10
10.1
Quasilinear Diffusion
Weak Turbulence Assumptions
The presence of a band of waves produces island overlap in phase space and
causes stochasticity. This leads to diffusion of particles in velocity space, and
the deformed distribution function affects wave-particle interaction. Quasilinear
theory takes into account such evolution of the distribution function, and is
formulated in the framework of weak turbulence.
The following assumptions are made in weak turbulence theory.
1. Perturbation amplitude is not so large, so that the use of 0th order orbit z = z0 + vt and spatially averaged distribution function f0 (⃗v , t) =
⟨f0 (⃗r, ⃗v , t)⟩ is justified.
2. Wave spectrum is sufficiently dense, so that coherence between modes is
destroyed by phase mixing.
3. |γ| ≪ ωr , |γ| ≪ kvrms .
10.2
Quasilinear Analysis
⃗ t) = ⃗ẑE cos(kz − ωt) and a particle with velocity v and
For a wave field E(z,
position z0 at t = 0, the 1st order equation of motion is
m
d
∆v = qE cos(kz0 + kvt − ωt).
dt
This can be integrated to
∆v =
qE
{sin [kz0 − (ω − kv)t] − sin kz0 } .
m(ω − kv)
which can be averaged over the initial position z0 to give
[
]2
[
]
(
)2
qE
qE
2 (ω − kv)t
2
(∆v) = 2
sin
≃π
tδ(ω − kv).
m(ω − kv)
2
m
The delta function represents the limit ωt ≫ 1 (but not so large that significant
trapping can occur). The velocity space diffusion function given by
Dv =
(∆v)2
π
=
2t
2
(
qE
m
)2
δ(ω − kv).
The 1st order Vlasov equation
∂f1
∂f1
q
∂f0
+v
+ E1 (ωk , k)
=0
∂t
∂z
m
∂v
can be solved to give the response to an initial disturbance g(v, z) = f1 (v, z, t =
0)
1
∂f0
ig(v, k)
iq
Ek (ωk , k)
+
f1 (ωk , k) = −
m ωk − kv
∂v
ωk − kv
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where g(v, k) is the Fourier transform of g(v, z). Time evolution of the 0th order
distribution function is
⟨ ⟩
⟨
⟩
⟨
⟩
⟨
⟩
∂f0 (v, t)
∂f
∂f
q
∂f
q
∂f1
=
=− v
−
E
≃−
E1
.
∂t
∂t
∂z
m
∂v
m
∂v
Substituting the expression for f1 yields
2πiq 2 ∑
∂f0 (v, t)
1
∂
2 ∂
= 2 2
f0 (v, t).
|Ek (t)|
∂t
m L
∂v ωk − kv ∂v
k̸=0
In the absence of external sources of wave energy, the wave electric field will
grow or decay with the linear theory growth rate γk so that
∂
|Ek (t)|2 = 2γk |Ek (t)|2 .
∂t
Fg. 1. Quasilinear flattening of the distribution function.
10.3
Wave-Associated Drag
The Fokker-Planck equation describes the evolution of the distribution function
∂f
= Q(f ) + C(f )
∂t
where Q(f ) describes quasilinear diffusion due to the wave electric field and
C(f ) describes relaxation due to Coulomb collisions. Defining the quantity ⟨q⟩
to mean q per unit time averaged over many occurrences, the Fokker-Planck
equation can be expressed as
∂f
1
= −∇v · (⟨∆⃗v ⟩ f ) + ∇v · [∇v · (⟨∆⃗v ∆⃗v ⟩ f )] .
∂t
2
The first term represents drag and the second term represents dispersion.
Quasilinear diffusion can be written in the form
(
)
∂ ∂f
∂ ∂D
∂2
∂f
=
D
=−
f + 2 (Df ) .
∂t
∂v ∂v
∂v ∂v
∂v
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Taking the velocity moment,
∫
∫
∫
∂f
d
∂D
∂f
n v = dv v
= dv
f = − dv D .
dt
∂t
∂v
∂v
For unshifted Maxwellian distribution function, v(∂f /∂v) < 0, so this equation
indicates that the drag is negative and tends to increase the average velocity v.
This negative drag provides the force for current drive.
10.4
Collisional Relaxation and Current Drive
Using the Lenard-Bernstein collision operator, Fisch has derived a model for
current drive
[ (
)]
∂f
∂
T ∂f
∂
∂f
=
ν vf +
+
DQL .
