Decay of an internal tide due to random topography in the ocean

We present a theoretical and numerical study of the decay of an internal wave caused by scattering at undulating sea-floor topography, with an eye towards building a simple model in which the decay of internal tides in the ocean can be estimated. As is well known, the interactions of internal waves with irregular boundary shapes lead to a mathematically ill-posed problem, so care needs to be taken to extract meaningful information from this problem. Here, we restrict the problem to two spatial dimensions and build a numerical tool that combines a real-space computation based on the characteristics of the underlying partial differential equation with a spectral computation that satisfies the relevant radiation conditions. Our tool works for finite-amplitude topography but is restricted to subcritical topography slopes. Detailed results are presented for the decay of the gravest vertical internal wave mode as it encounters finite stretches of either sinusoidal topography or random topography defined as a Gaussian random process with a simple power spectrum. A number of scaling laws are identified and a simple expression for the decay rate in terms of the power spectrum is given. Finally, the resulting formulae are applied to an idealized model of sea-floor topography in the ocean, which seems to indicate that this scattering process can provide a rapid decay mechanism for internal tides. However, the present results are restricted to linear fluid dynamics in two spatial dimensions and to uniform stratification, which restricts their direct application to the real ocean.