Simple analytic solutions of the linear delayed-action oscillator equation relevant to ENSO theory

The El Niño-Southern Oscillation (ENSO) is a major driver of climate variability in many parts of the world. Impressive progress has been made in the last 25 years in consolidating the scientific and mathematical basis to our understanding of ENSO. This includes the development and analysis of a hierarchy of models—including simple analogue models—to simulate and understand ENSO physics. The delayed-action oscillator (DAO) equation has been a particularly important analogue model in the historical development of our understanding of ENSO physics, and numerical solutions of this equation have been explored in detail in previous studies. Given this importance, it is surprising that no exact analytic solutions to the equation have been provided previously in the ENSO literature. This situation is rectified here by deriving and presenting analytic solutions to the linear DAO equation \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{{dT}}{{dt}} = aT - bT\left( {t - \tau } \right) $$\end{document} for parameter values relevant to ENSO. Here, T is an index for ENSO variability at time t; a, b, and τ (the delay time >0) are real parameters. A comparison between observations and (linear) theory suggests that ENSO behaves as a damped oscillator with a period of 3.8 years and a damping time-scale of 0.9 years. The parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \gamma = b\tau {e^{ - a\tau }} $$\end{document} is found to be crucial in understanding the behavior of the solution and the lowest frequency mode. For example, if γ > 1/e the solution is oscillatory. Exact analytic solutions to the DAO equation which are phase-locked to the annual cycle—as is the case for ENSO—are also obtained. The overall (annual average) stability of a phase-locked system and its intrinsic periodicities differ from the corresponding properties of the system with parameters set to their annual averages (i.e., the corresponding solution which is not phase-locked). Phase-locking therefore alters the growth rate and period of the lowest frequency mode.