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Oblique runup of non-breaking solitary waves on an inclined plane

Published online by Cambridge University Press:  26 January 2011

GEIR K. PEDERSEN
GEIR K. PEDERSEN*
Affiliation:
Department of Mathematics, University of Oslo, PO Box 1053, 0316, Oslo, Norway
*
†Email address for correspondence: [email protected]
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Abstract

When a wave of permanent form is obliquely incident on an inclined plane, the wave pattern becomes stationary in a frame of reference which moves along the shore. This enables a simplified mathematical description of the problem which is used herein as a basis for efficient and accurate numerical simulations. First, a nonlinear and weakly dispersive set of Boussinesq equations for the downstream evolution of such stationary patterns is derived. In the hydrostatic approximation, streamline-based Lagrangian versions of the evolution equations are developed for automatic tracing of the shoreline. Both equation sets are, in their present form, developed for non-breaking waves only. Finite difference models for both equation sets are designed. These methods are then coupled dynamically to obtain a single nonlinear model with dispersive wave propagation in finite depth and an accurate runup representation. The models are tested by runup of waves at normal incidence and comparison with a more general model for the refraction of a solitary wave on a slope. Finally, a set of runup computations for oblique solitary waves is performed and compared with estimates of oblique runup heights obtained from a combination of an analytic solution for normal incidence and optics. We find that the runup heights decrease in proportion to the square of the angle of incidence for angles up to 45°, for which the height is reduced by around 12% relative to that of normal incidence. In Appendix A, the validity of the downstream formulation is discussed in the light of solitary wave optics and wave jumps.

JFM classification

Geophysical and Geological Flows: Coastal engineering Waves/Free-surface Flows: Solitary waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 668 , 10 February 2011 , pp. 582 - 606
Copyright
Copyright © Cambridge University Press 2011

