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Description

We study stability of initial modulational perturbations of a class of stationary solutions of the two-dimensional Navier-Stokes equation. The goal of the modulational stability analysis is to understand the following unusual two-dimensional turbulence phenomenon: in the presence of small-scale eddies the transport of large-scale vector quantities is accompanied with depleted, and in some cases even ``negative", diffusion (negative eddy viscosity, inverse energy scattering). We model the eddies by cellular flows, exact solutions of the Euler equations (Reynolds number is infinity). In the case of finite Reynolds number these solutions are stationary, because specific suitable forcing is chosen. The stream function of the stationary solution satisfies $\Delta \phi= 2/\epsilon^2 \phi$. Formal multiple scale asymptotics gives rise to the homogenized equation for the modulational perturbations. This equation is nonlinear. It is Hadamard ill-posed for large Reynolds numbers. Using the dynamical systems approach we show that the time-semigroup is a contraction map for sufficiently small Reynolds number. This proves that the multiple scale asymptotics is rigorous for small Reynolds number. For any large Reynolds number we prove similar homogenization result for the linearized equation. Hence we confirm that for small Reynolds number modulational perturbations of cellular flows are (nonlinearly) stable, and for large Reynolds number modulational perturbations of cellular flows are linearly unstable.