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......@@ -19,7 +19,7 @@
*.bbl
*.blg
*Notes.bib
*.xdv
*.fls
*.fdb_latexmk
......@@ -36,6 +36,8 @@
venv/
**/.pdm-python
**/_minted-*
.vscode/*
......
......@@ -23,3 +23,5 @@
black:
black -l 82 py
all: format $(NAME).pdf
......@@ -14,6 +14,8 @@
export STRAT_TURB_TORO2022=/data/MyCore/2022strat-turb-toro
```
1. Install a virtual env with `pdm sync --clean` and activate it.
1. The command `make` should produce all Latex tables, figures and finally the
main pdf file `article.pdf`.
......
\documentclass[
aps,
prb,
superscriptaddress,
reprint,
onecolumn,
amsfonts,
amssymb,
amsmath,
]{revtex4-2}
% The result seems nicer with revtex4-2, but
% Debian 9 (Stretch) does not have revtex4-2
\usepackage[utf8]{inputenc}
\usepackage{hyperref} % Required for customising links and the PDF*
\hypersetup{pdfpagemode={UseOutlines},
bookmarksopen=true,
bookmarksopenlevel=0,
hypertexnames=false,
colorlinks=true, % Set to false to disable coloring links
citecolor=blue, % The color of citations
linkcolor=red, % The color of references to document elements (sections, figures, etc)
urlcolor=black, % The color of hyperlinks (URLs)
pdfstartview={FitV},
unicode,
breaklinks=true,
}
\usepackage{graphicx}
\usepackage{grffile}
\usepackage{color}
\usepackage{array}
\usepackage{hhline}
\usepackage[]{algorithm2e}
\usepackage{booktabs}
\usepackage{ulem}
\usepackage{siunitx}
\sisetup{
inter-unit-product = \ensuremath{{}\!\cdot\!{}},
detect-all,
separate-uncertainty = true,
exponent-product = \times,
space-before-unit = true,
output-decimal-marker = {,},
multi-part-units = brackets,
range-phrase = --,
% allow-number-unit-breaks,
list-final-separator = { et },
list-pair-separator = { et },
abbreviations
}
\usepackage{float}
\linespread{1.05}
\setlength{\tabcolsep}{7pt}
\newcommand{\cor}[1]{\textcolor{red}{#1}}
\newcommand{\todo}[1]{\textcolor{red}{TODO: #1}}
\newlength{\figwidth}
\setlength{\figwidth}{120mm}
% \setlength{\figwidth}{0.7\textwidth} % useful in single column
\newcommand{\R}{\mathcal{R}}
\newcommand{\eps}{\varepsilon}
\newcommand{\epsK}{{\varepsilon_{\!\scriptscriptstyle \rm kin}}}
\newcommand{\epsKK}{{\varepsilon_{\!\scriptscriptstyle K 2}}}
\newcommand{\epsKKKK}{{\varepsilon_{\!\scriptscriptstyle K 4}}}
\newcommand{\epsA}{{\varepsilon_{\!\scriptscriptstyle \text{pot}}}}
\newcommand{\xx}{\boldsymbol{x}}
\newcommand{\rr}{\boldsymbol{r}}
\newcommand{\kk}{\boldsymbol{k}}
\newcommand{\eek}{\boldsymbol{e}_{\boldsymbol{k}}}
\newcommand{\eeh}{\boldsymbol{e}_h}
\newcommand{\eep}{\boldsymbol{e}_{p\kk}}
\newcommand{\eet}{\boldsymbol{e}_{t\kk}}
\newcommand{\eex}{\boldsymbol{e}_x}
\newcommand{\eey}{\boldsymbol{e}_y}
\newcommand{\eez}{\boldsymbol{e}_z}
\newcommand{\cc}{\boldsymbol{c}}
\newcommand{\uu}{\boldsymbol{u}}
\newcommand{\vv}{\boldsymbol{v}}
\newcommand{\vvk}{\hat{\boldsymbol{v}}_{\kk}}
\newcommand{\vvp}{\hat{v}_{p\kk}}
\newcommand{\vvt}{\hat{v}_{t\kk}}
\newcommand{\vvs}{\hat{\boldsymbol{v}}_{s\kk}}
\newcommand{\bbk}{\hat{b}_{\kk}}
\newcommand{\ff}{\boldsymbol{f}}
\newcommand{\fk}{\hat{f}_{\kk}}
\newcommand{\ffk}{\boldsymbol{\hat{f}}_{\kk}}
\newcommand{\bomega}{\boldsymbol{\omega}}
\newcommand{\bnabla}{\boldsymbol{\nabla}}
