diff --git a/2022strat_turb_polo/input/article.tex b/2022strat_turb_polo/input/article.tex index 373a59148d2da6da2e5200a810b83f7738d5421d_MjAyMnN0cmF0X3R1cmJfcG9sby9pbnB1dC9hcnRpY2xlLnRleA==..a2355d9da5780035cbb1a70daad6758cea7c0ab4_MjAyMnN0cmF0X3R1cmJfcG9sby9pbnB1dC9hcnRpY2xlLnRleA== 100644 --- a/2022strat_turb_polo/input/article.tex +++ b/2022strat_turb_polo/input/article.tex @@ -169,7 +169,6 @@ \begin{abstract} -We present numerical simulations of stratified turbulence under the Oberbeck-Boussinesq -approximation with Schmidt number equal to unity and periodic boundary conditions. We use an anisotropic time correlated forcing on the poloidal component of the velocity field to excite internal -gravity waves, and the shear modes are removed from the dynamics. We vary the viscosity and the +We present a dataset of numerical simulations of stratified turbulence under the Oberbeck-Boussinesq +approximation with Schmidt number equal to unity and periodic boundary conditions. An anisotropic time correlated forcing acting on the poloidal component of the velocity field is used in order to excite internal gravity waves. We vary the viscosity and the \bv frequency in order to obtain flows at different horizontal Froude $F_h$ and the @@ -175,5 +174,5 @@ \bv frequency in order to obtain flows at different horizontal Froude $F_h$ and the -buoyancy Reynolds number $\R$. The datas are availabe \href{https://mycore.core-cloud.net/index.php/apps/files/?dir=/2022strat-turb-polo&fileid=2553105195}{https://mycore.core-cloud.net/index.php/apps/files/?dir=/2022strat-turb-polo\&fileid=2553105195}. Isotropy and mixing coefficients allows to distinguish between 5 regimes in the $F_h-\R$ parameters space. The dataset contains spectral energy budgets that are analyzed for simulations in each regime. This single dataset also allows to sum up results obtained in earlier separate studies. +buoyancy Reynolds number $\R$. The datas are availabe at \href{https://mycore.core-cloud.net/index.php/apps/files/?dir=/2022strat-turb-polo&fileid=2553105195}{https://mycore.core-cloud.net/index.php/apps/files/?dir=/2022strat-turb-polo\&fileid=2553105195}. Isotropy and mixing coefficients allows to distinguish between 5 regimes in the $F_h-\R$ parameters space. The spectral energy budgets are analyzed for simulations in each regime. \end{abstract} %---------------------------------------------------------------------------------------- @@ -199,7 +198,7 @@ \mintinline{python}{ns3d.strat} from the FluidSim Python package \cite{fluiddyn,fluidfft,fluidsim}. Using this solver, we integrate in a periodic domain of horizontal size $L_x = L_y = L_h = 3$ the three-dimensional Navier-Stokes equations under -the Boussinesq approximation with additional fourth-order hyperviscosity and hyperdiffusivity terms: +the Oberbeck-Boussinesq approximation with additional fourth-order hyperviscosity and hyperdiffusivity terms: \begin{align} \bnabla \cdot \vv &= 0 \\ \p_t\vv + (\vv \cdot \bnabla)\vv &= b\boldsymbol{e}_z - \bnabla p + @@ -207,10 +206,10 @@ \p_t{b} + (\vv \cdot \bnabla)b &= -N^2 v_z + \kappa\nabla^2{b} - \kappa_4 \nabla^4{b},\label{buoy} \end{align} where $\xx = (x,y,z)$ denotes the three spatial coordinates in the cartesian frame $(O, \eex, \eey, \eez)$, $\eez$ is the stratification axis, $\vv=(v_x, v_y, v_z)$ the velocity, $b$ the buoyancy, $p$ the total kinematic pressure, $N$ the \bv -frequency, $\nu$ the viscosity, $\kappa$ the diffusivity, $\kappa_4$ the +frequency, $\nu$ the viscosity, $\kappa$ the diffusivity, $\nu_4$ the hyperviscosity, $\kappa_4$ the hyperdiffusivity, and $\ff$ the velocity forcing. For all simulations the viscosity and the diffusivity are equal (Schmidt number $Sc = \nu/\kappa = 1$), as well as the hyperviscosity and hyperdiffusivity ($\nu_4 = \kappa_4$). Note that the buoyancy can be expressed as $b=-g\delta\rho/\rho_0$, with $g$ the gravitational acceleration, $\rho_0$ the mean density and $\delta\rho$ the departure from the stable linear density stratification. However, these three quantities do not enter separately @@ -211,8 +210,8 @@ hyperviscosity, $\kappa_4$ the hyperdiffusivity, and $\ff$ the velocity forcing. For all simulations the viscosity and the diffusivity are equal (Schmidt number $Sc = \nu/\kappa = 1$), as well as the hyperviscosity and hyperdiffusivity ($\nu_4 = \kappa_4$). Note that the buoyancy can be expressed as $b=-g\delta\rho/\rho_0$, with $g$ the gravitational acceleration, $\rho_0$ the mean density and $\delta\rho$ the departure from the stable linear density stratification. However, these three quantities do not enter separately -into the equations. The vertical size of the domain $L_z$, is varied with \bv frequency. We note $(n_x, n_y, n_z)$ the numbers of collocations points in the three spatial directions, and we fix $n_x = n_y = n_h$. \\ +into the equations. The vertical size of the domain $L_z$, is varied with the \bv frequency. We note $(n_x, n_y, n_z)$ the numbers of collocations points in the three spatial directions, and we fix $n_x = n_y = n_h$. \\ @@ -218,4 +217,5 @@ +\todo{continue here} In spectral space, the equations of motion reads \begin{align} (\p_t + \nu k^2 + \nu_4 k^4) \vvk - P_{\kk} \cdot \bbk \eez &= -i P_{\kk} \cdot \sum\limits_{\boldsymbol{p} + \boldsymbol{q} = \kk} ~ \hat{\boldsymbol{v}}_{\boldsymbol{p}} ~ \kk \cdot \hat{\boldsymbol{v}}_{\boldsymbol{q}} + \ffk \\ diff --git a/2022strat_turb_polo/input/intro.tex b/2022strat_turb_polo/input/intro.tex index 373a59148d2da6da2e5200a810b83f7738d5421d_MjAyMnN0cmF0X3R1cmJfcG9sby9pbnB1dC9pbnRyby50ZXg=..a2355d9da5780035cbb1a70daad6758cea7c0ab4_MjAyMnN0cmF0X3R1cmJfcG9sby9pbnB1dC9pbnRyby50ZXg= 100644 --- a/2022strat_turb_polo/input/intro.tex +++ b/2022strat_turb_polo/input/intro.tex @@ -6,7 +6,7 @@ heating or salt input, maintain the flow out of equilibrium, like in geophysical flows \cite{Vallis2017}. Contrary to flows at constant density which contain only eddies, stably stratified flows also support the propagation of internal gravity waves -\cite{Staquet-Sommeria2002}. Stratification is a key physical mechanism in many industrial an natural flows. As an example, stratification is essential to describe the dynamics of the atmosphere and oceans, in particular at mesoscales for which the effects of rotation is negligible. \\ +\cite{Staquet-Sommeria2002}. Stratification is a key physical mechanism in many industrial and natural flows. As an example, stratification is essential to describe the dynamics of the atmosphere and oceans, in particular at mesoscales for which the effects of rotation is negligible. \\ Most theoretical and numerical studies on stratified flows are performed using the Oberbeck-Boussinesq approximation and a linear average density profile. In this case, the @@ -10,8 +10,7 @@ Most theoretical and numerical studies on stratified flows are performed using the Oberbeck-Boussinesq approximation and a linear average density profile. In this case, the -intensity of the stratification is solely quantified by the \bv frequency $N$. The scaling analysis of the equations \cite{Brethouwer-Billant-Lindborg-Chomaz2007} shows that the dynamics is controled by three dimensionless parameters: the Schmidt number $Sc$; the buoyancy Reynolds number $\R$; and the -horizontal Froude number $F_h$ which quantify the stratification. Then, for a given fluid (with a fixed $Sc$) and far from the singularities of the flow, the statistical properties of the fields are supposed to depend only on $F_h$ and $\R$. When $F_h$ is large before unity, the effect of buoyancy is negligible and we expect to recover incompressible flows predictions, where the buoyancy behaves as a passive scalar. Yet, when $F_h$ is small, the buoyancy impacts the flow significantly and -three stratified regimes were predicted in \cite{Brethouwer-Billant-Lindborg-Chomaz2007}: +intensity of the stratification is solely quantified by the \bv frequency $N$. The scaling analysis of the equations shows that the dynamics is controled by three dimensionless parameters \cite{Brethouwer-Billant-Lindborg-Chomaz2007}: the Schmidt (or Prandtl) number $Sc$; the buoyancy Reynolds number $\R$; and the +horizontal Froude