Fall Term 2003
Plasma Transport Theory
Take Home FINAL EXAM
Prof. Molvig
Passed Out: Dec. 10, 2003
DUE: Dec. 15, 2003
1. Ware Pinch Effect: Compute the radial pinch velocity of trapped particles in two different
ways. Show that it is the same for all pitch angles. You may take the toroidal electric field to
be constant as a function of poloidal angle (as opposed to the inverse major radius dependence
that we used in class).
• Use the toroidal invariant,
Ivk
c
pφ =
−ψ
q
Ω
and the relation for the inductive electric field,
Eφ = −
1∂
Aφ
c ∂t
to show that the bounce average flow velocity, in ψ, is,
d
hψib = −cR0 Eφ
dt
Translate this into a spatial velocity.
• Use the toroidal invariant to compute the bounce average flux surface, ψ = hψib , without
electric field and then computer its change according to Vlasov operator,
¿
À
Bφ
d
q
hψib =
Eφ
b · ∇v ψ
dt
m
B
b
2. Magnetization Bootstrap Current: Make a drawing to show how the radial gradient
of banana centers can produce a parallel current. Show that this current is in the positive
toroidal direction so as to enhance the required confining current. Give a simple scaling
argument to show that the toroidal current density, in simple physical units may be expected
to scale like,
ρpe 3/2
²
Jφ ≈ qe nvT e
a
Get the same result by evaluating the integral we looked at in class,
¿ 2 µ
¶À
Z
I
2E
B
c
e
1−λ
dEdλwJ
f0
T31 = − T
2π
T
R2 B 2
B0
TR
b
You can be clever to avoid the energy integral and carry out the bounce average as a poloidal
angle integral using the circular tokamak approximation and retaining only leading order in
²1/2 terms.
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3. Simplified Implicit Transport Coefficient: The implicit transport coefficient for the
pinch effect (a collisional process involving circulating particles) can be written,
XZ
∂
i
dEdλf0 Dλψ g3
T13
=−
∂λ
σ
where, ∂g3 /∂λ, can be written,
∂
σewJ λ
g3 = −
q
∂λ
Dλλ τ b
where, q, is the geometric safety factor. The generalized Fokker-Planck coefficients, Dλλ , Dλψ ,
can be written,
¿
µ
¶À
B
B
Dλλ = 4ν ei (v) λ
wJ
1−λ
B0
B0
b
À
¿
I
mcI 0 ∂ω b
V
Dλλ + σν ei (v) 2λ vk
Dλψ = σ
2πe
∂λ
Ω J
Show that the transport coefficient can be written,
3 i c
i
nψ
= T I13
T13
4
2π
i , can be expressed,
where the dimensionless integral, I13
Ã
!
Z 1−²
1
∂ω
b
i
H
I13
+
= 4π
dλλ
B
∂λ
2 dθ
0
u B0 (1 − λB/B0 )
where the dimensionless velocity, u, and “bounce” frequency, ωb , are given as,
p
u ≡
1 − λB/B0
I
dθ
ωb ≡
u
i ∼ ²1/2 , and calculate an actual number.
Now evaluate the integrals to show that I13
Hint:
µ
¶
2²
π
E
'
1+²−λ
2
Where E is the complete elliptic integral of the second kind.
If this hint is really too obscure you may talk to Yong. . .
4. Diagonal Transport Coefficients: Prove that the implicit diagonal transport coefficients,
Tiii = − (αi , gi ), are positive definite, by actually carrying out the integration by parts to
write integral with positive definite integrand.
5. Onsager Symmetry of Transport Coefficients: Express the implicit off-diagonal transport coefficients, Tiji = − (αi , gj ), in a form that is manifestly symmetric by writing out the
integral in terms of the kinetic cross coefficient, Dλψ , appearing in the first order collision
operator,
∂
∂
∂
∂
Dψλ f0 +
Dλψ
f0
C1 =
∂ψ
∂λ
∂λ
∂ψ
New helpful comment: You need to show that the coefficient used to compute flux from first
order distribution function is the same coefficient used to compute the first order distribution
itself (by inverting zero order collision operator).
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