As the quality of the tidally-forced ENSO model improves, it’s instructive to evaluate its common-mode mechanism against other oceanic indices. So this is a re-evaluation of the Pacific Decadal Oscillation (PDO), in the context of non-autonomous solutions such as generated via LTE modulation. In particular, in this note we will clearly delineate the subtle distinction that arises when comparing ENSO and PDO. As background, it’s been frequently observed and reported that the PDO shows a resemblance to ENSO (a correlation coefficient between 0.5 and 0.6), but also demonstrates a longer multiyear behavior than the 3-7 year fluctuating period of ENSO, hence the decadal modifier.
A hypothesis based on LTE modulation is that decadal behavior arises from the shallowest modulation mode, and one that corresponds to even symmetry (i.e. cos not sin). So for a model that was originally fit to an ENSO time-series, it is anticipated that the modulation trending to a more even symmetry will reveal less rapid fluctuations — or in other words for an even f(x) = f(-x) symmetry there will be less difference between positive and negative excursions for a well-balanced symmetric input time-series. This should then exaggerate longer term fluctuations, such as in PDO. And for odd f(x) = -f(-x) symmetry it will exaggerate shorter term fluctuations leading to more spikiness, such as in ENSO.
To evaluate this hypothesis, all one needs to do is plot the fitted LTE modulation for + and – excursions along the same axis, as shown below
Removing the higher frequency LTE modulation, the even symmetry for PDO is even more apparent, while the underlying primary modulation frequency is shared by ENSO and PDO (expected, as same ocean basin implies same standing wave modes).
In addition to substantiating the premise of even vs odd non-autonomous modulation, this finding is perhaps revealing something even more fundamental about the LTE solution for PDO. As the high degree of even-symmetry (correlation coefficient > 0.92) implies, some symmetric physical mechanism or boundary condition must be driving the PDO behavior. So that positive (+) and negative (-) tidal forcing excursions have identical impacts off the equator (for PDO) but slightly asymmetrical on the equator (for ENSO). Indeed, this could be a foundational topological requirement or perhaps one that conserves energy and angular momentum at higher latitudes.
This even-symmetry finding is also a surprising result from a statistical signal processing and machine learning perspective, as the LTE modulation was not constrained to be even-symmetry during the fitting process. So not only could it reveal a natural symmetry as discussed in the previous paragraph, but it may lead to a leaner and more parsimonious modeling procedure since the number of degrees of freedom (DOF) can be reduced. As all the modulation phases need to be aligned it may actually cut the number of DOF in half!
As an addendum, the forcing applied on a monthly level is primarily the Mm tidal factor, with additional factors creating an 18-year Saros modulation, and 6-year sub-modulation.
This is the spectral representation when integrated with respect to an annual impulse.
The PDO data is from https://climexp.knmi.nl/getindices.cgi?WMO=UWData/pdo_ersst&STATION=PDO_ERSST based on ERSST anomalies.
UPDATE 4/25/2002
Making the PDO modulation perfectly even-symmetric by enforcing LTO(+) = LTO(-), this is the result
This has a weak 6 year repeat forcing pattern superimposed. An alternate fit with a 4.4-year repeat forcing pattern is below
Example of PDO (northern Pacific blob) and ENSO (la Nina conditions)
GeoEnergy Math 