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change biblio style + minor changes

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...@@ -322,7 +322,7 @@ ...@@ -322,7 +322,7 @@
numerical study of \cite{le_reun_inertial_2017}. More recently, it has been observed numerical study of \cite{le_reun_inertial_2017}. More recently, it has been observed
that resonance with the container modes also prevent to observe weak internal gravity that resonance with the container modes also prevent to observe weak internal gravity
wave turbulence, and that the introduction of slightly tilted panels at the top and at wave turbulence, and that the introduction of slightly tilted panels at the top and at
the bottom of the fluid domain allows to inhibit the emergence of this modes the bottom of the fluid domain allows to inhibit the emergence of these modes
\cite{lanchon_internal_2023}. All these works point out that some wave systems can \cite{lanchon_internal_2023}. All these works point out that some wave systems can
generate non-wave motions that can severely affect the wave dynamics and should be generate non-wave motions that can severely affect the wave dynamics and should be
suppressed in experiments aiming to observe wave turbulence. In stratified flows these suppressed in experiments aiming to observe wave turbulence. In stratified flows these
...@@ -384,7 +384,7 @@ ...@@ -384,7 +384,7 @@
\label{eq:Isotropic} \label{eq:Isotropic}
E_{\rm 1D}(k) \sim \epsK^{2/3} k^{-5/3} E_{\rm 1D}(k) \sim \epsK^{2/3} k^{-5/3}
\end{equation} \end{equation}
is expected. This happens when $\kb < \ko \ll k \ll k_{\eta}$ where $\ko$ is the is expected. It happens when $\kb < \ko \ll k \ll k_{\eta}$ where $\ko$ is the
Ozmidov wave-vector and $k_{\eta}$ is the Kolmogorov wave-vector. Ozmidov wave-vector and $k_{\eta}$ is the Kolmogorov wave-vector.
...@@ -892,7 +892,7 @@ ...@@ -892,7 +892,7 @@
$F_h$ \cite{linares_numerical_2020}. The kinetic energy flux $\Pikin(k_h)$ and the $F_h$ \cite{linares_numerical_2020}. The kinetic energy flux $\Pikin(k_h)$ and the
potential energy flux $\Pipot(k_h)$ start to show a plateau over almost a decade in potential energy flux $\Pipot(k_h)$ start to show a plateau over almost a decade in
these simulations (Figure~\ref{fig:spectra_1D}$\rm (c)$-$\rm (d)$). We observe that the these simulations (Figure~\ref{fig:spectra_1D}$\rm (c)$-$\rm (d)$). We observe that the
dissipation starts to be important at the Ozmidov scale, meaning that this simulations dissipation starts to be important at the Ozmidov scale, meaning that these simulations
lie between the LAST regime and the viscosity affected regime lie between the LAST regime and the viscosity affected regime
\cite{brethouwer_scaling_2007}. This explains why we do not observe an isotropic \cite{brethouwer_scaling_2007}. This explains why we do not observe an isotropic
turbulence range (with energy spectra $\sim k^{-5/3}$) in these simulations. The main turbulence range (with energy spectra $\sim k^{-5/3}$) in these simulations. The main
......
