results.tex 29.40 KiB
\subsection{Global energy distribution}
\label{subsec:global}
In this subsection, we study the global energy budget in the $(F_h, \R)$ plane. We note $E_{pot} = \sum\limits_{\kk} \frac{|\bbk|^2}{2N^2}$ the potential energy, $E_{polo} = \sum\limits_{\kk} \frac{|\vvp|^2}{2}$ the poloidal kinetic energy, $E_{toro} = \sum\limits_{\kk} \frac{|\vvt|^2}{2}$ the vortical modes (toroidal) kinetic energy, and $E = E_{pot} + E_{polo} + E_{toro}$ the total energy. Figure~\ref{fig:global-energy} shows the evolution of vortical modes energy ratio $E_{toro}/E$ and the relative difference between poloidal and potential energy $\tilde{\mathcal{D}} = (E_{polo} - E_{pot})/(E_{polo} + E_{pot})$ as a function of $F_h$ and $\R$. For waves, we expect to observe an equipartition between the poloidal kinetic energy and potential energy. Consequently, both $E_{toro}/E$ and $\tilde{\mathcal{D}}$ should be close to zero for a system mainly composed by internal gravity waves.
\begin{figure}[H]
\includegraphics[width=1.0\textwidth]{../tmp/fig_ratio_E_vs_Fh}
\caption{(a) Ratio of vortical modes energy for simulations without projection vs $F_h$. (b) Relative difference between poloidal and potential energies for simulations with or without vortical modes vs $F_h$.
\label{fig:global-energy}}
\end{figure}
We observe that vortical modes energy becomes dominant at high stratification (low $F_h$) if they are not removed from the dynamics in these simulations with $\R \geq 0.1$ (figure~\ref{fig:global-energy} (a)). The ratio $E_{toro}/E$ is the lowest at intermediate stratification $F_h \simeq 0.1-0.3$ and low values of $\R$. However, the same simulations are marked by a predominance of potential energy over poloidal energy, $\tilde{\mathcal{D}} < 0$ (figure~\ref{fig:global-energy} (b)), meaning that these weakly stratified flows do contain other structures than waves. When vortical modes are removed, we can obtain flows with a global balance between poloidal and potential energies ($\tilde{\mathcal{D}} \simeq 0$) at high stratification. Consequently, a flow governed by weak non-linear interactions between waves may be obtained at high stratification by removing vortical modes. In the two next subsections, we perform a spatiotemporal analysis of a couple of stongly stratified turbulent simulations with $N=40$ and $\R_i=20$ (with and without vortical modes).
\subsection{$(k_h, k_z)$ energy budget in the strongly stratified regime}
\label{subsec:khkz}
The kinetic energy and potential energy budgets for one Fourier mode read
\begin{equation}
\label{eq:seb}
\left\langle \frac{1}{2}\frac{\mathrm{d} |\vvk|^2}{\mathrm{d}t} \right \rangle = \mathcal{I}_{\kk} + \mathcal{T}_{kin, \kk} - \mathcal{B}_{\kk} - \varepsilon_{kin,\kk}, ~~~~ \text{and} ~~~~
\left\langle \frac{1}{2N^2}\frac{\mathrm{d}|\hat{b}_{\kk}|^2}{\mathrm{d}t} \right\rangle = \mathcal{T}_{pot,\kk} + \mathcal{B}_{\kk} - \varepsilon_{pot,\kk},
\end{equation}
where
\begin{align}
\nonumber
&\mathcal{I}_{\kk} = \left\langle Re \left( \ffk \cdot \vvk^* \right) \right \rangle, ~~~~
\mathcal{T}_{kin, \kk} = - \left\langle Re \left( \vvk^* \cdot \left[ \bar{\bar{P}}_{\kk} \cdot (\widehat{\vv \cdot \bnabla \vv})_{\kk} \right] \right) \right\rangle, \\
&\mathcal{T}_{pot, \kk} = - \left\langle Re \left( \hat{b}_{\kk}^* (\widehat{\vv \cdot \bnabla b})_{\kk} \right) / N^2 \right\rangle, ~~~~
\mathcal{B}_{\kk} = - \left\langle Re \left( \hat{v}_{z\kk}^* \hat{b}_{\kk} \right) \right\rangle,
\end{align}