number $F_h$ which quantify the stratification. Then, for a given fluid (with a fixed $Sc$) and far from the singularities of the flow, the statistical properties of stratified flows are supposed to depend only on $F_h$ and $\R$. When $F_h$ is large before unity, the effect of buoyancy is negligible and we expect to recover unstratified flows, where the buoyancy behaves as a passive scalar. Yet, when $F_h$ is small, the buoyancy impacts the flow significantly and three stratified regimes were predicted \cite{Brethouwer-Billant-Lindborg-Chomaz2007}: for small $\R$, the flow cannot develop small isotropic scales because viscosity is already important at larger anisotropic scales (we talk about a viscosity affected regime); if @@ -16,4 +15,4 @@ for small $\R$, the flow cannot develop small isotropic scales because viscosity is already important at larger anisotropic scales (we talk about a viscosity affected regime); if -$\R$ is large and $F_h < 1$, the flow is turbulent but weakly stratified; finally, +$\R$ is large and $F_h \lesssim 1$, the flow is turbulent but weakly stratified; finally, when $\R$ is large and $F_h \ll 1$, the flow is in a stronly stratified turbulent @@ -19,3 +18,3 @@ when $\R$ is large and $F_h \ll 1$, the flow is in a stronly stratified turbulent -state. The authors showed with DNS that the condition $\R \geq 10$ should be verified to attain this regime. Many geophysical data supports the idea that geophysical flows belong to this strongly stratified turbulent regime \cite{riley2008stratified}. \\ +state. Many geophysical data supports the idea that geophysical flows belong to this strongly stratified turbulent regime \cite{riley2008stratified}. \\ @@ -21,4 +20,6 @@ -The importance of internal waves in stratified flows, under rotation or not, was recognized early. For example, the empirical Garrett and Munk (GM) spectra \cite{Garrett-Munk1972, Garrett-Munk1975, Garrett-Munk1979} has been used for decades to describes the oceanic internal wave-field. Then, non-linear resonance between nearly linear internal waves were invoked to describe the internal gravity waves spectra in a more precise way \cite{Muller-Olbers1975, McComas-Muller1981Dynamics}. A more modern description of internal gravity waves interactions is given by the Wave Turbulence Theory \cite{Caillol-Zeitlin2000, Lvov-Tabak2001, Lvov-Polzin-Tabak2004, Lvov-Polzin-Tabak-Yokoyama2010}. Both the GM predictions in the high frequency limit \cite{Garrett-Munk1979}, and the Wave Turbulence Theory \cite{Lvov-Polzin-Tabak2004} in the $k_z \gg k_h$ limit are in agreement with a wave energy spatial spectra of the form +The importance of internal waves in stratified flows, under rotation or not, was recognized early. For example, the empirical Garrett and Munk (GM) spectra \cite{Garrett-Munk1972, Garrett-Munk1975, Garrett-Munk1979} has been used for decades to describes the oceanic internal wave-field. Then, non-linear resonance between nearly linear internal waves were invoked to describe the internal gravity waves spectra in a more precise way \cite{Muller-Olbers1975, McComas-Muller1981Dynamics, Caillol-Zeitlin2000, Lvov-Polzin-Tabak-Yokoyama2010}. A forward cascade of energy (from large to small scales) was predicted by theories involving saturated gravity waves \cite{Holloway1983, Dewan-Good1986, Smith-Fritts-Vanzandt1987, Dewan1997}. Yet, a slow growth of the energy contained in the horizontally invariant modes, called shear modes, is observed when $F_h$ is sufficiently small \cite{Smith2002generation}. Internal gravity waves is expected to be particularly important down to the buoyancy scale, under which they break and are eventually dissipated \cite{Weinstock1978}. \\ + +At scales smaller than the buoyancy scale, propagating gravity waves and non-propagative vortical modes, associated to vertical vorticity, strongly interact together and cannot be considered separately. In such case, it has been proposed that the 1D integrated energy spectra are given by \begin{equation} @@ -23,5 +24,5 @@ \begin{equation} -E_W(k_h, k_z) \propto k_h^{-2} ~ k_z^0 +E(k_z) \sim N^2 k_z^{-3} ~~~~ \text{and} ~~~~ E(k_h) \sim \eps^{2/3} k_h^{-5/3}, \end{equation} @@ -26,15 +27,6 @@ \end{equation} -\noindent where $k_h$ and $k_z$ are the horizontal and vertical wavevectors. Internal gravity waves is expected to be particularly important down to the buoyancy scale, under which they break and are eventually dissipated \cite{Weinstock1978}. A forward cascade of energy (from large to small scales) was predicted by theories involving saturated gravity waves \cite{Holloway1983, Dewan-Good1986, Smith-Fritts-Vanzandt1987, Dewan1997}. Yet, \cite{Smith2002generation} carried out numerical simulations with forcing at relatively small scales and showed that an inverse energy cascade (from small to large scales) occurs when $F_h$ is sufficiently small: they observed a slow growth of the energy contained in the horizontally invariant modes, called shear modes. \\ - -A theory of strongly stratified turbulence has emerged at the begining of the century \cite{Lindborg2006energy, Brethouwer-Billant-Lindborg-Chomaz2007}. In this description, one consider a second Froude number, based on a vertical characteristic lengthscale $L_v$: the vertical Froude number $F_v = U/(N L_v)$. $F_v$ quantifies the interaction between the different horizontal layers and therefore the importance of three dimensional processes. It was shown that the scaling $F_v \sim 1$ emerges through an invariance of the inviscid Boussinesq equations under the condition $F_h \ll 1$ \cite{Billant-Chomaz2001}. Therefore, in the limit $F_h \ll 1$, $F_v \sim 1$, propagating gravity waves and non-propagative vortical modes, associated to vertical vorticity, strongly interact together and cannot be considered separately. \cite{Lindborg2006energy} proposed that the horizontal kinetic energy and potential energy spectra are given by - -\begin{align} - E_h(k_z) &\sim N^2 k_z^{-3}, \\ - E_h(k_h) &= C_1 \varepsilon_K^{2/3} k_h^{-5/3}, \\ - E_A(k_h) &= C_2 \varepsilon_K^{2/3} k_h^{-5/3} \frac{\varepsilon_A}{\varepsilon_K}, -\end{align} -and measured $C_1 \simeq C_2 \simeq 0.5$. Yet, an anomalous energy excess around the buoyancy length scale was observed, and attributed to the nonlocal energy transfers from large horizontal scale to small scales resulting from the shear and gravitational instabilities \cite{Augier2015stratified}. In a recent numerical study \cite{Maffioli2017spectra}, a filter was applied in order to separate large and small horizontal scales when computing spatial spectra. They clearly showed that the $N^2 k_z^{-3}$ is verified only for large horizontal scales. It stresses the necessity to analyse the two-dimensional structure of the spatial spectra (in $k_h$ and $k_z$) when analysing stratified turbulence, and not simply integrate over one direction, which could introduce non-trivial dependencies on the resulting one-dimensional spectra depending on different physical mechanism occuring at different scales \cite{Yokoyama-Takaoka2019}. In fact, looking at the three-dimensional spatiotemporal spectra is necessary if we want to precisely quantify the propency and effects of waves \cite{Maffioli2020waves, Lam-Delache-Godeferd2020waves-eddies, Lam-Delache-Godeferd2021mixing}. \\ +where $k_h$ and $k_z$ are respectively the horizontal and vertical wave-vectors, and $\varepsilon$ the energy dissipation rate \cite{Lindborg2006energy}. Yet, an anomalous energy excess around the buoyancy length scale was observed, and attributed to the nonlocal energy transfers from large horizontal scale to small scales resulting from the shear and gravitational instabilities \cite{Augier2015stratified}. In a recent numerical study \cite{Maffioli2017spectra}, a filter was applied in order to separate large and small horizontal scales when computing spatial spectra. They clearly showed that the $N^2 k_z^{-3}$ is verified only for large horizontal scales. It stresses the necessity to analyse the two-dimensional structure of the spatial spectra (in $k_h$ and $k_z$) when analysing stratified turbulence, and not simply integrate over one direction, which could