13.5 Assignments 397
(e) Show that a steady-state equilibrium can develop where, because of collisions
with background electrons and ions, the fast beam distribution has the form
FT∼exp
(
−
(2+Z)mev^2 z
3 κTe
)
.
Is this consistent with the original assumption that the beam is fast compared to
the background plasma?
- An axisymmetric plasma has a magnetic field which can be expressed as
B=
1
2 π
(∇ψ×∇φ+μ 0 I∇φ)
where /φis the toroidal angle,ψ(r,z)is the poloidalflux, andIis the currentflowing
through a circle of radiusrat axial positionz.
(a) Show that the toroidal component of the vector potential is
Aφ(r,z,t)=
1
2 πr
ψ(r,z,t)
(b) Assume that the plasma obeys the resistive Ohm’s law
E+U×B=ηJ
and assume that the plasma is stationary so that the toroidal componentis simply
Eφ=ηJφ.
and Eq.(9.42) to show that
Jφ=−
r
2 πμ 0
∇·
(
1
r^2
∇ψ
)
so that the toroidal component of Ohm’s law is
Eφ=−
rη
2 πμ 0
∇·
(
1
r^2
∇ψ
)
.
(c) Assuming classical resistivityη∼Te−^3 /^2 sketch the temperature dependence
of|Eφ|as given above and also sketch the temperature dependence ofEDreicer
as given by Eq.(13.85). For a plasma with givenψand physical dimensions,
in what electron temperature limit do runaway electrons develop (high or low
temperature)? For a given temperature, do runaways develop with highψor low
ψ?For a given temperature andflux, do runaways develop in a large device or
in a small device? If the plasma density decays, will runaways develop?
