Anomalous eddy viscosity for two-dimensional turbulence

Abstract

We study eddy viscosity for generalized two-dimensional (2D) fluids. The governing equation for generalized 2D fluids is the advection equation of an active scalar q by an incompressible velocity. The relation between q and the stream function ψ is given by q = − (−∇2)α/2ψ. Here, α is a real number. When the evolution equation for the generalized enstrophy spectrum Qα(k) is truncated at a wavenumber kc, the effect of the truncation of modes with larger wavenumbers than kc on the dynamics of the generalized enstrophy spectrum with smaller wavenumbers than kc is investigated. Here, we refer to the effect of the truncation on the dynamics of Qα(k) with k < kc as eddy viscosity. Our motivation is to examine whether the eddy viscosity can be represented by normal diffusion. Using an asymptotic analysis of an eddy-damped quasi-normal Markovian (EDQNM) closure approximation equation for the enstrophy spectrum, we show that even if the wavenumbers of interest k are sufficiently smaller than kc, the eddy viscosity is not asymptotically proportional to k2Qα(k), i.e., a normal diffusion, but to k4−αQα(k) for α > 0 and k4Qα(k) for α < 0, i.e., an anomalous diffusion. This indicates that the eddy viscosity as normal diffusion is asymptotically realized only for α = 2 (Navier–Stokes system). The proportionality constant, the eddy viscosity coefficient, is asymptotically negative. These results are confirmed by numerical calculations of the EDQNM closure approximation equation and direct numerical simulations of the governing equation for forced and dissipative generalized 2D fluids. The negative eddy viscosity coefficient is explained using Fjørtoft’s theorem and a spreading hypothesis for the spectrum.