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

The predictability of turbulent flows remains a challenging problem for mathematicians, physicists and meteorologists. In this context, we consider the three-dimensional 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. This article is part of the theme issue 'Frontiers of turbulence and statistical physics'.