A Coupled-Mode Technique for the Run-Up of Non-Breaking Dispersive Waves on Plane Beaches

Volume 2: Ocean Engineering and Polar and Arctic Sciences and Technology ◽

10.1115/omae2006-92162 ◽

2006 ◽

Author(s):

K. A. Belibassakis ◽

G. A. Athanassoulis

Keyword(s):

Shallow Water ◽

Water Waves ◽

Wave Equations ◽

Neumann Boundary ◽

Breaking Waves ◽

Numerical Results ◽

Coupled Mode Theory ◽

Mode Theory ◽

Coupled Mode ◽

Run Up

A coupled-mode model is developed and applied to the transformation and run-up of dispersive water waves on plane beaches. The present work is based on the consistent coupled-mode theory for the propagation of water waves in variable bathymetry regions, developed by Athanassoulis & Belibassakis (1999) and extended to 3D by Belibassakis et al (2001), which is suitably modified to apply to a uniform plane beach. The key feature of the coupled-mode theory is a complete modal-type expansion of the wave potential, containing both propagating and evanescent modes, being able to consistently satisfy the Neumann boundary condition on the sloping bottom. Thus, the present approach extends previous works based on the modified mild-slope equation in conjunction with analytical solution of the linearised shallow water equations, see, e.g., Massel & Pelinovsky (2001). Numerical results concerning non-breaking waves on plane beaches are presented and compared with exact analytical solutions; see, e.g., Wehausen & Laitone (1960, Sec. 18). Also, numerical results are presented concerning the run-up of non-breaking solitary waves on plane beaches and compared with the ones obtained by the solution of the shallow-water wave equations, Synolakis (1987), Li & Raichlen (2002), and experimental data, Synolakis (1987).

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

Journal of Fluid Mechanics ◽

10.1017/s0022112099004978 ◽

1999 ◽

Vol 389 ◽

pp. 275-301 ◽

Cited By ~ 104

Author(s):

G. A. ATHANASSOULIS ◽

K. A. BELIBASSAKIS

Keyword(s):

Boundary Condition ◽

Water Waves ◽

Small Amplitude ◽

Coupled Mode Theory ◽

Bottom Boundary ◽

Mode Theory ◽

Coupled Mode ◽

Bottom Boundary Condition ◽

Additional Mode ◽

Sloping Bottom

Extended mild-slope equations for the propagation of small-amplitude water waves over variable bathymetry regions, recently proposed by Massel (1993) and Porter & Staziker (1995), are shown to exhibit an inconsistency concerning the sloping-bottom 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.

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