Emergent dynamical scaling in the inviscid limit of 3D stochastic Navier-Stokes equation with thermal noise

TL;DR

This paper explores the behavior of 3D stochastic Navier-Stokes equations with thermal noise, revealing an emergent dynamical scaling in the inviscid limit through theoretical and numerical analysis.

Contribution

It introduces the first analysis of the inviscid limit of stochastic Navier-Stokes equations with thermal noise, identifying a new dynamical scaling law.

Findings

Emergent τ∼k^{-1} scaling in the inviscid limit.

Crossover from τ∼1/(νk^2) to τ∼1/(u_{rms}k) scaling.

Non-trivial temporal correlations in Euler flows.

Abstract

In this work, we investigate the Navier-Stokes equation in the presence of thermal noise, both at finite viscosity (revisiting the seminal work by Forster-Nelson-Stephen) and in the inviscid limit, which has not yet been explored. We determine the space-time velocity correlations in this dynamics, using functional renormalisation group and direct numerical simulations. While spectrally truncated three-dimensional Euler flows reach a stationary equilibrium state, they exhibit non-trivial temporal correlations. We show that these non-trivial correlations persist for small but finite viscosity, yielding an emergent $\tau\sim k^{-1}$ dynamical scaling, where $\tau$ is the decorrelation time. We characterise the crossover from the scaling $\tau\sim 1/(\nu k^2)$, expected at large viscosity, to the scaling $\tau\sim 1/(u_{\rm rms}k)$ found in the inviscid limit.