For the solution to Laplace’s Tidal Equation described in Chapter 12, the spatial and temporal results are separable, leading to a ** non-linear** standing-wave time-series formulation:

sin(*kx*) sin(*A* sin(*wt*) )

By analogy to a * linear *standing-wave formulation, a solution such as

sin(*kx*) sin(*wt*)

with the following traveling wave solution (propagating in the +*x* direction):

sin(*kx-wt*)

becomes the following in the non-linear LTE solution mode:

sin(*kx* – *A* sin(*wt*) )

This is also a traveling wave, but with the characteristic property of being able to periodically reverse direction from +*x* to –*x* depending on the value of *A* and *w*. As an intuitive aid, a standing wave can be considered as the superposition of two traveling waves traveling in opposite directions:

sin(*kx* – *A* sin(*wt*) ) + sin(*kx* + *A* sin(*wt*) )

Here the cross terms cancel after applying the trig identity on sums, and the separable standing-wave result similar to the first equation results. But, whenever there is an imbalance of +x and -x travelling waves, a periodic reversing traveling-wave/standing-wave mix results. This is shown in the following animation, where a mix of nonlinear traveling-waves and standing-waves show the periodic reversal in direction quite clearly.

This reversal is actually observed in ocean measurements, as exemplified in this recent research article:

- Delpech, A., Cravatte, S., Marin, F., Ménesguen, C. & Morel, Y. Deep Eddy Kinetic Energy in the Tropical Pacific From Lagrangian Floats.
*Journal of Geophysical Research: Oceans***125**, e2020JC016313 (2020).

From their Figure 3, one can see this reversing process as the trajectory of a measured Argo float drift:

If that is not clear enough, the red arrows in the following annotated figure show the direction of the float motion. The drifting floats may not always exactly follow a trajectory as dictated by the velocity of a traveling wave, as this is partly a phase velocity with limited lateral volume displacement, but clearly a large wave-train such as a Tropical Instability Wave will certainly move a float. At least some of this is due to eddy behavior as the reversal is a natural consequence of a circular vortex motion of a large eddy.

Applying the LTE model to complete spatio-temporal data sets such as what Figure 3 is derived from would likely show an interesting match, adding value to the latest ENSO results, but this will require some digging into the data availability.

The topological/philosophical observation is whether the LTE solution, which occurs precisely along the equator, is essentially a mathematical reduction of a lower-dimensional eddy or vortex that acts as if it is circulating in both directions at once — thus producing a standing-wave in place. Something to consider for possible insight.

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Besides reversing, these waves also show characteristics of “jumping waves”, as they also seem to disappear and reappear at another location.

Two references discuss jumping waves:

“Diversity of the Madden-Julian Oscillation” Science Advances (2019)

https://www.researchgate.net/publication/334818470_Diversity_of_the_Madden-Julian_Oscillation

“Jumping solitary waves in an autonomous reaction–diffusion system with subcritical wave instability” Phys. Chem. Chem. Phys., 2006

https://pubs.rsc.org/en/content/articlelanding/2006/CP/B609214D#!divAbstract

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