./xz2./xz)./xgamma]).*(gamma.*pn2
-
5*Te-norm(jr2))/6/Te-norm(jr2)/Te-norm(jr2) ...
trapzyp([pn',xfltp.*masku.*xpn3.*xJ3./xz./xgamma]).*gAnma/2/betath/betath/Te-norm(jr2)/Te-norm(j
r2) ...
+ trapzyp([pn',xfitp.*masku.*xpn3.*xJi./xz2./xgamma]).*gamma/2/Te-norm(jr2) ...
+ trapzyp([pn',xfitp.*masku.*xpn.*(xgamma.*xsigma./xz -
1)./xgamma]).*pn2/Tenorm(jr2)
...
trapzyp( [pn' ,xf 1_tp. *masku. *xpn3. *xJ4. /xz. /xgamma])
.*gamma/12/betath/betath/Te-norm(jr2) /Te-norm(
jr2);
12_tp = trapzyp([pn',xfltp.*maskl./xgamma])/3/Te-norm(jr2) ...
+ trapzyp([pn',xfitp.*maskl]).*(-2*gamma/3/Te-norm(jr2) +
pn2/5/Te-norm(jr2)/Te-norm(jr2))
...
201
+ trapzyp( [pn' ,xf1_tp.*maskl./xgamma])
.*(gamma - sigma./z) ./pn2
...
trapzyp([pn',xfltp.*maskl.*(xgamma.*xpn2 - 5*Te-norm(jr2))./xgamma]).*(1 +
- trapzyp([pn',xf1_tp.*maskl]).*J2./z2/Te-norm(jr2)
+
3.0./z2
-
3*gamma.*sigma./z2./z)/6/Te-norm(jr2)/Te-norm(jr2) ...
+ trapzyp([pn',xf1_tp.*maskl]).*(J3./z/2/betath/betath/Te-norm(jr2)/Te-norm(jr2)
+ J1./z2/2/Te-norm(jr2)
-
J4./z/12/betath/betath/Te-norm(jr2)/Te.norm(jr2)) ...
+ trapzyp([pn',xf1_tp.*maskl.*xpn2./xgamma]).*(gamma.*sigma./z
-
1)./pn2/Te-norm(jr2);
XI-tp = 4*pi*((fM.*(fitp + II-tp./pn2 + 12-tp.*pn)./gamma)'*mhu).*Xmsom0s;
end
XItp-t = XI-tp(:,Xmask-tp);
XI.g_t = zeros(size(XI-tp-t));
Xf0_tp-t = XfQtp(:,Xmask-tp);
XfOg-th-t = XfOg-th(:,Xmask-tp);
for it.g = 2:nit-g,
XfOg = squeeze(XXfOg(:,:,jr2));%Load value only
XfOg-t = XfQg(:,Xmask-g);
if residug(jr2,it-g-1) >= precOg,
Xq-g-t = reshape(-MXP-tp-t*reshape(XfOtp-t'
,
(nmhu
-
nstp(jr2))*npn,1),(nmhu
-
ns-tp(jr2)),npn)';
Xq-g-t = Xq-g-t + (XILg-t + XI-tp-t).*Xpn2(:,Xmask-g);
Xq-g-t(1,:) = XfOg-th-t(,:);Boundary conditions at p = 0
Xq-g-t(:,(nmhu - ns-g(jr2))/2) = 0;%g(pn,mhu = -mhut) = 0
Xq-g-t(:,(nmhu - ns-g(jr2))/2 + 1) = 0;Xg(pn,mhu = mhut) = 0
XR-t = reshape(MXPDI-g-t*reshape(Xqg-t' ,(nmhu
ns-g(jr2)),npn)'
- ns-tp(jr2))*npn,1),(nmhu
-
-
reshape(MXPC-g-t*reshape(XfOgt'
,(nmhu - ns-g(jr2))*npn,1),(nmhu -
ns-g(jr2)),npn)';
if invproc == 1,XFull LU factorization
XY-t = reshape(MXL-g-t\reshape(XR.t'
,(nmhu - ns-tp(jr2))*npn,1),(nmhu
-
ns-g(jr2)),npn)';
%Solving the simultaneous linear equations
XS-t
= reshape(MXU-g-t\reshape(XY-t' , (nmhu - ns-tp(jr2))*npn,1),(nmhu
-
ns-g(jr2)),npn)';
%Solving the simultaneous linear equations
elseif invproc == 2,Xqmr iterative method
XS-t = reshape(qmr(MXPC-g-t,reshape(XRt'
6,(nmhu
-
,
(nmhu - ns-g(jr2))*npn,1),le-
.
