The solution to Laplace's Tidal Equation along the equator is

$$ I(t) = \Sigma^n_{i=0} A_i \sin (k_i f(t) + \theta_i) $$

where f(t) is the equatorial gravitational forcing, and the index terms are the separated standing wave components.

Since f(t) is cyclic as well, we want to determine what harmonics are generated by the sum. First, we isolate the phase factor and expand.

$$ I(t) = \Sigma A_i (\cos(\theta_i) \sin (k_i f(t)) + \sin(\theta_i) \cos(k_i f(t)) ) $$

The harmonics come about predominately from the Taylor's-series expansion of the sinusoidal factors.

$$ I(t) = \Sigma A_i (\cos(\theta_i) ( k_i f(t) - (k_i f(t))^3/3! + (k_i f(t))^5/5! + ...) + \sin(\theta_i) ( 1- (k_i f(t))^2/2! + (k_i f(t))^4/4! + ...) )$$

Notice that all the f(t) terms become power factors. This will essentially explode the power spectrum with multiplicative harmonics when there is significant saturation of the bounding sinusoidal terms. If the values are not close to saturated, only the first-order term remains, equivalent to the approximation

$$ sin(k f(t)) \sim k f(t) $$

The k pre-factors are related to the spatial standing-wave modes in the original derivation. The higher the value of k the greater the spatial wavenumber of the standing wave. Empirically, we observe one value of k that stands out, and that is likely the primary ENSO standing wave. The progressively higher values of k likely relate in some way to Tropical Instability Waves, which are thought to be related to ENSO according to a recent review article [1].

Suffice to say that this is very different math than we are used to applying for spectral analysis. How to optimize is a challenge. We can try one of these approaches:

* Applying a nonlinear search of the k values, while also modifying f(t), to best correlate to the ENSO data

* Group the terms of the same power factor and attempt a multi-linear regression to optimize values of k for a given f(t).

The first grinds away forever as it uses a gradient search. The second may be faster but it doesn't automatically adjust f(t). A combination of these may work best.

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[1] Timmermann, Axel, Soon-Il An, Jong-Seong Kug, Fei-Fei Jin, Wenju Cai, Antonietta Capotondi, Kim Cobb, et al. “El Niño–Southern Oscillation Complexity.” Nature 559, no. 7715 (July 2018): 535. https://doi.org/10.1038/s41586-018-0252-6.

$$ I(t) = \Sigma^n_{i=0} A_i \sin (k_i f(t) + \theta_i) $$

where f(t) is the equatorial gravitational forcing, and the index terms are the separated standing wave components.

Since f(t) is cyclic as well, we want to determine what harmonics are generated by the sum. First, we isolate the phase factor and expand.

$$ I(t) = \Sigma A_i (\cos(\theta_i) \sin (k_i f(t)) + \sin(\theta_i) \cos(k_i f(t)) ) $$

The harmonics come about predominately from the Taylor's-series expansion of the sinusoidal factors.

$$ I(t) = \Sigma A_i (\cos(\theta_i) ( k_i f(t) - (k_i f(t))^3/3! + (k_i f(t))^5/5! + ...) + \sin(\theta_i) ( 1- (k_i f(t))^2/2! + (k_i f(t))^4/4! + ...) )$$

Notice that all the f(t) terms become power factors. This will essentially explode the power spectrum with multiplicative harmonics when there is significant saturation of the bounding sinusoidal terms. If the values are not close to saturated, only the first-order term remains, equivalent to the approximation

$$ sin(k f(t)) \sim k f(t) $$

The k pre-factors are related to the spatial standing-wave modes in the original derivation. The higher the value of k the greater the spatial wavenumber of the standing wave. Empirically, we observe one value of k that stands out, and that is likely the primary ENSO standing wave. The progressively higher values of k likely relate in some way to Tropical Instability Waves, which are thought to be related to ENSO according to a recent review article [1].

Suffice to say that this is very different math than we are used to applying for spectral analysis. How to optimize is a challenge. We can try one of these approaches:

* Applying a nonlinear search of the k values, while also modifying f(t), to best correlate to the ENSO data

* Group the terms of the same power factor and attempt a multi-linear regression to optimize values of k for a given f(t).

The first grinds away forever as it uses a gradient search. The second may be faster but it doesn't automatically adjust f(t). A combination of these may work best.

---

[1] Timmermann, Axel, Soon-Il An, Jong-Seong Kug, Fei-Fei Jin, Wenju Cai, Antonietta Capotondi, Kim Cobb, et al. “El Niño–Southern Oscillation Complexity.” Nature 559, no. 7715 (July 2018): 535. https://doi.org/10.1038/s41586-018-0252-6.