Fundamentals of Plasma Physics

4.2 Two-fluid theory of unmagnetized plasma waves 127

Equation (4.24) shows that ifφ 1 = 0, the quantity1 +χe+χimust vanish. In other
words, in order to have a non-trivial normal mode it is necessary to have

1 +χe+χi= 0. (4.26)

This is called a dispersion relation and prescribes a functional relationbetweenω andk.
The dispersion relation can be considered as the determinant-like equation for the eigen-
valuesω(k)of the system of equations.
The normal modes can be identified by noting that Eq.(4.25) has two limiting behaviors
depending on how the wave phase velocity compares to

√

κTσ 0 /mσ, a quantity which is
of the order of the thermal velocity. These limiting behaviors are

1. Adiabatic regime:ω/k >>

√

κTσ 0 /mσandγ= (N+ 2)/N.Because plane waves

are one-dimensional perturbations (i.e., the plasma is compressed in theˆkdirection

only),N= 1so thatγ= 3.Hence the susceptibility has the limiting form

χσ = −

ω^2 pσ

ω^2 (1−γk^2 κTσ 0 /mσω^2 )

≃−

ω^2 pσ

ω^2

(

1 + 3

k^2

ω^2

κTσ 0

mσ

)

= −

1

k^2 λ^2 Dσ

k^2

ω^2

κTσ

mσ

(

1 + 3

k^2

ω^2

κTσ 0

mσ

)

. (4.27)

2. Isothermal regime:ω/k <<

√

κTσ 0 /mσandγ= 1.Here the susceptibility has the

limiting form

χσ=

ω^2 pσ

k^2 κTσ 0 /mσ

=

1

k^2 λ^2 Dσ

. (4.28)

Figure 4.1 shows a plot ofχσk^2 λ^2 Dσversusω/k

√

κTσ 0 /mσ. The isothermal and adia-
batic susceptibilities are seen to be substantially different and, in particular, do not coalesce
whenω/k

√

κTσ 0 /mσ→ 1 .This non-coalescence asω/k

√

κTσ 0 /mσ→ 1 indicates that
thefluid description, while valid in both the adiabatic and isothermal limits, fails in the
vicinity ofω/k∼

√

κTσ 0 /mσ. As will be seen later, the more accurate Vlasov descrip-
tion must be used in theω/k∼

√

κTσ 0 /mσregime.