...@@ -12,6 +12,4 @@ ...@@ -12,6 +12,4 @@
\usepackage{amssymb, amsmath} \usepackage{amssymb, amsmath}
\usepackage[top=0.8in,bottom=0.8in,left=2cm,right=2cm,a4paper]{geometry} \usepackage[top=0.8in,bottom=0.8in,left=2cm,right=2cm,a4paper]{geometry}
\usepackage[usenames,dvipsnames]{color}
\usepackage[normalem]{ulem}
...@@ -17,9 +15,5 @@ ...@@ -17,9 +15,5 @@
\usepackage{array} \title{\bf Coverletter}
\usepackage{soul}
\title{Coverletter}
\author{} \author{}
\date{} \date{}
...@@ -39,8 +33,10 @@ ...@@ -39,8 +33,10 @@
Dear Dr. Eckart Meiburg, Dear Dr. Eckart Meiburg,
We have now completed the revision of our manuscript FF10204, entitled "Internal \vspace{0.5cm}
gravity waves in stratified flows with and without vortical modes." We have attached to
this letter the revised version of the manuscript, a version with the changes \paragraph{}{We have now completed the revision of our manuscript FF10204, entitled
highlighted, and a separate point-by-point response to each referee. "Internal gravity waves in stratified flows with and without vortical modes." We have
attached to this letter the revised version of the manuscript, a version with the
changes highlighted, and a separate point-by-point response to each referee.}
...@@ -46,9 +42,9 @@ ...@@ -46,9 +42,9 @@
We have corrected the manuscript by improving the presentation of our methods. The \paragraph{}{We have corrected the manuscript by improving the presentation of our
corresponding section has been reorganized and strengthened. In particular, we provide methods. The corresponding section has been reorganized and strengthened. In
more precision about the forcing mechanism, how we use hyperviscosity and particular, we provide more precision about the forcing mechanism, how we use
hyperdiffusion, and how we varied the vertical resolution with respect to the hyperviscosity and hyperdiffusion, and how we varied the vertical resolution with
Brunt-Väisälä frequency. We have also clarified the manuscript to highlight the two respect to the Brunt-Väisälä frequency. We have also clarified the manuscript to
control parameters of this study: the horizontal Froude number and the buoyancy highlight the two control parameters of this study: the horizontal Froude number and
Reynolds number. the buoyancy Reynolds number.}
...@@ -54,7 +50,9 @@ ...@@ -54,7 +50,9 @@
Following the reviewers' suggestions, we have added visualizations of the flows \paragraph{}{Following the reviewers' suggestions, we have added visualizations of the
considered here and provided comments on the role of potential vorticity and potential flows considered here and provided comments on the role of potential vorticity and
enstrophy. We have also found additional studies that help to fill the parameter space potential enstrophy. We have also found additional studies that help to fill the
on the last figure of the manuscript, and we have commented on them in the discussion parameter space on the last figure of the manuscript, and we have commented on them in
section. the discussion section.}
\paragraph{}{Finally, we have corrected typos and small errors.}
...@@ -60,3 +58,3 @@ ...@@ -60,3 +58,3 @@
Finally, we have corrected typos and small errors. \vspace{0.5cm}
...@@ -62,3 +60,3 @@ ...@@ -62,3 +60,3 @@
We look forward to hearing from you. \paragraph{}{We look forward to hearing from you.}
...@@ -64,3 +62,3 @@ ...@@ -64,3 +62,3 @@
Best regards, \paragraph{}{Best regards,}
...@@ -66,5 +64,4 @@ ...@@ -66,5 +64,4 @@
Vincent Labarre, Pierre Augier, Giorgio Krstulovic, and Sergey Nazarenko \paragraph{}{Vincent Labarre, Pierre Augier, Giorgio Krstulovic, and Sergey Nazarenko}
\end{document} \end{document}
...@@ -12,7 +12,7 @@ ...@@ -12,7 +12,7 @@
\usepackage{graphicx} \usepackage{graphicx}
\usepackage{natbib} \usepackage{natbib}
\bibliographystyle{unsrtnat} \bibliographystyle{rusnat}
\usepackage[usenames,dvipsnames]{color} \usepackage[usenames,dvipsnames]{color}
\usepackage[normalem]{ulem} \usepackage[normalem]{ulem}
...@@ -21,7 +21,7 @@ ...@@ -21,7 +21,7 @@
\usepackage{soul} \usepackage{soul}
\title{Reply to referee 1} \title{\bf Reply to referee 1}
\author{} \author{}
\date{} \date{}
...@@ -56,14 +56,14 @@ ...@@ -56,14 +56,14 @@
\maketitle \maketitle
\noindent We thank the referee for his/her critical comments, that helped us to \noindent We thank the referee for their critical comments, that helped us to
significantly improve our manuscript. We realized that our Methods section was clearly significantly improve our manuscript. We realized that our Methods section was not
not comprehensive. This is partly due to the fact that this paper was written to cite completely comprehensive. This is partly due to the fact that this paper was written to
another article describing with another point of view the first dataset used in this cite another article describing with another point of view the first dataset used in
study (without projection). Finally, we submitted this manuscript first and did not this study (without projection). Finally, we submitted this manuscript first and did
correctly complete the Methods section. We tried to correct the manuscript by better not correctly complete the Methods section. We tried to correct the manuscript by
presenting our methods, and the corresponding section has been deeply reorganized and better presenting our methods, and the corresponding section has been deeply
strengthened. reorganized and strengthened.
The answer on all the comments are listed below. Corresponding corrections are made in The answer on all the comments are listed below. Corresponding corrections are made in
blue in the new draft. blue in the new draft.
