Fundamentals of Plasma Physics

252 Chapter 8. Vlasov theory of warm electrostatic waves in a magnetized plasma

8.6.1 Simple two-fluid model of drift waves

The plasma is assumed to have the exponential density gradient

n∼exp(−x/L) (8.100)

whereL is called the density-gradient scale length. We consider electrostatic, extremely
low frequency (ω<<ωci) waves having the potential perturbation

φ 1 = ̃φexp(ikyy+ ikzz−iωt). (8.101)

The parallel phase velocity is assumed to lie in the rangevTi<<ω/kz<<vTeso that
ions are adiabatic and electrons are isothermal (this is the same regime as ion acoustic
waves). The parallel and perpendicular wavelengths are both assumed to be much longer
than the electron Debye length, so that the plasma may be considered quasi-neutral and so
haveni≃ne.Sinceω/kz<<vTe,the parallel component of the electron equation of
motion is simply

0 ≃−qe

∂φ 1

∂z

−

1

ne

∂

∂z

(neκTe) (8.102)

which leads to a Boltzmann electron density,

ne=ne 0 exp(−qeφ 1 /κTe). (8.103)

Assumingφ 1 is small, linearization of this Boltzmann electron density gives the first-order
electron density
ne 1
ne 0

=−

qeφ 1

κTe

. (8.104)

Sinceω/kz >> vTi,ions may be considered cold to first approximation and the zero-
pressure limit of the ion equation of motion characterizes ion dynamics. Furthermore,
sinceω<<ωcithe lowest-order perpendicular ion motion is just theE×Bdrift so

ui 1 =

−∇φ 1 ×B

B^2

=−

ikyφ 1

B

x.ˆ (8.105)

Since theE×Bdrift is in thexdirection, it isin the direction of the density gradientand
so this drift leads to an ion density perturbation because of a convective interaction with
the equilibrium density gradient. The ion density perturbation is found by linearizing the
ion continuity equation

∂ni 1

∂t

+ui 1 ·∇ni 0 +ni 1 ∇·ui 1 = 0. (8.106)

After noting that∇·ui 1 = 0, substitution forui 1 in the convective term gives

ni 1

ni 0

=

kyφ 1

ωLB

. (8.107)

The density perturbation results from the ionE×Bdrift causing the ions with their density
gradient to slosh back and forth in thexdirection so that a stationary observer at a fixed
pointxsees oscillations of the ion density.
Quasi-neutrality means that the electron and ion density perturbations densities must be
almost exactly equal although electrons and ions are governed by entirely different physics.