8.4 Warm, magnetized, electrostatic dispersion with small, but finitek‖ 2398.4 Warm, magnetized, electrostatic dispersion with small, but finitek‖
An alternate limit for Eq.(8.36) would be to allowk‖to be finite but also havek^2 ⊥r^2 L<< 1
so that only the lowest-order finite Larmor radius terms are retained. Tokeep matters simple
and also to relate to cold plasma theory,k‖will be assumed to be sufficiently small that
ω/k‖>>vTe,vTi.Sincer^2 Lσ/λ^2 Dσ=ω^2 pσ/ω^2 cσ,the lowest-order perpendicular thermal
terms will beO(k^4 ⊥)and so perpendicular quantities up to fourth order must be retained.
This means that both then= 1and then= 2terms must be retained in the summation
overn.With these approximations and using 1 / 2 λ^2 Dσα^20 =ω^2 p/ω^2 ,Eq.(8.36) becomes
k^2 ‖+k^2 ⊥+∑
σ(
1 −k^2 ⊥rL^2)
(
1 +
k⊥^4 r^4 L
4)(
−
ω^2 pσ
ω^2+ iα 0√
π
λ^2 Dσe−α2
0)
+
k^2 ⊥ω^2 pσ
2 ω^2 cσ
−
2 ω^2 cσ
ω^2 −ω^2 cσ+iα 0√
π(
e−α(^21)
+e−α
(^2) − 1
)
+
k^4 ⊥r^2 Lω^2 pσ
8 ω^2 cσ
−
8 ω^2 cσ
ω^2 − 4 ω^2 cσ+iα 0√
π(
e−α2
2
+e−α(^2) − 2
)
= 0.
(8.47)
The Landau damping terms will be assumed to be negligible to keep matters simple and the
equation will now be grouped according to powers ofk^2 ⊥.Retaining only the lowest-order
finite-temperature perpendicular terms gives
k‖^2
(
1 −
∑
σω^2 pσ
ω^2)
+k^2 ⊥(
1 −
∑
σω^2 pσ
ω^2 −ω^2 cσ)
−k⊥^4∑
σ(
3 ω^4 pσλ^2 Dσ
(ω^2 −ω^2 cσ)(ω^2 − 4 ω^2 cσ))
= 0.
(8.48)
This is of the form
−k^4 ⊥ǫth+k^2 ⊥S+k‖^2 P= 0 (8.49)where the perpendicular fourth-order thermal coefficient is
ǫth=∑
σ(
3 ω^4 pσλ^2 Dσ
(ω^2 −ω^2 cσ)(ω^2 − 4 ω^2 cσ))
. (8.50)
Equation (8.49) is a quadratic equation ink^2 ⊥.The cold-plasma model used earlier in effect
setǫth= 0so that a wave propagating through an inhomogeneous plasma towards a hybrid
resonance whereS→ 0 would havek⊥^2 →∞.This non-physical prediction is resolved
by the warm plasma theory becauseǫth, while small, is finite and so preventsk^2 ⊥→∞
from occurring. What happens instead is that Eq. (8.49) has two qualitatively distinct roots,