are respectively the kinetic energy injection rate, the kinetic energy transfer, the potential energy transfer, and the conversion of kinetic energy to potential energy. In the last equations, $(\cdot)^*$ denotes the complex conjugate, $Re(\cdot)$ the real part, $\left\langle \cdot \right\rangle$ stands for the averaging operator, and $\bar{\bar{P}}_{\kk} = \mathbb{I} - \kk \otimes \kk / k^2$ is the projector onto the plane orthogonal to $\kk$. In this section, we study azimutal average of the energy budget. Namely, we computed quantities like:
\begin{equation}
F(k_h, k_z) = \frac{1}{\delta k_h ~ \delta k_z} \sum\limits_{k_h \leq k_h'< k_h + \delta k_h} ~ \sum\limits_{k_z \leq |k_z'| < k_z + \delta k_z} ~ F_{\kk},
\end{equation}
where $F$ can be $E_{pot}$, $E_{polo}$, $E_{toro}$, $\mathcal{T}_{pot}$, $\mathcal{T}_{kin}$, or $\mathcal{B}$. \\
Following \cite{yokoyama_energy-based_2019}, we use the following ratios of energies, like $E_{toro}(k_h, k_z)/E(k_h, k_z)$, in order to quantify the energy content scale by scale. In the same spirit, we also introduce the following quantities
\begin{align}
\label{eq:transferK}
\tilde{\mathcal{T}}_{kin} &= \mathcal{T}_{kin} /(|\mathcal{T}_{kin}| + |\mathcal{T}_{pot}| + |\mathcal{B}| + \epsK + \epsA), \\
\label{eq:transferA}
\tilde{\mathcal{T}}_{pot} &= \mathcal{T}_{pot} /(|\mathcal{T}_{kin}| + |\mathcal{T}_{pot}| + |\mathcal{B}| + \epsK + \epsA), \\
\label{eq:convK2toA}
\tilde{\mathcal{B}} &= \mathcal{B} /(|\mathcal{T}_{kin}| + |\mathcal{T}_{pot}| + |\mathcal{B}| + \epsK + \epsA)
\end{align}
that are useful for tracking the energy pathways. By construction, $\tilde{\mathcal{T}}_{pot}$, $\tilde{\mathcal{T}_{kin}}$, and $\tilde{\mathcal{B}}$ vary between $-1$ and $1$ depending on the amplitude and direction of the energy transfer or conversion. In statistically stationary state ($\langle\cdot \rangle = 0$) and in the inertial range ($\epsK = \epsA = \mathcal{P}_K = 0$), we should have $\mathcal{T}_{kin} = - \mathcal{T}_{pot} = \mathcal{B}$. For this reason, we will present only $\tilde{\mathcal{B}}$ in spectral energy budgets. As explained in the previous subsection, waves are marked by an equipartition between poloidal kinetic energy and potential energy. Then, we introduce the equipartition energy and the difference energy as
\begin{align}
E_{equi}(k_h, k_z) &= 2 \min \left\{ E_{polo}(k_h, k_z), E_{pot}(k_h, k_z)\right\}, \\
E_{diff}(k_h, k_z) &= E_{polo}(k_h, k_z) + E_{pot}(k_h, k_z) - E_{equi}(k_h, k_z),
\end{align}
in order to track the presence of waves in the $(k_h, k_z)$ plane. Indeed, $E_{diff}$ corresponds to the energy contained in $E_{pot}$ and $E_{polo}$ that cannot be in potential-kinetic equipartition. Therefore, $E_{diff}$ does not contain the waves' energy. On the other side, $E_{equi}$ corresponds to the energy contained in $E_{pot}$ and $E_{polo}$ that is in potential-kinetic equipartition. Consequently, $E_{equi}$ encompass the waves' energy. However, it is important to note that $E_{equi}$ can also contain the energy of structures that are not waves. A more careful separation between waves and non-waves structures requires a 4D spatiotemporal filtering \cite{lam_partitioning_2020, lam_energy_2021}, which is expansive in term of computational time and data storage. Yet, $E_{equi} /E$ and $E_{diff}/E$ help to track waves in the $(k_h, k_z)$ plane at a lower cost. We also compute the non-linearity parameter used for several physical systems, including stratified and rotating turbulence \cite{nazarenko_critical_2011, yokoyama_energy-based_2019} and MHD \cite{meyrand_direct_2016, cerri_turbulent_2022})
\begin{equation}
\chi_{\kk} = \frac{\tau_{L}}{\tau_{NL}} = \frac{(\epsK k^2)^{1/3}}{N k_h / k} = \frac{k}{k_h} \left( \frac{k}{k_O} \right)^{2/3}.
