General Bottom Topography Research Articles

Full Justification for the Extended Green–Naghdi System for an Uneven Bottom with/without Surface Tension

This paper is a continuation of a previous work on the extended Green–Naghdi system. We prolong the system, in arbitrary dimension, with/without surface tension, and for a general bottom topography. Confining the work to the one-dimensional case, wellposedness and consistency with respect to initial data and parameters are proved, taking into account the effect of surface tension. The results are local, but long term in the sense of dependence upon initial data. As a conclusion, our solution remains close to the exact solution of the full Euler system with a better (smaller) precision and therefore the full justification of the models.

A Modified Mild-Slope Model for the Hydrodynamic Analysis of Arrays of Heaving WECs in Variable Bathymetry Regions

A simplified model based on the Modified Mild-Slope Equation with inclusions is developed for modelling the scattering of waves from multiple heaving point absorbers arranged in an array in general bottom topography. The model is used, in conjunction with a 3D BEM, in order to estimate the parameters modelling the energy extraction of the devices using data obtained from the hydrodynamic responses and performance of the single floating WEC. Subsequently, the present model is used for specific examples to calculate the wave field and the hydrodynamic performance of arrays of heaving WECs in constant depth and variable bathymetry regions and illustrate the effect of bottom slope and variation on the calculated wave field in the domain and in the vicinity of the devices. The present simplified model provides a low-cost first estimation of the wave conditions in the domain, which could be exploited as a supporting tool for best arrangement and park design purposes.

Geometrically intrinsic modeling of shallow water flows

Shallow water models of geophysical flows must be adapted to geometric characteristics in the presence of a general bottom topography with non-negligible slopes and curvatures, such as a mountain landscape. In this paper we derive an intrinsic shallow water model from the Navier–Stokes equations defined on a local reference frame anchored on the bottom surface. The equations resulting are characterized by non-autonomous flux functions and source terms embodying only the geometric information. We show that the proposed model is rotational invariant, admits a conserved energy, is well-balanced, and it is formally a second order approximation of the Navier–Stokes equations with respect to a geometry-based order parameter. We then derive a numerical discretization by means of a first order upwind Godunov finite volume scheme intrinsically defined on the bottom surface. We study convergence properties of the resulting scheme both theoretically and numerically. Simulations on several synthetic test cases are used to validate the theoretical results as well as more experimental properties of the solver. The results show the importance of taking into full consideration the bottom geometry even for relatively mild and slowly varying curvatures.

Multi-DOF WEC Performance in Variable Bathymetry Regions Using a Hybrid 3D BEM and Optimization

In the present work a hybrid boundary element method is used, in conjunction with a coupled mode model and perfectly matched layer model, for obtaining the solution of the propagation/diffraction/radiation problems of floating bodies in variable bathymetry regions. The implemented methodology is free of mild-slope assumptions and restrictions. The present work extends previous results concerning heaving floaters over a region of general bottom topography in the case of generally shaped wave energy converters (WECs) operating in multiple degrees of freedom. Numerical results concerning the details of the wave field and the power output are presented, and the effects of WEC shape on the optimization of power extraction are discussed. It is demonstrated that consideration of heave in combination with pitch oscillation modes leads to a possible increase of the WEC performance.

A Nonlinear Coupled-Mode Model for Waves Propagating in Vertically Sheared Currents in Variable Bathymetry—Collinear Waves and Currents

A novel coupled-mode model is developed for the wave–current–seabed interaction problem, with application in wave scattering by non-homogeneous, sheared currents over general bottom topography. The formulation is based on a velocity representation defined by a series of local vertical modes containing the propagating and evanescent modes, able to accurately treat the continuity condition and the bottom boundary condition on sloping parts of the seabed. Using the above representation in Euler equations, a coupled system of differential equations on the horizontal plane is derived, with respect to the unknown horizontal velocity modal amplitudes. In the case of small-amplitude waves, a linearized version of the above coupled-mode system is obtained, and the dispersion characteristics are studied for various interesting cases of wave–seabed–current interaction. Keeping only the propagating mode in the vertical expansion of the wave potential, the present system is reduced to a one-equation, non-linear model, generalizing Boussinesq models. The analytical structure of the present coupled-mode system facilitates extensions to treat non-linear effects and further applications concerning wave scattering by inhomogeneous currents in coastal regions with general 3D bottom topography.

