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.

The generalized Langrangian mean theory provides exact equations for general wave-turbulence-mean flow interactions in three dimensions. For practical applications, these equations must be closed by specifying the wave forcing terms. Here an approximate closure is obtained under the hypotheses of small surface slope, weak horizontal gradients of the water depth and mean current, and weak curvature of the mean current profile. These assumptions yield analytical expressions for the mean momentum and pressure forcing terms that can be expressed in terms of the wave spectrum. A vertical change of coordinate is then applied to obtain glm2z-RANS equations (55) and (57) with non-divergent mass transport in cartesian coordinates. To lowest order, agreement is found with Eulerian-mean theories, and the present approximation provides an explicit extension of known wave-averaged equations to short-scale variations of the wave field, and vertically varying currents only limited to weak or localized profile curvatures. Further, the underlying exact equations provide a natural framework for extensions to finite wave amplitudes and any realistic situation. The accuracy of the approximations is discussed using comparisons with exact numerical solutions for linear waves over arbitrary bottom slopes, for which the equations are still exact when properly accounting for partial standing waves. For finite amplitude waves it is found that the approximate solutions are probably accurate for ocean mixed layer modelling and shoaling waves, provided that an adequate turbulent closure is designed. However, for surf zone applications the approximations are expected to give only qualitative results due to the large influence of wave nonlinearity on the vertical profiles of wave forcing terms.

This paper is the product of the wave modelling community and it tries to make a picture of the present situation in this branch of science, exploring the previous and the most recent results and looking ahead towards the solution of the problems we presently face. Both theory and applications are considered.The many faces of the subject imply separate discussions. This is reflected into the single sections, seven of them, each dealing with a specific topic, the whole providing a broad and solid overview of the present state of the art. After an introduction framing the problem and the approach we followed, we deal in sequence with the following subjects: (Section) 2, generation by wind; 3, non-linear interactions in deep water; 4, white-capping dissipation; 5, non-linear interactions in shallow water; 6, dissipation at the sea bottom; 7, wave propagation; 8, numerics. The two final sections, 9 and 10, summarize the present situation from a general point of view and try to look at the future developments. Keywords

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In this paper we couple collocated Boundary Element Methods (BEM) with unstructured analysis- Belibassakis et al. (2013) [4]. In this connection, this paper makes a step towards integrating modern CAD representations for ship-hulls with hydrodynamic solvers of improved accuracy and efficiency, which is a prerequisite for building efficient ship-hull optimizers.

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