Fundamentals of Plasma Physics

494 Chapter 17. Dusty plasmas

17.6 Dust ion acoustic waves

The presence of dust grains can also substantially modify the propagationproperties of
conventional ion acoustic waves. In a conventional electron-ion plasma, ion acoustic waves
can propagate only ifTe>>Ti. This is because strong ion Landau damping occurs when
the wave phase velocity is of the order of the ion thermal velocity and this is impossible
to avoid unlessTe>> Tisince the conventional ion acoustic phase velocity scales as
c^2 s=(γκTi+κTe)/mi.
As in the conventional ion acoustic wave, the derivation of dust ion acoustic waves
involves the linearized electron and ion equations

0 = −ne 0 eE 1 −κTe∇ne 1 (17.50)

ni 0 mi

∂ui

∂t

= +ni 0 eE 1 −γκTi∇ni 1 (17.51)

but here the possibility of finiteTihas been retained in order to allow consideration of
theTe∼Tiregime. The wave frequency is assumed to be sufficiently high so that the
dust grains are unable to respond to the wave;the dust grains are effectively considered
as being infinitely massive and therefore stationary. In this limit the dust grains contribute
to the equilibrium quasineutrality conditionni 0 =ne 0 +Zdnd 0 but since the dust grains
are assumed stationary their density cannot change and so the linearized quasi-neutrality
condition isni 1 =ne 1 .Thus, infinitely massive dust grains affect the equilibrium electron
density, but not the perturbed electron density.
EliminatingE 1 between Eqs.(17.50) and (17.51) gives

ni 0 mi

∂ui

∂t

=−

ni 0

ne 0

κTe∇ne 1 −γκTi∇ni 1. (17.52)

Sinceni 1 =ne 1 andni 0 /ne 0 =ni 0 /(ni 0 −Zdnd 0 ), Eq.(17.52) becomes

nimi

∂ui

∂t

=−

1

1 −Zdnd 0 /ni 0

κTe∇ni−γκTi∇ni. (17.53)

Taking the divergence and using Eqs. (17.11) and (17.32) gives

mi

∂^2 ni 1

∂t^2

=

(

1

1 −α

κTe+γκTi

)

∇^2 ni 1 (17.54)

which gives the dust ion acoustic wave (Shukla and Silin 1992), a wave with phase velocity

c^2 DIA=

1

1 −α

κTe

mi

+γ

κTi

mi

. (17.55)

Asαapproaches unity, the dust ion acoustic wave phase velocity becomes muchgreater
than the ion thermal velocity even in a plasma havingTi=Te. The dust ion acoustic wave
thus propagates without being attenuated by ion Landau damping in aTi=Teplasma
havingα≃ 1 , which corresponds to the left side of Fig.17.2. Thus, the presence of a large
dust grain density enables the propagation of ion acoustic waves that normally would be
damped in aTe=Tiplasma;this has been observed in experiments by Barkan, D’Angelo
and Merlino (1996).