Fundamentals of Plasma Physics

3.7 Non-adiabatic motion in symmetric geometry 95

or [

ρ

dU

dt

]

⊥

=

[

Jtotal×B−∇·

{

P⊥

←→

I+

(

P⊥−P‖

)ˆ

BBˆ

}]

⊥

(3.142)

which is just the perpendicular component of the double adiabatic MHD equation of mo-
tion. This demonstrates that if diamagnetic current is taken into account, the drift equations
for phenomena with characteristic frequenciesωmuch smaller thanωciand the double adi-
abatic MHD equations are equivalent descriptions of plasma dynamics. Thisanalysis also
shows that one has to be extremely careful when invoking single particle concepts to ex-
plain macroscopic behavior, because if diamagnetic effects are omitted, erroneous conclu-
sions can result.
The reason for the name polarization current can now be addressed by comparing this
current to the currentflowing through a parallel plate capacitor with dielectricε.The ca-
pacitance of the parallel plate capacitor isC=εA/dwhereAis the cross-sectional area
of the capacitor plates andwis the gap between the plates. The charge on the capacitor
isQ=CV whereV is the voltage across the capacitor plates. The current through the
capacitor isI= dQ/dtso

I=C

dV

dt

=

εA

d

dV

dt

. (3.143)

However the electric field between the plates isE=V/dand the current density isJ=
I/Aso this can be expressed as

J=ε

dE

dt

(3.144)

which gives the alternating current density in a medium with dielectricε.If this is compared
to the polarization current

Jp=

ρ

B^2

dE⊥

dt

(3.145)

it is seen that the plasma acts like a dielectric medium in the direction perpendicular to the
magnetic field and has an effective dielectric constant given byρ/B^2.

3.7 Non-adiabatic motion in symmetric geometry

Adiabatic behavior occurs when temporal or spatial changes in the electromagnetic field
from one cyclical orbit to the next are sufficiently gradual to be effectively continuous
and differentiable (i.e., analytic). Thus, adiabatic behavior corresponds to situations where
variations of the electromagnetic field are sufficiently gradual to be characterized by the
techniques of calculus (differentials, limits, Taylor expansions, etc.).
Non-adiabatic particle motion occurs when this is not so. It is therefore nosurprise that
it is usually not possible to construct analytic descriptions of non-adiabaticparticle motion.
However, there exist certain special situations where non-adiabatic motion can be described
analytically. Using these special cases as a guide, it is possible to develop an understanding
for what happens when motion is non-adiabatic.
One special situation is where the electromagnetic field isgeometrically symmetricwith
respect to some coordinateQjin which case the symmetry makes it possible to develop
analytic descriptions of non-adiabatic motion. This is because symmetry inQjcauses the
canonical momentumPjto be an exact constant of the motion. The critical feature is that
Pjremains constant no matter how drastically the field changes in time or space because