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.

20 pages, 9 figures

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Cited by in corpus (10)

• Cascades and Dissipative Anomalies in Compressible Fluid Turbulence
• Spontaneously stochastic solutions in one-dimensional inviscid systems
• Optimal subgrid scheme for shell models of turbulence
• Toward analytic theory of the Rayleigh-Taylor instability: lessons from a toy model
• 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
• RG approach to the inviscid limit for shell models of turbulence
• "Life after death" in ordinary differential equations with a non-Lipschitz singularity
• Spontaneous stochasticity in the fluctuating Navier-Stokes equations on a logarithmic lattice