Chaos in trapped particle orbits
R. B. White
Plasma Physics Laboratory, Princeton University, P.O. Box 451, Princeton, New Jersey 08543
Received 9 January 1998; revised manuscript received 25 March 1998
An analytic expression is obtained for the threshold for stochastic transport of high energy trapped particles
in a tokamak due to toroidal field ripple. The result permits a rapid evaluation of energetic particle loss in
general equilibria. The approach to chaos has two limiting cases: island overlap from neighboring precession
resonances, and the overlap of bounce resonance webs extending from resonant surfaces. In both cases the
physical resonance consists of motion through a distance which is an integer multiple of the field coil separa-
tion in one bounce. S1063-651X9813608-5
PACS numbers: 05.45.+b, 52.55.Dy, 52.20.Dq
I. INTRODUCTION
In magnetic fusion devices, the long time confinement of
high energy particles, whether 3.5 MeV alpha particle fusion
products or other ions introduced for heating purposes, is
essential to the attainment of a self sustaining nuclear burn. It
has been well established that the motion of such particles,
with energies much higher than the typical 10-keV back-
ground plasma, is neoclassical 1. That is, small scale fluc-
tuations present in plasmas, and thought to be responsible for
the anomalous thermal flux to the plasma edge, do not influ-
ence the orbits. Thus their motion is that of charged test
particles in a given magnetic field. Nevertheless, predicting
the degree of alpha particle confinement in a given device
with high accuracy requires many hours of computing time,
even using algorithms in which the rapid gyro motion has
been eliminated. This is because there are vastly differing
time scales for the three major contributing loss processes.
The slower loss processes correspond to the nonconservation
of an integral of the motion, and thus of a broken symmetry
of the underlying Hamiltonian describing the dynamics. The
simplest description is given by a toroidally symmetric equi-
librium, with neglect of particle-particle interactions.
Any axisymmetric equilibrium field can be expressed in
contravariant and covariant forms 2,3
B
0
=
p
+q
p
p
, 1
B
0
=g
p
+I
p
+h
p
,
p
, 2
with
p
the poloidal flux, the poloidal angle, and a
toroidal angle. The coordinate system is a straight field line
one, i.e., q (
p
) the safety factor gives the local helicity of
a field line q =d / d , and also relates the toroidal flux
and the poloidal flux
p
, through q =d / d
p
. The vari-
able is related to the geometric toroidal angle through
= + , with a function of
p
and , periodic in . The
magnetic field strength B
0
(
p
, ) is independent of the co-
ordinate . It is important to use general magnetic coordi-
nates so that results will be applicable to any equilibrium, but
for purposes of visualization
p
may simply be regarded as
the minor radius of the torus.
Guiding center motion in a field B is given by a Hamil-
tonian formalism 3–5
dP
dt
=-
H,
d
dt
=
P
H , 3
dP
dt
=-
H,
d
dt
=
P
H , 4
where P
0
=I
+ and P
=g
-
p
are the canonical mo-
menta with
=v
/ B , and
H=
1
2
P
+
p
g
2
B
2
+ B 5
is the Hamiltonian. Here and in the following we use units
given by the on-axis gyro frequency time and the major
radius distance. In these units = 2 E is the gyro radius,
which is the small parameter in the guiding center approxi-
mation. A very important feature of these equations is that
motion is given by B (
p
, ), the field magnitude only, and
real space metric quantities such as the Jacobian do not
enter.
Toroidal symmetry implies that P
is a constant of the
motion, and hence H=E implies that all orbits are closed
curves in the
p
, plane. By conservation of energy, orbits
are restricted to connected regions of space in which E
B . Normally the magnetic field decreases outwardly and
thus there are orbits which are trapped poloidally and ex-
ecute banana-shaped orbits. Although orbits close in the
p
, plane they do not close in space, but precess toroi-
dally. Because of the toroidal precession the banana tips de-
scribe constant Kolmogorov-Arnold-Moser KAM surfaces
6 in the
p
, plane.
It is easy to perform a Monte Carlo calculation to deter-
mine the prompt, or first, orbit loss, due to orbits which
intersect the outer boundary, the only loss process present
with an axisymmetric equilibrium, typically occurring on a
time scale of microseconds.
The most important symmetry breaking term to include in
the Hamiltonian is due to the perturbation of the magnetic
field strength due to the N toroidal field coils. It can be rep-
resented by a modulation of the field amplitude
B
p
, , =B
0
p
, 1 + cos N , 6
with the ripple strength a function of position, determined
by the coil geometry, and in all devices N 1.
PHYSICAL REVIEW E AUGUST 1998 VOLUME 58, NUMBER 2
PRE 58 1063-651X/98/582/17746/$15.00 1774 © 1998 The American Physical Society