After spending several years on formulating a model of ENSO (then and now) and then spending a day or two on the AMO model, it's obvious to try the other well-known standing wave oscillation — specifically, the Pacific Decadal Oscillation (PDO). Again, all the optimization infrastructure was in place, with the tidal factors fully parameterized for automated model fitting.

This fit is for the entire PDO interval:

What's interesting about the PDO fit is that I used the AMO forcing directly as a seeding input. I didn't expect this to work very well since the AMO waveform is not similar to the PDO shape except for a vague sense with respect to a decadal fluctuation (whereas ENSO has no decadal variation to speak of).

Yet, by applying the AMO seed, the convergence to a more-than-adequate fit was rapid. And when we look at the primary lunar tidal parameters, they all match up closely. In fact, only a few of the secondary parameters don't align and these are related to the synodic/tropical/nodal related 18.6 year modulation and the Ms* series indexed tidal factors, in particular the Msf factor (the long-period lunisolar synodic fortnightly). This is rationalized by the fact that the Pacific and Atlantic will experience maximum nodal declination at different times in the 18.6 year cycle.

Another parameter that does not match up is the Laplace Tidal Equation offset, \phi.

f(t) = sin ( A \cdot sin(\omega t) + \phi)

To understand how this equation is derived, see this post where it was originally formulated for QBO.

Here are the lunar forcings as applied to the best fit AMO and PDO models.

AMO model forcings

PDO model forcings

Total lunar forcing over a short interval

Biennial modulated forcing over the entire interval

They are essentially the same, and most importantly when one looks at the summed total lunar forcing, which averages any differences among the individual factors.

The list of factors were originally described here for ENSO, and the values for the AMO and PDO model fitting are shown in the table below. Note that only the factors highlighted in yellow show a significant difference between the AMO and PDO models.

The boxed parameters are separate from the purely lunar parameters and are simultaneously tuned to the differential equation structure used.  The highlighted \phi parameter leads to a stronger square-wave decadal modulation for the AMO time-series.

I will likely add this to my presentation "Biennial-Aligned Lunisolar-Forcing of ENSO: Implications for Simplified Climate Models" at the AGU in New Orleans set for December 14th. It's already set for the session titled: "Mechanisms and Impacts of Natural and Anthropogenic Tropical Pacific Decadal Variations and Trends", so that this additional info on PDO is applicable.

What makes this new finding important is that (1) the AMO and PDO can be cross-validated against each other and (2) the lunar factors modeled by one oceanic index can be used as a constraint in estimating the other. These indices are thus teleconnected by the common-mode of the lunar forcing. In other words, the moon's orbital forcing provides the teleconnection between the ENSO, QBO, AMO, and PDO indices.

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