The goal of this work is to understand the phenomenon of eddy viscosity
in two-dimensions. In the presence of small-scale eddies the transport
of large-scale vector quantities can be accompanied with depleted, and
even ``negative" diffusion at large Reynolds number. For stationary
flows, the eddy viscosity is a tensor in the equation that governs
the transport of the slow spatial modulations of highly oscillatory
cellular solutions (eddies) of the Navier-Stokes equations.
Using multiscale techniques we derive eddy viscosity of cellular flows -
special periodic stationary solutions of the Euler's equations. We
justify this derivation using bootstrapping and Littlewood-Paley type
arguments.
We design a numerical upscaling method that allows us to compute
eddy viscosity of cellular flows when ratio between the scales is small,
and compare it with the predictions of the multiscale analysis. For
cellular flows with closed streamlines we give rigorous bounds on eddy
viscosity at high Reynolds number by means of saddle point variational
principles for nonlocal, nonselfadjoint operators.
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