Stratified turbulence

Stratified turbulence#

\[\newcommand{\p}{\partial} \newcommand{\vv}{\boldsymbol{v}} \newcommand{\xx}{\boldsymbol{x}} \newcommand{\bnabla}{\boldsymbol{\nabla}} \newcommand{\Dt}{\text{D}_t} \newcommand{\R}{\mathcal{R}} \newcommand{\epsK}{\varepsilon_K}\]

Few experimental results#

_images/fig_zigzag_instab_simple.png — Fig. 6 (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 Billant and Chomaz [2000].#

_images/fig_praud_2d.png — Fig. 7 Streak photographs of an evolving, grid-generated turbulent flow. Taken from Praud *et al.* [2005].#

_images/fig_praud_3d.png — Fig. 8 Initial three-dimensional instability of the laminar wake represented in terms of isosurfaces of the vertical component of vorticity \(\omega_z\). Taken from Praud *et al.* [2005].#

_images/fig_augier_exp_phd.png — Fig. 9 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 Augier *et al.* [2014].#

Scaling analyses#

Inviscid stratified flows#

Let us follow Billant and Chomaz [2001] 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 \(F_h \equiv U / (N L_h) \).

\[\begin{split} \begin{align} \Dt \vv_h & = - \bnabla_h p \Rightarrow p \sim U^2,\\ \Dt v_z & = - \p_z p + b \Rightarrow ?,\\ \Dt b & = - N^2 v_z \Rightarrow b \sim N^2 L_h \frac{v_z}{U}.\\ \end{align} \end{split}\]

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:

\[\begin{split} \begin{align} \Dt \vv_h & = - \bnabla_h p,\\ {F_h}^2 \Dt v_z & = - \p_z p + b,\\ \Dt b & = - N^2 v_z,\\ \end{align} \end{split}\]

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#

Brethouwer et al. [2007]

Using the strongly stratified scaling presented above, the diffusive equations are written in non-dimensional form as

\[\begin{split} \begin{align} \Dt \vv_h & = - \bnabla_h p + \frac{1}{Re \alpha^2} (\p_{zz} + \alpha^2 \bnabla_h^2) \vv_h,\\ {F_h}^2 \Dt v_z & = - \p_z p + b + \frac{{F_h}^2}{Re \alpha^2} (\p_{zz} + \alpha^2 \bnabla_h^2) v_z,\\ \Dt b & = - N^2 v_z + \frac{1}{ScRe \alpha^2} (\p_{zz} + \alpha^2 \bnabla_h^2) b,\\ \bnabla_h \cdot \vv_h + \left(\frac{F_h}{\alpha}\right)^2 \p_z v_z & = 0,\\ \end{align} \end{split}\]

where \(Sc = \nu / \kappa\) is the Schmidt number. If we only keep the dominant terms when \(F_h \rightarrow 0\), it yields

\[\begin{split} \Dt \vv_h & = - \bnabla_h p + \frac{1}{Re \alpha^2} \p_{zz} \vv_h,\\ 0 & = - \p_z p + b,\\ \Dt b & = - v_z + \frac{1}{ScRe \alpha^2} \p_{zz} b,\\ \bnabla_h \cdot \vv_h + \left(\frac{F_h}{\alpha}\right)^2 \p_z v_z & = 0.\\ \end{split}\]

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 Riley et al. [1981] in the limit of \(Fr < 1\) (i.e. \(F_h < 1\) and \(F_v < 1\)).

\[\begin{split} (\p_t + \vv_h \cdot \bnabla) \vv_h & = - \bnabla_h p + \p_{zz} \vv_h,\\ 0 & = - \p_z p + b,\\ (\p_t + \vv_h \cdot \bnabla) b & = - v_z + \frac{1}{Sc} \p_{zz} b,\\ \bnabla_h \cdot \vv_h & = 0.\\ \end{split}\]

Large buoyancy Reynolds number limit

\[\begin{split} (\p_t + \vv_h \cdot \bnabla + v_z \p_z) \vv_h & = - \bnabla_h p,\\ 0 & = - \p_z p + b,\\ (\p_t + \vv_h \cdot \bnabla + v_z \p_z) b & = - v_z,\\ \bnabla_h \cdot \vv_h + \p_z v_z & = 0.\\ \end{split}\]

\(L_z \sim L_b = U/N\), \(E_P \sim E_K\). Strongly anisotropic with a 3D like advective term.

\(\Rightarrow\) “Strongly Stratified Turbulence” regime, latter called LAST regime (Layered Anisotropic Stratified Turbulence).