This paper is devoted to the study of the gravitational instability of a medium permeated by a uniform magnetic field along which a circularly polarized Alfven wave propagates. We concentrate on the case of perturbations purely transverse to the ambient field by means of direct numerical simulations of the MHD equations and of a linear stability analysis performed on a moderate amplitude asymptotic model. The Alfven wave provides an extra stabilizing pressure when the scale of perturbations is sufficiently large or small compared with the Jeans length $L_J$. However, there is a band of scales around $L_J$ for which the Alfven wave is found to have a destabilizing effect. In particular, when the medium is stable in absence of waves, the gravitational instability can develop when the wave amplitude lies in an appropriate range. This effect appears to be a consequence of the coupling between Alfven and magnetosonic waves. The prediction based on a WKB approach that the Alfven wave pressure tensor is isotropic and thus opposes gravity in all directions is only recovered for large amplitude waves for which the coupling between the different MHD modes is negligible.
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