438 Chapter 15. Wave-wave nonlinearities
If three waves satisfy the selection rulesω 3 = ω 1 +ω 2 (15.44)
k 3 = k 1 +k 2 , (15.45)they can only have thesamedispersion relation ifω^2 depends linearly onk^2 so that the
magnitude ofkis linearly proportional to the magnitude ofω.This is not so for the modes
listed in Eq.(15.43) except for thek^2 λ^2 De<< 1 limit of the ion acoustic wave;applying
the selection rules to thisω^2 =k^2 c^2 slimit of ion acoustic waves gives nonlinear inter-
actions between various sound wave harmonics, e.g., the first and second harmonic can
interact with the third harmonic. Since this limit corresponds to the well-known nonlinear
steepening of ordinary sound waves it will not be discussed further.
The situations which are of particular relevance to plasmas are wheremodes satisfying
different types of dispersion relations interact with each other, forexample an electromag-
netic wave interacting with an electron plasma wave and an ion acoustic wave. The various
possibilities for non-linear interactions of the three types of plasma waves are the follow-
ing:
pump hf daughter lf daughter common name
ω 3 ω 2 ω 1
em em Langmuir stimulated Raman backscatter
em em acoustic stimulated Brillouin backscatter
em em zero frequency self-focusing
em Langmuir Langmuir two-plasmon decay
em Langmuir acoustic parametric decay instability
Langmuir Langmuir acoustic electron decay instability
Langmuir Langmuir zero frequency caviton
Here the abbreviations em, Langmuir, and acoustic stand for electromagnetic wave,
electron plasma wave, and ion acoustic wave respectively and the ordering is by progres-
sively lower frequency, column by column taking into account thatω 3 > ω 2 > ω 1 and
electromagnetic waves have higher frequencies than Langmuir waves which inturn have
higher frequency than ion acoustic waves.
In each case the low frequency daughter modulates the density and beats with either the
pump or the high frequency daughter to provide a high frequency nonlinear current as given
by Eq.(15.41) while the pump and the high frequency daughter beat together to providea
ponderomotive force that couples to the low frequency wave. Becausec^2 >>vTe^2 >>c^2 s
the electromagnetic wave has a stronger dependence onk^2 than does the electron plasma
wave which in turn has a stronger dependence onk^2 than does the ion acoustic wave.
This means that ifk 1 ,k 2 ,andk 3 are all in the same direction and Eq.(15.45) is satisfied,
Eq.(15.44) cannot be satisfied sinceω 3 (|k 3 |) =ω 3 (|k 1 |+|k 2 |)>ω 1 (|k 1 |)+ω 2 (|k 2 |).
Thus, the frequency and wavenumber selection rules can only be satisfied ifk 1 ,k 2 ,andk 3
are not all in the same direction. The pump wave direction is taken todefine the positive
direction, i.e.,k 3 defines the positive direction so one of the daughter waves must propagate
in the direction opposite to the pump.
The electromagnetic wave in a given plasma always has a higher frequency than the
electron plasma or ion acoustic waves. As a consequence, the low frequency daughter can-