Skip to content
GitLab
Explore
Sign in
Register
Primary navigation
Search or go to…
Project
F
fluiddyn_papers
Manage
Activity
Members
Labels
Plan
Issues
Issue boards
Milestones
Wiki
Code
Merge requests
Repository
Branches
Commits
Tags
Repository graph
Compare revisions
Snippets
Build
Pipelines
Jobs
Pipeline schedules
Artifacts
Deploy
Releases
Container registry
Model registry
Operate
Environments
Monitor
Incidents
Analyze
Value stream analytics
Contributor analytics
CI/CD analytics
Repository analytics
Model experiments
Help
Help
Support
GitLab documentation
Compare GitLab plans
Community forum
Contribute to GitLab
Provide feedback
Terms and privacy
Keyboard shortcuts
?
Snippets
Groups
Projects
Show more breadcrumbs
fluiddyn
fluiddyn_papers
Commits
d144f653
Commit
d144f653
authored
2 years ago
by
Pierre Augier
Browse files
Options
Downloads
Patches
Plain Diff
2022strat_turb_toro: text
parent
21cd8d00
Branches
Branches containing commit
No related tags found
1 merge request
!7
2022strat_turb_toro: text
Changes
2
Hide whitespace changes
Inline
Side-by-side
Showing
2 changed files
2022strat_turb_toro/input/article.tex
+27
-17
27 additions, 17 deletions
2022strat_turb_toro/input/article.tex
2022strat_turb_toro/input/intro.tex
+26
-12
26 additions, 12 deletions
2022strat_turb_toro/input/intro.tex
with
53 additions
and
29 deletions
2022strat_turb_toro/input/article.tex
+
27
−
17
View file @
d144f653
...
...
@@ -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
...
...
This diff is collapsed.
Click to expand it.
2022strat_turb_toro/input/intro.tex
+
26
−
12
View file @
d144f653
...
...
@@ -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
...
...
This diff is collapsed.
Click to expand it.
Preview
0%
Loading
Try again
or
attach a new file
.
Cancel
You are about to add
0
people
to the discussion. Proceed with caution.
Finish editing this message first!
Save comment
Cancel
Please
register
or
sign in
to comment