A Novel Method for Estimating Wave Energy Converter Performance in Variable Bathymetry Regions and Applications

A numerical model is presented for the estimation of Wave Energy Converter (WEC) performance in variable bathymetry regions, taking into account the interaction of the floating units with the bottom topography. The proposed method is based on a coupled-mode model for the propagation of the water waves over the general bottom topography, in combination with a Boundary Element Method for the treatment of the diffraction/radiation problems and the evaluation of the flow details on the local scale of the energy absorbers. An important feature of the present method is that it is free of mild bottom slope assumptions and restrictions and it is able to resolve the 3D wave field all over the water column, in variable bathymetry regions including the interactions of floating bodies of general shape. Numerical results are presented concerning the wave field and the power output of a single device in inhomogeneous environment, focusing on the effect of the shape of the floater. Extensions of the method to treat the WEC arrays in variable bathymetry regions are also presented and discussed.

A coupled-mode model for water wave scattering by vertically sheared currents in variable bathymetry regions

A coupled-mode model is developed for treating the wave–current–seabed interaction problem, with application to wave scattering by non-homogeneous, sheared current with linear vertical velocity profile, over general bottom topography. The wave potential is represented by a series of local vertical modes containing the propagating and evanescent modes, plus additional terms accounting for the satisfaction of the boundary conditions. Using the above representation, in conjunction with a variational principle, a coupled system of differential equations on the horizontal plane is derived, with respect to the unknown modal amplitudes. In the case of small-amplitude waves, a linearized version of the above coupled-mode system is obtained, extending previous analysis by Belibassakis et al. (2011) to the propagation of water waves over variable bathymetry regions in the presence of vertically sheared currents. Keeping only the propagating mode in the vertical expansion of the wave potential, the present system reduces to a one-equation model, that is shown to extend known mild-slope mild vertical shear equation concerning wave–current interaction over slowly varying topography. After additional simplifications, the latter model is shown to be compatible with the extended mild-slope mild-shear equation by Touboul et al. (2016). Results are presented for various representative test cases demonstrating the usefulness of the present coupled mode system and the importance of various terms in the modal expansion, and compared against experimental data collected in wave flume validating the present method. The analytical structure of the present system facilitates extensions to model non-linear effects and applications concerning wave scattering by inhomogeneous currents in coastal regions with general 3D bottom topography.

A coupled-mode model for water wave scattering by horizontal, non-homogeneous current in general bottom topography

A coupled-mode model is developed for treating the wave–current–seabed interaction problem, with application to wave scattering by non-homogeneous, steady current over general bottom topography. The vertical distribution of the scattered wave potential is represented by a series of local vertical modes containing the propagating mode and all evanescent modes, plus additional terms accounting for the satisfaction of the free-surface and bottom boundary conditions. Using the above representation, in conjunction with unconstrained variational principle, an improved coupled system of differential equations on the horizontal plane, with respect to the modal amplitudes, is derived. In the case of small-amplitude waves, a linearised version of the above coupled-mode system is obtained, generalizing previous results by Athanassoulis and Belibassakis [J Fluid Mech 1999;389:275–301] for the propagation of small-amplitude water waves over variable bathymetry regions. Keeping only the propagating mode in the vertical expansion of the wave potential, the present system reduces to an one-equation model, that is shown to be compatible with mild-slope model concerning wave–current interaction over slowly varying topography, and in the case of no current it exactly reduces to the modified mild-slope equation. The present coupled-mode system is discretized on the horizontal plane by using second-order finite differences and numerically solved by iterations. Results are presented for various representative test cases demonstrating the usefulness of the model, as well as the importance of the first evanescent modes and the additional sloping-bottom mode when the bottom slope is not negligible. The analytical structure of the present model facilitates its extension to fully non-linear waves, and to wave scattering by currents with more general structure.

