A Generalized Transformed Eulerian Mean (TEM) Based on Non-neutral Waves in a Boussinesq Stratified Fluid

A general transformed Eulerian mean (TEM) set of equations are derived based on non-neutral (unstable/dissipative) waves including those excited by eddy forcing and/or diabatic heating in a Boussinesq stratified fluid. All the 9 components of the 3-dimensional eddy diffusion tensor are derived. The symmetric part of the eddy diffusion tensor is proportional to a growth rate of non-neutral waves, so that the sign can be negative when and where dissipative waves are dominant. The divergence of the inhomogeneous symmetric part as well as the anti-symmetric part contributes as Stokes drift, although the former induced velocity filed is generally divergent while the latter induced is exactly non-divergent by definition. Although the generalized TEM is derived assuming a WKB wave form and the Hilbert transformation is needed for the eddy forcing and heating terms, no explicit information of wave frequency and wave number is required for the application to real atmospheric or ocean data analyses. Some novel points of the generalized TEM are illustrated by using high-frequency sampled-data from a GCM experiment.