Derivation of an ENSO model using Laplace's Tidal Equations

Laplace developed his namesake tidal equations to mathematically explain the behavior of tides by applying straightforward Newtonian physics. In their expanded form, known as the primitive equations, Laplace's starting formulation is used as the basis of almost all detailed climate models. Since that's what they are designed to do, this post provides the details for solving Laplace's tidal equations in the context of the El Nino Southern Oscillation (ENSO) of the equatorial Pacific ocean. The derivation and results shown below essentially describe the framework of my presentation at this month's AGU meeting: Biennial-Aligned Lunisolar-Forcing of ENSO: Implications for Simplified Climate Models

The concise derivation for a model of ENSO depends on reducing Laplace's tidal equations along the equator. I could not find anyone taking a similar approach anywhere in the literature, even though it appears to be routinely obvious: (1) solve Laplace's tidal equations in a simplified context, then (2) apply the known tidal forcing and observe if the result correlates or matches the ENSO time series. In fact, it does, as I have shown before (and for QBO as well); but this is the first time that I have worked out the details in full for ENSO. Below is a two part solution.

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