This experiment has been realised by Jori E. Ruppert-Felsot, Emilie Regul, Eran Sharon et Harry L. Swinney at the Center for Nonlinear Dynamics, University of Texas using a rotating water tank with a forcing at the bottom. the vorticity is measured using particle image velocimetry (PIV) at a resolution N=128^{2}.

Time evolution of the total enstrophy Z

Time evolution of the maximal, minimum and mean vorticity

Time evolution of the total energy E

We split the total flow into coherent and incoherent flows using the CVS filtering method with the Coifman 12 wavelet.

Cut of the wavelet space with the treshold (red) separating coherent and incoherent wavelet modes

The percentage of retained coefficient is 4.9% N.

The percentage of retained enstrophy is 86.5% Z.

Total vorticity (min=-15.30 / max=33.53)

Coherent vorticity (min=-15.56 / max=34.09) and

incoherent vorticity (min=-3.25 / max=2.53)

Total velocity modulus (min=0 / max=2.20)

Coherent velocity modulus (min=0 / max=2.22) and

incoherent velocity modulus (min=0 / max=0.09)

Total streamfunction (min=-1.26 / max=1.28)

Coherent streamfunction (min=-1.26 / max=1.28) and

incoherent streamfunction (min=-0.02 / max=0.02)

Vorticity versus streamfunction for the total flow

Vorticity versus streamfunction for the coherent flow

Vorticity versus streamfunction for the incoherent flow

Probability density function (PDF) of the wavelet coefficients

Probability density function (PDF) of vorticity

Enstrophy spectrum in lin-lin coordinates

(total: green, coherent: red, incoherent: blue)

Enstrophy spectra in log-log coordinates

(total: green, coherent: red, incoherent: blue)

Total vorticity (min=-15.30 / max=33.53)

Coherent vorticity (min=-16.10 / max=33.12) and

incoherent vorticity (min=-5.28 / max=4.36)

Total velocity modulus (min=0 / max=2.20)

Coherent velocity modulus (min=0 / max=2.21) and

incoherent velocity modulus (min=0 / max=0.13)

Total streamfunction (min=-1.26 / max=1.28)

Coherent streamfunction (min=-1.28 / max=1.30) and

incoherent streamfunction (min=-0.06 / max=0.05)

Vorticity versus streamfunction for the total flow

Vorticity versus streamfunction for the coherent flow

Vorticity versus streamfunction for the incoherent flow

Probability density function (PDF) of vorticity

Enstrophy spectrum in lin-lin coordinates

(total: green, coherent: red, incoherent: blue)

Enstrophy spectra in log-log coordinates

(total: green, coherent: red, incoherent: blue)

This experiment has been performed by Kai Schneider, CMI, Université de Provence (Marseille), using a pseudo-spectral code integrating the two-dimensional Navier-Stokes equations without forcing. No-slip boundary conditions have been imposed in the cylindrical domain using volume penalisation.

The resolution is N=1024^{2}.

Time evolution of the total enstophy z

Time evolution of the maximal, minimal and mean vorticity

Cut of the wavelet space with the treshold (red) separating coherent and incoherent wavelet modes

The percentage of retained coefficient is 6.8% N.

The percentage of retained enstrophy is 98.9% Z.

Total vorticity (min=-123.63 / max=124.44)

Coherent vorticity (min=-123 / max=122.9) and

incoherent vorticity (min=-6.78 / max=7.09)

Total velocity modulus (min=0 / max=2.67)

**Figure: **Coherent velocity modulus (min=0 / max=2.67) and

incoherent velocity modulus (min=0 / max=0.03)

Total streamfunction (min=-1.18 / max=0.60)

**Figure: **Coherent streamfunction (min=-1.18 / max= 0.60) and

incoherent streamfunction (min=-3E-03 / max= 2E-03)

Vorticity versus streamfunction for the total flow

Vorticity versus streamfunction for the coherent flow

Vorticity versus streamfunction for the incoherent flow

Probability density function (PDF) of the wavelet coefficients

Probability density function (PDF) of the vorticity

**Figure: **Enstrophy spectrum in lin-lin coordinates

(total: green, coherent: red, incoherent: blue)

**Figure: **Enstrophy spectra in log-log coordinates

(total: green, coherent: red, incoherent: blue)

This experiment has been done using a pseudo-spectral code integrating the two-dimensional Saint-Venant equations in the Shallow-wather model without forcing. No-slip boundary conditions have been imposed in the cylindrical domain using volume penalisation.

The resolution is N=256^{2}.

Time evolution of the total enstophy z

Time evolution of the maximal, minimal and mean vorticity

Time evolution of the total energy E

We split the total flow into coherent and incoherent flows using the CVS filtering method with the Coifman 12 wavelet.

Cut of the wavelet space with the treshold (red) separating coherent and incoherent wavelet modes

The percentage of retained coefficient is 2.8% N.

The percentage of retained enstrophy is 81.4% Z.

Total vorticity (min=-123.63 / max=124.44)

Coherent vorticity (min=-123 / max=122.9) and

incoherent vorticity (min=-6.78 / max=7.09)

Total velocity modulus (min=0 / max=2.67)

**Figure: **Coherent velocity modulus (min=0 / max=2.67) and

incoherent velocity modulus (min=0 / max=0.03)

Total streamfunction (min=-1.18 / max=0.60)

**Figure: **Coherent streamfunction (min=-1.18 / max= 0.60) and

incoherent streamfunction (min=-3E-03 / max= 2E-03)

Vorticity versus streamfunction for the total flow

Vorticity versus streamfunction for the coherent flow

Vorticity versus streamfunction for the incoherent flow

Probability density function (PDF) of the wavelet coefficients

Probability density function (PDF) of the vorticity

**Figure: **Enstrophy spectrum in lin-lin coordinates

(total: green, coherent: red, incoherent: blue)

**Figure: **Enstrophy spectra in log-log coordinates

(total: green, coherent: red, incoherent: blue)

We plan to perform other numerical experiments:

- using a 2D Navier-Stokes code with forcing
- using a shallow water code with forcing.