260 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma
motion. The appropriately normalized lab frame distribution function is
fσ 0 (x,v) =n 0 σ
π^3 /^2 v^3 Tσexp[
−(v−uσzzˆ)^2 /v^2 Tσ−(vy/ωcσ+x)/L−(
vT
2 ωcL) 2 ]
(8.142)
where the normalization factorexp
(
−(vT/ 2 ωcL)^2)
has been inserted so that the zerothmoment offσ 0 gives the density. The necessity of this factor is made evident by completing
the squares for the velocities and writing Eq.(8.142) in the equivalent form
f(x,v)=
n 0
π^3 /^2 v^3 Texp
−
v^2 x+(
vy+v^2 T
2 ωcL) 2
+ (vz−uσz)^2v^2 T
−
x
L
. (8.143)
The termvy/ωcσwhich crept into Eq.(8.142) because of the seemingly abstract require-
ment of having the distribution function depend on the constant of the motionPycor-
responds to thefluid theory equilibrium diamagnetic drift. This correspondence is easily
seen by calculating the meany-direction velocity (i.e. first moment of Eq.(8.142)) and find-
ing that thevy/ωcσterm results in afluid velocityuywhich is precisely the diamagnetic
velocity given by Eq.(8.98).
We now insert Eq.(8.142) in Eq.(8.140) to calculate the perturbed distributionfunction.
The unperturbed particle orbits are the same as in Section 8.1 because the concepts of both
density gradient and axial current result from averaging over a distribution of many parti-
cles and so have no meaning for an individual particle. The integration over unperturbed
orbits is thus identical to that of Section 8.1, except that a different equilibrium distribution
function is used.
The essence of the calculation is in the term
∇φ 1 ·
∂fσ 0
∂v= iφ 1 k·
∂fσ 0
∂v= −iφ ̃eik·x(t)−iωt[
2 k·v
vTσ^2−
2 kzuσz
v^2 Tσ+
ky
ωcσL]
fσ 0