随机强迫二维Navier-Stokes方程快速平流极限中平稳测度向大尺度的集中

中文摘要 AI 辅助

我们研究了环面上随机强迫的二维不可压缩Navier-Stokes方程的快速平流极限（$\u03b5 \to 0$）。在分辨率$N = 128$下，对方形环面和薄环面上的四个不同强迫壳层进行了数值模拟，平流参数$\u03b5$变化约一个数量级。我们发现，随着$\u03b5 \to 0$，平稳分布越来越集中在最低傅里叶模态上，能量-拟能比$E/Ω$和最低模态中拟能的比例$R$均单调增加。在有限$\u03b5$下，集中程度通过有效谱值$B_1/B_0$依赖于强迫壳层，这与Sznitman和Widmayer的凝聚界一致。线性外推到$\u03b5 = 0$表明极限值可能不依赖于强迫，$B_1/B_0$仅控制收敛速率。同样的定性行为在薄环面上持续存在。

英文摘要

We investigate the fast-advection limit ($\varepsilon \to 0$) of the stochastically forced two-dimensional incompressible Navier-Stokes equations on the torus. Numerical simulations are performed at resolution $N = 128$ for four different forcing shells on both the square torus and a thin torus, with the advection parameter $\varepsilon$ varied over approximately one decade. We find that the stationary distributions become increasingly concentrated on the lowest Fourier modes as $\varepsilon \to 0$, with the energy-to-enstrophy ratio $E/Ω$ and the fraction $R$ of enstrophy in the lowest modes both increasing monotonically. At finite $\varepsilon$ the degree of concentration depends on the forcing shell through the effective spectral value $B_1/B_0$, in agreement with the condensation bound of Sznitman and Widmayer. Linear extrapolation to $\varepsilon = 0$ suggests that the limiting values may be forcing-independent, with $B_1/B_0$ governing only the rate of convergence. The same qualitative behaviour persists on the thin torus.

发表机构

• ETH Zurich（苏黎世联邦理工学院）

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