# Stratified turbulence ```{math} \newcommand{\p}{\partial} \newcommand{\vv}{\boldsymbol{v}} \newcommand{\xx}{\boldsymbol{x}} \newcommand{\bnabla}{\boldsymbol{\nabla}} \newcommand{\Dt}{\text{D}_t} \newcommand{\R}{\mathcal{R}} \newcommand{\epsK}{\varepsilon_K} ``` ```{role} raw-latex(raw) --- format: latex --- ``` {raw-latex}`\newcommand{\p}{\partial}` {raw-latex}`\newcommand{\vv}{\boldsymbol{v}}` {raw-latex}`\newcommand{\xx}{\boldsymbol{x}}` {raw-latex}`\newcommand{\bnabla}{\boldsymbol{\nabla}}` {raw-latex}`\newcommand{\Dt}{\text{D}_t}` {raw-latex}`\newcommand{\R}{\mathcal{R}}` {raw-latex}`\newcommand{\epsK}{\varepsilon_K}` ## Few experimental results ```{figure} ./figs/fig_zigzag_instab_simple.png --- width: 70% name: fig_zigzag_instab_simple.png --- (a) Photographies of the evolution of the zigzag instability of two columnar counter rotating vortices. The dipole propagates initially towards the reader. (b) Scheme of the non-linear evolution of the zigzag instability inllustrating the creation of pancake vortices. Taken from {cite:t}`BillantChomaz2000experimental`. ``` ```{figure} ./figs/fig_praud_2d.png --- width: 65% name: fig_praud_2d.png --- Streak photographs of an evolving, grid-generated turbulent flow. Taken from {cite:t}`PraudFinchamSommeria2005`. ``` ```{figure} ./figs/fig_praud_3d.png --- width: 60% name: fig_praud_3d.png --- Initial three-dimensional instability of the laminar wake represented in terms of isosurfaces of the vertical component of vorticity $\omega_z$. Taken from {cite:t}`PraudFinchamSommeria2005`. ``` ```{figure} ./figs/fig_augier_exp_phd.png --- width: 80% name: fig_augier_exp_phd.png --- Horizontal ((a), (c), (e)) and vertical ((b), (d), (f)) cross-sections of the velocity in the quasi-stationary regime for three values of the buoyancy Reynolds number $\R$. Taken from {cite:t}`Augier2014experimental`. ``` ## Scaling analyses ### Inviscid stratified flows Let us follow {cite:t}`BillantChomaz2001` by considering a flow with typical horizontal velocity $U$, typical horizontal scale $L_h$ and typical vertical scale $L_z$. We define the aspect ratio $\alpha \equiv L_z/L_h$. The most important non-dimensional parameter is the horizontal Froude number $F_h \equiv U / (N L_h) $. $$ \begin{align} \Dt \vv_h & = - \bnabla_h p \Rightarrow p \sim U^2,\\ \Dt v_z & = - \p_z p + b \Rightarrow ?,\\ \Dt b & = - N^2 v_z \Rightarrow b \sim N^2 L_h \frac{v_z}{U}.\\ \end{align} $$ We found the first two scaling laws. We can then compute the ratio $$ \frac{\Dt v_z}{b} \sim F_h^2, $$ which implies that the two terms of the right hand side of the vertical velocity equation have to be of the same order, which gives: $$ \frac{v_z}{U} \simeq \frac{{F_h}^2}{\alpha} $$ We can rewrite the equation using the dimensionless variables: $$ \begin{align} \Dt \vv_h & = - \bnabla_h p,\\ {F_h}^2 \Dt v_z & = - \p_z p + b,\\ \Dt b & = - N^2 v_z,\\ \end{align} $$ with $$\Dt = \p_t + \vv_h \cdot \bnabla + \left(\frac{F_h}{\alpha}\right)^2 v_z \p_z. $$ Dominant balance gives $\alpha\sim F_h \Rightarrow L_z \sim U/N$. This important scale is called the buoyancy scale $L_b \equiv U/N$. This scaling implies that the potential energy is