Reflection and dissipation of hydromagnetic waves in a viscous and thermally conducting isothermal atmosphere

Abstract

Abstract In this paper we consider two limiting cases of the vertical propagation of hydromagnetic waves in a viscous and thermally conducting isothermal atmosphere permeated by a uniform horizontal magnetic field. It will be assumed that the viscosity, thermal conductivity, and magnetic field strength are small, but that the effect of thermal conduction dominates in the first case, while in the second case it is the effects of the viscosity and the magnetic field which dominate. Since the equilibrium density decreases exponentially with height, all the diffusivities increase exponentially. In the first case it is convenient to consider the atmosphere divided into three distinct regions. In the lowest region the thermal diffusivity, kinematic viscosity and Alfvén speed are small whereas in the uppermost region all these quantities are large. In the middle region the thermal diffusivity is large while the Alfvén speed and the kinematic viscosity are still small. Therefore, in the lowest region the motion is adiabatic and in the middle region it is isothermal. The exponential increase of the thermal diffusivity with height transforms the oscillatory process from an adiabatic one in the lowest region, to an isothermal one in the middle region and creates a semitransparent reflecting layer between the two regions. The upper reflecting layer, which connects the two upper regions, is of a different type because it acts as an absorbing and reflecting layer. The existence of two reflecting layers will influence the reflection process in the two lower regions and the final conclusions depend on the relative effects of the kinematic viscosity with respect to that of the Alfvén speed and on the relative locations of the two reflecting layers. In the second case the viscosity and the magnetic field combine to create a reflecting and absorbing layer. Thus there are two regions connected by a reflecting and absorbing layer. Below this layer all the diffusivities are small and the motion is adiabatic, while above it the motion will decay to a constant value before it is influenced by the effects of the thermal conduction. Reflection coefficients are derived for both cases and it is shown that resonances will occur.