Resolving the Paradox of Oceanic Large-Scale Balance and Small-Scale Mixing

A puzzle of oceanic dynamics is the contrast between the observed geostrophic balance, involving gravity, pressure gradient, and Coriolis forces, and the necessary turbulent transport: in the former case, energy flows to large scales, leading to spectral condensation, whereas in the latter, it is transferred to small scales, where dissipation prevails. The known bidirectional constant-flux energy cascade maintaining both geostrophic balance and mixing tends towards flux equilibration as turbulence strengthens, contradicting models and recent observations which find a dominant large-scale flux. Analyzing a large ensemble of high-resolution direct numerical simulations of the Boussinesq equations in the presence of rotation and no salinity, we show that the ratio of the dual energy flux to large and to small scales agrees with observations, and we predict that it scales with the inverse of the Froude and Rossby numbers when stratification is (realistically) stronger than rotation. Furthermore, we show that the kinetic and potential energies separately undergo a bidirectional transfer to larger and smaller scales. Altogether, this allows for small-scale mixing which drives the global oceanic circulation and will thus potentially lead to more accurate modeling of climate dynamics.

Figure 1

(a) Horizontal cut of vertical vorticity: one observes large-scale filaments and small-scale gradients, together with intense localized vortex streets as seen, e.g., for $x=1700$ and $y=300$. (b) Time evolution of integral scales based on velocity (circles) and temperature (triangles) for the same run. After an initial transient phase, the former grows significantly in time, while the latter has a slow growth.

Figure 2

(a) Compensated kinetic energy spectra with (in the inset) the time evolution of kinetic energy (solid line) and of its (normalized) dissipation (dashed line); $30{\tau }_{\text{NL}}\approx {600}^{+}{N}^{-1}$ for $\mathrm{Fr}\approx 0.047$, $N/f=7$, and $\mathrm{Re}\approx 2\times {10}^4$. The large scales follow a $\sim k^{-5/3}$ spectrum, whereas at small scales, $E_V(k)\sim k^{-2.5}$; the spectra cross at $k_F\approx 10.5$. (b) Total energy flux, normalized by ${\epsilon }_V=⟨\mathbf{u}·{\mathbf{F}}_v⟩$, for $2\le N/f\le 10.5$ and runs with similar Froude and Reynolds numbers ($0.045\le \mathrm{Fr}\le 0.047$, $\mathrm{Re}\approx 2\times {10}^4$).

Figure 3

Scatter plots of $R_{\mathrm{\Pi }}$ as a function of (a) Fr and (b),(c) $\mathrm{Fr}\times \mathrm{Ro}$ in lin-log and log-log coordinates, respectively. (a) Points are labeled by their final ${\mathcal{R}}_B$; the six runs with ${2048}^3$ grids have $16500\le \mathrm{Re}\le 39000$ (black symbols) while the others use a ${1024}^3$ grid and $6400\le \mathrm{Re}\le 10000$ (blue symbols). (b) The six runs with low ${\mathcal{R}}_B$ shown in (a) with empty symbols are eliminated in (b) and (c); the same symbols are used, but colors now indicate three ranges for $N/f$. The vertical green bar gives a plausible interval of $R_{\mathrm{\Pi }}$ values for the ocean [12, 13]. The inset gives the slope of the variation of $R_{\mathrm{\Pi }}$ with $\mathrm{Fr}\times \mathrm{Ro}$ for various $N/f$. Error bars on $R_{\mathrm{\Pi }}$ are based on the standard deviations associated with the averages of the fluxes over about a decade of scales. (c) Scatter plot of the ratio of kinetic energy fluxes (green symbols) or potential energy fluxes (black symbols) $R_{{\mathrm{\Pi }}_{V,P}}$ for flows with a bidirectional energy transfer with negative flux for $k<k_F$ and positive for $k>k_F$. The inset shows energy fluxes for velocity (solid line) and temperature (dashed line), for the same flow as in Fig. 2 ($N/f=7$).