In Chapter 12 of the book, we presented a math model for the equatorial Pacific ocean dipole known as ENSO (El Nino /Southern Oscillation).  We argued that the higher wavenumber (×15 of the fundamental) characteristic of ENSO was related to the behavior known as Tropical Instability Waves (TIW). Taken together, the fundamental and TIW components provide enough detail to model ENSO at the monthly level. However if we drill deeper, especially with respect to the finer granularity SOI measure of ENSO, there are rather obvious cyclic factors in the 30 to 90 day range that can add even further detail. The remarkable aspect is that these appear to be related to the behavior known as the Madden-Julian Oscillation (MJO), identified originally as a 40-50 day oscillation in zonal wind [1].

The key to making an association between SOI and MJO is to analyze the daily measure of SOI and compare that to a high-resolution (pentad=5-day) time-series of MJO. Consider if the MJO traveling wave (see Fig 1) is kicked off by the ENSO equatorial disturbance, then an index such as SOI should lead MJO by ~20 days if MJO is traveling at 5 to 15 m/s and it has to fully propagate to be precisely measured.

Indeed, the SOI leads MJO by ~21 days according to this optimal overlay:

As an aside, one can now see why the high resolution of SOI is necessary. The issue with the monthly SOI readings is that if the MJO period is ~45 days, that’s above the Nyquist frequency, so it would be challenging to isolate these features via sampling the monthly time-series (I have tried and had little success).

Fig 3 below shows that the correlation is ~0.47 at lead of 21 days and damps quickly with lead/lag shifts of a few days. The 2nd-order satellite wings are at +78 and -25, which would put the MJO period at around 57 to 46 days, which is a sanity check for the MJO typical range.

Revisiting the high resolution SOI model, but applying it to the MJO time-series, Fig 4 below shows that a good correlation can be achieved by assuming a nearly identical tidal forcing and similar Laplace Tidal Equation (LTE) modulation parameters.

As a comparison, the high resolution SOI model is shown below. The daily SOI data goes back to 1991, so the MJO data provides the continuation of the time-series prior to that date (but wasn’t applied to the fitting procedure). The parameters and forcing for two models (MJO & SOI) are very similar, yet that should be expected from the known temporal correlation and short lag applied to MJO.

Now comes a novel signal processing algorithm based on the LTE model. We know from the Mach-Zehnder-like modulation of the LTE analytical result that the forcing level replaces the time parameter as the sinusoidal input. So instead of using time for the Fourier spectral analysis, if we use the forcing level then the wavenumber parameters should obviously be revealed in the amplitude spectra. We can then use these directly in the model as a means to quickly tune the fit. This is shown in Fig 6 below for the SOI LTE modulation. The spectra to the left is related to the low frequency aspect of ENSO, while the spikes on the right correspond to the 40 to 50 day MJO cycles.

This also applies to the MJO time series with similarly isolated spikes shown in Fig 7. These are sharply delineated wavenumbers that drive the response via the forcing level .

Note that in the the Fourier domain of the actual temporal signal, the addition of a strong single LTE modulation will create a spread response in the spectrum as shown in Fig 8, creating in effect the broad 30 to 90 day spectral spread of MJO. This is just Mach-Zehnder in action, with the temporal => forcing transformation providing a handy adjoint conjugation. So it’s much easier to fit a model to data via Fig. 6 or Fig. 7 than by Fig. 8.

To elaborate further, the reason that these spikes show up so clearly is that they are driven by slight changes in the forcing level, which the Fourier series representation captures by grouping only level shifts, see Fig 9. It thus isolates the standing waves that can develop during the relatively flat tops of the annually-driven step response modulation to tidal forcing. The step change is likely the mechanism that provides a synchronization to the high wavenumber responses and thus long-term coherence between the model and data.

This only explains the math. According to most “just-so” explanations of the MJO behavior [2], a specific location provides the launching pad for the traveling wave to start.

and here’s video I made of a recurring flame soliton running along a groove in a burning oak log. This likely has some of the same math as the MJO, QBO, and Kelvin waves that race around the equator.

References

1. Madden R. and P. Julian, 1971: Detection of a 40-50 day oscillation in the zonal wind in the tropical Pacific, J. Atmos. Sci., 28, 702-708.
2. Wei, Y., Ren, H.-L., Mu, M. & Fu, J.-X. Nonlinear optimal moisture perturbations as excitation of primary MJO events in a hybrid coupled climate model. Climate Dynamics 54, 675–699 (2020).