A New Method to Compute Transition Probabilities in Multi‐Stable Stochastic Dynamical Systems: Application to the Wind‐Driven Ocean Circulation

Abstract

Abstract The Kuroshio Current in the North Pacific displays path changes on an interannual‐to‐decadal time scale. In an idealized barotropic quasi‐geostrophic model of the double‐gyre wind‐driven circulation under stochastic wind‐stress forcing, such variability can occur due to transitions between different equilibrium states. The high‐dimensionality of the problem makes it challenging to determine the probability of these transitions under the influence of stochastic noise. Here we present a new method to estimate these transition probabilities, using a Dynamical Orthogonal (DO) field approach. In the DO approach, the solution of the stochastic partial differential equations system is decomposed using a Karhunen–Loève expansion and separate problems arise for the ensemble mean state and the so‐called time‐dependent DO modes. The original method is first reformulated in a matrix approach which has much broader application potential to various (geophysical) problems. Using this matrix‐DO approach, we are able to determine transition probabilities in the double‐gyre problem and to identify transition paths between the different states. This analysis also leads to the understanding which conditions are most favorable for transition.

Keywords

• model order reduction techniques
• wind‐driven ocean circulation
• transition probabilities and paths
• multi‐stable stochastic dynamical systems
• numerical methods