Skip to main content
SpringerLink
Log in

High-order finite difference modeling of tsunami generation in a compressible ocean from offshore earthquakes

  • ORIGINAL PAPER
  • Published:
Computational Geosciences Aims and scope Submit manuscript

Abstract

To study the full seismic, ocean acoustic, and tsunami wavefields generated by subduction zone earthquakes, we have developed a provably stable and accurate finite difference method that couples an elastic solid to a compressible fluid subject to gravitational restoring forces. We introduce a linearized dynamic traction-free boundary condition for the moving sea surface that is valid for small amplitude perturbations about an ocean initially in hydrostatic balance. We derive an energy balance for the continuous problem and then use high-order summation-by-parts finite difference operators and weak enforcement of boundary conditions to derive an equivalent discrete energy balance. The discrete energy balance is used to prove stability of the numerical scheme, and stability and accuracy are verified through convergence tests. The method is applied to study tsunami generation by time-dependent rupture on a thrust fault in an elastic solid beneath a compressible ocean. We compare the sea surface evolution in our model to that predicted by the standard tsunami modeling procedure, which assumes seafloor uplift occurs instantaneously and neglects compressibility of the ocean. We find that the leading shoreward-traveling tsunami wave in our model has a noticeably smaller amplitude than that predicted by the standard approach.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Carpenter, M.H., Gottlieb, D., Abarbanel, S.: Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes. J. Comput. Phys. 111(2), 220–236 (1994)

    Article  Google Scholar 

  2. Carpenter, M.H., Kennedy, C.A.: Fourth-order 2N-storage Runge-Kutta schemes. NASA TM 109112 (1994)

  3. Gustafsson, B.: The convergence rate for difference approximations to mixed initial boundary value problems. Math. Comput. 29(130), 396–406 (1975)

    Article  Google Scholar 

  4. Gustafsson, B., Kreiss, H.O., Oliger, J.: Time-dependent problems and difference methods, vol. 121. Wiley, New York (2013)

    Book  Google Scholar 

  5. Kajiura, K.: The leading wave of a tsunami. Bull. Earthq. Res. Inst. 43, 535–571 (1963)

    Google Scholar 

  6. Kozdon, J.E., Dunham, E.M.: Rupture to the trench: dynamic rupture simulations of the 11 March 2011 Tohoku earthquake. Bull. Seism. Soc. Am. 103(2B), 1275–1289 (2013)

    Article  Google Scholar 

  7. Kozdon, J.E., Dunham, E.M.: Constraining shallow slip and tsunami excitation in megathrust ruptures using seismic and ocean acoustic waves recorded on ocean-bottom sensor networks. Earth Planet. Sci. Lett. 396, 56–65 (2014)

    Article  Google Scholar 

  8. Kozdon, J.E., Dunham, E.M., Nordström, J.: Interaction of waves with frictional interfaces using summation-by-parts difference operators: weak enforcement of nonlinear boundary conditions. J. Sci. Comput. 50(2), 341–367 (2012)

    Article  Google Scholar 

  9. Kozdon, J.E., Dunham, E.M., Nordström, J.: Simulation of dynamic earthquake ruptures in complex geometries using high-order finite difference methods. J. Sci. Comput. 55(1), 92–124 (2013)

    Article  Google Scholar 

  10. Kreiss, H.O.: Initial boundary value problems for hyperbolic systems. Commun. Pure Appl. Math. 23(3), 277–298 (1970)

    Article  Google Scholar 

  11. Kreiss, H.O., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: Mathematical aspects of finite elements in partial differential equations, pp. 195–212. Academic Press (1974)

  12. Kundu, P., Cohen, I., Dowling, D.: Fluid mechanics. Academic Press, New York (2012)

    Google Scholar 

  13. Maeda, T., Furumura, T.: FDM simulation of seismic waves, ocean acoustic waves, and tsunamis based on tsunami-coupled equations of motion. Pure Appl. Geophys. 170(1-2), 109–127 (2013)

    Article  Google Scholar 

  14. Maeda, T., Furumura, T., Noguchi, S., Takemura, S., Sakai, S., Shinohara, M., Iwai, K., Lee, S.J.: Seismic and tsunami wave propagation of the 2011 off the pacific coast of Tohoku earthquake as inferred from the tsunami-coupled finite difference simulation. Bull. Seism. Soc. Am. 103(2B), 1456–1472 (2013)

    Article  Google Scholar 

  15. Maeda, T., Furumura, T., Sakai, S., Shinohara, M.: Significant tsunami observed at ocean-bottom pressure gauges during the 2011 off the Pacific coast of Tohoku earthquake. Earth Planets Space 63, 803–808 (2011)

    Article  Google Scholar 

  16. Marano, K.D., Wald, D.J., Allen, T.I.: Global earthquake casualties due to secondary effects: a quantitative analysis for improving rapid loss analyses. Nat. Hazards 52(2), 319–328 (2010)

    Article  Google Scholar 

  17. Matsumoto, H., Inoue, S., Ohmachi, T.: Some features of water pressure change during the 2011 Tohoku earthquake. Proceedings of the International Symposium on Engineering Lessons Learned from the 2011 Great East Japan Earthquake (2012)

  18. Mori, N., Takahashi, T.: The 2011 Tohoku earthquake tsunami joint survey group: nationwide post event survey and analysis of the 2011 Tohoku earthquake tsunami, vol. 54 (2012)

  19. Saito, T.: Dynamic tsunami generation due to sea-bottom deformation: analytical representation based on linear potential theory. Earth Planets Space 65, 1411–1423 (2013)

    Article  Google Scholar 

  20. Segall, P.: Earthquake and volcano deformation. Princeton University Press, Princeton (2010)

    Book  Google Scholar 

  21. Sells, C.C.L.: The effect of a sudden change of shape of the bottom of a slightly compressible ocean. Philos. Trans. R. Soc. Lond. Ser. A, Math. Phys. Sci. 258(1092), 495–528 (1965)

    Article  Google Scholar 

  22. Strand, B.: Summation by parts for finite difference approximations for d/dx. J. Comp. Phys. 110(1), 47–67 (1994)

    Article  Google Scholar 

  23. Trugman, D.T., Dunham, E.M.: A 2d pseudodynamic rupture model generator for earthquakes on geometrically complex faults. Bull. Seism. Soc. Am. 104(1), 95–112 (2014)

    Article  Google Scholar 

  24. Yamamoto, T.: Gravity waves and acoustic waves generated by submarine earthquakes. Int. J. Soil Dyn. Earthq. Eng. 1(2), 75–82 (1982)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Geophysics, Stanford University, 397 Panama Mall, Stanford, CA, 94305, USA

    Gabriel C. Lotto

  2. Department of Geophysics and Institute for Computational and Mathematical Engineering, Stanford University, 397 Panama Mall, Stanford, CA, 94305, USA

    Eric M. Dunham

Authors
  1. Gabriel C. Lotto
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Eric M. Dunham
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gabriel C. Lotto.

Additional information

This work was supported by the National Science Foundation (EAR-1255439) and the Alfred P. Sloan Foundation (BR2012-097).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lotto, G.C., Dunham, E.M. High-order finite difference modeling of tsunami generation in a compressible ocean from offshore earthquakes. Comput Geosci 19, 327–340 (2015). https://doi.org/10.1007/s10596-015-9472-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10596-015-9472-0

Keywords

Search

Navigation