Fundamentals of Plasma Physics

14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 405

where thequasi-linear velocity space diffusion coefficientis

DQL=

2ie^2

ε 0 m^2

∫

dk

E(k,t)

ω−kv

. (14.36)

The factoriin the velocity diffusion tensor seems surprising at first becausef 0 is a real
quantity. However, theifactor is entirely appropriate and is intimately related to the parity
properties ofω(k)since

i

ω(k)−kv

=

i[ωr(k)−kv]+ωi(k)

[ωr(k)−kv]^2 +ω^2 i(k)

. (14.37)

The denominator in this expression is an even function ofk, the term[ωr(k)−kv]in the
numerator is an odd function ofk, andωi(k)is an even function ofk.SinceE(k,t)is an
even function ofk, integration overkin Eq.(14.36) annihilates the imaginary component
(this component is an odd function ofk)and so

DQL=

e^2

ε 0 m^2

∫

dk

2 ωi(k)E(k,t)

[ωr(k)−kv]^2 +ω^2 i(k)

. (14.38)

In summary, the self-consistent coupled system of equations for the non-linear evolution
of the equilibrium consists of Eq.(14.35), (14.38), and

∂

∂t

E(k,t)=2ωi(k)E(k,t) (14.39)

which is obtained from Eq.(14.33). The real and imaginary parts of the frequencyωr(k),
ωi(k)appear as parameters in these equations and are determined from the linear wave
dispersion relation which in turn depends onf 0. This linear wave dispersion relation is
obtained as in Section 4.5 by writingE 1 =−∂φ 1 /∂xand then combining Eq.(14.18) with
Poisson’s equation

k^2 φ ̃ 1 (k)=−

e

ε 0

∫

dvf ̃ 1 (14.40)

to obtain

k^2 ̃φ 1 (k)=−

e^2

ε 0 m

∫

dv

∂fσ 0

∂v

k ̃φ 1 (k)

ω−kv

(14.41)

which can be expressed as

1+

ω^2 p

k

∫

dv

∂f ̄ 0 /∂v

ω−kv

=0 (14.42)

using Eq.(14.4). Thus, given the instantaneous value off 0 (v,t)=n 0 f ̄ 0 (v,t),the complex
frequency is determined from Eq.(14.42) and then, given the instantaneous value of the
wave spectral energy, both the evolution off 0 and the wave spectral energy are determined
from Eqs.(14.35) and (14.39).