Fundamentals of Plasma Physics

14.4 Assignments 427

Show that the steady-state solution to this equation has the form (Fisch1978)

FT∼exp







−

∫w

dw

(2+Z)w

(

w^3 D ̄QL+

3

4

)









SketchFTfor the cases where (i)D ̄QL>> 1 over a certain range of velocities
(i.e., where wave spectrum is resonant with particles) and (ii)D ̄QL<< 1 .How
could this be used to drive a toroidal current in a tokamak? Using the appro-
priate form ofDQLshow that the normalized quasilinear diffusion for resonant
particles is

D ̄QL(w)=^8 π

(^2) E(ωr/v,t)
wmeω^2 pelnΛ

.

If the wave is electrostatic then show that

E(ωr/v,t)∼k|φ|^2

wherekis the wavenumber of the waves resonantly interacting with the particles
so that

D ̄QL(w)∼^8 π

(^2) k|φ| 2
mewω^2 pelnΛ

.

What sort of waves would be most efficient at driving current, large or smallk?
Would this current drive work best in high or low density plasmas?