Measuring fluxes between wave and geostrophic features in rotating non-hydrostatic flows with variable stratification

A challenge in physical oceanography is quantifying the energy content of waves and balanced flows and the fluxes that connect these reservoirs with their sources and sinks. Methodological limitations have prevented decompositions for realistic flows with non-hydrostatic motions and variable stratification. We present a framework that separates the flow into wave and geostrophic components using the principle that waves have no Eulerian available potential vorticity signature. Starting from new expressions for available energy and potential vorticity conservation, we construct a basis of wave and geostrophic modes, complete and orthogonal with respect to quadratic approximations of the conserved quantities. Using the resulting non-hydrostatic projection operators, the nonlinear equations of motion are expressed as coupled wave and geostrophic equations, quantifying cascade and transfer fluxes of wave and geostrophic energy. We apply the method to non-hydrostatic mid-ocean simulations with geostrophic mean-flow, near-inertial, and tidal forcing. From these experiments, we construct source-sink-reservoir diagrams for exact and quadratic fluxes, quantifying the fluxes between geostrophic and wave components. Because the cascade fluxes obey total energy conservation, we construct energy flow diagrams within the wave and geostrophic reservoirs and diagnose nonlocal transfers. The simulations show a geostrophic inverse cascade, a forward wave cascade, and a direct transfer of geostrophic to wave energy, with no indication of a forward geostrophic cascade. The mean-flow-only simulation shows weak spontaneous wave emission during spin-up, which diminishes to zero. Finally, we evaluate the decomposition by comparing linearized and fully conserved available potential vorticity, finding that errors become significant at scales below 15\,km.