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

TL;DR

The study investigates spontaneous stochasticity in the 3D incompressible Navier–Stokes equations with small-scale stochastic forcing by employing a fluctuating hydrodynamics framework on a 3D logarithmic lattice (LogLatt). By analyzing the triple scaling limit $Re^{-1}\to0$, $\Theta=\theta Re^{-\alpha}\to0$, and $\delta\to0$, it examines the large-scale statistics of Fourier modes to determine whether the limit is a universal stochastic process. The results show spontaneous stochasticity in both an irregular input scenario and in a regular scenario after finite-time Euler blowup, with limiting large-scale PDFs that are non-delta and largely independent of $\theta$ and $\alpha$, supporting universality. Together, these findings suggest that the LogLatt framework captures essential stochastic features of turbulent NS dynamics and offers a practical platform for testing theories of stochastic limits in fluid flows.

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.

Figures (4)

• Figure 1: If we have a continuous and periodic domain (left) and apply the Fourier transform, we obtain a discrete domain in the wave vector space (center). Then, logarithmic lattices (right) are toy models for the Fourier space, characterized by the fact that the spacing factor is no longer constant for all wave vectors. Instead, it increases geometrically as the wave vectors grow in norm.
• Figure 2: Probability density functions (PDFs) of the real part of the first component of the vorticity field, evaluated at a fixed large scale wave vector $k_* = (\lambda,1,1)$ and time $t_\ast=0.1$, for different values of Reynolds numbers and dimensionless temperature $\theta$. Solid lines in the last row represent the Gaussian PDFs with the same mean and variance.
• Figure 3: Time evolution of the supremum norm of the vorticity $\Vert \bm{\omega}(t)\Vert_{\infty}$. The vertical axis is plotted using logarithmic scale, which allows us to observe the growth of the norm across several orders of magnitude. In the left panel, we see the rapid growth of the vorticity norm for the solution of the deterministic Euler equations, exhibiting the asymptotics $\Vert \bm{\omega}(t) \Vert_\infty \sim (t_b-t)^{-1}$ as it approaches the blowup time $t_b \approx 0.112$. The solid line in the inset shows the same graph in loglog scale with the horizontal coordinate $t_b-t$, compared to the scaling $\sim (t_b-t)^{-1}$ shown by the dashed line. In the right panel, we can see the evolution of the same vorticity norm for the solutions of the deterministic Navier-Stokes equations at different Reynolds numbers. The inset shows final-time values as functions of $Re$ in loglog scale compared to the power law $\propto Re^{0.4}$.
• Figure 4: PDFs of the real part of the first component of the vorticity field, evaluated at a fixed large scale wave vector $k_* = (\lambda,1,1)$ and at two different instants of time, $t_1=0.075$ (before Euler's blowup) and $t_2=0.2$ (after Euler's blowup), in order to illustrate the two different statistical behaviors in the solutions of our model. The left panel corresponds to $t_1$, where we observe convergence to a deterministic solution. In contrast, the right panel correspond to $t_2$, where convergence to a spontaneously stochastic solution takes place.