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):


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.

3 thoughts on “Reversing Traveling Waves

  1. 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.


  2. 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)

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

    “We describe a new type of solitary waves, which propagate in such a manner that the pulse periodically disappears from its original position and reemerges at a fixed distance”


  3. Another paper

    “Wave, vortex and wave-vortex dipole (instability wave): Three flavors of the intra-seasonal variability of the North Equatorial Undercurrent”


    Intra-seasonal variations (ISVs) have been frequently observed in the North Equatorial Undercurrent (NEUC) jets, yet their dynamical nature remains elusive. Based on field observations and model results, we identify three flavors of the ISVs: wave (accounting for 10%), wave-vortex dipole (20%) and vortex (70%). The wave flavor refers to a Rossby wave, the wave-vortex dipole flavor consists of a pair of counter-rotating vortices, and the vortex flavor refers to a sub-thermocline monopole eddy. To our knowledge, the wave-vortex dipole is identified for the first time and is found characterized by a unique dynamical feature: it manifests both as a second baroclinic mode-like Rossby wave and as a pair of dipole-like sub-thermocline eddies. It is further identified as a second baroclinic mode-like Rossby wave-initiated instability wave in an equilibrium being fueled by baroclinic conversion. These results indicate that mesoscale instability waves are an important component of subthermocline intra-seasonal and mesoscale variations.

    Plain Language Summary

    At the depths of 200–1000 m of the Pacific Ocean between 10°N and 18°N, the North Equatorial Undercurrent consisting of several jets flows eastward with mean velocity of ∼ 5 cm s-1. But it is not simply a steady current; instead, eastward and westward perturbations interlace, with an oscillation period of 70–120 days, which is called intra-seasonal variations (ISVs). These variations are known to exist from previous field observations, but their dynamical nature remains elusive. Here we present that they are possibly associated with three distinct flavors of subsurface mesoscale activities with a length scale of O(100 km). They are called, ‘vortex wave’, ‘wave-vortex dipole (WVD)’ and ‘vortex’. Among them, the WVD flavor is an instability wave, dynamically intermediate between vortex wave and vortex, which might be an important component of oceanic intra-seasonal and mesoscale variations.

    Supporting info available

    Click to access downloadSupplement


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