Normal and Anomalous Self-Similarity of Decaying Two-Dimensional Turbulence

Abstract

A new similarity theory is proposed for decaying two-dimensional Navier--Stokes turbulence, including the viscous range. It requires the length scale \Lambda \sim t^1/2, where t is time, and encompasses all Reynolds numbers. The theory is called anomalous self-similarity, as the multiplicative constant determining the amplitude of the motion is dimensional. In the high Reynolds number limit, the theory explains the decay laws E \sim t⁰, Q \sim t^-1 found numerically by Chasnov (1997) and Das {\it et al.} (2001), where E is energy and Q is enstrophy. For a particular (low) choice of Reynolds number, the self-similarity is normal rather than anomalous, and the theory recovers the decay laws E \sim t^-1, Q \sim t^-2 of Chasnov \& Herring (1998). The inviscid limit is shown to be singular, in that similarity theory based on the inviscid equations predicts an upscale energy flux for all wavenumbers, in violation of basic physical constraints. This may be part of the reason for the failure of Batchelor's (1969) decay laws E \sim t⁰, Q \sim t^-2.