8.5 Analysis of linear mode conversion 247
whereAis a constant. The inverse transform is
φ(ξ) =A
∫
C
ef(p)dp (8.84)
where
f(p) =
p^3
3
−
μ
p
−ln(p) +pξ. (8.85)
- Boundary conditions: Since Eq.(8.79) is a fourth-order ordinary differential equation,
fourindependent solutions must exist with four associated independent choices for the
contourCin Eq.(8.84). The appropriate contour is determined by the imposed physi-
cal boundary conditions which must be equal in number to the order of the equation.
We consider a specific physical problem where an external antenna located atξ>> 0
generates a cold plasma wave propagating to the left. This physical prescription im-
poses the following four boundary conditions:
(a) To the right ofξ = 0there is a cold plasma wave withenergypropagating
to the left (i.e., towards the hybrid layer). It is important to take into account
the fact that the group velocity is orthogonal to the phase velocity for these
waves in which case the wave is backwards in either thexor thezdirection.
The dispersion of these waves is given in Eq.(6.102) and the group velocity in
Eq.(6.104). A plot ofS(ω)andP(ω)shows that∂S/∂ω> 0 and∂P/∂ω> 0.
Thus, for the situation whereS > 0 andP < 0 thexcomponent of the wave
phase and group velocities have opposite sign. Hence, the cold wave phase
velocity must propagate to therightto be consistent with the boundary condition
that cold wave energy is propagating to theleft.
(b) To the right ofξ= 0there isnowarm plasma wave propagating to the left (for
the hot wave, energy and phase propagate in the same direction).
(c) To the left ofξ= 0the cold plasma evanescent mode vanishes asξ→−∞.
(d) To the left ofξ= 0the hot plasma evanescent mode vanishes asξ→−∞.
The question is: what happens at the hybrid layer? Possibilities include absorption of
the incoming cold plasma wave (unlikely since there is no dissipation in this problem),
reflection of the incoming cold plasma wave at the hybrid layer, or mode conversion.
- Calculation of saddle points: The saddle points are the roots of
f′(p) =p^2 +ξ+
μ
p^2
−
1
p
= 0. (8.86)
For largeξ,these roots separate into two large roots (hot mode) given when the first
two terms in Eq.(8.86) are balanced with each other and two small roots (cold mode)
given by balancing the second and third terms. The large roots satisfyp^2 =−ξwhile
the small roots satisfyp^2 =−μ/ξ.For largeξ,the fourth term is small compared to
the dominant terms for both large and small roots. The quantityfcan be approximated