\end{equation}
It represents the ratio between the period of the linear waves and the eddy turnover time, estimated by dimensional analysis. \\
Figure \ref{fig:seb-no-proj} shows the spectral energy budget of the simulation $(N=40, \R_i=20)$ with vortical modes. We observe that $E_{equi}$ is the dominant energy in almost all the inertial range. $E_{diff}$ dominates only in the forcing region. $E_{toro}$ is never negligible away of the forcing region, and very important for small wave-vectors. For this simulation, the conversion between potential energy and kinetic energy $\tilde{\mathcal{B}}$ is important only when $\chi_{\kk} < 10$ and $k < k_O$. $\tilde{\mathcal{B}} \geq 0$ in the forcing region and when $k_z < k_h < k_O$, while $\tilde{\mathcal{B}} < 0$ if $\chi_{\kk} < 10$ and $k_h \leq k_z \leq k_O$. Interestingly, $E_{equi}$ is dominant around the line $\chi_{\kk} = 3$ for this simulation with $F_h \ll 1$ and $\R \gg 1$, even beyond the Ozmidov scale. This indicates that the ratio of temporal scales $\chi_{\kk}$ is important when identifying ranges in anisotropic turbulence, as explained in \cite{yokoyama_energy-based_2019}.
\begin{figure}[H]
\includegraphics[width=1.0\textwidth]{../tmp/fig_seb_regimes_L_projNone}
\caption{Spectral energy budget for simulation $N=40$, $\R_i=20$, with vortical modes. The gray dotted and dashed lines correspond to $\chi_{\kk} = 1/3$ and $\chi_{\kk} = 3$, the continuous black line to $k_z = k_h$, the black dotted line to $k_b$, the black dashed line to $k_O$, and the green dashed line to the dissipative scale. The orange dashed lines correspond to the maximal vertical and horizontal wave-vectors of the forcing. \label{fig:seb-no-proj}}
\end{figure}
Figure \ref{fig:seb-proj} represents the spectral energy budget of the simulation $(N=40, \R_i=20)$ without vortical modes. Obviously, no energy is contained in vortical modes in this case. One difference with the case were vortical modes are not removed is that $E_{diff}$ is less concentrated around the line $\chi_{\kk} = 3$. This could be due to the fact that this line is closer to the dissipative scale when we remove vortical modes due to a small increase of $F_h$ (and therefore $k_b$ and $k_O$). Another important difference is that we clearly see that $\tilde{E}_d$ can be important when $k \geq k_b$, in particular when $k_z \ll k_h$ or $k_z \gg k_h$ due, for example, to wave breaking. This is a first indication that the wave dominated range cannot lie above $k_b$. Moreover, $E_{equi}$ is dominant for $k \leq k_b$, which is another indication that the waves dominated region lies in the buoyancy range. The scaled conversion to potential energy is remarkably similar to the case where vortical modes are present (figure \ref{fig:seb-no-proj}). This is not surprising since the vertical velocity is fully contained in waves modes, and not in vortical modes. Despite the fact that $\tilde{\mathcal{B}}$ is important only when $\chi_{\kk} < 10$ and $k < k_O$, we observe that it is whether positive or negative. Consequently, if waves certainly exist in this range, they cannot subsist because kinetic energy is converted into potential energy or vice-versa. Nevertheless, we observe that $\tilde{\mathcal{B}}$ fluctuates in time in the buoyancy range. The region with $\chi_{\kk} > 10$ is marked by wave-breaking and strong vortices. Therefore, these simulations are unlikely to correspond to a WWT regime, even in the buoyancy range and without vortical modes.