\newcommand{\Dt}{\mbox{D}_t}
\newcommand{\p}{\partial}
\newcommand{\mean}[1]{\langle #1 \rangle}
\newcommand{\epsP}{\varepsilon_{\!\scriptscriptstyle P}}
\newcommand{\epsm}{\varepsilon_{\!\scriptscriptstyle m}}
\newcommand{\CKA}{C_{K\rightarrow A}}
\newcommand{\D}{\mbox{D}}
\newcommand{\diff}{\text{d}}
\newcommand{\bv}{Brunt-V\"ais\"al\"a }
\newcommand{\kmax}{k_{\max}}
\newcommand{\thk}{\theta_{\kk}}
\newcommand{\phk}{\varphi_{\kk}}
\newcommand{\thf}{\theta_f}
\newcommand{\ok}{\omega_{\kk}}
\newcommand{\oemp}{\omega_{\text{emp}}}
\newcommand{\odoppler}{\delta \omega_{\text{doppler}}}
\newcommand{\Ivelo}{I_{\rm kin}}
\newcommand{\Idiss}{I_{\text{diss}}}
\newcommand{\Pikin}{\Pi_{\rm kin}}
\newcommand{\Pipot}{\Pi_{\text{pot}}}
\newcommand{\Etoro}{E_{\rm toro} }
\newcommand{\Epolo}{E_{\rm polo} }
\newcommand{\Epot}{E_{\rm pot} }
\newcommand{\Eequi}{E_{\rm equi} }
\newcommand{\Ediff}{E_{\text{diff}}}
\newcommand{\Ewave}{E_{\rm wave}}
\newcommand{\Ewaver}{\tilde{E}_{\rm wave}}
\setlength\parindent{0pt}
\begin{document}
\title{Internal gravity waves in stratified flows with and without vortical modes}
\author{Vincent Labarre}
\email[]{vincent.labarre@oca.eu}
\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}
\author{Giorgio Krstulovic}
\email[]{giorgio.krstulovic@oca.eu}
\affiliation{Universit\'{e} C\^{o}te d'Azur, Observatoire de la C\^{o}te d'Azur, CNRS,
Laboratoire Lagrange, Nice, France.}
\author{Sergey Nazarenko}
\email[]{sergey.nazarenko@unice.fr}
\affiliation{Universit\'{e} C\^{o}te d'Azur, CNRS, Institut de Physique de Nice -
INPHYNI, Nice, France}
\begin{abstract}
The comprehension of stratified flows is important for geophysical and astrophysical
applications. The Weak Wave Turbulence theory aims to provide a statistical description
of internal gravity waves propagating in the bulk of such flows. Yet, internal gravity
waves are usually perturbed by other structures present in stratified flow, namely the
shear modes and the vortical modes. In order to check whether a weak internal gravity
wave turbulence regime can occur, we perform direct numerical simulations of stratified
turbulence without shear modes, and with or without vortical modes at various Froude
and buoyancy Reynolds numbers. We observe that removing vortical modes naturally helps
to have a better overall balance between poloidal kinetic energy, involved in internal
gravity waves, and potential energy. However, conversion between kinetic energy and
potential energy do not necessarily show fluctuations around zero in our simulations,
as we would expect for a system of statistically stationary waves. A spatiotemporal
analysis reveals that removing vortical modes helps to concentrate the energy around
the wave frequency, but it is not enough to observe a weak wave turbulence regime. Yet,
we observe that internal gravity waves whose frequency are large compared to the eddy
turnover time are present, and we also find evidences for slow internal gravity waves
interacting by Triadic Resonance Instabilities in our strongly stratified flows
simulations. Finally, we propose conditions that should be fulfilled in order to
observe a weak internal gravity waves turbulence regime in real flows.
