A delay differential model of ENSO variability – Part 2: Phase locking, multiple solutions and dynamics of extrema

Abstract. We consider a highly idealized model for El Niño/Southern Oscillation (ENSO) variability, as introduced in an earlier paper. The model is governed by a delay differential equation for sea-surface temperature T in the Tropical Pacific, and it combines two key mechanisms that participate in ENSO dynamics: delayed negative feedback and seasonal forcing. We perform a theoretical and numerical study of the model in the three-dimensional space of its physically relevant parameters: propagation period τ of oceanic waves across the Tropical Pacific, atmosphere-ocean coupling κ, and strength of seasonal forcing b. Phase locking of model solutions to the periodic forcing is prevalent: the local maxima and minima of the solutions tend to occur at the same position within the seasonal cycle. Such phase locking is a key feature of the observed El Niño (warm) and La Niña (cold) events. The phasing of the extrema within the seasonal cycle depends sensitively on model parameters when forcing is weak. We also study co-existence of multiple solutions for fixed model parameters and describe the basins of attraction of the stable solutions in a one-dimensional space of constant initial model histories.