Formation of small scales via Alfven wave propagation in compressible nonuniform media

In weakly dissipative media governed by the magnetohydrodynamics (MHD) equations, any efficient mechanism of energy dissipation requires the formation of small scales. In a previous paper (Paper I) the possibility of producing small scales has been numerically studied in the case of MHD disturbances propagating in an incompressible and inhomogeneous medium, for a strictly two-dimensional slab geometry, the ignorable coordinate being the one that is perpendicular both to the magnetic field and to the inhomogeneity direction. In this paper we extend the work of Paper I, to study the properties of Alfvén waves propagating in a compressible medium, in the presence of an inhomogeneity transverse to the direction of wave propagation. The study has been performed using a 2.5-dimensional pseudospectral numerical code. The simulations show that, when an Alfvén wave propagates in a compressible nonuniform medium, the two dynamical effects responsible for the formation of small scales in the incompressible case are still at work: energy pinching and phase mixing. Moreover, the interaction between the initial Alfvén wave and the inhomogeneity gives rise to the formation of compressible perturbations (fast and slow waves or a static entropy wave). Some of these compressive fluctuations are subject to the steepening of the wave front and become shock waves, which are extremely efficient in dissipating their energy, their dissipation being independent of the Reynolds number. A rough estimate of the typical times which the various dynamical processes take to produce small scales and then to dissipate the energy show that for Reynolds numbers of the order of S = 108 or greater, the steepening of compressive fluctuations can represent an efficient mechanism to dissipate the wave energy and do not require large values of the wave amplitude.