Deep-Water Internal Waves with Weak Dispersion

There is experimental evidence that in density-stratified fluids, contaminants can become trapped near the pycnocline, the interface below the fluid surface where the density changes sharply. Such experiments provide one explanation for the underwater "oil plumes" observed following the recent Deepwater Horizon oil leak in the Gulf of Mexico. Modeling the dynamics of deformations of the pycnocline (so-called internal waves) is a step toward providing a description of how the contaminants move vertically in the fluid column. Ideally, one would like to accurately and easily predict whether contaminants will interact with biological populations at various depths, and perhaps whether contaminants will reach the surface mixing layer.

The Benjamin-Ono equation is a model for gravity-driven internal waves in certain density-stratified fluids. It has the features of being a nonlocal equation (the dispersion term involves the Hilbert transform of the disturbance profile) and also of having a Lax pair and an associated inverse-scattering algorithm for the solution of the Cauchy initial-value problem. We will review known phenomena associated with this equation in the limit when the dispersive effects are nominally small, and compare with the better-known Korteweg-de Vries (KdV) equation. Then we will present a new result (joint with former AIM PhD student Zhengjie Xu) establishing the zero-dispersion limit of the solution of the Benjamin-Ono Cauchy problem for certain initial data, in the topology of weak convergence. Our methodology is a novel analogue of the Lax-Levermore method in which the equilibrium measure is given more-or-less explicitly rather than via the solution of a variational problem. The proof relies on aspects of the method of moments from probability theory.

The weak limit is given by a remarkably simple formula that is easy to implement, far easier than the analogous formula for the KdV equation. As it is a weak limit, it only captures the local mean value of wild oscillations that can form as a result of dispersive regularization of shock waves. It remains an open problem to rigorously obtain formulae for the upper and lower envelopes of the oscillatory wave packet, a result that would certainly have further application in the modeling of internal waves.