El Niño–Southern Oscillation (ENSO) is the largest natural climate cycle on Earth, affecting weather from Australia to the Andes. Its phases, El Niño and La Niña, alternate irregularly, and scientists build mathematical models to understand this irregularity. This work sits alongside studies of mixed-mode oscillations in systems with multiple time scales, which have already helped explain oscillations in population and neurobiology models. However, for ENSO, a three-scale analysis was performed for the first time, opening the way to a more realistic description.
The authors used a model of three ordinary differential equations describing the temperatures of the western and eastern tropical Pacific and the thermocline depth. By introducing small parameters δ and ρ, they separated the variables into fast, intermediate, and slow. They then applied geometric singular perturbation theory: they examined critical manifolds on which the system behaves as slow-fast and classified where these manifolds lose normal hyperbolicity. Numerical simulations with δ=ρ=0.01 confirmed the analytical findings.
The analysis revealed six regimes in the (c, k) parameter plane, each with its own type of oscillation. For example, in regime V1, trajectories exhibit plateaus: long periods of almost no change, followed by abrupt jumps. In regime V2, both plateaus and plateau-free relaxation oscillations are possible depending on parameter a. For the first time, it is shown that plateau-free relaxation oscillations also exist in the two-scale limit, which had not been documented before. Key numbers: parameter c varies from 1 to c0, k from 0 to 1, and a can be on the order of 25.55 in one example.
This work differs from neighboring studies in that it is the first to systematically classify the geometry of a three-scale ENSO model. Unlike works that analyzed only two-scale systems or used numerical methods, here a complete analytical description of possible oscillatory regimes is given. This allows predicting system behavior under parameter changes without expensive simulations and links the climate model to Rosenzweig–MacArthur models in ecology.
The topic may develop toward studying the regime a≫1, where scale separation breaks down, and toward a precise description of the transition from relaxation oscillations to mixed-mode oscillations with small amplitudes. Also promising is the study of canard trajectories along the repelling branch of the slow manifold, which could provide new mechanisms for irregularity.
The results will influence the modeling of climate oscillations, especially the understanding of El Niño irregularity, and ecological dynamics, where similar geometry occurs in predator-prey models.
The next step is to study the system's behavior for large values of parameter a and to test predictions on more realistic climate models with observational data.
In climatology, the question of the causes of El Niño irregularity remains open: whether it results from chaos or stochastic forcing. This work shows that even a deterministic model with three scales can produce complex, seemingly irregular oscillations, bringing us closer to an answer.
The name 'El Niño' was given by Peruvian fishermen in honor of the Christ child, because the warm current usually arrived around Christmas.