论文标题
Navier-Stokes方程的一致随机大规模表示
A consistent stochastic large-scale representation of the Navier-Stokes equations
论文作者
论文摘要
在本文中，我们分析了位置不确定性（LU）下建模框架中定义的不可压缩Navier-Stokes方程的随机表示的理论特性。这种由雷诺传送定理的随机版本构建的设置结合了所谓的传输噪声，涉及几个特定的附加功能，例如大规模扩散术语，类似于经典的子网格模型，以及由小规模囊泡组成部分的空间不均匀性引起的修改后的对流术语。在一系列研究中，对这种形式主义进行了数值评估，对地球物理流近似和数据同化特别感兴趣。在这项工作中，我们更具体地将其理论分析关注。我们通过古典论点证明了以Lu形式的随机Navier-Stokes方程的Martingale解决方案的存在。我们证明它们是路径方向且独特的2D流。然后，我们证明，如果噪声强度为零，这些解决方案会收敛到尺寸$ 3 $的子序列，以确定性navier-Stokes方程的解决方案。与已建立的大型模拟策略的网格收敛属性类似，该结果使我们能够对Lu Navier-Stokes方程的解释进行一些保证，以作为确定性Navier-Stokes方程的一致大型模型。
In this paper we analyze the theoretical properties of a stochastic representation of the incompressible Navier-Stokes equations defined in the framework of the modeling under location uncertainty (LU). This setup built from a stochastic version of the Reynolds transport theorem incorporates a so-called transport noise and involves several specific additional features such as a large scale diffusion term, akin to classical subgrid models, and a modified advection term arising from the spatial inhomogeneity of the small-scale velocity components. This formalism has been numerically evaluated in a series of studies with a particular interest on geophysical flows approximations and data assimilation. In this work we focus more specifically on its theoretical analysis. We demonstrate, through classical arguments, the existence of martingale solutions for the stochastic Navier-Stokes equations in LU form. We show they are pathwise and unique for 2D flows. We then prove that if the noise intensity goes to zero, these solutions converge, up to a subsequence in dimension $3$, to a solution of the deterministic Navier-Stokes equation. similarly to the grid convergence property of well established large-eddies simulation strategies, this result allows us to give some guarantee on the interpretation of the LU Navier-Stokes equations as a consistent large-scale model of the deterministic Navier-Stokes equation.
