Self-Similarity of Decaying Two-Dimensional Turbulence governed by the Charney—Hasegawa—Mima Equation

Abstract

In decaying two-dimensional Navier—Stokes turbulence, Batchelor’s simi larity hypothesis fails due to the existence of coherent vortices. However, it has recently been shown that in decaying two-dimensional turbulence governed by the Charney—Hasegawa—Mima (CHM) equation

$$\frac{\partial }{{\partial t}}\left( {{\nabla ^2}\varphi - {\lambda ^2}\varphi } \right) + J\left( {\varphi ,{\nabla ^2}\varphi } \right) = D$$

where D is a damping, the one-point probability density functions of various physical fields are well described by Batchelor’s similarity hypothesis for wave numbers k « λ (the so-called AM regime) [1]. In this report, we extend this analysis to the dynamics of spectral energy transfers. It is shown that the energy flux exhibits self-similar scaling, and the relation between the energy spectrum and the energy flux predicted by the similarity theory holds well for scales larger than that of the energy maximum. However, this relation breaks down for scales smaller than that of the energy maximum, where the observed downscale energy transfers would, according to the similarity theory, require negative energy spectra.