Fundamentals of Plasma Physics

406 Chapter 14. Wave-particle nonlinearities

14.2.2Conservation properties of the quasilinear diffusion equation

Conservation of particles
Conservation of particles occurs automatically because Eq.(14.35) has theform of a
derivative in velocity space. Thus, the zeroth moment of Eq.(14.35) issimply

∂

∂t

∫

dvf 0 =

∫

dv

∂

∂v

(

DQL

∂f 0

∂v

)

=

[

DQL

∂f 0

∂v

]v=∞

v=−∞

=0 (14.43)

and so the quasi-linear diffusion equation conserves the densityn=

∫

dvf 0.

Conservation of momentum

Examination of momentum conservation requires taking the first moment of Eq.(14.35),

∂

∂t

(nmu)=m

∫

dvv

∂

∂v

(

DQL

∂f 0

∂v

)

=−m

∫

dvDQL

∂f 0

∂v

. (14.44)

Using Eq.(14.36) this becomes

∂

∂t

(nmu) = −m

∫

dv

2ie^2

ε 0 m^2

∫

dk

E(k,t)

ω−kv

∂f 0

∂v

= −2iω^2 p

∫

dkE(k,t)

∫

dv

1

ω−kv

∂f ̄ 0

∂v

. (14.45)

However, the linear dispersion relation Eq.(14.41) shows that

ω^2 p

∫

dv

1

ω−kv

∂f ̄ 0

∂v

=−k (14.46)

and so
∂
∂t

(nmu)=2i

∫

dkE(k,t)k=0 (14.47)

which vanishes because the integrand is an odd function ofk.Thus, the constraint provided
by the linear dispersion relation shows that the quasi-linear velocity diffusion equation also
conserves momentum.
Conservation of energy
Consideration of energy conservation starts out in a similar manner but leads to some
interesting, non-trivial results. The mean particle kinetic energy is defined to be

WP=

∫

dv

mv^2

2

f 0. (14.48)

The time evolution ofWPis obtained by taking the second moment of the quasi-linear
diffusion equation, Eq.(14.35),

∂WP

∂t

=

∫

dv

mv^2

2

∂

∂v

DQL

∂f 0

∂v

= −

∫

dvmvDQL

∂f 0

∂v

. (14.49)