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Self-similar solution in the Leith model of turbulence : anomalous power law and asymptotic analysis
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Grebenev, V. N., Nazarenko, Sergey, Medvedev, S. B., Schwab, I.V. and Chirkunov, Yu A. (2013) Self-similar solution in the Leith model of turbulence : anomalous power law and asymptotic analysis. Journal of Physics A: Mathematical and Theoretical, Volume 47 (Number 2). Article number 025501. doi:10.1088/1751-8113/47/2/025501 ISSN 1751-8113.
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Official URL: http://dx.doi.org/10.1088/1751-8113/47/2/025501
Abstract
We consider a Leith model of turbulence (Leith C 1967 Phys. Fluids 10 1409) in which the energy spectrum obeys a nonlinear diffusion equation. We analytically prove the existence of a self-similar solution with a power-law asymptotic on the low-wavenumber end and a sharp boundary on the high-wavenumber end, which propagates to infinite wavenumbers in a finite-time t*. We prove that this solution has a power-law asymptotic with an anomalous exponent x*, which is less than the Kolmogorov value, x* > 5/3. This is a result that was previously discovered by numerical simulations in Connaughton and Nazarenko (2004 Phys. Rev. Lett. 92 044501). We also prove the convergence to this self-similar solution of the spectrum evolving from an arbitrary finitely supported initial data as t → t*.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | Journal of Physics A: Mathematical and Theoretical | ||||
Publisher: | IOP Publishing Ltd | ||||
ISSN: | 1751-8113 | ||||
Official Date: | 2013 | ||||
Dates: |
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Volume: | Volume 47 | ||||
Number: | Number 2 | ||||
Page Range: | Article number 025501 | ||||
DOI: | 10.1088/1751-8113/47/2/025501 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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