Chen and Zhang (Physica D 453, 2023) derived, with full mathematical rigor, low-order stochastic conceptual models for the El Niño–Southern Oscillation (ENSO) — including stochastic versions of the discharge–recharge and delayed oscillators — by eigen-decomposition and projection of a spatially-extended linear stochastic coupled atmosphere–ocean system forced by a two-state Markov-jump representation of wind bursts. The derivation is restricted to eastern-Pacific (EP) El Niño events: the starting model omits the ocean zonal-advection feedback and the reduction machinery is linear, so central-Pacific (CP) events and ENSO diversity lie outside its scope, and the projected multiplicative-noise amplitudes had to be retuned by hand to compensate truncation error. The open problem is to extend this rigorous model-reduction program to ENSO diversity: formulate a spatially-extended stochastic coupled system that includes zonal advection and other CP-relevant mechanisms together with the essential nonlinearity; develop the nonlinear analog of the eigen-decomposition-based reduction, including systematic projection of state-dependent (jump or continuously SST-dependent) noise with controlled truncation error rather than ad hoc retuning; and validate the resulting low-order nonlinear stochastic conceptual models against observed EP and CP statistics. Later work by the same group produces diversity-capturing conceptual models by direct construction or data-driven learning, but none derives them by rigorous reduction from a spatially-extended system, so the source question remains open.

Background

The El Niño–Southern Oscillation (ENSO) is the leading mode of interannual climate variability, produced by coupled feedbacks between sea-surface temperature (SST), surface winds, and the upper-ocean thermocline (the density interface whose depth controls how cold subsurface water enters the surface layer) across the tropical Pacific. Events are diverse: in eastern-Pacific (EP) El Niño events the warm SST anomaly is centered near the South American coast, whereas central-Pacific (CP, or Modoki) events peak near the date line; the two types differ in dynamics, teleconnections, and predictability. The ocean zonal-advection feedback — anomalous east–west currents, driven by wind stress, that advect the warm pool and shift the region of strong air–sea coupling — is one of the mechanisms held responsible for CP events, and CP dynamics is generally acknowledged to involve stronger nonlinearity than EP dynamics.

Conceptual (low-order) models reduce ENSO to a few ordinary or delay differential equations for basin-scale observables; classical examples are the recharge oscillator, in which the equatorial thermocline depth slowly discharges and recharges upper-ocean heat in interaction with the sea-surface temperature anomaly, and the delayed oscillator, in which wind-driven Rossby waves (westward-propagating planetary waves) reflect at the western basin boundary and return as a delayed negative feedback on the temperature anomaly. They occupy the bottom of the model hierarchy, below intermediate-complexity models (ICMs) with explicitly resolved equatorial wave dynamics and full general circulation models (GCMs). Because ENSO predictability is limited by atmospheric stochasticity, modern conceptual models are stochastic: intermittent westerly wind bursts perturb the ocean, and since warm SST favors convection and hence bursts, the noise is multiplicative (state-dependent), often idealized as a two-state Markov jump process switching between quiescent and active wind phases.

Chen and Zhang (Physica D 453, 2023) made this construction rigorous in one direction of the hierarchy. They started from a spatially-extended stochastic coupled atmosphere–ocean system — a linear deterministic core describing equatorial Kelvin and Rossby wave dynamics (the eastward- and westward-propagating trapped oceanic wave motions that carry wind-induced thermocline-depth signals across the tropical Pacific) coupled to SST, forced by the Markov-jump wind-burst noise — and derived low-order stochastic conceptual models, including stochastic versions of the discharge–recharge and delayed oscillators, by eigen-decomposition and Galerkin projection (expanding the fields in the system's eigenmodes and projecting the dynamics onto a few retained modes), with the memory effects of the noise obtained explicitly. The framework was validated against observed large-scale EP ENSO statistics. Its limits are stated plainly in the paper: the starting model omits the zonal-advection feedback and is linear, so it cannot generate CP events; and the projected multiplicative noise amplitude σp(TW)\sigma_p(T_W), which depends on western-Pacific SST TWT_W, had to be retuned (from σp1=2.6\sigma_{p1}=2.6 to σp1=1.8\sigma_{p1}=1.8) by hand to compensate the truncation error of the low-order representation. The conclusion states that incorporating zonal advection into the starting spatially-extended model and adding suitable nonlinearity to the linear low-order conceptual models to describe CP El Niño "is remained as a future work".

