On the importance of stochasticity in closures of turbulence^(a)

¹ Department of Physics and INFN, University of Rome “Tor Vergata” - Rome, Italy
² LTCI, Télécom Paris, IP Paris - Paris, France
³ Inria and École Polytechnique, IP Paris - Paris, France
⁴ Department of Applied Mathematics & Statistics, The Johns Hopkins University - Baltimore, MD, USA
⁵ Instituto de Matemtica Pura e Aplicada - IMPA - Rio de Janeiro, Brazil

Received: 23 February 2026
Accepted: 1 April 2026

Abstract

Deterministic closures for coarse-grained turbulence models help reproduce mean statistics, but often fail to capture the finite-time growth of uncertainty. Using the framework of shell models as a quantitative multi-scale testbed, we compare fully resolved simulations with large-eddy simulations using either stochastic or deterministic subgrid closures. While in the fully resolved system a single microscopic perturbation is rapidly amplified by strongly chaotic dynamics, truncation produces a strong delay and suppression of variance growth when uncertainty is introduced through initial condition perturbations only. We show that a data-driven Langevin-type stochastic closure restores the correct timing and magnitude of variance growth across scales, demonstrating that sustained stochasticity is essential for predictability in reduced turbulent dynamics.