\begin{figure}[H]
\includegraphics[width=1.0\textwidth]{../tmp/fig_seb_regimes_L_projpoloidal}
\caption{Spectral energy budget for simulation $N=40$, $\R_i=20$, without vortical modes. See figure \ref{fig:seb-no-proj} for definitions of the lines. \label{fig:seb-proj}}
\end{figure}
\subsection{Spatiotemporal analysis in the buoyancy range}
\label{subsec:khkzomega}
From the previous subsection, it is expected that waves can dominate (but cannot necessarily sustain) not too far from the buoyancy range ($k \leq k_b$), even for the simulations without vortical modes. To assess further the presence and degree of non-linearity of waves, we performed a spatiotemporal analysis for small $k$. On figure \ref{fig:omega-L}, we represent relevant quantities computed from temporal spectra for several $(k_h, k_z)$. The first one is the normalized temporal spectral $E_{equi}(k_h, k_z, \omega)/ \max\limits_{\omega} E_{equi}(k_h, k_z, \omega)$ which indicates how the energy is distributed among temporal scales for a given spatial scale. The second one is the ratio $E_{equi}(k_h, k_z, \omega)/E(k_h, k_z, \omega)$ which indicates how much energy is contained in $E_{equi}$ for a given $(k_h, k_z, \omega)$. For a system of internal gravity waves, we should observe a pic of $E_{equi}$ around $\omega = \ok$ (for the energy to be concentrated around the linear frequency), and $E_{equi}/E$ should be close to unity (for the energy to be concentrated in waves modes). \\
For these simulations of strongly stratified turbulence, we observe that the concentration around the linear frequency mostly depends on the ratio $k/k_b$. As expected, the pics of energy around $\omega = \ok$ is more pronounced when $k \ll k_b$ (figure \ref{fig:omega-L} (a) and (b)). This is an indication that linear waves effectivelly dominate at $k \ll k_b$, with or without vortical modes. When $k \simeq k_b$, energy starts to spread at all temporal scales: we observe a plateau up to $\ok$, then the energy decreases, indicating that structures at temporal scales smaller than the linear waves are more dissipated. We note that removing vortical modes from the dynamics has no strong effect on the concentration of $E_{equi}$ in temporal scales: despite the fact that spectra are more picked for some modes at intermediate $k/k_b$, but this is not the case in general. This could be due to the fact that removing vortical modes forbid interactions between modes, and tends to make discretization effects more important. Yet, simulations without vortical modes naturally have bigger $E_{equi}/E$ at all spatial and temporal scales (figure \ref{fig:omega-L} (d)) when compared to simulations with vortical modes (figure \ref{fig:omega-L} (c)). \\
\begin{figure}[H]
\includegraphics[width=1.0\textwidth]{../tmp/fig_spectra_slices_omega_kh_kz_regimes_L}
\caption{Spatiotemporal analysis of the simulations with $N=40$ and $\R_i=20$. $E_{equi}(k_h, k_z, \omega)/E(k_h, k_z, \omega)$ for the simulation with vortical modes (a) and for the simulation without vortical modes (b). $E_{equi}(k_h, k_z, \omega)/ \max\limits_{\omega} E_{equi}(k_h, k_z, \omega)$ for simulation with vortical modes (c) and for simulation without vortical modes (d). Only some couples $(k_h, k_z)$ are shown. Lines' colors corresponds to different values of $k/k_b$. The vertical black lines corresponds to $\omega = \ok$.
