236 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma
where the subscriptσhas been omitted from theαnto keep the algebra uncluttered. Sub-
stitution of these expansions back into the dispersion relation gives
D(ω,k) = 1+
∑
σe−k(^2) ⊥r (^2) Lσ
k^2 λ^2 Dσ
I 0
(
k^2 ⊥r^2 Lσ)
(
−
1
2 α^20 σ+iα 0 σ√
πexp(−α^20 σ))
+
∑∞
n=1In(
k^2 ⊥rLσ^2)
−
2 n^2 ω^2 cσ
ω^2 −n^2 ω^2 cσ+iα 0 σ√
π(
e−α(^2) n
+e−α
(^2) −n
)
= 0. (8.36)
Equation (8.36) generalizes the magnetized cold plasma electrostatic dispersion to finite
temperature and shows how Landau damping appears both at the wave frequencyωand
also at cyclotron harmonics, i.e., in the vicinity ofnωcσ. Two important limits of Eq.(8.36)
are discussed in the following sections.
8.3 Bernstein waves
Suppose the wave phase is uniform in the direction along the magnetic field. Such a situa-
tion would occur if the antenna exciting the wave is an infinitely long wire aligned parallel
to a magnetic field line (in reality, the antenna would have to be sufficiently long to be-
have as if infinite). This situation corresponds to havingk‖→ 0 in which case the Landau
damping terms and the 1 / 2 α^20 σterm vanish. The dispersion relation Eq.(8.36) consequently
reduces to
1=∑
σe−λσ
λσ∑∞
n=12 n^2 ω^2 pσ
ω^2 −n^2 ω^2 cσ
In(λσ) (8.37)where
λσ=k^2 ⊥r^2 Lσ (8.38)
and
r^2 Lσ=ω^2 pσλ^2 Dσ
ω^2 cσ=
κTσ
mσω^2 cσ(8.39)
is the Larmor radius. The waves resulting from this dispersion were first derived by Bern-
stein (1958) and are called Bernstein waves or cyclotron harmonic waves;these waves
are often called hot plasma waves because their existence depends on the plasma having
a finite temperature. Both electron and ion Bernstein waves exist;early measurements of
electron Bernstein waves in laboratory experiments were reported byCrawford (1965) and
by Leuterer (1969) and of ion Bernstein waves were reported by Schmitt (1973). These
waves have also been routinely observed by spacecraft and fitting of the measurements to
the wave dispersion has been used, for example by Moncuquet, Meyervernet and Hoang
(1995), to infer the electron temperature in the magnetized plasma of Jupiter’s moon Io.
Electron Bernstein waves involveωbeing in the vicinity of an electron cyclotron harmonic,
i.e.,ω^2 ∼O(n^2 ω^2 ce),while for ion wavesωis in the vicinity of an ion cyclotron harmonic.