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Commit d144f653 authored by Pierre Augier's avatar Pierre Augier
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2022strat_turb_toro: text

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......@@ -173,7 +173,7 @@
of horizontal size $L_x = L_y = 3$ the three-dimensional Navier-Stokes equations under
the Boussinesq approximation:
\begin{align}
\p_t\vv + (\vv \cdot \bnabla)\vv = b\boldsymbol{e}_z - \frac{1}{\rho_0}\bnabla p +
\p_t\vv + (\vv \cdot \bnabla)\vv = b\boldsymbol{e}_z - \bnabla p +
\nu_2\nabla^2\vv + \ff_{\text{toro}},\label{ns} \\
\p_t{b} + (\vv \cdot \bnabla)b = -N^2 v_z + \kappa_2\nabla^2{b},\label{buoy}
\end{align}
......@@ -177,7 +177,7 @@
\nu_2\nabla^2\vv + \ff_{\text{toro}},\label{ns} \\
\p_t{b} + (\vv \cdot \bnabla)b = -N^2 v_z + \kappa_2\nabla^2{b},\label{buoy}
\end{align}
where $\vv$ is the velocity, $b$ the buoyancy, $p$ the pressure and $N$ the \bv
where $\vv$ is the velocity, $b$ the buoyancy, $p$ a rescaled pressure and $N$ the \bv
frequency. For all simulations the viscosity $\nu_2$ and the diffusivity $\kappa_2$ are
equal (Schmidt number $Sc = \nu_2/\kappa_2 = 1$). Note that the buoyancy can be
expressed as $b=-g\delta\rho/\rho_0$, with $g$ the gravitational acceleration, $\rho_0$
......@@ -191,5 +191,5 @@
k_x$ are truncating to limit aliasing. The precise shape of the truncation actually
corresponds to the Fluidsim parameter
\mintinline{python}{params.oper.truncation_shape="no_multiple_aliases"}. (ii) All shear
modes (for which $|\mathbf{k_h}| = 0$) are truncated (Fluidsim parameter
modes (i.e. modes for which $|\mathbf{k_h}| = 0$) are truncated (Fluidsim parameter
\mintinline{python}{params.oper.NO_SHEAR_MODES = True}). If we do not truncate them,
......@@ -195,7 +195,12 @@
\mintinline{python}{params.oper.NO_SHEAR_MODES = True}). If we do not truncate them,
they tend to grow very slowly so the simulations do not really reach a statistically
stationary flow. Finally, (iii) vertically invariant vertical velocity (internal waves
at $\omega = N$) is also forbidden \mintinline{python}{params.no_vz_kz0 = True}). Note
they tend to grow very slowly and the simulations do not really reach a statistically
stationary flow. This was observed in many numerical studies of stratified turbulence
[TODO citations]. A quasi statistically stationary is finally reached with very strong
shear modes dominated the dynamics. All other structures at other scales are strongly
distorted by the shear modes.
Finally, (iii) vertically invariant vertical velocity (corresponding to internal waves
at $\omega = N$) is also forbidden (\mintinline{python}{params.no_vz_kz0 = True}). Note
that in all experiments in tanks both shear modes and vertically invariant vertical
velocity are also blocked to zero.
......@@ -199,17 +204,22 @@
that in all experiments in tanks both shear modes and vertically invariant vertical
velocity are also blocked to zero.
The term $\ff_{\text{toro}}$ is a large scale ($k_z = 0$ and $3 \leq k_h/\delta k_h
\leq 5$) time correlated toroidal forcing computed in spectral space such that the
kinetic energy injection rate is constant and equal to unity. In physical space, large
columnar vortices of horizontal length scale of typically $L_f = 1$ associated with
vertical vorticity are constantly forced. In few time units, a statistically stationary
state is reached (remember that there is no shear mode in these simulations). In this
state, the time averaged total energy dissipation rate $\eps$ is equal to the kinetic
energy injection rate $P_K = 1$. The kinetic energy dissipation rate $\epsK$ is just a
function of the mixing coefficient $\Gamma = \epsA / \eps$ and is in any case of order
unity. By construction, there are transfers of energy from the large forced scales to
small dissipative scales.
The forcing term $\ff_{\text{toro}}$ is a large scale ($k_z = 0$ and $3 \leq k_h/\delta
k_h \leq 5$) time correlated toroidal forcing computed in spectral space such that the
kinetic energy injection rate $P_K$ is constant and equal to unity. The exact method to
normalize the injection rate correspond to the Fluidsim parameter
\mintinline{python}{params.forcing.normalized.type = "2nd_degree_eq"}. In physical
space, large columnar vortices of horizontal length scale of typically $L_f = 1$
associated with vertical vorticity are constantly forced. In few time units, a
statistically stationary state is reached (remember that there is no shear mode in
these simulations). In this state, the time averaged total energy dissipation rate
$\eps$ is equal to the kinetic energy injection rate $P_K = 1$. The kinetic energy
dissipation rate $\epsK$ is just a function of the mixing coefficient $\Gamma = \epsA /
\eps$ and is in any case of order unity. By construction, there are transfers of energy
from the large forced scales to small dissipative scales. The fact that a stationnary
state is reached implies that there are efficient mechanisms to transfer energy from
the large forced scales to structures with scales sufficiently small to dissipate the
energy.
