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\title{Reply to referee 2}
\author{}
\date{}

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\noindent The authors thank the referee for his/her useful and constructive comments
that helped us to improve the manuscript. Below we have responded to all questions.
Corresponding corrections are made in blue in the new draft.

\subsection*{Major comments}

\begin{enumerate}

\itemit{The methods section is vague about the choice of vertical domain size and
vertical resolution. It should be stated clearly here how $L_z$ is varied with $N$ and
why, and how $n_z$ is chosen.}

We add the following explanations in the method section:

\Add{``We decrease $L_z$ with $N$. Typically, $L_z \propto 1/N$, while $L_h$ is kept
constant for all simulations. More precisely, the aspect ratio $L_z/L_h$ is $1/2$ for
$N \leq 20$, $1/4$ for $N \leq 60$, and $1/8$ for $N \geq 80$.  This choice is
motivated by the fact that the unforced and undissipated Boussinesq equations are
self-similar in the limit $F_h \rightarrow 0$, with similarity variable $zN/U$, where
$U$ is the typical velocity \cite{billant_self-similarity_2001}. In that way, we
simulate few layers (of height $L_b = U/N$) for all our simulations. We chose $n_z$ in
order to have an isotropic mesh in physical space, i.e. $L_z/n_z = L_h/n_h$.''.}
\itemit{The poloidal forcing is meant to excite only waves but you do not discuss its
impact on the potential vorticity (PV), which governs the vortical mode.  In fact I was
surprised that you do not discuss PV in your paper more generally, given how important
a concept it is in geophysical fluid mechanics.}

PV is known to be much less dynamically important in stratified flows than in
stratified and rotating flows. The Ertel PV $\bOmega \cdot \bnabla (N^2 z + b) =
\Omega_z (N^2 + \partial_z b) + \bOmega_h \cdot \bnabla_h b$ is not quadratic and is
not equal to the Charney PV. There is no PV cascade like in quasi-geostrophic flows.

The forcing in poloidal velocity produces and removes PV but we do not know what can be
deduced from that.

We add some facts concerning Potential Vorticity (PV) and Potential Enstrophy (PE) in
the method section:

\Add{``We also note $\bOmega = \bnabla \times \vv$ the vorticity. Without dissipation
and forcing, equations (7-9) conserve the total energy $E = \int ~ \left[
\frac{\vv^2}{2} + \frac{b^2}{2N^2} \right] ~ \diff x \diff y \diff z$ and the potential
vorticity $\Pi = \bOmega \cdot \left( N^2 \eez + \bnabla b \right)$ is a Lagrangian
invariant \cite{bartello_geostrophic_1995}. It follows that the spatial average of any
function of $\Pi$ is conserved. As a special case, the potential enstrophy}
\Add{\begin{align*} V &\equiv \frac{1}{2} \int ~ \Pi^2 ~ \diff x \diff y \diff z
\\         &= \frac{1}{2} \int ~ N^4 \Omega_z^2 ~ \diff x \diff y \diff z + \int ~
N^2 \Omega_z \bOmega \cdot \bnabla b ~ \diff x \diff y \diff z + \frac{1}{2} \int ~
\left( \bOmega \cdot \bnabla b \right)^2 ~ \diff x \diff y \diff z \\ &\equiv V_2 +
V_3 + V_4 \end{align*}}
\Add{is an invariant of equations (7-9) if no dissipation and no forcing. For a flow
without vertical vorticity $V_2 = V_3 =0$.''}

We also give the relation between vortical modes and vertical vorticity, $\bOmegakz = i
Pierre Augier's avatar
Pierre Augier committed
\vvt \sin \thk$ in the text. We also refer to the study of \cite{waite_potential_2013}
in the discussion section:

\Add{``We observe that \cite{waite_potential_2013} also attained very small $\R$. In
this study, the authors forced vortical modes so their simulations are not well suited
for WWT. Yet, they showed that potential enstrophy tends to be quadradic (i.e. $V
\simeq V_2$) for $F_h, \R \ll 1$ and that $V_2 \propto \int ~ \Omega_z^2 ~ \diff x
\diff y \diff z$ increases with $\R$ in their simulations. It suggests that vortical
modes energy increases with $\R$ when $\R \lesssim 1$, as explained by
\cite{lam_energy_2021}''}
\itemit{I felt overwhelmed by the great number of non-dimensional parameters introduced
here; even with all these definitions one could still invent more, ie parameters based
on the hyperviscosities.  Basic physical insight is lost for me here, in particular as
I would like to see parameters based on controllable parameters such as the energy
input. It’s a personal choice/failing but I lost interest in the detailed numerical
diagnostics once this parameter zoo unfolded.}

We have removed the definition of the Reynolds number from equation (17) and put it in
the text (\Add{``We can also use the Reynolds number $Re = U_h^4/(\nu \epsK) = \R
F_h^{-2}$ to characterize the flow.''}) in order to make the choice of the two main
dimensionless numbers more explicit.
Only few of our simulations remain affected by hyperviscosity. They correspond to
$\kmax \eta \lesssim 1$. We checked that these simulations don't change the results
presented in this study (see answer to reviewer 1) so the relevant dimensionless
numbers are only the horizontal Froude and the buoyancy Reynolds numbers defined in
equation (17).

