Abstract
We investigate the theoretical and numerical computation of rare transitions in simple geophysical turbulent models. We consider the barotropic quasi-geostrophic and two-dimensional Navier–Stokes equations in regimes where bistability between two coexisting large-scale attractors exist. By means of large deviations and instanton theory with the use of an Onsager–Machlup path integral formalism for the transition probability, we show how one can directly compute the most probable transition path between two coexisting attractors analytically in an equilibrium (Langevin) framework and numerically otherwise. We adapt a class of numerical optimization algorithms known as minimum action methods to simple geophysical turbulent models. We show that by numerically minimizing an appropriate action functional in a large deviation limit, one can predict the most likely transition path for a rare transition between two states. By considering examples where theoretical predictions can be made, we show that the minimum action method successfully predicts the most likely transition path. Finally, we discuss the application and extension of such numerical optimization schemes to the computation of rare transitions observed in direct numerical simulations and experiments and to other, more complex, turbulent systems.
Highlights
Many turbulent flows related to climate dynamics undergo sporadic random transitions [1]; after long periods of apparent statistical stationarity close to one of the dynamical attractors they spontaneously switch to another dynamical attractor
We show how the numerical optimization algorithm predicts a rare transition that remains in the set of zonal jet states, greatly simplifying the accompanying theory
Using equilibrium theory derived in [28], we showed how the numerically predicted transition agreed with those computed through the relaxation equations of the corresponding dual system when the equilibrium hypothesis holds
Summary
Many turbulent flows related to climate dynamics undergo sporadic random transitions [1]; after long periods of apparent statistical stationarity close to one of the dynamical attractors they spontaneously switch to another dynamical attractor. Through the action functional A with the a priori given attractors one can predict the most probable rare transition by considering the local action minimizers or by solving the instanton equations with appropriate boundary conditions This problem is extremely difficult, and for turbulent flows, can only be achieved in the simplest of circumstances (see section ). Through the equilibrium Langevin dynamics theory, we can analytically predict the most probable rare transition paths between two attractors by considering an effective potential landscape and relaxation (unforced) trajectories of a dual dynamics. We know how to describe and compute the the instantons corresponding to the phase transitions between zonal flows They are none other the reversed trajectories for the relaxation paths for the dual dynamics. We observe that the theoretically predicted attractors (black dashed curves) overlay perfectly to the numerically founds attractors (coloured curves)
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