...@@ -74,8 +74,10 @@ ...@@ -74,8 +74,10 @@
rule for dealiasing? Is it related to the use of hyperviscosity and hyperdiffusion?} rule for dealiasing? Is it related to the use of hyperviscosity and hyperdiffusion?}
Instead of the exact 2/3 rule for cubic truncation, we use a 0.8 spherical truncation. Instead of the exact 2/3 rule for cubic truncation, we use a 0.8 spherical truncation.
We performed with Jason Reneuve an extensive study on dealiasing methods (also
including phase shifting). It is not yet published but a notable result is that for
In a non published work with Jason Reneuve, we performed an extensive study on
dealiasing methods (also including phase shifting). One notable result is that for
simulations with $\kmax \eta$ slightly smaller than one, we obtain better results to simulations with $\kmax \eta$ slightly smaller than one, we obtain better results to
reproduce known flows with a 0.8 spherical truncation rather than with the exact 2/3 reproduce known flows with a 0.8 spherical truncation rather than with the exact 2/3
cubic truncation. First, a spherical truncation is better than a cubic truncation cubic truncation. First, a spherical truncation is better than a cubic truncation
...@@ -84,14 +86,13 @@ ...@@ -84,14 +86,13 @@
flow (for $\kmax \eta$ close to one, more wavenumbers to represent the dissipative flow (for $\kmax \eta$ close to one, more wavenumbers to represent the dissipative
range, in which the energy spectra decrease very quickly). On the one hand, for range, in which the energy spectra decrease very quickly). On the one hand, for
spherical truncation, 2/3 removes too much modes (Only $16\% = (4/3) \pi ((2/3)0.5)^3 spherical truncation, 2/3 removes too much modes (Only $16\% = (4/3) \pi ((2/3)0.5)^3
\%$ of the grid points are kept!). On the other hand, the coefficient needs to be \%$ of the grid points are kept). On the other hand, the coefficient needs to be
smaller than 0.94 to kill the double aliases, which are the most annoying ones. A value smaller than 0.94 to kill the double aliasing, which is the most annoying ones. A value
of 0.8 seems to be a good compromize for which most of the aliases are removed. of 0.8 seems to be a good compromize for which most of the aliasing is removed. These
arguments are valid without hyperdiffusion, which was not used for our study with Jason
These arguments are valid without hyperdiffusion, which was not used for our study with Reneuve. However, using a bit of hyperdiffusion of course helps to limit the effect of
Jason Reneuve. However, using a bit of hyperdiffusion of course helps to limit the the aliasing. Note also than anyway most simulations of our datasets are proper DNS
effect of the aliases. Note also than anyway most simulations of our datasets are with $\kmax\eta$ larger or close to one.
proper DNS with $\kmax\eta$ larger or close to one.
Since these ideas will be published elsewhere (with serious checks) and are about a Since these ideas will be published elsewhere (with serious checks) and are about a
small detail of our numerical methods, we do not think that it makes sense to include small detail of our numerical methods, we do not think that it makes sense to include
...@@ -129,7 +130,7 @@ ...@@ -129,7 +130,7 @@
spectra close to $N$.}} spectra close to $N$.}}
Yet, the referee raise an important point: forcing waves at higher frequencies would The second question is interesting: forcing waves at higher frequencies would indeed
help to satisfy the timescale separation required by WWT. In our opinion, this would help to satisfy the timescale separation required by WWT. In our opinion, this would
require additional studies. require additional studies.
...@@ -191,7 +192,7 @@ ...@@ -191,7 +192,7 @@
under-resolved, with $\kmax$ close to $1/\eta$ and in the diffusive range linked to under-resolved, with $\kmax$ close to $1/\eta$ and in the diffusive range linked to
standard dissipation. standard dissipation.
Regarding your question about the importance of hyperviscosity and the potential ``blow Regarding the question about the importance of hyperviscosity and the potential ``blow
up without it'', we can mention that, except for very small $\kmax\eta$ values, up without it'', we can mention that, except for very small $\kmax\eta$ values,
removing hyperdiffusion would not lead to simulation blow up but we would observe a removing hyperdiffusion would not lead to simulation blow up but we would observe a
tendency to thermalisation, with accumulation of energy at the larger wavenumbers. tendency to thermalisation, with accumulation of energy at the larger wavenumbers.