\label{fig:omega-L}}
\end{figure}
A way to measure the dispersion of temporal scales is to compute the empirical frequency $\omega_{emp}$ and the deviation from linear waves frequency $\delta \ok$, computed with the waves energy spectra. Here, we estimate them using the equipartition energy spectra such that
\begin{equation}
\omega_{emp}(k_h, k_z) = \frac{\sum\limits_{\omega} ~ \omega ~ E_{equi}(k_h, k_z, \omega)}{\sum\limits_{\omega} ~ E_{equi}(k_h, k_z, \omega)} ~~ \text{and} ~~
\delta \ok(k_h, k_z) = \sqrt{ \frac{\sum\limits_{\omega} ~ (\omega - \ok)^2 ~ E_{equi}(k_h, k_z, \omega)}{\sum\limits_{\omega} ~ E_{equi}(k_h, k_z, \omega)}}.
\end{equation}
The quantity $\delta \ok / \ok$ is another estimation of the ratio between the period of linear waves and the characteristic time of non linear interactions. Contrary to the non-linearity parameter $\chi_{\kk}$, which is derived by dimensional analysis, $\delta \ok/ \ok$ requires the knowledge of the spatio-temporal spectra. $\delta \ok/ \ok$ is called non-linear broadening, and is particularly important in the the WWT theory: for $\delta \ok/ \ok \gg 1$, waves' dynamics is strongly affected by non-linear interactions, while when $\delta \ok/ \ok \ll 1$, waves can propagate with only weak non-linear perturbations and the theory could hold \cite{nazarenko_wave_2011}. For a system of non-interacting linear waves, both $|\omega_{emp}-\ok|/N$ and $\delta \ok/\ok$ should be zero. Figure \ref{fig:nonlinear-L} shows this quantities in the $(k_h, k_z)$ plane. It shows that the empirical frequency is close to $\ok$, and that the nonlinear broadening is small if $k \ll k_b$ and $\chi_{\kk} \leq 1/3$. In fact, we observed that the limit of the region where waves dominate depends both on $\chi_{\kk}$ and $k/k_b$. This is slightly different than the conclusions drawn in \cite{yokoyama_energy-based_2019}, where this region was expected to be delimited by $\chi_{\kk} \leq 1/3$ only. However, it is important to remember that we use a different forcing mechanism in our simulations, a smaller aspect ratio, and shear modes are truncated. $\omega_{emp}$ deviates significantly from $\ok$ and $\delta \ok/\ok$ becomes large when $k \gg k_b$ or $\chi_{\kk} > 3$. As we observed in figure \ref{fig:omega-L}, removing vortical modes does not have a significant direct effect on $\omega_{emp}$ and $\delta \ok/\ok$. However, the simulations without vortical modes tend to have smaller $F_h$, which increases the size of the buoyancy range when compared to the simulations with vortical modes.
\begin{figure}[H]
\includegraphics[width=1.0\textwidth]{../tmp/fig_nonlinear_regimes_L}
\caption{$\omega_{emp}/N$ vs $\omega_{\kk}/N$ for simulation with vortical modes (a) and without vortical modes (b). $|\omega_{emp} - \ok|/N$ for simulation with vortical modes (c) and without vortical modes (d). $\delta \ok/\ok$ for simulation with vortical modes (e) and without vortical modes (f). The simulations are the same that in figure \ref{fig:omega-L} and legend is the same that in figures \ref{fig:seb-no-proj} and \ref{fig:seb-proj}.
