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We present a dataset of numerical simulations of stratified turbulence under the Oberbeck-Boussinesq
approximation with Schmidt number equal to unity and periodic boundary conditions. An anisotropic time correlated forcing acting on the poloidal component of the velocity field is used in order to excite internal gravity waves. We vary the viscosity and the
\bv frequency in order to obtain flows at different horizontal Froude $F_h$ and the
buoyancy Reynolds number $\R$. The datas are availabe at \href{https://mycore.core-cloud.net/index.php/apps/files/?dir=/2022strat-turb-polo&fileid=2553105195}{https://mycore.core-cloud.net/index.php/apps/files/?dir=/2022strat-turb-polo\&fileid=2553105195}. Isotropy and mixing coefficients allows to distinguish between 5 regimes in the $F_h-\R$ parameters space. The spectral energy budgets are analyzed for simulations in each regime. \todo{improve abstract: add results...}
buoyancy Reynolds number $\R$. The datas are availabe at \href{https://mycore.core-cloud.net/index.php/apps/files/?dir=/2022strat-turb-polo&fileid=2553105195}{https://mycore.core-cloud.net/index.php/apps/files/?dir=/2022strat-turb-polo\&fileid=2553105195}. Isotropy and mixing coefficients allows to distinguish between 5 regimes in the $F_h-\R$ parameters space. The spectral energy budgets are analyzed for simulations in each regime.
\end{abstract}
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We performed direct numerical simulations of a stratified turbulent flows in a periodic domain, using a forcing mechanism designed to excite internal gravity waves. We added fourth order hyper-viscosity and hyper-diffusivity for most simulations to stabilize simulations. However, the resolution was increased sufficiently such that it has a negligible effect on the measures of considered quantities and spectra in the inertial range, except for some simulations at high $F_h$ and $\R$. We obtained a dataset of more than $40$ simulations spanning the $(F_h, \R)$ parameters space, which allows for parametric studies of stratified flows. We showed that regimes already predicted in the literature \cite{Brethouwer-Billant-Lindborg-Chomaz2007, Maffioli2016mixing, Garanaik-Venayagamoorthy2019mixing} can be classified using a large-scale isotropy coefficient, a small-scale isotropy coefficient, and the mixing coefficient. These turbulent regimes are: passive scalar, weakly stratified, optimal mixing, Layered Anisotropic Stratified Turbulence (LAST), and viscosity affected. \\
The optimal regime, occuring at $\R \geq 10$, delimits the weakly stratified and LAST regimes. It was already known than it occurs for an optimum of potential energy \cite{Maffioli2016mixing}, we also observed that this regime corresponds to the case where the large-scale isotropy coefficient is near $0.5$, and also corresponds to a minimum of toroidal energy. \todo{Check if it true with toroidal forcing}. Our simulations are consistent with the scaling predictions of \cite{Maffioli2016mixing, Garanaik-Venayagamoorthy2019mixing} for the mixing coefficient. However, we observed a different limit $\lim\limits_{F_h \rightarrow 0} \Gamma \simeq 0.42$ which is different from the value measured in \cite{Maffioli2016mixing} with a vertical vorticity, uncorrelated forcing. Also the position of the optimal mixing appeared to be slightly different ($F_h \simeq 0.13$ in our simulations vs $F_h \simeq 0.3$ in \cite{Maffioli2016mixing}). This enforced the idea that the forcing mechanism is important in the precise evaluation of the mixing in stratified flows \cite{Howland2020mixing}. \\
We performed direct numerical simulations of a stratified turbulent flows in a periodic domain, using a forcing mechanism designed to excite internal gravity waves. We added fourth order hyper-viscosity and hyper-diffusivity for most simulations to stabilize simulations. However, the resolution was increased sufficiently such that it has a negligible effect on the measures of considered quantities and spectra in the inertial range, except for some simulations at high $F_h$ and $\R$. We obtained a dataset of more than $40$ simulations spanning the $(F_h, \R)$ parameters space, which allows for parametric studies of stratified flows. We showed that regimes already predicted in the literature \cite{Brethouwer-Billant-Lindborg-Chomaz2007, Maffioli2016mixing, Garanaik-Venayagamoorthy2019mixing} can be classified using a large-scale isotropy coefficient, a small-scale isotropy coefficient, and the mixing coefficient. These turbulent regimes are: passive scalar, weakly stratified, Layered Anisotropic Stratified Turbulence (LAST), and viscosity affected. \\
