Lecture 2 ENSO toy models

Let us consider first a heuristic derivation of an equation for the sea surface temperature in the East Pacific, which will be followed by a more rigorous derivation in the following sections. Assume that the East Pacific SST affects the atmospheric heating and thus the central Pacific wind speed. The resulting wind stress, in turn, excites equatorial Kelvin and Rossby ocean waves. These waves affect the East Pacific thermocline depth and hence the East Pacific equatorial SST, and the whole feedback loop may be quantified as follows. Let τK and τR be the basin crossing times of equatorial Kelvin and Rossby waves, correspondingly. Now, a positive central Pacific equatorial thermocline depth anomaly heq(xc) at time t − 2 τK excites an eastward propagating downwelling Kelvin wave at the central Pacific that arrives after about 1 2 τK ≈ 1 month to the eastern Pacific and deepens the thermocline there. Similarly, a negative off-equatorial depth anomaly (that is, a shallowing signal of the thermocline) in the central Pacific ho f f−eq(xc) at a time t− [ 1 2τR +τK ] ( 2 τR +τK ≈ 6 months) excites a westward propagating Rossby wave at the central Pacific that is reflected off the western boundary as an equatorial Kelvin waves and eventually arrives to the eastern Pacific at time t, shallows the thermocline there and causes cooling of the SST. We add a nonlinear damping term that can stabilize the system, and write an equation for the eastern Pacific temperature that includes the Kelvin wave, Rossby wave and local damping terms as follows