Benjamin-Feir instability and the propagation of swell across the Pacific

BF-swell The paper “The Benjamin-Feir instability and propagation of swell across the Pacific“, co-authored by Diane Henderson and Harvey Segur, has been published in Mathematics and Computers in Simulation.  About 40 years ago, Snodgrass and other oceanographers (1966) tracked ocean swell propagating across the entire Pacific Ocean. At about the same time, several investigators showed that a uniform train of plane waves of finite amplitude on deep water is unstable. Comparing these two results leads to the following question: in light of this instability, how did the waves tracked by oceanographers travel coherently more than 10,000 km across the Pacific Ocean? A possible explanation is provided in recent work that re-examined the Benjamin–Feir instability in the presence of linear damping. The conclusion was that even small amounts of damping can stabilize the instability before nonlinear effects become important. In addition, the theoretical predictions agree well with results from laboratory experiments. In this paper we re-examine ocean data from 1966 to estimate whether the oceanic damping that was measured could have controlled the Benjamin–Feir instability for the swell that was tracked. We find that for one set of ocean swell, dissipation controls the Benjamin–Feir instability enough to allow coherent wave propagation across the Pacific. For a second set of ocean swell, it does not. For a third set of ocean swell, an integral that the theory predicts to be constant is not constant in the data; it decreases and correspondingly the spectral peak shifts to a lower frequency—this is frequency downshifting. For this case the theory is not an adequate model, so the corresponding Benjamin–Feir analysis can be misleading. Thus, our results from the historical records are inconclusive: we can assert neither that dissipation of ocean swell is always negligible, nor that it is always important. But our results show that dissipation can control the Benjamin–Feir instability for small-amplitude waves and that downshifting occurs in ocean swell with relatively small wave slopes.  A link to the paper is here.

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