\end{abstract}
%----------------------------------------------------------------------------------------
% Print the title
\maketitle
%----------------------------------------------------------------------------------------
% ARTICLE CONTENTS
%----------------------------------------------------------------------------------------
\section{Introduction}
\label{sec:introduction}
\input{introduction.tex}
\section{Methods}
\label{sec:methods}
\input{methods.tex}
\section{Results}
\label{sec:results}
\input{results.tex}
\section{Discussions and conclusions}
\label{sec:conclusions}
\input{conclusions.tex}
\begin{acknowledgments}
This project was supported by the Simons Foundation through the Simons collaboration on
wave turbulence. Part of the computations have been done on the ``Mesocentre SIGAMM''
machine, hosted by Observatoire de la Cote d'Azur. 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}
\appendix
\section{Forcing scheme}
\label{appendix:forcing}
The forcing used in our simulations is prescribed by the following algorithm:
\begin{algorithm}[H]
$t=0$: generate two random numbers $f_{0\kk}, f_{1\kk} \sim \mathcal{P}$ $\forall \kk$ in the forcing region; $t_0 = 0$; \\
\While{$t \leq T$}{
$t \rightarrow t +\delta t$; \\
\If{$t - t_0 \geq T_c$}{
$t_0 \rightarrow t$; \\
$f_{0 \kk} \rightarrow f_{1 \kk}$; \\
genererate $f_{1\kk} \sim \mathcal{P}$
}{}
$\ffk = \left\{f_{0\kk} - \dfrac{(f_{1\kk} - f_{0\kk})}{2} \left[\cos\left(\dfrac{\pi(t-t_0)}{T_c} \right)+ 1 \right]\right\}~ \eep$ \\
$\ffk$ are normalized to ensure $P_K = 1$
}
%\caption{}
\end{algorithm}
where $\mathcal{P}$ is the probability law which has for distribution
\begin{equation}
p_X(x) = \begin{cases}
\frac{1}{4} ~~ \text{if} ~ Re(x) \in[-1:1] ~ \text{and} ~ Im(x) \in[-1:1] \\
0 ~~ \text{otherwise}
\end{cases}
\end{equation}
and $\delta t$ is the time increment at each time step.
\section{List of simulations}
\begin{table}
\begin{tabular}{m{5cm}m{4cm}m{4cm}}
\textbf{Control parameters} & \textbf{With vortical modes} & \textbf{Without vortical modes}
\end{tabular}
\input{../tmp/table_params.tex}
%\hfill
\input{../tmp/table_better_simuls.tex}
%\hfill
\input{../tmp/table_better_simuls_proj.tex}
\label{table-better-simuls}
\end{table}
\newpage
%\appendix\section{A great appendix}
%\label{appendix}
\bibliography{biblio}
\end{document}
This diff is collapsed.
We performed direct numerical simulations of stratified turbulence without shear modes,
and with or without vortical modes, and at various Froude and buoyancy Reynolds
numbers, in order to check in which conditions a weak internal gravity wave turbulent
regime could occur. We observed that removing vortical modes helps to have a better
overall balance between poloidal kinetic energy and potential energy. However, the
spatial spectra of kinetic and potential energy appear to behave very similarly for
simulations with or without vortical modes in our simulations.
We performed a spectral energy budget in the two-dimensional spectral space. We
observed that the conversion between kinetic energy and potential energy did not show
fluctuations around zero in our simulations, as we would expect for a system of
statistically stationary waves. We also observe that the conversion between potential
energy and kinetic energy becomes small when $\chi_d > 1$. Due to the anisotropy of
stratified flows, this means that it exists a range where waves are efficiently
dissipated by viscosity, but not necessarily the vortices. \\ A spatiotemporal
analysis in the buoyancy range showed that removing vortical mode helps to concentrate
the energy in temporal scales. However, increasing stratification is necessary to
approach a WWT regime. As in \cite{yokoyama_energy-based_2019}, we observed that waves
are present in a region delimited by the non-linearity parameter $\chi_L \leq 1/3$ even
when vortical modes are absent. We also observed evidences for the presence of slow
waves ($k_z \gg k_h$) subject to TRI when $\chi_L \geq 1/3$. However, the non-linear
broadening is shown to be large for $k_z \gg k_h$, therefore breaking a WWT theory's
assumption in this region of the spectral space. This shows that the presence of
vortical modes does not appear to be the only barrier to internal gravity waves
turbulence in real flows. Using a simple diagnostic to quantify wave's energy contained
in a stratified flow, we showed that a WWT regime should be attained at very high
stratification if the vortical modes are removed from the dynamics.
