Stochastic anomaly and large Reynolds number limit in hydrodynamic turbulence models

TL;DR

This paper investigates the high Reynolds number limit in hydrodynamic turbulence by introducing a vanishing random viscosity, demonstrating that the flow remains stochastic in the limit, supported by numerical simulations of the Sabra model.

Contribution

It proposes a novel stochastic approach to the high Reynolds number limit in turbulence and confirms the conjecture through numerical analysis of the Sabra model.

Findings

The probability distribution limit exists as Reynolds number approaches infinity.

The flow remains stochastic at finite times despite deterministic initial conditions.

Numerical simulations show the solution becomes stochastic after a blowup in the Sabra model.

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.