The linearized problem of two-dimensional gravity waves in a viscous incompressible stratified fluid occupying the upper half-space z > 0 is investigated. It is assumed that the dynamic viscosity coefficient μ is constant and that the density distribution ρ(z) is exponential. This leads to a fourth-order differential equation in the z co-ordinate, the coefficients of which depend on ρ(z) and on a dimensionless parameter ε which is proportional to μ/σ, σ being the frequency of the oscillation. The problem is solved for small ε. It is found that there is a region in which the solutions behave like certain solutions of the inviscid problem (with ε = 0). However, when the solutions of the inviscid problem are wave-like in z, they do not satisfy the radiation condition. This is because the viscosity, in addition to damping the motion for large z, reflects waves. The appropriate solution of the inviscid problem consists, therefore, of an incident and a reflected wave. As μ → 0, the ratio of the amplitudes of the reflected and the incident wave approaches exp (− 2π2H/Λ), where Λ is the vertical wavelength, and H the density scale height. The solution, however, does not have a limit since the reflecting layer shifts, altering the phase of the reflected wave. The results of the analysis are supplemented by a number of numerically computed solutions, which are then used to discuss the validity of the linearization.

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.

The problem considered is that of small amplitude vertical oscillations of a viscous isothermal atmosphere. The viscosity coefficient μ is assumed to be constant. It is shown that in any fixed z interval (z being the vertical coordinate) for frequencies below a critical frequency, the solution approaches (as μ → 0) the solution of the inviscid problem for which the kinetic energy in a column of fluid is finite. For frequencies higher than the critical one, however, viscosity not only damps the motion at large altitudes, but also reflects waves downward. The solution does not go to a Iimit as μ → 0, but the ratio of the amplitudes of the reflected and the incident wave approaches exp(−πβ), where β is the ratio of the vertical scale height to the wavelength. For small β, therefore, the solution of the inviscid problem satisfying the radiation condition is not accurate.

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