Fundamentals of Plasma Physics

15.3 Application to waves 437

frequency modes, then the ponderomotive force contains a term at the difference (i.e.,

beat) frequency between the frequencies of the two modes. Thus, at low frequencies

the electron equation of motion, Eq.(15.34) becomes

∂ ̃ue

∂t

=

qe

me

E ̃−^1

men

∇P ̃e−

1

2

∇

(

̃uhe

) 2

(15.39)

where only the beat frequency component in

(

u ̃he

) 2

is used. The ponderomotive force

provides a mechanism for high-frequency waves to couple to low frequency waves.

It acts as an effective pressure scaling asmen

(

u ̃he

) 2

/ 2 and so in a sense, the quiver

velocityu ̃heacts as a thermal velocity. The ratio of ion radiation pressure to the ion

pressure is smaller than the corresponding ratio for electrons by a factorofme/mi

because the ion quiver velocity is smaller by this factor. Thus, ion ponderomotive

force is ignored since it is so small.

2. Beating of a low frequency wave with a high frequency wave to drive another high
   frequency wave (modulation).By writing Ampere’s law as

∇×B ̃=μ 0

∑

σ

( ̃nσqσ ̃uσ+n 0 qσ ̃uσ)+μ 0 ε 0

∂E ̃

∂t

(15.40)

it is seen that densityfluctuations provide a nonlinear component to the current den-

sity. The nonlinear term can be put on the right hand side to emphasize its role as a

nonlinear driving term so that Ampere’s law becomes

∇×B ̃−μ 0 ε 0

∂ ̃E

∂t

−μ 0

∑

σ

n 0 qσ ̃uσ=−μ 0

∑

σ

̃nσqσ ̃uσ. (15.41)

The nonlinear term is assumed to be a product of a high frequency wave and a low

frequency wave. The linearized continuity equation gives

∂n ̃σ

∂t

=−nσ∇· ̃uσ (15.42)

showing thatn ̃σ∼ ̃uσ/ωso that the productnlσuhσis much larger than the product

nhσulσwherelandhrefer to low and high frequency waves. Thus, the dominant effect

of a low frequency wave is to modulate the density profile seen by a high frequency

wave.

15.3.2Possible types of wave interaction

As discussed in Sec.4.2, three distinct types of waves can propagate in anunmagne-
tized uniform plasma and these waves have the dispersion relations:

ω^2 = ω^2 pe+k^2 c^2 ,electromagnetic wave

ω^2 = ω^2 pe

(

1+3k^2 λ^2 de

)

,electron plasma wave

ω^2 =

k^2 c^2 s

1+k^2 λ^2 De

,ion acoustic wave. (15.43)