ns-g(jr2))*npn,MXL-g-t,MXU-g-t),(nmhu
- ns-g(jr2)),npn)';
elseif invproc == 2,%bicgstab iterative method
XS-t = reshape(bicgstab(MXPC-g-t,reshape(XRt'
202
,(nmhu
- ns-g(jr2))*npn,1),1e-
6,(nmhu - nsg(jr2)),npn)';
ns-g(jr2))*npn,MXL-g-t,MXU-g-t),(nmhu
elseif invproc == 2,%cgs iterative method
XSt = reshape(cgs(MXPC-g-t,reshape(XRt'
,
(nmhu - ns-g(jr2))*npn,1),1e-
6,(nmhu - nsg(jr2)),npn)';
ns-g(jr2))*npn,MXL-g-t,MXU-g-t),(nmhu
elseif invproc == 2,%pcg iterative method
XS-t = reshape(pcg(MXPC-g-t,reshape(XR-t'
,
(nmhu
- nsg(jr2))*npn,1),1e-
6,(nmhu ns-g(jr2))*npn,MXL-g-t,MXU-g-t),(nmhu
- ns-g(jr2)),npn)';
end
Xfig-t
=
XfO-g-t
+
XS-t;
%Expansion of the function g
Xf2_g(1:npn,i:nmhu/2
- ns-g(jr2)/2) = Xflg-t(:npn,1:(nmhu - ns-g(jr2))/2);
Xf2_g(1:npn,nmhu/2 + ns-g(jr2)/2 + 1:nmhu) = Xflg-t(:npn,(nmhu - ns-g(jr2))/2 +
1:nmhu
-
nsg(jr2));
for is = 1:ns-g(jr2),
Xf2_g(:npn,Xmask-g((nmhu - ns-g(jr2))/2) + is)
end
Xf1_g = Xf2_g;
= 0;
norml.g(jr2,it-g) = 2*pi*trapzyp([pn',pn2'.*trapzyp([mhu',Xfig'])']);
residu-g(jr2,it-g) = sqrt(abs(2*pi*trapzyp([pn',pn2'.*trapzyp([mhu',(Xflg'
-
XfOg').^2.*Xflg'])'])/normlg(jr2,it-g)));
if displaymode >= 1,
infoyp(2, ['Normalized minor radius:
ratio:
',num2str(rho(jr2)),',
Inverse aspect
',...
num2str(ia(jr2)),', iteration : ',int2str(it-g),' /
,int2str(nit-g),...
',
Residu = ',num2str(residu-g(jr2,it-g))]);
end
if display-mode == 2,
if jr2 == ir-display,
figure(101),subplot(121) ,plot(-pn,Xf1_g(: ,1), 'r-' ,pn,Xfl1_g(: ,nmhu),'r-');
%g(perp = 0)
axis('square');set(gca,'xlim',[-10,10]);
%faxis = axis;axis([faxis(1)/4,faxis(2)/4,faxis(3),faxis(4)]);zoom
title(
['Iteration:
',int2str(itg)]);xlabel('pn-
refV);ylabel('g(pperp=0)')
hold on;zoom on
figure(101),subplot(122),semilogy([1:it-g]
,residu-g(jr2,1:it-g),...
'r+',[1:itg],precog*ones(size([1:it-g])),'b--');
axis('square');axis([0,it-g,precg/10,1]);
title('Convergence - g');xlabel('Iteration');ylabel('Residu')
203
on
hold on;zoom on
pause(O.1);
end
end
%First order Legendre correction for g
XI-g-t = zeros(size(XI-tp-t));
if trunc == 1,
f1-g = 1.5*trapzyp([mhu',Xf1_g'.*(mhu'*ones(1,npn))]);
xflg = f1_g'*ones(1,length(f1_g));
I1_g
=
-
trapzyp([pn',xflg.*masku.*xpn3./xgammal)/3/Te-norm(jr2)
trapzyp( [pn' ,xf1_g. *masku. *xpn3]). *gamma*2/3/Te-norm(jr2)
...
...
trapzyp([pn',xf1_g.*masku.*xpn3.*xpn2./xgamma]).*gamma/5/Te-norm(jr2)/Te-norm(jr2)
...
+ trapzyp([pn',xf1_g.*masku.*(xgamma - xsigma./xz).*xpn./xgamma])
-
...
trapzyp([pn',xf1_g.*masku.*xpn3.*xJ2./xgamma./xz2]).*gamma/Te-norm(jr2)
+ trapzyp([pn',xflg.*masku.*xpn3.*(1
3*xgamma.*xsigma./xz2./xz)./xgamma]).*(gamma.*pn2 -
+ 3.0./xz2 -
...
5*Tenorm(jr2))/6/Te-norm(jr2)/Te-norm(jr2)
...
trapzyp([pn',xf1_g.*masku.*xpn3.*xJ3./xz./xgamma]).*gamma/2/betath/betath/Te-norm(jr2)/Te-norm(jr
2) ...
trapzyp([pn',xflg.*masku.*xpn3.*xJi./xz2./xgamma]).*gamma/2/Te-norm(jr2) ...
+ trapzyp([pn',xfig.*masku.*xpn.*(xgamma.*xsigma./xz
-
1)./xgamma]).*pn2/Tenorm(jr2) ...
trapzyp([pn',xflg.*masku.*xpn3.*xJ4./xz./xgamma]).*gamma/12/betath/betath/Te-norm(jr2)/Te-norm(j
r2);
12_g = trapzyp([pn',xflg.*maskl./xgamma])/3/Te-norm(jr2) ...
+ trapzyp([pn',xf1_g.*maskl]).*(-2*gamma/3/Te-norm(jr2) +
pn2/5/Te-norm(jr2)/Te-norm(jr2))
...
+ trapzyp([pn',xf
1_g.*maskl./xgamma]).*(gamma
- sigma./z)./pn2
- trapzyp([pn',xf1_g.*maskl]).*J2./z2/Te-norm(jr2)
...
...
+ trapzyp([pn',xflg.*maskl.*(xgamma.*xpn2 -
5*Tenorm(jr2))./xgamma]).*(1 + 3.0./z2
3*gamma.*sigma./z2./z)/6/Te-norm(jr2)/Te-norm(jr2)
...
trapzyp([pn',xflg.*maskl]).*(J3./z/2/betath/betath/Te-norm(jr2)/Te-norm(jr2)
+
J1./z2/2/Tenorm(jr2)
4./z/12/betath/betath/Te-norm(jr2)/Te-norm(jr2))
...
