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Next Section: The Dispersion Relation Title/Abstract Page: Alfvén Waves in Dusty Previous Section: Introduction | Contents Page: Volume 14, Number 2 |
As in Pilipp et al. (1987) we consider small-amplitude waves in a static
uniform molecular cloud
consisting of neutral atomic and molecular species, the ionized atomic and
molecular species, the electrons, neutral dust grains, and negatively
charged dust grains. The cloud is assumed cold, so that the gas pressures
of all the species may be neglected. A 4-fluid model of the plasma is used, which employs the linearized
fluid momentum equations for plasma ions (singly charged), electrons,
neutral molecules and charged dust grains: (We neglect the motion of
neutral dust grains that was included
by Pilipp et al. (1987))
where 
is the wave electric field, 
is the species mass and 
is the species velocity in the wave.
is the collision frequency of a particle of species s with the particles
of species t. We have neglected electron inertia and momentum exchange between
ions and electrons, but have included ion and neutral molecule inertia terms
because we are mainly interested in the frequency regime above the dust
cyclotron frequency, where the ion and neutral molecule dynamics are important.
To complete the system of equations, Maxwell's equations ignoring the
displacement current are used, with the conduction current density given by
where equilibrium charge neutrality is expressed by (1).
The background magnetic field 
is
assumed to be in the z-direction, and the steady electron, ion and dust
densities are 
,
and 
. The parameter 
measures the charge imbalance in the
plasma, with the remainder of the charge residing on the dust particles, so
that the total system is charge neutral.
The equations are linearized, so that since we can define the transverse
direction of wave field variation to
be the x-axis, without loss of generality the wave fields are assumed to
vary as 
. Thus our analysis
differs from that of Pilipp et al. (1987) in that oblique
propagation (i.e. non-zero 
) is allowed for.
We assume for simplicity that
the charge on the dust particles is not affected by the wave, i.e. we
neglect the dust charging effects discussed by Vladimirov (1994a,b).
Define the Alfvén speed
, where 
, the
ion-cyclotron frequency 
and the dust-cyclotron
frequency 
.
Eliminating and 
,
and using the assumed time dependence, we obtain, for 
,
where 
is the wave magnetic field, 
and 
.
The neglect
of gas pressure in all species implies that 
.
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Next Section: The Dispersion Relation Title/Abstract Page: Alfvén Waves in Dusty Previous Section: Introduction | Contents Page: Volume 14, Number 2 |
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