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\begin{document}

\title{Regimes in stratified turbulence forced in vertical vorticity analyzed
from a new comprehensive open dataset}
\affiliation{Laboratoire des Ecoulements G\'eophysiques et Industriels, Universit\'e
Grenoble Alpes, CNRS, Grenoble-INP, F-38000 Grenoble, France}
\author{Vincent Reneuve}
\affiliation{Universit\'{e} C\^{o}te d'Azur, Observatoire de la C\^{o}te
d'Azur, CNRS, Laboratoire Lagrange, Nice, France.}
\author{Jason Reneuve}
\affiliation{Laboratoire des Ecoulements G\'eophysiques et Industriels, Universit\'e
Grenoble Alpes, CNRS, Grenoble-INP, F-38000 Grenoble, France}

\email[]{pierre.augier@univ-grenoble-alpes.fr}

\begin{abstract}


\end{abstract}

%----------------------------------------------------------------------------------------

% Print the title
\maketitle

%----------------------------------------------------------------------------------------
%	ARTICLE CONTENTS
%----------------------------------------------------------------------------------------

\section{Introduction}

\input{intro.tex}

\section{Numerical setup}
\label{sec:num}

The numerical simulations presented in this article are performed using the
pseudospectral solver \mintinline{python}{ns3d.strat} from the FluidSim Python package
\cite{fluiddyn,fluidfft,fluidsim}. Using this solver, we integrate in a periodic domain
of horizontal size $L_x = L_y = 3$ the three-dimensional Navier-Stokes equations under
the Boussinesq approximation:
\begin{align}
\p_t\vv + (\vv \cdot \bnabla)\vv = b\boldsymbol{e}_z - \frac{1}{\rho_0}\bnabla p +
\nu_2\nabla^2\vv + \ff_{\text{toro}},\label{ns} \\
\p_t{b} + (\vv \cdot \bnabla)b = -N^2 v_z + \kappa_2\nabla^2{b},\label{buoy}
where $\vv$ is the velocity, $p$ the pressure, $N$ the \bv frequency and
$b=-g\delta\rho/\rho_0$ the buoyancy, with $\rho_0$ the mean density and $\delta\rho$
the departure from the stable linear density stratification.
The aspect ratio of the numerical domain depends on the stratification strength.

Some modes in the Fourier space are disabled because they cause numerical and physical
problems and/or are not consistent with experiments in which the flow is bounded with
walls. (i) Dealiasing... (ii) Shear modes.... Finally, (iii) vertically invariant
vertical velocity (internal waves at $\omega = N$).

The term $\ff_{\text{toro}}$ is a large scale ($k_z = 0$ and $3 \leq k_h/\delta k_h
\leq 5$) time correlated toroidal forcing computed in spectral space such that the
kinetic energy injection rate is constant and equal to unity. In physical space, large
columnar vortices associated with vertical vorticity are constantly forced. In few time
units, a statistically stationary state is reached (remember that there is no shear
mode in these simulations). In this state, the time averaged total energy dissipation
rate $\eps$ is equal to the kinetic energy injection rate $P_K = 1$. The kinetic energy
dissipation rate $\epsK$ is just a function of the mixing coefficient $\Gamma = \epsA /
\eps$ and is in any case of order unity. By construction, there are transfers of energy
from the large forced scales to small dissipative scales.

Coarse, badly resolved simulations to reach the steady state.

For some simulations, a fourth-order hyperviscosity term is added. The fourth-order
viscosity $\nu_4$ is left as a free parameter and adapted to the resolution of
simulations in order to ensure that dissipative scales are well resolved. Similarly,
the equation of motion \eqref{buoy} for the buoyancy field $b$ presents both second and
fourth order diffusive terms, with corresponding diffusion coefficients $\kappa_2$ and
$\kappa_4$. We can then build two different Prandtl numbers
$\text{Pr}_i=\nu_i/\kappa_i$ for $i=2,4$. In all simulations, both those Prandtl
numbers are set to unity, such that $\kappa_2=\nu_2$ and $\kappa_4=\nu_4$. The use of
both normal and hyperviscosity is an important tool for the comparison with
experiments. More specifically, we use the measure of the turbulent kinetic
dissipations $\epsKK$ and $\epsKKKK$ based on both viscosities, and the ratio
$\epsKK/\epsK$ where $\epsK=\epsKK+\epsKKKK$, as an indicator of how close the
simulations we perform are to proper DNS. For a set of physical parameters, the needed
hyperviscosity decreases when the resolution is increased and the ratio $\epsKK/\epsK$
grows towards unity.

%% Method: simulations 1 couple (N, R_i)

\input{../tmp/table_methods_1couple.tex}

Table \ref{table-methods-1couple} shows ...

