C
Quasi-linear Theory
• Velocity-space diffusion rate
(∇v)2
∝ E2
2t
where E is the amplitudes of the linear-theory modes
Squares of E → “Quasi-linear”
Dv =
• Motivation of the development of quasi-linear theory.
– Microinstabilities form an important part of wave theory.
– And the questions arise, how will the mode amplitude grow?
– What is the instability saturation mechanism?
– Bibliography
∗ W.E. Drummond and D.Pines (1961): “Nonlinear stability
of Plasma Oscillations, General Atomic” GA-2386 (1961)
∗ A.A.Vedenov, E.P.Velikhov and R.Z. Sagdeev (1961): “Nonlinear Oscillations of Rare field Plasma” Nuclear Fusion 1,
82 (1961)
∗ Yu.A.Romanov and G.Filippov (1961): “The Interaction of
Fast Electron Beams with Longitudinal Plasma Waves” sov.phys.JETP 13, 87 (1961)
– In which it was found that a temporally growing micro-instability
acts back on the zero-order velocity distribution function.
– Its effect on the distribution function is to produce velocity-space
diffusion.
– The diffusion tends to flatten f0 (~v ) in this region and drive the
instability growth rate to zero.
– This saturation process can be viewed as a continuous diffusion
through which the zero-order distribution function evolves slowly
in time.
– Two assumptions in quasi-linear theory.
∗ The amplitudes of the perturbations in the plasma are not
so large as to invalidate the use of zero-order orbits and
of the spatially averaged distribution function, f0 (~v , t) =<
f0 (~r, ~v , t) >
∗ The effective wave spectrum should be sufficiently dense.
Any appreciable coherence between modes will be destroyed
by phase mixing.
• Electromagnetic Quasi-linear Theory
³
´
∂f
~v· E
~ + ~v × B
~ f = 0 (∗f = f (~r, ~v , t))
~ + q∇
+ ~v · ∇f
∂t
m
Averaging over a number of space and time periods of the rapid fluc~ 0 , we also average over the
tuations. In addition, in presence of B
180
Figure 38: Velocity distribution for “bump-on-tail” instability. Real part of
unstable frequencies are such that v = ω0 (k)/k lies in region where vdf0 /dv
is positive (opposite sense to Landau damping). Quasi-linear diffusion due
to these modes tends to flatten out the bump.
gyro-angle in velocity space:
¿ À
¿
À
´ E
∂f0 (~v , t)
∂f
∂f
q D~ ³ ~
~ f
=
= − ~v
−
∇v · E + ~v × B
∂t
∂t
∂z
m
slow evolution of f0 (~v , t) = hf (~r, ~v , t)i

D E
∂f

Space
averaging
:

∂z = 0

< E >= 0
D
³
´ E


 Higher order contributions to ∇
~v· E
~ + ~v × B
~ f are neglected
∂f0
∂t
¿Z 2π
´ À
q
dφ ~ ³ ~
~
≃ −
∇v · E1 + ~v × B1 f1
m
2π
0
Z
´
q X 1 2π dφ ~ ³ ~
∇v · Ek + ~v × B~k f−k
= −
m
V 0 2π
modes
where V is the volume and we used Fourier formalism of
Z
1 ∞
dωA(−ω)B(ω)
< A(t)B(t) >= lim
T →∞ T −∞
Z
1 ∞ 3
d k(−k)B(k)
< A(z)B(z) >= lim
V →∞ V −∞
181
f−k = f (ω−k , −~k, ~v ) = f ∗ (ωk , ~k, ~v ) (∵ f1 (~r, ~v , t) is real.)
since B~1 =
~k
ω
× E~1
~k~v
~k · ~v
)+
] · E~k
E~k + ~v × B~k = [1(1 −
ω
ω
Also,
~v = x̂v⊥ cos φ + ŷv⊥ sin φ + ẑvk = ρ̂v⊥ + ẑvk
~ v = ρ̂ ∂ + φ̂ 1 ∂ + ẑ ∂
∇
∂v⊥
v⊥ ∂vφ
∂vk
~k = x̂k⊥ cos θ + ŷk⊥ sin θ + ẑkk
= ρ̂k⊥ cos(φ − θ) − φ̂k⊥ sin(φ − θ) + ẑkk
~ = x̂Ex + ŷEy + ẑEz
E
= ρ̂(−Ex cos φ + Ey cos φ) + ẑEz
where ρ̂ is the unit vector in the direction of v~⊥ , and φ̂ = ẑ × ρ̂.
