Anomalous eddy viscosity for two-dimensional turbulence

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 psi is given by q = -(-del(2))(alpha/2)psi. Here, alpha is a real number. When the evolution equation for the generalized enstrophy spectrum Q(alpha)(k) is truncated at a wavenumber k(c), the effect of the truncation of modes with larger wavenumbers than k(c) on the dynamics of the generalized enstrophy spectrum with smaller wavenumbers than k(c) is investigated. Here, we refer to the effect of the truncation on the dynamics of Q(alpha)(k) with k < k(c) 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 k(c), the eddy viscosity is not asymptotically proportional to k(2)Q(alpha)(k), i.e., a normal diffusion, but to k(4-alpha)Q(alpha)(k) for alpha > 0 and k(4)Q(alpha)(k) for alpha < 0, i.e., an anomalous diffusion. This indicates that the eddy viscosity as normal diffusion is asymptotically realized only for alpha = 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 Fjortoft's theorem and a spreading hypothesis for the spectrum. (C) 2015 AIP Publishing LLC.