Vertical oscillations in a viscous and thermally conducting isothermal atmosphere

The presence of dissipation in an isothermal atmosphere may cause upwardpropagating small amplitude waves to be reflected downward. For an atmosphere with small dynamic viscosity μ this was demonstrated in Yanowitch (1967b); this will be referred to as case II. Here two problems will be investigated: (i) a thermally conducting atmosphere with small conductivity k (case III) and (ii) a viscous and thermally conducting atmosphere with small k and μ, and a small ratio μ/k, i.e. small Prandtl number (case IV). It will be shown that the validity of the model in case III is questionable. The solution for case IV is determined from the conditions that the average rate of energy dissipation and of entropy increase in a column of fluid be finite, but a radiation condition is required in case III. The solution for case III does not approximate the one for case IV uniformly, and the reflexion coefficient for case IV does not tend to the one for case III as the Prandtl number Pr → 0, but varies periodically with log Pr. Numerical results show that when the Prandtl number is not small the reflexion coefficient can be approximated by the asymptotic value obtained from case II.