\label{fig:nonlinear-L}}
\end{figure}
\subsection{Simulations at higher aspect ratio, in the buoyancy range}
\label{subsec:ratio-one}
We saw that waves dominate if $k \ll k_b$, $\chi_{\kk} < 1/3$, and when vortical modes are removed. A first conclusion to draw from that is that performing simulation at high buoyancy Reynolds number is not mandatory to observe a waves dominated region in the spectral space. If we are interested in observing an internal gravity waves system, it is therefore tempting to reduce $\R$ in numerical simulations or experiments. Yet, our previous simulations also reveal that $\tilde{\mathcal{B}}(k_h, k_z)$ in the region $k < k_b, \chi_{\kk} <1/3$ is positive or negative, but does not fluctuate around zero, as a system of statistically stationary waves should do. This is an indication that waves certainly do not survive for a long time in these simulations. It appears that this remains true for all the simulations with aspect ratio one that we performed. Yet, this fact could be due to the small aspect ratio of the simulations. To understand this, we can consider the linear inviscid Boussinesq equations without forcing (equations (\ref{eq:Continuity}-\ref{eq:Buoyancy}) with $\nu = \kappa = 0$, $\ff = \boldsymbol{0}$, and no nonlinear terms) and scale the variables as follow
\begin{align}
&t \rightarrow \frac{t}{N}, ~~ x \rightarrow \frac{x}{L_h}, ~~ y \rightarrow \frac{y}{L_h}, ~~ z \rightarrow \frac{z}{L_z}, \\
&v_x \rightarrow \frac{v_x}{U_h}, ~~ v_y \rightarrow \frac{v_y}{U_h}, ~~ v_z \rightarrow \frac{v_z}{U_z}, \\
&p \rightarrow \frac{p}{P}, ~~ b \rightarrow \frac{b}{B},
\end{align}
Where $P$ and $B$ are nominal values for the pressure and the buoyancy. Then, an order of magnitude analysis leads to (i) $U_h/L_h = U_z/L_z$ for equation (\ref{eq:Continuity}); (ii) $P = U_h L_h N$ for the horizontal part of equation (\ref{eq:Impulsion}); (iii) $B=NU_z$ for equation (\ref{eq:Buoyancy}). For a system of internal gravity waves, the vertical velocity is due to buoyancy variations, which constitute the forcing. Due to the vertical component of equation (\ref{eq:Impulsion}), the buoyancy forcing should be compensated by the temporal evolution of the vertical velocity, or by the vertical pressure gradient, giving
\begin{equation}
B = \max \left\{ \frac{P}{L_z}, N U_z \right\} ~~ \Rightarrow ~~
N U_z = \max \left\{ N U_z \left(\frac{L_h}{L_z}\right)^2, N U_z \right\},
\end{equation}
which can hold only if $L_h \leq L_z$. The last condition simply means that if the relevant vertical lengthscale is too small compared to the horizontal one, the vertical pressure gradient becomes too important which breaks the weak non-linearity assumption. Consequently, performing numerical simulations or experiments with not too small aspect ratio is important when trying to observe WWT in stratified flows. For this reason, we performed additional numerical simulations with aspect ratio one (tables \ref{table-better-simuls-ratio-one} and \ref{table-better-simuls-proj-ratio-one}). In order to excite waves more easily, we also decided to move the forcing region at larger angles $\thk$ and smaller $k_h/\delta k_h$, for which $\chi_{\kk}$ is smaller. \\
We observed that not all simulations are in statistically steady state, i.e. the energy injection rate stays bigger that the energy dissipation rate at the end of some runs (not shown) \todo{Still working on this... It happens only for simulations with $N\geq50$. However, the energy is almost constant in all runs.}. Simulations with $N \leq 20$, and the simulation with $N=80$ without vortical modes are all in steady state. However, we still observe a very slow increases of the energy in other simulations with $N \geq 50$. Because of the computational cost, we did not run simulations for longer times.
\begin{figure}[H]
\includegraphics[width=1.0\textwidth]{../tmp/fig_ratio_E_vs_Fh_ratio_one}
\caption{(a) Ratio of vortical modes energy for simulations with aspect ratio one and with vortical modes vs $F_h$. (b) Relative difference between poloidal and potential energies for simulations with aspect ratio one, with or without vortical modes vs $F_h$.
\label{fig:ratio-one-global-energy}}
\end{figure}
Figure \ref{fig:ratio-one-global-energy} represents the global energy budget for simulations with aspect ratio one. Here, all the simulations have the same hyperviscosity. Because $\nu = 0$ in these simulations, we indicate $\R_4$ in color scale instead of $\R$ in figure \ref{fig:global-energy}. Since $\nu_4$ is kept constant, decreasing $F_h$ tends to decrease $\R_4$ and the ratio $k_b \eta_4$, meaning that dissipation occurs near or in the buoyancy range at highest stratification. This is consistent with previous numerical simulations where it was observed that the energy of eddies dominate for $\R \gg 1$, and that energy of waves dominate for $\R \ll 1$ \cite{lam_energy_2021}. It is also consistent with recent experiments that suggest that the WWT regime should exist for $\R$ not too large \cite{rodda_experimental_2022}. This explain why, contrary to simulations with smaller aspect ratio (figure \ref{fig:global-energy}), $\tilde{E}_{toro}$ stays relatively small ($\leq 0.5$) even at low $F_h$ in these new simulations. Moreover, we observe that $\tilde{\mathcal{D}} \simeq 0$ which means that these new simulations are also characterized by a good global balance between poloidal energy and potential energy, removing vortical modes or not.
