A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions

Abstract

Extended mild-slope equations for the propagation of small-amplitude water waves over variable bathymetry regions, recently proposed by Massel (1993) andStaziker (1995), are shown to exhibit an inconsistency concerning the slopingbottom boundary condition, which renders them non-conservative with respect to wave energy. In the present work, a consistent coupled-mode theory is derived from a variational formulation of the complete linear problem, by representing the vertical distribution of the wave potential as a uniformly convergent series of local vertical modes at each horizontal position. This series consists of the vertical eigenfunctions associated with the propagating and all evanescent modes and, when the slope of the bottom is different from zero, an additional mode, carrying information about the bottom slope. The coupled-mode system obtained in this way contains an additional equation, as well as additional interaction terms in all other equations, and reduces to the previous extended mild-slope equations when the additional mode is neglected. Extensive numerical results demonstrate that the present model leads to the exact satisfaction of the bottom boundary condition and, thus, it is energy conservative. Moreover, it is numerically shown that the rate of decay of the modal-amplitude functions is improved from O(n −2 ), where n is the mode number, to O(n −4 ), when the additional sloping-bottom mode is included in the representation. This fact substantially accelerates the convergence of the modal series and ensures the uniform convergence of the velocity field up to and including the boundaries.

Wave modelling – The state of the art

“…It can be shown that for small bottom amplitudes and in the presence of currents the reflection also conserves the wave action . This is also a consequence of the existence of a Hamiltonian for waves over variable topography, see, e.g., Henyey et al (1988), Athanassoulis and Belibassakis (1999). The reflection of linear waves is well known for simple bottom profiles (such as a step, a rectangular canyon or a ramp).…”

“…For general topographies of finite amplitudes, several extensions of the mild slope equation have been presented, e.g. by Athanassoulis and Belibassakis (1999).…”

“…A general representation of the variation in the wave field at the scale of the wavelength requires a phase resolving model that accounts for the interference patterns, particularly in areas of crossing wave rays (e.g., Berkhof, 1972;Dalrymple and Kirby, 1988;Athanassoulis and Belibassakis, 1999). However, the representation of the effect of diffraction on scales larger than the wavelength can be included in (7.1) by a proper modification of C θ (e.g.…”

Propagation of water waves through shearing currents in general bathymetry

“…To see this we calculate the first variation δ F of the above functional (see also, Athanassoulis & Belibassakis, 1999). Making use of the Green's theorem and the properties of the modal representations (2.4) in the two constant-depth strips, the variational equation (3.2) takes the form: and thus, it is needed only in subareas where the bottom surface is not flat, making the series (4.1) compatible with the Neumann bottom boundary condition (2.8c) there, while, at the same time, it significantly accelerates the convergence of the local-mode series.…”

“…For more details about the role and significance of this term we refer to Athanassoulis & Belibassakis (1999, Sec. 4), Belibassakis et al (2001), where this idea is first introduced and discussed for wave propagation/diffraction problems in variable bathymetry regions.. By using the local-mode series representation (4.1) in the variational principle (3.3), and by following exactly the same procedure as in Athanassoulis & Belibassakis (1999), the following coupled-mode system (CMS) with respect to the pressure mode amplitudes is obtained: …”