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Modulational Stability of Cellular FlowsOptimization
|Speaker: ||Dr. Alexei Novikov, CALTECH|
|Location: ||693 Kerr|
|Start time: ||Fri, Apr 26 2002, 4:10PM|
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
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
number these solutions are stationary, because specific suitable forcing
chosen. The stream function of the stationary solution satisfies
$\Delta \phi= 2/\epsilon^2 \phi$. Formal multiple scale asymptotics gives
to the homogenized equation for the modulational perturbations. This
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
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
of cellular flows are (nonlinearly) stable, and for large Reynolds number
modulational perturbations of cellular flows are linearly unstable.