The Averaged Hydrostatic Boussinesq Ocean Equations in Generalized Vertical Coordinates

Due to their limited resolution, numerical ocean models need to be interpreted as representing filtered or averaged equations. How to interpret models in terms of formally averaged equations, however, is not always clear, particularly in the case of hybrid or generalized vertical coordinate models, which limits our ability to interpret the model results and to develop parameterizations for the unresolved eddy contributions. We here derive the averaged hydrostatic Boussinesq equations in generalized vertical coordinates for an arbitrary thickness-weighted average. We then consider various special cases and discuss the extent to which the averaged equations are consistent with existing ocean model formulations. As previously discussed, the momentum equations in existing depth-coordinate models are best interpreted as representing Eulerian averages (i.e., averages taken at fixed depth), while the tracer equations can be interpreted as either Eulerian or thickness-weighted isopycnal averages. Instead we find that no averaging is fully consistent with existing formulations of the parameterizations in semi-Lagrangian discretizations of generalized vertical coordinate ocean models such as MOM6. A coordinate-following average would require "coordinate-aware" parameterizations that can account for the changing nature of the eddy terms as the coordinate changes. Alternatively, the model variables can be interpreted as representing either Eulerian or (thickness-weighted) isopycnal averages, independent of the model coordinate that is being used for the numerical discretization. Existing parameterizations in generalized vertical coordinate models, however, are not always consistent with either of these interpretations, which, respectively, would require a three-dimensional divergence-free eddy tracer advection or a form-stress parameterization in the momentum equations.