Propagation of Alfven waves in the dusty interstellar medium Can we neglect dust inertia in molecular clouds?

Context. Alfven waves are fundamental magnetized modes that play an important role in the dynamics of magnetized flows such as the interstellar medium (ISM).Aims. In a weakly ionized medium, their propagation critically depends on the ionization rate as well as on the charge carriers. Depending on the gas density, these may be ions, electrons, or dust grains. The latter are particularly well known to have a drastic influence on the magnetic resistivities in the dense ISM, such as collapsing dense cores. Yet, in most calculations, for numerical reasons, the grain inertia is usually neglected.Methods. We carried out an analytical investigation of the propagation of Alfven waves both in a single-size and multi-size grain medium such as the ISM and we obtained exact expressions giving wavenumbers as a function of wave frequencies. These expressions were then solved analytically or numerically by taking into account or neglecting grain inertia.Results. At long wavelengths, neglecting grain inertia is a very good approximation, however, the situation is rather different for wavelengths shorter than a critical value, which broadly scaled as 1/n, with n being the gas density. More precisely, when inertia is neglected, the waves do not propagate at short wavelengths or, due to the Hall effect, they develop for one circular polarization only, namely, a whistler mode such that Script capital R-e(& omega;) & PROP; k(2). The other polarization presents a zero group velocity, namely, Script capital R-e(& omega;) & PROP; k(0). When grain inertia is accounted for, the propagation of the two polarizations tend to be more symmetrical and the whistler mode is only present at density higher than & SIME;10(8) cm(-3). At a lower density, it is replaced by a mode having Script capital R-e(& omega;) & PROP; k(& SIME;1.2). Interestingly, one of the polarization presents a distribution, instead of a single & omega; value. Importantly, for short wavelengths, wave damping is considerably reduced when inertia is properly accounted for.Conclusions. To properly handle the propagation of Alfven waves at short wavelengths, it is necessary to self-consistently treat grain inertia. We discuss the possible consequences this may have in the context of diffuse and dense molecular gas regarding turbulence, magnetic braking, and protoplanetary disk formation as well as cosmic ray propagation in the dense ISM.