8.6 Drift waves 261which causes Eq.(8.140) to become
fσ 1 (x,v,t) = −qσ ̃φfσ 0
mσ∫t−∞dt′eik·x(t′)−iωt′[
2ik·v
v^2 Tσ−
2ikzuσz
v^2 Tσ+
iky
ωcσL]
= −
qσ ̃φfσ 0
mσ∫t−∞dt′
2e−iωt
′v^2 Tσ[
d
dt′eik·x(t′)]
−
(
2ikzuσz
v^2 Tσ−
iky
ωcσL)
eik·x(t
′)−iωt′
= −
2 qσφf ̃σ 0
mσvTσ 2
eik·x(t)−iωt+ i(ω−kzuσz−ω∗σ)∫t−∞dt′eik·x(t′)−iωt′
;
(8.145)
here
kyv^2 Tσ
2 ωcσL
=
kyκT
qBL=−ω∗σ (8.146)has been used as a generalization of Eq.(8.110). The rest of the analysis is as before and
gives the dispersion relation
1 +
∑
σ1
k^2 λ^2 Dσ[
1 +
(ω−kzuσz−ω∗σ)
kzvTσ∑∞
n=−∞In(Λσ)e−ΛσZ(αnσ)]
= 0 (8.147)
where as beforeΛσ=k^2 ⊥r^2 Lσandαnσ= (ω−nωcσ)/kzvTσ.It is seen that the density
gradient and the axial current both provide a Doppler shift in theα 0 σterm. In both cases
the Doppler shift is given by the wavenumber in the appropriate directiontimes the mean
fluid velocity in that direction.
Because Eq.(8.147) contains so much detailed physics in addition to the sought-after
collisionless drift wave^4 , some effort and guidance is required to disentangle the drift wave
information from all the other information. This is done by taking appropriate asymptotic
limits of Eq.(8.147) and guidance is obtained using the results from the two-fluid analysis.
These results showed that drift waves exist in the regime where:
- vTi<<ω/kz<<vTeso that the ions are adiabatic and the electrons are isothermal,
- ω∼ω∗e<<ωci,ωce,
- k^2 ⊥r^2 Le∼ 0 because the extremely small electron mass means that the electron Larmor
orbit radius is negligible compared to the perpendicular wavelength even though the
electrons are warm, - k^2 ⊥r^2 Li< 1 since the ions were assumed cold,
- k^2 ⊥ρ^2 s∼ 1 ,
- k^2 λ^2 De<< 1 so that the waves are quasi-neutral,
(^4) Eq.(8.147) also describes Bernstein waves, the electrostatic limit of magnetized cold plasma waves, and mode
conversion.