of the same order of magnitude as the kinetic energy. ### Viscous stratified flows {cite:t}`Brethouwer+2007` Using the strongly stratified scaling presented above, the diffusive equations are written in non-dimensional form as $$ \begin{align} \Dt \vv_h & = - \bnabla_h p + \frac{1}{Re \alpha^2} (\p_{zz} + \alpha^2 \bnabla_h^2) \vv_h,\\ {F_h}^2 \Dt v_z & = - \p_z p + b + \frac{{F_h}^2}{Re \alpha^2} (\p_{zz} + \alpha^2 \bnabla_h^2) v_z,\\ \Dt b & = - N^2 v_z + \frac{1}{ScRe \alpha^2} (\p_{zz} + \alpha^2 \bnabla_h^2) b,\\ \bnabla_h \cdot \vv_h + \left(\frac{F_h}{\alpha}\right)^2 \p_z v_z & = 0,\\ \end{align} $$ where $Sc = \nu / \kappa$ is the Schmidt number. If we only keep the dominant terms when $F_h \rightarrow 0$, it yields $$ \Dt \vv_h & = - \bnabla_h p + \frac{1}{Re \alpha^2} \p_{zz} \vv_h,\\ 0 & = - \p_z p + b,\\ \Dt b & = - v_z + \frac{1}{ScRe \alpha^2} \p_{zz} b,\\ \bnabla_h \cdot \vv_h + \left(\frac{F_h}{\alpha}\right)^2 \p_z v_z & = 0.\\ $$ At this point, we need to define the buoyancy Reynolds number $$\R = Re F_h^2 = \frac{\epsK}{\nu N^2} = \left(\frac{L_b}{L_\nu}\right)^2,$$ where $L_\nu = \sqrt{\nu L_h / U}$. - Small buoyancy Reynolds number limit $L_z \sim L_\nu$. Viscous stratified turbulence with 2D like advective term. Recover equations found by {cite:t}`Riley+1981` in the limit of $Fr < 1$ (i.e. $F_h < 1$ and $F_v < 1$). $$ (\p_t + \vv_h \cdot \bnabla) \vv_h & = - \bnabla_h p + \p_{zz} \vv_h,\\ 0 & = - \p_z p + b,\\ (\p_t + \vv_h \cdot \bnabla) b & = - v_z + \frac{1}{Sc} \p_{zz} b,\\ \bnabla_h \cdot \vv_h & = 0.\\ $$ - Large buoyancy Reynolds number limit $$ (\p_t + \vv_h \cdot \bnabla + v_z \p_z) \vv_h & = - \bnabla_h p,\\ 0 & = - \p_z p + b,\\ (\p_t + \vv_h \cdot \bnabla + v_z \p_z) b & = - v_z,\\ \bnabla_h \cdot \vv_h + \p_z v_z & = 0.\\ $$ $L_z \sim L_b = U/N$, $E_P \sim E_K$. Strongly anisotropic with a 3D like advective term. $\Rightarrow$ "Strongly Stratified Turbulence" regime, latter called LAST regime (Layered Anisotropic Stratified Turbulence). ## Few numerical results - {cite:t}`RileyDeBruynKops2003` [pdf](https://people.umass.edu/debk/Papers/riley03.pdf) - {cite:t}`Lindborg2006` [pdf](https://www.researchgate.net/profile/Erik-Lindborg/publication/231939096_The_energy_cascade_in_a_strongly_stratified_fluid/links/640efb4ea1b72772e4f4417b/The-energy-cascade-in-a-strongly-stratified-fluid.pdf) ```{figure} ./figs/fig_table_Lindborg2006.png --- width: 70% name: fig_table_Lindborg2006 --- {cite:t}`Lindborg2006` simulation parameters. $No$ is the number of time steps and $\Delta t$ is the time step in the statistically stationary state. For run 6 no stationary state was reached and the time step was therefore decreasing during the whole run. ``` ```{figure} ./figs/fig_Lindborg2006_spectra.png --- width: 70% name: fig_Lindborg2006_spectra --- Compensated horizontal kinetic and potential energy spectra from runs 8 and 9. ``` - {cite:t}`HebertDeBruynKops2006` [pdf](https://people.umass.edu/debk/Papers/hebert06b.pdf) - {cite:t}`Brethouwer+2007` [pdf](http://yakari.polytechnique.fr/Django-pub/documents/chomaz2007rp-1pp.pdf) - http://legi.grenoble-inp.fr/people/Pierre.Augier/docs/2023/gafdem/pres.html