+ trapzyp([pn',xf1_g.*maskl.*xpn2./xganmma]).*(gamma.*sigma./z
1)./pn2/Tenorm(jr2);
XI-g = 4*pi*((fM.*(flg
-
+ I1_g./pn2 + 12_g.*pn)./gamma)'*mhu).*Xsssl;
XI-g-t = XI-g(:,Xmask-g);
end
204
XXfOg(:,:,jr2) = Xflg;
else
residu-g(jr2,it-g) = residu-g(jr2,itg-1);
end
end
c-g(jr2) = -1/ne-norm(jr2)
* 2*pi*trapzyp([mhu',trapzyp([pn',(Xfog
+
XfOtp).*Xpn2])']);XDensity conservation for g
XfOg = XfOg + c-g(jr2)*XfO;
XXfOg(:,:,jr2) = XfOg;
end
else
XXfOtp = zeros(size(XXf0));
XXfOg = zeros(size(XXf0));
end
clear fM XfM Xsssl XmsomOs Xlambda MXP-tp-t MXPC-g-t MXPDI-g-t Xmask-g MXL-g-t MXUg-t
clear Xdf0drho XfO-tp XfOtp-t ir jri jr2 jr3 dr drm drp
clear Xf0 XfOm XfOp XfO-g XfOg-t XfO-g-th-t Xq-g Xq-g-t Xf1_g Xf2_g Xflg-t XRt XSt XY_t
clear fl-g xflg
clear fitp
I1_g 12_g XI-g XI-g-t
xfltp
Ii-tp
12_tp XILtp XI-g-tp
if display == 2,
if ~isempty(find(rdke==ir-display))
=A
XfM = squeeze(XXfM(:,:,ir-display-1));
XfO = squeeze(XXfO(:,:,ir..display-1));
figure(102),subplot(321),semilogy(-pn,XfM(:,2),'g--',pn,XfM(:,nmhu-1),'g--',pn,XfO(:,2),'r-',...
pn,XfO(:,nmhu-1),'r-');%f(perp = 0)
axis('square');axis([-max(pn),max(pn),ie-20,10]);zoom
title(['Iteration:
on
',int2str(nit-f)]);xlabel('pn-ref');ylabel('f(pperp=o)')
hold on;zoom on
figure(102),subplot(322),plot([i:nit-f] ,currl(ir-display-1,:),'r-');
axis('square');if max(max(curr1(ir-display-1,:)))
-=
0,axis([O,nit_f,O,1.1*max(max(currl(ir-display-1,:)))]);end
title('Convergence');xlabel('Iteration');ylabel('Current')
hold on;zoom on
pause(O.1);
XfM = squeeze(XXfM(:,:,ir-display));
Xf0 = squeeze(XXfO(:,:,ir-display));
figure(102),subplot(323),semilogy(-pn,XfM(:,2),'g--',pn,XfM(:,nmhu-1),'g--',pn,XfO(:,2),'r-',...
pn,XfO(:,nmhiu-1),'r-');%f(perp = 0)
axis('square');axis([-max(pn),max(pn),ie-20,10]);zoom
on
title(['Iteration: ',int2str(nit-f)]);xlabel('pn-ref');ylabel('f(pperp=o)')
205
hold on;zoom on
figure(102),subplot(324),plot([1:nit-f],currl(ir-display,:),'r-');
axis('square');if max(max(currl(irdisplay,:)))
-=
0,axis([O,nit_f,0,1.1*max(max(curr1(irdisplay,:)))]);end
title('Convergence');xlabel('Iteration');ylabel('Current')
hold on;zoom on
pause(O.1);
XfM
=
squeeze(XXfM(:,:Xirdisplay+1));
XfO = squeeze(XXfO(:,:,ir-display+1));
figure(102),subplot(325),semilogy(-pn,XfM(: ,2), 'g--' ,pn,XfM(: ,nmhu-1) ,'g-pn,XfO(:,2),'r-',...
pn,XfO(:,nmhu-1),'r-');Xf(perp = 0)
axis('square');axis([-max(pn) ,max(pn) ,le-20,10]) ,zoom on
title(['Iteration:
',int2str(nit-f)]);xlabel('pn-ref');ylabel('f(pperp=O)')
hold on;zoom on
figure(102),subplot(326),plot([l:nitf],currl(ir-display+,:),'r-');
axis('square');if max(max(curr1(ir_display+1,:)))
-=
0,axis([0,nit_.f,Q,1.1*max(max(currl(irdisplay+1,:)))]);end
title('Convergence');xlabel('Iteration');ylabel('Current')
hold on;zoom on
pause(O.1);
else
XfM = squeeze(XXfM(:,:,ir_display));
Xf0 = squeeze(XXf0(:,:,irdisplay));
figure(102),subplot(121),semilogy(-pn,XfM(:,2),
'g--',pn,XfM(: ,nmhu-1),
'g--
pn,XfO(:,2),'r-',...
