Fall Term 2003
Plasma Transport Theory, 22.616
Problem Set #6
Prof. Molvig
Passed Out: Nov. 18, 2003
DUE: Nov. 25, 2003
1. Ambipolar Potential in a Magnetized Plasma Column:
Compute the electrostatic floating potential, Φ, of a strongely magnetized plasma column,
ν ii ¿ Ωi , in steady state as required by ambipolarity. What is the sign of this potential?
How does the potential energy, eΦ, compare to the thermal energies, Te , and, Ti ? What is
the net rotation of the plasma in this steady state?
2. Self-Adjoint Property of Collision Operator:
Show that the linearized collision operator (for like-particle collisions),
¶
µ
³ ´ Γ ∂ Z
∂ b0
∂ b
3 0
0
b
· d v f0 f0 U ·
f−
f
Cl f =
2 ∂v
∂v
∂v0
with the usual definitions,
Γ ≡
is self-adjoint,
¢
¡
U v, v0 ≡
Z
4πe4 ln Λ
m2 µ
¶
1
(v − v0 ) (v − v0 )
I
−
|v − v0 |
|v − v0 |2
³ ´ Z
g Cl fb = d3 v fbCl (b
g)
d3 vb
for any two functions, fb, and, gb, that are reasonably well behaved and don’t diverge exponentially at infinity. The distribution, f0 , is the local Maxwellian.
3. Conservation Laws for Linearized Collision Operator:
Show that the linearized collision operator automatically satisfies the conservation laws for
any fb,
⎤
⎡
Z
1
³ ´
d3 v ⎣ mv ⎦ Cl fb = 0
1
2
2 mv
4. Ambipolarity and Impurity Diffusion:
We have seen that reasonable impurity levels can lead to a situation in which particle transport is dominated by ion-impurity collisions rather than ion-electron collisions. In this case,
classical transport is still ambipolar to leading order in the gyroradius, but this comes about
by opposite flows in the ion and impurity channels. Conventional wisdom has it that impurities flow IN and ions OUT, leading to the deliterious accumulation of impurities in the core
— a decidedly bad situation.
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It is, however, possible that a very good situation could result wherein ions flow IN and impurities OUT, leading to an ideal fueling situation for main ions and an impurity and/or
helium ash removal scheme (if impurities are alphas). Take what you know about ambipolarity and density and temperature profiles to either: 1) design a system where this good
arrangement happens or, 2) show that some basic constraints prohibits it and that the bad
situation invariably prevails if classical impurity transport dominates.
5. Diamagnetic Fluxes:
The first order, in ρ, distribution function takes the form,
µ
¶¶
µ 2
vy
mv
5
A1 + A2
−
fM
f1 =
Ω
2T
2
where the thermodynamic forces are defined in terms of pressure gradient and temperature
gradient,
A1 ≡ d ln P/dx
A2 ≡ d ln T /dx
Show that the diamagnetic flow is proportional to the pressure gradient alone,
Z
nmV∗y = d3 vmvy f1 ∝ A1
with no separate temperature gradient driven flow. Write the complete expression for nmV∗y .
6. Generalized Flux-Friction Relations:
Carry outRthe integrals and linearization of the collision operator to show that the energy flux
moment, d3 v 12 mv 2 v, of the leading order kinetic equation,
Ωv × b ·
∂
f ' C (f, f )
∂v
leads to the heat-flux, heat-friction relation,
Z
³ ´
1
ΩQx = d3 v mv 2 vy Cl fb
2
where, Qx , is the energy flux,
Qx ≡
Z
1
d3 v mv 2 vx f
2
and, fb, is the first order in ρ/a piece of the distribution, defined as,
³
´
f = f0 1 + fb
7. Like-Particle (Ion) Collision Fluxes:
Assume that the kinetic equation for the distribution of ion guiding centers can be written in
the form,
∂f
= C0 (f, f ) + C2 (f, f )
∂t
2
where C0 is the “velocity” operator,
Γ ∂
·
C0 (f, f ) ≡
2 ∂v
Z
3 0
d v U·
µ
∂
∂
−
∂v ∂v0
¶
ff 0
and C2 is the spatial diffusion operator (including the back reaction effect of the field particles),
¶
µ
Z
¢ ∂f 0
¡
∂
∂f 0
3 0
0
C2 (f, f ) =
d v D v, v
f −f
∂X
∂X
∂X
¡
¢
Γ
D v, v0 ≡
ey · U · ey
Ω2
Show that the transport equations result from the particle and energy moments of the second
order kinetic equation and are given by,
and,
∂
∂n
+
Γi = 0
∂t
∂X
¶
µ
Z
¢ ∂f0 0
¡
∂f00
3 3 0
0
Γi = − d vd v D v, v
f − f0
∂X 0
∂X
∙
¸
∂ 3
∂ 5
nTi +
Ti Γi + qi = 0
∂t 2
∂X 2
qi = −Ti
Z
3
3 0
d vd v
µ
mv 2 5
−
2Ti
2
¶
¢
¡
D v, v0
µ
∂f0 0
∂f 0
f0 − f0 0
∂X
∂X
¶
Finally carry out the calculations to show that, Γi = 0, for any values of, A1 = d ln Pi /dx,
and, A2 = d ln Ti /dx, and that the heat flux can be expressed as,
qi
= −A2
Ti
Z
3
¡
1
2
3 0 2 mv
d vd v
− 12 mv 02
2Ti2
¢2
¡
¢
D v, v0 f0 f00
In terms of the overall transport matrix,
∙
∙
¸
¸
¸∙
1 −D T12 d ln Pi /dx
Γi
=
qi /Ti
n T21 −χi d ln Ti /dx
you have just shown that for ion-ion collisions that, D = T12 = T21 = 0, with only, χi
non-zero.
Extra Credit: Do the integrals to recover result from class!
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