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

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.

14 pages, 4 figures

References in corpus (12)

• Chaos and predictability of homogeneous-isotropic turbulence
• Thermal Fluctuations in the Dissipation Range of Homogeneous Isotropic Turbulence
• From the butterfly effect to intrinsic randomness: the spontaneous growth of singular shear flows
• Spontaneously stochastic solutions in one-dimensional inviscid systems
• Chaotic blowup in the 3D incompressible Euler equations on a logarithmic lattice
• Spontaneous stochasticity of velocity in turbulence models
• A Renormalization Group Approach to Spontaneous Stochasticity
• Fluid dynamics on logarithmic lattices
• Spontaneous stochasticity and renormalization group in discrete multi-scale dynamics
• Statistical determinism in non-Lipschitz dynamical systems
• RG analysis of spontaneous stochasticity on a fractal lattice: stability and bifurcations
• Logarithmic lattice models for flows with boundaries