Joint downscale fluxes of energy and potential enstrophy in rotating stratified Boussinesq flows

We employ a coarse-graining approach to analyze nonlinear cascades in Boussinesq flows using high-resolution simulation data. We derive budgets which resolve the evolution of energy and potential enstrophy simultaneously in space and in scale. We then use numerical simulations of Boussinesq flows, with forcing in the large-scales, and fixed rotation and stable stratification along the vertical axis, to study the inter-scale flux of energy and potential enstrophy in three different regimes of stratification and rotation: (i) strong rotation and moderate stratification, (ii) moderate rotation and strong stratification, and (iii) equally strong stratification and rotation. In all three cases, we observe constant fluxes of both global invariants, the mean energy and mean potential enstrophy, from large to small scales. The existence of constant potential enstrophy flux ranges provides the first direct empirical evidence in support of the notion of a cascade of potential enstrophy. The persistent forward cascade of the two invariants reflects a marked departure of these flows from two-dimensional turbulence.

💡 Research Summary

The paper presents a comprehensive investigation of simultaneous downscale cascades of kinetic‑potential energy and potential enstrophy in rotating, stably stratified Boussinesq turbulence. Using a coarse‑graining (low‑pass filtering) framework, the authors derive exact, scale‑local budget equations for the large‑scale energy and for the quadratic invariant “potential enstrophy” (the square of potential vorticity, PV). The filtering operation introduces sub‑grid stress tensors that represent the influence of eliminated small‑scale motions; the associated sub‑grid fluxes, (\Pi_\ell) for energy and (\Pi^Q_\ell) for potential enstrophy, are defined so that they are Galilean invariant, vanish when the filter scale reaches the maximum resolved wavenumber, and therefore constitute true measures of inter‑scale transfer.

Three direct‑numerical simulations (DNS) are analyzed, each performed on a 640³ grid with eighth‑order hyper‑viscosity and large‑scale stochastic forcing concentrated at wavenumber (k_f\approx4). The simulations explore three asymptotic regimes of the nondimensional parameters: (i) strong rotation, moderate stratification (Rs, (f/N\gg1)), (ii) moderate rotation, strong stratification (rS, (f/N\ll1)), and (iii) equally strong rotation and stratification (RS, (f=N)). In all cases the mean potential enstrophy (\langle Q\rangle) is shown to be overwhelmingly dominated (≥97 %) by the quadratic part of PV, i.e., the linear PV approximation (q = f\partial_z\theta - N\omega_z). This satisfies the condition identified by Kurien, Smith, and Wingate (2006) under which PV becomes effectively quadratic and viscous effects are confined to the smallest scales.

The authors compute space‑averaged fluxes (\langle\Pi_\ell\rangle) and (\langle\Pi^Q_\ell\rangle) as functions of the filter scale (\ell) (or equivalently wavenumber (k)). In the inertial‑range band (k\approx6)–30, both fluxes are positive and essentially constant, indicating a steady forward cascade of energy and of potential enstrophy from the forcing scales down to the dissipative scales. The constancy of (\langle\Pi^Q_\ell\rangle) provides the first direct empirical confirmation of the “2/3‑law” derived analytically by KSW06, which predicts a constant potential‑enstrophy flux in the linear‑PV regime. This result overturns earlier findings (e.g., Herring et al., 1994) that potential enstrophy, being quartic in the general case, is contaminated by viscous diffusion at all scales and cannot sustain an inertial range.

The simultaneous presence of constant energy and constant potential‑enstrophy fluxes distinguishes rotating‑stratified Boussinesq turbulence from two‑dimensional Navier‑Stokes turbulence, where the dual cascade (inverse energy, forward enstrophy) is a hallmark. Here, despite the existence of two quadratic invariants, both cascade forward, reflecting the three‑dimensional nature of the flow even under strong rotation or stratification.

Methodologically, the paper demonstrates the power of the coarse‑graining approach: it yields exact, locally defined fluxes that respect physical symmetries and can be evaluated pointwise in space, unlike traditional spectral shell‑to‑shell transfers that average over the entire domain. The authors also discuss the implications for large‑eddy simulation (LES) modeling: sub‑grid models must be capable of reproducing not only the energy flux but also the potential‑enstrophy flux to faithfully represent the dynamics of geophysical flows.

In summary, the study (1) validates the theoretical prediction of a forward, scale‑local cascade of potential enstrophy in the linear‑PV regime, (2) shows that energy and potential enstrophy cascade together with constant fluxes across a substantial inertial range in all three regimes of rotation and stratification, and (3) provides a robust framework for analyzing and modeling multiscale interactions in rotating‑stratified turbulence, with direct relevance to atmospheric and oceanic dynamics.