∂t
∂v
m ∂v
∂v
∂v
For DQL = 0, the distribution function relaxes toward Maxwellian distribution.
For finite DQL , the solution is expressed as
[ ∫
]
mv
f (v) = (const.) exp − dv
Teff (v)
where
Teff (v) = T +
mDQL (v)
.
ν(v)
If the range of velocities over which DQL is finite (characterized by the full width
w) is small compared to vth , one can assume
DQL
ω w
ω w
= const. for
− ≤v≤ + .
ν
k
2
k
2
In this case,
∫
ω ( ω ) mwDQL
dv vf ≃ q f
k
k
νTeff
−∞
¯
∫ ∞
( ω )2 ( ω ) mwD
mv 2 ∂f ¯¯
QL
P =
dv
≃m
f
.
¯
2
∂t
k
k
T
eff
−∞
rf
∞
j=q
The current drive efficiency can be expressed as
j
q k
1
≃
.
P
m ω ν (ω/k)
10.5
Stochasticity
Quasilinear theory assumes a random distribution of phases between initial particle motions and the electromagnetic field, and that phase-sensitive nonlinear
effects such as trapping may be neglected.
The motion of a single particle in the field of one plane wave is described by
m
dv
= qE sin(kz − ωt)
dt
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Fg. 2. Contours of f (v∥ , v⊥ ) for lower-hybrid current drive.
Fg. 3. Contours of f (v∥ , v⊥ ) for electron cyclorton current drive.
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which has an exact solution. In the frame moving with the wave, particle energy
is conserved. In the laboratory frame
m(
ω )2 q
G=
v−
+ E cos(kz − ωt) = const.
2
k
k
Solving for v,
√
ω
2G 2qE
v= ±
−
cos(kz − ωt).
k
m
mk
The phase-space plot for this motion shows island structures with an island full
width w0
√¯
¯
¯ qE ¯
¯
¯
w0 = 2∆v = 4 ¯
mk ¯
defined by the separatrix at G = |qE/k|. The orbits within the separatrices
correspond to oscillating particles trapped in the wave potential. The orbits
outside the separatrices correspond to passing particles which are not reflected
by the wave potential.
Fg. 4. Phase-space plot of particle motion in a single plane wave.
When there are two plane waves with ω/k ̸= ω1 /k1
m
dv
= qE sin(kz − ωt) + qE1 sin(k1 z − ω1 t).
dt
For small values of E and E1 , effects of these wave on particle motion are nearly
independent.
Fg. 5. Phase-space plot of particle motion for two small-amplitude waves.
For larger values of E and E1 such that the trapped-particle islands overlap,
the trajectories fill up the stochastic volume between the outermost separatrices
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of the two sets of islands. The main points are summarized as follows. (a)
Resonance between the perturbation field E1 and the harmonics of particle
motion leads to the formation of secondary trapped-particle islands. (b) Many
secondary island chains appear on each side of a primary separatrix, and overlap
of these secondary island chains produces a stochastic layer of finite thickness.
(c) The fraction of phase space in the primary island occupied by stochasticity
varies roughly as E1 exp[−2πω2 /(kw0 )].
The superposition of many plane waves of the same amplitude and wavelength, evenly spaced in phase velocity, is described by
dx
dt
= v
dv
dt
= ϵ2
N
∑
cos(x − nt) → 2πϵ2 cos x
∞
∑
δ(t − 2πn)
n=−∞
n=−N
where the limit N → ∞ is taken. This can be represented by the Standard (or
Chirikov-Taylor) Mapping
um+1
= um + vm
vm+1
= vm − 2πϵ2 sin(2πum+1 ).
For small values of ϵ, this mapping shows invariant surfaces, with the appearance
of primary islands of full width 4ϵ centered on the lines v = n, where n is
an integer. For larger values of ϵ, the primary islands grow and approach an
overlap condition. Secondary island chains start to become visible. Significant
stochasticity appears around ϵ = 0.18, and is very strong at ϵ = 0.25.
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Fg. 6. Standard Mapping for ϵ = 0.125. Vertical axis is v, and horizontal axis
is x. Stochastic area is very small.
Fg. 7. Standard Mapping for ϵ = 0.200. Stochastic area is large, and vertically
adjacent island chains are linked by stochasticity.
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