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Following \cite{Yokoyama-Takaoka2019}, we performed a spectral analysis of each regime in the $(k_h, k_z)$ plane. In addition to the relative distribution of potential, poloidal, and toroidal energy, we also looked at the spectral energy budget. Our simulations gives slightly different results. It seems than the anisotropy of the modes, as well as the non-linearity parameter $\chi_{\kk}$ influence whether we observe a conversion of kinetic to potential energy or the converse, and delimits the region where toroidal energy is important. In the LAST regime, when $\chi_{\kk} \gg 3$, conversion to kinetic energy becomes negligible, and potential energy represents approximately half oh the total energy. On the contrary, when $\chi_{\kk} < 3$, the poloidal energy dominates and represents almost half of the total energy. Potential energy is converted into kinetic energy along the line $\chi_{\kk} = 3$, while kinetic energy is converted into potential energy for $k_z < k_h < k_{\rm O}$. Spectral energy budgets and slices of $(k_h,k_z)$ spectra supports the scenario of an inverse energy cascade of toroidal energy, which get stronger with stratification, as first observed in \cite{Smith2002generation} and investigated in \cite{Herbert2016cascade}. In the buoyancy range, the $(k_h, k_z)$ spectra are consistent with $\sim k_h^2 k_z^0$, suggesting the presence of internal gravity waves. However the toroidal energy remains of the same order than poloidal and potential energy, so eddies should play an important role in this range. \\
When the buoyancy Reynolds number is large ($\R \geq 10$), the optimal mixing occurs between the weakly stratified and the LAST regimes. This optimal mixing happens for a maximal potential energy ratio \cite{Maffioli2016mixing}, which is also a minimum of toroidal (i.e. vertical vorticity modes) energy ratio in our simulations. The dependency of the mixing ratio are consistent with the scaling predictions of \cite{Maffioli2016mixing, Garanaik-Venayagamoorthy2019mixing}. However, we observed a different limit $\lim\limits_{F_h \rightarrow 0} \Gamma \simeq 0.42$ which is different from the value measured in \cite{Maffioli2016mixing} with a vertical vorticity, uncorrelated forcing. Also the position of the optimal mixing appeared to be slightly different ($F_h \simeq 0.13$ in our simulations vs $F_h \simeq 0.3$ in \cite{Maffioli2016mixing}). This enforced the idea that the forcing mechanism is important in the precise evaluation of the mixing in stratified flows \cite{Howland2020mixing}. \\
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We have not explored the low $\R$ regime which might be interesting for geophysical applications. Other regimes of stratified turbulence might appear in this regime, presumably a WWT regime
Following \cite{Yokoyama-Takaoka2019}, we performed a spectral analysis of each regime in the $(k_h, k_z)$ plane for a simulation in the LAST regime. In addition to the potential, poloidal, and toroidal energy ratios, we also looked at the energy transfer, buoyancy, and dissipation rate ratios. We observe that conversion to potential energy becomes negligible when the parameter $\gamma_{\kk}$ (\ref{eq:gamma}) becomes greater than one. Dissipation is important when $\gamma_{\kk}$ is large mostly ar small vertical scales, or when eddie turnover times is comparable to the linear wave frequency $\chi_{\kk} \simeq 1$ (\ref{eq:chi}) and $k\geq k_{\rm b}$. This suggests two mechanisms for dissipation in stratified turbulence, occurring at different scales: dissipation by vertical shears at small vertical scales and wave-breaking at small horizontal scales. \\
To use the data do \mintinline{python}{hg clone https://foss.heptapod.net/fluiddyn/fluidsim}
