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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:
\p_t\vv + (\vv \cdot \bnabla)\vv = b\boldsymbol{e}_z - \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 a divergence-free velocity field, $b$ the buoyancy, $p$ a rescaled
pressure and $N$ the constant and uniform \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 (i.e.\ 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 and the simulations do not reach a truly
statistically stationary flow. This was observed in many numerical studies of
stratified turbulence \cite[for example,][]{brethouwer_billant_lindborg_chomaz_2007,
augier2015stratified, Maffioli2016mixing}. When the shear modes are kept, a state for
which the mean energy grows slowing is finally reached. This state is associated with
very strong shear modes and the dynamics of all other structures is strongly distorted
by the shear modes.

Finally, (iii) vertically invariant vertical velocity (corresponding to 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 forcing 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 $P_K$ is constant and equal to unity. The exact method to
normalize the injection rate \cite[]{Maffioli2016mixing, AugierMohananLinborg2019}
corresponds to the Fluidsim parameter
\mintinline{python}{params.forcing.normalized.type = "2nd_degree_eq"}. 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. 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 of order unity. By construction, there are
transfers of energy from the large forced scales to small dissipative scales. The fact
that a stationnary state is reached implies that there are efficient mechanisms to
transfer energy from the large forced scales to structures with scales sufficiently
small to dissipate the energy.

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 couples $(N,\ \R_i)$ for quite large $N$ and $\R_i$, the required
resolution for proper DNS become very large. To decrease the computational cost of the
comprehensive dataset spanning the $(N,\ \R_i)$ space with more than 40 couples, we
do two compromises.
First, the aspect ratio of the numerical domain is decreased when the stratification is
increased. Indeed, strongly stratified flows are strongly anisotropic and dominated by
thin layers. The aspect ratio is chosen such that the vertical spectra are not
dominated by the first vertical mode and that there are few vertical layers in the
domain.
Second, for each couple, we run first simulations at coarse resolution to reach a
steady state. For these "approach" simulations, fourth-order hyperviscous and
hyperdiffusive terms are added, with fourth-order viscosity $\nu_4$ and diffusivity
$\kappa_4$ computed so that dissipative scales are well resolved.
\begin{equation}
\nu_4 = \kappa_4 = P ^{1 / 3} {\kmax} ^ {-10 / 3},
\end{equation}
where $\kmax$ is the largest wavenumber taken into account truncation. The effect of
hyperdiffusion can be quantified with the ratio $\epsKK/\epsK$ where is
$\epsK=\epsKK+\epsKKKK$ with $\epsKK$ and $\epsKKKK$ the kinetic energy dissipations
due to standard and hyper viscosities, respectively.

%% Method: simulations 1 couple (N, R_i)
\input{../tmp/table_methods_1couple.tex}

\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}}
Table \ref{table-methods-1couple} presents physical and numerical parameters for the
simulations carried out for one particular couple $(N,\ R_i) = (40,\ 20)$. 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 time evolution of some components of the energy for these
simulations are shown in figure~\ref{fig:method-N40-R20}(a). The resolution is
increased step by step until the product $\kmax\eta$ is larger than or very close to
one. The times of these resolution increases are indicated in
figure~\ref{fig:method-N40-R20}(a) by dashed vertical lines (for example $t_{640}$ is
the beginning of the simulation with $n_h=640$). Some quantities time averaged during
the statistical stationnary state of each simulation are plotted as a function of
$\kmax\eta$ (proportional to $n_h$) in figure~\ref{fig:method-N40-R20}(b). Here $\eta =
...$ is the Kolmogorov length scale.
Figure~\ref{fig:method-N40-R20}(b, top) shows the ratio $\epsKK/\epsK$ for the same
simulations. This ratio is an indicator of how close the simulations are to proper DNS.
For a resolution approximately half smaller than the required resolution for a proper
DNS (typically $\kmax\eta \simeq 0.5$, in this case $n_h=1344$), the standard
dissipation $\epsKK$ starts to dominate the total dissipation ($\epsKK/\epsK > 0.5$).
This means that the standard dissipation range starts to be resolved. For the
simulations for $\kmax\eta \sim 1$, most of the dissipation is due to standard
viscosity and it is reasonable to qualify such simulations as ``quasi-DNS".

Figure ~\ref{fig:method-N40-R20}(b, bottom) shows the effect of increasing the
resolution on 3 time averaged quantities that will be used in the result section:
$\Gamma = \epsA/\epsK$ is the mixing coefficient, $\Ivelo$ is a large scale isotropy
coefficient computed as a ratio of velocities and $\Idiss$ is a small scale isotropy
coefficient computed as a ratio of dissipation terms (the exact definitions will be
given in the next section). We see that as soon as $\kmax\eta \geq 0.5$ these
quantities depend very weakly on the resolution, which shows that we can get a good
evaluation of them by using only badly resolved quasi-DNS for $\kmax\eta \geq 0.5$.
Note that the variations for $\Ivelo$ between simulations are rather associated with
lacks of statistics (simulations not long enough) than with too coarse resolutions.

\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}}
Finally, figure~\ref{fig:method-N40-Ri20-spectra} shows the one-dimensional compensated
spectra for the same simulations. We see that the spectra gently converge when one
increases $\kmax\eta$ towards the spectra for the DNS ($\kmax\eta > 1$). As soon as
$\kmax\eta > 0.5$, the differences between the spectra are very small in the inertial
range and are mainly noticeable in the precise shape of the spectra in the dissipation
range.

