# HG changeset patch
# User vlabarre <Vincent.Labarre@oca.eu>
# Date 1654785642 -7200
#      Thu Jun 09 16:40:42 2022 +0200
# Node ID 352a5973d2c48554b9c8dea12b2d94446f63f06d
# Parent  a09997bb9da32176055baf09c3ba04b9ff6cd251
draft for introduction and method

diff --git a/2022strat_turb_polo/input/article.tex b/2022strat_turb_polo/input/article.tex
--- a/2022strat_turb_polo/input/article.tex
+++ b/2022strat_turb_polo/input/article.tex
@@ -105,6 +105,14 @@
 \newcommand{\diff}{\text{d}}
 \newcommand{\bv}{Brunt-V\"ais\"al\"a }
 \newcommand{\kmax}{k_{\max}}
+\newcommand{\vk}{\hat{\boldsymbol{v}}_{\kk}}
+\newcommand{\vpk}{\hat{v}_{p\kk}}
+\newcommand{\vtk}{\hat{v}_{t\kk}}
+\newcommand{\epk}{\boldsymbol{e}_{p\kk}}
+\newcommand{\etk}{\boldsymbol{e}_{t\kk}}
+\newcommand{\thk}{\theta_{\kk}}
+\newcommand{\thf}{\theta_f}
+\newcommand{\ok}{\omega_{\kk}}
 
 \newcommand{\todo}[1]{\textcolor{red}{TODO: #1}}
 
@@ -130,7 +138,7 @@
 
 \begin{document}
 
-\title{Regimes in stratified turbulence forced by gravity waves analyzed
+\title{Regimes in stratified turbulence forced by horizontally divergent modes analyzed
 from a new comprehensive open dataset}
 
 \author{Vincent Labarre}
@@ -138,11 +146,12 @@
 \affiliation{Universit\'{e} C\^{o}te d'Azur, Observatoire de la C\^{o}te d'Azur, CNRS, Laboratoire Lagrange, Nice, France.}
 
 \author{Pierre Augier}
+\email[]{pierre.augier@univ-grenoble-alpes.fr}
 \affiliation{Laboratoire des Ecoulements G\'eophysiques et Industriels, Universit\'e
 Grenoble Alpes, CNRS, Grenoble-INP, F-38000 Grenoble, France}
 
 
-% TODO: add Giorgio, Sergey, and other authors?
+% TODO: add Giorgio, Sergey, and other authors? Discuss the order of the authors.
 \author{Giorgio Krstulovic}
 \affiliation{Universit\'{e} C\^{o}te d'Azur, Observatoire de la C\^{o}te d'Azur, CNRS, Laboratoire Lagrange, Nice, France.}
 
@@ -153,7 +162,7 @@
 
 \begin{abstract}
 
-We present numerical simulations of stratified turbulence under the Oberbeck-Boussinesq approximation with Prandtl $Pr=1$ and periodic boundary conditions, performed using the pseudo spectral solver \texttt{ns3d.strat} of the \texttt{Fluidsim} framework. The particularities of this dataset are
+We present numerical simulations of stratified turbulence under the Oberbeck-Boussinesq approximation with Schmidt number equal to unity and periodic boundary conditions, performed using the pseudo spectral solver \texttt{ns3d.strat} of the \texttt{Fluidsim} framework. The particularities of this dataset are
 
