Free Internal Waves in Polytropic Atmospheres
  Free internal waves in polytropic atmospheres are studied (polytropic atmosphere is such one that the temperature of gas linearly depends on altitude). We suppose gas to be ideal and incompressible. Also, we regard the atmosphere of constant height with the "rigid lid" condition on its top to filter internal waves. If temperature, density and pressure of such undisturbed atmosphere do not depend on latitude and longitude then the internal waves are harmonic with apriori unknown eigenfrequencies, the problem permits separation of variables and reduces to the system of two ODE's. The first ODE (the Laplace's tidal equation) is analyzed by author earlier. The second ODE determines the vertical structure of the waves to be considered and has analytical solution for polytropic atmospheres. There are 6 dimensionless numbers, 2 for the Laplace's tidal equation and 4 for the vertical structure equation. The solution is a countable set of the eigenfrequencies and eigenfunctions of the vertical structure equation; every eigenfrequency/eigenfunction corresponds to its own countable set of the eigenfrequencies and eigenfunctions of the Laplace's tidal equation. Parametric analysis of the problem has been done. It shows that there exists the solution weakly depending on altitude-temperature variations and the atmosphere's height for parameters modelling the Earth's troposphere (with the "rigid lid" between the troposphere and the tropopause). The natural periods of internal waves have been obtained for this case. 
Mikhail I. Ivanov
astro-ph.EP physics.ao-ph