4.2 Two-fluid theory of unmagnetized plasma waves 129
standard form for the high-frequency, electrostatic, unmagnetized plasma wave
ω^2 =ω^2 pe+3k^2
κTe 0
me
. (4.31)
This most basic of plasma waves is called the electron plasma wave, the Langmuir
wave (Langmuir 1928), or the Bohm-Gross wave (Bohm and Gross 1949).
- Case whereω/k <<
√
κTe 0 /me,
√
κTi 0 /mi
Here both electrons and ions are isothermal and the dispersion becomes
1+
∑
σ
1
k^2 λ^2 Dσ
=0. (4.32)
This has no frequency dependence, and is just the Debye shielding derived in Chapter
- Thus, whenω/k <<
√
κTe 0 /me,
√
κTi 0 /mithe plasma approaches the steady-
state limit and screens out any applied perturbation. This limit shows why ions cannot
provide Debye shielding for electrons, because if the test particle were chosen to be
an electron then its nominal speed would be the electron thermal velocityand from
the point of view of an ion the test particle motion would constitute a disturbance
with phase velocity√ ω/k∼vTewhich would then violate the assumptionω/k <<
κTi 0 /mi.
- Case where
√
κTi 0 /mi<<ω/k <<
√
κTe 0 /me
Here the ions act adiabatically whereas the electrons act isothermally so that the dis-
persion becomes
1+
1
k^2 λ^2 De
−
ω^2 pi
ω^2
(
1+3
k^2
ω^2
κTi 0
mi
)
=0. (4.33)
It is conventional to define the ‘ion acoustic’ velocity
c^2 s=ω^2 piλ^2 De=κTe/mi (4.34)
so that Eq.(4.33) can be recast as
ω^2 =
k^2 c^2 s
1+k^2 λ^2 De
(
1+3
k^2
ω^2
κTi 0
mi
)
. (4.35)
Sinceω/k >>
√
κTi 0 /mi, this may be solved iteratively by first assumingTi 0 =
0 giving
ω^2 =
k^2 c^2 s
1+k^2 λ^2 De
. (4.36)
This is the most basic form for theion acoustic wavedispersion and in the limit
k^2 λ^2 De>> 1 ,becomes simplyω^2 =c^2 s/λ^2 De=ω^2 pi.To obtain the next higher
order of precision for the ion acoustic dispersion, Eq.(4.36) may be used to eliminate
k^2 /ω^2 from the ion thermal term of Eq.(4.35) giving
ω^2 =
k^2 c^2 s
1+k^2 λ^2 De
+3k^2
κTi 0
mi
. (4.37)
