Fundamentals of Plasma Physics

410 Chapter 14. Wave-particle nonlinearities

resonant particle velocity range respectively map to the lower and upper bounds of the
values ofkfor whichE(k,t)is finite. For these particles, the delta function approximation
Eq.(14.55) can be used to evaluate the quasilinear diffusion coefficient given by Eq.(14.38)
and obtain

DQL,res(v) ≃

2 πe^2

ε 0 m^2

∫

dkδ(ωr−kv)E(k,t)

=

2 πe^2

ε 0 m^2

∫

dkδ(ωr/v−k)

E(k,t)

v

=

2 πe^2

ε 0 m^2

E(ωr/v,t)

v

. (14.65)

Using this coefficient the quasilinear velocity diffusion for the resonant particles is

∂f 0 ,res

∂t

=

2 πe^2

ε 0 m^2

∂

∂v

(

E(ωr/v,t)

v

∂f 0 ,res

∂v

)

. (14.66)

It is seen from Eq.(14.62), the generalized formula for Landau damping, that

π

(

∂f 0

∂v

)

v=ωr/k

= 2ωi

k^2

ω^3 r

n 0 (14.67)

and so Eq.(14.66) becomes

∂f 0 ,res

∂t

=

2 e^2

ε 0 m^2

∂

∂v

(

E(ωr/v,t)

v

2 ωi

k^2

ω^3 r

n 0

)

≃

2 ωp

m

∂

∂v

(

E(ωr/v,t)

v^3

2 ωi

)

=

2 ωp

m

∂

∂t

∂

∂v

(

E(ωr/v,t)

v^3

)

(14.68)

whereωr≃ωp=

√

n 0 e^2 /ε 0 mhas been used and also, because the particles are resonant,
v≃ωr/k.This can be integrated with respect to time to obtain

f 0 ,res(v,t)−f 0 ,res(v,0)=

2 ωp

m

∂

∂v

(

E(ωr/v,t)−E(ωr/v,0)

v^3

)

. (14.69)

By definitionE(ωr/v,t)vanishes forvlying outside the resonant particle velocity range
vminres < v < vmaxres because outside this range there is no wave energy with which the
particles can resonate (see Fig.14.1). Hence, integration of Eq.(14.69) over the velocity
rangevminres<v<vmaxres of the resonant particles gives

∫vmaxres

vresmin

dvf 0 ,res(v,t)=

∫vmaxres

vresmin

dvf 0 ,res(v,0) (14.70)

which shows that the number of resonant particles is conserved.
Equation (14.64) showed that the resonant particle energy is not conserved and can be
exchanged with the wave energy. Thus, the zeroth moment of the resonant particles is