Our work suggests that the buoyancy Reynolds should not be too large in order to
observe a WWT regime. To understand this, we propose the following conditions in order
to limit the generation of non waves structures, and to guaranty the weak non-linearity
for the evolution of all energetic modes:
\begin{itemize}
\item The waves and eddies should be dissipated in the buoyancy range which leads to
the conditions
\begin{equation}
\label{eq:DissInBuoyancy}
k_{\rm d} \leq k_{\rm b} ~~~~ \Rightarrow ~~~~ \R \leq F_h^{2/3} ~~~~ \text{and} ~~~~ k_{\eta} \leq k_{\rm b} ~~~~ \Rightarrow ~~~~ \R \leq F_h.
\end{equation}
if we assume that dissipation of eddies occurs at the Kolmogorov wave-vector $k_{\eta}
= (\epsK/\nu^3)^{1/4}$. \item The non-linearity parameter $\chi_{\kk}$ should remain
small for all energetic modes. Still assuming that dissipation occurs at $k \lesssim
k_\eta$, this leads to the condition
\begin{equation}
\chi_{\rm max} \equiv \max\limits_{\substack{\kk \\ k \leq k_{\eta}}} \chi_{\kk}\leq 1 ~~~~ \Rightarrow ~~~~ \max\limits_{\substack{\kk \\ k \leq k_{\eta}}} \frac{1}{\sin \thk} \left( \frac{k}{k_{\rm O}} \right)^{2/3} = \frac{k_\eta}{k_{\rm h, min}} \left( \frac{k_\eta}{k_{\rm O}} \right)^{2/3} \leq 1
\end{equation}
where $k_{\rm h, min}$ represents the minimal horizontal wave-vector. If we consider a
flow of finite size with no shear modes, $k_{\rm h, min} = 2\pi / L_h$ such that
\begin{equation}
\label{eq:smallchi}
\chi_{\rm max} = \frac{k_\eta L_h}{2 \pi} \left( \frac{k_\eta}{k_{\rm O}} \right)^{2/3} \leq 1 ~~~~ \Rightarrow ~~~~ Re^{3/4} \R^{1/2} \lesssim 1,
\end{equation}
where $Re = (\epsK^3 L_h^4 / \nu^4)^{1/3}$ is the turbulent Reynolds number.
\end{itemize}
Finally, we require the flow to be in a turbulent state, i.e. the Reynolds number to be
sufficiently large to observe an anomalous dissipation $\epsK \sim U_h^3 / L_h$. Then
we have
\begin{equation}
\label{eq:LargeRe}
Re = \R F_h^{-2} \gg 1
\end{equation}
and the condition (\ref{eq:smallchi}) becomes
\begin{equation}
\label{eq:SmallChi}
\chi_{\rm max} \leq 1 ~~~~ \Rightarrow ~~~~ \R = Re F_h^2 \lesssim F_h^{6/5}.
\end{equation}
Tu sum up, we propose that a weak internal gravity waves turbulence regime could occur
only if $F_h \ll 1$ and conditions (\ref{eq:DissInBuoyancy}), (\ref{eq:LargeRe}), and
(\ref{eq:SmallChi}) are fulfilled. On Figure~\ref{fig:discussion}, we represent our
simulations in the $(F_h,\R)$ plane, as well as the ones obtained by
\cite{reun_parametric_2018}. For the simulations of \cite{reun_parametric_2018}, the
Froude number is computed using our definition (\ref{eq:FhR}), and the buoyancy
Reynolds number as $\R = Re F_h^2$ in order to compare to our work. Namely, we have
used Table~1 in \cite{reun_parametric_2018} with $U_h = u_{\rm rms}$, $\epsK =
\varepsilon_k$, and $Re = Re_0$. This leads to different values of $F_h$ and $\R$ than
the ones presented by the authors. We observe that our simulations do not lie in the
region corresponding (\ref{eq:DissInBuoyancy}), (\ref{eq:LargeRe}), and
(\ref{eq:SmallChi}). On the contrary, \cite{reun_parametric_2018} attained such a
region in their simulations, giving a tangible explanation on why they obtained
signatures of weak wave turbulence, while other studies, including the present study,
did not.
