Wave-Activity Conservation Laws for the Three-Dimensional Anelastic and Boussinesq Equations with a Plane-Parallel Background Flow

Wave-activity conservation laws are key to understanding wave propagation in inhomogeneous environments. Their most general formulation follows from the Hamiltonian structure of geophysical fluid dynamics. On the large scales, the Eliassen-Palm wave activity is a well-known example and is central to theoretical analysis. On the mesoscale, while such conservation laws have been worked out in two dimensions (x-z), their application to a plane-parallel background flow in three dimensions fails because of a degeneracy created by the absence of a background potential vorticity gradient. Consideration of a background state depending only on altitude is motivated by the parameterization of unresolved scales in climate models where there is an imposed separation of horizontal length and time scales, but vertical coupling within each column. Here we show how this degeneracy can be overcome and wave-activity conservation laws derived for three-dimensional disturbances. Explicit expressions for pseudoenergy and pseudomomentum in the anelastic and Boussinesq models are derived, and it is shown how the previously derived relations for the two-dimensional anelastic and Boussinesq models can be treated as a limiting case of the three-dimensional problem. The relationship E = cP between pseudoenergy E and pseudomomentum P, where c is the horizontal phase speed, has important applications to gravity-wave parameterization.