 % TODO: Elaborate these points in the abstract when the article is nearly finished
 \begin{itemize}
@@ -191,15 +200,15 @@
 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:
+the Boussinesq approximation with additional hyperviscosity and hyperdiffusivity terms:
 \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}
+\nu_2\nabla^2\vv - \nu_4(\nabla^2)^2\vv + \ff_{\text{polo}},\label{ns} \\
+\p_t{b} + (\vv \cdot \bnabla)b = -N^2 v_z + \kappa_2\nabla^2{b} - \kappa_4(\nabla^2)^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
+where $\vv$ is the velocity, $b$ the buoyancy, $p$ the total pressure, $N$ the \bv
+frequency, $\nu_2$ is the viscosity, $\kappa_2$ is the diffusivity, $\kappa_4$ is the hyperviscosity and $\kappa_4$ is the hyperdiffusivity. For all simulations the viscosity and the diffusivity are
+equal (Schmidt number $Sc = \nu_2/\kappa_2 = 1$), as well as the hyperviscosity and hyperdiffusivity ($\nu_4 = \kappa_4$). 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
@@ -213,54 +222,43 @@
 \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
+they tend to grow very slowly so it is longer to reach a statistically steady state. 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.
+
+Geometrically, a divergent free velocity velocity field can be decomposed into a poloidal $\vpk$ part and toroidal part $\vtk$. More precisely, the Fourier transform of the velocity field can be written 
+\begin{equation}
+	\vk = \vpk \epk + \vtk \epk
+\end{equation}
+where $\kk$ is the wavevector, $\etk \equiv (\eez \times \kk) / |\eez \times \kk|$ is the toroidal unitary vector, and $\epk$ is such that $(\kk/|\kk|, \epk, \etk)$ forms an orthonormal basis. 
 
-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 this dataset, we were motivated by forcing internal waves modes in which only the poloidal part is involved \cite{Augier2011}. Then, we add a forcing term $\ff_{\text{polo}}$ acting on $\vpk$ at large scale ($1.25 \leq |\kk|/\delta k_z \leq 5$) and small angle ($|\thk - \thf| \leq \delta \thf/2$ where $\sin \thf = 0.3$, $\sin \delta \thf = 0.1$ and $\thk$ is the angle between the wavevector $\kk$ and the stratification axis $\eez$).
+
+This anisotropic forcing is motivated by the fact that the dispersion relation of internal gravity waves, $\ok = N \sin \thk$, implies that the angle of the frequency fully determine the frequency. Consequently, in order to force internal waves at given temporal scales, we have to fix the direction of the forced modes. The forcing is time correlated with a correlation time corresponding to the frequency of the forced waves $T_c = 2\pi / N \sin \thf$. It is computed in spectral space such that the kinetic energy injection rate is constant and equal to unity. \\
+
+The time needed to reach a statistically steady state increases with $N$. In fact, we observe that the energy tends to concentrate in modes with small horizontal variation $k_h /\delta k_h =1$ (remember that there is no shear mode in these simulations). In this state, the time averaged total energy dissipation rate $\eps = \epsA + \epsK$ 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 $\Gamma = \epsA / \epsK = (1 - \epsK) / \epsK$ and is in any case of order
+unity. As usual in 3D stratified turbulence, there is a transfers of energy from the large forced scales to small dissipative scales \cite{?}. \\
+
+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.
+resolution for proper DNS become too large. To decrease the computational cost of the comprehensive dataset, we combined two strategies. Firstly, the aspect ratio of the numerical domain is varied depending only on the stratification strength, going from $1/2$ for $N<20$ to $1/8$ for $N \geq 80$. Secondly, we used coarse simulations to devellop the large scales present in 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.
+For most simulations, we used the fourth-order hyperviscosity and hyperdifusivity terms. Here, $\nu_4$ 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. The product of the maximal wavevector $\kmax$ with the Kolmogorov scale $\eta \equiv \nu_2^3 / \epsKK$ is also computed as diagnostic to estimate que quality of the simulation. For a set of physical parameters, the needed hyperviscosity decreases when the resolution is increased and the ratio $\epsKK/\epsK$ grows towards unity and $\kmax \eta$ increases (eventually beyond unity). \\
 
 %% Method: simulations 1 couple (N, R_i)
 
 \input{../tmp/table_methods_1couple.tex}
 
-Table \ref{table-methods-1couple} shows ...
+Table \ref{table-methods-1couple} shows quantities computed for the simulations performed for the 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 results presented in this article are obtained from periods of the simulations 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 horizontal resolution $n_h=320$, and increase the resolution step by step only when a sufficiently stationary state has been reached. The vertical resolution $n_z$ is fixed by the aspect ratio (which depends only on $N$). When such a state is observed, specific outputs are turned on and the simulation is ran further for several units time in order to produce substantial data to
+analyze, before increasing the resolution again if needed.
+
 
 \begin{figure}
 \centerline{
@@ -274,6 +272,8 @@
 \label{fig:method-N40-R20}}
 \end{figure}
 
