How Much of a Baroclinic Coastal Kelvin Wave Gets over a Ridge?

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Abstract
This paper considers the interaction between a bottom-trapped low-frequency, reduced-gravity maid Kelvin wave propagating along a coastal wall, and a smooth ridge extending away from the coastline. Although the full problem appears intractable, it is shown that simple bounds may be placed on the amplitude of the Kelvin wave after it has passed the region of topography. The upper bound is found to be a good estimate for cases examined here. For a ridge of width one or two deformation radii, the reduction in amplitude of the Kelvin wave, induced by scattering along the ridge, is roughly equal to the fractional depth remaining in the undisturbed fluid layer at the highest point of the ridge; the reduction in energy is of course given by the square of this quantity. The bounds are found by considering the (approximate) conservation of mass between the incoming and transmitted Kelvin waves and the range of topographic waves on the rider, and also the (exact) conservation of energy. The full, and very complicated, interaction problem near the intersection of the ridge and the coastal wall does not need to be solved. The effects of changing width of the topography are examined. Narrow ridges (with widths much ten than a deformation radius) permit Kelvin waves to pass them without loss of amplitude in the limit of vanishing width. Broad ridges (with widths much larger than a deformation radius) can have two effects depending on size of the frequency. When the frequency is small enough so that a small but finite number of topographic modes are possible, there is no loss in amplitude of the Kelvin wave. For smaller frequencies, where there are many topographic modes possible, a finite amount of amplitude is lost to topographic waves. Thus ridges of width of order a deformation radius or wider are the most efficient scatterers of coastal wave energy; four successive ridges of height one-half that of the resting depth of the layer would reduce the transmitted wave energy to less than 2% of its initial value. Poor numerical resolution can strongly overestimate the transmitted wave amplitude. Since present general circulation models cannot resolve all the various modes discussed here, this overestimation must occur, and may be quite drastic. Additionally, the effects of mild numerical damping are discussed, and compared with the ideal fluid case. When the damping is Laplacian, the short topographic waves are damped, with two results: the flow field resembles that for a step topography, and the transmitted wave amplitude is very strongly over-estimated, despite the diffusion.

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