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Commit 80afc55fd866 authored by Pierre Augier's avatar Pierre Augier
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Small corrections for rebut1

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vortical modes energy increases with $\R$ when $\R \lesssim 1$, as explained by
\cite{lam_energy_2021}. Yet, the study of the wave energy ratio is not done while
keeping $F_h$ or $\R$ constant in \cite{lam_energy_2021}. To our knowledge, additional
studies are needed to confirm if the wave energy ratio is a decreasing function of $\R
< 1$ at a constant $F_h \ll 1$, and if a threshold below which both shear and vortical
studies are needed to confirm if the wave energy ratio is for $\R\ll 1$ and $F_h \ll 1$
a decreasing function of $\R$, and if a threshold below which both shear and vortical
modes are stable exists.}
\begin{figure}
......
......@@ -106,7 +106,7 @@
required by WWT?}
Forcing low frequency waves is mainly motivated by geophysical applications: in the
oceans, waves are generated by tidal flow over topography leading to internal tides
oceans, tidal flows over topography generate low frequency waves (internal tides)
\cite[]{mackinnon_climate_2017}. This forcing is particularly important when compared
to other mechanisms, and motivate many studies on stratified flows. For example,
radiation and dissipation of internal waves generated by the barotropic $M_2$
......@@ -174,7 +174,7 @@
be analyzed carefuly. We checked that these simulations do not change the results
presented here.}}
We can show here a re-plotting of Figure 3 of our manuscript while keeping only DNS
We show here a re-plotting of Figure 3 of our manuscript while keeping only DNS
($\kmax\eta \geq 1$):
\begin{figure}[h]
......@@ -236,8 +236,8 @@
waves with a positive average. In our opinion it is nicer if the forcing is time
correlated and do not force wave modes at $\omega > \omega_k$ (which is why we use a
time correlated forcing), but anyway the system responds with its own dynamics to a
forcing, and as long as poloidal velocity (or buoyancy) are forced, internal waves are
forced. Note that for large $N$, the forcing is weak compared to linear terms. As a
forcing, and as long as poloidal velocity (or buoyancy) are forced, internal waves can
be forced. Note that for large $N$, the forcing is weak compared to linear terms. As a
result, we clearly see in the spatio-temporal poloidal spectra that the waves ($\omega
\simeq \omega_k$) strongly dominate the forced wavenumbers.
......@@ -249,10 +249,10 @@
The flow is forced at large spatial scales $ \left\{\kk ~ | ~ 5 \leq k/\Delta k_h \leq
20 \right\}$ and small angle $\left\{\kk ~ | ~ |\ok /N - \sin \theta_f| \leq 0.05
\right\}$ where $\sin \theta_f = 0.3$, meaning that relatively slow internal waves are
forced. \Remove{The correlation time of the forcing is equal to the period of the
forced waves $T_c = 2\pi /(N \sin \theta_f)$.} The forcing scheme is described in
Appendix~A. \Add{It is not harmonic, and not given by a stochastic differential
equation either. Instead, the phase and the amplitude of the forced wave numbers are
randomly changed every renewal time $T_c$ equal to the period of the forced waves $2\pi
/(N \sin \theta_f)$.}}
forced. The forcing scheme is described in Appendix~\ref{appendix:forcing}. \Add{It is
neither harmonic nor given by a stochastic differential equation. Instead, a time
correlated forcing is computed via generations of pseudo random numbers and time
interpolations.} Its correlation time is equal to the period of the forced waves $T_c =
2\pi /(N \sin \theta_f)$. \Add{The forcing is normalized such that the kinetic
injection rate $P_K$ is always equal to 1.}}
......@@ -258,5 +258,5 @@
and in Appendix~A.
and in Appendix~A, which has also been improved.
\itemit{Figure 7: could you show the spatial structure of the dominant vortical mode by
......@@ -291,7 +291,7 @@
\caption{\Add{Snapshots of the buoyancy fields for simulations $(N,\R_i)=(40,20)$ with
$\rm (a)$ and without $\rm (b)$ vortical modes.} \label{fig:buoyancy_fields}}
\end{figure}
We also add the following text to describe this figure:
We also added the following text to describe this figure:
\addtoman{\Add{Figure~4 shows the buoyancy fields for these two simulations with the
same color scale. We observe that the flow is layered in the vertical direction and
......@@ -317,7 +317,7 @@
For studies motivated by geophysical flows, the focus is more on large buoyancy
Reynolds $\R$ number, or at least not very small. This is why most of recent
simulations are for $1 \lesssim \R$ (see e.g. \cite{kimura_energy_2012,
simulations are for $\R \gtrsim 1$ (see e.g. \cite{kimura_energy_2012,
bartello_sensitivity_2013, maffioli_vertical_2017}). Some older studies could have fill
the gap between our study and the one of \cite{reun_parametric_2018}, possibly because
the authors were forced to use larger viscosity at that time
......@@ -344,11 +344,11 @@
increases with $\R$ in their simulations. It suggests that vortical modes energy
increases with $\R$ when $\R \lesssim 1$, as explained by [42]. Yet, the study of the
wave energy ratio is not done while keeping $F_h$ or $\R$ constant in [42]. To our
knowledge, additional studies are needed to confirm if the wave energy ratio is a
decreasing function of $\R < 1$ at a constant $F_h \ll 1$, and if a threshold below
which both shear and vortical modes are stable exists.}}
knowledge, additional studies are needed to confirm if the wave energy ratio is for
$\R\ll 1$ and $F_h \ll 1$ a decreasing function of $\R$, and if a threshold below which
both shear and vortical modes are stable exists.}}
\itemit{The shear modes are always suppressed while the vortical modes are either left
or suppressed. Do the authors expect surprising behaviours with the shear modes but
without the vortical modes?}
......@@ -350,9 +350,9 @@
\itemit{The shear modes are always suppressed while the vortical modes are either left
or suppressed. Do the authors expect surprising behaviours with the shear modes but
without the vortical modes?}
We know thanks to a study on 2D turbulence \cite{linares_numerical_2020} that shear
We know thanks to a study on 2D turbulence \cite[]{linares_numerical_2020} that shear
modes can grow without toroidal modes. Therefore we can anticipate accumulation of
energy in shear modes, which should become very strong and distore the waves, at least
for large $\R$.
......
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