\begin{figure}[H]
\includegraphics[width=1.0\textwidth]{../tmp/fig_seb_ratio_one_transition}
\caption{Dimensionless conversion to potential energy $\tilde{\mathcal{B}}$ for simulations with aspect ratio one at different stratifications. $N=10$ with vortical modes $(a)$ and without vortical modes $(b)$. $N=20$ with vortical modes $(c)$ and without vortical modes $(d)$. $N=80$ with vortical modes $(e)$ and without vortical modes $(f)$. See figure \ref{fig:seb-no-proj} for definitions of lines.
\label{fig:ratio-one-B}}
\end{figure}
Figure \ref{fig:ratio-one-B} shows the dimensionless conversion to potential energy $\tilde{\mathcal{B}}$ for simulations with aspect ratio one. For low stratification, i.e. $N=10$ (figures \ref{fig:ratio-one-B} $(a)$ and $(b)$), the behaviour of $\tilde{\mathcal{B}}$ is very similar to what we observed for strongly stratified turbulent simulations at lower aspect ratio: despite the fluctuations in the buoyancy range, $\tilde{\mathcal{B}}$ tends to be positive or negative, indicating that the waves' kinetic energy is significantly converted into potential energy or conversely. For intermediate stratification, i.e. $N=20$ (figures \ref{fig:ratio-one-B} $(c)$ and $(d)$), fluctuations of $\tilde{\mathcal{B}}$ grow significantly in the buoyancy range. This coincide with the point at which the forcing region starts to lie in the region $\chi_{\kk} \leq 1/3$. For higher stratification, i.e. $N=80$ (figures \ref{fig:ratio-one-B} $(e)$ and $(f)$), fluctuations of $\tilde{\mathcal{B}}$ are very important in the buoyancy range. The temporal window for averaging is kept constant and is of 2 units time for all runs. We see that at this stratification, the forcing region fully lie in $\chi_{\kk} \leq 1/3$, suggesting that injected energy can be converted into waves even more efficiently. This also means that the energy injected energy will less feed non waves structures which could cascade among scales and perturb the all spectra. In other words, when the forcing region fully lie in $\chi_{\kk} \leq 1/3$, we can start to observe a sub-inertial range between the forcing region and $\chi_{\kk} \leq 1/3$, $k\leq k_b$ in which waves dominate and can survive for longer times. An indication of such a transition is also visible on Figure 6 of \cite{yokoyama_energy-based_2019} which shows the ratio between poloidal energy and total kinetic energy for various stratifications. The authors stressed out the importance the line $\chi_{\kk} < 1/3$ and discussed it in detail. However, it seems that they did not pointed out that $E_{polo}/(E_{polo} + E_{toro})$ starts to be really close to one only if the forcing region lies in $\chi_{\kk} \leq 1/3$. As explained in the introduction, it is important to have a large region with small values of $\chi_{\kk}$ in WWT perspective. Our simulations illustrate that it is also important to have the forcing in that region. Note that one could also think that this transition is due to the fact that the Ozmidov wave-vector becomes larger than the dissipative scale. Yet, our simulations with smaller aspect ratio show that it is not the case. For more, $k_{\eta} / k_O$ is known to be related to $\R$, and determine whether small isotropic turbulence range can develop or not \cite{bartello_sensitivity_2013}.
\begin{figure}[H]
\includegraphics[width=1.0\textwidth]{../tmp/fig_nonlinear_ratio_one}
\caption{Non-linear broadening for simulations with aspect ratio one at different stratifications. $N=10$ with vortical modes $(a)$ and without vortical modes $(b)$. $N=20$ with vortical modes $(c)$ and without vortical modes $(d)$. $N=80$ with vortical modes $(e)$ and without vortical modes $(f)$. See figure \ref{fig:seb-no-proj} for definitions of lines.
