Based on first principles, a theoretical model for El Niño-Southern Oscillation (ENSO) is derived that consists of prognostic equations for sea surface temperature (SST) and for thermocline variation. Considering only the large-scale, equatorially symmetric, standing basin mode yields a minimum dynamic system that highlights the cyclic, chaotic, and season-dependent evolution of ENSO.
For a steady annual mean basic state, the dynamic system exhibits a unique limit cycle solution for a fairly restricted range of air-sea coupling. The limit cycle is a stable attractor and represents an intrinsic interannual oscillation of the coupled system. The deepening (rising) of the thermocline in the eastern (western) Pacific leads eastern Pacific warming by a small fraction of the cycle, which agrees well with observation and plays a critical role in sustaining the oscillation. When the nonlinear growth of SST anomalies reaches a critical amplitude, the delayed response of thermocline adjustment provides a negative feedback, turning over warming to cooling or vice versa.
When the basic state varies annually, the limit cycle develops a strange attractor and the interannual oscillation displays inherent deterministic chaos. On the other hand, the transition phase of the oscillation tends to frequently occur in boreal spring when the basic state is most unstable. The strongest boreal spring instability is due to the weakest mean upwelling and largest vertical temperature difference across the mixed layer base. The former minimizes the negative feedback of mean upwelling, whereas the latter maximizes the positive feedback of anomalous upwelling effects on SST; both favor spring instability. It is argued that the season-dependent coupled instability may be responsible for the tendencies of ENSO phase locking with season and period-locking to integer multiples of the annual period, which, in turn, create irregularities in oscillation period and amplitude.