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\caption{$(k_h, k_z)$ spectral energy budget for simulation L (table~\ref{table-simuls-regimes}). The magenta dotted line corresponds to $\gamma_{\kk} = 1$, the cyan dotted line to $\chi_{\kk} = 1$, and the cyan dashed line to $k = k_{\rm b}$. See equations (\ref{eq:transfers}), and (\ref{eq:chi}-\ref{eq:gamma}) for definitions. \label{fig:sebkhkz-L}}
\end{figure}
\todo{continue here}
Figure~\ref{fig:sebkhkz-L} represents the $(k_h, k_z)$ spectral energy budget for simulation L. The main differences with the optimal regime are: (i) the forcing region fully lies in the buoyancy range. (ii) The potential energy is less dominant at large $k_z$. (iii) Toroidal energy is now dominant at very large horizontal scales $k_h \ll k_{\rm b}$, supporting the fact that the toroidal energy tends to accumulate at small $k_h$ when we increases stratification \cite{Smith2002generation, }. Figure~\ref{fig:spectra-L} represents slices of $(k_h, k_z)$ spectra for simulation L. When compared to simulation O, the simulation L naturally has larger buoyancy and Ozmidov ranges since it has a lower $F_h$. Horizontal and vertical spectra can be dissociated clearly. In the buoyancy range, spectra are close to $k_h^2~k_z^0$ away from the forcing region, with almost an equipartition between potential, poloidal and toroidal energies. Slopes of vertical slices of the spectra decreases between $k_{\rm b}$ and $k_{\rm O}$. For $k_z > k_{\rm b}$, horizontal slices goes roughly as $k_h^{-5/3}$. At small-scales ($k_h$ or $k_z \gg k_{\rm O}$), spectra are consistent with an equipartition of energy, i.e. $~ k_h^1 ~k_z^0$, as in the weak, and optimal regimes (not shown). In the LAST regime, the dissipation occurs at wavevectors larger than $k_{\rm O}$. Yet, our resolution does not allow to attain sufficiently large $\R$ to observe an isotropic inertial range after $k_{\rm O}$.
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Figure~\ref{fig:sebkhkz-L} represents the $(k_h, k_z)$ spectral energy budget for simulation L. Despite the fact that we force the poloidal velocity, we observe that (i) the toroidal velocity is nowhere negligible, and (ii) the potential energy is dominant in the forcing region showing an efficient conversion to potential energy. Poloidal energy is dominant at large $\thk$ (i.e. $k_h \gg k_z$), while potential energy is dominant at small $\thk$ (i.e. $k_h \ll k_z$). The energy transfer, conversion to potential energy, and energy dissipation rate are clearly anisotropic in this strongly stratified simulation. We observe that the energy transfert $\tilde{\mathcal{T}}$ fluctuates a lot in the buoyancy range ($k \leq k_{\rm b}$), and around the critical balance line $\chi_{\kk} \simeq 1$ \cite{nazarenko_critical_2011}, taking positive or negative values. This might be due to interactions between waves and 3D eddies of similar frequencies in this region of spectral space. Small horizontal scales $k_h \geq k_{\rm O}$ is a region of convergence of energy fluxes. The conversion to potential energy $\tilde{\mathcal{B}}$ is naturally negligible when $\gamma_{\kk} \geq 1$, as we expect from the definition of $\gamma_{\kk}$. We observe that $\tilde{\mathcal{B}}$ tends to be whether positive (in the forcing region and at large $\thk$) or negative (at small $\thk$ outside the forcing region). The energy dissipation $\tilde{\varepsilon}$ seems to be due to two mechanisms: (i) For a given small $k_h$, dissipation starts to be important at $k_z$ such that $\gamma_{\kk} \simeq 1$. This suggests that dissipation due to vertical shear is responsible for the destruction of kinetic energy at these scales. (ii) For a given small $k_z$, dissipation starts to be important at $k_h$ such that $\chi_{\kk} \simeq 1$ (around the Ozmidov wave-vector $k_{\rm O}$). A plausible mechanism to explain this is the dissipation of 3D eddies generated by wave-breaking at the Ozmidov scale. Finally dissipation is important around the critical balance line $\chi_{\kk} \simeq 1$ outside the buoyancy range ($k \geq k_{\rm b}$), suggesting that wave-breaking may also appear in this region of spectral space
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