\begin{figure}
\includegraphics[width=0.66\textwidth]{../tmp/fig_discussion_regimes}
\caption{Simulations of the present study and \cite{reun_parametric_2018} in the $(F_h,
\R)$ parameters space. The lines corresponding to conditions (\ref{eq:DissInBuoyancy}),
(\ref{eq:LargeRe}), and (\ref{eq:SmallChi}) are represented by colored full lines (see
legend). The blue dashed line corresponds to $Re = 500$. \label{fig:discussion}}
\end{figure}
If the last claims are true, they constitute very prohibitive conditions for observing
a weak wave turbulence regime in numerical simulations. To illustrate this point, let
us consider a turbulent flow at $Re = 1000$ that typically requires a $\sim 1024^3$
resolution. Then (\ref{eq:SmallChi}) stipulates that the flow need to attain $F_h \leq
Re^{-5/4} = 1.78 \times 10^{-4}$ for the non-linearity parameter to be small for all
energetic modes. Yet, the kinetic time for a system of internal gravity waves is
expected to grow as $F_h^{-2}$, meaning that very long simulations are required to
attain the steady state, even for relatively small domains.
2022strat_polo_proj/input/fig_poloidal-toroidal.png

76.7 KiB

image diff could not be displayed: it is too large. Options to address this: view the blob.
% WWT
As eddies in classical hydrodynamic turbulence, waves in nonlinear systems interact and
transfer conserved quantities along scales in a cascade process. The Weak-Wave
Turbulence (WWT) theory aims to provides a statistical description of the system when
the non-linearity is small \cite{zakharov_kolmogorov_1992,nazarenko_wave_2011,
nazarenko_wave_2015}. The applications of this theory encompass capillary-gravity waves
\cite{falcon_experiments_2022}, Alfvén waves in magnetohydrodynamics (MHD)
\cite{galtier_weak_2000}, inertial and internal waves in rotating and stratified fluids
\cite{caillol_kinetic_2000, galtier_weak_2003, medvedev_turbulence_2007}, Kelvin waves
in superfluids \cite{lvov_weak_2010}, elastic plates \cite{during_weak_2006},
gravitational waves \cite{galtier_turbulence_2017}, density waves in Bose-Einstein
condensates \cite{dyachenko_optical_1992}, or 2D acoustic waves
\cite{griffin_energy_2022}.
% Weak non-linearity and small non-linear broadening
Two key hypotheses are used in WWT. The first one is weak non-linearity of the
dynamical equations. This condition is often translated in terms of the separation of
spatial scales: the considered scale on one side and the saturation scale on the other
side. The saturation scale is here defined as the scale at which the weak non-linearity
does not holds anymore, inducing wave breaking. This hypothesis is quite similar to the
separation between the integral scale and the dissipative scale in the classical
picture of three-dimensional hydrodynamic turbulence. The second hypothesis corresponds
to a time scales separation, and is more specific to WWT. It requires that the linear
time, given by the wave period $\tau_{\rm L}$, is small compared to the non-linear time
of interactions between waves $\tau_{\rm NL}$ \cite{nazarenko_wave_2011}.
For many physical systems, at least one the two hypotheses is broken in some range of
scales, which reduce the validity of WWT \cite{biven_breakdown_2001}. Yet, when scales
separations are observed in space and time, a weak wave turbulent range can appear in
spatiotemporal spectra of waves' energy. Besides experimental and numerical
difficulties for obtaining scales' separations, testing WWT in isotropic systems is
conceptually simpler. For this reason, many progress have been made in the experimental
and numerical verification of WWT for elastic plates \cite{miquel_nonstationary_2011,
yokoyama_identification_2014}, capillary-gravity waves \cite{pan_direct_2014,
falcon_experiments_2022}, density waves in Bose-Einstein condensates
\cite{zhu_testing_2022}, or 2D acoustic waves \cite{griffin_energy_2022}. Anisotropic
turbulence is, generally speaking, more difficult to investigate due to the reduced
number of symmetries of the considered flows. For example, it has been shown that
studying two-dimensional spatial spectra instead of one-dimensional integrated spectra
is essential to investigate stratified turbulence \cite{yokoyama_energy-based_2019}.
Then, the anisotropy makes the problem multidimensional in Fourier space, which makes
the notion of spectral energy fluxes more difficult than in isotropic turbulence
\cite{yokoyama_energy-flux_2021}. An additional difficultly is that scales separations
required by WWT can also be anisotropic. For linearly stratified flows, weak
non-linearity of the Navier-Stokes equations requires the spatial scale's separation
\begin{equation}
k/k_{\rm b} \ll 1,
\end{equation}
where $k_{\rm b} = N/U_h$ is the buoyancy wave-vector, $k$ is the wave-vector modulus,
$U_h$ is the rms of the horizontal velocity, and $N$ is the \bv frequency. This
condition should be fulfilled in order to avoid wave-breaking. The temporal scale
separation between linear waves and eddies leads to the different condition
\begin{equation}
\tau_{\rm L}/\tau_{\rm NL} = \frac{(\epsK k^2)^{1/3}}{N k_h/k} \ll 1,
\end{equation}
where $\epsK$ is the kinetic energy dissipation rate, $k_h$ is the horizontal
wave-vector modulus, $\tau_{\rm L} = 2\pi k /(N k_h)$ is the period of the linear
waves, and $\tau_{\rm NL} = 2\pi/(k^2 \epsK)^{1/3}$ is the eddy turnover time.