\todo{Include product kmax * Taylor microscale}

\begin{figure}
\centerline{
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_E_vs_time_N40_Ri20}
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_means_vs_kmaxeta_N40_Ri20}
}
\caption{(a) Energy versus time for simulations at different resolutions for $N=40$ and
$\R_i=20$. (b) Averaged quantities versus $\kmax\eta$ for the same simulations.
\label{fig:method-N40-R20}}
\end{figure}

\begin{figure}% [H]
\centerline{
\includegraphics[width=0.98\textwidth]{%
../tmp/fig_spectra_1couple}
}
\caption{Horizontal (a) and vertical (b) spectra for simulations at different
resolutions for $N=40$ and $\R_i=20$. \label{fig:method-N40-Ri20-spectra}}
\end{figure}

%% Method: resolution and hyperdiffusivity for the better simulations for each couple (N, R_i)

\begin{figure}
\centerline{
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_kmaxeta_vs_FhR}
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_epsK2overepsK_vs_FhR}
}
\caption{. \label{fig:method-resolution-hyperdiffusivity}}
\end{figure}

\input{../tmp/table_better_simuls.tex}

The remaining free parameters are the resolution $n_x\times n_y\times n_z$ and the
hyperviscosity $\nu_4$ which is adapted to the other parameters so that the dissipative
scales are well-resolved. The simulations were performed on a local cluster at LEGI for
resolutions up to $n_h = 640$ and on the national CINES cluster Occigen for finer
resolutions. Parameters and dimensionless numbers for each simulations are summarized
in table~\ref{table-better-simuls}. The turbulent nondimensional numbers are computed
from the statistically stationary flows as $F_h = \epsK / ({U_h}^2 N)$, $\R_2 = \epsK /
(\nu_2 N^2)$ and $\R_4 = \epsK{U_h}^2 / (\nu_4 N ^ 4)$, where $\epsK$ is the mean
kinetic energy dissipation and $U_h$ the rms horizontal velocity. The results presented
in this article are obtained from periods of the simulation when a steady state has
been approximately reached. Because the time scales of the flows studied here are very
long, finding such steady-state period can be very difficult and computationally
costly. In order to reach an approximately steady state in a reasonable time, we start
all the simulations at a reduced resolution $240\times240\times40$, and increase the
resolution step by step only when a sufficiently stationary state has been reached.
When such a state is observed, specific outputs are turned on and the simulation is ran
further for 10 to 20 minutes of equation time in order to produce substantial data to
analyze, before increasing the resolution again if needed.
\subsection{Large and small scale isotropy coefficients}

%% Large scale isotropy

\begin{figure}
\centerline{
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_isotropy_velo_vs_Fh}
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_isotropy_velo_vs_R}
}
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\caption{Large scale isotropy coefficient $I_{velo}$. \label{fig:large-scale-isotropy}}
Figure~\ref{fig:large-scale-isotropy} ...

%% Small scale isotropy

\begin{figure}
\centerline{
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_isotropy_diss_vs_Fh}
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_isotropy_diss_vs_R}
}
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\caption{Small scale isotropy coefficient $I_{diss}$. \label{fig:small-scale-isotropy}}
Figure~\ref{fig:small-scale-isotropy} ...

%% Isotropy coefficient: summary

\begin{figure}
\centerline{
\includegraphics[width=0.7\textwidth]{%
../tmp/fig_isotropy_coef_vs_FhR}
}
\caption{Large and small scale isotropy coefficients. Red letters correspond to
simulations of table~\ref{table-simuls-regimes} analyzed in
subsection~\ref{spectra-seb-regimes}. \label{fig:isotropy-coefficients}}
Figure~\ref{fig:isotropy-coefficients} ...

\subsection{Ratio of integral scales, velocities and energies}

\begin{figure}
\centerline{
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_ratio_velo_vs_Fh}
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_ratio_length_scales_vs_Fh}
}
\centerline{
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_ratio_EA_EK_vs_Fh}
}
\caption{Ratio of ... \label{fig:ratio-vs-Fh}}
\end{figure}

Figure~\ref{fig:ratio-vs-Fh} ...
\subsection{Mixing coefficient}

\begin{figure}
\centerline{
\includegraphics[width=0.7\textwidth]{%
../tmp/fig_mixing_coef_vs_FhR}
}
\caption{Mixing coefficient. \label{fig:mixing-coefficients-vs-FhR}}
\end{figure}

Figure~\ref{fig:mixing-coefficients-vs-FhR} ...


\begin{figure}
\centerline{
\includegraphics[width=0.7\textwidth]{%
../tmp/fig_mixing_coef_vs_Fh}
}
\caption{Mixing coefficient versus the horizontal Froude number. The colors represent
$\R_2$. Red letters correspond to simulations of table~\ref{table-simuls-regimes}
analyzed in subsection~\ref{spectra-seb-regimes}.
\label{fig:mixing-coefficients-vs-Fh}}
\end{figure}

Figure~\ref{fig:mixing-coefficients-vs-Fh} ...