pn,XfO(:,nmhu-1),'r-');%f(perp = 0)
axis('square');axis([-max(pn),max(pn),le-20,10]),zoom
title(['Iteration:
on
',int2str(nit-f)]);xlabel('pn-ref');ylabel('f(pperp=O)')
hold on;
figure(102),subplot(122),plot([i:nit-] ,currl(irdisplay,:),'r-');
axis('square');if max(max(currl(ir-display,:)))
-=
0,axis([O,nit-f,0,1.1*max(max(curr(ir_display,:)))]);end
title('Current convergence');xlabel('Iteration');ylabel('Current')
hold on;
pause(O.1);
end
end
clear ir XfM Xf0
for jr = 1:nr,
XDlh = squeeze(XXDlh(:,:,jr));
XDlhim = squeeze(XXDlhim(:,:,jr));
XDlhjm = squeeze(XXDlhjm(:,:,jr));
206
-
Xmhu-psib
= sign(Xmhu).*sqrt(1
- xpsib(jr)*(i
- Xmhu.~2));
Xmhu-psibm = sign(Xmhum).*sqrt(1 - xpsib(jr)*(i - Xmhum.~2));
Xmhu-psibp = sign(Xmhup).*sqrt(1 - xpsib(jr)*(1 - Xmhup.^2));
Xmhu-psib(Xmhu-psib==O)
= eps;
Xmhu-psibm(Xmhu-psibm==O)
=
Xmhu-psibp(Xmhu-psibp==O)
=
eps;
eps;
XDcy = squeeze(XXDcy(:,:,jr));
XDcyim = squeeze(XXDcyim(:,:,jr));
XDcyjm = squeeze(XXDcyjm(:,:,jr));
XA
squeeze(XXA(:,:,jr));
=
XAm = squeeze(XXAm(:,:,jr));
XF = squeeze(XXF(:,:,jr));
XFm = squeeze(XXFm(:,:,jr));
XBt = squeeze(XXBt(:,:,jr));
Xdeltaip = XXdeltaip(:,:,jr);
Xdeltaim = XXdeltaim(:,:,jr);
Xdeltajp = XXdeltajp(:,:,jr);
Xdeltajm = XXdeltajm(:,:,jr);
Xsssl = squeeze(XXsssl(:,:,jr));
XfO = squeeze(XXfO(:,:,jr));
XfM = squeeze(XXfM(:,:,jr));
XfOtp = squeeze(XXfOtp(:,))
Xfg
= squeeze(XXfOg(:,:,jr));
XfOtot = Xf0 + XfOg + XfO-tp;%Electron momentum distribution function from drift kinetic
equation
%
Bulk heating calculations
fO = 0.5*trapzyp([mhu',XfO']);
fM = XfM(:,1)';
vth20 = 4*pi*trapzyp([pn',pn'.*pn'.*v'.*v'.*fO'])/3;
vth2M = 4*pi*trapzyp([pn',pn'.*pn'.*v'.*v'.*fM'])/3;
Te-out-norm(jr) = (vth20/vth2M)*Te-norm(jr) ;%Bulk heating calculations
% Calculation of bounce-averaging coefficients
if bounce-mode >= 1,
m = 30;% m-1 is the order of the series expansion for lambda and lambda-tp (m>0,
enough)
Xmmask = (Xmhu2>mhubounce2(jr));
[lambda,Delta,lambda-tp,sstar-tp]
= bouncejd(mhubounce2(jr),mhu,m);
Xlambda = ones(npn,1)*lambda;
Xlambda-tp = ones(npn,1)*lambdatp;
207
m=2 is
XDelta = ones(npn,1)*Delta;
Xsstar-tp = ones(npn,i)*sstar-tp;
[lambdam,Deltam,lambda-tpm,sstar-tpm]
Xlambdam = ones(npn,i)*lambdam;
Xlambda-tpm = ones(npn,1)*lambda.tpm;
XDeltam = ones(npn,1)*Deltam;
Xsstar-tpm = ones(npn,1)*sstar-tpm;
=
bouncejd(mhubounce2(jr),mhum,m);
else
Xmmask = ones(npn,nmhu);
Xlambda = ones(npnnmhu);
Xlambda-tp = ones(npn,nmhu);
XDelta = zeros(npn,nmhu);
Xsstar-tp = ones(npn,nmhu);
Xlambdam
ones(npnnmhu);
Xlambdatpm = ones(npn,nmhu);
XDeltam = zeros(npn,nmhu);
Xsstar-tpm = ones(npn,nmhu);
end
Xdphib = -XDelta;
Xdphibm = -XDeltam;
%
Current densities calculations
if bounce-mode >= 1,
normf(jr) = 2*pi*trapzyp([pn',pn2'.*trapzyp([mhu',XfO'])']);
normf-fsav(jr) = 2*pi*trapzyp([pn',pn2'.*trapzyp([mhu',XfO'*(i+ia(jr)).*Xlambda'])']);
curr_0(jr)
= 2*pi*trapzyp([pn',pn2'.*(pn'./gamma).*trapzyp([mhu',Xmhu'.*Xf0'])']);
currOfsa(jr) =
2*pi*trapzyp([pn',pn2l.*(pn'./gAmma').*trapzyp(Emhu',Xmhu'.*Xf0'*(1+ia(jr)).*Xmmask'])']);
curr-tp(jr) = 2*pi*trapzyp([pn',pn2'.*(pn'./gamma').*trapzyp([mhu',Xmhu'.*Xfotp'])']);
currtp-fsav(jr) =
2*pi*trapzyp([pn',pn2'.*(pn'./gamma').*trapzyp([mhu',Xmhu'.*Xfotp'*(1+ia(jr)).*Xlambda-tp']),]);
curr-g(jr) =
2
*pi*trapzyp([pn',pn2'.*(pn'./gamma').*trapzyp([mhu',Xmhu'.*Xfog'])
curr-g-fsav(jr) =
2*pi*trapzyp([pn',pn2'.*(pn'./gamma').*trapzyp([mhu',Xmhu'.*XfOg'*(1+ia(jr)).*Xmmask'])']);
X
curr-tot(jr) = currO(jr) + curr.tp(jr) + curr-g(jr);
curr-tot-fsav(jr) = curr_0_fsav(jr) + curr-tp-fsav(jr) + curr-g-fsav(jr);%
else
normf(jr) = 2*pi*trapzyp([pn',pn2'.*trapzyp([mhu',XfO'])']);
normf-fsav(jr) = normf(jr);
208
]);