...@@ -231,7 +232,7 @@ ...@@ -231,7 +232,7 @@
differential equation either. The phase and the amplitude of the forced wavenumbers are differential equation either. The phase and the amplitude of the forced wavenumbers are
randomly changed in time every $T_c$ such that the kinetic energy injection rate is randomly changed in time every $T_c$ such that the kinetic energy injection rate is
equal to one. However, even if we think that your question about the effect of a purely equal to one. However, even if we think that your question about the effect of a purely
harmonic forcing deserves additional studies, we don't think that we need to have a harmonic forcing deserves additional studies, we do not think that we need to have a
purely harmonic forcing to force waves. The forcing injects or removes energy to the purely harmonic forcing to force waves. The forcing injects or removes energy to the
waves with a positive average. In our opinion it is nicer if the forcing is time 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 correlated and do not force wave modes at $\omega > \omega_k$ (which is why we use a
...@@ -249,12 +250,12 @@ ...@@ -249,12 +250,12 @@
The flow is forced at large spatial scales $ \left\{\kk ~ | ~ 5 \leq k/\Delta k_h \leq The flow is forced at large spatial scales $ \left\{\kk ~ | ~ 5 \leq k/\Delta k_h \leq
20 \right\}$ and small angle $\left\{\kk ~ | ~ |\ok /N - \sin \theta_f| \leq 0.05 20 \right\}$ 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 \right\}$ where $\sin \theta_f = 0.3$, meaning that relatively slow internal waves are
forced. The forcing scheme is described in Appendix~\ref{appendix:forcing}. \Add{It is forced. The forcing scheme is described in Appendix~A. \Add{It is neither harmonic nor
neither harmonic nor given by a stochastic differential equation. Instead, a time given by a stochastic differential equation. Instead, a time correlated forcing is
correlated forcing is computed via generations of pseudo random numbers and time computed via generations of pseudo random numbers and time interpolations.} Its
interpolations.} Its correlation time is equal to the period of the forced waves $T_c = correlation time is equal to the period of the forced waves $T_c = 2\pi /(N \sin
2\pi /(N \sin \theta_f)$. \Add{The forcing is normalized such that the kinetic \theta_f)$. \Add{The forcing is normalized such that the kinetic injection rate $P_K$
injection rate $P_K$ is always equal to 1.}} is always equal to 1.}}
and in Appendix~A, which has also been improved. and in Appendix~A, which has also been improved.
...@@ -264,8 +265,8 @@ ...@@ -264,8 +265,8 @@
More generally, some visualisations of the typical flows considered, with and without More generally, some visualisations of the typical flows considered, with and without
vortical modes, would be helpful in my opinion.} vortical modes, would be helpful in my opinion.}
Here is a 3D figure showing the toroidal velocity of the dominant vortical mode. The Figure~3 shows the toroidal velocity of the dominant vortical mode. It is obtained from
figure of the dominant vortical mode is from the simulation at resolution $n_h = 320$. the simulation at resolution $n_h = 320$.
\begin{figure}[h] \begin{figure}[h]
\centering \centering
\includegraphics[width=0.8\linewidth]{vt_filter} \includegraphics[width=0.8\linewidth]{vt_filter}
...@@ -283,9 +284,9 @@ ...@@ -283,9 +284,9 @@
In our opinion, the last figure is not useful enough to be included in the manuscript. In our opinion, the last figure is not useful enough to be included in the manuscript.
Yet, following reviewer's suggestion, we include the following figure (now figure 4) Yet, following reviewer's suggestion, we include the following figure (now Figure 4 in
showing the buoyancy field for simulations with $(N,\R_i) = (40,20)$, which are studied the manuscript) showing the buoyancy field for simulations with $(N,\R_i) = (40,20)$,
in detail in our manuscript. which are studied in detail in our manuscript.
\begin{figure}[h] \begin{figure}[h]
\centering \includegraphics[width=1\linewidth]{../../input/figure4} \centering \includegraphics[width=1\linewidth]{../../input/figure4}
\caption{\Add{Snapshots of the buoyancy fields for simulations $(N,\R_i)=(40,20)$ with \caption{\Add{Snapshots of the buoyancy fields for simulations $(N,\R_i)=(40,20)$ with
...@@ -316,7 +317,7 @@ ...@@ -316,7 +317,7 @@
these two extremes?} these two extremes?}
For studies motivated by geophysical flows, the focus is more on large buoyancy 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 Reynolds $\R$ number, or at least not very small. For this reason, most of recent
simulations are for $\R \gtrsim 1$ (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 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 gap between our study and the one of \cite{reun_parametric_2018}, possibly because
...@@ -325,7 +326,7 @@ ...@@ -325,7 +326,7 @@
waite_stratified_2011}. Unfortunately, it not always easy to correctly quantify these waite_stratified_2011}. Unfortunately, it not always easy to correctly quantify these
numbers from the data given in the articles, or to compare it with our simulations. numbers from the data given in the articles, or to compare it with our simulations.
