Fundamentals of Plasma Physics

444 Chapter 15. Wave-wave nonlinearities

where

λ=

ωpeqek 1

√

mimeω 2 ω 3

. (15.81)

Defining the mode frequencies as

ω^23 = ω^2 pe+k^23 c^2

ω^22 = ω^2 pe+3k^23 κTe/me

ω^21 = k^21 c^2 s (15.82)

the coupled equations become
(
∂^2
∂t^2

+ω^23

)

E ̃ 3 = −λψ ̃E ̃ 2

(

∂^2

∂t^2

+ω^22

)

E ̃ 2 = −λψ ̃E ̃ 3

(

∂^2

∂t^2

+ω^21

)

ψ ̃ = −λE ̃ 2 ·E ̃ 3 (15.83)

which is identical to the coupled oscillator system described by Eq.(15.3) ifmis set to
unity in Eq.(15.3) andE ̃ 2 is parallel toE ̃ 3 .Using Eq.(15.30), the nonlinear growth rate is
found to be

γ =

λE 3

4

√

ω 1 ω 2

=

1

4

ωpi

ω 2

√

ω 3

ω 1

k 1

qeE 3

meω 3

. (15.84)

15.4 Non-linear dispersion formulation and instability threshold

An equivalent way of considering the effect of nonlinearity is to derive a so-called nonlin-
ear dispersion relation (Nishikawa 1968b, Nishikawa 1968a). This method has the virtue
that both wave damping and frequency mismatches can easily be incorporated. To see how
damping can be introduced, consider an electrical circuit consisting of an inductor, capaci-
tor and resistor all in series. The circuit equation is

L

d^2 Q

dt^2

+R

dQ

dt

+

Q

C

=0 (15.85)

whereQis the charge stored in the capacitor and the current isI= dQ/dt.The general
solution isQ∼e−iωtwhereωsatisfies

ω^2 +iω

R

L

−

1

LC

=0. (15.86)

Solving forωgives the usual damped harmonic oscillator solution

ω=−

iR

2 L

±

√

1

LC

−

R^2

4 L^2

. (15.87)