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A coupled-mode model for the scattering of water waves by shearing currents in variable bathymetry

Published online by Cambridge University Press:  26 April 2007

K. A. BELIBASSAKIS
K. A. BELIBASSAKIS*
Affiliation:
School of Naval Architecture and Marine Engineering, National Technical University of Athens, Heroon Polytechniou 9, Zografos 15773, Athens, [email protected]
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Abstract

A coupled-mode model is presented for wave–current–seabed interaction, with application to the problem of wave scattering by ambient shearing currents in variable bathymetry regions. We consider obliquely incident waves on a horizontally non-homogeneous current in a variable-depth strip, which is characterized by straight and parallel bottom contours. The flow associated with the current is assumed to be directed along the bottom contours and it is considered to be steady and known. In a finite subregion containing the bottom irregularity, we assume that the horizontal current profile is general and smoothly varying. Outside this region, the current is assumed to be uniform (or zero). Based on a variational principle, in conjunction with a rapidly convergent local-mode series expansion of the wave pressure field in the finite subregion containing the current variation and the bottom irregularity, a new coupled-mode system of equations is obtained, governing the scattering of waves in the presence of variable bathymetry and longshore shearing currents. By keeping only the propagating mode in the local-mode series, a new one-equation model is derived, having the property to reduce to the modified mild-slope equation when the current is zero, and to the enhanced mild-shear equation when the bottom is flat. An important aspect of the present model is that it can be further elaborated to treat shearing currents with general, depth-dependent vertical structure, and to include the effects of weak nonlinearity.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 578 , 10 May 2007 , pp. 413 - 434
Copyright
Copyright © Cambridge University Press 2007

