Ocean Response to Wind Variations, Warm Water Volume, and Simple Models of ENSO in the Low-Frequency Approximation

Physical processes that control ENSO are relatively fast. For instance, it takes only several months for a Kelvin wave to cross the Pacific basin (Tk ≈ 2 months), while Rossby waves travel the same distance in about half a year. Compared to such short time scales, the typical periodicity of El Niño is much longer (T ≈ 2–7 yr). Thus, ENSO is fundamentally a low-frequency phenomenon in the context of these faster processes. Here, the author takes advantage of this fact and uses the smallness of the ratio ɛk = Tk/T to expand solutions of the ocean shallow-water equations into power series (the actual parameter of expansion also includes the oceanic damping rate). Using such an expansion, referred to here as the low-frequency approximation, the author relates thermocline depth anomalies to temperature variations in the eastern equatorial Pacific via an explicit integral operator. This allows a simplified formulation of ENSO dynamics based on an integro-differential equation. Within this formulation, the author shows how the interplay between wind stress curl and oceanic damping rates affects 1) the amplitude and periodicity of El Niño and 2) the phase lag between variations in the equatorial warm water volume and SST in the eastern Pacific. A simple analytical expression is derived for the phase lag. Further, applying the low-frequency approximation to the observed variations in SST, the author computes thermocline depth anomalies in the western and eastern equatorial Pacific to show a good agreement with the observed variations in warm water volume. Ultimately, this approach provides a rigorous framework for deriving other simple models of ENSO (the delayed and recharge oscillators), highlights the limitations of such models, and can be easily used for decadal climate variability in the Pacific.