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El Ni\~no-Southern Oscillation (ENSO) is the most predominant interannualvariability in the tropics, significantly impacting global weather and climate.In this paper, a framework of low-order conceptual models for the ENSO issystematically derived from a spatially-extended stochastic dynamical systemwith full mathematical rigor. The spatially-extended stochastic dynamicalsystem has a linear, deterministic, and stable dynamical core. It also exploitsa simple stochastic process with multiplicative noise to parameterize theintraseasonal wind burst activities. A principal component analysis based onthe eigenvalue decomposition method is applied to provide a low-orderconceptual model that succeeds in characterizing the large-scale dynamical andnon-Gaussian statistical features of the eastern Pacific El Ni\~no events.Despite the low dimensionality, the conceptual modeling framework containsoutputs for all the atmosphere, ocean, and sea surface temperature componentswith detailed spatiotemporal patterns. This contrasts with many existingconceptual models focusing only on a small set of specified state variables.The stochastic versions of many state-of-the-art low-order models, such as therecharge-discharge and the delayed oscillators, become special cases withinthis framework. The rigorous derivation of such low-order models provides aunique way to connect models with different spatiotemporal complexities. Theframework also facilitates understanding the instantaneous and memory effectsof stochastic noise in contributing to the large-scale dynamics of the ENSO.

Rigorous Derivation of Stochastic Conceptual Models for the El Ni\~no-Southern Oscillation from a Spatially-Extended Dynamical System