182
~ v · (E
~ + ~v × B)
~ k f−k = 1 ∂ v⊥ [(E
~ + ~v × B)
~ k f−k ]ρ + 1 ∂ v⊥ [(E
~ + ~v × B)
~ k f−k ]φ
∇
v⊥ ∂v⊥
v⊥ ∂vφ
∂
~ + ~v × B)
~ k f−k ]k
v⊥ [(E
+
∂vk
~
~
~ k ]ρ f−k
~ + ~v × B)
~ f−k ] = [(1(1 − k · ~v ) + k~v ) · E
(1)[(E
k
ρ
ω
ω
1
= [1 − (k⊥ v⊥ cos(φ − θ) + kk vk )](Ekx cos φ + Eky sin φ)f−k
ω
1
+ [v⊥ (Ekx cos φ + Eky sin φ) + vk Ez ]kperp cos(φ − θ)f−k
ω
1 ∂
1
∂
∴
v⊥ [· · · ]ρ =
([· · · ]ρ ) +
[· · · ]ρ
v⊥ ∂v⊥
v⊥
∂v⊥
k⊥
1
[· · · ]ρ + [−
cos(φ − θ)(Ekx cos φ + Eky sin φ)f−k ]
=
v⊥
ω
∂f−k
1
+ [1 − (k⊥ v⊥ cos(φ − θ) + kk vk )](Ekx cos φ + Eky sin φ)
ω
∂v⊥
1
+ (Ekx cos φ + Eky sin φ)k⊥ cos(φ − θ)f−k
ω
1
∂f−k
+ [v⊥ (Ekx cos φ + Eky sin φ) + vk Ez ]k⊥ cos(φ − θ)
ω
∂v⊥
~k~v
~k · ~v
~ k ]φ f−k
~ + ~v × B)
~ f−k ] = [(1(1 −
)+
)·E
(2)[(E
k
φ
ω
ω
1
= [1 − (k⊥ v⊥ cos(φ − θ) + kk vk )](−Ekx sin φ + Eky cos φ)f−k
ω
1
+ [v⊥ (Ekx cos φ + Eky sin φ) + vk Ez ](−k⊥ sin(φ − θ))f−k
ω
1 ∂
1
1
∂f−k
∴
[· · · ]φ =
[1 − (k⊥ v⊥ cos(φ − θ) + kk vk )](−Ekx sin φ + Eky cos φ)
v⊥ ∂vφ
v⊥
ω
∂vφ
1 1
∂f−k
+
[v⊥ (Ekx cos φ + Eky sin φ) + vk Ez ](−k⊥ sin(φ − θ))
v⊥ ω
∂vφ
~
~
~ + ~v × B)
~ f−k ] = [(1(1 − k · ~v ) + k~v ) · E
~ k ]k f−k
(3)[(E
k
k
ω
ω
1
= [1 − (k⊥ v⊥ cos(φ − θ) + kk vk )]Ekz f−k
ω
1
+ [v⊥ (Ekx cos φ + Eky sin φ) + vk Ekz ]kk f−k
ω
1
∂f−k
∂
[· · · ]k = kk Ekz f−k + [1 − (k⊥ v⊥ cos(φ − θ) + kk vk )]Ekz
∴
∂vk
ω
∂vk
1
1
∂f−k
+ Ekz kk f−k + [v⊥ (Ekx cos φ + Eky sin φ) + vk Ekz ]kk
ω
ω
∂vφ
Thus,
~ v · [(E
~ + ~v × B)
~ f−k ] = cos(φ − θ)[(Ek + + Ek − )Sf−k − Ekz T f−k ]
∇
k
1 ∂
∂f−k
+
(· · · )
− i sin(φ − θ)(Ek + + Ek − )Sf−k + Ekz
∂vk
v⊥ ∂vφ
183
where
kk v⊥ ∂f−k
kk vk 1 ∂
)
(v⊥ f−k ) +
ω v⊥ ∂v⊥
ω ∂vk
k⊥ vk 1 ∂
k⊥ v⊥ ∂f−k
−
(v⊥ f−k )
=
ω ∂vk
ω v⊥ ∂v⊥
Sf−k = (1 −
T f−k
1
Ek± = (Ekx ± iEky )e∓iθ
2
and
ùE
(
kx cos φ
Ekx ≡ Ex (~k), Eky ≡ Ey (~k), Ekz ≡ Ez (~k)
+ Eky sin φ = cos(φ − θ)(Ek+ + Ek− ) − i sin(φ − θ)(Ek+ − Ek− )
Ans.)
