1988 : A chaotic model of El Niño

Christophe LETELLIER

El Niño is a phenomenon sometimes occuring in the Pacific Ocean and which affects weather all over the world. It results from the coupling between atmosphere and ocean inducing a positive feedback.

Weather strongly depends on ocean temperatures : when ocean temperature increases, more clouds are formed, then inducing more rain falls in the corresponding part of the world. In the Pacific Ocean, near the equator, the solar activity warms the surface water. In common conditions, strong winds along the equator push the warm surface water near South America westward toward Indonesia. By the same time, the cooler water underneath rises up toward the surface of the ocean near South America.

However, in the fall and winter of some years, these winds are much weaker than usual and the warm surface water along the equator piles up along the coast of South America and then moves north towards California and south toward Chile. Many fish that live in the normally cooler waters off the coast of South America move away or die. The fishermen call this condition of warm coastal waters and poor fishing "El Niño" meaning "the Christ Child," because it sometimes arises at Christmas time. The event occurs aperiodically, with intervals of between 2 and 11 years. Roughly, nine events were recorded since 1945 (especially in 1957, 1965, 1972 and 1982) but El Niño is known for at least 400 years [1].

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Geoffrey K. Vallis

A continuous model was reduced by Geoffrey K. Vallis to the chaotic low-dimensional system  [2]

    \dot{x} = By - C(x+p) \\[0.1cm]
    \dot{y} = -y + xz \\[0.1cm]
    \dot{z} = -z -xy +1 \, .

For certain parameter values (B=102, C=3, p=0), a chaotic attractor (Fig. 1) is obtained. The similarity with the Lorenz system is related to the fact that the two systems describe convection, a single cell of Rayleigh-Bénard convection for the Lorenz system and the wind produced by a horizontal temperature gradient in the Vallis system. The ratio of time scales of decay of sea surface temperature anomalies to a frictional time scale, C, corresponds to the Prandlt number in the Lorenz system. Parameter B governing the strength of the air-sea interaction and the vertical temperature difference corresponds to the reduced Rayleigh number in the Lorenz system. When p=0, the Vallis system is equivariant under a rotation symmetry {\cal R}_z (\pi) as the Lorenz system is. A first-return map to a two-component Poincaré section reveals 6 monotonous branches split by three non-differentiable critical points and two differentiable ones. The attractor is thus structured by two tears and two folds. This attractor in fact corresponds to an attractor solution to the Lorenz system for R=91, \sigma=10 and b=8/3 [3]

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Fig. 1. Chaotic behavior solution to the Vallis system.

[1] W. H. Quinn, V. T. Neal & S. E. Antunez de Mayolo, El Niño occurrences over the past four and a half centuries, Journal of Geophysical Researches, 92, 14449-14462, 1987.

[2] G. K. Vallis, Conceptual models of El Niño and the southern oscillation, Journal of Geophysical Researches, 93, 13979-13991, 1988.

[3] C. Letellier, P. Dutertre & G. Gouesbet, Characterization of the Lorenz system taking into account the equivariance of the vector field, Physical Review E, 49 (4), 3492-3495, 1994. Here.

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