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References

REFRENCES

Athanassoulis, G. A. & Belibassakis, K. A. 1999 A consistent coupled-mode theory for the propagation of small-amplitude water waves over variable bathymetry regions. J. Fluid. Mech. 389, 275–301.CrossRefGoogle Scholar
Bai, K. J. & Yeung, R. W. 1974 Numerical solution to free-surface flow problems.Proc. 10th Naval Hydrodyn. Symp. Office of Naval Research, Cambridge, MA.Google Scholar
Belibassakis, K. A., Athanassoulis, G. A. & Gerostathis, Th. 2001 A coupled-mode model for the refraction–diffraction of linear waves over steep three-dimensional bathymetry. Appl. Ocean Res. 23, 319–336.CrossRefGoogle Scholar
Berkhoff, J. C. W. 1972 Computation of combined refraction–diffraction. Proc. 13th Intl Conf. on Coastal Engineering, pp. 796–814. ASCE.CrossRefGoogle Scholar
Chamberlain, P. G. & Porter, D. 1995 The modified mild-slope equation. J. Fluid Mech. 291, 393–407.CrossRefGoogle Scholar
Chen, H. S. & Mei, C. C. 1974 Oscillations and wave forces in a man-made harbor in the open sea, Proc. 10th Naval Hydrodyn. Symp. Office of Naval Research, Cambridge, MA.Google Scholar
Chen, W., Panchang, V. & Demirbilek, Z. 2005 On the modeling of wave–current interaction using the elliptic mild-slope wave equation. Ocean Engng 32, 2135–2164.CrossRefGoogle Scholar
Davies, A. G & Heathershaw, A. D. 1984 Surface-wave propagation over sinusoidal varying topography. J. Fluid Mech. 144, 419–433.CrossRefGoogle Scholar
Dysthe, K. B. 2000 Modelling a rogue wave – speculations or a realistic possibility. In Rogue Waves 2000 (ed. Olagnon, M. & Athanassoulis, G.), pp. 255–264. Editions Ifremer, Plouzane, France.Google Scholar
Evans, D. V. 1975 The transmission of deep-water waves across a vortex sheet. J. Fluid Mech. 68, 389–401.CrossRefGoogle Scholar
Faulkner, D. 2000 Rogue waves – defining their characteristics for marine design. In Rogue Waves 2000 (ed. Olagnon, M. & Athanassoulis, G.), pp. 3–18. Editions Ifremer, Plouzane, France.Google Scholar
Guazzeli, E., Rey, V. & Belzons, M. 1992 Higher-order Bragg reflection of gravity surface waves by periodic beds. J. Fluid Mech. 245, 301–317.CrossRefGoogle Scholar
Jonsson, I. G. 1990 Wave–current interactions. In The Sea (ed. LeMehaute, B. & Hanes, D.M.), pp. 65–120. John Wiley.Google Scholar
Kirby, J. T. 1984 A note on linear surface wave–current interaction over slowly varying topography. J. Geophys. Res. 89, 745–74.CrossRefGoogle Scholar
Kirby, J. T. 1988 Current effects on resonant reflection of surface water waves by sand bars. J. Fluid Mech. 186, 501–520.CrossRefGoogle Scholar
Kirby, J. T. 1993 A note on Bragg scattering of surface waves by sinusoidal bars. Phys. Fluids A 5, 380–386.CrossRefGoogle Scholar
Kirby, J. T., Dalrymple, R. A. & Seo, S. N. 1987 Propagation of obliquely incident water waves over a trench. Part 2. Currents flowing along the trench. J. Fluid Mech. 176, 95–116.CrossRefGoogle Scholar
Liu, P. L.-F. 1990 Wave transformation. In The Sea (ed. LeMehaute, B. & Hanes, D. M.). John Wiley.Google Scholar
Liu, Y. & Yue, D. K. P. 1998 On generalized Bragg scattering of surface waves by bottom ripples. J. Fluid Mech. 356, 297–326.CrossRefGoogle Scholar
McKee, W. D. 1987 Waver waves propagation across a shearing current. Wave Motion, vol. 9, pp. 209–215.CrossRefGoogle Scholar
McKee, W. D. 1996 A model for surface wave propagation across a shearing current. J. Phys. Oceanogr. 26, 276–278.2.0.CO;2>CrossRefGoogle Scholar
McKee, W. D. 2003 The propagation of water waves across a shearing current. Rep. Dept of Appl. Math., Univ. of New South Wales AMR 03/26.Google Scholar
Massel, S. R. 1993 Extended refraction–diffraction equation for surface waves. Coastal Engng 19, 97–126.CrossRefGoogle Scholar
Mei, C. C. 1983 The Applied Dynamics of Ocean Surface Waves. John Wiley (2nd Reprint, 1994, World Scientific).Google Scholar
Mei, C. C. 1985 Resonant reflection of surface water waves by periodic sand-bars.J. Fluid Mech. 152, 315–335.CrossRefGoogle Scholar
Mei, C. C., Hara, T. & Naciri, M. 1988 Note on Bragg scattering of water waves by parallel bars on the seabed. J. Fluid Mech. 186, 147–162.CrossRefGoogle Scholar
Miles, J. W. & Chamberlain, P. G. 1998 Topographical scattering of gravity waves.J. Fluid Mech. 361, 175–188.CrossRefGoogle Scholar
O'are, T. J. & Davies, A. G. 1993 A comparison of two models for surface-wave propagation over rapidly varying topography. Appl. Ocean Res. 15, 1–11.CrossRefGoogle Scholar
Peregrine, D. H. 1976 Interaction of waves and currects. Adv. Appl. Mech. 16, 9–17.CrossRefGoogle Scholar
Smith, J. 1983 On surface gravity waves crossing weak current jets. J. Fluid Mech. 134, 277–299.CrossRefGoogle Scholar
Smith, J. 1987 On surface waves crossing a step with horizontal shear. J. Fluid Mech. 175, 395–412.CrossRefGoogle Scholar
Smith, J. 2001 Observations and theories of Langmuir circulation: a story of mixing. In Fluid Mechanics and the Environment: Dynamical Approaches (ed. Lumney, J. L.). Springer.Google Scholar
Thomas, G. P. & Klopman, G. 1997 Wave–current interaction in the nearshore region. In Gravity Waves in water of Finite Depth (ed. Hunt, J. N.).Computational Mechanics Publications.Google Scholar