Turbulence generation by large-scale extreme vertical drafts and the modulation of local energy dissipation in stably stratified geophysical flows

We observe the emergence of strong vertical drafts in direct numerical simulations of the Boussinesq equations in a range of parameters of geophysical interest. These structures, which appear intermittently in space and time, generate turbulence and enhance kinetic and potential energy dissipation, providing a possible explanation for the observed variability of the local energy dissipation in the bulk of oceanic flows and the modulation of its probability distribution function. We show how, due to the extreme drafts, in runs with Froude numbers observable in geophysical scenarios, roughly $10\%$ of the domain flow can account for up to $50\%$ of the global volume dissipation, reminiscent of estimates based on oceanic models.

Figure 1

Kurtosis $K_w$ of the vertical velocity $w$ as a function of time, in units of the turnover time ${\tau }_{\mathrm{NL}}$, for runs $P5$ (top panel), $P4$ (middle panel), as well as (bottom panel) $P8$ (solid line) and $P9$ (dashed line). Red and blue identify respectively the portions of the signal across relative maxima and minima used to compute the spectra in Fig. 2. $T^{*}\approx 290{\tau }_{\mathrm{NL}}$ is the reference time used in Fig. 3 (see above top panel). The inset in the middle panel shows the small-scale kinetic (V, dashed line) and potential (P, solid line) energies at late times for run P5, obtained through the integration of the corresponding spectra from $k=30$ to $k_{\max }\simeq 512/3$.

Figure 2

Kinetic (top) and potential (bottom) energy spectra computed for run $P5$ at the relative maxima (red) and minima (blue) of $K_w$ (see Fig. 1). Insets show $\alpha$- or $\beta$-compensated spectra respectively for $E_V$ or $E_P$, with $\alpha ,\beta$ either $5/3$ or 2.

Figure 3

Left: PDFs of the kinetic energy dissipation computed in the regions indicated by the shaded areas in the middle panel for run $\mathrm{P}5$. Middle: Profile of the normalized kinetic energy dissipation ${\widehat{\varepsilon }}_V(z,t)$ in individual horizontal planes, at fixed time $T^{*}\approx 290{\tau }_{\mathrm{NL}}$ (see Fig. 1), and as a function of $z$. Right: Vertical profile of the by-plane kurtosis $K_w(z,t)$.

Figure 4

Temporal evolution in units of ${\tau }_{\mathrm{NL}}$ for run $P5$ of (top) the vertical profile of the by-plane kurtosis $K_w(z,t)$, and (middle) the normalized kinetic energy dissipated in horizontal planes ${\widehat{\varepsilon }}_V(z,t)$. The bottom panel gives the joint PDF of $K_w(z,t)$ and ${\widehat{\varepsilon }}_V(z,t)$ for all P5 data, the solid red line indicating averages of $\widehat{{\varepsilon }_V}$ conditioned on $K_w$. Inset: Evolution of the distance correlation coefficient dCor between $K_w(z,t)$ and ${\widehat{\varepsilon }}_V(z,t+\varphi )$, as a function of a temporal lag $\varphi$ between the two signals. Note that the highest correlation obtains for $\varphi \approx {\tau }_{\mathrm{NL}}/3$ (not for $\varphi =0$), thus indicating causation.

Figure 5

Kinetic and potential energy dissipation efficiency as a function of $K_w$ (for high-Re runs P1–P7), measured in terms of the minimal domain volume percentage $V_{{\varepsilon }_V}(\%)$ and $V_{{\varepsilon }_P}(\%)$ needed to achieve $50\%$ of respectively the global kinetic and potential energy dissipation.

Figure 6

Time evolution of the vertical gradients of the scaled density fluctuations, ${\partial }_z\zeta$, and of two different combinations of the velocity field gradients ($Q$ and $R$, two quantities relevant for the evolution of fluid elements and of dissipation in reduced Euler models [73, 74]), in a reduced model of field gradients in stratified flows. Parameters are compatible with those of run P5.