This paper brings several ideas about the “butterfly effect” in turbulent fluids into one framework. The authors show that the different ways small changes can grow — from the familiar exponential sensitivity of chaos to a stronger notion called spontaneous stochasticity — can be understood by measuring how perturbations grow as a function of their size. The key tool is the Finite‑Size Lyapunov Exponent (FSLE), which gives a growth rate for disturbances at each scale instead of focusing only on infinitesimal changes.

To make this concrete, the researchers work with simplified models of turbulence. They use the Sabra shell model, a well‑known toy model that represents a hierarchy of scales by a sequence of wavenumbers and associated velocity variables. The model includes nonlinear terms that mimic advection, a viscosity that dissipates energy at small scales, and forcing at large scales to drive a cascade of energy down to dissipation. The authors extend this model to include thermal noise (following ideas from fluctuating hydrodynamics) and they also use the Kraichnan model to illustrate related behavior for particle pairs, known as Lagrangian spontaneous stochasticity.

Why the FSLE matters here is that it links three different regimes. At very small scales, classical chaos applies and infinitesimal perturbations grow roughly exponentially, δ(t) ≈ δ0 exp(λt), which leads to the standard Lyapunov predictability time Tp ≈ (1/λ) ln(Δ/δ0) when both the initial uncertainty δ0 and the tolerated error Δ are infinitesimal. At larger scales in a multiscale turbulent cascade, disturbances do not simply follow that single exponential law. The FSLE captures this scale dependence and shows how predictability is controlled by how disturbances cascade across scales. Using these models the authors bridge the small‑scale Lyapunov regime with large‑scale predictability and interpret the latter in terms of Eulerian spontaneous stochasticity — a form of intrinsic unpredictability that can persist even as background noise goes to zero. They also use the FSLE and the Kraichnan model to illustrate the closely related Lagrangian spontaneous stochasticity for particle pairs.