Physically, when $\tau_{\rm L}/\tau_{\rm NL} \ll 1$, the waves are faster than the
typical time of their non-linear interactions. Due to the anisotropic dispersion
relation of internal gravity waves, time scales separation will be less valid for small
$k_h$, and even impossible for modes with $k_h=0$. Consequently, separation of times
scales can be violated even if a large separation of spatial scales is observed (i.e.
$k/k_{\rm b} \ll 1$). It is therefore more difficult to observe signatures of wave
turbulence in anisotropic systems like MHD, or rotating and stratified flows.
Stratified flows are not only an interesting conceptual problem, but they are also
important for the understanding of geophysical flows \cite{staquet_internal_2002,
vallis_atmospheric_2017}. In particular, understanding the role of waves in mixing is
required for sub-grid parameterizations in climate models \cite{mackinnon_climate_2017,
gregg_mixing_2018}. It is therefore not surprising that stratified flows received a
particular attention from both the ``strong'' turbulence community
\cite{billant_self-similarity_2001, waite_stratified_2004, waite_stratified_2006,
lindborg_energy_2006, brethouwer_scaling_2007, waite_stratified_2011,
kimura_energy_2012, bartello_sensitivity_2013, brunner-suzuki_upscale_2014,
augier_stratified_2015, maffioli_vertical_2017}, and the WWT community
\cite{caillol_kinetic_2000, lvov_hamiltonian_2001, lvov_weak_2010,
dematteis_downscale_2021, dematteis_origins_2022}. It turns out that internal waves are
effectively important in the dynamics and mixing in stratified flows
\cite{maffioli_signature_2020, lam_partitioning_2020, lam_energy_2021}, and that,
three-wave resonant interactions are responsible for slow, net energy transfers between
different wave-numbers \cite{brouzet_internal_2016, davis_succession_2020,
rodda_experimental_2022}. Internal waves can be excited, for example, by tides
\cite{reun_parametric_2018} or by linearly unstable waves attractor
\cite{brouzet_internal_2016}. However, many questions and issues remain about the
applicability of WWT to stratified flows. In particular, for which values of the
control parameters a weak wave turbulence regime could occur.
A recent study showed that a wave dominated region should lie in the spectral region
defined by $\tau_{\rm L} / \tau_{\rm NL} < 1/3$ \cite{yokoyama_energy-based_2019}, in
agreement with observations made for MHD \cite{meyrand_direct_2016}. Yet, non-waves
structures like shear modes (purely vertical shear) and vortical modes (vertical
vorticity) still make the observation of a system of weakly interacting internal
gravity waves delicate. The same problem appear in rotating flows, in which the
geostrophic modes plays the role of the non-propagative structure. It has been
ingeniously shown that tidal forcing is very efficient at triggering weakly nonlinear
internal gravity waves \cite{reun_parametric_2018}, in a way reminiscent of the
inertial waves' excitation by libration in rotating fluids \cite{le_reun_inertial_2017,
reun_experimental_2019}. Experimentalists bypassed this difficulty to observe weak
inertial wave turbulence in a rotating tank, by using honeycomb grids at top and bottom
boundaries to dissipate geostrophic modes efficiently \cite{brunet_shortcut_2020,
monsalve_quantitative_2020}, in a similar fashion to the numerical study of
\cite{le_reun_inertial_2017}. These works point out that some waves systems have a
tendency to generate non-wave motions that can severely affect the waves' dynamics and
should be suppressed in experiments aiming to observe wave turbulence. In stratified
flows these modes correspond to vertical shear and vertical vorticity.
