Transverse oscillations of the incompressible MHD mode in the visco-resistive plasmas: an explanation of Alfvénic to Landau-type characteristics

Abstract

ABSTRACT In this paper, we characterize transverse oscillations as either Alfvénic or Landau-type in an incompressible non-ideal magnetohydrodynamic (MHD) fluid. We consider shear viscosity and magnetic diffusivity as dissipation mechanisms to derive a general dispersion relation for the incompressible MHD waves. The solutions of this dispersion relation for k as a function of ω – denoting by the source for any value of θ up to which magnetic tension acts as restoring force and dominates over internal friction forces – result in four roots, as follows. Two roots, which have a high phase velocity $c_{\rm A}\cos\theta $ are identified as almost undamped propagating Alfvén waves. The other two roots, which have a phase velocity $(2c_{\rm A}\cos\theta)/(\sqrt{\eta/\nu} + \sqrt{\nu/\eta})$, result in Alfvénic-type disturbances of a much shorter decay length than the wavelength. In contrast, when internal frictional forces start dominating over magnetic tension (i.e. for the propagation perpendicular to the background magnetic field, where the tension in the magnetic field becomes zero), the solutions of the dispersion are akin to Landau-type transverse oscillations. Transverse waves of this type were initially reported by Landau in an ordinary viscous fluid. However, our study corresponds to MHD visco-resistive fluid. The prediction for these lateral propagating transverse waves to be of Landau type may be very useful to explain the heating of observed filamentary structures across the magnetic field on a very small spatial scale in the solar coronal plasma, wherein the heating rate is directly proportional to the operating frequency of the driver, while its damping length is inversely proportional to the square root of the frequency.