Pure and Applied Geophysics

, Volume 168, Issue 6–7, pp 1199–1222 | Cite as

Validation and Verification of a Numerical Model for Tsunami Propagation and Runup

Article

Abstract

A robust numerical model to simulate propagation and runup of tsunami waves in the framework of non-linear shallow water theory is developed. The numerical code adopts a staggered leapfrog finite-difference scheme to solve the shallow water equations formulated for depth-averaged water fluxes in spherical coordinates. A temporal position of the shoreline is calculated using a free-surface moving boundary algorithm. For large scale problems, the developed algorithm is efficiently parallelized employing a domain decomposition technique. The developed numerical model is benchmarked in an exhaustive series of tests suggested by NOAA. We conducted analytical and laboratory benchmarking for the cases of solitary wave runup on simple beaches, runup of a solitary wave on a conically-shaped island, and the runup in the Monai Valley, Okushiri Island, Japan, during the 1993 Hokkaido-Nansei-Oki tsunami. In all conducted tests the calculated numerical solution is within an accuracy recommended by NOAA standards. We summarize results of numerical benchmarking of the model, its strengths and limits with regards to reproduction of fundamental features of coastal inundation, and also illustrate some possible improvements.

Keywords

Numerical modeling tsunami inundation 

Notes

Acknowledgments

We would like to thank C.E. Synolakis, V.V. Titov, J. Stroh and others for all their valuable advice, critique and reassurances along the way. We are thankful to reviewers and the editor for valuable suggestions making the manuscript easier to read and understand. This study was supported by NOAA grants 27-014d and 06-028a through Cooperative Institute for Arctic Research. Numerical calculations for this work are supported by a grant of High Performance Computing resources from the Arctic Region Supercomputing Center at the University of Alaska Fairbanks as part of the US Department of Defense HPC Modernization Program.

