Internal wave attractors over random, small-amplitude topography

We consider whether small-amplitude topography in a two-dimensional ocean may contain internal wave attractors. These are closed orbits formed by the characteristics (or wave beam paths) of the linear, inviscid, steady-state Boussinesq equations, and their existence may imply enhanced scattering and energy decay for the internal tide when dissipation is present. We develop a numerical code to detect attractors over arbitrary topography, and apply this to random, Gaussian topography with different covariance functions. The rate of attractors per length of topography increases with the fraction of supercritical topography, but surprisingly, it also increases as the amplitude of the topography is decreased, while the supercritical fraction is held constant. This can partly be understood by appealing to Rice's formula for the rate of zero crossings of a stochastic process. We compute the rate of attractors for a covariance function typical of ocean bathymetry away from large features and find it is about 10 attractors per 1000 km. This could have implications for the overall energy budget of the ocean.