Theoretical and Computational Fluid Dynamics

, Volume 27, Issue 1–2, pp 177–199 | Cite as

On the use of the finite fault solution for tsunami generation problems

  • Denys DutykhEmail author
  • Dimitrios Mitsotakis
  • Xavier Gardeil
  • Frédéric Dias
Original Article


The present study is devoted to the problem of tsunami wave generation. The main goal of this work is twofold. First of all, we propose a simple and computationally inexpensive model for the description of the sea bed displacement during an underwater earthquake, based on the finite fault solution for the slip distribution under some assumptions on the dynamics of the rupturing process. Once the bottom motion is reconstructed, we study waves induced on the free surface of the ocean. For this purpose, we consider three different models approximating the Euler equations of the water wave theory. Namely, we use the linearized Euler equations (we are in fact solving the Cauchy–Poisson problem), a Boussinesq system, and a novel weakly nonlinear model. An intercomparison of these approaches is performed. The developments of the present study are illustrated on the July 17, 2006 Java event, where an underwater earthquake of magnitude 7.7 generated a tsunami that inundated the southern coast of Java.


Water waves Tsunami waves Co-seismic displacements Moving bottom Tsunami generation 


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  1. 1.
    Ammon C.J., Kanamori H., Lay T., Velasco A.A.: The 17 July 2006 Java tsunami earthquake. Geophys. Res. Lett. 33, L24308 (2006)CrossRefGoogle Scholar
  2. 2.
    Basher R.: Global early warning systems for natural hazards: systematic and people-centred. Philos. Trans. R. Soc. A 364(1845), 2167–2182 (2006)CrossRefGoogle Scholar
  3. 3.
    Bona J.L., Chen M., Saut J.-C.: Boussinesq equations and other systems for small-amplitude long waves in nonlinear dispersive media. I: derivation and linear theory. J. Nonlinear Sci. 12, 283–318 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Bassin C., Laske G., Masters G.: The current limits of resolution for surface wave tomography in North America. EOS Trans. AGU 81, F897 (2000)Google Scholar
  5. 5.
    Boussinesq J.: Théorie des ondes et des remous qui se propagent le long d’un canal rectangulaire horizontal, en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. 17, 55–108 (1872)Google Scholar
  6. 6.
    Cauchy A.-L.: Mémoire sur la théorie de la propagation des ondes à à la surface d’un fluide pesant d’une profondeur indéfinie. Mém. Présentés Divers Savans Acad. R. Sci. Inst. France 1, 3–312 (1827)Google Scholar
  7. 7.
    Chazel F.: On the Korteweg-de Vries approximation for uneven bottoms. Eur. J. Mech. B/Fluids 28(2), 234–252 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Coifman R.R., Meyer Y.: Nonlinear harmonic analysis and analytic dependence. Proc. symp. Pure Math. 43, 71–78 (1985)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Craig W., Sulem C.: Numerical simulation of gravity waves. J. Comput. Phys. 108, 73–83 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Craig W., Sulem C., Sulem P.-L.: Nonlinear modulation of gravity waves: a rigorous approach. Nonlinearity 5(2), 497–522 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Cooley J.W., Tukey J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dias F., Bridges T.J.: The numerical computation of freely propagating time-dependent irrotational water waves. Fluid Dyn. Res. 38, 803–830 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Dias F., Dutykh D.: Dynamics of tsunami waves. In: Ibrahimbegovic, A., Kozar, I. (eds) Extreme Man-Made and Natural Hazards in Dynamics of Structures, Springer, Berlin (2007)Google Scholar
  14. 14.
    Dutykh D., Dias F.: Dissipative Boussinesq equations. C. R. Mecanique 335, 559–583 (2007)zbMATHCrossRefGoogle Scholar
  15. 15.
