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 viscosity \(0<\nu \ll 1\). 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 \(\delta \approx 1\).

中文翻译：

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环上随机强迫Navier-Stokes方程中准平稳态的选择

与相关联的随机被迫涡度方程二维不可压缩Navier-Stokes方程上\（D_ \增量：= [0,2 \ PI \增量] \倍[0,2 \ PI] \）被认为是用于\（\ delta \ approx 1 \），周期性边界条件和粘度\（0 <\ nu \ ll 1 \）。已知确定性涡度方程的一个明确的拟平稳状态族在有噪声和无噪声的情况下，在溶液的长期演化中都起着重要作用。最近的结果显示参数\（\ delta \）在选择哪个准平稳状态中最重要的过程中起着中心作用。在本文中，我们旨在建立一个有限维模型，该模型捕获了随机涡度方程的这种选择机制。这是通过将傅立叶空间中的涡度方程投影到与最低的八个傅立叶模式相对应的中心流形上来完成的。通过蒙特卡洛模拟，表明涡度方程和模型在长期动力学的关键方面是一致的。在进行此比较之后，通过求平均值和均化技术对模型执行扰动分析，以确定\（\ delta \ approx 1 \）的感兴趣统计量的前导动态。