Fundamentals of Plasma Physics

14.2 Vlasov non-linearity and quasi-linear velocity space diffusion 411

conserved, but the second moment is not. Change of the resonant particle energy while
conserving their number is achieved by adjusting the number of resonant particles that
are slightly slower and slightly faster thanω/k.For example, if there is a decrease in the
number of resonant particles havingv≃(ω/k)−there must be a corresponding increase in
the number of resonant particles havingv≃(ω/k)+. This process provides a net transfer
of energy from the wave to the resonant particles, involves wave Landau damping, and
requires having∂f 0 ,res/∂v < 0. The result, increasing the number of resonant particles
havingv≃(ω/k)+while decreasing the number havingv≃(ω/k)−flattensf 0 ,res(v)
and so makes it plateau-like as indicated in Fig.14.1(b).
Behavior of the non-resonant particles
The quasi-linear diffusion coefficient for the non-resonant particles comes from the
principle part of the integral in Eq.(14.38), i.e.,

DQL,non−res=

e^2

ε 0 m^2

P

∫

dk

2 ωi(k)E(k,t)

[ωr(k)−kv]^2 +ω^2 i(k)

. (14.71)

The vast majority of the non-resonant particles have velocities much slowerthan the wave,
so for the non-resonant particles it can be assumedv<<ωr/kin which case

DQL,non−res ≃

e^2

ε 0 m^2

∫

dk

2 ωi(k)E(k,t)

ω^2 r(k)

≃

1

mn 0

∫

dk 2 ωi(k)E(k,t). (14.72)

Since this non-resonant particle velocity space diffusion coefficient is velocity-independent,
it can be factored from velocity integrals or derivatives. Equation (14.11) showed that the
change inf 0 is orderǫ^2 whereǫ<< 1 .This means that changes inf 0 are small compared
tof 0. On the other hand, there is no zero-order wave energy since the wave energy scales
asE 12 and so is entirely constituted of terms which are orderǫ^2 .Thus, the wave energy
spectrum can change substantially (for example, disappear altogether) whereas there is a
only slight corresponding change tof 0 .Thus, Eq.(14.35), becomes

∂f 0 ,non−res

∂t

=

1

mn 0

∂

∂v

∫

dk 2 ωi(k)E(k,t)

∂f 0 ,non−res

∂v

≃

1

mn 0

(

d

dt

∫

dkE(k,t)

)

∂^2 f 0 ,non−res

∂v^2

(14.73)

which can be integrated to give

f 0 ,non−res(v,t)−f 0 ,non−res(v,0)=

1

mn 0

(∫

dk [E(k,t)−E(k,0)]

)

∂^2 f 0 ,non−res

∂v^2

.

(14.74)

Since the number of resonant particles is conserved, the number of non-resonant particles
must also be conserved.
If an initial wave spectrum becomes damped att=∞then it is possible to write

f 0 ,non−res(v,∞)=f 0 ,non−res(v,0)−

1

mn 0

[∫

dkE(k,0)

]

∂^2 f 0 ,non−res(v,0)

∂v^2

.

(14.75)