Negative eddy viscosity in isotropically forced two-dimensional flow: linear and nonlinear dynamics

Abstract

The existence of two-dimensional flows with an isotropic and negative eddy viscosity is demonstrated. Such flows, when subject to a very weak large-scale perturbation of wavenumberkwill amplify it with a rate proportional tok2, independent of the direction.Specifically, it is assumed that the basic (unperturbed) flow is space-time periodic, possesses a centre of symmetry (parity-invariance) and has three- or six-fold rotational invariance to ensure isotropy of the eddy-viscosity tensor.The eddy viscosities emerging from the multiscale analysis are calculated by two different methods. First, there is an expansion in powers of the Reynolds number which can be carried out to large orders, and then extended analytically (thanks to a meromorphy property) beyond the disk of convergence. Secondly, there is a spectral method. The two methods typically agree within a fraction of 1%.A simple example, the ‘decorated hexagonal flow’, of a time-independent flow with isotropic negative eddy viscosity is given. Flows with randomly chosen Fourier components and the required symmetry have typically a 30% chance of developing a negative eddy viscosity when the Reynolds number is increased.For basic flow driven by a prescribed external force and sufficiently strong largescale flow, the analysis is extended to the nonlinear régime. It is found that the largescale dynamics is governed by a Navier-Stokes or a Navier-Stokes-Kuramoto-Sivashinsky equation, depending 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 equation. With chiral forcing, circular vortex patches are strongly enhanced or attenuated, depending on their cyclonicity.