1
1
[Ekx e−iθ + iEky e−iθ ] + [Ekx eiθ − iEky eiθ ]
2
2
1
[Ekx (cos θ − i sin θ) + iEky (cos θ − i sin θ)]
=
2
1
[Ekx (cos θ + i sin θ) − iEky (cos θ + i sin θ)]
+
2
= Ekx cos θ + Eky sin θ
Ek+ + Ek− =
Ek+ − Ek− = −iEkx sin θ + iEky cos θ
= −i(Ekx sin θ − Eky cos θ)
cos(φ − θ)(Ek+ + Ek− ) − i sin(φ − θ)(Ek+ − Ek− )
= (cos φ cos θ + sin φ sin θ)(Ekx cos θ + Eky sin θ)
−i(sin φ cos θ − cos φ sin θ)(−i)(Ekx sin θ − Eky cos θ)
= Ekx cos φ + Eky sin φ)
But,
fk = −
q X X −i(n−m)(φ−θ)
k⊥ v⊥
k⊥ v⊥
)Jn (
)
e
Jm (
m n m
Ω
Ω
×{cos(φ − θ + Ωτ )[(Ek+ + Ek− )U ′ − Ekz V ]
∂f0
}
−i sin(φ − θ + Ωτ )(Ek+ − Ek− )U ′ + Ekz
∂vk
Z
∞
0
dτ exp [(ω − kk vk − nΩ)τ ]
Remember f1 obtained for the calculation of the first-order Vlasov equation
with the simple replacement of φ by φ − θ and τ by −τ
where
kk
∂f0
∂f0
∂f0
1
U′ =
+ (v⊥
− vk
)= U
∂v⊥
ω
∂vk
∂v⊥
ω
k⊥
∂f0
∂f0
V =
(v⊥
− vk
)
ω
∂vk
∂v⊥
nΩ ∂f0
1
nΩ ∂f0
)
+
vz
= W
W ′ = (1 −
ω ∂vk ωv⊥ ∂vk
ω
184
1
2π
Z
2π
0
~v · [(E~k + ~v × B~k )∗ fk ]
dφ∇
∞
X
n
{[(Ek+ + Ek− )S − Ekz T ] Jn (λ) + (Ek+ − Ek− )SJn′ (λ)
λ
n=−∞
Z ∞
q
∂ ∗
} {(−
+Ekz Jn (λ)
dτ ei(ω−kk vk −nΩ)τ )}
∂vk
m 0
n
×{ Jn (λ)[(Ek+ + Ek− )U ′ − Ekz V ]
λ
∂f0
+Jn′ (λ)(Ek+ − Ek− )U ′ + Jn (λ)Ekz
}
∂vk
P P
P R
φ integral n m → n , dτ → nλ Jn , Jn′
But,
=
ù
n
∂f0
− V +
λ
∂vk
nΩ k⊥ v⊥ ∂f0 k⊥ vk ∂f0
∂f0
(
−
)+
k⊥ v⊥ ω ∂vk
ω ∂v⊥
∂vk
nΩ ∂f0 nΩvk ∂f0
)
+
= W′
= (1 −
ω ∂vk
ωv⊥ ∂v⊥
vk ′ ω − kk vk − nΩ ∂f0
vk ∂f
=
U +
(
−
)
v⊥
ω
∂vk v⊥ ∂v⊥
