The analytical solution to Laplace’s Tidal Equation along the 1-D equatorial wave guide not only appears odd, but it acts odd, showing a Mach-Zehnder-like modulation which can be quite severe. It essentially boils down to a sinusoidal modulation of a forcing, belonging to a class of non-autonomous functions. The standing wave comes about from deriving a separable spatial component.

*sin*(* f(t) *) *sin*(*kx*)

The underlying structure of the solution shouldn’t be surprising, since as with Mach-Zehnder, it’s fundamentally related to a path integral formulation known from mathematical physics. As derived via quantum mechanics (originally by Feynman), one temporally integrates an energy Hamiltonian over a path allowing the wave function to interfere with itself over all possible wavenumber (*k*) and spatial states (*x*).

Because of the imaginary value *i* in the exponential, the result is a sinusoidal modulation of some (potentially complicated) function. Of course, the collective behavior of the ocean is not a quantum mechanical result applied to fluid dynamics, yet the topology of the equatorial waveguide can drive it to *appear *as one, see the breakthrough paper “Topological Origin of Equatorial Waves” for a rationale. (The curvature of the spherical earth can also provide a sinusoidal basis due to a trigonometric projection of tidal forces, but this is rather weak — not expanding far beyond a first-order expansion in the Taylor’s series)

Moreover, the rather strong interference may have a physical interpretation beyond the derived mathematical interpretation. In the past, I have described the modulation as wave breaking, in that the maximum excursions of the inner function *f*(*t*) are folded non-linearly into itself via the limiting sinusoidal wrapper. This is shown in the figure below for progressively increasing modulation.

In the figure above, I added an extra dimension (roughly implying a toroidal waveguide) which allows one to visualize the wave breaking, which otherwise would show as a progressively more rapid up-and-down oscillation in one dimension.

Perhaps coincidentally (or perhaps not) this kind of sinusoidal modulation also occurs in heuristic models of the *double-gyre* structure that often appears in fluid mechanics. In the excerpt below, note the *sin*(*f*(*t*)) formulation.

The interesting characteristic of the structure lies in the more rapid cyclic variations near the edge of the gyre, which can be seen in an animation (Jupyter notebook code here).

Whether the equivalent of a double-gyre is occurring via the model of the LTE 1-D equatorial waveguide is not clear, but the evidence of double-gyre wavetrains (Lagrangian coherent structures, Kelvin–Helmholtz instabilities), occurring along the equatorial Pacific is abundantly clear through the appearance of tropical instability (TIW) wavetrains.

These so-called coherent structures may be difficult to isolate for the time being, especially if they involve subtle interfaces such as thermocline boundaries :

Mercator analysis does show higher levels of waveguide modulation, so perhaps this will be better discriminated over time *(see figure below with the long wavelength ENSO dipole superimposed along with the faster TIW wavenumbers in dashed line, with the double-gyre pairing in green + dark purple)*, and something akin to a 1-D gyre structure will become a valid description of what’s happening along the thermocline. In other words, the wave-breaking modulation due to the LTE modulation is essentially the same as the vortex gyre mapped into a 1-D waveguide.