Fundamentals of Plasma Physics

436 Chapter 15. Wave-wave nonlinearities

replacingδj.Hence, so long as the selection rulesω 3 =ω 1 +ω 2 andk 3 =k 1 +k 2 are
satisfied, a high frequency waveω 3 should decay spontaneously into two low frequency
wavesω 1 ,ω 2 providing there is a suitable coupling coefficient.

15.3.1Examples of nonlinearities

Nonlinearities can be divided into two general types: (1) two high frequency waves beating
together to give a low frequency driving term and (2) a high frequency wave beating with a
low frequency wave to give a high frequency driving term. As shown below, the first type
of nonlinearity has the form of a radiation pressure (also known as a ponderomotive force)
while the second type of nonlinearity has the form of a density modulation.

1. Beat of two high-frequency waves driving a low frequency wave (ponderomotive force).
   Since ion motion is negligible in a high frequency wave, all that is requiredwhen con-
   sidering nonlinearities is the electron equation of motion,

∂ue

∂t

=−ue·∇ue+

qe

me

(E+ue×B)−

1

mene

∇Pe. (15.34)

For simplicity, it is assumed no equilibrium magnetic field exists, so theonly magnetic

field is the wave magnetic field. Tildes are used to denote linear quantities to avoid

confusion with the subscripts 1 , 2 , 3 which denote the low frequency daughter, high

frequency daughter and pump wave respectively. Because the pressuregradient term

in the electron equation of motion provides only a small correction at highfrequencies,

the linear motion for either the pump or the high frequency daughter must thus be a

solution of

∂ ̃ue

∂t

=

qe

me

̃E. (15.35)

The electron quiver velocity is defined to be

̃uhe=

qe

me

∫t

E ̃dt′ (15.36)

which is the solution to Eq.(15.35) for both electron plasma waves and electromag-

netic waves. The main non-linear terms in Eq.(15.34) are− ̃ue·∇ ̃ue+qe

(

̃ue×B ̃

)

/me;

non-linearity in the pressure gradient is ignored because, by assumption, this term is

already small. In order to obtainB ̃,Faraday’s law is integrated with respect to time

giving

B ̃=−∇×

∫t

E ̃dt′=−me

qe

∇× ̃uhe. (15.37)

The two nonlinear terms can be combined using Eq.(15.37) to form what is called the

ponderomotive force or radiation pressure,

− ̃ue·∇ ̃ue+

qe

me

(

̃ue×B ̃

)

= − ̃uhe·∇ ̃uhe−

(

̃uhe×∇× ̃uhe

)

= −

1

2

∇

(

u ̃he

) 2

. (15.38)

If only one high frequency mode exists and beats with itself, then the ponderomotive

force−∇

(

u ̃he

) 2

/ 2 is at zero frequency but if the beating is between two distinct high