An Exact Thickness-Weighted Average Formulation of the Boussinesq Equations

Abstract The author shows that a systematic application of thickness-weighted averaging to the Boussinesq equations of motion results in averaged equations of motion written entirely in terms of the thickness-weighted velocity; that is, the unweighted average velocity and the eddy-induced velocity do not appear in the averaged equations of motion. This thickness-weighted average (TWA) formulation is identical to the unaveraged equations, apart from eddy forcing by the divergence of three-dimensional Eliassen–Palm (EP) vectors in the two horizontal momentum equations. These EP vectors are second order in eddy amplitude and, moreover, the EP divergences can be expressed in terms of the eddy flux of the Rossby–Ertel potential vorticity derived from the TWA equations of motion. That is, there is a fully nonlinear and three-dimensional generalization of the one- and two-dimensional identities found by Taylor and Bretherton. The only assumption required to obtain this exact TWA formulation is that the buoyancy field is stacked vertically; that is, that the buoyancy frequency is never zero. Thus, the TWA formulation applies to nonrotating stably stratified turbulent flows, as well as to large-scale rapidly rotating flows. Though the TWA formulation is obtained by working on the equations of motion in buoyancy coordinates, the averaged equations of motion can then be translated into Cartesian coordinates, which is the most useful representation for many purposes.