We introduce many physical quantities in equations (22-29). This constitutes a
technical part that is rather difficult to read. Yet, all the presented quantities are
useful (and used in the manuscript) to discuss the spectral energy budget. In our
opinion, removing some of this diagnostic would make the manuscript unclear and less
precise.

- TODO: Move end of page 7 later...


\itemit{You talk of “removing vortical modes”.  How is that done numerically?  Is it a
similar process as in Holmes-Cerfon et al. in JFM 2013?}

The mentioned paper account for rotation (Coriolis parameter $f \neq 0$) and employ a
damping of geostrophic modes. Here, we project the nonlinear terms in the velocity
equation on the poloidal manifold, forbidding energy transfer to vortical modes. Shear
modes are removed in the same way. We add the following sentence in the method section:

\Add{``For simulations without vortical modes, we also set the toroidal projection of
nonlinear transfers to zero. \citet{smith_generation_2002} used a similar procedure in
order to distangle the roles of potential vorticity modes and waves modes in rotating
stratified turbulence.''}
\itemit{In the discussion session you say that the energy conversion rate does not
fluctuate around zero.  As only the kinetic energy is forced, shouldn’t the KE->PE
conversion rate be nonzero a priori?}

The reviewer is right: the \ul{spatially integrated} KE->PE conversion rate is nonzero
because the energy is injected under the form of kinetic energy. Yet, we refer here to
the KE->PE conversion spectra, shown in Figure 10. If WWT apply, we expect a range of
$(k_h,k_z)$ for which the KE->PE conversion is zero in average, even if the spatially
integrated KE->PE conversion rate is not. We have modify this sentence in order to be
clarify this point:

\Add{``The spectral energy budget reveals that \Add{there is no range in $(k_h,k_z)$
for which the} conversion between kinetic energy and potential energy \Remove{do not
show fluctuations} \Add{fluctuates} around zero, as we would expect for a system of
statistically stationary waves.''}
\subsection*{Minor comments}

\begin{enumerate}

\itemit{(1) why introduce a symbol $k_b$ here, which is non-standard anyway?  This
conditions is $k U_h/N << 1$, which is a standard criterion for horizontal phase
speeds, not scale separation as stated here.}

The buoyancy scale $L_{\rm b} = U_h/N$ and its inverse, the buoyancy wave vector
$k_{\rm b} = N/U_h$, are well established concepts in the study of stratified
turbulence (see e.g.
\cite{billant_self-similarity_2001,kimura_energy_2012,maffioli_vertical_2017,yokoyama_energy-based_2019}).
$L_{\rm b}$ corresponds physically to the typical vertical displacement of fluid
particles with respect to their floatability level, which is also the size of the
layers observed in stratified turbulent flows \cite{waite_stratified_2011}.
Consequently, the condition $k U_h/N = k / k_{\rm b}\ll 1$ can be seen as a
length-scale separation.
\itemit{(3) maybe it could be stated how these are derived from each other?  Also, the
folklore in oceanography is that the internal wave energy flows downscale in wavenumber
but upscale in frequency. Do the quoted WWT theories agree with that?}

We explain how the change of coordinates $(k_h, k_z) \rightarrow (\omega, k_z)$ is done
after equation (3) by adding \Add{``... where the change of coordinates is defined by
the dispersion relation $E(\omega, k_z) = E(k_h,k_z) \left( \partial \ok / \partial k_h
\right)^{-1}$.''}. The question of energy transfers in anisotropic wave systems is a
subtle question. For example, the direction of the energy cascade depends on the
considered triad for inertial wave turbulence \cite[]{david_locality_2023}. Generally
speaking, the transfers also depend on the exponents of the energy spectrum. In the
case of stratified flows, \cite{dematteis_origins_2022} argued that the global energy
flux is direct in both $k_z$ and $\omega$ for some spectra due to the dominance of
Induced Diffusion (ID) contribution. Yet, the Parametric Subharmonic Instability (PSI)
mechanism is responsible for inverse cascade in $\omega$ as explained in section III.D
(see Figure 15). To our opinion, a complete prediction for energy transfers (based on
WWT or a more general theory) remains to be find.
- TODO: oceans: forced waves (inertial waves, M2 and spontaneous generation from
vortices) $\omega/N$ small.


\itemit{typo p4 hyper viscosity}

This typo is corrected.
\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
velocity when we removing vortical modes. However, the reviewer's remark make us
realize that (20b) is wrong. Indeed, the dissipation still occurs in the three spatial
directions we remove vortical modes, so definition (19) does not need to adapted. We
have suppressed (20b) and corresponding text, and corrected Figure 2. It changes
$I_{\rm diss}$ 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
injection frequencies, as is typical for turbulent spectra.  Why is that?}

The energy tends to be accumulated at small $k_h/k$ (see 2-dimensional spectra on
Figure 8, and Figure 14), corresponding to slow internal gravity waves and/or vortical
modes. It follows that small frequencies have more energy than forced frequencies such
that no ``hump'' is observed on the spectra shown in Figure 4 (now Figure 5).
\itemit{The forcing renewal time $T_c$ is chosen as a wave period; why is that? Why not
force with white noise in time?}

We chose $T_c$ to be equal to the wave period to excite frequencies of the same order
than the linear internal gravity $\ok = N k_h/k$. On the contrary, a white noise would
excite all frequencies, favoring the appearance of vortical modes and non linear
structures.

\end{enumerate}

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