Results

It is instructive to use (18) to reproduce the result of Pilipp et al.\ (1987) for parallel propagation with the effects of dust and collisions, and compare it with the case where collisions between particles are neglected. As in Pilipp et al. (1987), a dense molecular cloud is considered, in which the density of molecular hydrogen cm, , the magnetic field is G and the dominant ion species is . It is assumed that 1% of the mass is contained in spherical dust grains of radius cm composed of material with a mass density of 1 g cm. We then have . The temperature of the cloud is assumed to be 20 K, so that the thermal pressure is approximately of the magnetic pressure, and we are justified in neglecting the pressure gradient terms in (2)-(5). At very low frequencies and for highly electron density depleted plasmas (), dust-acoustic modes can couple to the Alfvén waves (Birk et al. 1996), but we do not consider such situations here.

We choose to plot the real part of the wavenumber (solid curve) and the imaginary part of the wavenumber (dotted curve) against the real frequency in Figures 1-6, as was done by Pilipp et al. (1987). We can thus determine the wavelength and damping length of the waves from a localized source oscillating at a given real frequency. The inverse determination of the real part of the frequency and the damping time of a wave of given real wavelength and angle of propagation is not so straightforward because the dispersion relation (17) is of eighth order in . However physically relevant solutions may be readily found numerically, and for lightly damped modes the ratio of imaginary to real frequency for real wavenumber is the same as the ratio of imaginary to real wavenumber for real frequency.

  
Figure 1: The wavenumber k plotted against frequency for propagation parallel to the magnetic field, for the right hand polarized mode. The frequency is normalized to , and the wavenumber to . The dust parameter is . (a) Collisions between species are included. (b) Collisions are neglected.

Figure 1a shows the plot of the real and imaginary parts of against frequency for the right hand polarized mode. The frequency is normalized to the dust cyclotron frequency (), and the wavenumber is normalized to (), where is the Alfvén speed using the total mass density of the ions, neutral molecules and dust grains. We also have . The collision frequencies are the same as used in Pilipp et al. (1987). Figure 1b shows the corresponding result when all the collision frequencies are set to zero. The wave is now either purely propagating or purely evanescent, with the cutoff and resonance obtained from (19) evident. With collisions included (Figure 1a), the wave is heavily damped between the resonance and cutoff frequencies. Figure 2 shows the corresponding dispersion relations for the left hand polarized mode. This mode does not experience the dust-induced resonance and cutoff behaviour.

  
Figure 2: As for Figure 1, but for the left hand polarized mode.

We turn now to the oblique propagation case, for propagation with the wavenumber k at an angle to the magnetic field, with and . We have from (17) the following quadratic equation in k describing the two independent elliptically polarized modes:
 

  
Figure 3: As for Figure 1, but for propagation at an angle of to the magnetic field.

Figure 3 shows the real and imaginary parts of k obtained from (20), plotted against frequency for , for the right hand polarized mode. With collisions included (Figure 3a), the real and imaginary parts of k are comparable in size for frequencies between the resonance and cutoff, i.e. the relative damping of the wave is less than for the parallel propagating wave. The left hand polarized mode at (Figure 4) does not show much difference to the parallel propagating case.

  
Figure 4: As for Figure 3, but for the left hand polarized mode.

If is fixed and there are no collisions, there is a resonance frequency at which , given from (17) by the condition
 
For no collisions and small this frequency is given to first order in by
 
which is the generalization to the dusty plasma of the well-known Alfvén resonance in dust-free plasmas (eg. Hasegawa and Chen 1976). There are also two cutoffs in determined by the vanishing of the two factors in the numerator of (17). In the collision-free case, and for small and , the cutoffs occur at , i.e. there is a cutoff-resonance-cutoff triplet of frequencies. (Cramer and Vladimirov 1996 discuss the collisionless case further).

  
Figure 5: The wavenumber perpendicular to the magnetic field plotted against frequency for fixed wavenumber parallel to the magnetic field , where the same normalizations as in Figure 1 are used. (a) Dust is present () and collisions between species are included. (b) Dust is absent () and collisions are included.

If collisions are included, there is no longer a pure resonance in at , but the real part of attains a maximum there. This is shown in Figure 5a, which is a plot of the real and imaginary parts of against frequency for fixed . For our parameters, . For comparison, Figure 5b shows the corresponding plot for the case of no dust (). Also for comparison, Figure 6 shows the corresponding plots for no collisions between the species: Figure 6a for dust present and Figure 6b for no dust. In this case the cutoff-resonance-cutoff behaviour of the dispersion relation which is absent for the dust-free plasma is clearly shown.

  
Figure 6: As for Figure 5, but collisions between species are not included.

We see from Figure 5 that the presence of dust still radically alters the dispersion relation in the vicinity of the Alfvén resonance in the presence of collisions: the cutoff-resonance-cutoff triplet is still discernible and strong absorption occurs over a wide range of frequencies about the Alfvén resonant frequency. Another view of the Alfvén resonance absorption mechanism is gained by considering a wave of fixed frequency and propagating into a plasma of increasing ion density in the x-direction. The wave will encounter the cutoff-resonance-cutoff triplet at successive spatial points in the density gradient, and wave energy will be absorbed at the resonance position where (22) is satisfied. It has been shown by Cramer and Vladimirov (1996) that in the collisionless case the resonance absorption in such a nonuniform plasma can be considerably enhanced by the presence of the dust, because the wave may be cutoff downstream of the resonance. The same will occur in the collisional case considered here, the only difference being that the resonance absorption occurs via the collisional damping processes.

© Copyright Astronomical Society of Australia 1997