In Mathematical Geoenergy, Chapter 12, a biennially-impulsed lunar forcing is suggested as a mechanism to drive ENSO. The current thinking is that this lunar forcing should be common across all the oceanic indices, including AMO for the Atlantic, IOD for the Indian, and PDO for the non-equatorial north Pacific. The global temperature extreme of the last year had too many simultaneous concurrences among the indices for this not to be taken seriously.

NINO34

PDO

AMO

IOD – East

IOD-West

Each one of these uses a nearly identical annual-impulsed tidal forcing (shown as the middle green panel in each), with a 5-year window providing a cross-validation interval. So many possibilities are available with cross-validation since the tidal factors are essentially invariantly fixed over all the climate indices.

The approach follows 3 steps as shown below

The first step is to generate the long-period tidal forcing. I go into an explanation of the tidal factors selected in a Real Climate comment here.

Then apply the lagged response of an annual impulse, in this case alternating in sign every other year, which generates the middle panel in the flow chart schematic (and the middle panel in the indexed models above).

Finally, the Laplace’s Tidal Equation (LTE) modulation is applied, with the lower right corner inset showing the variation among indices. This is where the variability occurs — the best approach is to pick a slow fundamental modulation and generate only integer harmonics of this fundamental. So, what happens is that different harmonics are emphasized depending on the oceanic index chosen, corresponding to the waveguide structure of the ocean basin and what standing waves are maximally resonant or amplified.

Note that for a dipole behavior such as ENSO, the LTE modulation will be mirror-inverses for the maximally extreme locations, in this case Darwin and Tahiti

A machine learning application is free to scrape the following GIST GitHub site for model fitting artifacts.

https://gist.github.com/pukpr/3a3566b601a54da2724df9c29159ce16

Another analysis that involved a recursively cycled fit between AMO and PDO. It switched fitting AMO for 2.5 minutes and then PDO for 2.5 minutes, cycling 50 times. This created a common forcing with an optimally shared fit, forcing baselined to PDO.

PDO

AMO

NINO34

IOD-East

IOD-West

Darwin

Tahiti

The table above shows the LTE modulation factors for Darwin and Tahiti model fits. The highlighted blocks show the phase of the modulation, which should have a difference of π radians for a perfect dipole and higher harmonics associated with it. (The K0 wavenumber = 0 has no phase, but just a sign). Of the modes that are shared 1, 45, 23, 36, 18, 39, 44, the average phase is 3.09, close to π (and K0 switches sign).

1.23-(-1.72) = 2.95 
1.47-(-2.05) = 3.52
-2.89-(0.166) = -3.056 
-0.367-(-2.58) = 2.213 
1.59-(-2.175) = 3.765 
0.27 - (-2.84) = 3.11 
-1.87 -1.14 = -3.01 

Average (2.95+3.52+3.056+2.213+3.765+3.11+3.01)/7 = 3.0891

Contrast to the IOD East/West dipole. Only the K0 (wavenumber=0) shows a reversal in sign. The LTE modulation terms are within 1 radian of each other, indicating much less of a dipole behavior on those terms. It’s possible that these sites don’t span a true dipole, either by its nature or from siting of the measurements.

Cross-validating a large interval span on PDO

using CC

using DTW metric, which pulls out more of the annual/semi-annual signal

adding a 3rd harmonic

Complement of the fitting interval, note the spectral composition maintains the same harmonics, indicating that the structure mapped to is stationary in the sense that the tidal pattern is not changing and the LTE modulation is largely fixed.

This is the resolved tidal forcing, finer than the annual impulse sampling used on the models above.

Below can see the primary 27.5545 lunar anomalistic cycle, mixed with the draconic 27.2122/13.606 cycle to create the 6/3 year modulation and the 206 day perigee-syzygy cycle (or 412 full cycle, as 206 includes antipodal full moon or new moon orientation).

(click on any image to magnify)