## 1. IntroductionMagnetohydrodynamic (MHD) waves play an important role in the dynamics of various astrophysical plasma systems. They are key building elements in theories of acceleration of the solar and stellar winds, coronal heating and the formation of inhomogeneities in and support of molecular clouds. Also, in recent times, MHD waves have been observed in the solar wind and the solar corona. Therefore, investigation of the main features of MHD wave dynamics is a significant part of plasma astrophysics. The behaviour of MHD waves in homogeneous plasmas is well understood. However, in astrophysical situations, the wavelength is often of the same order as the characteristic spatial scale of some macro parameters of the plasma (e.g. density, magnetic field, temperature, steady flow). Interaction of MHD waves with plasma inhomogeneities generates a number of very interesting physical phenomena, such as mode coupling, appearance of trapped modes and guided propagation, wave dispersion, resonant absorption and phase mixing (see, e.g. Roberts 1991 and Goossens 1991), which can dramatically affect the MHD wave dynamics and, in particular, cause enhanced dissipation and heating of astrophysical plasmas. Alfvén wave phase mixing has been studied extensively as a possible mechanism for coronal heating (Heyvaerts & Priest 1983; Browning 1991; Malara et al. 1996; Nakariakov et al. 1997, 1998; De Moortel et al. 1999). Briefly, the idea of the mechanism is simple: when the medium has a density gradient perpendicular to the magnetic field, the Alfvén speed is a function of the transverse coordinate. Consequently, on each magnetic field line, Alfvén waves propagate with their local Alfvén speed. After a certain time, perturbations by Alfvén waves of neighbouring magnetic field lines become out of phase, i.e. there is phase mixing. This effect leads to the generation of smaller and smaller transverse spatial scales. The generation of transverse gradients in the wave leads to a strong increase in the dissipation of Alfvén wave energy due to viscosity and/or resistivity, because the dissipation is proportional to the wave number squared. Throughout this paper we will refer to this model of Alfvén wave phase mixing, which ignores compressible effects and nonlinear wave coupling, as the classical phase mixing model. For a compressible plasma, an alternative sink of Alfvén wave energy in an inhomogeneous plasma is conversion to the magnetosonic mode, and this mode can propagate across the magnetic field. Alfvén waves are subject to dissipation due to the shear component of the viscosity tensor, while the fast magnetosonic modes are subject to dissipation due to the viscosity tensor's volume component. In the near collisionless plasma of the corona the volume component of the viscosity is much higher than the shear component. As a result fast magnetosonic waves can be dissipated much faster than Alfvén waves and the coupling of Alfvén into fast waves may be an efficient mechanism for dissipating Alfvén wave energy. This process has been called indirect heating of the plasma through phase mixing (Nakariakov et al. 1997). According to weakly nonlinear theory the amplitude of this fast wave component grows linearly in time. Also, because the fast waves propagate across magnetic field lines they can give rise to heating of the plasma away from the initial phase mixing region. This effect leads to thermal transport across the magnetic field, which is very important in the thermodynamics of strongly magnetized plasmas where the usual thermal conduction across the field is depressed. Similar effects were found for phase mixing on inhomogeneous steady flows in plasmas (Nakariakov et al. 1998). In addition, in the almost collisionless plasmas of solar and stellar corona, the obliquely propagating compressive waves can be dissipated by Landau damping (see, e.g. Wentzel 1989). The analytical results of Nakariakov et al. (1997), showing the secular () growth of the fast wave perturbations during the initial stage of the phase mixing development, have been obtained under the simplifying assumptions of no dissipation, zero plasma beta and weakly nonlinear coupling. Also, magnetosonic perturbations were assumed to be initially absent from the system. The last assumption restricts the analysis to the early stage of the system's evolution, when the magnetosonic waves are of sufficiently low amplitude. The analytical treatment of the developed stage of phase mixing has not yet been undertaken. However, useful information on MHD wave behaviour can also be obtained by full MHD numerical simulation. There have been several numerical studies undertaken in this direction. Malara et al. (1996) have, through other interesting phenomena, found nonlinear generation of fast magnetosonic waves in the phase mixing region (their runs 7-9). The efficiency of the generation has been estimated as proportional to the square of the Alfvén wave amplitude. However, the relatively low Reynolds numbers (2000) used in the simulations, did not allow the authors to simulate significant phase mixing and, so, to observe the effect of indirect heating. The same effect has also been observed by Ofman & Davila (1997) numerically simulating nonlinear Alfvén waves in coronal holes. On the slopes of the density inhomogeneity associated with the coronal hole, the generation of fast waves has been clearly seen (see, e.g., Fig. 2 of the paper). Also the Lundquist number, the measure of dissipation of short scale perturbations, was low () in the simulations. Poedts et al. (1997) studied both phase mixing and resonant dissipation in the Solar corona. These simulations showed phase mixing in expanding coronal holes (see Fig. 4 of the paper) but they did not run long enough to reach the strongly phase mixed regime or study nonlinearly generated waves. In this study, we consider the developed stage of Alfvén wave phase mixing, aiming to determine if the amplitude of the generated fast mode continues to grow linearly or saturates. In addition, effects of weak finite plasma- are incorporated into the model. We perform 2.5D, time dependent numerical simulations of the ideal MHD set of equations, using the Lagrangian-Eulerian remap technique. The simulations allow us to investigate the effects of the sharp transverse structuring generated by phase mixing. © European Southern Observatory (ESO) 2000 Online publication: December 5, 2000 |