Amplitude Equations for Electrostatic Waves: multiple species
Abstract
The amplitude equation for an unstable electrostatic wave is analyzed using an expansion in the mode amplitude
A(t)
. In the limit of weak instability, i.e.
γ→
0
+
where
γ
is the linear growth rate, the nonlinear coefficients are singular and their singularities predict the dependence of
A(t)
on
γ
. Generically the scaling
|A(t)|=
γ
5/2
r(γt)
as
γ→
0
+
is required to cancel the coefficient singularities to all orders. This result predicts the electric field scaling
|
E
k
|∼
γ
5/2
will hold universally for these instabilities (including beam-plasma and two-stream configurations) throughout the dynamical evolution and in the time-asymptotic state. In exceptional cases, such as infinitely massive ions, the coefficients are less singular and the more familiar trapping scaling
|
E
k
|∼
γ
2
is recovered.