Fundamentals of Plasma Physics

8.6 Drift waves 253

Equating the electron and ion perturbed densities, respectively given by Eqs.(8.104) and
(8.107), provides the most basic form of the drift wave dispersion relation,

ω=−

kyκTe

qeLB

=kyude, (8.108)

where
ude=−κTe/qeLB (8.109)
is the electron diamagnetic drift velocity given by Eq.(8.97). Equation (8.108) describes a
normal mode where Boltzmann electron density perturbations caused by isothermal elec-
tron motion along field lines neutralize ion density perturbations causedby ionE×B
motion sloshing thex-dependent equilibrium density profile in thexdirection. This ba-
sic dispersion relation provides no information about the mode stabilitybecause collisions
and Landau damping have not yet been considered. The wave phase velocityω/kyis equal
to the electron diamagnetic drift velocity given by Eq.(8.98). This dispersion relation has
the interesting feature that it does not depend on the mass of either species;this is because
neither the electron Boltzmann dependence nor the ionE×Bdrift depend on mass. This
basic dispersion is the building block for the more complicated models to be considered
later, and it is conventional to define the ‘drift frequency’

ω∗=kyude (8.110)

which will appear repeatedly in the more complicated models. The basic dispersion is
therefore simplyω=ω∗(the asterisk should not be confused with complex conjugate).

8.6.2 Two-fluid drift wave model with collisions

The next level of sophistication involves assuming that the plasma is mildly collisional so
that the electron equation of motion is now

me

due

dt

=qe(−∇φ+ue×B)−

1

ne

∇(neκTe)−νeime(ue−ui). (8.111)

The collision frequency is assumed to be small compared to the electron cyclotron fre-
quency. Becauseω/kz<<vTe,the electron inertia term on the left hand side is negligible
compared to the electron pressure gradient term on the right hand side and sothe parallel
component of the electron equation reduces to

0 =−qe

∂φ

∂z

−

κTe

ne

∂ne

∂z

−νeimeuez; (8.112)

the ion parallel velocity has been dropped since it is negligible compared to the electron
parallel velocity. After linearization and assuming perturbed quantities have a space-time
dependence given by Eq.(8.101), Eq.(8.112) can be solved to give the parallel electron
velocity

uez=−

ikzκTe

νeime

(

qeφ 1

κTe

+

ne 1

ne 0

)

. (8.113)

It is seen that in the limit of no collisions, this reverts to the Boltzmann result, Eq.(8.104).
As in the previous section, quasi-neutrality is invoked so that the dispersion relation is
obtained by equating the perturbed electron and ion densities. Equation (8.113) together