Fundamentals of Plasma Physics

8.6 Drift waves 257

which is true provided the ratioνei/ωceis sufficiently small for a given geometric factor
ky/k^2 zL. The imaginary part of the frequency is

ωi=−

Di

(∂Dr/∂ω)ω=ωr

=−

(ω−ω∗)τ‖

ω∗/ω^2 r

=ω∗

k^2 yρ^2 sω∗τ‖

(

1 +k^2 yρ^2 s

) 3 (8.135)

and has several important features:

1. ωiis positive so collisional drift waves arealways unstable.


2. ωiis proportional toτ‖so that modes with the smallestkzand hence the fastest par-
   allel phase velocities are the most unstable, subject to the provisothatωτ‖<< 1 is
   maintained. This increase of growth rate withk−z^1 or equivalently parallel wavelength
   means that drift waves typically have the longest possible parallel wavelength consis-
   tent with boundary conditions. For example, a linear plasma of finite axialextent with
   grounded conducting end walls has the boundary condition thatφ 1 = 0at both end
   walls;the longest allowed parallel wavenumber isπ/hwherehis the axial length of
   the plasma (i.e., half a wavelength is the minimum number of waves that can be fitted
   subject to the boundary condition).


3. Collisions make the wave unstable sinceτ‖∼νei.


4. ωi is proportional to the factork^2 yρ^2 s/(1+ky^2 ρ^2 s)^3 which has its maximum value when
   kyρsis of order unity.


5. ωiis proportional toL−^2 ;for a realistic cylindrical plasma (cf. Fig.8.3) the density
   is uniform near the axis and has a gradient near the edge (e.g., a Gaussian density
   profile wheren(r)∼exp(−r^2 /L^2 )). The density gradient is localized near the edge
   of the plasma and so drift waves should have the largest growth rate near the edge.
   Drift waves in real plasmas are typically observed to have maximum amplitude in the
   region of maximum density gradient.
   The free energy driving drift waves is the pressure gradient and so the driftwaves might
   be expected tend to deplete their energy source byflattening out the pressure gradient.
   This indeed happens and in particular, the nonlinear behavior of drift waves reduces the
   pressure gradient. Drift waves pump plasma from regions of high pressure to regions of
   low pressure. This pumping can be calculated by considering the time-averaged, nonlinear,
   x-directed particleflux associated with drift waves, namely

Γx=〈Ren 1 Reu 1 x〉=

1

2

Re (n 1 u∗ 1 x). (8.136)

The species subscriptσhas been omitted here, because to lowest order both species have
the sameu 1 xgiven by theE×Bdrift, i.e.,

u 1 x= ˆx·

−∇φ 1 ×B

B^2

=−ikyφ 1 /B. (8.137)

Being careful to remember thatω∗ is a real quantity (this is a conventional, but confus-
ing notation), Eqs.(8.121) and (8.137) are substituted into Eq.(8.136) to obtain the wave-
induced particleflux