Talk by S. Nazarenko, July 18, 2006 Differential Model for 2D Turbulence Sergey Nazarenko, Warwick, UK In collaboration with Victor Lvov, Weizmann JETP.

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Presentation transcript:

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

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: C Q /C P =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’05

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 – 2 nd 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.