\label{fig:ratio-one-nonlinear}}
\end{figure}
Figure \ref{fig:ratio-one-nonlinear} shows the non-linear broadening for simulations with aspect ratio one and different stratifications. The more we increase $N$, the lower is the value of $\delta \ok/\ok$, and the largest is the waves' dominated region. We observed the same tendency for simulations at smaller smaller aspect ratios.
\begin{figure}[H]
\includegraphics[width=1.0\textwidth]{../tmp/spectra_kh_kz_ratio_one}
\caption{Slices of $E_{equi}(k_h,k_z) \times k_h^{5/3}$ for simulations with aspect ratio one at different stratifications. $N=10$ with vortical modes $(a)$ and without vortical modes $(b)$. $N=20$ with vortical modes $(c)$ and without vortical modes $(d)$. $N=80$ with vortical modes $(e)$ and without vortical modes $(f)$. The lines' colors correspond to different values of $k_z/k_b$. The black dotted line is $k_b$, the black dashed line is $k_O$, the green dashed line is the dissipative scale, and the orange dashed lines is the maximal horizontal wave-vectors of the forcing.
\label{fig:ratio-one-spectra}}
\end{figure}
Figure \ref{fig:ratio-one-spectra} shows slices of the compensated spatial spectra of $E_{equi}$ for simulations with aspect ratio one and different stratifications. The first important point is that simulations without vortical modes give very similar $E_{equi}$ spectra than simulations with vortical modes in the inertial range: panels $(a)$, $(c)$, and $(e)$ are very similar to panels $(b)$, $(d)$, and $(f)$ respectively. The second important point is that spectra change drastically with stratification: for a constant $k_z$, $E_{equi}(k_h,k_z)$ goes from something close to $k_h^{-5/3}$ for $N=10$ to something slightly shallower than $k_h^{-4}$ for $N=80$ and $k_z \leq k_h \leq k_b$. These simulations at high stratifications are therefore not consistent with the vertically homogeneous stationary solution of the internal gravity wave kinetic equation, obtained recently by \cite{dematteis_downscale_2021, dematteis_origins_2022}. In their work, they predicted a wave energy spectra of the form $E_{waves}(k_h,k_z) = n(k_h, k_z) \ok \sim k_h^{-2.69} k_z^{-1}$ in the hydrostatic limit. This theoretical solution is obtained by considering contributions from both extreme scale separated interactions and interactions that are quasi-collinear in horizontal wave-number. For $k_h \leq k_z$, $E_{equi}(k_h,k_z)$ does not follow a power law in the buoyancy range and it is not separable, i.e. $E_{equi}(k_h, k_z) \neq f_h(k_h) f_z(k_z)$ (otherwise, all lines of different colors on Figure \ref{fig:ratio-one-spectra} should have been parallel). The same observation were made for simulations of the reduced equations with rotation in the hydro-static limit \cite{lvov_nonlinear_2009}.
\begin{figure}[H]
\includegraphics[width=1.0\textwidth]{../tmp/spectra_kh_kz_ratio_one_2D}
\caption{$E_{equi}(k_h,k_z)$ for simulations with aspect ratio one at different stratifications. $N=10$ with vortical modes $(a)$ and without vortical modes $(b)$. $N=20$ with vortical modes $(c)$ and without vortical modes $(d)$. $N=80$ with vortical modes $(e)$ and without vortical modes $(f)$. The gray dotted and dashed lines correspond to $\chi_{\kk} = 1/3$ and $\chi_{\kk} = 3$, the continuous black line to $k_z = k_h$, the black dotted line to $k_b$, the black dashed line to $k_O$, and the green dashed line to the dissipative scale. The orange line corresponds to the maximal wave-vector modulus of the forcing.
\label{fig:ratio-one-spectra-2D}}
\end{figure}