El Niño–Southern Oscillation (ENSO) remains the main source of interannual climate variability, affecting rainfall, hurricanes, and crops worldwide. Observations show that warm events (El Niño) and cold events (La Niña) most often reach their maximum in boreal winter. However, it has remained unclear which mechanisms are responsible for this seasonal locking. This work continues the study of a simple delay model proposed earlier, and fits within attempts to explain ENSO complexity through the interaction of nonlinear dynamics and seasonal forcing. Unlike neighboring works that often limit themselves to linear stability analysis or individual trajectories, here a systematic numerical study is carried out in a three-dimensional parameter space.
The authors consider a delay differential equation where the rate of change of the sea surface temperature anomaly depends on its past value (through a hyperbolic tangent function) and on seasonal forcing (a cosine with a one-year period). The delay corresponds to the travel time of Kelvin and Rossby waves across the Pacific Ocean. The equation is solved numerically using a specialized solver for delay equations (dde_solver). Three parameters are explored: the delay (from 0 to 2 years), the ocean-atmosphere coupling strength, and the amplitude of seasonal forcing. To study multiple solutions, the initial history (a constant temperature value over the delay interval) is varied in the range from -10 to 10.
The model shows strong phase locking of local extrema to the seasonal cycle: maxima (warm events) most often occur in the first half of the year (around phase 0.23–0.27, i.e., boreal winter), and minima (cold events) in the second half (around phase 0.73–0.77, i.e., boreal summer). This agrees with observations for El Niño but contradicts them for La Niña, which in reality also tend to occur in winter. Under weak seasonal forcing (b < 1.5), the phase of extrema can change abruptly: for example, the maximum jumps to summer. Coexistence of multiple stable solutions for the same parameters is also found: in one case, 61 different solutions with a period of 10 years were found. The boundaries of the basins of attraction of these solutions in the space of initial conditions can be fractal. As the delay increases, the period of solutions grows in jumps (1, 3, 5, ... years), separated by narrow zones of aperiodic, presumably chaotic behavior.
The work shows that even the simplest delay model reproduces a key feature of ENSO: the seasonal locking of warm events, but not of cold ones. This indicates that an additional mechanism is needed for a realistic description of La Niña, such as a positive feedback with a separate delay, as in more complex models. Unlike neighboring works that often focus on linear stability or individual scenarios, here a systematic analysis over a wide parameter range is presented, revealing multiple solutions and sensitivity to initial conditions. This underscores that ENSO may be an intrinsically unstable system capable of abrupt changes in behavior under climate change.
The model can be extended by adding a positive feedback with a separate delay to correct the unrealistic phase of cold events. The influence of stochastic noise, which in reality interacts with deterministic dynamics, can also be explored. Further analysis of fractal basin boundaries and transitions to chaos will help better understand ENSO predictability.
The results are important for understanding the predictability of El Niño and La Niña, as well as for assessing possible changes in their behavior under global warming. They also have implications for the general theory of delay differential equations applied in climatology and other fields.
A next step could be to compare the model with observational data to refine parameters and test predictions about phase locking. It is also worth investigating how the addition of noise affects multiple solutions and phase transitions.
The work is directly related to the unresolved problem of explaining the asymmetry between El Niño and La Niña: why warm events are stronger and rarer, while cold events are weaker and more frequent, and why both types of events are locked to winter. The model reproduces the locking for warm events but not for cold ones, indicating a gap in understanding the physics of La Niña.
The name 'El Niño' (Spanish for 'the boy child') comes from the warm current off the coast of Peru often appearing around Christmas, in honor of the baby Jesus.