Spontaneous Stochasticity of Velocity in Turbulence Models

Abstract

We analyze the phenomenon of spontaneous stochasticity in fluid dynamics formulated as the nonuniqueness of solutions resulting from viscosity at infinitesimal scales acting through intermediate on large scales of the flow. We study the finite-time onset of spontaneous stochasticity in a real version of the GOY shell model of turbulence. This model allows high-accuracy numerical simulations for a wide range of scales (up to ten orders of magnitude) and demonstrates non-chaotic dynamics, but leads to an infinite number of solutions in the vanishing viscosity limit after the blowup time. Thus, the spontaneous stochasticity phenomenon is clearly distinguished from the chaotic behavior in turbulent flows. We provide the numerical and theoretical description of the system dynamics at all stages. This includes the asymptotic analysis before and after the blowup leading to universal (periodic and quasi-periodic) renormalized solutions, followed by nonunique stationary states at large times.