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Evolution of Surface Gravity Waves Over a Submarine Canyon Rudy Magne, Kostas Belibassakis, Thomas Herbers, Fabrice Ardhuin, William O’Reilly, Vincent Rey

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Evolution of Surface Gravity Waves Over a Submarine Canyon Rudy Magne, Kostas Belibassakis, Thomas Herbers, Fabrice Ardhuin, William O’Reilly, Vincent Rey Evolution of surface gravity waves over a submarine canyon Rudy Magne, Kostas Belibassakis, Thomas Herbers, Fabrice Ardhuin, William O’Reilly, Vincent Rey To cite this version: Rudy Magne, Kostas Belibassakis, Thomas Herbers, Fabrice Ardhuin, William O’Reilly, et al.. Evo- lution of surface gravity waves over a submarine canyon. Journal of Geophysical Research. Oceans, Wiley-Blackwell, 2007, 112, pp.C01002. 10.1029/2005JC003035. hal-00089326 HAL Id: hal-00089326 https://hal.archives-ouvertes.fr/hal-00089326 Submitted on 17 Aug 2006 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JOURNAL OF GEOPHYSICAL RESEARCH, VOL. ???, XXXX, DOI:10.1029/, Evolution of surface gravity waves over a submarine canyon R. Magne1,5, K. A. Belibassakis2, T. H. C. Herbers3, Fabrice Ardhuin1, W. C. O’Reilly4, and V. Rey5 Abstract. The effects of a submarine canyon on the propagation of ocean surface waves are examined with a three-dimensional coupled-mode model for wave propagation over steep topography. Whereas the classical geometrical optics approximation predicts an abrupt transition from complete transmission at small incidence angles to no transmission at large angles, the full model predicts a more gradual transition with partial reflection/transmission that is sensitive to the canyon geometry and controlled by evanescent modes for small incidence angles and relatively short waves. Model results for large incidence angles are compared with data from directional wave buoys deployed around the rim and over Scripps Canyon, near San Diego, California, during the Nearshore Canyon Experiment (NCEX). Wave heights are observed to decay across the canyon by about a factor 5 over a dis- tance shorter than a wavelength. Yet, a spectral refraction model predicts an even larger reduction by about a factor 10, because low frequency components cannot cross the canyon in the geometrical optics approximation. The coupled-mode model yields accurate re- sults over and behind the canyon. These results show that although most of the wave energy is refractively trapped on the offshore rim of the canyon, a small fraction of the wave energy ‘tunnels’ across the canyon. Simplifications of the model that reduce it to the standard and modified mild slope equations also yield good results, indicating that evanescent modes and high order bottom slope effects are of minor importance for the energy transformation of waves propagating across depth contours at large oblique an- gles. 1. Introduction O’Reilly and Guza [1991, 1993] compared Kirby’s [1986] refraction-diffraction model to a spectral geometrical optics Waves are strongly influenced by the bathymetry when refraction model based on the theory of Longuet-Higgins they reach shallow water areas. Munk and Traylor [1947] [1957]. The two models generally agreed in simulations of conducted a first quantitative study of the effects of bottom realistic swell propagation in the Southern California Bight. topography on wave energy transformation over Scripps and However, both models assume a gently sloping bottom, and La Jolla Canyons, near San Diego, California. Wave refrac- their limitations in regions with steep topography are not tion diagrams were constructed using a manual method, and well understood. Booij [1983], showed that the MSE is compared to visual observations. Fairly good agreement was valid for bottom slopes as large as 1/3 for normal wave in- found between predicted and observed wave heights. Other cidence. To extend its application to steeper slopes, Mas- effects such as diffraction were found to be important else- sel [1993 ; see also Chamberlain and Porter, 1995] modified where, for sharp bathymetric features (e.g. harbour struc- the MSE by including terms of second order in the bottom tures or coral reefs), prompting Berkhoff [1972] to intro- slope, that were neglected by Berkhoff [1972]. This modified duce an equation that represents both refraction and diffrac- mild slope equation (MMSE) includes terms proportional to tion. Berkhoff’s equation is based on a vertical integration the bottom curvature and the square of the bottom slope. Chandrasekera and Cheung of Laplace’s equation and is valid in the limit of small bot- [1997] observed that the cur- vature terms significantly change the wave height behind a tom slopes. It is widely known as the mild slope equation shoal, whereas the slope-squared terms have a weaker in- (MSE). A parabolic approximation of this equation was pro- fluence. Lee and Yoon [2004] noted that the higher order posed by Radder [1979], and further refined by Kirby [1986] bottom slope terms change the wavelength, which in turn