\subsection{Spatial spectra and spectral energy budget}
\input{../tmp/table_simuls_regimes.tex}

Table~\ref{table-simuls-regimes} ...

\begin{figure}
\centerline{
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_spectra_regime_D}
\includegraphics[width=0.48\textwidth]{%
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../tmp/fig_ratio_spectra_regime_D}
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\caption{Spectra and ratio of spectra for simulation D (see
table~\ref{table-simuls-regimes}) corresponding to the dissipative regime.
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\label{fig:spectra-D}}
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Figure~\ref{fig:spectra-D} ...
\begin{figure}
\centerline{
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\includegraphics[width=0.55\textwidth]{%
../tmp/fig_seb_regime_D}
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\caption{Spectral energy budget for simulation D (see table~\ref{table-simuls-regimes})
corresponding to the dissipative regime. \label{fig:seb-D}}
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Figure~\ref{fig:seb-D} ...

\begin{figure}
\centerline{
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_spectra_regime_L}
\includegraphics[width=0.48\textwidth]{%
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../tmp/fig_ratio_spectra_regime_L}
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\caption{Spectra and ratio of spectra for simulation L (see
table~\ref{table-simuls-regimes}) corresponding to the LAST regime.
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\label{fig:spectra-L}}
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Figure~\ref{fig:spectra-L} ...
\begin{figure}
\centerline{
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\includegraphics[width=0.55\textwidth]{%
../tmp/fig_seb_regime_L}
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\caption{Spectral energy budget for simulation L (see table~\ref{table-simuls-regimes})
corresponding to the LAST regime. \label{fig:seb-L}}
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Figure~\ref{fig:seb-L} ...

\begin{figure}
\centerline{
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_spectra_regime_O}
\includegraphics[width=0.48\textwidth]{%
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../tmp/fig_ratio_spectra_regime_O}
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\caption{Spectra and ratio of spectra for simulation O (see
table~\ref{table-simuls-regimes}) corresponding to the optimal stratified turbulence
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regime. \label{fig:spectra-O}}
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Figure~\ref{fig:spectra-O} ...
\begin{figure}
\centerline{
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\includegraphics[width=0.55\textwidth]{%
../tmp/fig_seb_regime_O}
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\caption{Spectral energy budget for simulation O (see table~\ref{table-simuls-regimes})
corresponding to the optimal stratified turbulence regime. \label{fig:seb-O}}
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Figure~\ref{fig:seb-O} ...

\begin{figure}
\centerline{
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_spectra_regime_W}
\includegraphics[width=0.48\textwidth]{%
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../tmp/fig_ratio_spectra_regime_W}
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\caption{Spectra and ratio of spectra for simulation W (see
table~\ref{table-simuls-regimes}) corresponding to the weakly stratified turbulence
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regime. \label{fig:spectra-W}}
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Figure~\ref{fig:spectra-W} ...
\begin{figure}
\centerline{
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\includegraphics[width=0.55\textwidth]{%
../tmp/fig_seb_regime_W}
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\caption{Spectral energy budget for simulation W (see table~\ref{table-simuls-regimes})
corresponding to the weakly stratified turbulence regime. \label{fig:seb-W}}
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Figure~\ref{fig:seb-W} ...

\begin{figure}
\centerline{
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_spectra_regime_P}
\includegraphics[width=0.48\textwidth]{%
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../tmp/fig_ratio_spectra_regime_P}
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\caption{Spectra and ratio of spectra for simulation P (see
table~\ref{table-simuls-regimes}) corresponding to the passive scalar turbulence
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regime. \label{fig:spectra-P}}
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Figure~\ref{fig:spectra-P} ...
\begin{figure}
\centerline{
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\includegraphics[width=0.55\textwidth]{%
../tmp/fig_seb_regime_P}
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\caption{Spectral energy budget for simulation P (see table~\ref{table-simuls-regimes})
corresponding to the passive scalar turbulence regime. \label{fig:seb-P}}
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Figure~\ref{fig:seb-P} ...
\begin{figure}% [H]
\centerline{
\includegraphics[width=0.98\textwidth]{%
../tmp/fig_spectra_1strat}
}
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\caption{Horizontal (a) and vertical (b) spectra for simulations at different buoyancy
Reynolds number for $N=40$. \label{fig:spectra-1strat}}
\end{figure}

Figure~\ref{fig:spectra-1strat} ...

\begin{figure}% [H]
\centerline{
\includegraphics[width=0.98\textwidth]{%
../tmp/fig_spectra_1R}
}
\caption{Horizontal (a) and vertical (b) spectra for simulations at different
stratification for $\R_i=20$. \label{fig:spectra-1R}}
\end{figure}

Figure~\ref{fig:spectra-1R} ...

\section{Conclusions and perspectives}

We performed numerical simulations of a stratified turbulent flow, using a forcing
mechanism...

\begin{acknowledgments}
This project has received funding from the European Research Council (ERC)
under the European Union's Horizon 2020 research and innovation program (Grant
No. 647018-WATU). It was also partially supported by the Simons Foundation
through the Simons collaboration on wave turbulence. Part of this work was
performed using resources provided by \href{https://www.cines.fr/}{CINES} under
GENCI allocation number A0080107567.
\end{acknowledgments}

%\appendix\section{A great appendix}
%\label{appendix}

\bibliography{biblio}
\end{document}