Acoustic-gravity waves in a viscous and thermally conducting isothermal atmosphere (part II: For small Prandtl number) - Hadi Y. Alkahby (International Journal of Mathematics and Mathematical Sciences)
In part one of these series we investigated the effect of Newtonian cooling on acoustic-gravity waves in an isothermal atmosphere for large Prandtl number. It was shown that the atmosphere can be divided into two regions connected by an absorbing and reflecting layer, created by the exponential increase of the kinematic viscosity with height, and if Newtonian cooling coefficient goes to infinity the temperature perturbation associated with the wave will be eliminated. In addition all linear relations among the perturbation quantities will be modified. In this paper we will consider the effect of Newtonian cooling on acoustic-gravity waves for small Prandtl number in an isothermal atmosphere. It is shown that if the Newtonian cooling coefficient is small compared to the adiabatic cutoff frequency the atmosphere may be divided into three distinct regions. In the lower region the motion is adiabatic and the effect of the kinematic viscosity and thermal diffusivity are negligible, while the effect of these diffusivities is more pronounced in the upper region. In the middle region the effect of the thermal diffusivity is large, while that of the kinematic viscosity is still negligible. The two lower regions are connected by a semitransparent reflecting layer as a result of the exponential increase of the thermal diffusivity with height. The two upper regions are joined by an absorbing and reflecting barrier created but the exponential increase of the kinematic viscosity. If the Newtonian cooling coefficient is large compared to the adiabatic cutoff frequency, the wavelengths below and above the lower reflecting layer will be equalized. Consequently the reflection produced by the thermal conduction is eliminated completely. This indicates that in the solar photosphere the temperature fluctuations may be smoothed by the transfer of radiation between any two regions with different temperatures. Also the heat transfer by radiation is more dominant than the conduction process.