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Leith-type model of 2D turbulence and its predictions

Sergey V. Nazarenko
Mathematics Institute
University of Warwick, UK

Abstract

In 1967, Leith proposed a nonlinear differential equation to describe evolution of the spectrum of 3D turbulence in Fourier space. This equation preserves the scalings of the original fluid, it conserves energy and it has steady state solutions corresponding to the Kolmogorov energy cascade and to the thermodynamic energy equipartition. This model is attractive because of its simplicity. It allows to study unsteady formation of the Kolmogorov spectrum, study states in which the cascade and the thermal components and simultaneously present, study the effect of surface friction and other additional effects.

In the present work, we present a Leith-type model for 2D turbulence, which is different from the original Leith model in that it has two cascade solutions, - the direct cascade of enstrophy and the inverse cascade of energy. This model allows to make a prediction for the Kolmogorov exponents for the forward and the inverse cascades which is in good agreement with existing experimental and numerical data.