vk ′
=
U
v⊥
= −
∗ J (λ)
The Coefficient of Ekz
n
n
∂
− T+
λ
∂vk
k⊥ vk 1 ∂
∂
nΩ k⊥ v⊥ ∂
(
v⊥ ) +
−
k⊥ v⊥ ω ∂vk
ω v⊥ ∂v⊥
∂vk
nΩvk 1 ∂
nΩ ∂
= (1 −
)
+
v⊥
ω ∂vk
ωv⊥ v⊥ ∂v⊥
ω − kk vk − nΩ ∂
vk 1 ∂
vk
S+
(
−
v⊥ )
=
v⊥
ω
∂vk v⊥ v⊥ ∂v⊥
vk
=
S
v⊥
= −
1
2π
Z
2π
0
−→ −i
~v·
dφ∇
h³
~k + ω
~k
E
~ ×B
´∗
fk
i
q X
1
Ak U ′
SA∗k
m
ω − kk vk − nΩ
n
hn
i
i
o
hn
where Ak = v⊥ Ek+
Jn (λ) + Jn′ (λ) + Ek−
Jn (λ) − Jn′ (λ) + vk Ekz Jn (λ)
λ
λ
= v⊥ Ek+ Jn−1 (λ) + v⊥ Ek− Jn+1 (λ) + vk Ekz Jn (λ)
185
Thus, a remarkably compact expression for “quasi-linear evolution”
∞
∂f0 (v⊥ , vk , t)
πq 2 X 1 X
Lδ(ωk − kk vk − nΩ)|A|2 Lf0
= 2
∂t
m
ωk2 −∞
modes
L is the operator, such that
L = (ωk − kk vk )
1 ∂
∂
1 ∂
∂
+ kk
= nΩ
+ kk
v⊥ ∂v⊥
∂vk
v⊥ ∂v⊥
∂vk
We used Plemelj relation
1
=P
ωkv
µ
1
ωkv
¶
− iπδ(ω − kv)
n
Jn (x) + Jn′ (x)
x
n
Jn+1 (x) = Jn (x) − Jn′ (x)
x
Jn−1 (x) =
Thus,
µ
·
¶¸
∞ ½
³ e ´2 X
∂f0 (~v , t)
1 ∂
∂f0
2 nΩ ∂f0
nΩδ(ω − kk vk − nΩ)|A|
=π
+ kk
∂t
mω −∞ v⊥ ∂v⊥
v⊥ ∂v⊥
∂vk
·
µ
¶¸¾
∂f0
nΩ ∂f0
∂
+ kk
kk δ(ω − kk vk − nΩ)|A|2
+
∂vk
v⊥ ∂v⊥
∂vk
Let E + = Ex + iEy (left-hand polarization)
E − = Ex − iEy (right-hand polarization)
1
Ak =
2
∴
µ
¶
vk
1
+ −iθ
− iθ
v⊥ E e Jn−1 + v⊥ E e Jn+1 + 2 Ez Jn ⇒ A
v⊥
2
·
µ
¶¸
∞ ½
³ e ´2 X
∂f0
nΩ ∂f0
1 ∂
∂f0
nΩδ(ω − kk vk − nΩ)|A|2
=π
+ kk
∂t
2mω −∞ v⊥ ∂v⊥
v⊥ ∂v⊥
∂vk
·
¶¸¾
µ
∂f0
nΩ ∂f0
∂
+ kk
kk δ(ω − kk vk − nΩ)|A|2
+
∂vk
v⊥ ∂v⊥
∂vk
where
A = v⊥ E + e−iθ JN −1 + v⊥ E − eiθ JN +1 + 2
186
vk
Ez JN
v⊥