Fundamentals of Plasma Physics

78 Chapter 3. Motion of a single plasma particle

while the component perpendicular toBis

m

[

dv⊥gc(t)

dt

+v^2 ‖gcB̂·∇B̂

]

= q





E⊥(xgc(t))

+vgc(t)×B(xgc(t))

+〈vL(t)×(rL(t)·∇)B〉⊥



. (3.71)

Equation (3.71) is of the generic form

m

dv⊥gc

dt

=F⊥+qvgc×B (3.72)

where

F⊥ = q

[

E⊥(xgc(t)) +〈vL(t)×(rL(t)·∇)B〉⊥

]

−mv^2 ‖gcB̂·∇B.̂

(3.73)

Equation (3.72) is solved iteratively based on the assumption thatv⊥gchas a slow time
dependence. In the first iteration, the time dependence is neglected altogether so that the
LHS of Eq.(3.72) is set to zero to obtain the ‘first guess’ for the perpendicular drift to be

v⊥gc≃vF≡

F⊥×B

qB^2

.

Next,vpis defined to be a correction to this first guess, wherevpis assumed small and
incorporates effects due to any time dependence ofv⊥gc.To determinevp, we write
v⊥gc=vF+vPso, to second order Eq. (3.72) becomes,

m

d(vF+vP)

dt

=F⊥+q(vF+vP)×B. (3.74)

In accordance with the slowness condition, it is assumed that|dvP/dt|<<|dvF/dt|so
Eq.(3.74) becomes

0 =−m

dvF

dt

+qvP×B. (3.75)

Crossing this equation withBgives the general polarization drift

vP=−

m

qB^2

dvF

dt

×B. (3.76)

The most important example of the polarization drift is whenvFis theE×Bdrift in a
uniform, constant magnetic field so that

vP = −

m

qB^2

d

dt

(

E×B

B^2

)

×B (3.77)

=

m

qB^2

dE

dt

.

To calculate the middle term on the RHS of Eq.(3.73), it is necessary toaverage over
cyclotron orbits (also called gyro-orbits or Larmor orbits). This middleterm is defined as
the ‘gradB’ force
F∇B =q〈vL(t)×(rL(t)·∇)B〉. (3.78)