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A weakly nonlinear theory of continental shelf waves

In this paper three systems of evolution equations are presented which describe the free propagation of long continental shelf waves in the linear and weakly nonlinear regime. Two different degrees of nonlinearity are considered: for the first the Korteweg-de Vries equation is found to govern the dynamics of the system in the case of a single energy-containing mode (theories by Smith 1972; Grimshaw 1977), whereas, for the second nonlinear range, a nonlinear hyperbolic equation is derived. The nonlinear interactions among shelf-wave modes are also considered: they are modelled through nonlinear coupling terms in the evolution equations. This theory allows the timescale for the development of dispersive and nonlinear effects to be determined for each parameter range. The amplitude ranges corresponding to linear and nonlinear shelf waves are evaluated for the Oregon and the East Australian shelves, and some qualitative conclusions on the importance of nonlinear effects are derived. Finally the case of a shelf with longshore variation in topography is analysed and coupling terms in the evolution equations appear. They account for the scattering of energy between the various modes due to the linear and nonlinear interactions of the wave with the topographic changes.