Stochastic anomaly and large Reynolds number limit in hydrodynamic turbulence models

Abstract

In this work we address the open problem of high Reynolds number limit in hydrodynamic turbulence, which we modify by considering a vanishing random (instead of deterministic) viscosity. In this formulation, a small-scale noise propagates to large scales in an inverse cascade, which can be described using qualitative arguments of the Kolmogorov-Obukhov theory. We conjecture that the limit of the resulting probability distribution exists as $\mathrm{Re} \to \infty$, and the limiting flow at finite time remains stochastic even if forcing, initial and boundary conditions are deterministic. This conjecture is confirmed numerically for the Sabra model of turbulence, where the solution is deterministic before and random immediately after a blowup. Then, we derive a purely inviscid problem formulation with a stochastic boundary condition imposed in the inertial interval.