Number of degrees of freedom and energy spectrum of surface quasi-geostrophic turbulence

Chuong Van Tran, Luke Austen Kazimierz Blackbourn, Richard Kirkness Scott

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We study both theoretically and numerically surface quasi-geostrophic turbulence regularized by the usual molecular viscosity, with an emphasis on a number of classical predictions. It is found that the system's number of degrees of freedom N, which is defined in terms of local Lyapunov exponents, scales as Re-3/2, where R e is the Reynolds number expressible in terms of the viscosity, energy dissipation rate and system's integral scale. For general power-law energy spectra k(-alpha), a comparison of N with the number of dynamically active Fourier modes, i.e. the modes within the energy inertial range, yields alpha = 5/3. This comparison further renders the scaling Re-1/2 for the exponential dissipation rate at the dissipation wavenumber. These results have been predicted on the basis of Kolmogorov's theory. Our approach thus recovers these classical predictions and is an analytic alternative to the traditional phenomenological method. The implications of the present findings are discussed in conjunction with related results in the literature. Support for the analytic results is provided through a series of direct numerical simulations.

Original languageEnglish
Pages (from-to)427-440
Number of pages14
JournalJournal of Fluid Mechanics
Early online date5 Sept 2011
Publication statusPublished - Oct 2011


  • Geostrophic turbulence
  • Quasi-geostrophic flows


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