    Dutykh D., Dias F.: Viscous potential free-surface flows in a fluid layer of finite depth. C. R. Acad. Sci. Paris Ser. I 345, 113–118 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Dutykh D., Dias F.: Water waves generated by a moving bottom. In: Anjan, K. (eds) Tsunami and Nonlinear Waves, Springer (Geo Sc.), Heidelberg (2007)Google Scholar
  17. 17.
    Dutykh D., Dias F.: Tsunami generation by dynamic displacement of sea bed due to dip-slip faulting. Math. Comput. Simul. 80(4), 837–848 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Dutykh D., Dias F.: Influence of sedimentary layering on tsunami generation. Comput. Appl. Mech. Eng. 199(21–22), 1268–1275 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Dutykh D., Dias F., Kervella Y.: Linear theory of wave generation by a moving bottom. C. R. Acad. Sci. Paris Ser. I 343, 499–504 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Dias F., Dyachenko A.I., Zakharov V.E.: Theory of weakly damped free-surface flows: a new formulation based on potential flow solutions. Phys. Lett. A 372, 1297–1302 (2008)zbMATHCrossRefGoogle Scholar
  21. 21.
    Delis A.I., Kazolea M., Kampanis N.A.: A robust high-resolution finite volume scheme for the simulation of long waves over complex domains. Int. J. Numer. Method Fluids 56, 419–452 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Dougalis V.A., Mitsotakis D.E., Saut J.-C.: On some Boussinesq systems in two space dimensions: theory and numerical analysis. Math. Model. Numer. Anal. 41(5), 254–825 (2007)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Dougalis V.A., Mitsotakis D.E., Saut J.-C.: On initial-boundary value problems for a Boussinesq system of BBM-BBM type in a plane domain. Discret. Cont. Dyn. Syst. 23(4), 1191–1200 (2009)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Dougalis V.A., Mitsotakis D.E., Saut J.-C.: Initial-boundary-value problems for Boussinesq systems of Bona-Smith type on a plane domain: theory and numerical analysis. J. Sci. Comput. 44, 109–135 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Dormand J.R., Prince P.J.: A family of embedded Runge–Kutta formulae. J. Comput. Appl. Math. 6, 19–26 (1980)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Dutykh D., Poncet R., Dias F.: The VOLNA code for the numerical modeling of tsunami waves: generation, propagation and inundation. Eur. J. Mech. B/Fluids 30(6), 598–615 (2011)MathSciNetCrossRefGoogle Scholar
  27. 27.
    de Saint-Venant A.J.C.: Théorie du mouvement non-permanent des eaux, avec application aux crues des rivières et à à l’introduction des marées dans leur lit. C. R. Acad. Sci. Paris 73, 147–154 (1871)zbMATHGoogle Scholar
  28. 28.
    Dutykh, D.: Mathematical Modelling of Tsunami Waves. PhD thesis, École Normale Supérieure de Cachan (2007)Google Scholar
  29. 29.
    Dutykh D.: Group and phase velocities in the free-surface visco-potential flow: new kind of boundary layer induced instability. Phys. Lett. A 373, 3212–3216 (2009)zbMATHCrossRefGoogle Scholar
  30. 30.
    Dutykh D.: Visco-potential free-surface flows and long wave modelling. Eur. J. Mech. B/Fluids 28, 430–443 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  31. 31.
    Fructus D., Clamond D., Kristiansen O., Grue J.: An efficient model for three dimensional surface wave simulations. Part I: free space problems. J. Comput. Phys. 205, 665–685 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Frigo, M., Johnson, S.G.: The design and implementation of FFTW3. Proc. IEEE 93(2), 216–231 (2005). Special issue on “Program Generation, Optimization, and Platform Adaptation”Google Scholar
  33. 33.
    Fritz H.M., Kongko W., Moore A., McAdoo B., Goff J., Harbitz C., Uslu B., Kalligeris N., Suteja D., Kalsum K., Titov V.V., Gusman A., Latief H., Santoso E., Sujoko S., Djulkarnaen D., Sunendar H., Synolakis C.: Extreme runup from the 17 July 2006 Java tsunami. Geophys. Res. Lett. 34, L12602 (2007)CrossRefGoogle Scholar
  34. 34.
    González F.I., Bernard E.N., Meinig C., Eble M.C., Mofjeld H.O., Stalin S.: The NTHMP tsunameter network. Nat. Hazards 35, 25–39 (2005)CrossRefGoogle Scholar
