Pure and Applied Geophysics

, Volume 172, Issue 3–4, pp 985–1002 | Cite as

Tsunami Generation Above a Sill

  • Themistoklis S. Stefanakis
  • Frédéric Dias
  • Costas Synolakis


The generation of surface waves by seafloor displacement is a classic problem that arises in the study of tsunamis. The generation of waves in a two-dimensional domain of uniform depth by uplift or subsidence of a portion of a flat bottom boundary has been elegantly studied by Hammack (Tsunamis: a model of their generation and propagation, Ph.D. thesis, California Institute of Technology, 1972), for idealized motions. The physical problem of tsunami generation is more complex; even when the final displacement is known from seismic analysis, the deforming seafloor includes relief features such as mounts and trenches. Here, following Kajiura (J Oceanogr Soc Jpn 28:260–277, 1972), we investigate analytically the effect of bathymetry on the surface wave generation, by solving the forced linear shallow water equation. While Kajiura’s geometry consisted of a step-type bottom bathymetry with a rectangular uplift to understand the effect of the continental shelf on tsunami generation, our model bathymetry consists of an uplifting cylindrical sill initially resting on a flat bottom, a geometry which helps evaluate the effect of seamounts on tsunami generation. We find that as the sill height increases, partial wave trapping reduces the wave height in the far field, while amplifying it above the sill.


Tsunamis tsunami generation 



This work was funded by EDSP of ENS-Cachan, the Cultural Service of the French Embassy in Dublin, the ERC under the research project ERC-2011-AdG 290562-MULTIWAVE, SFI under the programme ERC Starter Grant-Top Up, Grant 12/ERC/E2227 and the Strategic, and Major Initiatives scheme of University College Dublin.