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References

REFERENCES

Briggs, M. J., Synolakis, C. E., Harkins, G. S. & Green, D. R. 1995 Laboratory experiments of tsunami runup on a circular island. Pure Appl. Geophys. 144 (3–4), 569–593.CrossRefGoogle Scholar
Brocchini, M. 1998 The run-up of weakly-two-dimensional solitary pulses. Nonlinear Processes Geophys. 5, 27–38.CrossRefGoogle Scholar
Brocchini, M., Kennedy, A., Soldini, L. & Mancinelli, A. 2004 Topographically controlled, breaking-wave-induced macrovortices. Part 1. Widely separated breakwaters. J. Fluid Mech. 507, 289–307.CrossRefGoogle Scholar
Brocchini, M. & Peregrine, D. H. 1996 Integral flow properties of the swash zone and averaging. J. Fluid Mech. 317, 241–273.CrossRefGoogle Scholar
Bühler, O. & Jacobson, T. E. 2001 Wave-driven currents and vortex dynamics on barred beaches. J. Fluid Mech. 449, 313–339.CrossRefGoogle Scholar
Carrier, G. F. & Greenspan, H. P. 1958 Water waves of finite amplitude on a sloping beach. J. Fluid Mech. 4, 97–109.CrossRefGoogle Scholar
Carrier, G. F. & Noiseux, C. F. 1983 The reflection of obliquely incident tsunamis. J. Fluid Mech. 133, 147–160.CrossRefGoogle Scholar
Carrier, G. F., Wu, T. T. & Yeh, H. 2003 Tsunami run-up and draw-down on a plane beach. J. Fluid Mech. 475, 79–99.CrossRefGoogle Scholar
Choi, B. H., Pelinovsky, E., Kim, D. C., Didenkulova, I. & Woo, S.-B. 2008 Two- and three-dimensional computation of solitary wave runup on non-plane beach. Nonlinear Processes Geophys. 15, 489–502.CrossRefGoogle Scholar
Didenkulova, I. 2008 New trends in the analytical theory of long sea wave runup. In Applied Wave Mathematics: Selected Topics in Solids, Fluids, and Mathematical Methods, pp. 265–296. Springer.Google Scholar
Didenkulova, I., Pelinovsky, E., Soomere, T. & Zahibo, N. 2007 Runup of nonlinear asymmetric waves on a plane beach. In Tsunami and Nonlinear Waves (ed. Kundu, A.), pp. 175–190. Springer.CrossRefGoogle Scholar
Glimsdal, S., Pedersen, G. & Langtangen, H. P. 2004 An investigation of overlapping domain decomposition methods for one-dimensional dispersive long wave equations. Adv. Water Resour. 27 (11), 1111–1133.CrossRefGoogle Scholar
Grilli, S., Svendsen, I. & Subramanya, R. 1997 Breaking criterion and characteristics for solitary waves on slopes. ASCE J. Waterway Port Coastal Ocean Engng 123 (3), 102–112.CrossRefGoogle Scholar
Guza, R. T. & Davis, R. E. 1974 Excitation of edge waves by waves incident on a beach. J. Geophys. Res. 79, 1285–1291.CrossRefGoogle Scholar
Hall, J. V. & Watts, J. W. 1953 Laboratory investigation of the vertical rise of solitary waves on impermeable slopes. Tech. Memo. 33. Beach Erosion Board, US Army Corps of Engineers.Google Scholar
Imamura, F. 1996 Review of tsunami simulation with a finite difference method. In Long-Wave Runup Models (ed. Yeh, H., Synolakis, C. E. & Liu, P. L.-F.), pp. 25–42. World Scientific.Google Scholar
Jensen, A., Pedersen, G. & Wood, D. J. 2003 An experimental study of wave run-up at a steep beach. J. Fluid. Mech. 486, 161–188.CrossRefGoogle Scholar
Kânoglu, U. & Synolakis, C. E. 1998 Long-wave runup on piecewise linear topographies. J. Fluid Mech. 374, 1–28.CrossRefGoogle Scholar
Keller, H. B., Levine, A. D. & Whitham, G. B. 1960 Motion of a bore over a sloping beach. J. Fluid Mech. 7, 302–316.CrossRefGoogle Scholar
Kennedy, A. B., Chen, Q., Kirby, J. T. & Dalrymple, R. A. 2000 Boussinesq modeling of wave transformation, breaking, and run-up. Part I. 1D. ASCE J. Waterway Port Coastal Ocean Engng 126 (1), 39–47.CrossRefGoogle Scholar
Ko, K. & Kuehl, H. H. 1979 Cylindrical and spherical Korteweg–deVries solitary waves. Phys. Rev. Lett. 22, 1343–1348.Google Scholar
Koshimura, S., Imamura, F. & Shuto, N. 1999 Propagation of obliquely incident tsunamis on a slope. Part I. Amplification of tsunamis on a continental slope. Coast. Engng J. 41 (2), 151–164.CrossRefGoogle Scholar
LeVeque, R. J. & George, D. L. 2008 High-resolution finite volume methods for the shallow water equations with bathymetry and dry states. In Advanced Numerical Models for Simulating Tsunami Waves and Runup (ed. Liu, P. L.-F., Yeh, H. & Synolakis, C. E.), vol. 10, pp. 43–74. World Scientific.CrossRefGoogle Scholar
Li, Y. & Raichlen, F. 2001 Solitary wave runup on plane slopes. ASCE J. Waterway Port Coastal Ocean Engng 127 (1), 33–44.CrossRefGoogle Scholar
Liu, P. L.-F., Cho, Y.-S., Briggs, M. J., Kânoglu, U. & Synolakis, C. E. 1995 Runup of solitary waves on a circular island. J. Fluid Mech. 302, 259–285.CrossRefGoogle Scholar