The most commonly used WWT prediction is certainly the scale invariant stationary
solutions to the kinetic equation, which gives the expected spatial waves' energy
spectra in the statistically steady state. The solutions are usually of two types: the
thermodynamic equilibrium solution (Rayleigh-Jeans spectra), and the non equilibrium
solution(s) that are linked to the cascade(s) of the dynamical invariant(s) of the
system among scales (Kolmogorov-Zakharov spectra) \cite{nazarenko_wave_2011}. It is
important to note that these solutions can be considered as valid only if the collision
integral in the waves' kinetic equation converges. The importance of the nonlinear
interactions among internal gravity waves was recognized early, leading to several
derivations of waves' kinetic equations (see \cite{muller_nonlinear_1986,
lvov_resonant_2012}). Derivation of the Kolmogorov-Zakharov spectra in the limit of
large vertical wave-numbers $k_z \simeq k \gg k_h$ can be found in
\cite{caillol_kinetic_2000, lvov_hamiltonian_2001}. This solution corresponds to an
energy cascade, and is given by the energy spectra
\begin{equation}
E(k_h, k_z) \sim k_h^{-5/2} k_z^{-3/2}.
\end{equation}
Yet, it was noted that this solution comes through a spurious cancellation between
oppositely signed divergent contributions in their respective collision integrals.
Therefore, the existence of this solution is fortuitous. Later, it was shown that power
law solutions $E(k_h,k_z) \sim k_h^{-\alpha_h} ~ k_z^{-\alpha_z}$ have convergent
collision integral's contributions only on the segment $\alpha_h \in ]2, 3[,
\alpha_z=1$ \cite{lvov_oceanic_2010, dematteis_downscale_2021}. The collision integral
was then computed numerically for $\alpha_z=1$, and it was deduced that the only scale
invariant stationary solution to the kinetic equation (zero of the collision integral)
was close to
\begin{equation}
\label{eq:WWTpredictions}
E(k_h, k_z) \sim k_h^{-2.69} k_z^{-1}.
\end{equation}
It was also shown that, on the segment $\alpha_z =1$, the dominant contributions to the
collision integral corresponds to non-local transfers first identified by McComas
\cite{mccomas_resonant_1977}, notably the Parametric Subharmonic Instability (PSI)
\cite{mccomas_resonant_1977, muller_nonlinear_1986} observed in oceans
\cite{mackinnon_parametric_2013} and experiments \cite{rodda_experimental_2022},
consisting in the resonant interaction of a primary wave and two smaller-scale waves of
nearly half the frequency.
When $k/k_{\rm b} \gg 1$, non-linearity is not small. Based on the idea that there
exists a range of spatial scales where buoyancy force has the same order of magnitude
than inertia, the following 1D integrated energy spectra
\begin{equation}
\label{eq:CriticalBalance}
E_{\rm 1D}(k_h) \sim \eps^{2/3} k_h^{-5/3} ~~~~ \text{and} ~~~~ E_{\rm 1D}(k_z) \sim N^2 k_z^{-3}
\end{equation}
where predicted \cite{lindborg_energy_2006}. In the WWT context, these predictions can
also be obtained using critical balance arguments \cite{nazarenko_critical_2011,
nazarenko_wave_2011}. At even smaller scales where stratification is negligible, an
isotropic range with the classical energy spectra
\begin{equation}
\label{eq:Isotropic}
E_{\rm 1D}(k) \sim \eps^{2/3} k^{-5/3}
\end{equation}
can develop. This happens when $k_{\rm b} < k_{\rm O} \ll k \ll k_{\eta}$ where $k_{\rm
O}$ is the Ozmidov wave-vector and $k_{\eta}$ is the Kolmogorov wave-vector.
The present study first deals with the existence and properties of a weak internal
gravity waves turbulence regime. To this end, we perform numerical simulations of
stratified turbulence at various stratifications and viscosity. In the same spirit than
experiments in a rotating tank presented in \cite{brunet_shortcut_2020,
monsalve_quantitative_2020}, we removed shear modes in all of our simulations. For each
values of the control parameters, we performed two ``twins'' simulations: one where the
vortical modes remain, and one where vortical modes are removed from the dynamics by a
projection in spectral space \cite{craya_contribution_1957}. Note that, in order to get
rid of non waves structures, numerical simulations of reduced dynamical equations (i.e.
without non-waves structures) of stratified rotating flows in the hydrostatic balance
where already done \cite{lvov_nonlinear_2009}. Despite the simplifications, these
simulations reproduced some key features of oceanic internal-wave spectra: accumulation
of energy at near-inertial waves and realistic frequency and horizontal wave-number
dependencies of spatiotemporal spectra. In the present work, we do not account for
rotation, and do not use the hydrostatic balance approximation. Also, having twins
simulations (i.e. with or without vortical modes) allows to have a better comparison to
understand the role of the vortical modes on the dynamics of stratified flows.
