Chinese Annals of Mathematics, Series B

, Volume 40, Issue 1, pp 1–38

# Mathematical Analysis of the Jin-Neelin Model of El Niño-Southern-Oscillation

• Yining Cao
• Mickaël D. Chekroun
• Aimin Huang
• Roger Temam
Article

## Abstract

The Jin-Neelin model for the El Niño–Southern Oscillation (ENSO for short) is considered for which the authors establish existence and uniqueness of global solutions in time over an unbounded channel domain. The result is proved for initial data and forcing that are sufficiently small. The smallness conditions involve in particular key physical parameters of the model such as those that control the travel time of the equatorial waves and the strength of feedback due to vertical-shear currents and upwelling; central mechanisms in ENSO dynamics.

From the mathematical view point, the system appears as the coupling of a linear shallow water system and a nonlinear heat equation. Because of the very different nature of the two components of the system, the authors find it convenient to prove the existence of solution by semi-discretization in time and utilization of a fractional step scheme. The main idea consists of handling the coupling between the oceanic and temperature components by dividing the time interval into small sub-intervals of length k and on each sub-interval to solve successively the oceanic component, using the temperature T calculated on the previous sub-interval, to then solve the sea-surface temperature (SST for short) equation on the current sub-interval. The passage to the limit as k tends to zero is ensured via a priori estimates derived under the aforementioned smallness conditions.

## Keywords

El Niño–Southern Oscillation Coupled nonlinear hyperbolic-parabolic systems Fractional step method Semigroup theory

## 2000 MR Subject Classification

35K55 35L50 35M33 47D03 76U05

## Notes

### Acknowledgements

MDC is grateful to David Neelin for the numerous inspiring discussions about the JN model and ENSO modeling in general, and to Dmitri Kondrashov for the useful discussions regarding the numerical integration of the JN model.

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© Fudan University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

## Authors and Affiliations

• Yining Cao
• 1
Email author
• Mickaël D. Chekroun
• 2
• Aimin Huang
• 1
• Roger Temam
• 1
1. 1.Department of Mathematics and the Institute for Scientific Computing and Applied MathematicsIndiana UniversityBloomingtonUSA
2. 2.Department of Atmospheric and Oceanic SciencesUniversity of CaliforniaLos AngelesUSA