262 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma
- |uzi|<<|uze|since the parallel equilibrium velocities satisfymeuez+miuiz= 0;
there is no parallelflow, just a parallel current.
BecauseΛe∼ 0 ,all electron terms in the summation overnvanish except for the
n= 0term. Sinceω/kz << vTe,the small-argument limit of the plasma dispersion
function is used for electrons and it is seen that the imaginary term isthe dominant term
for the electrons. In contrast sincevTi<<ω/kz, the large-argument limit of the plasma
dispersion function is used for the ions and also equilibrium parallel motionis neglected for
ions because of the large ion mass. For then= 0harmonics the termsZ(α 1 )+Z(α− 1 )→
−kzvTi[1/(ω−nωci) + 1/(ω+nωci)]and hence cancel each other sinceω<<ωci. On
making these approximations, Eq.(8.147) reduces to
1 +
Te
Ti[
1 −e−ΛiI 0 (Λi)]
−
ω∗e
ωe−ΛiI 0 (Λi) + iπ^1 /^2
(ω−kzuze−ω∗e)
kzvTe= 0. (8.148)
The real part of this dispersion gives
ωr=ω∗e
e−ΛiI 0 (Λi)1 +Te
Ti
[1−e−ΛiI 0 (Λi)]
(8.149)
or in the limit of smallΛi
ωr=ω∗e{
1 −Λi
1 +k⊥^2 ρ^2 s}
; (8.150)
this corresponds to thefluid dispersion and moreover shows how finite ion temperature
affects the dispersion. Using the method of Eq.(8.135), the imaginary part of the frequency
is obtained from Eq.(8.148) as
ωi = −ω^2 π^1 /^2
ω∗ee−ΛiI 0 (Λi)(ωr−kzuze 0 −ω∗e)
kzvTe=
π^1 /^2 (ω∗e)
2kzvTe{
(1−Υ)(1 +Te/Ti) +kzuze 0 Υ/ωr
[1 + (1−Υ)Te/Ti]^3}
(8.151)
whereΥ = e−ΛiI 0 (Λi ̇).In the limitΛi→ 0 ,it is seen thatΥ→ 1 −Λiso that the
imaginary part of the frequency becomes
ωi=
π^1 /^2 (ω∗e)
2kzvTe[1 +k^2 ⊥ρ^2 s]^3{
k^2 ⊥(ρ^2 s+r^2 Li) +kzuze 0 /ω}
. (8.152)
The imaginary part of the frequency is therefore always positive so there is always in-
stability. Two collisionless destabilization mechanisms are seen to exist,normal Landau
dampingrepresented by the term involvingk⊥^2 (ρ^2 s+r^2 Li)in the curly brackets, andcurrent,
represented bykzuz 0 /ω.Since collisions can also destabilize drift waves there are at least
three mechanisms by which drift waves can be destabilized. Comparisonof Eq.(8.152)
with Eq.(8.135) shows which of these mechanisms will be dominant in a given plasma.
Modes with long parallel wavelengths are the most unstable for drift waves driven unstable
by Landau damping, just as for collisional drift waves.