Introduction

The physics of dusty plasmas has recently been studied intensively because of its importance for a number of applications in space and laboratory plasmas (Goertz 1989, Mendis and Rosenberg 1994). Dusty plasmas are characterized as a low-temperature ionised gas whose constituents are electrons, ions and micron-sized dust particulates. The latter are usually negatively charged due to the attachment of the background plasma electrons on the surface of the dust grains via collisions. The presence of dust particles (grains) changes the plasma parameters and affects the collective processes in such plasma systems. In particular, the charged dust grains can effectively collect electrons and ions from the background plasma. Thus in the state of equilibrium the electron and ion densities are determined by the neutrality condition which is given by
 
where is the concentration of plasma electrons (with the charge -e), ions (for simplicity, we consider singly charged ions), and dust particles, respectively; the charge of the dust particle can vary significantly depending on plasma parameters. Dust can be a major component of interstellar molecular clouds, and the effects of the dust on small amplitude magnetohydrodynamic waves (Pilipp et al. 1987) and on shock waves (Pilipp and Hartquist 1994) in the clouds have been investigated previously. For interstellar molecular clouds this charge is negative (i.e., ). For HII regions there can be several hundred electrons per grain, while for HI regions there are only a few per grain (Spitzer 1978).

In this paper we consider the effect of the charged dust on the propagation of low frequency hydromagnetic waves in a magnetized interstellar cloud. It is important to know the speeds and damping rates of the hydromagnetic waves in interstellar clouds because the waves transport angular momentum during cloud collapse to form protostars. There is some possible direct evidence of these hydromagnetic waves in such clouds, since they have been postulated to be the cause of the observed large widths of CO emission lines from molecular clouds (Arons and Max 1975). Recently Balsara (1996) has investigated the damping of hydromagnetic waves in a partially ionised but dust-free self-gravitating plasma. It is of interest to know the effect of dust in the clouds on the speeds and damping rates of the waves. Alfvén wave propagation parallel to the magnetic field in a dusty interstellar cloud was first investigated by Pilipp et al. (1987), and was applied to the problem of wave propagation in the dustless and dusty regions of stellar outflows such as Ori (Havnes et al. 1989). It was argued that waves which propagate with no losses out to the zone of dust formation are dissipated rapidly in the dusty region. More recently, Rao (1993) and Shukla and Rahman (1996) focussed on the very low frequency waves that are affected by the motion of the dust grains themselves and by non-zero pressures, while Mendis and Rosenberg (1992) and Shukla (1992) considered the circularly polarized electromagnetic waves propagating parallel to the magnetic field in a plasma with static dust grains, in particular the case of frequency much less than the ion-cyclotron frequency. Birk et al. (1996) have derived generalized magnetohydrodynamic equations for partially ionized dusty plasmas including thermal pressure gradient terms, and have used them to investigate low-frequency Alfvén-like modes and tearing instabilities, in particular for dusty plasmas with a large proportion of the negative charge on the dust grains. The effects of dust on the propagation of shock waves in interstellar clouds, including the rotation of the magnetic field in an obliquely propagating shock, have been considered by Pilipp and Hartquist (1994).

Even if the proportion of negative charge on the dust grains compared to that carried by free electrons is quite small (typically in interstellar clouds), it can have a large effect on hydromagnetic Alfvén waves propagating at frequencies well below the ion-cyclotron frequency, as was shown by Pilipp et al. (1987). In particular, the right hand circularly polarized mode experiences a cutoff due to the presence of the dust. In this paper we generalize the previous work by considering the general dispersion relation for waves in a magnetised plasma with dust grains, propagating at an arbitrary angle with respect to the magnetic field. The wave frequencies are supposed to be much less than the ion-cyclotron frequency. The character of the waves at frequencies less than the ion-cyclotron frequency is radically altered when a proportion of the negative charge resides on the dust grains, in analogy with the case of electromagnetic waves in plasmas in solids with unequal electron and hole numbers (Baynham and Boardman 1971). With a negligibly small charge on the dust grains, the waves have the usual shear and compressional Alfvén wave properties, while for a non-zero charge on the grains the waves are better described as circularly polarized whistler or helicon waves extending to low frequencies (Mendis and Rosenberg 1994).

Waves that propagate obliquely to the magnetic field in a nonuniform plasma can encounter the Alfvén resonance, where the wavenumber perpendicular to the magnetic field direction becomes infinite (in the limit of zero electron temperature) and wave energy can be absorbed. This process is believed to be responsible for the heating of the solar corona (Ionson 1978), and has been used as an auxiliary heating mechanism for laboratory plasmas (Hasegawa and Chen 1976). Alfvén resonance absorption of waves in dusty interstellar molecular clouds could play a role in their energy balance and in the magnetic braking of protostellar clouds. We show here that the behaviour of the waves near the Alfvén resonance is strongly modified by the presence of dust.

The paper is organized as follows. In Sec. II the general dispersion relation for Alfvén waves propagating at arbitrary angles with respect to the external magnetic field is derived. In Sec. III, we present results of the dispersion relation for various situations and demonstrate the modification of the Alfvén resonance absorption in the presence of dust. In Sec. IV we briefly discuss the results obtained.

© Copyright Astronomical Society of Australia 1997