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Résumé |
\scriptsize{We will present the proof of a new formula for the transition rates between two basins of attraction of dynamical systems with weak noises, for non gradient dynamics, in arbitrary dimensions. This formula extend both the large deviation results of Freidlin-Wentzell theory (by computing the prefactor of the large deviation estimate), and the Eyring-Kramers formula valid only for gradient dynamics. We discuss applications to turbulent flows. For the computation of transitions rates for the metastable turbulent dynamics of atmosphere jets, described by the stochastic barotropic quasi-geostrophic model, we will discuss analytical results (based on averaging) and numerical results (based on the adaptive multilevel splitting algorithm). }\\~\\
\scriptsize{\textbf{Extended abstract}:
Many natural and experimental turbulent flows display a bistable behavior: rare and abrupt dynamical transitions between two very different subregions of the phase space. The most prominent natural examples are probably the Earth magnetic field reversals (over geological timescales), the Kuroshio bistability, or the Dansgaard-Oeschger events that have affected the Earth climate during the last glacial period, and are probably due to several attractors of the turbulent ocean dynamics. Recent results show that similar bistability occur also in the turbulent dynamics of atmosphere jets. Those abrupt transitions are extremely rare events that change drastically the nature of the flow and are thus of paramount importance.
The first part of the talk will be mainly mathematical. We will review the Freidlin-Wentzell theory that describes large deviations for dynamical systems with weak noise, and the Eyring-Kramers relation for the transition rates between two attractors for gradient dynamics. We will present our recent generalization of the Eyring-Kramers relation for non gradient dynamics. This result will be useful for any non-equilibrium dynamics, including turbulent flows, in the regimes of rare transitions.
The second part will focus on turbulent flows, including experimental and numerical studies. Most of this part will focus on theoretical, mathematical, and numerical works in the framework of the quasi-geostrophic barotropic model. This is the simplest turbulence model to set up the theoretical and numerical tools to study these phenomena. From a numerical point of view, those events can not be studied directly because they are too rare. We will first discuss the use of a rare event algorithm, Adaptive Multilevel Splitting, in order to sample from direct numerical simulation such rare transitions.
In quasigeostrophic models, the classical eddy-mean flow interactions are involved to explain the evolution of the large scale flow. The issue is then to understand and predict those eddy-mean flow configurations that lead to the rare fluctuations that trigger rare transitions. We will present the proper mathematical framework to analyze those rare events: large deviation theory for dynamical systems with a separation of time scales. We will discuss the development of this framework and the computation of rare transitions for the barotropic quasi-geostrophic model.} |
