A Boolean delay equation model of ENSO variability

Boolean delay equations (BDEs) provide a mathematical framework to formulate and analyze conceptual models of complex multi-component systems. This framework is used here to construct a simple conceptual model for the El-Niño/Southern Oscillation (ENSO) phenomenon. ENSO involves the coupling of atmospheric and oceanic processes that are far from being completely understood. Our BDE model uses Boolean variables to represent key atmospheric and oceanic quantities and equations that involve logical operators to describe their evolution. Two distinct time-delay parameters, one for the local atmosphere–ocean coupling effects and the other for oceanic wave propagation, are introduced. Over a range of physically relevant delay values, this truly minimal model captures two essential features of ENSO’s interannual variability — its regularity and its tendency to phase-lock to the annual cycle. Oscillations with average cycle length that is an integer multiple of the seasonal cycle are prevalent and range from 2 to 7 years. Transition zones — where the average period lengths are noninteger rational multiples of the forcing period — exhibit Devil’s staircases, a signature of the quasi-periodic (QP) route to chaos. Our BDE model thus validates results from previous studies of the interaction of the seasonal cycle with ENSO’s “delayed oscillator”. It gives therewith support to the view that the observed irregularity results predominantly from low-order chaotic processes rather than from stochastic weather noise. Moreover, in the transition zone between the two integer periodicities of 2 and 3 years, a heretofore unsuspected, self-similar “fractal sunburst” pattern emerges in phase-parameter space. This pattern provides a distinct and more complex scenario than the QP route to chaos found in earlier, more detailed ENSO models. Period selection in this 2–3-year transitional region seems to play a key role in ENSO’s irregularity, as well as in the appearance of the observed quasi-biennial mode of variability.