\cite{waite_stratified_2004} forced vortical modes so these it is not well suited for \cite{waite_stratified_2004} forced vortical modes so these it is not well suited for
observing internal gravity wave turbulence. We don't see how to extract $\R$ from observing internal gravity wave turbulence. We do not see how to extract $\R$ from
\cite{waite_stratified_2004} and \cite{waite_stratified_2006} tables. In \cite{waite_stratified_2004} and \cite{waite_stratified_2006} tables. In
\cite{lindborg_energy_2006} and \cite{waite_stratified_2011}, different values of the \cite{lindborg_energy_2006} and \cite{waite_stratified_2011}, different values of the
viscosity where used on the vertical and horizontal so these studies are more difficult viscosity where used on the vertical and horizontal so these studies are more difficult
...@@ -366,8 +367,8 @@ ...@@ -366,8 +367,8 @@
geostrophic flows, there must be some threshold below which both shear and vortical geostrophic flows, there must be some threshold below which both shear and vortical
modes are stable and won't grow if not directly forced?} modes are stable and won't grow if not directly forced?}
This is indeed an important information that must be mentioned. Consequently, we add It is indeed an important information that must be mentioned. Consequently, we add the
the following sentences in the discussion: following sentences in the discussion:
\addtoman{\Add{It is worth mentioning that [46] did not have to remove shear nor \addtoman{\Add{It is worth mentioning that [46] did not have to remove shear nor
vortical modes in their simulations to observe signatures of internal wave turbulence. vortical modes in their simulations to observe signatures of internal wave turbulence.
...@@ -404,8 +405,7 @@ ...@@ -404,8 +405,7 @@
\itemit{Ambiguous notation between real part and Reynolds} \itemit{Ambiguous notation between real part and Reynolds}
We replace the notation for the real part by $\Re$ and for the imaginary part by $\Im$ We replace the notation for the real part by $\Re$ to avoid ambiguous notation.
to avoid ambiguous notation.
\end{itemize} \end{itemize}
......
...@@ -12,7 +12,7 @@ ...@@ -12,7 +12,7 @@
\usepackage{graphicx} \usepackage{graphicx}
\usepackage{natbib} \usepackage{natbib}
\bibliographystyle{unsrtnat} \bibliographystyle{rusnat}
\usepackage[usenames,dvipsnames]{color} \usepackage[usenames,dvipsnames]{color}
\usepackage[normalem]{ulem} \usepackage[normalem]{ulem}
...@@ -21,7 +21,7 @@ ...@@ -21,7 +21,7 @@
\usepackage{soul} \usepackage{soul}
\title{Reply to referee 2} \title{\bf Reply to referee 2}
\author{} \author{}
\date{} \date{}
...@@ -59,10 +59,10 @@ ...@@ -59,10 +59,10 @@
\begin{document} \begin{document}
\maketitle \maketitle
\noindent The authors thank the referee for his/her useful and constructive comments \noindent The authors thank the referee for their useful and constructive comments that
that helped us to improve the manuscript. We realized that our Methods section was helped us to improve the manuscript. We realized that our Methods section was not
clearly not comprehensive. This is partly due to the fact that this paper was written completely comprehensive. It is partly due to the fact that this paper was written to
to cite another article describing with another point of view the first dataset used in cite another article describing with another point of view the first dataset used in
this study (without projection). Finally, we submitted this manuscript first and did this study (without projection). Finally, we submitted this manuscript first and did
not correctly complete the Methods section. We tried to correct the manuscript by not correctly complete the Methods section. We tried to correct the manuscript by
better presenting our methods, and the corresponding section has been deeply better presenting our methods, and the corresponding section has been deeply
...@@ -171,8 +171,8 @@ ...@@ -171,8 +171,8 @@
which are commonly used in other studies. which are commonly used in other studies.
Only few of our simulations remain (weakly) affected by hyperviscosity. They correspond Only few of our simulations remain (weakly) affected by hyperviscosity. They correspond
to $0.45< \kmax \eta \lesssim 1$. We checked that these simulations don't change the to $0.45< \kmax \eta \lesssim 1$. We checked that these simulations do not change the
results presented in this study (see our answers to reviewer 1) so the relevant results presented in this study (see our answers to reviewer 1) so the relevant
dimensionless numbers are only the horizontal Froude and the buoyancy Reynolds numbers dimensionless numbers are only the horizontal Froude and the buoyancy Reynolds numbers
defined in equation (17). defined in equation (17).
