differential model for 2d turbulence

Talk by S. Nazarenko, July 18, 2006

Differential Model for 2D Turbulence

Sergey Nazarenko, Warwick, UK

In collaboration with Victor Lvov, Weizmann

JETP Letters, 2006, Vol. 83, No. 12, pp. 541–545.

Talk by S. Nazarenko, July 18, 2006

Leith’68 model of 3D turbulence

Kolmogorov solution:

Thermodynamic energy equipartition:

Talk by S. Nazarenko, July 18, 2006

“Warm” cascade

Analytical solution with both cascade and thermodynamic components, Connaugton & Nazarenko’2004.

Describes the bottleneck phenomenon.

Talk by S. Nazarenko, July 18, 2006

“warm cascade” (Connaughton, Nazarenko, 2004)

Cascade scaling at low k Thermodynamic at large k

Talk by S. Nazarenko, July 18, 2006

“gelation” and anomalous wake

Self-similar solution reaching infinite k in finite time Spectrum in the wake is steeper than Kolmogorov

Talk by S. Nazarenko, July 18, 2006

Setup of Kolmogorov

After reaching infinite k, the Kolmogorov spectrum sets up as a reflected from infinity wave

Typical for all finite capacity spectra Previously seen in Weak MHD

turbulence (Galtier, Nazarenko, Newell, Pouquet, 2000)

Talk by S. Nazarenko, July 18, 2006

Talk by S. Nazarenko, July 18, 2006

Talk by S. Nazarenko, July 18, 2006

Talk by S. Nazarenko, July 18, 2006

Talk by S. Nazarenko, July 18, 2006

Talk by S. Nazarenko, July 18, 2006

Superfluid turbulence

Turbulent superfluid and normal components coupled via mutual friction, Lvov, Nazarenko, Volovik’2005; Vinen 2005; Lvov, Nazarenko, Skrbek’2006.

Talk by S. Nazarenko, July 18, 2006

Systems with dual cascades

Gravity wave turbulence on water surface, Hasselmann & Hasselmann’85; Dyachenko, Newell, Pushkarev, Zakharov’91

Talk by S. Nazarenko, July 18, 2006

Differential model for 2D turbulence (DM2D)

Lvov and Nazarenko’2006.

Talk by S. Nazarenko, July 18, 2006

Invariants of DM2D

Talk by S. Nazarenko, July 18, 2006

Energy and Enstrophy Fluxes

Talk by S. Nazarenko, July 18, 2006

Cascade solutions

Talk by S. Nazarenko, July 18, 2006

Predictions for Kolmogorov constants

Ihihara & Kaneda’2001; Danilov & Gurarie’2001 DNS:

CQ/CP=1.9/6=0.32

Lvov, Pomyalov, Proccacia’2002

Talk by S. Nazarenko, July 18, 2006

Effect of friction

Change of scaling like in superfluids?

Change of scaling due to friction in passive scalar (Chertkov’98) and 2D turbulence Boffetta et al’2005)

Talk by S. Nazarenko, July 18, 2006

Nastrom-Gage spectrum

Nastrom & Gage’84,

Friction?Gkioulekas’0

5

Talk by S. Nazarenko, July 18, 2006

Not here…

Now, the -3 exponent is in resonance with the inverse cascade exponent.

Hence a log rather than power-law correction.

Talk by S. Nazarenko, July 18, 2006

Direct cascade with friction

Talk by S. Nazarenko, July 18, 2006

Inverse cascade with friction

Talk by S. Nazarenko, July 18, 2006

Summary of friction effects

There is no Nastrom-Gage shape Friction arrests both cascades at finite scales.

Talk by S. Nazarenko, July 18, 2006

Lilly’89 model Get rid of the thermodynamic

solutions – 2nd order equation:

NG spectrum, Lilly’89

Talk by S. Nazarenko, July 18, 2006

Summary

Differential models: put something in in order to get more useful stuff out.

Time evolution. Setup of cascades. Rate of total energy and enstrophy decay.

Mixed solutions with simultaneous cascades and thermal components.

Friction effects and other modifications.