Fundamentals of Plasma Physics

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

1. 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.


2. 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).


3. 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.