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Selection of quasi-stationary states in the stochastically forced Navier-Stokes equation on the torus. (English) Zbl 1446.37064
The two-dimensional (in space) periodic Navier-Stokes equation with stochastic forcing is considered on \([0,2\pi\delta]\times[0,2\pi]\) with \(\delta\approx 1\). A finite-dimensional stochastic differential equation model is developed to study the dynamics of the vorticity equation with noise when the viscosity is small. In fact, eight Fourier modes are involved in the computations. Special solutions, the so-called bar states (i.e., unidirectional Kolmogorov flows), and dipoles are obtained as long-time quasi-stationary asymptotic states for that model. These solutions approximate the Euler system in the limit of inviscid flows.
MSC:
37L55 Infinite-dimensional random dynamical systems; stochastic equations
37N10 Dynamical systems in fluid mechanics, oceanography and meteorology
37H10 Generation, random and stochastic difference and differential equations
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76D06 Statistical solutions of Navier-Stokes and related equations
76D17 Viscous vortex flows
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