NEGATIVE EDDY VISCOSITY IN ISOTROPICALLY FORCED 2-DIMENSIONAL FLOW - LINEAR AND NONLINEAR DYNAMICS

The existence of two-dimensional flows with an isotropic and negative eddy viscosity is demonstrated. Such flows, when subject to a very wea k large-scale perturbation of wavenumber k will amplify it with a rate proportional to k2, independent of the direction. Specifically, it is assumed that the basic (unperturbed) flow is space-time periodic, pos sesses a centre of symmetry (parity-invariance) and has three- or six- fold rotational invariance to ensure isotropy of the eddy-viscosity te nsor. The eddy viscosities emerging from the multiscale analysis are c alculated by two different methods. First, there is an expansion in po wers of the Reynolds number which can be carried out to large orders, and then extended analytically (thanks to a meromorphy property) beyon d the disk of convergence. Secondly, there is a spectral method. The t wo methods typically agree within a fraction of 1%. A simple example, the 'decorated hexagonal flow', of a time-independent flow with isotro pic negative eddy viscosity is given. Flows with randomly chosen Fouri er components and the required symmetry have typically a 30% chance of developing a negative eddy viscosity when the Reynolds number is incr eased. For basic flow driven by a prescribed external force and suffic iently strong large-scale flow, the analysis is extended to the nonlin ear regime. It is found that the large-scale dynamics is governed by a Navier-Stokes or a Navier-Stokes-Kuramoto-Sivashinsky equation, depen ding on the sign and strength of the eddy viscosity. When the driving force is not mirror-symmetric, a new 'chiral' nonlinearity appears. In special cases, the large-scale equation reduces to the Burgers equati on. With chiral forcing, circular vortex patches are strongly enhanced or attenuated, depending on their cyclonicity.