currO(jr) = 2*pi*trapzyp([pn',pn2'.*(pn'./gamma').*trapzyp([mhu',Xmhu'.*Xf0'])']);
curr_0_fsav(jr) = currO(jr);
curr-tp(jr) = 0;
curr-tp-fsav(jr) = 0;
curr-g(jr) = 0;
curr-g-fsav(jr) = 0;
curr-tot(jr) = curr_0(jr) + curr-tp(jr) + curr-g(jr);
curr-tot-fsav(jr) = curr_0_fsav(jr) + curr-tp.fsav(jr) + curr-g(jr);%-fsav-s(jr);
end
%
Flux Calculations
Xpnm(i,:) = eps;
Xmhum(Xmhum==0) = eps;
Xmhu2m(Xmhu2m==0) = eps;
= eps;
Xlambdam(Xlambdam==0)
Xmhup(Xmhup==0)
%
= eps;
For fO
XfOim = zeros(size(Xf0));
Xf0im(2:npn,:) = Xdeltaim(2:npn,:).*XfO(i:npn-i,:) + (1-Xdeltaim(2:npn,:)).*XfO(2:npn,:);
XfOim(i,:) = XfO(i,:);
XdfOdpim = gradientjd(XfO,dpn,i);
XdfOdmim = (gradientjd(XfOim',dmhu,i)'+gradientjd(XfOim',dmhu,2)')/2;
XfOjm = zeros(size(XfO));
XfOjm(:,2:nmhu) = Xdeltajm(:,2:nmhu).*XfO(:,1:nmhu-1) + (1Xdeltaim(: ,2:nmhu)).*XfO(: ,2:nmhu);
Xfojm(:,1) = XfO(:,1);
XdfOdpjm = (gradientjd(XfOjm,dpn,i)+gradientjd(XfOjm,dpn,2))/2;
XdfOdpjm(npn,:) = 2*XdfOdpjm(npn,:);
Xdf0dmjm = gradientjd(Xf0',dmhu,1)';
if bouncemode >= 1,
XSpOcoll = - XAm.*Xdf0dpim - XFm.*XfOim;%Collision momentum flux, not bounce averaged
XSmhuOcoll = sqrt(X1mmhu2m).*XBt.*XdfOdmjm./Xpn;%Collision angular flux, not bounce
averaged
XSpOcoll-bav =
-
XAm.*XdfOdpim - XFm.*XfOim;%Collision momentum flux
XSmhu&_coll-bav
=
sqrt(Ximmhu2m).*(XBt.*XdfOdmjm.*(1
+ Xdphibm. /Xmhu2m)./Xpn);%Collision
angular flux
XSpOohm-bav = epsi(jr).*Xmhu.*Xfaim.*Xmmask./Xlambda;XOhmic electric field momentum flux
XSmhu0_ohmbav = - epsi(jr)*sqrt(Xmmbihu2m). *Xf0jm. *Xmmask./Xlambdam; %Ohmic electric field
angular flux
XSpOlh-bav = - XDlhim.*Xmhu.*(Xmhu.*XdfOdpim + Xlmmhu2.*XdfOdmim./Xpnm)./Xlambda;XLH
wave field momentum flux
209
XSmhuO-lhbav = sqrt(Ximmhu2m).*XDlhjm.*(Xmhum.*XdfOdpjm +
Ximmhu2m.*XdfOdmjm./Xpn)./Xlambdam;LH wave field angular flux
XSpO-cy-bav = - XDcyim.*Xmhu./Xmhu-psib.*Xlmmhu2.*(xpsib(jr)*XdfOdpim +
(Xlmmhu2*xpsib(jr) xomega-cy(jr)./Xgammam).*XdfOdmim./Xpnm./Xmhu)./Xlambda;X/*EC wave field momentum flux
XSmhuO-cy-bav = sqrt(Xlmmhu2m) .*XDcyjm. *Xmhum./Xmhu..psibm. *(XdfOdpjm. *(Xlmmhu2m*xpsib(jr)
xomega-cy(jr)./Xgamma) + XdfOdmjm.*(Xlmmhu2m*xpsib(jr) xomegacy(jr) ./Xgamma) .2. /Xpn./Xmhum/xpsib(jr)) ./Xmhum. /Xlambdam; /EC
wave field
angular flux
XSpObav = XSpOcoll-bav + XSpOohm-bav + XSpOlh-bav + XSpQcy-bav;%Total momentum flux
XSmhuObav = XSmhu0_collbav + XSmhuOohmbav + XSmhuOlhbav + XSmhuOcy-bav; %Total
angular flux
XXSp'(:,:,jr) = XSpAbav;
XXSmhu(:,:,jr) = XSmhuObav;
%
For g
Xf0_gim = zeros(size(XfOg));
XfOgim(2:npn,:) = Xdeltaim(2:npn,:).*XfQg(l:npn-,:) + (1Xdeltaim(2:npn,:)).*XfOg(2:npn,:);
XfOgim(1,:) = XfO-g(1,:);
XdfOgdpim = gradientjd(XfOg,dpn,1);
XdfOgdmim = (gradientjd(XfO-gim',dmhu,i)'+gradientjd(XfOgim',dmhu,2)')/2;
Xf0_gjm = zeros(size(XfQg));
XfOgjm(:,2:nmhu) = Xdeltajm(:,2:nmhu).*XfOg(:,1:nmhu-1) + (1Xdeltaim(:,2:nmhiu)).*XfOg(:,2:nmhu);
Xfogjm(:,1) = XfOg(:,1);
Xdf0_gdpjm = (gradientjd(XfO-gjm,dpn,1)+gradientjd(XfO-gjm,dpn,2))/2;
XdfOgdpjm(npn,:) = 2*XdfOgdpjm(npn,:);
XdfOgdmjm = gradientjd(XfOg',dmhu,1)';
XSpOg-coll
= -
XAm.*XdfOgdpim - XFm.*Xfc-gim;%Collision momentum flux, not bounce
averaged
XSmhuO-g-coll
=
sqrt(X1mmhu2m).*XBt.*XdfOgdmjm./Xpn;%Collision
angular flux , not bounce
averaged
XSpOg-coll-bav =
XSmhu0_g-coll-bav
-
=
XAm.*Xdf0_gdpim - XFm.*XfOgim;XCollision momentum flux
sqrt(Xlmmhu2m).*(XBt.*XdfOgdmjm.*(1 +
Xdphibm./Xmhu2m) ./Xpn) ;XCollision angular flux
XSpOg-ohm-bav = epsi(jr). *Xmhu.*Xf0_gim.*Xmmask. /Xlambda;%Ohmic
electric field momentum
flux
XSmhuO-g-ohm-bav
= -
epsi (jr)*sqrt(Xmmhlu2m). *Xf 0_gjm. *Xmmask. /Xlambdam; %Ohmic electric
field angular flux
210
XSpOg-lh_bav = - XDlhim.*Xmhu.*(Xmhu.*Xdf&_gdpim + ...
