Inviscid limit and an effective energy-enstrophy diffusion process

TL;DR

This paper investigates the inviscid limit of a stochastic Galerkin-Navier-Stokes system, demonstrating that the enstrophy-energy process converges to a diffusion process in a 2D cone, with implications for mode attrition.

Contribution

It establishes the inviscid limit of the enstrophy-energy laws for a stochastic Navier-Stokes type system, linking it to a diffusion process in a 2D cone, and provides quantitative bounds on mode attrition.

Findings

The enstrophy-energy process converges to a diffusion process in the inviscid limit.

Quantitative bounds on mode attrition show suppression of higher modes.

The results hold regardless of the stirring strength.

Abstract

In this article we consider a stationary $N$-dimensional Galerkin-Navier-Stokes type evolution with Brownian forcing and random stirring (of arbitrarily small strength). We show that the stationary diffusion in an open two-dimensional cone constructed in a companion article, stands as the inviscid limit of the laws of the ``enstrophy-energy'' process of the $N$-dimensional diffusion process considered here, this regardless of the strength of the stirring. With the help of the quantitative condensation bounds of the companion article, we infer quantitative inviscid condensation bounds, which for suitable forcings show an attrition of all but the lowest modes in the inviscid limit.