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John W. Miles

Abstract

Laplace's tidal equations are augmented by dissipation in a bottom boundary layer that is intermediate in character between those of Ekman and Stokes. Laplace's tidal equation for a global ocean remains second-order and self-adjoint, but the operator and eigenvalues are complex with imaginary parts are O(E½), where E = ν/2ωh 2 (ν is the vertical component of the kinematic eddy viscosity, ω the rotational speed of the Earth, and h the depth of the global ocean). The imaginary part of the eigenvalue is expressed as a quadratic integral of the corresponding Hough function. The Q for a free oscillation is expressed as the ratio of two quadratic integrals that represent the mean energy and dissipation rates. Approximate calculations for the semidiurnal tides (with azimuthal wave number 2) are given.

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John W. Miles

Abstract

A solitary wave of amplitude a evolving in water of depth d over a bottom of gradual slope δ and turbulent friction coefficient Cf is found to have the asymptotic (as d ↓0) relative amplitude a/d = α1 = 15α/4Cf α1b where αb, is the relative amplitude above which breaking occurs. It is argued that an initially sinusoidal wave of sufficiently small amplitude evolves over a shoaling bottom into a periodic sequence of solitary waves with relative amplitude α1. This prediction is supported by observations (Wells, 1978) of the evolution of swell over mud flats.

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