Fundamentals of Plasma Physics

14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 407

Using Eq.(14.36) gives

∂WP

∂t

= −

∫

dvmv

2ie^2

ε 0 m^2

∫

dk

E(k,t)

ω−kv

∂f 0

∂v

= 2iω^2 p

∫

dk

E(k,t)

k

∫

dv

ω−kv−ω

ω−kv

∂f ̄ 0

∂v

= −ω^2 p

∫

dk

2iω(k)E(k,t)

k

∫

dv

1

ω−kv

∂f ̄ 0

∂v

. (14.50)

Invoking Eq.(14.42) to substitute for the velocity integral results in

∂WP

∂t

=

∫

dk2i[ωr(k)+iωi(k)]E(k,t). (14.51)

Sinceωr(k)is an odd function ofkandωi(k)is an even function ofk, only the term
involvingωisurvives thekintegration and so

∂WP

∂t

+

∫

dk 2 ωi(k)E(k,t)=0. (14.52)

Using Eq.(14.39) this becomes

∂

∂t

[

WP+

∫

dkE(k,t)

]

=0 (14.53)

which can now be integrated to give

WP+

∫

dkE(k,t)=WP+WE=const.

showing that the sum of the particle and electric field energies is conserved. The particle
energy and the electric field energy therefore need not be individually conserved – only
the sum of these two types of energy is conserved. This result allows for energyexchange
between the particles and the electric field.

14.2.3Energy exchange with resonant particles

More detailed insight is obtained by considering the role of resonant particles, i.e., those
particles having velocityv≈ω/kas indicated by the shaded region in Fig.14.1. This is
done by using Eq.(14.37) to re-write the top line of Eq.(14.50) as

∂WP

∂t

= −

2 e^2

ε 0 m

∫

dkE(k,t)

∫

dvv

i[ωr(k)−kv]+ωi(k)

[ωr(k)−kv]^2 +ω^2 i(k)

∂f 0

∂v

= − 2 ω^2 p

∫

dkE(k,t)

∫

dv

vωi(k)

[ωr(k)−kv]^2 +ω^2 i(k)

∂f ̄ 0

∂v

(14.54)

where becauseωr(k)−kvis an odd function ofk, only theωi(k)numerator term survives
thekintegration. The velocity integral can be decomposed into a resonant portionwhich
is the velocity range whereωr≃kvand the remaining or non-resonant portion. In the
resonant portion, it is possible to approximate

ωi

(ωr−kv)^2 +ω^2 i

≃πδ(ωr−kv)=

π

k

δ(v−

ωr

k

) (14.55)