448 Chapter 15. Wave-wave nonlinearities
and substitution forx ̄andx ̄−∆in Eq.(15.113) results in the expression for the growth
rate
(Γ 2 +y)(Γ 1 +y)(
1+
∆^2
(2y+Γ 1 + Γ 2 )^2)
=
1
ω 1 ω 2(
λZ 3
4) 2
. (15.116)
Onset of instability onset corresponds toybeing close to zero and at marginal stability
Eq.(15.116) becomes
(Γ 2 +Γ 1 )y+Γ 1 Γ 2 =λ^2 Z 3216 ω 1 ω 2(
1+
∆^2
(Γ 2 +Γ 1 )^2
). (15.117)
The growth rate near threshold therefore is
y=1
(Γ 2 +Γ 1 )
λ^2 Z 3216 ω 1 ω 2(
1+
∆^2
(Γ 2 +Γ 1 )^2
)−Γ 1 Γ 2
. (15.118)
This shows that the pump amplitude threshold for instability is
Z 3 =
4
√
ω 1 ω 2 Γ 1 Γ 2
λ√
1+
∆^2
(Γ 2 +Γ 1 )^2
; (15.119)
this threshold is proportional to the geometric mean of the linear damping rate of the two
modes. If the pump amplitude is below threshold then the nonlinear instability does not
occur. The lowest threshold occurs when∆ = 0,i.e., when the frequency selection rule
is exactly satisfied. Instability when the pump amplitude exceeds a threshold is routinely
observed in actual experimental situations as was discussed earlier, i.e., decay instability of
a pump wave is observed to begin only when the pump wave amplitude exceeds threshold
(for example, see Stenzel and Wong (1972)). For a pump amplitude well above threshold,
Eq.(15.116) becomes
y =√
1
ω 1 ω 2(
λZ 3
4) 2
−
∆^2
4
(15.120)
which shows that frequency mismatch reduces the growth rate. The situation reduces to
Eqs.(15.84) and (15.30) when∆ = 0.Thus the non-linear dispersion relation formalism
extends the Manley-Rowe coupled oscillator model to include the effects of both dissipa-
tion and frequency mismatch.
15.5 Digging a hole in the plasma via ponderomotive force
If the low frequency daughter mode is at zero frequency, then the ion acoustic mode ceases
to be a wave. Instead, it becomes a density depletion caused by the ponderomotive force