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

TL;DR

The paper develops a wave–vortex decomposition grounded in available energy and APV to quantify energy fluxes between waves and geostrophic motions in rotating, non-hydrostatic flows with variable stratification. It constructs a complete, energetically orthogonal basis of eigenmodes and reformulates the nonlinear dynamics in wave–vortex space, enabling exact-like and quadratic energy flux budgets and triad analyses. Applying the method to realistic mid-ocean simulations with mean-flow, near-inertial, and tidal forcing reveals an inverse geostrophic cascade, a forward wave cascade, and a strong geostrophic-to-wave transfer, with no evidence for a forward geostrophic cascade. The framework provides diagnostics for energy pathways and offers a principled way to evaluate PV-based decompositions and closure-scale dynamics in complex stratified flows.

Abstract

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.

Figures (15)

• Figure 1: Vertical component of relative vorticity $\zeta$ for mean flow forcing only (MF, left) and mean flow & wave forcing (MFW, right) simulations under steady-state conditions.
• Figure 2: The area-averaged depth-integrated energy (top) and enstrophy (bottom) time series for the two simulations. The initial spin-up period at $256^2 \times 43$ resolution is 3000 days, at which time the resolutions are doubled to $512^2 \times 86$, as indicated by the dashed vertical line. The gray box highlights the period of steady-state analysis, from day 3050--3250.
• Figure 3: Rotary spectrum of the horizontal velocity field $u + i v$ at a synthetic mooring in the simulations.
• Figure 4: Energy sources, sinks and reservoirs for the MF and MFW simulations. Energy and energy fluxes shown in brackets are computed from the exact expression \ref{['eqn:volume-integrated-exact-energy']} and shown for each forcing and the total system energy. The energy and energy fluxes in and between the wave and geostrophic reservoirs are computed with the quadratic approximation and are shown without brackets. All values are averages over the analysis period and energy reservoirs include the gain or loss over the period in parentheses, which is required to close the energy flux budgets. Fluxes with less than $0.01$ GM/yr are not shown in the diagram.
• Figure 5: Energy sources (solid contours), sinks (dashed contours), and advective flux (arrows) in the MF (left) and MFW (right) simulations.