Problem Statement

Extend the rigorous model-reduction program of Chen and Zhang (2023) from the eastern-Pacific linear core to ENSO diversity. Specifically: (1) formulate a spatially-extended stochastic coupled atmosphere–ocean system, of the same type as the source's starting model, that includes the ocean zonal-advection feedback (and any further mechanisms required for central-Pacific events) together with the nonlinearity those events require; (2) develop the nonlinear analog of the source's eigen-decomposition-based reduction and apply it with mathematical rigor to derive finite-dimensional low-order stochastic conceptual models from this system — including a systematic treatment of the state-dependent multiplicative noise, whether Markov-jump or continuously SST-dependent, under projection, and explicit control or quantification of the truncation error in the deterministic and stochastic coefficients so that ad hoc retuning of noise amplitudes is no longer needed; and (3) validate the derived low-order models against observations, showing that they reproduce the statistics distinguishing EP from CP El Niño (for example event frequencies, amplitudes, and the spatial and temporal distributions of the Niño3 and Niño4 SST indices, area-averaged SST anomalies over the eastern- and central-equatorial Pacific, respectively). All three parts are jointly required: neither the extended system and its rigorous nonlinear reduction without the corresponding empirical validation, nor an empirical validation of a model not obtained by that reduction, resolves the question. In part (3), "reproduce the statistics" means quantitative agreement of the observed EP/CP-distinguishing statistics — assessed against the same kind of long-record satellite-era SST reanalysis the source validates with (the OISST Niño3 and Niño4 indices, 1979–present) — including the probability-density, autocorrelation, and spatial-pattern statistics of those indices, evaluated within the derived model's own ensemble uncertainty rather than as a merely qualitative exhibition of diversity. A construction or data-driven learning of a diversity-capturing conceptual model that is not derived by rigorous reduction from a spatially-extended system does not answer the question; the open target is the mathematically controlled connection between spatially-extended dynamics and nonlinear stochastic low-order models.

References

  1. Nan Chen and Yinling Zhang, Rigorous Derivation of Stochastic Conceptual Models for the El Niño–Southern Oscillation from a Spatially-Extended Dynamical System, Physica D: Nonlinear Phenomena 453 (2023) 133842. DOI: 10.1016/j.physd.2023.133842; arXiv:2212.14179. https://arxiv.org/abs/2212.14179
  2. Yinling Zhang, Charlotte Moser and Nan Chen, A Stochastic Conceptual Model for the Coupled ENSO and MJO, preprint (2024). arXiv:2411.05264. https://arxiv.org/abs/2411.05264
  3. Yinling Zhang, Nan Chen, Jérôme Vialard and Xianghui Fang, A Physics-Informed Auto-Learning Framework for Developing Stochastic Conceptual Models for ENSO Diversity, Journal of Climate 37 (23) (2024) 6323–6347. DOI: 10.1175/JCLI-D-24-0092.1. https://doi.org/10.1175/JCLI-D-24-0092.1
  4. Jinyu Wang, Xianghui Fang, Nan Chen, Bo Qin, Mu Mu and Chaopeng Ji, A Dual-Core Model for ENSO Diversity: Unifying Model Hierarchies for Realistic Simulations, preprint (2025). arXiv:2503.20164. https://arxiv.org/abs/2503.20164
  5. Yinling Zhang, Nan Chen and Charlotte Moser, A Simple Intermediate Coupled MJO–ENSO Model: Multiscale Interactions and ENSO Complexity, Journal of Advances in Modeling Earth Systems 17 (12) (2025) e2025MS005374. DOI: 10.1029/2025MS005374. https://doi.org/10.1029/2025MS005374
  6. Mickaël D. Chekroun, Niccolò Zagli and Valerio Lucarini, Kolmogorov modes and linear response of jump-diffusion models, Reports on Progress in Physics 88 (12) (2025) 127601. DOI: 10.1088/1361-6633/ae2206. https://doi.org/10.1088/1361-6633/ae2206
  7. Qiu Yang, Andrew J. Majda and Nan Chen, ENSO Diversity in a Tropical Stochastic Skeleton Model for the MJO, El Niño, and Dynamic Walker Circulation, Journal of Climate 34 (9) (2021) 3481–3501. DOI: 10.1175/JCLI-D-20-0447.1. https://doi.org/10.1175/JCLI-D-20-0447.1