8.6 Drift waves 259and potentialfluctuations resulting in an imaginary term in the dispersion relation.Also,
since Vlasov analysis showed that a relative motion between electrons and ions (i.e., a
net current) destabilizes ion acoustic waves, it is reasonable to expect that current-driven
destabilization could occur for drift waves.
The Vlasov analysis of drift waves is less intuitive than thefluid analysis, but the reward
for abstraction is a more profound model. As in thefluid analysis, the plasma is assumed to
have a uniform equilibrium magnetic fieldB=Bzˆ, anx-directed density gradient given
by Eq.(8.100), and a perturbed potential given by Eq.(8.101). In a sense the Vlasovanalysis
is simpler than thefluid analysis, because the Vlasov analysis involves a modest extension
of the warm plasma electrostatic Vlasov model discussed in Section 8.1. As before, the
perturbed distribution function is evaluated by integrating along the unperturbed orbits,
i.e.,
fσ 1 (x,v,t) =∫t−∞dt′[
qσ
mσ∇φ 1 ·∂fσ 0
∂v]
x=x(t′),v=v(t′). (8.140)
The new feature here is that the equilibrium distribution function must incorporate the
assumed density gradient.
A first instinct would be to accomplish this by simply multiplying the Maxwellian dis-
tribution function of Section 8.1 by the factorexp(−x/L)so that the assumed equilibrium
distribution function would befσ 0 (x,v) = (πvτσ)−^3 /^2 exp(−v^2 /v^2 Tσ−x/L). This ap-
proach turns out to be wrong becausefσ 0 (x,v) = (πvτσ)−^3 /^2 exp(−v^2 /vTσ^2 −x/L)is
nota function of the constants of the motion and so is not a solution of the equilibrium
Vlasov equation.
What is needed is some constant of the motion which includes the parameterx.The
equilibrium distribution function could then be constructed from this constant of the motion
and arranged to have the desiredx−dependence. The appropriate constant of the motion is
the canonical momentum in theydirection, namely
Py=mσvy+qσAy=mσvy+qσBzx=(
vy
ωcσ+x)
qσBz (8.141)sinceBz=∂Ay/∂x.Multiplying the original Maxwellian by the factorexp[−(vy/ωcσ+
x)/L]produces the desired spatial dependence while simultaneously satisfying the require-
ment that the distribution function depends only on constants of the motion. Note thatPy
is a constant of the motion becauseyis an ignorable coordinate.
It turns out thatz-directed currents can also destabilize collisionless drift waves, much
like currents provide free energy which can destabilize ion-acousticwaves. Since it takes
little additional effort to include this possibility, az-directed current will also be assumed
so that electrons and ions are assumed to have distinct mean velocitiesuσz. In a frame
moving with velocityuσzthe speciesσis thus assumed to have a distribution function
∼exp[−mv′^2 / 2 −(vy/ωcσ+x)/L]wherev′=v−uσzˆzis the velocity in the moving
frame. This is a valid solution to the equilibrium Vlasov equation, since both the energy
mv′^2 / 2 and they-direction canonical momentum(vy/ωcσ+x)/Lare constants of the