Spontaneous stochasticity in the fluctuating Navier-Stokes equations on a logarithmic lattice

TL;DR

This paper investigates spontaneous stochasticity in 3D Navier-Stokes equations with small-scale forcing on a logarithmic lattice, demonstrating that solutions remain stochastic as viscosity and noise vanish, indicating universal behavior.

Contribution

It introduces a novel numerical approach on logarithmic Fourier lattices to verify spontaneous stochasticity in turbulent flows, highlighting universality in the limiting stochastic process.

Findings

Spontaneous stochasticity observed in simulations with increasing Reynolds numbers.

Solutions remain stochastic in the zero-viscosity and zero-noise limit.

Limiting solutions exhibit universal stochastic behavior.

Abstract

The predictability of turbulent flows remains a challenging problem for mathematicians, physicists, and meteorologists. In this context, we consider the 3D incompressible Navier-Stokes equations with small-scale random forcing on logarithmic lattices in Fourier space. Our goal is to probe the phenomenon of spontaneous stochasticity in this system, which means that its solutions remain stochastic in the limit of vanishing viscosity and noise. For this, we consider numerical simulations with increasing Reynolds numbers and vanishing noise amplitudes. Through measurements of statistics of individual large-scale Fourier modes, we verify the spontaneous stochasticity in two different setups: from rough initial data, and after a finite-time blowup of a strong solution. The convergence of probability density functions for distinct parameters suggests that the limiting solution is a universal stochastic process.