Fundamentals of Plasma Physics

240 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma

namely

k⊥^2 =

S±

√

S^2 + 4ǫthk^2 ‖P

2 ǫth

. (8.51)

hot wave

going out

cold wave

going in

mode

coalescence

S
0

x

kx^2

Figure 8.2: Linear mode conversion of cold mode into hot mode near hybrid resonance

Far from the hybrid resonanceSis large so that the two modes are well separated and
given by

k⊥^2 =−

P

S

k‖^2 , cold plasma wave

k^2 ⊥=

S

ǫth

, hot plasma wave

(8.52)

but asS→ 0 these two modes coalesce. The hot plasma wave results from balancing the
first and middle terms in Eq. (8.49) and the cold plasma wave results frombalancing the
middle and last terms. The actual mode coalescence occurs where the square root term in
Eq.(8.51) vanishes, i.e., where

S^2 =− 4 ǫthk^2 ‖P. (8.53)

The WKB approximation fails at the mode coalescence, because heredk⊥/dx=∞and
so a more sophisticated analysis is required. This analysis, which will bediscussed in the
next section, shows that the hot and cold waves are strongly coupled at the coalescence
region. In particular, it will be demonstrated that a cold wave propagating towards the
hybrid resonance, as shown in Fig.8.2, will be linearly converted into a hot wave which
then propagates back out from the resonance.