A coupled-mode system with application to nonlinear water waves propagating in finite water depth and in variable bathymetry regions

A non-linear coupled-mode system of horizontal equations is presented, modelling the evolution of nonlinear water waves in finite depth over a general bottom topography. The vertical structure of the wave field is represented by means of a local-mode series expansion of the wave potential. This series contains the usual propagating and evanescent modes, plus two additional terms, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The present coupled-mode system fully accounts for the effects of non-linearity and dispersion, and the local-mode series exhibits fast convergence. Thus, a small number of modes (up to 5–6) are usually enough for precise numerical solution. In the present work, the coupled-mode system is applied to the numerical investigation of families of steady travelling wave solutions in constant depth, corresponding to a wide range of water depths, ranging from intermediate depth to shallow-water wave conditions, and its results are compared vs. Stokes and cnoidal wave theories, as well as with fully nonlinear Fourier methods. Furthermore, numerical results are presented for waves propagating over variable bathymetry regions and compared with nonlinear methods based on boundary integral formulation and experimental data, showing good agreement.

A boundary element method for the hydrodynamic analysis of floating bodies in variable bathymetry regions

In this work, a hybrid technique is presented for the hydrodynamic analysis of floating bodies in variable bathymetry regions. Our method is based on the coupled-mode theory for the propagation of water waves in general bottom topography, developed by Athanassoulis and Belibassakis [Athanassoulis GA, Belibassakis KA. A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions. J Fluid Mech 1999;389:275–301.] and extended to 3D by Belibassakis et al. [Belibassakis KA, Athanassoulis GA, Gerostathis TP. A coupled-mode model for the refraction–diffraction of linear waves over steep three-dimensional bathymetry. Appl Ocean Res 2001;23:319–336.], which is free of any mild-slope assumption, in conjunction with a boundary integral representation of the near field in the vicinity of the floating body. Both 2D and 3D problems have been considered. In all cases the near field is represented by boundary integral representation involving simple (Rankine) sources. In the 2D case, the far-field is modelled by complete (normal-mode) series expansions derived by separation of variables in the constant-depth half-strips. In the 3D case, the far-field is modelled by an integral representation involving the appropriate Green's function for harmonic water waves over a bottom with different depths at infinity, developed by Belibassakis and Athanassoulis [Belibassakis KA, Athanassoulis GA. Three-dimensional Green's function for harmonic water waves over a bottom with different depths at infinity. J Fluid Mech 2004;510:267–302.]. The numerical solution is obtained by means of a low-order panel method materialising the hybrid technique. Numerical results are presented concerning floating bodies of simple geometry, lying over sloping and undulating seabeds. With the aid of systematic comparisons with benchmark solutions the convergence and accuracy of the present method in 3D has been studied, and the effects of bottom slope and curvature on the hydrodynamic characteristics (hydrodynamic coefficients and responses) of the floating bodies are illustrated and discussed.

Extension of second-order Stokes theory to variable bathymetry

In the present work second-order Stokes theory has been extended to the case of a generally shaped bottom profile connecting two half-strips of constant (but possibly different) depths, initiating a method for generalizing the Stokes hierarchy of second- and higher-order wave theory, without the assumption of spatial periodicity. In modelling the wave–bottom interaction three partial problems arise: the first order, the unsteady second order and the steady second order. The three problems are solved by using appropriate extensions of the consistent coupled-mode theory developed by the present authors for the linearized problem. Apart from the Stokes small-amplitude expansibility assumption, no additional asymptotic assumptions have been introduced. Thus, bottom slope and curvature may be arbitrary, provided that the resulting wave dynamics is Stokes-compatible. Accordingly, the present theory can be used for the study of various wave phenomena (propagation, reflection, diffraction) arising from the interaction of weakly nonlinear waves with a general bottom topography, in intermediate water depth. An interesting phenomenon, that is also very naturally resolved, is the net mass flux induced by the depth variation, which is consistently calculated by means of the steady second-order potential. The present method has been validated against experimental results and fully nonlinear numerical solutions. It has been found that it correctly predicts the second-order harmonic generation, the amplitude nonlinearity, and the amplitude variation due to non-resonant first-and-second harmonic interaction, up to the point where the energy transfer to the third and higher harmonics can no longer be neglected. Under the restriction of weak nonlinearity, the present model can be extended to treat obliquely incident waves and the resulting second-order refraction patterns, and to study bichromatic and/or bidirectional wave–wave interactions, with application to the transformation of second-order random seas in variable bathymetry regions.

