The real butterfly effect: from the pop culture to mathematics and physics

Abstract

The "butterfly effect", introduced over half a century ago by Edward Lorenz, has shifted from a cornerstone of dynamical systems to a popular metaphor, yet its true physical manifestation in fully developed turbulence spans a spectrum of phenomena from standard chaotic sensitivity to the recently established concept of Eulerian spontaneous stochasticity. This paper presents an attempt at a systematic synthesis that brings these different but interconnected ideas together within the unifying framework of the Finite Size Lyapunov Exponent (FSLE). The FSLE describes the growth rate of perturbations as a function of their scale, enabling a comprehensive characterization of the multiscale physics of turbulent flows. Using the FSLE and the Sabra shell model, extended to include thermal noise, we bridge the classical, small-scale Lyapunov regime with predictability at large scales and its interpretation in terms of Eulerian spontaneous stochasticity. Moreover, using the FSLE and the Kraichnan model, we also illustrate the closely related phenomenon of Lagrangian spontaneous stochasticity. To complete the spectrum of butterfly effects, we also examine the "literal butterfly" scenario of localized, sub-dissipative perturbations. Ultimately, this synthesis clarifies the physical mechanisms that dictate the fundamental boundaries of forecasting in high-Reynolds-number flows.