8.7 Assignments 263
There are many more varieties of drift waves besides the basic versionsdiscussed here.
The common feature is that the gradient of pressure perpendicular to the magnetic field
provides a new mode and free energy from the pressure gradient or from parallelcurrent
can be tapped by this mode so that the mode is spontaneously unstable.
8.7 Assignments
- Electrostatic ion cyclotron waves: Using the electrostatic hot plasma dispersion rela-
tion show that there exists a mode in the vicinity of the ion cyclotron frequency having
the dispersion
ω^2 =ω^2 ci+k^2 ⊥c^2 s (8.153)
Show that this mode can be driven unstable by an axial current. - Mode conversion: Work through the algebra of the linear mode conversion problem
and plot the contours of Ref(p)for both positive and negativeξ. Hint: It is easier to
plot the contours off(p)if one uses polar coordinates in the complexpplane so that
p=reiθ. Then the real part of the termp^3 is just Re(r^3 e3iθ) =r^3 cos(3θ).Choose
a set of contour paths forξ< 0 which satisfy the boundary condition that there are no
modes growing exponentially with increasing distance fromξ= 0.Then determine
which combination of these contours does not give an inward propagating hot plasma
wave whenξ> 0 (this was another boundary condition). - Prove that the distribution function
fσ 0 (x,v) =
n 0 σ
π^3 /^2 v^3 Tσ
exp
[
−(v−uσzzˆ)^2 /v^2 Tσ−(vy/ωcσ+x)/L−
(
vT
2 ωcL
) 2 ]
provides a mean drift in theydirection corresponding to the diamagnetic drift.