Butterfly Effects in Turbulence: A Unified View through the Finite Size Lyapunov Exponent
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 sensitivity on initial conditions to the recently established concept of Eulerian spontaneous stochasticity. Here, we propose 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.
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BibTeX
@misc{valadao2026butterfly,
title = {{Butterfly Effects in Turbulence: A Unified View through the Finite Size Lyapunov Exponent}},
author = {Victor de Jesus Valadão and Erik Aurell and Guido Boffetta and Massimo Cencini and Stefano Musacchio and Angelo Vulpiani},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2607.24715},
eprint = {2607.24715},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.24715},
}
RIS
TY - UNPB
TI - Butterfly Effects in Turbulence: A Unified View through the Finite Size Lyapunov Exponent
AU - Victor de Jesus Valadão
AU - Erik Aurell
AU - Guido Boffetta
AU - Massimo Cencini
AU - Stefano Musacchio
AU - Angelo Vulpiani
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2607.24715
UR - https://arxiv.org/abs/2607.24715
AB - 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 sensitivity on initial conditions to the recently established concept of Eulerian spontaneous stochasticity. Here, we propose 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.
ER -