Log-Normal Turbulence Dissipation in Global Ocean Models

Data from turbulent numerical simulations of the global ocean demonstrate that the dissipation of kinetic energy obeys a nearly log-normal distribution even at large horizontal scales $\mathcal{O}(10\mathrm{km})$. As the horizontal scales of resolved turbulence are larger than the ocean is deep, the Kolmogorov-Yaglom theory for intermittency in 3D homogeneous, isotropic turbulence cannot apply; instead, the down-scale potential enstrophy cascade of quasigeostrophic turbulence should. Yet, energy dissipation obeys approximate log-normality—robustly across depths, seasons, regions, and subgrid schemes. The distribution parameters, skewness and kurtosis, show small systematic departures from log-normality with depth and subgrid friction schemes. Log-normality suggests that a few high-dissipation locations dominate the integrated energy and enstrophy budgets, which should be taken into account when making inferences from simplified models and inferring global energy budgets from sparse observations.

Figure 1

Probability distribution functions (PDFs) for the logarithm of dissipation (symbols), and for the best-fit Gaussian function for each set of dissipation values (curves). The dissipation PDFs are shown for the surface (black), 500 m depth (red), and 3000 m depth (blue).

Figure 2

(Top) Map of surface dissipation. (Bottom left) Surface dissipation map for a $30°\times 30°$ region denoted by a black box. (Bottom right) PDFs of ${\log }_{10}(ϵ)$, as in Fig. 1, for the global ocean (blue), the $30°\times 30°$ region (black), and the $10°\times 10°$ region (red). Also shown are the range of the mean and amplitude of Gaussian fits for the nine $10°\times 10°$ child regions within the black box [illustrating $\log ({\eta }_n)$ distributions from (4)].

Figure 3

Skewness (left) and kurtosis (right) of the global ${\log }_{10}(ϵ)$ PDF as a function of the depth for several simulations. For the QG harmonic simulation, the ${\log }_{10}(ϵ)$ statistics for both January and July are shown, as well as the statistics of the QGPE dissipation [${\log }_{10}(\eta )$; gray crosses]. The statistics are consistently near the Gaussian values.