Computation of rare transitions in the barotropic quasi-geostrophic equations

TL;DR

This paper combines theoretical analysis and numerical methods to compute the most probable paths of rare transitions between coexisting states in geophysical turbulence models, advancing understanding of transition mechanisms.

Contribution

It introduces a numerical optimization approach using minimum action methods to predict rare transition paths in geophysical turbulent systems, validated against theoretical predictions.

Findings

Minimum action methods successfully predict transition paths.

Theoretical and numerical results agree in equilibrium cases.

Approach applicable to complex turbulent systems and experiments.

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 compute rare transitions observed in direct numerical simulations, experiments and to other, more complex, turbulent systems.