  35. 35.
    Guyenne P., Nicholls D.P.: A high-order spectral method for nonlinear water waves over moving bottom topography. SIAM J. Sci. Comput. 30(1), 81–101 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  36. 36.
    Hammack, J.L.: Tsunamis—A Model of Their Generation and Propagation. PhD thesis, California Institute of Technology (1972)Google Scholar
  37. 37.
    Hammack J.: A note on tsunamis: their generation and propagation in an ocean of uniform depth. J. Fluid Mech. 60, 769–799 (1973)zbMATHCrossRefGoogle Scholar
  38. 38.
    Hairer E., Nørsett S.P., Wanner G.: Solving Ordinary Differential Equations: Nonstiff Problems. Springer, Berlin (2009)Google Scholar
  39. 39.
    Ioualalen M., Asavanant J., Kaewbanjak N., Grilli S.T., Kirby J.T., Watts P.: Modeling the 26 december 2004 Indian Ocean tsunami: case study of impact in Thailand. J. Geophys. Res. 112, C07024 (2007)CrossRefGoogle Scholar
  40. 40.
    Imamura, F.: Long-wave Runup Models. Chapter Simulation of Wave-Packet Propagation Along Sloping Beach by TUNAMI-code, pp. 231–241. World Scientific (1996)Google Scholar
  41. 41.
    Imamura, F., Yalciner, A.C., Ozyurt, G.: Tsunami Modelling Manual (2006)Google Scholar
  42. 42.
    Ji, C.: Preliminary result of the 2006 July 17 magnitude 7.7—south of Java, Indonesia earthquake. Technical Report. (2006)
  43. 43.
    Ji C., Wald D.J., Helmberger D.V.: Source description of the 1999 Hector Mine, California earthquake; Part I: wavelet domain inversion theory and resolution analysis. Bull. Seismol. Soc. Am. 92(4), 1192–1207 (2002)CrossRefGoogle Scholar
  44. 44.
    Kajiura K.: The leading wave of tsunami. Bull. Earthq. Res. Inst. Tokyo Univ. 41, 535–571 (1963)Google Scholar
  45. 45.
    Kervella Y., Dutykh D., Dias F.: Comparison between three-dimensional linear and nonlinear tsunami generation models. Theor. Comput. Fluid Dyn. 21, 245–269 (2007)zbMATHCrossRefGoogle Scholar
  46. 46.
    Lamb H.: Hydrodynamics. Cambridge University Press, Cambridge (1932)zbMATHGoogle Scholar
  47. 47.
    Love A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover Publications, New York (1944)zbMATHGoogle Scholar
  48. 48.
    Madsen P.A., Bingham H.B., Schaffer H.A.: Boussinesq-type formulations for fully nonlinear and extremely dispersive water waves: derivation and analysis. Proc. R. Soc. Lond. A 459, 1075–1104 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Mei C.C.: The Applied Dynamics of Ocean Surface Waves. World Scientific, Singapore (1994)Google Scholar
  50. 50.
    Mindlin R.D.: Force at a point in the interior of a semi-infinite medium. Physics 7, 195–202 (1936)zbMATHCrossRefGoogle Scholar
  51. 51.
    Mitsotakis D.E.: Boussinesq systems in two space dimensions over a variable bottom for the generation and propagation of tsunami waves. Math. Comput. Simul. 80, 860–873 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Mansinha L., Smylie D.E.: Effect of earthquakes on the Chandler wobble and the secular polar shift. J. Geophys. Res. 72, 4731–4743 (1967)CrossRefGoogle Scholar
  53. 53.
    Mansinha L., Smylie D.E.: The displacement fields of inclined faults. Bull. Seismol. Soc. Am. 61, 1433–1440 (1971)Google Scholar
  54. 54.
    Ozgun Konca, A.: Preliminary result 06/07/17 (Mw 7.9) , Southern Java earthquake. Technical Report. (2006)
  55. 55.
    Okada Y.: Surface deformation due to shear and tensile faults in a half-space. Bull. Seismol. Soc. Am. 75, 1135–1154 (1985)Google Scholar
  56. 56.
    Okada Y.: Internal deformation due to shear and tensile faults in a half-space. Bull. Seismol. Soc. Am. 82, 1018–1040 (1992)Google Scholar
  57. 57.
    Ohmachi T., Tsukiyama H., Matsumoto H.: Simulation of tsunami induced by dynamic displacement of seabed due to seismic faulting. Bull. Seismol. Soc. Am. 91, 1898–1909 (2001)CrossRefGoogle Scholar
  58. 58.
    Peregrine D.H.: Long waves on a beach. J. Fluid Mech. 27, 815–827 (1967)zbMATHCrossRefGoogle Scholar