and Dalrymple and Kirby [1988]. affects the refraction. In spite of these improvements, an important restriction of these equations is that the vertical structure of the wave field is described by the Airy solu- 1Centre Militaire d’Oc´eanographie, Service tion of waves over a horizontal bottom. Hence the MMSE Hydrographique et Oc´eanographique de la Marine, 29609 cannot describe the wave field accurately over steep bot- Brest, France. tom topography. Thus, Massel [1993] introduced an addi- 2Department of Naval Architecture and Marine tional infinite series of local modes (’evanescent modes’ or Engineering, National Technical University of Athens, PO ’decaying waves’), that allows a local adaptation of the wave Box 64033 Zografos, 15710 Athens, Greece. field [see also Porter and Staziker, 1995], and converges to 3 ccsd-00089326, version 1 - 17 Aug 2006 Department of Oceanography, Naval Postgraduate the exact solution of Laplace’s equation, except at the bot- School, Monterey, CA 93943, USA. tom interface. Indeed, the vertical velocity at the bottom 4Integrative Oceanography Division, Scripps Institution of is still zero, and is discontinuous in the limit of an infinite Oceanography, La Jolla, CA 92093, USA. number of modes. Recently, Athanassoulis and Belibassakis 5Laboratoire de Sondages Electromagn´etique de [1999] added a ’sloping bottom mode’ to the local mode se- l’Environnement Terrestre, Universit´ede Toulon et du Var, ries expansion, which properly satisfies the Neuman bottom La Garde, France. boundary condition. This approach was further explored by Chandrasekera and Cheung, [2001] and Kim and Bai, [2004]. Although the sloping-bottom mode yields only small correc- Copyright 2006 by the American Geophysical Union. tions for the wave height, it significantly improves the accu- 0148-0227/06/$9.00 racy of the velocity field close to the bottom. Moreover, this 1 X-2 MAGNE ET AL.: WAVES OVER SUBMARINE CANYONS ∞ mode enables a faster convergence of the series of evanescent modes, by making the convergence mathematically uniform. + ϕn(x)Zn(z; x), (1) n=1 As these steep topography models are becoming available, X one may wonder if this level of sophistication is necessary to accurately describe the transformation of ocean waves over where ϕ0(x)Z0(z; x) is the propagating mode and natural continental shelf topography. It is expected that if ϕn(x)Zn(z; x) are the evanescent modes. The additional such models are to be useful anywhere, it should be around term ϕ−1(x)Z−1(z; x) is the sloping-bottom mode, which steep submarine canyons. Surprisingly, a geometrical optics permits the consistent satisfaction of the bottom boundary refraction model that assumes weak amplitude gradients on condition on a sloping bottom. The modes allow for the lo- the scale of the wavelength, usually corresponding to gentle cal adaptation of the wave potential. The functions Zn(z; x) bottom slopes, was found to yield accurate predictions of which represent the vertical structure of the nth mode are swell transformation over Scripps canyon [Peak, 2004]. The given by, practical limitations of mild slope approximations for natu- ral seafloor topography are clearly not well established. cosh[k0(x)(z + h(x))] The goal of the present paper is to understand the prop- Z0(z,x)= , (2) cosh(k0(x)h(x)) agation of waves over a submarine canyon, including the practical imitations of geometrical optics theory for the as- sociated large bottom slopes. Numerical models will be cos[kn(x)(z + h(x))] used to sort out the relative importance of refraction, and Zn(z,x)= , n = 1, 2, ..., (3) diffraction effects. Observations of ocean swell transforma- cos(kn(x)h(x)) tion over Scripps and La Jolla Canyons, collected during the Nearshore Canyon Experiment (NCEX), are compared 3 2 with predictions of the three-dimensional (3D) coupled- z z Z−1(z,x)= h(x) + , (4) mode model. This model is called NTUA5 because its h(x) h(x) present implementation will be limited to a total of 5 modes " # [Belibassakis et al., 2001]. This is the first verification of a NTUA-type model with field observations, as previous where k0 and kn are the wavenumbers obtained from the model validations were done with laboratory data. This ap- dispersion relation (for propagating and evanescent modes), plication of NTUA5 to submarine canyons is not straight- evaluated for the local depth h = h(x): forward since the model is based on the extension of the two-dimensional (2D) model of Athanassoulis and Belibas- 2 ω = gk0 tanh k0h = −gk tan k h, (5) sakis [1999], and requires special care in the position of the n n offshore boundary and the numerical damping of scattered waves along the boundary. Further details on these and soft- with ω the angular frequency ware developments, and a comparison with results of the As discussed in Athanassoulis and Belibassakis [1999], al- SWAN model [Booij et al., 1999] for the same NCEX case ternative formulations of Z−1 exist, and the extra sloping- are given by Gerosthathis et al.
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