Xlmmhu2.*Xdf0_gdmim./Xpnm)./Xlambda;X.LH wave field momentum flux
XSmhu0g-lh-bav = sqrt(Ximmhu2m).*XDlhjm.*(Xmhum.*XdfOgdpjm + ...
Xlmmhu2m.*XdfOgdmjm./Xpn)./Xlambdam;LH wave field angular flux
XSpOg-cy-bav = - XDcyim.*Xmhu./Xmhu-psib.*Xmmhu2.*(xpsib(jr)*XdfO-gdpim +
(Xlmmihu2*xpsib(jr) - xomegacy(jr) ./Xgammam).*Xdf Ogdmim./Xpnm./Xmhu) ./Xlambda; XEC
wave field momentum flux
XSmhu0g-cy-bav =
sqrt(Xmmhu2m) .*XDcyjm.*Xmhum./Xmhu-psibm.*(XdfO-gdpjm.*(Ximmi2m*xpsib(jr)
xomega-cy(jr)./Xgamma) + XdfO.gdmjm.*(Ximmhu2m*xpsib(jr) xomegacy(jr) ./Xgamma) .2./Xpn./Xmhum/xpsib(jr)) ./Xmhum. /Xlambdam; /EC wave field
angular flux
XSpO-g-bav = XSpO-g-coll-bav + XSp0g-ohm-bav + XSpOg-lh-bav + XSpO-g-cy-bav;%Total
momentum flux
XSmhuOg_bav = XSmhu0g-collbav + XSmhu0g-ohm-bav + XSmhu0g-lh-bav +
XSmhuOg-cy-bav;%Total angular flux
XXSpOg(:,:,jr)
= XSpOg-bav;
XXSmhuO-g(:,:,jr) = XSmhuO.g_bav;
%
For f-tp
XfOtpim = zeros(size(XfOtp));
XfOtpim(2:npn,:)
= Xdeltaim(2:npn,:).*XfO-tp(1:npn-,:)
+ (1-
Xdeltaim(2:npn,:)).*XfOtp(2:npn,:);
XfOtpim(,:) = Xfotp(,:);
XdfOtpdpim = gradientjd(XfOtp,dpn,i);
XdfOtpdmim = (gradientjd(XfOtpim',dmhu,i)'+gradientjd(XfO-tpim',dmhu,2)')/2;
Xf&-tpjm = zeros(size(XfOtp));
XfOtpjm(:,2:nmhu) = Xdeltajm(:,2:nmihu).*XfOtp(:,1:nmihu-1)
+ (1-
Xdeltaim(:,2:nmhiu)).*XfOtp(:,2:nmhu);
XfOtpjm(:,i) = XfOtp(:,i);
XdfOtpdpjm = (gradientjd(XfOtpjm,dpn,1)+gradientjd(XfOtpjm,dpn,2))/2;
XdfOtpdpjm(npn,:) = 2*Xdf0_tpdpjm(npn,:);
XdfOtpdmjm = gradientjd(XfOtp',dmhu,i)';
% for EC bounce averaged fluxes for f-tp
XfOtp-cy = XfOtp.*Xmhu-psib./Xmhu/xpsib(jr);
XfOtp-cyim = zeros(size(XfOtp-cy));
XfO-tp-cyim(2:npn,:)
= Xdeltaim(2:npn,:).*XfOtp-cy(1:npn-,:)
+ (1-
Xdeltaim(2:npn,:)).*XfOtp(2:npn,:);
XfOtp-cyim(,:) = Xfotp-cy(,:);
XdfOtp-cydpim = gradientjd(XfOtp-cy,dpn,1);
XdfOtp-cydmim = (gradientjd(XfOtp-cyim',dmhu,1)'+gradientjd(XfOtp-cyim',dmhu,2)')/2;
211
Xf0_tp-cyjm = zeros(size(XfOtp-cy));
XfOtp-cyjm(:,2:nmhu)
= Xdeltajm(:,2:nmhu).*XfOtp-cy(:,1:nmhu-1)
+
(1-
Xdeltaim(: ,2:nmhu)).*XfOtp_ cy(: ,2:nmhu);
Xfotpcyjm(:,i) = Xfotp-cy(:,i);
XdfOtp-cydpjm = (gradientjd(XfOtp-cyjm,dpn,1)+gradientjd(XfO-tp-cyjm,dpn,2))/2;
XdfOtp-cydpjm(npn,:) = 2*XdfO-tp-cydpjm(npn,:);
XdfOtp-cydmjm
=
gradientjd(XfOtp-cy',dmhu,i)';
XSpOtp-coll
=
-
XAm.*XdfOtpdpim - XFm.*XfO-tpim;%Collision momentum flux, not bounce
XSmhuOtp-coll
=
sqrt(Xmmhu2m).*XBt.*Xdf
averaged
0_tpdmjm./Xpn;%Collision
angular flux, not
bounce averaged
XSpOtp-coll-bav
=
Xmmask./Xlambda/(+ia(jr)).*(XAm.*XdfQtpdpim +
-
XFm.*Xf0_tpim);%Collision momentum flux
XSmhuOtp-coll-bav
= Xmmask./Xlambda/(i+ia(jr)).*sqrt(Xlmmhu2m).*(XBt.*Xdf0_tpdmjm.*(1
Xdphibm./Xmhu2m) ./Xpn) ;%Collision angular flux
XSpOtp-ohm-bav = epsi (jr). *Xmhu. *XfO._tpim. *Xmmask. *Xlambda-tp. /Xlambda;
%Ohmic
electric
field momentum flux
XSmhu0tp-ohm-bav
=
epsi(jr)*srt(Xmmhu2m).*Xf
-
0_tpjm.