References

  1. Angot, P., Bruneau, C., Fabrie, P., 1999. A penalization method to take into account obstacles in incompressible viscous flows. Nümerische Mathematik 81(4), 497–520.Google Scholar
  2. Arakawa, A., Lamb, V., 1977. Computational design of the basic dynamical processes of the UCLA general circulation model. In: Methods in Computational Physics. Vol. 17. Academic Press, pp. 174–267.Google Scholar
  3. Balay, S., Buschelman, K., Eijkhout, V., Gropp, W. D., Kaushik, D., Knepley, M. G., McInnes, L. C., Smith, B. F., Zhang, H., 2004. PETSc Users Manual. Tech. Rep. ANL-95/11 - Revision 2.1.5, Argonne National Laboratory.Google Scholar
  4. Balzano, A., 1998. Evaluation of methods for numerical simulation of wetting and drying in shallow water flow models. Coastal Engineering 34, 83–107.Google Scholar
  5. Briggs, M., Synolakis, C., Harkins, G., Green, D., 1995. Laboratory experiments of tsunami runup on a circular island. Pure and Applied Geophysics 144, 569–593.Google Scholar
  6. Buzbee, B., Dorr, F., George, J., Golub, G., 1971. The direct solution of the discrete poisson equation on irregular regions. SIAM Journal on Numerical Analysis 8, 722–736.Google Scholar
  7. Chaudhry, M., 1993. Open-Channel Flow. Prentice-Hall, 483 pp.Google Scholar
  8. Chow, V., 1959. Open Channel Hydraulics. McGraw-Hill, 680 pp.Google Scholar
  9. Courant, R., Friedrichs, K., Lewy, H., 1928. \"Uber die partiellen differenzengleichungen der mathematischen physic. Mathematische Annalen 100, 32–74.Google Scholar
  10. Dalrymple, R., Rogers, B., 2006. Numerical modeling of water waves with the SPH method. Coastal Engineering 53, 141–147.Google Scholar
  11. Dissez, A., Sous, D., Vincent, S., Caltagirone, J., Sottolichio, A., 2005. A novel implicit method for coastal hydrodynamics modeling: application to the Arcachon lagoon. Comptes Rendus Mecanique 333, 796–803.Google Scholar
  12. Fischer, G., 1959. Ein numerisches verfahren zur errechnung von windstau und gezeiten in randmeeren. Tellus 11, 60–76.Google Scholar
  13. Fletcher, C., 1991. Computational Techniques for Fluid Dynamics 1. Springer-Verlag, 401 pp.Google Scholar
  14. George, D. L., LeVeque, R. J., 2006. Finite volume methods and adaptive refinement for global tsunami propagation and inundation. Science of Tsunami Hazards 24(5), 319–328.Google Scholar
  15. Glowinski, R., Pan, T.-W., Periaux, J., 1994. Fictitious domain method for dirichlet problems and applications. Computer Methods in Applied Mechanics and Engineering 111, 283–303.Google Scholar
  16. Goring, D., 1979. Tsunamis—the propagation of long waves onto a shelf. PhD Thesis, California Institute of Technology, Pasadena, California.Google Scholar
  17. Goto, C., Ogawa, Y., Shuto, N., Imamura, F., 1997. Numerical method of tsunami simulation with the leap-frog scheme. Manuals and Guides 35, UNESCO: IUGG/IOC TIME Project.Google Scholar
  18. Gropp, W., Lusk, E., Skjellum, A., 1999. Using MPI: Portable Parallel Programming with the Message-Passing Interface. The MIT press, 406 pp.Google Scholar
  19. Hammack, J., 1972. Tsunamis—A model for their generation and propagation. Tech. Rep. KH-R-28, W.M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology.Google Scholar
  20. Hansen, 1956. Theorie zur errechnung des wasserstands und der stromungen in randemeeren. Tellus 8, 287–300.Google Scholar
  21. Imamura, F., 1995. Tsunami numerical simulation with the staggered leap-frog scheme. Tech. rep., School Disaster Control Research Center, Tohoku University, Manuscript for TUNAMI code, 33 pp.Google Scholar
  22. Imamura, F., 1996. Review of tsunami simulation with a finite difference method. In: Yeh, H., Liu, P., Synolakis, C. (Eds.), Long-Wave Runup Models. World Scientific, pp. 25–42.Google Scholar
  23. Kato, K., Tsuji, Y., 1994. Estimation of fault parameters of the 1993 Hokkaido-Nansei-Oki earthquake and tsunami characteristics. Bulletin of the Earthquake Research Institute 69, 39–66, University of Tokyo.Google Scholar
  24. Kennedy, A., Chen, Q., Kirby, J., Dalrymple, R., 2000. Boussinesq modeling of wave transformation, breaking, and runup, Part I:1D. Journal of Waterway, Port, Coastal and Ocean Engineering 126(1), 39–47.Google Scholar
  25. Khadra, K., Parneix, S., Angot, P., Caltagirone, J., 2000. Fictitious domain approach for numerical modelling of Navier-Stokes equations. International Journal for Numerical Methods in Fluids 341, 651–684.Google Scholar
  26. Kirby, J., Wei, G., Chen, Q., Kennedy, A., Dalrymple, R., 1998. FUNWAVE 1.0, fully nonlinear boussinesq wave model documentation and users manual. Tech. Rep. Research Report No. CACR-98-06, Center for Applied Coastal Research, University of Delaware.Google Scholar
  27. Kowalik, Z., Murty, T., 1993a. Numerical modeling of ocean dynamics. World Scientific, 481 pp.Google Scholar
  28. Kowalik, Z., Murty, T., 1993b. Numerical simulation of two-dimensional tsunami runup. Marine Geodesy 16, 87–100.Google Scholar
  29. Linsley, R., Franzini, J., 1979. Water Resources Engineering. McGraw-Hill, New York, 716 pp.Google Scholar
