El Niño–Southern Oscillation models

Definition

An El Niño–Southern Oscillation model (ENSO model) is a dynamical model of the coupled ocean and atmosphere of the tropical Pacific. Its state includes the sea surface temperature anomaly $T$ of the eastern equatorial Pacific, and it is built from:

1. a positive feedback, in which a warm anomaly $T > 0$ weakens the easterly winds and the weaker winds increase $T$;
2. a negative feedback, delayed or slower, carried by the adjustment of the ocean thermocline through equatorial Kelvin waves and Rossby waves.

Two standard conceptual ENSO models are the following. The delayed oscillator is the delay differential equation $$\frac{dT}{dt} = T - T^3 - \alpha \, T(t - \delta), \qquad 0 < \alpha < 1, \ \delta > 0.$$ It is in nondimensional form, and the delayed term is the effect of waves reflected at the western boundary. The recharge oscillator is the linear system of ordinary differential equations $$\frac{dT}{dt} = R T + \gamma h, \qquad \frac{dh}{dt} = -r h - a T.$$ Here $h$ is the depth anomaly of the thermocline in the western Pacific, and $\gamma, r, a > 0$. The constant $R$ is the net growth rate of $T$ from clause 1 and damping.