We think that the results presented in figures~\ref{fig:method-N40-R20} and
\ref{fig:method-N40-Ri20-spectra} show that it is reasonable to evaluate the evolution
with the input parameters of a lot of quantities that are not related to the precise
dynamics in the dissipation range with a mix of quasi-DNS ($\kmax\eta \geqslant 0.5$)
and ``coarse DNS" ($\kmax\eta \simeq 1$). It is of course important to keep in mind the
limitations of these simulations and to not overinterpret results that could be due to
the limited resolution. Note that the DNS by
\cite{brethouwer_billant_lindborg_chomaz_2007} and \cite{Maffioli2017vertical}
correspond to $\kmax\eta \simeq 1$ and 1.5, respectively. In contrast, other studies
use more resolved DNS, for example $\kmax\eta \simeq 3$ for \cite{Portwood2016robust}.
It is interesting to consider the difference in terms of numerical costs for the same
input parameters between a quasi-DNS, a coarse DNS and a ``finely-resolved DNS". If one
consider for simplicity that the fast Fourier transform scale linearly with the number
of points and that the time step scales inversely proportional to the grid mech, a
finely-resolved DNS for $\kmax\eta = 3$ ($8064\times8064\times1008$ grid points for
$N,\ \R_i = 40, 20$) would be approximately 1300 ($=6^4$) and 80 ($=3^4$) times
more expensive than a quasi-DNS for $\kmax\eta = 0.5$ ($1344\times1344\times168$ grid
points) and a coarse DNS for $\kmax\eta = 1$ ($2688\times2688\times336$ grid points),
respectively.
%% 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{$\kmax\eta$ (a) and $\epsKK / \epsK$ (b) in a space $(F_h,\ \R)$ for the
simulations listed in table~\ref{table-better-simuls}. Each point corresponds to a
simulation. \label{fig:method-resolution-hyperdiffusivity}}
Table \input{../tmp/table_better_simuls.tex} and
figure~\ref{fig:method-resolution-hyperdiffusivity} describe the finest simulations of
our dataset for each couple $(N,\ \R_i)$. We see that for most couples, the finest
simulation corresponds to a coarse DNS with $\kmax\eta \simeq 1$.
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 results presented in this article are obtained
from periods of the simulation when a steady state has been approximately reached. In
the following the spatial spectra and the spectral energy budget are taken from coarse
DNS for which $\kmax\eta \simeq 1$.
This articles is accompanied by 2 datasets. The first dataset shared through Zenodo
(citation) is of moderate size (less than 30 GB) and can easily be downloaded as a
whole for future analyses. It contains for each simulation (i) the Fluidsim simulation
folder, (ii) a executed notebook showing a short description of the flow and how to
load and plot the Fluidsim results and (iii) a 3D representation of the statistically
stationnary flow. A Fluidsim simulation folder can be loaded with the program Fluidsim
to load and plot some data saved during the simulation. The different datafiles contain
input parameters of the simulation, space averaged quantities, spatial spectra,
spectral energy budget, temporal spectra and spatio-temporal spectra. The second
dataset is available through a LEGI server (address?) and contains for each simulations
a raw Fluidsim restart file.

TODO: Fluidsim versions...


\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 $\Ivelo$ versus $F_h$ in (a) and $\R$ in (b).
\label{fig:large-scale-isotropy}}
Figure~\ref{fig:large-scale-isotropy} ...

\begin{equation}
\Ivelo \equiv \frac{3 \mean{{v_z}^2}}{\mean{{v_x}^2 + {v_y}^2 + {v_z}^2}}
\end{equation}

%% 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 $\Idiss$ versus $F_h$ in (a) and $\R$ in (b).
\label{fig:small-scale-isotropy}}
Figure~\ref{fig:small-scale-isotropy} ...

\begin{equation}
\Idiss \equiv \frac{1 - r_{\text{diss}}}{1 - r_{\text{diss,\,iso}}},
\end{equation}
where $r_{\text{diss}}$ is the ratio of the kinetic energy dissipation associated with
vertical gradients over the total kinetic energy dissipation $\epsK$, and
$r_{\text{diss,\,iso}} = 1/3$ is the value for this ratio for isotropic turbulence.


%% 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 and velocities}

\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}
}
\caption{Ratio of integral scales and velocities \label{fig:ratio-vs-Fh}}
\end{figure}

Figure~\ref{fig:ratio-vs-Fh} ...
\subsection{Mixing coefficient and energy ratio}
\begin{figure}
\centerline{
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_mixing_coef_vs_FhR}
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_ratio_EA_EK_vs_FhR}
\caption{Mixing coefficient (a) and ratio $E_A/E_K$ (b).
\label{fig:mixing-coefficients-vs-FhR}}
\end{figure}

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


\begin{figure}
\centerline{
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_mixing_coef_vs_Fh}
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_ratio_EA_EK_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_V}
\includegraphics[width=0.48\textwidth]{%
../tmp/fig_ratio_spectra_regime_V}
\caption{Spectra and ratio of spectra for simulation V (see
table~\ref{table-simuls-regimes}) corresponding to the viscosity affected regime.
\label{fig:spectra-V}}
Figure~\ref{fig:spectra-V} ...
\begin{figure}
\centerline{
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\includegraphics[width=0.55\textwidth]{%
../tmp/fig_seb_regime_V}
\caption{Spectral energy budget for simulation V (see table~\ref{table-simuls-regimes})
corresponding to the viscosity affected regime. \label{fig:seb-V}}
Figure~\ref{fig:seb-V} ...

\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}