*Xmmask. *Xlambdatp. /Xlambdam;%Ohmic
electric field angular
flux
XSpO-tp-lh-bav
=
-
XDlhim.*Xmhu.*(Xmhu.*XdfOtpdpim +
Xlmmhu2.*XdfOtpdmim./Xpnm)./Xlambda/(+ia(jr))
;%LH wave field momentum flux
XSmhu0_tp-lh-bav = sqrt(Ximmhu2m).*XDlhjm.*(Xmhum.*Xdf0_tpdpjm
Xlmmhu2m.*XdfOtpdmjm./Xpn)./Xlambdam/(+ia(jr))
XSpO-tp-cy-bav
=
-
+ ...
;%LH wave field angular flux
XDcyim.*Xmhu./Xmhu-psib.*Xlmmhu2.*(xpsib(jr)*XdfOtp-cydpim
(Xlmmhu2*xpsib(jr)
xomegacy(jr) ./Xgammam) .*Xdf
+
-
0_tpcydmim.
/Xpnm. /Xmhu) ./Xlambda; XEC wave field momentum flux
XSmhuOtp-cy-bav =
sqrt(Xmmhu2m).*XDcyjm.*Xmhum./Xmhu-psibm.*(XdfO-tp-cydpjm.*(Ximmhil2m*xpsib(jr)
xomega-cy(jr)./Xgamma)
+ XdfOtp-cydmjm.*(Xmmhu2m*xpsib(jr)
-
-
xomega-cy(jr)./Xgamma).^2./Xpn./Xmhum/xpsib(jr))./Xmhum./Xlambdam;%EC
wave field
angular flux
XSpOtp-bav = XSpOtp-coll-bav + XSpOtp-ohm-bav + XSp0tp-lh-bav + XSpO-tp-cy-bav;%Total
momentum flux
XSmhu0tp-bav = XSmhu0_tp-coll-bav + XSmhuOtp-ohm-bav + XSmhuOtp-lh-bav +
XSmhuOtp-cy-bav;%Total angular flux
XXSpOtp(:,:,jr) = XSpQtp-bav;
XXSmhuO-tp(:,:,jr) = XSmhu0tp-bav;
else
212
XSpOcoll = - XAm.*XdfOdpim - XFm.*XfOim;%Collision momentum flux, not bounce averaged
XSmhuOcoll = sqrt(X1mmhu2m).*XBt.*XdfOdmjm./Xpn;%Collision angular flux, not bounce
averaged
XSpOohm = epsi(jr).*Xmhu.*XfOim;%Ohmic electric field momentum flux
XSmhuOohm = - epsi(jr)*sqrt(Ximmhu2m).*XfOjm;XOhmic electric field angular flux
XSpOlh =
momentum flux
XSmhuOlh
-
XDlhim.*Xmhu.*(Xmhu.*XdfOdpim
=
sqrt(Ximmhu2m).*XDlhjm.*(Xmhum.*XdfOdpjm + X1mmhiu2m.*Xdf0dmjm./Xpn);%LH wave
+
Xlmmhu2.*XdfOdmim./Xpnm); LH wave field
field angular flux
XSpOcy
=
-
XDcyim.*Xlmmhu2.*(XdfOdpin
+ (Ximmhiu2
-
xomega-cy(jr)./Xgammam).*XdfOdmim./Xpnm./Xmhu); /*EC wave field momentum flux
XSmhuOcy = sqrt(Xlmmhu2m) .*XDcyjm. *(Xdf Odpjm. *(Xlmmhu2m xomega-cy(jr)./Xgamma) + XdfOdmjm.*(Ximmhu2mxomegacy(jr) ./Xgamma) .2./Xpn./Xmhum) ./Xmhum;;EC wave field angular flux
XSpO = XSpO-coll + XSpOohm + XSpO-lh + XSpOcy;%Total momentum flux
XSmhuO = XSmhuOcoll + XSmhuOohm + XSmhuOlh + XSmhuOcy;%Total angular flux
X=
XXSp(:,:,jr) = XSp;
XXSmhuQ(:,:,jr) = XSmhu;
XXSpOg(:,:,jr) = 0;
XXShuOg(:,:,jr) = 0;
XXSpOtp(:,:,jr) = 0;
XXSmhuOtp(:,:,jr) = 0;
end
% Power densities calculations
dPcollO = trapzyp([pnm',Xpn2m.*Xpnm.*XSpOcoll./Xgammam]);
PcollO(jr) = -2*pi*trapzyp([mhu',dPcollO']) ;Power loss on collisions
if bounce-mode >= 1,
dPcollOg = trapzyp([pnm',Xpn2m.*Xpnm.*XSpOg-coll./Xgammam]);
PcollO-g(jr) = -2*pi*trapzyp([mhu',dPcoll&.g']);%Power loss on collisions
dPcollOtp = trapzyp([pnm',Xpn2m.*Xpnm.*XSpOtp-coll./Xgammam]);
PcollOtp(jr) = -2*pi*trapzyp([mhu',dPcollOtp']);%Power
Pcolltot(jr)
= PcollO(jr)
+ Pcoll0_g(jr)
loss on collisions
+ PcollOtp(jr);
else
Pcolltot(jr) = PcollO(jr);
end
213
% Power densities calculations (flux-surface averaged)
if bounce_mode >= 1,
