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

Spectrally truncated three-dimensional (3D) Euler flows reach an absolute equilibrium. Yet they show non-trivial temporal correlations, similar to the ones of turbulent flows. In this work, we investigate Navier-Stokes equation in equilibrium conditions, both at finite viscosity and in the inviscid limit. We determine the space-time velocity correlations, using functional renormalisation group and direct numerical simulations. We characterise the crossover in the dynamical scaling from $\tau\sim 1/(\nu k^2)$ at finite viscosity to $\tau\sim 1/(u_{\rm rms}k)$ in the inviscid limit, where $\tau$ is the decorrelation time. We show that, although the static properties of spectrally truncated Euler flows radically differ from their hydrodynamical limit, they share a similar emergent $k^{-1}$ dynamical scaling.