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.

Detailed theoretical and numerical results are presented for the eddy viscosity of three-dimensional forced spatially periodic incompressible flow.As shown by Dubrulle & Frisch (1991), the eddy viscosity, which is in general a fourth-order anisotropic tensor, is expressible in terms of the solution of auxiliary problems. These are, essentially, three-dimensional linearized Navier–Stokes equations which must be solved numerically.The dynamics of weak large-scale perturbations of wavevector k is determined by the eigenvalues – called here ‘eddy viscosities’ – of a two by two matrix, obtained by contracting the eddy viscosity tensor with two k-vectors and projecting onto the plane transverse to k to ensure incompressibility. As a consequence, eddy viscosities in three dimensions, but not in two, can become complex. It is shown that this is ruled out for flow with cubic symmetry, the eddy viscosities of which may, however, become negative.An instance is the equilateral ABC-flow (A = B = C = 1). When the wavevector k is in any of the three coordinate planes, at least one of the eddy viscosities becomes negative for R = 1/v > R_c [bsime ] 1.92. This leads to a large-scale instability occurring for a value of the Reynolds number about seven times smaller than instabilities having the same spatial periodicity as the basic flow.