dPcollQ-fsav = trapzyp([pnm',Xpn2m.*Xpnm.*(i+ia(jr)).*Xlambda.*XSpQcoll-bav./Xgammam]);
PcollO-fsav(jr) = -2*pi*trapzyp([mhu',dPcollOfsav']);%Power loss on collisions
dPcollQ-g-fsav =
trapzyp([pnm',Xpn2m.*Xpnm.*(+ia(jr)).*Xlambda.*XSpO.g-coll-bav./Xgammam]);
PcollOg-fsav(jr) = -2*pi*trapzyp([mhu',dPcollOg-fsav']);%Power loss on collisions
dPcollOtp-fsav =
trapzyp([pnm' ,Xpn2m.*Xpnm.*(i+ia(jr)).*Xlambda.*XSpOtp-collbav./Xgammam]);
PcollO-tp-fsav(jr) = -2*pi*trapzyp([mhu',dPcollQ-tp-fsav']) ;%Power loss on collisions
dPabsO-lh_fsav = trapzyp([pnm',Xpn2m.*Xpnm.*(i+ia(jr)) .*Xlambda.*XSpQlhbav./Xgammam]);
PabsOlh-fsav = 2*pi*trapzyp([mhu',dPabsOlh-fsav']);%LH absorbed power
dPabsO-g-lh-fsav =
trapzyp([pnm',Xpn2m.*Xpnm.*(i+ia(jr)).*Xlambda.*XSpO-g-lh-bav./Xgammam]);
PabsOg-lh-fsav = 2*pi*trapzyp([mhu',dPabsOg-lh-fsav']);%LH absorbed power
dPabsO-tp-lh-fsav =
trapzyp([pnm',Xpn2m.*Xpnm.*(i+ia(jr)).*Xlambda.*XSpOtp-1h-bav./Xgammam]);
PabsOtp-lh-fsav = 2*pi*trapzyp([mhu',dPabsOtp_lh_fsav']);%LH absorbed power
dPabsOcy-fsav = trapzyp([pnm',Xpn2m.*Xpnm.*(i+ia(jr)).*Xlambda.*XSpOcy-bav./Xgammam]);
PabsOcy-fsav = 2*pi*trapzyp([mhu',dPabsOcy-fsav']);%EC absorbed power
dPabsOg-cy-fsav =
trapzyp([pnm',Xpn2m.*Xpnm.*(i+ia(jr)).*Xlambda.*XSpOg-cy-bav./Xgmmam);
PabsOg-cy-fsav = 2*pi*trapzyp([mhu',dPabsO-gcy-fsav']);%EC
absorbed power
dPabsO-tp-cy-fsav =
trapzyp([pnm',Xpn2m.*Xpnm.*(1+ia(jr)).*Xlambda.*XSpOtpcy-bav./Xgammam]);
PabsOtp-cy-fsav = 2*pi*trapzyp([mhu',dPabsOtp-cy-fsav']);%EC
absorbed power
PabsOfsav(jr) = PabsOlh-fsav + PabsOcy-fsav;XLH + EC absorbed power
PabsQg-fsav(jr) = PabsO-g-lh-fsav + PabsOg_cy-fsav;%LH + EC absorbed power
PabsOtp-fsav(jr) = PabsOtp-lh-fsav + PabsOtp-cy-fsav;XLH + EC absorbed power
Pcolltotfsav(jr) = PcollOfsav(jr) + PcollOg-fsav(jr) + PcollOtp-fsav(jr);
Pabstotfsav(jr) = PabsOfsav(jr) + PabsOg-fsav(jr) + PabsO-tp-fsav(jr);
else
dPabsO-lhfsav = trapzyp([pnm',Xpn2m.*Xpnm.*XSpOlh./Xgammam]);
PabsOlhfsav = 2*pi*trapzyp([mhu',dPabsOlh-fsav']);%LH
214
absorbed power
dPabsOcy-fsav = trapzyp([pnm',Xpn2m.*Xpnm.*XSpOcy./Xgammam]);
PabsOcy-fsav = 2*pi*trapzyp([mhu',dPabsOcy-fsav']);%EC absorbed power
PabsOfsav(jr) = PabsOlh_fsav + PabsOcy-fsav;%LH + EC absorbed power
PabsOg-fsav(jr) = 0;
PabsO.tp-fsav(jr) = 0;
Pcoll0.fsav(jr) = PcollO(jr);
Pcolltotfsav(jr) = Pcoltot(jr);
Pabstot-fsav(jr)
=
PabsOfsav(jr) + PabsO-g-fsav(jr) + PabsOtp-fsav(jr);
end
if bouncemode >= 1,
RR-fsav(jr,:)
= 2*pi*pn2.*trapzyp([mhu',XSpObav'.*(i+ia(jr)).*Xlambda']
);Runaway rate
else
RR-fsav(jr,:) = 2*pi*pn2.*trapzyp([mhu',XSpO']);%Runaway rate
end
if epsi(jr)
0,
currB-fsav(jr) = currtot.fsav(jr) - 0.5*RR-fsav(jr,npn-1)*max(v2)/epsi(jr);
%Bulk current (useful for non-zero dc electric field only) ******
-=
else
currB-fsav(jr) = curr-tot.fsav(jr);
end
end
215
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