+Figure \ref{fig:method-N40-R20} shows the energies signal for simulations at different resolutions for the couple $(N, \R_i) = (40, 20)$ and the averaged quantities vs $\kmax\eta$ for the same simulations. The coarse simulations with $n \geq 20$ where runned for $2 N$ simulation times at resolution $n_h=320$, and then restarted for $N/2$ simulations times at resolution $n_h=640$. We observed that the flow takes a small time to adjust when changing the resolution, but this time remains small before the total simulation time. \todo{Discuss of the simulation times for $n_h = 1280, 1920,$ and $2560$}. As expected, the ratio $\epsKK / \epsK$ approach unity when $\kmax \eta$ increases and the dimensionless quantities tend to constant values, meaning that hyperviscosity and hyperdiffusivity become negligeable for sufficiently large simulations (typically when the criteria $\kmax \eta > 1$ is attained). Note that this convergence is also visible in spatial spectra, as illustrated on figure \ref{fig:method-N40-Ri20-spectra}.
+
 \begin{figure}% [H]
 \centerline{
 \includegraphics[width=0.98\textwidth]{%
@@ -283,7 +283,16 @@
 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)
+
+%% Method: synthesis of simulations at different (N, R_i)
+
+\input{../tmp/table_better_simuls.tex}
+
+In the remaining of this manuscript, will keep only the most resolved simulation for each $(N, \R_i)$. Parameters and dimensionless numbers for these simulations are summarized in
+table~\ref{table-better-simuls}. Figure \ref{fig:method-resolution-hyperdiffusivity} allows to see how our simulations fill the $(F_h, \R_2)$ parameter space, and appreciate which of those simulations can be considered as DNS. We see that most of it verify $\kmax \eta \geq 1$ and $\epsKK / \epsK \simeq 1$, except for very low $F_h$ and/or high $\R_2$. The simulations were performed on local clusters for resolutions up to $n_h =
+640$: the Mesocentre SIGAMM machine, hosted by Observatoire de la Cote d'Azur, and the Université Côte d'Azur's Center for High-Performance Computing. For larger resolutions ($n_h \geq 1280$), we used the cluster Jean-Zay of the french national center IDRIS. 
+
+
 
 \begin{figure}
 \centerline{
@@ -292,33 +301,20 @@
 \includegraphics[width=0.48\textwidth]{%
 ../tmp/fig_epsK2overepsK_vs_FhR}
 }
-\caption{. \label{fig:method-resolution-hyperdiffusivity}}
+\caption{$\kmax \eta$ (a) and $\epsKK/\epsK$ (b) vs $(F_h, \R_2)$ for our dataset. \todo{Expand the $F_h$ range in the figures.} \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}
 \label{sec:res}
 
 \subsection{Large and small scale isotropy coefficients}
 
+
+
 %% Large scale isotropy
 
 \begin{figure}
@@ -333,6 +329,8 @@
 
 Figure~\ref{fig:large-scale-isotropy} ...
 
+Large scale isotropy mostly depends on $F_h$, but much less on $\R$
+
 %% Small scale isotropy
 
 \begin{figure}
@@ -347,6 +345,8 @@
 
 Figure~\ref{fig:small-scale-isotropy} ...
 