introduce non-trivial dependencies on the resulting one-dimensional spectra depending on different physical mechanism occuring at different scales \cite{Yokoyama-Takaoka2019}. In fact, looking at the three-dimensional spatiotemporal spectra is necessary if we want to precisely quantify the propency and effects of waves \cite{Maffioli2020waves, Lam-Delache-Godeferd2020waves-eddies, Lam-Delache-Godeferd2021mixing}. \\ At scales smaller than the Ozmidov scale, influence of stratification is expected to be negligible. So, if $\R$ is sufficiently large, it is expected to recover isotropic turbulence \cite{Lumley1965, Ozmidov1965}: \begin{equation} @@ -38,5 +30,5 @@ At scales smaller than the Ozmidov scale, influence of stratification is expected to be negligible. So, if $\R$ is sufficiently large, it is expected to recover isotropic turbulence \cite{Lumley1965, Ozmidov1965}: \begin{equation} - E_K(k) = C_K \varepsilon_K^{2/3} k^{-5/3} + E(k) \sim \eps^{2/3} k^{-5/3}. \end{equation} @@ -42,5 +34,5 @@ \end{equation} -where $C_K \simeq 1$ is the Kolmogorov constant. This transition is visible on the vertical spectra in numerical simulations \cite{Augier2015stratified}, and oceanic spectra \cite{Nastrom-Gage-Jasperson1984, Nastrom-Gage1985, riley2008stratified}. Numerical simulations acheaving sufficiently large $\R$ to escape the viscosity affected regime exhibit downscale energy transfers, from large to small horizontal scales with horizontal spectrum of kinetic energy scaling like $\varepsilon_K^{2/3} k_h^{-5/3}$ \cite{Lindborg2006energy, Waite2006stratified, Brethouwer-Billant-Lindborg-Chomaz2007, Almalkie-deBruynKops2012, Kimura-Herring2012spectra, Bartello-Tobias2013-buoyancyreynolds}. \\ +This transition is visible on the vertical spectra in numerical simulations \cite{Augier2015stratified}, and oceanic spectra \cite{Nastrom-Gage-Jasperson1984, Nastrom-Gage1985, riley2008stratified}. Numerical simulations acheaving sufficiently large $\R$ to escape the viscosity affected regime exhibit downscale energy transfers, from large to small horizontal scales with horizontal spectrum of kinetic energy scaling like $\eps^{2/3} k_h^{-5/3}$ \cite{Lindborg2006energy, Waite2006stratified, Brethouwer-Billant-Lindborg-Chomaz2007, Almalkie-deBruynKops2012, Kimura-Herring2012spectra, Bartello-Tobias2013-buoyancyreynolds}. \\ @@ -44,7 +36,7 @@ -The study of stratified flows is crucial for predicting oceanic spectra, but it is also very important for the improvement of sub-grid parameterizations in climate models \cite{Gregg-etal2018mixing}. If the transport terms have no net effect, the eddy diffusivity can be approximated as $\kappa_T \simeq \epsK \Gamma / N^2$, where $\Gamma = \epsA /\epsK$ is the mixing ratio, $\epsK$ is the kinetic energy dissipation rate, and $\epsA$ is the potential energy dissipation rate \cite{Caulfield2020review}. The evolution of the mixing coefficient as a function of $F_h$ in purely stratified flows was investigated in \cite{Maffioli2016mixing}. The authors showed that the optimal mixing efficiency coincides with an optimal ratio of potential energy to kinetic energy, observed that $\Gamma \propto F_h^{-2}$ for $F_h > 1$, and $\Gamma \propto F_h^0$ if $\mathcal{R} \gg 1$. Later, \cite{Garanaik-Venayagamoorthy2019mixing} completed the description by explaining the evolution of $\Gamma$ in the weakly stratified regime, corresponding to $F_h \in [0.3,1]$. To sum up, the predictions in the limit $\R \gg 1$ is +The study of stratified flows is crucial for predicting oceanic spectra, but also for the improvement of sub-grid parameterizations in climate models \cite{Gregg-etal2018mixing}. If the transport terms have no net effect, the eddy diffusivity can be approximated as $\kappa_T \simeq \epsK \Gamma / N^2$, where $\Gamma = \epsA /\epsK$ is the mixing ratio, $\epsK$ is the kinetic energy dissipation rate, and $\epsA$ is the potential energy dissipation rate \cite{Caulfield2020review}. The evolution of the mixing coefficient as a function of $F_h$ in purely stratified flows was investigated \cite{Maffioli2016mixing, Garanaik-Venayagamoorthy2019mixing}. To sum up, the predictions in the limit $\R \gg 1$ is \begin{equation} \label{eq:Maffioli-Garanaik-Venayagamoorthy} @@ -56,7 +48,6 @@ \end{equation} -\noindent The precise variations of the mixing coefficient appeared to also depend on the nature of the forcing. In \cite{Howland2020mixing}, the authors have shown that the mixing coefficient indeed depends on wether the forcing is applied to waves, or to horizontal motions. They observed that forcing waves allowed to attain larger values of the mixing coefficient for similar $F_h$. In recent studies \cite{Lam-Delache-Godeferd2020waves-eddies, Lam-Delache-Godeferd2021mixing}, waves and eddies are partitioned to study mixing properties of these two components separately. \\ - +\noindent The precise variations of the mixing coefficient appeared to also depend on the nature of the forcing. For example, the mixing coefficient indeed depends on wether the forcing is applied to waves, or to horizontal motions \cite{Howland2020mixing}. In recent studies \cite{Lam-Delache-Godeferd2020waves-eddies, Lam-Delache-Godeferd2021mixing}, waves and eddies are partitioned to study mixing properties of these two components separately. \\ @@ -61,7 +52,6 @@ -Testing the above mentionned theories in various regimes, is challenging numerically and experimentally. It would be usefull -to have comprehensive open datasets to investigate various aspect of stratified -turbulence, and allow simpler comparison to litterature. The goal of this manuscript is to present such a dataset where both $F_h$ and $\R$ are varied. The particularity of this dataset is that it is -designed to excite the kinetic part of internal gravity waves. It contains averaged quantities, spatial spectra, temporal spectra, and spatiotemporal spectra. We present simulations in which the vortical +Having comprehensive open datasets of stratified +turbulence would allow to investigate various aspect of these flows and allow simpler comparison to litterature. The goal of this manuscript is to present such a dataset where both $F_h$ and $\R$ are varied. The particularity of this dataset is that it is +designed to excite the kinetic part of internal gravity waves. It contains averaged quantities, spatial spectra, and spatiotemporal spectra. We present simulations in which the vortical modes are forced in a similar way in a separate study \todo{Quote the manuscript with toroidal forcing}. The remaining of the @@ -67,5 +57,5 @@ modes are forced in a similar way in a separate study \todo{Quote the manuscript with toroidal forcing}. The remaining of the -manuscript is as follows. In section \ref{sec:num}, we present the numerical setup, and the methodology used to contruct the dataset. In \ref{sec:res}, we present our results and compare them to previous studies. In a first subsection \ref{sec:res-isotropy}, we study the large and small scale isotropy coefficients, that allows to distinguish between 4 different turbulent regimes. A second subsection \ref{sec:res-mixing} is dedicated to the study of the evolution of the mixing coefficient, and component of energies as a function of $F_h$ in the limit of large $\R$. In a third subsection \ref{sec:res-spectral}, we analyse the spectral energy budget in different regimes. In section \ref{sec:conclusion}, we give concluding remarks. +manuscript is as follows. In section \ref{sec:num}, we present the numerical setup, and the methodology used to contruct the dataset. In \ref{sec:res}, we present our results and compare them to previous studies. In a first subsection \ref{sec:res-isotropy}, we study the large and small scale isotropy coefficients that allows to distinguish between 5 different turbulent regimes. A second subsection \ref{sec:res-mixing} is dedicated to the study of the evolution of the mixing coefficient, and component of energies as a function of $F_h$ in the limit of large $\R$. In a third subsection \ref{sec:res-spectral}, we analyse the spectral energy budget in different regimes. In section \ref{sec:conclusion}, we give concluding remarks.