Fundamentals of Plasma Physics

17.5 LargePlimit: dust acoustic waves 493

Substitution forni 1 andne 1 in Eq.(17.38) gives

nd 0 md

∂ud 1

∂t

= −κTi

1 −ne 0 /ni 0

1+

ne 0 Ti

ni 0 Te

Zd∇nd 1

≃−κTiZd∇nd 1 (17.45)

sincene 0 <<ni 0 and typicallyTi/Te<<ni 0 /ne 0.
The linearized dust continuity equation is

∂nd 1

∂t

+nd 0 ∇·ud 1 =0 (17.46)

so taking the divergence of Eq.(17.45) and substituting Eq.(17.46) gives

∂^2 nd 1

∂t^2

=

ZdκTi

md

∇^2 nd 1 (17.47)

which is a wave with phase velocity

c^2 da=

ZdκTi

md

. (17.48)

This wave is called the dust acoustic wave and its phase velocity is extremely low because
of the large dust grain mass.
This analysis could be extended to include finitekλDterms as obtained by using the
full Poisson’s equation instead of making the simplifying assumption of perfectneutrality.
A Vlasov approach could also be invoked to demonstrate the effect of Landau damping.
Rather than work through the details of these extensions, one can argue that theα≃ 1 dusty
plasma is effectively a two-component plasma where the heavy particles are the negatively
charged dust grains and the light particles are the positively charged ions. Thepreviously
derived results from both two-fluid theory and Vlasov theory could then be invoked by
simply identifying the heavy and light particles in the manner stated above. Thus by making
the identification,

ion acoustic wave dust acoustic wave

inertia provided by ions dust grains

restoring force provided by electron pressure ion pressure

taking into account finitek^2 λ^2 Dterms will result in a dispersive dust acoustic wave

ω^2 =

k^2 c^2 da

1+k^2 λ^2 Di

. (17.49)

Similarly, just as an electronflow velocity faster than the ion acoustic phase velocity would
destabilize ion acoustic waves via inverse Landau damping, a Landau analysis would show
that an ionflow velocity faster than the dust acoustic phase velocity would destabilize dust
acoustic waves.
Destabilized dust acoustic waves have been observed in an experiment byBarkan, Mer-
lino and D’Angelo (1995). The phase velocity of these waves was of the order of 10 cm/s
which is an order of magnitude less than a human walking speed.