+Large scale isotropy mostly depends on $\R_2$, but much less on $F_h$
+
 %% Isotropy coefficient: summary
 
 \begin{figure}
@@ -576,19 +576,9 @@
 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 supported by the Simons Foundation
-through the Simons collaboration on wave turbulence. Part of this work was
-performed using resources provided by . \\
-
-
-Part of the computations have been done on the ``Mesocentre SIGAMM'' machine, hosted by Observatoire de la Cote d'Azur. \\
-
-This work was supported by the French government, through the UCAJEDI Investments in the Future project managed by the National Research Agency (ANR) under reference number ANR-15-IDEX-01. The authors are grateful to the OPAL infrastructure from Université Côte d’Azur and the Université Côte d’Azur’s Center for High-Performance Computing for providing resources and support. \\
-
-This work was granted access to the HPC/AI resources of [CINES/IDRIS/TGCC] under the
-allocation 2022- A0122A13417 made by GENCI.
+This project was supported by the Simons Foundation
+through the Simons collaboration on wave turbulence. It has received funding from the European Research Council (ERC)
+under the European Union's Horizon 2020 research and innovation program (Grant No. 647018-WATU). Part of the computations have been done on the ``Mesocentre SIGAMM'' machine, hosted by Observatoire de la Cote d'Azur. This work was supported by the French government, through the UCAJEDI Investments in the Future project managed by the National Research Agency (ANR) under reference number ANR-15-IDEX-01. The authors are grateful to the OPAL infrastructure from Université Côte d’Azur and the Université Côte d’Azur’s Center for High-Performance Computing for providing resources and support. This work was granted access to the HPC/AI resources of IDRIS under the allocation 2022-A0122A13417 made by GENCI.
 
 
 \end{acknowledgments}
diff --git a/2022strat_turb_polo/input/intro.tex b/2022strat_turb_polo/input/intro.tex
--- a/2022strat_turb_polo/input/intro.tex
+++ b/2022strat_turb_polo/input/intro.tex
@@ -3,25 +3,16 @@
 When density variation is present in a flow, fluid parcels feel the buoyancy that tends to push them to their floatability levels. When the denser fluid is below, we talk about stable stratification, and the flow is then organized as thin layers whose extend along the stratification axis (i.e. the direction of the average density gradient) is related to the intensity of the average density gradient. In practice, density differences arises because of the variation of a scalar which causes density variation. This can occurs as long as sources and sink, like radiative heating or salt input, maintain the flow out of equilibrium. Contrary to flows at constant density which contain only eddies, stably stratified flows also support the propagation of internal gravity waves. \\
 
 
-Stratification is a key physical mechanism in many industrial, geophysical and astrophysical flows. As an example, stratication is essential to describe the dynamics  of the atmosphere and oceans, in particular at mesoscales at which the effects of rotation is negligeable. Most studies analytical and numerical studies are performed using the Oberbeck-Boussinesq approximation. In this case, the intensity of the stratification is solely quantified by the \bv frequency.  \\
-
-
-Chomaz 2000 -> Brethouwer 2007 did the scaling analysis of the equations to show that the dynamics is mainly controled by $3$ dimensionless parameters: the Prandtl $Pr \equiv \nu / \kappa$, $\R$, $F_h$. 
+Stratification is a key physical mechanism in many industrial, geophysical and astrophysical flows. As an example, stratification is essential to describe the dynamics  of the atmosphere and oceans, in particular at mesoscales at which the effects of rotation is negligeable. Most studies analytical and numerical studies are performed using the Oberbeck-Boussinesq approximation and a constant stratification. In this case, the intensity of the stratification is solely quantified by the \bv frequency.  \\
 
-
-
+The scaling analysis of the equations to show that the dynamics is mainly controled by $3$ dimensionless parameters \cite{Chomaz2000, ..., Brethouwer2007}: the Schmidt number $Sc \equiv \nu / \kappa$ where $\nu$ and $\kappa$ are the viscosity and diffusivity; the buoyancy Reynolds number $\R$; and the horizontal Froude number $F_h$. Then, for a given fluid (with a fixed $Sc$) and far away from the singularities, the flow is supposed to depends only on $F_h$ and $\R$. When $F_h>1$ is flow is weakly stratified and we expect to recovers incompressible turbulence. Yet, when $F_h < 1$, the stratifiction impacts the flow significantly and three stratified regimes where predicted: if $\R < 1$, the flow is dominated by viscosity; if $\R > 1$ and $10^{-2} <F_h < 1$, the flow is turbulent but weakly stratified; finally, when $\R > 1$ and $F_h < 10^{-2}$, the flow is in stronly stratified turbulent state. \\
 
