The El Niño–Southern Oscillation (ENSO) remains one of the most powerful climate systems on the planet: its warm phase El Niño and cold phase La Niña cause droughts in Australia, India, and Africa, floods in South America, and serious damage to fisheries and agriculture. Despite decades of research, ENSO's irregularity and amplitude modulation are difficult to predict, especially rare extreme events. This work aligns with research on nonlinear climate dynamics, where neighboring studies examine critical transitions in cellular systems, stochastic background signals from black holes, and neural network training on the Ising model. However, this article focuses on a specific mechanism for the generation of extreme events in a low-dimensional ocean-atmosphere model, distinguishing it from more general approaches to complex systems.
The authors use a three-dimensional slow-fast model proposed by Timmermann and colleagues, where the thermocline depth in the western Pacific changes slowly, while sea surface temperatures in the eastern and western parts change rapidly. They numerically integrate the system using a fourth-order Runge–Kutta method with a step size of 0.01 and vary the strength of zonal advection (heat transport along the equator) from 0.0961 to 0.17. To uncover the mechanism, they construct bifurcation diagrams of maximum eastern ocean temperature, compute the critical manifold (the surface to which fast dynamics are attracted), and analyze stability via Jacobian eigenvalues. Extreme events are identified using the mean excess function, which sets a threshold above which an event is considered extreme. Statistical properties are assessed from long time series (up to $10^{12}$ iterations), fitting Weibull, Gamma, and lognormal distributions to inter-event intervals.
As the parameter increases from 0.0961 to 0.17, the system transitions from a small limit cycle through bounded chaos to a sudden expansion of the attractor at a critical value near 0.09848. This expansion occurs due to an interior crisis when a period-doubling cascade merges with a period-adding cascade. After the crisis, rare large temperature spikes emerge with variable amplitude and return time, which the authors classify as extreme events. In contrast, for larger parameter values, chaotic mixed-mode oscillations (El Niño) occur with nearly uniform height and more frequent occurrence. A channel-like structure in phase space plays a key role: a narrow channel causes trajectories to linger longer and creates large variability, while a wide channel speeds up passage and makes events more regular. The distribution of event heights is multimodal with an upper-bounded tail (the estimated shape parameter of the generalized Pareto distribution is negative). Inter-event intervals are best described by a lognormal distribution, and the autocorrelation function shows short-term dependence, indicating potential for partial prediction.
This work refines the picture of extreme event generation in ENSO, showing that they are not merely chaotic mixed-mode oscillations but arise through a specific interior crisis involving a channel-like structure. This distinguishes it from neighboring studies that either consider more general critical transitions in other systems or do not single out a separate class of extreme events with variable amplitude. The proposed mechanism explains why rare giant El Niño events may be fundamentally less predictable than ordinary ones and underscores the importance of the slow-fast timescale. For climate science, this means that simple low-dimensional models can reproduce the complex statistics of extreme events, opening the way for more targeted analysis of observational data.
The topic could develop in several directions. First, it is worth testing whether the discovered channel-like mechanism appears in more realistic ENSO models with more variables and in observational data. Second, one could investigate how external forcing (e.g., anthropogenic warming) shifts the critical parameter and alters the frequency of extreme events. Third, the methods for analyzing interior crises and phase-space channels could be transferred to other climate systems, such as monsoons or the North Atlantic Oscillation. Finally, improving statistical models for predicting return times of extreme events based on the identified dependence structure could enhance practical value.
The results will impact climatology, especially the understanding and prediction of El Niño, as well as nonlinear dynamics, where the interior crisis with a channel mechanism may be universal for many systems with rare catastrophic events.
The next step will be to compare model predictions with real sea surface temperature time series from the tropical Pacific to find traces of the channel-like structure and interior crisis. It is also important to assess how changes in the slow parameter (thermocline relaxation rate) affect the threshold for extreme events over a wider range of climatic conditions.
This work directly addresses the unsolved problem of predictability of extreme climate events: despite successes in ENSO modeling, rare strong events remain difficult to forecast. The interior crisis and channel-like structure mechanism provides a new explanation for this unpredictability, linking it to global instability in phase space rather than just local chaos. This resonates with a general problem in nonlinear dynamics: how to distinguish truly random fluctuations from deterministic chaos with rare outliers. Furthermore, the question of how well low-dimensional models can reproduce the statistics of complex climate systems remains open and important for the entire field.
The threshold for an extreme event in the model is about 25.5°C for sea surface temperature in the eastern Pacific, close to real values at which El Niño is considered strong. The time series for statistics was generated over $10^{12}$ iterations, corresponding to thousands of years of model time.