  59. 59.
    Press F.: Displacements, strains and tilts at tele-seismic distances. J. Geophys. Res. 70, 2395–2412 (1965)CrossRefGoogle Scholar
  60. 60.
    Rabinovich A.B., Lobkovsky L.I., Fine I.V., Thomson R.E., Ivelskaya T.N., Kulikov E.A.: Near-source observations and modeling of the Kuril Islands tsunamis of 15 November 2006 and 13 January 2007. Adv. Geosci. 14, 105–116 (2008)CrossRefGoogle Scholar
  61. 61.
    Synolakis C.E., Bernard E.N.: Tsunami science before and beyond Boxing Day 2004. Philos. Trans. R. Soc. A 364, 2231–2265 (2006)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Synolakis, C.E., Bernard, E.N., Titov, V.V., Kanoglu, U., Gonzalez, F.I.: Standards, Criteria, and Procedures for NOAA Evaluation of Tsunami Numerical Models. Technical Report, NOAA/Pacific Marine Environmental Laboratory (2007)Google Scholar
  63. 63.
    Saito T., Furumura T.: Three-dimensional tsunami generation simulation due to sea-bottom deformation and its interpretation based on the linear theory. Geophys. J. Int. 178, 877–888 (2009)CrossRefGoogle Scholar
  64. 64.
    Sokolnikoff I.S., Specht R.D.: Mathematical Theory of Elasticity. McGraw-Hill, New York (1946)Google Scholar
  65. 65.
    Stoker J.J.: Water Waves, the Mathematical Theory with Applications. Wiley, New York (1958)Google Scholar
  66. 66.
    Synolakis C.: India must cooperate on tsunami warning system. Nature 434, 17–18 (2005)CrossRefGoogle Scholar
  67. 67.
    Tanaka M.: The stability of solitary waves. Phys. Fluids 29(3), 650–655 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Tkalich P., Dao M.H., Soon C.E.: Tsunami propagation modeling and forecasting for early warning system. J. Earthq. Tsunami 1(1), 87–98 (2007)CrossRefGoogle Scholar
  69. 69.
    Titov, V.V., González, F.I.: Implementation and Testing of the Method of Splitting Tsunami (MOST) Model. Technical Report ERL PMEL-112, Pacific Marine Environmental Laboratory, NOAA (1997)Google Scholar
  70. 70.
    Titov V.V., Gonzalez F.I., Bernard E.N., Eble M.C., Mofjeld H.O., Newman J.C., Venturato A.J.: Real-time tsunami forecasting: challenges and solutions. Nat. Hazards 35, 41–58 (2005)CrossRefGoogle Scholar
  71. 71.
    Todorovska M.I., Hayir A., Trifunac M.D.: A note on tsunami amplitudes above submarine slides and slumps. Soil Dyn. Earthq. Eng. 22, 129–141 (2002)CrossRefGoogle Scholar
  72. 72.
    Titov V.V., Synolakis C.E.: Numerical modeling of tidal wave runup. J. Waterw Port Coast. Ocean Eng. 124, 157–171 (1998)CrossRefGoogle Scholar
  73. 73.
    Todorovska M.I., Trifunac M.D.: Generation of tsunamis by a slowly spreading uplift of the seafloor. Soil Dyna. Earthq. Eng. 21, 151–167 (2001)CrossRefGoogle Scholar
  74. 74.
    Volterra V.: Sur l’équilibre des corps élastiques multiplement connexes. Ann. Scientifiques de l’Ecole Normale Supérieure 24(3), 401–517 (1907)MathSciNetzbMATHGoogle Scholar
  75. 75.
    Whitham G.B.: Linear and Nonlinear Waves. Wiley, New York (1999)zbMATHCrossRefGoogle Scholar
  76. 76.
    Weinstein S.A., Lundgren P.R.: Finite fault modeling in a tsunami warning center context. Pure Appl. Geophys. 165, 451–474 (2008)CrossRefGoogle Scholar
  77. 77.
    Xu L., Guyenne P.: Numerical simulation of three-dimensional nonlinear water waves. J. Comput. Phys. 228(22), 8446–8466 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  78. 78.
    Yalciner, A.C.: July 17, 2006 Indonesia Java tsunami. Technical Report. (2008)
  79. 79.
    Zakharov V.E.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 9, 1990–1994 (1968)Google Scholar

Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Denys Dutykh
    • 1
    Email author
  • Dimitrios Mitsotakis
    • 2
  • Xavier Gardeil
    • 1
  • Frédéric Dias
    • 3
  1. 1.LAMA, UMR 5127 CNRS, Université de SavoieLe Bourget-du-Lac CedexFrance
  2. 2.IMA, University of MinnesotaMinneapolisUSA
  3. 3.School of Mathematical SciencesUniversity College DublinDublin 4Ireland

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