440 Chapter 15. Wave-wave nonlinearities
the low frequency beat component of
(
̃uhe) 2
can be considered as modulating the effective
electron temperature. Specifically,
n ̃eκTe+nκT ̃e= ̃neκTe+nme(
u ̃he) 2
2
(15.53)
and so
nc ̃^2 s→nc ̃s^2 +n( ̃c^2 s)= ̃nc^2 s+nκT ̃e
mi= ̃nc^2 s+nme(
u ̃he) 2
2 mi(15.54)
which is consistent with Eq.(15.52).
Low frequency wave is an electron plasma wave In this case the low frequency wave
frequency is aboveωpe so that the ions form a stationary background. Now only the
electron dynamic response matters and the density perturbation is not quasi-neutral. The
electron equation of motion for the low frequency wave is
∂ ̃ue
∂t=
qe
meE ̃−^3 κTe
men∇ ̃ne−1
2
∇
(
u ̃he) 2
(15.55)
where the 3 comes from the adiabatic pressure perturbation. Combining this with thetime
derivative of the electron continuity equation,
∂^2 ̃ne
∂t^2+n∇·(
∂ ̃ue
∂t)
=0, (15.56)
gives
∂^2 ̃ne
∂t^2
+n∇·(
qe
meE ̃−^3 κTe
men∇n ̃e−1
2
∇
(
̃uhe) 2
)
=0. (15.57)
This may be simplified by invoking Poisson’s equation
∇·E ̃=
1
ε 0
̃neqe (15.58)to obtain
∂^2 ̃ne
∂t^2
+ ω^2 pen ̃e−3 κTe
me∇^2 n ̃e=n
2∇^2
(
u ̃he) 2
. (15.59)
The left hand side is the electron plasma wave equation (Langmuir wave) and the right
hand side provides the nonlinear drive (or coupling) due to ponderomotive force. Again,
using Eq.(15.53) it is seen that the ponderomotive force term acts like a modulation of the
electron temperature such thatnκT ̃e→nκT ̃ e+nκT ̃ewhereT ̃eis due to the high frequency
electron quiver velocity.
High frequency wave is an electron plasma wave In this case the wave is electro-
static and so there is no high frequency oscillating magnetic field. The non-linear continuity
equation is
∂n ̃e
∂t
+n∇· ̃ue=−∇·( ̃ne ̃ue) (15.60)
and taking a time derivative this becomes
∂^2 ̃ne
∂t^2
+n∇·∂ ̃ue
∂t=−
∂
∂t
∇·( ̃ne ̃ue). (15.61)