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
- 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