  30. Liu, P.-F., Cho, Y.-S., Briggs, M., Kanoğlu, U., Synolakis, C., 1995. Runup of solitary waves on a circular island. Journal of Fluid Mechanics 302, 259–285.Google Scholar
  31. Liu, P.-F., Woo, S., Cho, Y., 1998. Computer programs for tsunami propagation and inundation. Tech. rep., Cornell University, 104 pp.Google Scholar
  32. Liu, P. L.-F., Synolakis, C., Yeh, H., 1991. Report on the international workshop on long-wave runup. Journal of Fluid Mechanics 229, 675–688.Google Scholar
  33. Liu, P. L.-F., Yeh, H., Synolakis, C., 2007. Advanced Numerical Models for Simulationg Tsunami Waves and Runup. Vol. 10 of Advances in Coastal and Ocean Engineering. World Scientific, Proceedings of the Third International Workshop on Long-Wave Runup Models, Catalina, 2004 Benchmark problems, pp. 223–230.Google Scholar
  34. Lynett, P., Borrero, J., Liu, P.-F., Synolakis, C., 2003. Field survey and numerical simulations: a review of the 1998 Papua New Guinea earthquake and tsunami. Pure and Applied Geophysics 160, 2119–2146.Google Scholar
  35. Lynett, P., Wu, T.-R., Liu, P.-F., 2002. Modeling wave runup with depth-integrated equations. Coastal Engineering 46(2), 89–107.Google Scholar
  36. Mader, C., Lukas, S., 1984. SWAN-A Shallow Water, Long Wave Code. Tech. Rep. HIG-84-4, Hawaii Institute of Geophysics, University of Hawaii.Google Scholar
  37. Marchuk, G. I., Kuznetsov, Y. A., Matsokin, A. M., 1986. Fictitious domain and domain decomposition methods. Soviet Journal of Numerical Analysis and Mathematical Modelling 1(1), 3–35.Google Scholar
  38. Hokkaido Tsunami Survey Group, 1993. Tsunami devastates Japanese coastal regions. EOS, Transactions AGU 74 (37), 417–432.Google Scholar
  39. Meyer, R., Taylor, A., 1972. Waves on beaches and resulting sediment transport. Academic Press, Ch. Run-up on beaches, pp. 357–411.Google Scholar
  40. Okada, Y., 1985. Surface deformation due to shear and tensile faults in a half-space. Bulletin of the Seismological Society of America 75, 1135–1154.Google Scholar
  41. Paglieri, L., Ambrosi, D., Formaggia, L., Quarteroni, A., Scheinine, A., 1997. Parallel computation for shallow water flow: A domain decomposition approach. Parallel Computing 23, 1261–1277.Google Scholar
  42. Peregrine, D., 1967. Long waves on a beach. Journal of Fluid Mechanics 27(4), 815–827.Google Scholar
  43. Peregrine, D., Williams, S., 2001. Swash overtopping a truncated plane beach. Journal of Fluid Mechanics 440, 391–399.Google Scholar
  44. Shen, M., Meyer, R., 1963. Climb of a bore on a beach. Part 3. Run-up. Journal of Fluid Mechanics 16, 113–125.Google Scholar
  45. Shuto, N., 1991. Tsunami Hazard. Kluwer Academic Publishers, Netherlands, Ch. Numerical simulation of tsunamis, pp. 171–191.Google Scholar
  46. Svendsen, I., 2005. Introduction to Nearshore Hydrodynamics. World Scientific Publishing Company, 722pp.Google Scholar
  47. Synolakis, C., 1986. The Runup of Long Waves. Ph.D. thesis, California Institute of Technology, Pasadena, California, 228 pp.Google Scholar
  48. Synolakis, C., 1987. The runup of solitary waves. Journal of Fluid Mechanics 185, 523–545.Google Scholar
  49. Synolakis, C., Bernard, E., 2006. Tsunami science before and beyond Boxing Day 2004. Philosophical Transactions of the Royal Society A 364, 2231–2265.Google Scholar
  50. Synolakis, C., Bernard, E., Titov, V., K\^anoğlu, U., González, F., 2007. Standards, criteria, and procedures for NOAA evaluation of tsunami numerical models. OAR PMEL-135 Special Report, NOAA/OAR/PMEL, Seattle, Washington,, 55 pp.Google Scholar
  51. Synolakis, C., Bernard, E., Titov, V., K\^anoğlu, U., González, F., 2008. Validation and verification of tsunami numerical models. Pure and Applied Geophysics 165, 2197–2228.Google Scholar
  52. Takahashi, T., Takahashi, T., Shuto, N., Imamura, F., Ortiz, M., 1995. Source models for the 1993 Hokkaido-Nansei-Oki earthquake tsunami. Pure and Applied Geophysics 144, 747–768.Google Scholar
  53. Titov, V., Synolakis, C., 1995. Evolution and runup of breaking and nonbreaking waves using VTSC-2. Journal of Waterway, Port, Coastal and Ocean Engineering 121(6), 308–316.Google Scholar
  54. Titov, V., Synolakis, C., 1998. Numerical modeling of tidal wave runup. Journal of Waterway, Port, Coastal and Ocean Engineering 124, 157–171.Google Scholar
  55. Yeh, H., 1991. Tsunami bore runup. Natural Hazards 4, 209–220.Google Scholar
  56. Yeh, H., Liu, P.-F., Synolakis, C., 1996. Long-Wave Runup Models. World Scientific, 403 pp.Google Scholar
  57. Yeh, H., Liu, P. L.-F., Briggs, M., Synolakis, C. E., 1994. Propagation and amplification of tsunamis at coastal boundaries. Nature 372, 353–355.Google Scholar
  58. Zelt, J., 1991. The runup of breaking and nonbreaking solitary waves. Coastal Engineering 125, 205–246.Google Scholar
  59. Zhang, Y., Baptista, A., 2008. An efficient and robust tsunami model on unstructured grids. Part I: Inundation benchmarks. Pure and Applied Geophysics 165, 2229–2248.Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • D. J. Nicolsky
    • 1
  • E. N. Suleimani
    • 1
  • R. A. Hansen
    • 1
  1. 1.University of Alaska FairbanksFairbanksUSA

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