Fundamentals of Plasma Physics

15.5 Digging a hole in the plasma via ponderomotive force 449

and so Eq.(15.49) reduces to

1

n

∇ ̃n=−

1

2

me

κTe

∇

(

̃uhe

) 2

(15.121)

which can be integrated to give a Boltzmann-like relation

n ̃

n

=−

1

2

me

〈(

̃uhe

) 2 〉

κTe

. (15.122)

Since the high frequency daughter wave is the same as the pump wave there is now only one
high frequency wave and so there is no need to have subscripts distinguishing themodes.
The consequence is that the high frequency wave propagates in a plasma having adensity
depletion dug out by the ponderomotive force. For example the Langmuir wave equation
in this case would become



∂

2

∂t^2

+ 2Γ

∂

∂t

+ω^2 pe



 1 −^1

2

me

〈(

u ̃he

) 2 〉

κTe



−^3 κTe

me

∇^2



 ̃uhe=0 (15.123)

where ̃uhehas been used as the linear variable instead ofE ̃and a linear damping term
involvingΓ,the linear damping rate has been introduced.
An undamped linear Langmuir wave in a uniform plasma satisfies the dispersion rela-
tionω^2 =ω^2 pe(1+3k^2 λ^2 De)where kλDe<< 1 so that the wave frequency is very close
toωpe.It is reasonable to presume that the nonlinear wave also oscillates at a frequency
very close toωpeand so what is important is the deviation of the frequency fromωpe.To
investigate this, the electronfluid velocity is assumed to be of the form

̃uhe(x,t) = Re

[

A(x,t)e−iωpet

]

=

1

2

{

A(x,t)e−iωpet+A∗(x,t)eiωpet

}

(15.124)

in which case 〈
(
̃uhe

) 2 〉

=

1

2

|A|^2 (15.125)

and the time dependence ofA(x,t)characterizes the extent to which the wave frequency
deviates fromωpe.Because this deviation is small, Achanges slowly compared toωpe,
and so in analogy to Eq.(3.22) it is possible to approximate

∂^2

∂t^2

[

A(x,t)e−iωpet

]

≃−ω^2 peA(x,t)e−iωpet−2iωpe

∂A

∂t

e−iωpet (15.126)

and

2Γ

∂

∂t

[

A(x,t)e−iωpet

]

≈−i2ωpeΓAe−iωpet.

Substitution of Eq. (15.126) into Eq.(15.123) gives

2iωpe

∂A

∂t

+i2ωpeΓA+

ω^2 pe

4

me|A|^2

κTe

A+c^2 ∇^2 A=0. (15.127)