Geophysical Turbulence and the Duality of the Energy Flow Across Scales

The ocean and the atmosphere, and hence the climate, are governed at large scale by interactions between pressure gradient and Coriolis and buoyancy forces. This leads to a quasigeostrophic balance in which, in a two-dimensional-like fashion, the energy injected by solar radiation, winds, or tides goes to large scales in what is known as an inverse cascade. Yet, except for Ekman friction, energy dissipation and turbulent mixing occur at a small scale implying the formation of such scales associated with breaking of geostrophic dynamics through wave-eddy interactions or frontogenesis, in opposition to the inverse cascade. Can it be both at the same time? We exemplify here this dual behavior of energy with the help of three-dimensional direct numerical simulations of rotating stratified Boussinesq turbulence. We show that efficient small-scale mixing and large-scale coherence develop simultaneously in such geophysical and astrophysical flows, both with constant flux as required by theoretical arguments, thereby clearly resolving the aforementioned contradiction.

Figure 1

(a) Horizontal ($xy$) and (b) vertical ($xz$) two-dimensional cuts of the vertical velocity for Run 15a at the latest time, with ${\mathcal{R}}_B\approx 80$ and a small-scale spectrum slightly steeper than a Kolmogorov law. The axes are labeled in terms of grid spacing, and the forcing scale corresponds to roughly 145 in these units. Observe the large-scale structures, with a size of up to a third of the overall flow (or more in the filaments), arising from the inverse cascade, together with superimposed intense small scale eddies (e.g., at $x\approx 200$, $y\approx 1100$ in the top figure). Notice also the different structures in the two cuts, indicative of the persistent anisotropy of the flow.

Figure 2

(a) Kinetic energy spectra for Run 10d (red line), 10e (blue line), and 15a (black line), all with $N/f=2$ and increasing ${\mathcal{R}}_B={\mathrm{ReFr}}^2$. The straight lines with different power laws are given as indications. In the bottom inset are shown the temporal evolution of the kinetic energy for the same runs (solid lines), together with their (scaled) dissipation (dashed lines) $5\times 2\nu ⟨|\omega {|}^2⟩$, with $\omega =\nabla \times \mathbf{u}$ the vorticity. The spectra, not averaged in time, are shown at $t/{\tau }_{NL}\sim 22$, whereas the peak of dissipation occurs for all the runs around $t/{\tau }_{NL}\sim 1.3$, time after which the energy starts to grow, with ${\tau }_{NL}=L_F/U_0$ the turnover time. (b) Total (kinetic plus potential) energy fluxes normalized by energy input ${ϵ}_V=⟨\mathbf{u}·\mathbf{F}⟩$ for the same runs, as well as for runs 10a (magenta dashed line), 10b (green dashed line), and 10c (cyan dashed line) for which $N/f=4$.