254 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma
with the continuity equation gives the perturbed electron density, provided the electron
perpendicular velocity is also known. Since it has been assumed thatωce >> νei,the
magnetic force term in Eq.(8.111) is much greater than the perpendicular component of
the collision term. The perpendicular component of the electron equation of motion is
therefore
0 =qe(−∇⊥φ+ue×B)−1
ne∇⊥(neκTe). (8.114)
In previous derivations the continuity equation was typically linearized right away, but
here it turns out to be computationally advantageous to postpone linearizationand instead
solve Eq.(8.114) as it stands for the perpendicular electronflux to obtain
Γe⊥=neue⊥=−
ne∇φ×B
B^2−
1
qeB^2∇(neκTe)×B. (8.115)Using the vector identities∇·(∇φ×B) = 0and∇·[∇(neκTe)×B] = 0it is seen that
the divergence of the perpendicular electronflux is
∇·Γe⊥=−∇ne·∇φ×B
B^2. (8.116)
The electron continuity equation can be expressed as
∂ne
∂t+
∂
∂z(neuez) +∇·Γe⊥= 0, (8.117)and, after substitution for the perpendicular electronflux, this becomes
∂ne
∂t+
∂
∂z(neuez)−∇ne·
∇φ×B
B^2= 0. (8.118)
At this stage of the derivation it is now appropriate to linearize the continuity equation
which becomes
∂ne
∂t
+ne 0
∂uez 1
∂z−
dne 0
dx
xˆ·∇φ 1 ×B
B^2= 0. (8.119)
Substituting forφ 1 using Eq.(8.101), and using Eq.(8.113) foruz 1 gives
ne 1
ne 0(
−iω+k^2 zκTe
νeime)
+
qeφ 1
Te(
k^2 zκTe
νeime+
ikyTe
qeBL)
= 0 (8.120)
which may be solved to give the sought-after electron density perturbation
ne 1
ne 0=−
qeφ 1
Te(
−iω∗+1
τ‖)
(
−iω+1
τ‖). (8.121)
Here
τ‖=νeime/kz^2 Te (8.122)
is the nominal time required for electrons to diffuse a distance of the orderof a parallel
wavelength. This is because the parallel collisional diffusion coefficient scales asD‖∼
(random step)^2 /(collision period)∼Te/νeimeand according to the diffusion equation