Selection of Quasi-stationary States in the Stochastically Forced Navier–Stokes Equation on the Torus

Abstract

The stochastically forced vorticity equation associated with the two dimensional incompressible Navier-Stokes equation on $D_\delta:=[0,2\pi\delta]\times [0,2\pi]$ is considered for $\delta\approx 1$, periodic boundary conditions, and viscocity $0<\nu\ll 1$. An explicit family of quasi-stationary states of the deterministic vorticity equation is known to play an important role in the long-time evolution of solutions both in the presence of and without noise. Recent results show the parameter $\delta$ plays a central role in selecting which of the quasi-stationary states is most important. In this paper, we aim to develop a finite dimensional model that captures this selection mechanism for the stochastic vorticity equation. This is done by projecting the vorticity equation in Fourier space onto a center manifold corresponding to the lowest eight Fourier modes. Through Monte Carlo simulation, the vorticity equation and the model are shown to be in agreement regarding key aspects of the long-time dynamics. Following this comparison, perturbation analysis is performed on the model via averaging and homogenization techniques to determine the leading order dynamics for statistics of interest for $\delta\approx1$.