Investigating irregular behavior in a model for the EI Niño Southern oscillation with positive and negative delayed feedback

Conceptual climate models are very useful for investigations into the roles of underlying processes of a climate system. We consider the conceptual model introduced by Tziperman et al. for the El Nino Southern Oscillation (ENSO) system. It describes the interactions of both a positive and a negative delayed feedback, created by an ocean-atmosphere coupling, as well as seasonal forcing. The delayed effects of the feedback mechanisms are due to the finite velocities of oceanic waves; they are incorporated explicitly into the model, which takes the form of a delay differential equation (DDE). Tziperman et al. demonstrated that, for a certain set of parameter values, this model can generate a chaotic solution that reproduces irregular behavior with certain characteristic features of the ENSO system. However, it was not clear how robust the observed chaotic behavior is against changes in model parameters and which dynamical processes are involved in its creation. Associated are the more general issues of how prominent the chaotic behavior is in the model and what ingredients are required for the observed behavior. In this paper, we conduct a bifurcation analysis with dedicated continuation software, giving an overview of the possible dynamical behavior of the ENSO model, including transitions to chaos, across large ranges of relevant parameters. Our starting point is a simpler ENSO model, a special case without positive delayed feedback and without asymmetry in the ocean-atmosphere coupling, first studied by Ghil et al. We transition to the full version of the ENSO model of Tziperman et al. by gradually introducing these two additional model features. We study their individual and combined influence on the dynamics by means of bifurcation analysis with state-of-the-art numerical tools for DDEs. Our findings show that asymmetry in the coupling function is particularly important for reproducing characteristic features more realistically. We can confirm that the chaotic behavior seen in previous studies is a result of simultaneous resonances and that, more specifically, chaotic solutions appear through the emergence of period-doubling cascades within overlapping resonance tongues. We also establish the coexistence of different routes to chaos depending on the path taken through parameter space. Such transitions to chaos are demonstrated to be robust against the seasonal forcing strength. Finally, we investigate the sensitivity of the model to changes of the delay times across a range of realistic values. We find that the observed behavior is sensitive to changes of the delay associated with the negative feedback but not the positive feedback.