Ocean Sciences Research Spotlight

The Dance of Surface Waves and Ocean Circulation

One mathematical model best describes the complex interplay between an ocean's surface waves and its underlying circulation.

Source: Journal of Geophysical Research: Oceans


An ocean’s surface waves are commonly sculpted into their crested shapes by wind that has transferred its energy to the water. But the interaction between surface waves and the ocean’s underlying circulation isn’t so well understood, despite the fact that this interplay has a critical role in regulating the Earth’s climate and weather systems.

For 4 decades, two mathematical models have competed to explain how the two motions of surface waves and ocean circulation mix. The first theory, introduced in 1962, relies on a momentum equation to model the interaction. It uses a term known as radiation stress, which can best be understood as changes in the distribution of momentum carried by the waves. The second theory, introduced in 1976, uses an equation term known as the vortex force, which accounts for the effect of waves on currents.

Last year, George Mellor asked the crucial question: Can both theories be correct? If so, a good mathematician should be able to derive one from the other. Alas, in a 2015 paper, Mellor demonstrated that it could not be done: One theory had to be false.

In a new paper Mellor compares the equations underlying each theory and finds that the model invoking a vortex force is incompatible with the other. In other words, the equation with a vortex force term was incorrect because it didn’t stand up to physical or mathematical scrutiny. This leaves the other, older model invoking radiation stress as the more likely explanation. Thus, the dizzying dance just beneath the ocean’s surface can be mathematically treated as a simple combination of currents and waves. (Journal of Geophysical Research: Oceans, doi:10.1002/2016JC011768, 2016)

—Shannon Hall, Freelance Writer

Citation: Hall, S. (2016), The dance of surface waves and ocean circulation, Eos, 97, https://doi.org/10.1029/2016EO055941. Published on 19 July 2016.
© 2016. The authors. CC BY-NC-ND 3.0
  • Fabrice Ardhuin

    Dear colleagues,
    There is no doubt that the interaction of waves and currents is complext and it seems that more pedagogy will be required. We have argued in the scientific litterature before, and a blog is probably not the best place to do so. It is interesting however, that it is much more demanding to expose errors or exagerated claims (that lead to misinterpretation) that to let pass your way and just ignore those publications.

    So, I guess that I will be involved in writing a formal rebuttal once again, but frankly the reason why this new paper by George L. Mellor is incorrect is already fully explained in our 2008 (Ardhuin et al. J. Phys. Ocenogr.) and 2011 (Bennis and Ardhuin, J. Phys. Oceanogr.) rebuttals.

    1) First of all, as very well explained also by Lane et al. (JPO 2007), the vortex force and radiation stress are consistent… but they do not apply to the same momentum: radiation stresses apply to a momentum that contains the wave momentum, whereas the vortex force only apply to the mean flow momentum and actually expresses the exchange of momentum between the waves and currents: when the waves are refracted by the current, their momentum changes and the vortex force expresses the “recoil” effect on the mean flow. This was very well presented by Garrett (1976).

    2) A great confusion introduced by Mellor (2008) and all subsequent papers by this author (including the JPO 2015 and the latest JGR 2016) is that he uses a different average for the pressre term than for the other terms of the momentum equation. This is inconsistent: with the same reasoning you could end up saying that the Lagrangian mean velocity is equal to the Eulerian mean velocity: this is not the case in particular for the Stokes drift, for which the Eulerian mean has a parabolic profile (in the case of monochromatic waves) confined between crests and troughs, while the Lagrangian mean has a exponential profile (in deep water). As a result Mellor (2008) ends up with a w^2 term in the mean pressure which should not be there if he was doing a proper average moving up and down with the wave motion like the other terms.

    3) Yes there is a vertical Stokes drift in general! It happens to be zero for monochromatic waves of constant amplitude over a flat bottom but this is not true in general, in particular in the case of a sloping bottom: see figure 3 in Ardhuin et al. (Ocean Modelling 2008).