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