8.6 Drift waves 255the distance diffused in timetis given byz^2 ∼ 4 D‖t. Equation (8.121) shows that in the
limitτ‖→ 0 ,the electron density perturbation reverts to being Boltzmann, but for finite
τ‖, a phase lag occurs betweenne 1 andφ 1.
The next step is to calculate the ion density perturbation. The ion equation ofmotion
is
midui
dt=qi(−∇φ+ui×B)−νiemi(ui−ue); (8.123)because the ions are heavy, the left hand side inertial term must now be retained. The ions
are assumed to be cold and the ion inertial term is assumed to be much larger than the
collisional term so that the parallel component of the linearized Eq.(8.123) is
uiz 1 =kzqiφ 1
ωmi=
kzc^2 s
ωqiφ 1
κTe. (8.124)
Again assuming that the inertial term is much larger than the collisionalterm, the linearized
perpendicular ion equation becomes
midui 1
dt=qi(−∇φ 1 +ui 1 ×B) (8.125)which can be solved to give the perpendicular velocity as a sum of anE×Bdrift and a
polarization drift, i.e.,
ui⊥=−∇φ×B
B^2+
mi
qiB^2d
dt(−∇φ). (8.126)The polarization drift is retained in the ion but not the electron equation, because polariza-
tion drift is proportional to mass. The perpendicular ionflux is
Γi⊥=−ni∇φ×B
B^2+
nimi
qiB^2d
dt(−∇⊥φ) (8.127)which has a divergence
∇·Γi⊥=−∇ni·∇φ×B
B^2−∇·
(
nimi
qiB^2d
dt(∇⊥φ))
. (8.128)
Substitution of Eq.(8.128) and (8.124) into the ion continuity equation, linearizing, and
solving for the ion density perturbation gives
ni 1
ni 0=
[
k^2 zc^2 s
ω^2+
qe
qikyκTe
ωqeLB−
miκTe
q^2 iB^2k^2 y]
qiφ 1
Te=
[
k^2 zc^2 s
ω^2−
qe
qiω∗
ω
−ky^2 ρ^2 s]
qiφ 1
Te(8.129)
whereρ^2 s≡κTe/miω^2 ciis a fictitious ion Larmor orbit defined using the electron instead
of ion temperature. The fictitious lengthρ^2 sis analogous to the fictitious velocityc^2 s=
κTe/mithat occurred in the analysis of ion acoustic waves.