Generalized Eliassen-Palm and Charney-Drazin Theorems for Waves on Axismmetric Mean Flows in Compressible Atmospheres

Abstract

The theorems, which exhibit the role of wave dissipation, excitation and transience in the forcing of mean flow changes of second order in wave amplitude by arbitrary, small-amplitude disturbances, are obtained 1) for the primitive equations in pressure coordinates on a sphere, and 2) in a more general form (applicable for instance to nonhydrostatic disturbances in tornadoes or hurricanes) establishing that no approximations beyond axisymmetry of the mean flow are necessary. It is shown how the results reduce to those found by Boyd (1976) for the case of sinusoidal, hydrostatic waves with exponentially growing or decaying amplitude, and it is explained why the approximation used by Boyd in the thermodynamic equation is not needed. The reduction to Boyd's results entails the use of a virial theorem. This theorem amounts to a generalization of the ?equipartition? law derived in an earlier paper (Andrews and Mclntyre, 1976). That derivation appeared to rely on an assumption about relative phases of disturbance Fourier components; the present derivation shows that no such assumption is in fact necessary.