A Simple Model of Mass-driven Abyssal Circulation over a General Bottom Topography

Abstract The classic Stommel–Arons problem is revisited in the context of a basin with a general bottom topography containing an equator. Topography is taken to be smoothly varying and, as such, there are no vertical side walls in the problem. The perimeter of the abyssal basin is thus defined as the curve along which the layer depth vanishes. Because of this, it is not required that the component of horizontal velocity perpendicular to the boundary curve vanish on the boundary. Planetary geostrophic dynamics leads to a characteristic equation for the interface height field in which characteristics typically originate from a single point located on the eastern edge of the basin at the equator. For a simple choice of topography it is possible to solve the problem analytically. In the linear limit of weak forcing, the solution exhibits an intensified flow on the western edge of the basin. This flow is pan of the interior solution and is thus not a traditional, dissipative western boundary current. When the ...

Stable and unstable barotropic shelf waves in a coastal current

Abstract We consider the linearized stability of a barotropic coastal current flowing parallel to a straight coastline over a continental shelf and slope whose depth varies monotonically with distance from the coast. Some necessary conditions for stability and various semi-circle theorems are reviewed for general current profiles and bottom topography. A criterion for topography to be a destabilizing influence is derived. Some general results for stable waves are also described. Analytic solutions are obtained for a piece-wise linear current profile and the exponential depth profile (Buchwald and Adams, 1968). Dispersion diagrams are obtained for a monotonic current profile, where it is shown that the effect of topography is destabilizing, and for a triangular current profile. The dispersion diagrams generally contain a finite number (usually one or two) of unstable waves, and a set of stable waves, which may be infinite in number. The results are applied to some specific coastal regimes.

Coastal-Trapped and Frontal-Trapped Waves in a Baroclinic Western Boundary Current

Abstract Four types of stable, subinertial Rossby-like waves have been found to propagate alongshore in a model baroclinic boundary current similar to the Gulf Stream as it flows along the continental margin of the southeastern United States. The two-layer model incorporates a general bottom topography with continental shelf and slope, and a thermal-wind mean current confined to the upper layer. The four wave components are the familiar continental shelf wave, the quasi-geostrophic edge wave and complementary mode edge wave, plus a new frontal-trapped wave, which has wave amplitude maxima within the cyclonic side of the Stream. The dispersion diagram for a particular topography/density/current setting may be interpreted as a composite of the four components’ dispersion curve families. As in other similar problems a mode coupling between two components allows the characteristics of the two to be interchanged. The cases studied show the phase speeds of the various components to be affected by the location o...

Geomorphology of Aldabra Atoll

1. Regional setting Aldabra Atoll (latitude 9° 24' S, longitude 46° 20' E) is situated 420 km northwest of Madagascar and 640 km from the East African mainland, in the southwest Indian Ocean (figure 1). It forms one of a group of slightly elevated coral reefs to the north of Madagascar, and is thus distinguished from the sea-level coral reefs of the Farquhar group, the Amirantes, and the central Indian Ocean. The raised reefs of Aldabra, Assumption, Cosmoledo and Astove are situated on the summits of mountains approximately 4000 m high, rising from a fairly flat sea floor between 4000 and 4300 m deep. Aldabra and Assumption cap two neighbouring peaks, which are distinct at depths shallower than 2500 m, and Cosmoledo and Astove another pair, distinct above the 2000 m level (figure 2). The general bottom topography round these islands is based on surveys by H.M.S. Owen in 1962. More detailed surveys have been made of Aldabra itself, by H.M.S. Owen in 1962 and H.M.S. Vidal in 1967, and these soundings are contoured in figure 3.