The manuscript is organized as follows. In section \ref{sec:methods}, we present our
methodology including a presentation of the code and the simulations. Our results are
presented in section \ref{sec:results}. Subsection \ref{subsec:global} is devoted to
the study of flow regimes and global energy ratios in the control parameters space for
our simulations. It allows us to observe that WWT is naturally more likely to occurs at
high stratification and without vortical modes. Subsections \ref{subsec:khkz} and
\ref{subsec:khkzomega} deal with the spatiotemporal analysis of a couple of strongly
stratified simulations in order to investigate further the presence of linear waves in
spatial scales. In the last subsection \ref{subsec:waves_energy}, we propose a simple
diagnostic to evaluate the predominancy of waves in the parameter space. We give
concluding remarks in section \ref{sec:conclusions}.
......@@ -1059,6 +1059,28 @@
Publisher: Ubiquity Press},
}
@article{fluiddyn,
doi = {10.5334/jors.237},
year = {2019},
publisher = {Ubiquity Press, Ltd.},
volume = {7},
author = {Augier, Pierre and Mohanan, Ashwin Vishnu and Bonamy, Cyrille},
title = {{FluidDyn}: A Python Open-Source Framework for Research and Teaching in Fluid Dynamics
by Simulations, Experiments and Data Processing},
journal = {Journal of Open Research Software}
}
@article{fluidfft,
doi = {10.5334/jors.238},
year = {2019},
publisher = {Ubiquity Press, Ltd.},
volume = {7},
author = {Mohanan, Ashwin Vishnu and Bonamy, Cyrille and Augier, Pierre},
title = {{FluidFFT}: Common {API} ({C}$++$ and {P}ython)
for Fast Fourier Transform {HPC} Libraries},
journal = {Journal of Open Research Software}
}
@thesis{linares_numerical_2020,
title = {Numerical study of 2D stratified turbulence forced by internal gravity waves},
url = {https://tel.archives-ouvertes.fr/tel-02612797},
......@@ -1118,3 +1140,62 @@
keywords = {stratified turbulence, internal waves, rotating turbulence},
}
@article {nikurashin_legg_mechanism_2011,
author = "Maxim Nikurashin and Sonya Legg",
title = "A Mechanism for Local Dissipation of Internal Tides Generated at Rough Topography",
journal = "Journal of Physical Oceanography",
year = "2011",
publisher = "American Meteorological Society",
address = "Boston MA, USA",
volume = "41",
number = "2",
doi = "https://doi.org/10.1175/2010JPO4522.1",
pages= "378 - 395",
url = "https://journals.ametsoc.org/view/journals/phoc/41/2/2010jpo4522.1.xml"
}
@article{waite_potential_2013,
title={Potential enstrophy in stratified turbulence},
volume={722}, DOI={10.1017/jfm.2013.150},
journal={Journal of Fluid Mechanics},
publisher={Cambridge University Press},
author={Waite, Michael L.},
year={2013},
pages={R4}
}
@article{david_locality_2023,
title={Locality of triad interaction and Kolmogorov constant in inertial wave turbulence},
volume={955},
DOI={10.1017/jfm.2022.1073},
journal={Journal of Fluid Mechanics},
publisher={Cambridge University Press},
author={David, Vincent and Galtier, Sébastien},
year={2023},
pages={R2}
}
@article {bartello_geostrophic_1995,
author = "Peter Bartello",
title = "Geostrophic Adjustment and Inverse Cascades in Rotating Stratified Turbulence",
journal = "Journal of Atmospheric Sciences",
year = "1995",
publisher = "American Meteorological Society",
address = "Boston MA, USA",
volume = "52",
number = "24",
doi = "https://doi.org/10.1175/1520-0469(1995)052<4410:GAAICI>2.0.CO;2",
pages= "4410 - 4428"
}
@article{deBruynKops1998,
title={Direct numerical simulation of laboratory experiments in isotropic turbulence},
author={de Bruyn Kops, SM and Riley, JJ},
journal={Physics of Fluids},
volume={10},
number={9},
pages={2125--2127},
year={1998},
publisher={American Institute of Physics}
}
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