-
-
+Despite intensive studies, many questions still need to be adressed, more or less related to geophysical applications: What is the mixing efficiency in stratified flows? How energy is distributed amoung spatial and temporal scales? How anisotropic large and small scales are? What are the consequences of the presence of internal gravity waves \cite{Maffioli2020}? Is Wave Turbulence Theory suitable to describe large scales of stratified flows? These question are difficult to adress because of numerical and experimental limitations that make difficult to inversigate large range in the $(F_h ,\R)$ parameters space. \\
 
-\begin{itemize}	 	 
-	\item Despite intensive studies, many questions still need to be adressed, more or less related to geophysical applications: mixing , spatial and temporal spectra, spatiotemporal analysis, ..., isotropy of large and small scales, presence and role of waves, both when $F_h$ and $\R$ (and $Pr$) vary.	
-	
-	\item Simuls1 (small but more simulations): Lindborg 2006 was perhaps the first to investigate the effect of varying the horizontal Froude number. But his simulations where performed using hyperviscosity of order 8. At that time, limitations did not allow to performed large simulations $512^2 \times 64$ box
-	and one in a $768^2 \times 48$ box.	
-	\item Simuls2 (large but few simulations): Since then, Many simulations, but most foccus on very large domain with purely DNS (Mininni + Centrale Lyon + ...), but for a reduced number of simulations. These very large simulations are necessary to investigate the very small scales of stratified turbulence, but does not allows to have an overview of the different regimes in such flows. 
-	\item => In this context, it would be usefull to have comprehensive open dataset to investigate stratified turbulence, and to restart these simulations for further studies.	
-	\item 
-	\item Here, we present intermediate simulations where both the Froude and the floatability Reynolds are varied. They have the advantage to be sufficiently small to perform a parametric study, and large enough to study transition to turbulent regimes. We force the velocity part involed in internal gravity waves
-	\item The remaining of the paper is as follows. In section \ref{sec:num}, we present our numerical setup and the dataset. In section \ref{sec:res}, we present several figures plotted using the data set that illustrate: the large and small scale isotropy coefficients, the ratio of integral scales, the velocities and energies, the mixing coefficient, the spatial spectra and spectral energy budget.	
-\end{itemize}
\ No newline at end of file
+The first simulations used for parametric studies \cite{Waite-Bartello2004, Waite-Bartello2006, Lindborg2006} where limited by the computationnal power at that time.  For example, in \cite{Lindborg2006}, the maximal resolutions were $512^2 \times 64$ and $768^2 \times 48$. To observe an inertial range, these simulations where performed using hyperviscosity of order $8$. \\
+
+Nowadays, direct numerical up to resolution $8192^2 \times 2048$ are used \cite{Maffioli2017, Mininni2017, Maffioli2020}. However, most of these simulations are used to study specific problems with a reduced number of simulations. These very large simulations are necessary to investigate the very small scales of stratified turbulence, but does not allows to have an overview of the different regimes in such flows. \\
+ 
+In this context, it would be usefull to have comprehensive open dataset to investigate various aspect of stratified turbulence. The goal of this manuscript is to present such a dataset. Intermediate resolution simulations (up to $2560^2  \times 640$) where both the Froude and the floatability Reynolds are varied. They have the advantage to be sufficiently small to perform a parametric study, and large enough to study transition to turbulent regimes. We force the horizontally divergent modes of the velocity which is involed in internal gravity waves. In a companion paper, we present simulations in which the vortical modes (vertical vorticity) is forced in a similar way. \\
+
+The remaining of the paper is as follows. In section \ref{sec:num}, we present our numerical setup and the dataset. In section \ref{sec:res}, we present several figures plotted using the data set that illustrate: the large and small scale isotropy coefficients, the ratio of integral scales, the velocities and energies, the mixing coefficient, the spatial spectra and spectral energy budget.