...@@ -175,12 +175,11 @@ ...@@ -175,12 +175,11 @@
results presented in this study (see our answers to reviewer 1) so the relevant results presented in this study (see our answers to reviewer 1) so the relevant
dimensionless numbers are only the horizontal Froude and the buoyancy Reynolds numbers dimensionless numbers are only the horizontal Froude and the buoyancy Reynolds numbers
defined in equation (17). defined in equation (17).
We introduce many physical quantities in equations (21-32). This constitutes a We introduce many physical quantities in equations (21-32). It constitutes a technical
technical part that is rather difficult to read. Yet, all the presented quantities are part that is rather difficult to read. Yet, all the presented quantities are useful
useful (and used in the manuscript) to discuss the spectral energy budget. In our (and used in the manuscript) to discuss the spectral energy budget. In our opinion,
opinion, removing some of this diagnostic would make the manuscript unclear and less removing some of this diagnostic would make the manuscript unclear and less precise.
precise.
\itemit{You talk of “removing vortical modes”. How is that done numerically? Is it a \itemit{You talk of “removing vortical modes”. How is that done numerically? Is it a
...@@ -243,7 +242,7 @@ ...@@ -243,7 +242,7 @@
relation $E(\omega, k_z) = E(k_h,k_z) \left( \partial \ok / \partial k_h relation $E(\omega, k_z) = E(k_h,k_z) \left( \partial \ok / \partial k_h
\right)^{-1}$.}} \right)^{-1}$.}}
The question of energy transfers in anisotropic wave systems is a subtle question. For The question of energy transfers in anisotropic wave systems is a subtle issue. For
example, the direction of the energy transfer depends on the considered triad for example, the direction of the energy transfer depends on the considered triad for
inertial wave turbulence \cite[]{david_locality_2023}. Generally speaking, the inertial wave turbulence \cite[]{david_locality_2023}. Generally speaking, the
transfers also depend on the exponents of the energy spectrum. In the case of transfers also depend on the exponents of the energy spectrum. In the case of
...@@ -268,11 +267,11 @@ ...@@ -268,11 +267,11 @@
\itemit{(20) is an interesting definition. But I don't understand (20b)} \itemit{(20) is an interesting definition. But I don't understand (20b)}
(20a) is relevant because we remove one over two degree of freedom of the horizontal (20a) is relevant because we remove one over two degree of freedom of the horizontal
velocity when we removing vortical modes. However, the reviewer's remark make us velocity when we removing vortical modes. However, this remark make us to realize that
realize that (20b) is wrong. Indeed, the dissipation still occurs in the three spatial (20b) is wrong. Indeed, the dissipation still occurs in the three spatial directions if
directions if we remove vortical modes, so definition (19) does not need to adapted. We we remove vortical modes, so definition (19) does not need to adapted. We have
have suppressed (20b) and corresponding text, and corrected Figure 2. It changes suppressed (20b) and corresponding text, and corrected Figure 2. It changes $I_{\rm
$I_{\rm diss}$ by a factor 4/3 in Figure 2~b, so the conclusions remain the same. diss}$ by a factor 4/3 in Figure 2~b, so the conclusions remain the same.
\itemit{Figure 4: I find it surprising that there is no “hump” in the spectrum at the \itemit{Figure 4: I find it surprising that there is no “hump” in the spectrum at the
...@@ -288,9 +287,12 @@ ...@@ -288,9 +287,12 @@
force with white noise in time?} force with white noise in time?}
We chose $T_c$ to be equal to the wave period to excite frequencies of the same order We chose $T_c$ to be equal to the wave period to excite frequencies of the same order
than the linear internal gravity $\ok = N k_h/k$. In contrast, a white noise would than the linear internal gravity $\ok = N k_h/k$. Indeed, a white noise in time forcing
excite all frequencies, favoring the appearance of vortical modes and non linear could also be a good choice, provided that it is supported in a narrow sector in
structures. wave-vector space. We expect that the results are independent of the particularities of
the forcing. Furthermore, dealing with stochastic forcing is slightly more delicate in
terms of numerical integration.
\end{enumerate} \end{enumerate}
......
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