Effects of stochastic drifts and time variation on particle diffusion in magnetic turbulence
M. Vlad,* F. Spineanu,* and J. H. Misguich
Association Euratom-Commissariat a ` l’Energie Atomique sur la Fusion, De ´partement de Recherches sur la Fusion Controle ´e,
Centre d’Etudes de Cadarache, 13108 Saint-Paul-lez-Durance Cedex, France
R. Balescu
Association Euratom-Etat Belge sur la Fusion, Physique Statistique et Plasmas, CP231, Universite ´ Libre de Bruxelles,
Campus Plaine Bd.du Triomphe, 1050 Bruxelles, Belgium
Received 20 November 1995
The effect on the guiding center trajectories of the stochastic drifts due to the curvature of the stochastic
magnetic lines is studied in the first part of this paper. It is shown that the subdiffusive t behavior of the mean
square displacement of the particles cannot exist in a realistic magnetic configuration. The particles undergo a
diffusive process even in the absence of the perpendicular collisional diffusion. The anomalous diffusion
coefficient is estimated. The second part of this work deals with time-dependent stochastic perturbations of the
confining magnetic field. It is shown that the time variation has a strong effect of decorrelating the particles
from the magnetic lines. The diffusion coefficient is determined as a function of the correlation time of the
stochastic field. The influence of a cross field collisional diffusion on the results of the two problems presented
here is also estimated. S1063-651X9610405-0
PACS numbers: 52.25.Fi, 52.25.Gj, 52.35.Ra, 05.40.+j
I. INTRODUCTION
A large number of theoretical and experimental studies
1–16 led to the conclusion that the fluctuations of the mag-
netic field observed in fusion plasmas tokamak, stellarator,
etc. provide a major contribution to the enhanced particle
and energy transport. Even a very small stochastic compo-
nent of the magnetic field in the radial direction combined
with the high velocity motion of the particles along field
lines determines high radial displacements. However, these
two processes do not produce a radial diffusion but rather a
slower time growth of the dispersion of the trajectories pro-
portional to t ). It was shown 3,5,9,15 that only when a
supplementary mechanism acts to decouple the particles
from the magnetic lines is this subdiffusive behavior domi-
nated asymptotically by a diffusion process. Such a decou-
pling mechanism, provided by the collisions producing a
small cross field diffusion that is strongly enhanced due to
the fast parallel motion along the stochastic magnetic lines,
is presented in detail in Ref. 5see also Ref. 9.
In the present paper we study two other mechanisms of
decoupling the particles from the magnetic lines: these are
not related to collisions.
First, we show that there exists an intrinsic decorrelation
mechanism in any space-dependent stochastic magnetic field.
It consists in the stochastic drifts determined by the gradients
of the magnetic field, which are always present along a sto-
chastic magnetic line. As a consequence, the subdiffusive
t behavior of the mean square displacement cannot exist
even when the perpendicular collisional diffusion can be ne-
glected. We evaluate the diffusion coefficient determined by
the stochastic drifts and show that, in weakly collisional
plasmas, this mechanism may provide the major contribution
to trajectory dispersion. This problem was previously studied
by Coronado, Vitela, and Akcasu 6 starting from a model
similar to ours and by Mynick and Krommes in Ref. 10 for
the magnetic configuration of the tokamak in a more general
context including stochasticity criteria. However, both stud-
ies consider only a constant velocity motion of the particles
along perturbed magnetic lines. In our model the parallel
velocity is a stochastic function of time determined by colli-
sions. We show that, even in the weakly collisional limit, the
results of the two models are different and that the constant
velocity is not a good approximation of the physical prob-
lem. This is in agreement with the conclusion of several
papers 4,8,14 that the fluctuating nature of the particle ve-
locity along field lines is essential for an appropriate descrip-
tion of plasma transport processes in magnetic turbulence.
The decorrelation of the particles from the magnetic lines
can also be produced by the time variation of the stochastic
magnetic field. We show that this mechanism is very effi-
cient when the correlation time of the fluctuating field
c
is
of the order of the inverse of the collision frequency and
that particle diffusion is strongly enhanced in these condi-
tions. We determine the diffusion coefficient as a function of
c
and demonstrate that the collisional cross field diffusivity
has no decisive influence on the shape of this curve. This
problem was previously treated in several papers 6,9,11,12.
Our results are in qualitative agreement with the heuristic
analysis presented in Ref. 9 and with part of the conclu-
sions obtained in Ref. 11 from numerical calculations of
the confinement time in the case of a single coherent pertur-
bation of the magnetic field. In Ref. 6, the problem is
treated in a simplified frame stochastic magnetic field de-
pending only on time or space-time fluctuating field but with
constant velocity parallel motion that prevents a direct com-
parison with our results. In Ref. 12, the time dependence of
*
On leave of absence from the Institute of Atomic Physics, P.O.
Box MG-7, Magurele, Bucharest, Romania.
PHYSICAL REVIEW E MAY 1996 VOLUME 53, NUMBER 5
53 1063-651X/96/535/530213/$10.00 5302 © 1996 The American Physical Society