Differential Model for 2D Turbulence

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Differential Model for 2D Turbulence

• Sergey Nazarenko, Warwick, UK
• In collaboration with Victor Lvov, Weizmann
• JETP Letters, 2006, Vol. 83, No. 12, pp. 541545.

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Leith68 model of 3D turbulence

• Kolmogorov solution

Thermodynamic energy equipartition
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Warm cascade

• Analytical solution with both cascade and
  thermodynamic components, Connaugton
  Nazarenko2004.
• Describes the bottleneck phenomenon.

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warm cascade (Connaughton, Nazarenko, 2004)

• Cascade scaling at low k
• Thermodynamic at large k

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gelation and anomalous wake

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

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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)

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Superfluid turbulence

• Turbulent superfluid and normal components
  coupled via mutual friction, Lvov, Nazarenko,
  Volovik2005 Vinen 2005 Lvov, Nazarenko,
  Skrbek2006.

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Systems with dual cascades

• Gravity wave turbulence on water surface,
  Hasselmann Hasselmann85 Dyachenko, Newell,
  Pushkarev, Zakharov91

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Differential model for 2D turbulence (DM2D)

• Lvov and Nazarenko2006.

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Invariants of DM2D
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Energy and Enstrophy Fluxes
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Cascade solutions
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Predictions for Kolmogorov constants
Lvov, Pomyalov, Proccacia2002

• Ihihara Kaneda2001 Danilov Gurarie2001
  DNS
• CQ/CP1.9/60.32

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Effect of friction

• Change of scaling like in superfluids?
• Change of scaling due to friction in passive
  scalar (Chertkov98) and 2D turbulence Boffetta
  et al2005)

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Nastrom-Gage spectrum

• Nastrom Gage84,
• Friction?
• Gkioulekas05

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Not here

• Now, the -3 exponent is in resonance with the
  inverse cascade exponent.
• Hence a log rather than power-law correction.

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Direct cascade with friction
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Inverse cascade with friction
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Summary of friction effects

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

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Lilly89 model

• Get rid of the thermodynamic solutions 2nd
  order equation

NG spectrum, Lilly89
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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.