The propagation of planetary waves over a random topography

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Abstract
The purpose of this paper is to consider the effect of one-dimensional random depth variation on the propagation of planetary waves in a homogeneous layer of fluid having a free upper surface. We begin by determining the dispersion relation for the coherent part of the wave using the vorticity equation for the transport stream function and a previously described perturbation method. Then, from the resulting first-order expressions for the wavenumber, we obtain the phase speeds for the two possible planetary-wave solutions. These are compared with the corresponding phase speeds of planetary waves over a smoothly varying topography; the validity limits of the approximations are discussed. For the most physically realizable situation, of random depth correlation lengths much shorter than a typical wavelength, we find that the phase speed of the shorter (longer) wave component is less (greater) over a randomly varying topography than over a smoothly varying topography. In the case of the shorter waves, greatest relative changes in phase speed occur when the associated fluid motions are at right angles to the ‘strike’ of the roughness elements, while for both long and short waves there is no relative change in phase speed if fluid motions are parallel to the roughness contours. Moreover, both types of waves are shown to lose energy in the direction of energy propagation as a result of scattering. Numerical values are then obtained using hydrographic charts of the western North Pacific, and show that the randomness may significantly decrease the phase speed of the shorter planetary-wave component. Finally, we give a brief descriptive explanation of the results based on the effect of the topography on the wave restoring mechanism.