Fundamentals of Plasma Physics

3.7 Non-adiabatic motion in symmetric geometry 107

should be construed assin[ky(t)−ωt] so that, taking into account the time dependence
ofyon the right hand sided, Eq.(3.179) becomes

vp=ˆy

mkφ

qB^2

d

dt

sin(ky−ωt) = ˆy

mkφ

qB^2

(

k

dy

dt

−ω

)

cos(ky−ωt). (3.181)

However,dy/dt=vpsincevpis the motion in theydirection. Equation (3.181) becomes
an implicit equation forvpand may be solved to give

vp=−ˆy

ωmkφ

qB^2

cos(ky−ωt)

[1−αcos(ky−ωt)]

(3.182)

where

α=

mk^2 φ

qB^2

(3.183)

is a non-dimensional measure of the wave amplitude (McChesney, Stern and Bellan 1987,
White, Chen and Lin 2002).
Ifα > 1 ,the denominator in Eq.(3.182) vanishes whenky−ωt= cos−^1 α−^1 and
this vanishing denominator would result in an infinite polarization drift. However, the
derivation of the polarization drift was based on the assumption that thetime derivative
of the polarization drift was negligible compared to the time derivative ofvE, i.e., it was
explicitly assumeddvp/dt <<dvE/dt.Clearly, this assumption fails whenvpbecomes
infinite and so the iteration scheme used to derive the particle driftsfails. What is happening
is that whenα∼ 1 , the particle displacement due to polarization drift becomes∼k−^1.
Thus the displacement of the particle from its gyrocenter is of the order of a wavelength. In
such a situation it is incorrect to represent the its actual location by its gyrocenter because
the particle experiences the wave field at the particle’s actual location, not at its gyrocenter.
Because the wave field is significantly different at two locations separated by∼k−^1 ,it is
essential to evaluate the wave field evaluated at the actual particle location rather than at
the gyrocenter.
Direct numerical integration of the Lorentz equation in this large-amplitude limit shows
that whenαexceeds unity, particle motion becomes chaotic and cannot be described by
analytic formulae. Onset of chaotic motion resembles heating of the particles since chaos
and heating both broaden the velocity distribution function. However, chaotic heating is
not a true heating because entropy is not increased — the motion is deterministic and not
random. Nevertheless, this chaotic (or stochastic) heating is indistinguishable for practical
purposes from ordinary collisional thermalization of directed motion.
An alternate way of looking at this issue is to consider the Lorentz equations for two
initially adjacent particles, denoted by subscripts 1 and 2 which arein a wave electric field
and a uniform, steady-state magnetic field (Stasiewicz, Lundin and Marklund 2000). The
respective Lorentz equations of the two particles are

dv 1

dt

=

q

m

[E(x 1 ,t)+v 1 ×B]

dv 2

dt

=

q

m

[E(x 2 ,t)+v 1 ×B]. (3.184)

Subtracting these two equations gives an equation for the difference between thevelocities
of the two particles,δv=v 1 −v 2 in terms of the differenceδx=x 1 −x 2 in their positions,