Fundamentals of Plasma Physics

426 Chapter 14. Wave-particle nonlinearities

14.3.6Higher order echoes

If one expands the Vlasov equation to higher than second-order, then higher order products
of ballistic terms can result. For spatial echoes high order products of the form

[

eiω^1 (x−L)/v

]N

×

[

e−iω^2 x/v

]M

(14.153)

will appear and will phase mix away except at locations where

Nω 1 (x−L) =Mω 2 x (14.154)

so there will be a higher order echo at the location

xecho =

Mω 2 L

Nω 1 −Mω 2

14.4 Assignments

1. Competition between collisions and quasilinear diffusion
   (a) LetvTe=

√

2 κTe/meandw = vz/vTeand show that the Fokker-Planck

equation in problem 1 of Chapter 13 can be written in dimensionless form as

∂FT

∂τ

≃

∂

∂w

(

FT

2+Z

w^2

+

3

4 w^3

∂FT

∂w

)

.

whereτ=νtand

ν=

nee^4 lnΛ

4 πε^20 m^2 ev^3 Te

is an effective collision frequency.

Now show that the quasilinear diffusion equation can similarly be writtenin

dimensionless form as

∂FT

∂τ

=

∂

∂w

D ̄QL∂FT

∂w

where

D ̄QL(w)=DQL

Dcol

and

Dcol=

2 νκTe

me

is an effective collisional velocity-space diffusion coefficient.

(b) Suppose that both collisions and quasilinear diffusion are operative sothat

∂FT

∂τ

=

∂

∂w

(

D ̄QL∂FT

∂w

+FT

2+Z

w^2

+

3

4 w^3

∂FT

∂w

)

.