# Is the equatorial LTE solution a 1-D gyre?

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 :

“For instance, boundaries of coherent material eddies embracing and transporting volumes of ocean water with different salinity or temperature are notoriously difficult to identify.”

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.

# Reversing Traveling Waves

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(kxA 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(kxA 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:

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.