Ripple-induced energetic particle loss in tokamaks
R. B. White,
a)
R. J. Goldston, M. H. Redi, and R. V. Budny
Plasma Physics Laboratory, Princeton University, P.O. Box 451, Princeton, New Jersey 08543
͑Received 22 February 1996; accepted 29 April 1996͒
The threshold for stochastic transport of high energy trapped particles in a tokamak due to toroidal
field ripple is calculated by explicit construction of primary resonances, and a numerical
examination of the route to chaos. Critical field ripple amplitude is determined for loss. The
expression is given in magnetic coordinates and makes no assumptions regarding shape or up-down
symmetry. An algorithm is developed including the effects of prompt axisymmetic orbit loss, ripple
trapping, convective banana flow, and stochastic ripple loss, which gives accurate ripple loss
predictions for representative Tokamak Fusion Test Reactor ͓R. Hawryluk, Plasma Phys. Controlled
Fusion 33, 1509 ͑1991͔͒ and International Thermonuclear Experimental Reactor ͓K. Tomabechi,
Proceedings of the 12th International Conference on Plasma Physics and Controlled Nuclear
Fusion Research ͑International Atomic Energy Agency, Vienna, 1989͒, Vol. 3, p. 214͔ equilibria.
The algorithm is extended to include the effects of collisions and drag, allowing rapid estimation of
alpha particle loss in tokamaks. © 1996 American Institute of Physics. ͓S1070-664X͑96͒01808-3͔
I. INTRODUCTION
Loss of alpha particles or other high energy particles due
to field ripple caused by the discrete toroidal field coils is an
important consideration in the design of magnetic fusion de-
vices. Collisionless losses are due to prompt axisymmetric
orbit loss, ripple trapping, ripple-induced convective banana
flow, and stochastic ripple loss. This work extends a previous
calculation
1
due to Goldston, White, and Boozer ͑GWB͒,
where stochastic threshold was estimated using phase deco-
rrelation arguments, to explicitly calculate the resonance lo-
cations and widths, and explore the route to chaos. An ex-
pression for the stochastic threshold is found, and an
algorithm is developed for energetic particle loss including
the effects of prompt axisymmetric orbit loss, ripple trap-
ping, convective banana flow, and stochastic ripple loss
which gives reasonable accuracy in the estimation of colli-
sionless loss in a tokamak. The calculation is carried out in
general magnetic coordinates, giving expressions which can
be directly applied to high pressure and noncircular equilib-
ria without up-down symmetry, and the results are illustrated
for alpha particle loss in the Tokamak Fusion Test Reactor
2
͑TFTR͒ and the International Thermonuclear Experimental
Reactor
3
͑ITER͒.
Any axisymmetric equilibrium field can be expressed in
contravariant and covariant form through the equations
4,5
B
ជ
0
ϭٌ
ជ
ϫٌ
ជ
⌿
p
ϩqٌ
ជ
⌿
p
ϫٌ
ជ
, ͑1͒
B
ជ
0
ϭgٌ
ជ
ϩIٌ
ជ
ϩhٌ
ជ
⌿
p
, ͑2͒
with ⌿
p
the poloidal flux,
the poloidal angle, and
a
straight-field-line toroidal angle. The coordinate system is a
straight field line one, i.e., q(⌿
p
) ͑the safety factor͒ gives
the local helicity of a field line q ϭd
/d
. The variable
is
related to the geometric toroidal angle
through
ϭ
ϩ
,
with
a function of ⌿
p
and
, periodic in
. The magnetic
field strength B
0
(⌿
p
,
) is independent of the coordinate
.
The perturbation of the magnetic field strength due to the
N toroidal field coils is represented by a modulation of the
field amplitude
B
͑
⌿
p
,
,
͒
ϭB
0
͑
⌿
p
,
͒
͑
1ϩ
␦
cos
͑
N
͒͒
, ͑3͒
with the ripple strength,
␦
, a function of position, determined
by the coil geometry. The discrete coils also modulate the
direction of the field, but the principal result for induced loss
is due to the mirroring effect of the magnitude of B.
Particles trapped poloidally execute banana-shaped or-
bits, conserving energy E and magnetic moment
with
Eϭm
v
ʈ
2
/2ϩ
B. Guiding center motion in this field is given
by a Hamiltonian formalism incorporated into the code
OR-
BIT
and described elsewhere.
5–7
In the following we use units
given by the on-axis gyrofrequency ͑time͒, and the major
radius of the magnetic axis ͑distance͒. In these units
ϭ
ͱ
2E is the gyroradius, which is the small parameter in the
guiding center approximation.
Axisymmetry, or the absence of ripple, makes the toroi-
dal canonical momentum P
ϭg
v
ʈ
/BϪ⌿
p
an integral of the
motion, and this along with energy conservation means that
all orbits are closed curves in the ⌿
p
,
plane. The banana
tips describe constant Kolmogorov, Arnold, and Moser
͑KAM͒ surfaces
8
in the ⌿
p
,
plane. The KAM theory guar-
antees that for small ripple this phase space changes topo-
logically only in a small region proportional to ͱ
␦
where
resonances produce islands. No diffusion can occur until
these islands grow to overlap and produce chaotic wandering
of orbits. To understand this process it is necessary to inves-
tigate resonances in the banana tip motion.
In Section II the route to chaos is investigated, and a
reliable expression for stochastic threshold is found. The
complications of up-down asymmetry and ripple wells are
treated in Section III. In Section IV we develop the algorithm
for loss, and in Section V comparisons of predicted loss with
guiding center calculations are given. In Section VI we dis-
cuss collisions and drag, and the conclusions are summarized
in Section VII.
a͒
Electronic mail: rwhite@pppl.gov
3043Phys. Plasmas 3 (8), August 1996 1070-664X/96/3(8)/3043/12/$10.00 © 1996 American Institute of Physics