Colloquium

The butterfly effect is often equated with the sensitive dependence of deterministic chaotic systems on initial conditions. However, this represents only one aspect of the unpredictability envisioned by Lorenz, who hypothesized that multiscale fluid flows could spontaneously lose their deterministic nature, giving rise to intrinsic randomness. This phenomenon, known as spontaneous stochasticity, is fundamentally distinct from chaos: in spontaneously stochastic systems, solutions diverge in finite time, even if they are initially arbitrarily close. Despite its significance, spontaneous stochasticity has long eluded detailed physical observation.

 In this talk, we demonstrate that spontaneous stochasticity is an inherent feature across various hydrodynamical settings and is intimately linked to the singular and dissipative nature of turbulent flows. For instance, we examine the Kelvin-Helmholtz instability in an initially singular shear layer, where the emergent macroscopic flow exhibits universal statistical properties that are triggered by, yet independent of, microscopic perturbative details.

 We further extend this perspective to surface quasi-geostrophic (SQG) turbulence, which models the active transport of temperature in a strongly stratified and rotating environment. In this system, spontaneous stochasticity and irreversibility are intertwined. However, our numerical simulations reveal that SQG turbulence exhibits a tempered form of spontaneous stochasticity, characterized in particular by a singular, yet continuous dependence upon initial conditions.