Liu, P. L.-F., Yeh, H. & Synolakis, C. E. (Ed.) 2008 Advances in Coastal and Ocean Engineering, vol. 10. World Scientific.Google Scholar
Lynett, P. J., Wu, T.-R. & Liu, P. L.-F. 2002 Modeling wave runup with depth-integrated equations. Coast. Engng 46, 89–107.CrossRefGoogle Scholar
Løvholt, F. L. & Pedersen, G. 2009. Instabilities of Boussinesq models in non-uniform depth. Numer. Meth. Fluid Mech. 61 (6), 606–637, doi:10.1002/fld.1968.CrossRefGoogle Scholar
Løvholt, F. L., Pedersen, G. & Gisler, G. 2008 Oceanic propagation of a potential tsunami from the La Palma Island. J. Geophys. Res. 113, C09026, doi:10.1029.Google Scholar
Madsen, P. A. & Schäffer, H. A. 1999 A review of Boussinesq-type equations for surface gravity waves. In Advances in Coastal and Ocean Engineering, vol. 5, pp. 1–95. World Scientific.Google Scholar
Meyer, R. E. & Taylor, A. D. 1972 Run-up on Beaches, pp. 95–122. Academic Press.Google Scholar
Miles, J. W. 1977 a Diffraction of solitary waves. Z. Angew. Math. Phys. 28, 889–902.CrossRefGoogle Scholar
Miles, J. W. 1977 b Obliquely interacting solitary waves. J. Fluid Mech. 79, 157–169.CrossRefGoogle Scholar
Miles, J. W. 1977 c Resonantly interacting solitary waves. J. Fluid Mech. 79, 171–179.CrossRefGoogle Scholar
Miles, J. W. 1980 Solitary waves. Annu. Rev. Fluid Mech. 12, 11–43.CrossRefGoogle Scholar
Pedersen, G. 1985 Run-up of Periodic Waves on a Straight Beach. Preprint Series in Applied Mathematics, ISBN 82-553-0952-7 1/85. Department of Mathematics, University of Oslo, Norway.Google Scholar
Pedersen, G. 1988 Three-dimensional wave patterns generated by moving disturbances at transcritical speeds. J. Fluid Mech. 196, 39–63.CrossRefGoogle Scholar
Pedersen, G. 1994 Nonlinear modulations of solitary waves. J. Fluid Mech. 267, 83–108.CrossRefGoogle Scholar
Pedersen, G. 1996 Refraction of solitons and wave jumps. In Waves and Nonlinear Processes in Hydrodynamics (ed. Gjevik, B., Grue, J. & Weber, J. E.), pp. 139–150. Kluwer.CrossRefGoogle Scholar
Pedersen, G. 2008 a A Lagrangian model applied to runup problems. In Advanced Numerical Models for Simulating Tsunami Waves and Runup (ed. Liu, P. L.-F., Yeh, H. & Synolakis, C. E.), pp. 311–314. Advances in Coastal and Ocean Engineering, vol. 10. World Scientific.CrossRefGoogle Scholar
Pedersen, G. 2008 b Modeling run-up with depth-integrated equation models. In Advanced Numerical Models for Simulating Tsunami Waves and Runup (ed. Liu, P. L.-F., Yeh, H. & Synolakis, C. E.), pp. 3–42. Advances in Coastal and Ocean Engineering, vol. 10. World Scientific.CrossRefGoogle Scholar
Pedersen, G. & Gjevik, B. 1983 Run-up of solitary waves. J. Fluid Mech. 135, 283–299.CrossRefGoogle Scholar
Pedersen, G. & Løvholt, F. L. 2008 Documentation of a Global Boussinesq Solver. Preprint Series in Applied Mathematics 1. Department of Mathematics, University of Oslo, Norway. Available at: http://www.math.uio.no/eprint/appl_math/2008/01-08.html.Google Scholar
Peregrine, D. H. 1972 Equations for water waves and the approximation behind them. In Waves on Beaches (ed. Meyer, R. E.), pp. 357–412. Academic Press.Google Scholar
Peregrine, D. H. 1983 Wave jumps and caustics in the propagation of finite-amplitude water waves. J. Fluid Mech. 136, 435–452.CrossRefGoogle Scholar
Peregrine, D. H. 1998 Surf zone currents. Theor. Comput. Fluid Dyn. 10, 295–309.CrossRefGoogle Scholar
Pritchard, D. & Dickinson, L. 2007 The near-shore behaviour of shallow-water waves with localized initial conditions. J. Fluid Mech. 591, 413–436.CrossRefGoogle Scholar
Reutov, V. A. 1976 Behaviour of perturbations of solitary and periodic waves on the surface of a heavy liquid. Fluid Dyn. 11, 778–781.CrossRefGoogle Scholar
Ryrie, S. C. 1983 Longshore motion generated on beaches by obliquely incident bores. J. Fluid Mech. 129, 193–212.CrossRefGoogle Scholar
Shen, M. C. & Meyer, R. E. 1963 Climb of a bore on a beach. Part 3. Run-up. J. Fluid Mech. 16, 113–125.CrossRefGoogle Scholar
Synolakis, C. E. 1987 The run-up of solitary waves. J. Fluid Mech. 185, 523–545.CrossRefGoogle Scholar
Tadepalli, S. & Synolakis, C. 1994 The run-up of N-waves on sloping beaches. Proc. R. Soc. Lond. A 445, 99–112.Google Scholar
Thacker, W. C. 1981 Some exact solutions to the nonlinear shallow-water wave equations. J. Fluid Mech. 107, 499–508.CrossRefGoogle Scholar
Titov, V. V. & Synolakis, C. E. 1998 Numerical modeling of tidal wave runup. ASCE J. Waterway Port Coastal Ocean Engng 124 (4), 157–171.CrossRefGoogle Scholar
Wei, G., Kirby, J. T., Grilli, S. T. & Subramanya, R. 1995 A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. J. Fluid Mech. 294, 71–92.CrossRefGoogle Scholar