  1. Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions (Dover, 1965).Google Scholar
  2. Bartholomeusz, E.F. (1958), The reflexion of long waves at a step, Math. Proc. Cambridge, 54:106–118.Google Scholar
  3. Didenkulova, I., Nikolkina, I., Pelinovsky, E., and Zahibo, N. (2010), Tsunami waves generated by submarine landslides of variable volume: analytical solutions for a basin of variable depth, Nat. Hazard Earth Sys., vol. 10, 2407–2419.Google Scholar
  4. Di Risio, M., De Girolamo, P., Bellotti, G., Panizzo, A., Aristodemo, F., Molfetta, M. G. and Petrillo, A. F. (2009) Landslide-generated tsunamis runup at the coast of a conical island: New physical model experiments, J. Geophys. Res., 114(C01009).Google Scholar
  5. Dutykh, D, Dias, F. and Kervella, Y. (2006) Linear theory of wave generation by a moving bottom, C. R. Acad. Sci. Paris, Ser. I, 343:499–504.Google Scholar
  6. Dutykh, D., Poncet, R. and Dias, F. (2011), The VOLNA code for the numerical modelling of tsunami waves: generation, propagation and inundation, Eur. J. Mech. B/Fluids, 30:598–615.Google Scholar
  7. 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. and Synolakis, C.E. (2007), Extreme runup from the 17 July 2006 Java tsunami, Geophys. Res. Lett., 34:L12602.Google Scholar
  8. Hammack, J.L. (1972), Tsunamis - A Model of Their Generation and Propagation, PhD thesis, California Institute of Technology.Google Scholar
  9. Hammack, J.L. (1973), A note on tsunamis: their generation and propagation in an ocean of uniform depth, J. Fluid Mech., 60:769–799.Google Scholar
  10. Hill, E.M., Borrero, J.C., Huang, Z., Qiu, Q., Banerjee, P., Natawidjaja, D.H., Elosegui, P., Fritz, H.M., Suwargadi, B.W., Pranantyo, I.R., Li, L., Macpherson, K.A., Skanavis, V., Synolakis, C.E. and Sieh, K. (2012), The 2010 Mw 7.8 Mentawai earthquake: Very shallow source of a rare tsunami earthquake determined from tsunami field survey and near-field GPS data, J. Geophys. Res.-Sol. Ea., 117(B6):B06402.Google Scholar
  11. Jamin, T., Gordillo, L., Ruiz-Chavarria, G., Berhanu, M., and Falcon, E. (2013), Generation of surface waves by an underwater moving bottom: Experiments and application to tsunami modeling, submitted.Google Scholar
  12. Kajiura, K. (1963), The leading wave of a tsunami, Bulletin of Earthquake Engineering Research Institute, 41:535–571.Google Scholar
  13. Kajiura, K. (1972), The directivity of energy radiation of the tsunami generated in the vicinity of a continental shelf, Journal of the Oceanographical Society of Japan, 28:260–277.Google Scholar
  14. Kanamori, H. (1972), Mechanisms of tsunami earthquakes, Physics of the Earth and Planetary Interiors, 6:346–359.Google Scholar
  15. Kanamori, H. and Kikuchi, M. (1993), The 1992 Nicaragua earthquake: a slow tsunami earthquake associated with subducted sediments, Nature, 361:714–716.Google Scholar
  16. Kânoğlu, U. and Synolakis, C.E. (1998), Long wave runup on piecewise linear topographies, J. Fluid Mech., 374:1–28.Google Scholar
  17. Kerr, R.A. (2005), Model shows islands muted tsunami after latest Indonesian quake, Science, 308(5720):341.Google Scholar
  18. Kervella, Y., Dutykh, D. and Dias, F. (2007), Comparison between three-dimensional linear and nonlinear tsunami generation models, Theor. Comput. Fluid Dyn., 21:245–269.Google Scholar
  19. Knowles, J.K. (1966), On Saint-Venant’s principle in the two-dimensional linear theory of elasticity, Arch. Rat. Mech. Anal., 21:1–22.Google Scholar
  20. Lautenbacher, C.C. (1970), Gravity wave refraction by islands, J. Fluid Mech., 41:655–672.Google Scholar
  21. Levin, B.V., Nosov, M.A. Physics of Tsunamis (Springer, 2008).Google Scholar
  22. Lin, I.-C. and Tung, C.C. (1982) A preliminary investigation of tsunami hazard, Bulletin of the Seismological Society of America, 72(6):2323–2337.Google Scholar
  23. Liu, P.L.-F., Lynett, P. and Synolakis, C.E. (2003), Analytical solutions for forced long waves on a sloping beach, J. Fluid Mech., 478:101–109.Google Scholar
  24. Longuet-Higgins, M. S. (1967), On the trapping of wave energy round islands, J. Fluid Mech., 29:781–821.Google Scholar
  25. Mei, C. C., The applied dynamics of water waves (World Scientific, 1989).Google Scholar
  26. Okada, Y. (1992), Internal deformation due to shear and tensile faults in a half-space, Bull. Seism. Soc. Am., 82:1018–1040.Google Scholar
  27. Okal, E.A. and Synolakis, C.E. (2007), Far-field tsunami hazard from mega-thrust earthquakes in the Indian Ocean, Geophysical Journal International, 172:995–1015.Google Scholar
  28. Renzi, E. and Sammarco, P. (2010), Landslide tsunamis propagating around a conical island, J. Fluid Mech., 650:251–285.Google Scholar
  29. Sammarco, P. and Renzi, E. (2008), Landslide tsunamis propagating along a plane beach, J. Fluid Mech., 598:107–119.Google Scholar
  30. Stoker, J.J. Water Waves (Wiley, 1957).Google Scholar
  31. Synolakis, C.E., Liu, P.L.-F., Carrier, G. and Yeh, H. (1997), Tsunamigenic seafloor deformations, Science, 278(5338):598–600.Google Scholar
  32. Tinti, S., Bortolucci, E. and Chiavettieri, C. (2001), Tsunami excitation by submarine slides in shallow-water approximation, Pure Appl. Geophys., 158, 759–797.Google Scholar
  33. Tuck, E. O. and Hwang, L.-S. (1972), Long wave generation on a sloping beach, J. Fluid Mech., 51:449–461.Google Scholar
  34. Zhang Y. and Zhu, S. (1994), New solutions for the propagation of long water waves over variable depth, J. Fluid Mech., 278:391–406.Google Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Themistoklis S. Stefanakis
    • 1
    • 2
  • Frédéric Dias
    • 1
    • 2
  • Costas Synolakis
    • 3
  1. 1.CMLA, ENS Cachan, CNRSCachanFrance
  2. 2.UCD School of Mathematical SciencesUniversity College DublinDublin 4Ireland
  3. 3.Viterbi School of EngineeringUniversity of Southern CaliforniaLos AngelesUSA

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