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\begin{document}

\title{Regimes in stratified turbulence forced by gravity waves analyzed
from a new comprehensive open dataset}

\author{Vincent Reneuve}
\affiliation{Universit\'{e} C\^{o}te d'Azur, Observatoire de la C\^{o}te
% TODO: add other authors
\author{Pierre Augier}
\affiliation{Laboratoire des Ecoulements G\'eophysiques et Industriels, Universit\'e
Grenoble Alpes, CNRS, Grenoble-INP, F-38000 Grenoble, France}
d'Azur, CNRS, Laboratoire Lagrange, Nice, France.}

% TODO: change corresponding author
\email[]{pierre.augier@univ-grenoble-alpes.fr}

\begin{abstract}

Blabla 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}
\end{align}
where $\vv$ is the velocity, $b$ the buoyancy, $p$ the pressure and $N$ the \bv
frequency. For all simulations the viscosity $\nu_2$ and the diffusivity $\kappa_2$ are
equal (Schmidt number $Sc = \nu_2/\kappa_2 = 1$). Note that the buoyancy can be
expressed as $b=-g\delta\rho/\rho_0$, with $g$ the gravitational acceleration, $\rho_0$
the mean density and $\delta\rho$ the departure from the stable linear density
stratification. However, these three quantities do not enter separately into the
equations.

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) All modes with wavenumber modulus larger than $\kmax = 0.8 (n_x/2) \delta
k_x$ are truncating to limit aliasing. The precise shape of the truncation actually
corresponds to the Fluidsim parameter
\mintinline{python}{params.oper.truncation_shape="no_multiple_aliases"}. (ii) All shear
modes (for which $|\mathbf{k_h}| = 0$) are truncated (Fluidsim parameter
\mintinline{python}{params.oper.NO_SHEAR_MODES = True}). If we do not truncate them,
they tend to grow very slowly so the simulations do not really reach a statistically
stationary flow. Finally, (iii) vertically invariant vertical velocity (internal waves
at $\omega = N$) is also forbidden \mintinline{python}{params.no_vz_kz0 = True}). Note
that in all experiments in tanks both shear modes and vertically invariant vertical
velocity are also blocked to zero.

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 of horizontal length scale of typically $L_f = 1$ 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.

The main physical input parameters are the \bv frequency and the viscosity. Since both
forcing length and energy injection rate are in practice equal to 1, we can define an
input Reynolds number $Re_i = 1/\nu_2$ and an input horizontal Froude number $F_{hi} =
1/N$. For stratified turbulence, it is actually more convenient to take as input
parameters the \bv frequency and an input buoyancy Reynolds number $\R_i = Re_i
F_{hi}^2$. The input Reynolds number is thus computed as $Re_i = \R_i N^2$.

For some couple $(N,\ \R_i)$ for quite large $N$ and $\R_i$, the required
resolution for proper DNS become too large. To decrease the computational cost of the
comprehensive dataset, we use three ...

The aspect ratio of the numerical domain is varied depending on the stratification
strength.

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. 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 ...

\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 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.

\section{Results}


\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}
}
\caption{Large scale isotropy coefficient $I_{velo}$. \label{fig:large-scale-isotropy}}
\end{figure}

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}
}
\caption{Small scale isotropy coefficient $I_{diss}$. \label{fig:small-scale-isotropy}}
\end{figure}

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}}
\end{figure}

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}
\label{spectra-seb-regimes}

\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]{%
../tmp/fig_ratio_spectra_regime_D}
}
\caption{Spectra and ratio of spectra for simulation D (see
table~\ref{table-simuls-regimes}) corresponding to the dissipative regime.
\label{fig:spectra-D}}
\end{figure}

Figure~\ref{fig:spectra-D} ...

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

Figure~\ref{fig:seb-D} ...

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

Figure~\ref{fig:spectra-L} ...

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

Figure~\ref{fig:seb-L} ...


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

Figure~\ref{fig:spectra-O} ...

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

Figure~\ref{fig:seb-O} ...




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

Figure~\ref{fig:spectra-W} ...

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

Figure~\ref{fig:seb-W} ...


\begin{figure}
\centerline{
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_spectra_regime_P}
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_ratio_spectra_regime_P}
}
\caption{Spectra and ratio of spectra for simulation P (see
table~\ref{table-simuls-regimes}) corresponding to the passive scalar turbulence
regime. \label{fig:spectra-P}}
\end{figure}

Figure~\ref{fig:spectra-P} ...


\begin{figure}
\centerline{
\includegraphics[width=0.55\textwidth]{%
../tmp/fig_seb_regime_P}
}
\caption{Spectral energy budget for simulation P (see table~\ref{table-simuls-regimes})
corresponding to the passive scalar turbulence regime. \label{fig:seb-P}}
\end{figure}

Figure~\ref{fig:seb-P} ...


\begin{figure}% [H]
\centerline{
\includegraphics[width=0.98\textwidth]{%
../tmp/fig_spectra_1strat}
}
\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}