In Chapter 12 of the book, we describe the forcing mechanism behind the El Nino / Southern Oscillation (ENSO) behavior and here we continue to evaluate the rich dynamic behavior of the Southern Oscillation Index (SOI) — the pressure dipole measure of ENSO. In the following, we explore how the low-fidelity version of the SOI can reveal the high-frequency content via the solution to Laplace’s Tidal Equations.

The essential idea is that the higher frequency standing wave modes reinforce the primary dipole standing wave modes by sharpening the temporal waveform. The sharpening can occur either by (1) squaring off a low-pass filtered version of ENSO or (2) identifying bifurcating wave-breaking terms under a high-resolution view. The latter is possible by characterizing the daily observations of the SOI data set over the last 27 years.

First, we look at the low-resolution view of ENSO, which is a low-pass filtered time-series composed of SOI and NINO3.4 data sets. The entire historical series is used as the primary fit, with a portion of the daily time series providing some selectivity of the high-frequency terms. The rest of the daily SOI time-series is thus cross-validated via an implicit interpolation method. In other words, the implicit interpolation is provided by the low-resolution view with extrapolated cross-validating calibration providing a short fitting interval of the high-resolution daily view.

(click on images to enlarge)

The objective fitting function is an average of the correlation coefficient for the low-frequency data and a selected short interval of the daily SOI data.

Next, we look at characterizing the raw monthly SOI data and use that as a guide to better guess the high-frequency terms. The raw unfiltered data constrains the fit somewhat more and continues to generate good agreement in the out-of-band cross-validating daily time-series via the same interpolating method.

Calibrating interval 2010-2018

Calibrating interval 1992-2007

Calibrating interval 2004-2006

For those familiar with standing wave modes, a much greater state-space is available to select from the high-frequency modes. Thus, the selected high-*f* fitting terms may not be optimal, so the characterization is still open to further iteration.

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These are using Fourier transform fits to lower frequencies, for both SOI and NINO34. These are essentially extrapolating what the high-frequency time series would be based on the non-linear harmonic interactions. The SOI shows stronger fluctuations as that does show higher fluctuation in the monthly data than the NINO34 index does.

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This was constrained via a spectral plus optimally fit against the high-res daily SOI

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