The main physical input parameters are the \bv frequency and the viscosity. Since both
forcing length and energy injection rate are in practice equal to 1, we can define an
......
......@@ -6,10 +6,10 @@
Oceanic and atmospheric measurements tend to show that some kind of anisotropic
turbulence is important for some intermediate scales.
Which measurements? Which scales in the oceans and the atmosphere? (submesoscales
/mesoscales).
Which measurements? Which scales in the oceans and the atmosphere? (submesoscales /
mesoscales).
These turbulent processes are fundamental for the mixing, and to better parametrize the
mixing coefficient $\Gamma$. This coefficient directly appears in the equations solved
for simulations of the oceans.
......@@ -11,13 +11,15 @@
These turbulent processes are fundamental for the mixing, and to better parametrize the
mixing coefficient $\Gamma$. This coefficient directly appears in the equations solved
for simulations of the oceans.
There exist in strongly stratified flows 2 types of structures with very different
dynamics.
Important for modeling to have a robust dynamical explanation for such measurements.
The most common explanations are based on a field of Internal Gravity Waves.
There exist in stratified flows different types of flows with very different dynamics.
1. horizontal vortices associated with vertical vorticity $\omega_z$. Very weak
vertical velocity and horizontal divergence. So-called "vortices".
2. internal gravity waves (IGW). Dispersion relation. No vertical vorticity.
......@@ -18,9 +20,14 @@
1. horizontal vortices associated with vertical vorticity $\omega_z$. Very weak
vertical velocity and horizontal divergence. So-called "vortices".
2. internal gravity waves (IGW). Dispersion relation. No vertical vorticity.
3. Kelvin-Helmholtz billow are strongly non-linear structures dominated by horizontal
vorticity.
4. Quasi isotropic turbulence.
Toroidal / poloidal decomposition. Craya-Herring.
Note that the association between poloidal field and IGW only works in particular
......@@ -36,6 +43,8 @@
GM spectra. Description compatible with internal inertial gravity waves.
However, simulations of stratified turbulence do not support these hypotheses.
However, simulations and experiments of stratified flows do not support these
hypotheses. It has not been possible to produce a flow made only of IGW consistent with
oceanic and atmospheric observations.
Simulations of stratified turbulence: different choices of initial conditions and/or
......@@ -40,4 +49,5 @@
Simulations of stratified turbulence: different choices of initial conditions and/or
forcing. Waves or vortices.
forcing. Waves or vortices. Or result of a simulation of isotropic turbulence, for
which the toroidal and the poloidal energy are statistically equal.
......@@ -43,5 +53,5 @@
A lot of studies have been dedicated to stratified turbulence forced by "vortices"
A lot of studies have been dedicated to stratified turbulence forced only in "vortices"
(vertical vorticity).
It has been understood that there is a regime of SST ("LAST") with strongly anisotropic
......@@ -45,5 +55,5 @@
(vertical vorticity).
It has been understood that there is a regime of SST ("LAST") with strongly anisotropic
flows associated with a downscale cascade.
flows associated with a downscale energy cascade.
......@@ -49,9 +59,12 @@
Parameters to reach this regime : very small $F_h$ and large $\R = Re {F_h}^2 = \epsK /
(\nu N^2)$.
Parameters to reach this regime : very small $F_h = U/(NL_h)$ (strongly influenced by
stratification) and large buoyancy Reynolds number $\R = Re {F_h}^2 = \epsK / (\nu
N^2)$ (effect of viscosity small at large horizontal scales).
Exact definition for the "turbulent" Froude and buoyancy Reynolds numbers.
Discordance between authors about the precise values but something like $F_h < 0.02$
and $\R > 20$.
Due to experimental and numerical limitations, the relation $\R = Re {F_h}^2$ makes it
very difficult to produce the LAST regime. Typically, there are no simulation or
......@@ -52,12 +65,13 @@
Discordance between authors about the precise values but something like $F_h < 0.02$
and $\R > 20$.
Due to experimental and numerical limitations, the relation $\R = Re {F_h}^2$ makes it
very difficult to produce the LAST regime. Typically, there are no simulation or
experiment for simultaneously $F_h < 0.01$ and $\R > 100$.
experiment for simultaneously $F_h < 0.01$ and $\R > 100$, since it would require very
large Reynolds number $Re = \R / {F_h}^2 > 10^6$.
Moreover in most datasets, $\R$ tends to scale as ${F_h}^2$, meaning that the data tend
to align on a line in the $(F_h,\ \R)$ space.
Since it is very difficult to cover this space, it is difficult to analyze the
......@@ -59,9 +73,9 @@
Moreover in most datasets, $\R$ tends to scale as ${F_h}^2$, meaning that the data tend
to align on a line in the $(F_h,\ \R)$ space.
Since it is very difficult to cover this space, it is difficult to analyze the
transition of regimes in stratified turbulence. If one observes a change of behavior
transition of regimes in stratified turbulence. When one observes a change of behavior
between simulations, it is because of the variations in $\R$ or in $F_h$?
Needs and Ambitions: build and publish open datasets of several clean forced-dissipated
......
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