Weakly nonlinear cubic interactions in coastal resonance

Abstract

In this paper the qualitative nonlinear influence of advection and continuity on the resonance characteristics of co-oscillating coastal basins is investigated. For this purpose a weakly nonlinear analysis was carried out on the shallow-water equations describing a coastal basin resonating with exterior water-level oscillations. It extends previous work on an almost-enclosed basin with one single (Helmholtz) mode to arbitrarily shaped ‘shallow’ basins with an infinite number of modes. In line with that work, it is necessary to assume friction to be sufficiently weak such that it is in balance with the nonlinear effects, instead of occurring in the linearized equations. The main result of this paper is the system of Landau equations describing the slow evolution of the amplitudes of the oscillatory eigenmodes of the basin, disregarding a zero-frequency eigenmode, should it exist. The dynamics of the zero-frequency mode neglecting oscillatory eigenmodes have been discussed by others, but a consistently balanced model for small amplitudes incorporating both the zero-frequency and oscillatory modes is not yet available. The behaviour of this system, describing the dynamics of the oscillatory eigenmodes only, is analysed. On the longer time scale, it gives rise to a ‘bent resonance curve’, multiple equilibria (several tidal regimes under the same tidal forcing), sudden regime changes and even chaotic dynamics (when these regime changes occur in an irregular way).