Pure and Applied Geophysics

, Volume 175, Issue 4, pp 1341–1354 | Cite as

Effect of Dynamical Phase on the Resonant Interaction Among Tsunami Edge Wave Modes

  • Eric L. Geist


Different modes of tsunami edge waves can interact through nonlinear resonance. During this process, edge waves that have very small initial amplitude can grow to be as large or larger than the initially dominant edge wave modes. In this study, the effects of dynamical phase are established for a single triad of edge waves that participate in resonant interactions. In previous studies, Jacobi elliptic functions were used to describe the slow variation in amplitude associated with the interaction. This analytical approach assumes that one of the edge waves in the triad has zero initial amplitude and that the combined phase of the three waves φ = θ1 + θ2 − θ3 is constant at the value for maximum energy exchange (φ = 0). To obtain a more general solution, dynamical phase effects and non-zero initial amplitudes for all three waves are incorporated using numerical methods for the governing differential equations. Results were obtained using initial conditions calculated from a subduction zone, inter-plate thrust fault geometry and a stochastic earthquake slip model. The effect of dynamical phase is most apparent when the initial amplitudes and frequencies of the three waves are within an order of magnitude. In this case, non-zero initial phase results in a marked decrease in energy exchange and a slight decrease in the period of the interaction. When there are large differences in frequency and/or initial amplitude, dynamical phase has less of an effect and typically one wave of the triad has very little energy exchange with the other two waves. Results from this study help elucidate under what conditions edge waves might be implicated in late, large-amplitude arrivals.


Tsunamis edge waves nonlinear resonance earthquake stochastic slip 



The author very much appreciates the constructive comments received on the manuscript by Efim Pelinovsky, Kenny Ryan, and an anonymous reviewer, as well as the thoughtful insights of Editor Alexander Rabinovich.


  1. Abe, K., & Ishii, H. (1987). Distribution of maximum water levels due to the Japan Sea tsunami on 26 May 1983. Journal of the Oceanographical Society of Japan, 43, 169–182.CrossRefGoogle Scholar
  2. Alber, M. S., Luther, G. G., Marsden, J. E., & Robbins, J. M. (1998). Geometric phases, reduction and Lie–Poisson structure for the resonant three-wave interaction. Physica D: Nonlinear Phenomena, 123, 271–290. Scholar
  3. Andrews, D. J. (1980). A stochastic fault model 1. Static case. Journal of Geophysical Research, 85, 3867–3877.CrossRefGoogle Scholar
  4. Armstrong, J. A., Bloembergen, N., Ducuing, J., & Pershan, P. S. (1962). Interactions between light waves in a nonlinear dielectric. Physical Review, 127, 1918–1939.CrossRefGoogle Scholar
  5. Benjamin, L. R., Flament, P., Cheung, K. F., & Luther, D. S. (2016). The 2011 Tohoku tsunami south of Oahu: High-frequency Doppler radio observations and model simulations of currents. Journal of Geophysical Research: Oceans, 121, 1133–1144. Scholar
  6. Bretherton, F. P. (1964). Resonant interactions between waves. The case of discrete oscillations. Journal of Fluid Mechanics, 20, 457–479.CrossRefGoogle Scholar
  7. Bricker J. D., Munger S., Pequignet C., Wells J. R., Pawlak G., & Cheung K. F. (2007). ADCP observations of edge waves off Oahu in the wake of the November 2006 Kuril Islands tsunami. Geophysical Research Letters, 34.
  8. Bustamante M. D., & Kartashova E. (2009a). Dynamics of nonlinear resonances in Hamiltonian systems. Europhysics Letters, 85.
  9. Bustamante M. D., & Kartashova E. (2009b). Effect of the dynamical phases on the nonlinear amplitudes’ evolution. Europhysics Letters, 85.
  10. Carrier, G. F. (1995). On-shelf tsunami generation and coastal propagation. In Y. Tsuchiya & N. Shuto (Eds.), Tsunami: Progress in prediction, disaster prevention and warning (pp. 1–20). Dordrecht: Kluwer.Google Scholar
  11. Catalán, P. A., Aránguiz, R., González, G., Tomita, T., Cienfuegos, R., González, J., et al. (2015). The 1 April 2014 Pisagua tsunami: Observations and modeling. Geophysical Research Letters, 42, 2918–2925. Scholar
  12. Cortés, P., Catalán, P. A., Aránguiz, R., & Bellotti, G. (2017). Tsunami and shelf resonance on the northern Chile coast. Journal of Geophysical Research: Oceans, 122, 7364–7379. Scholar
  13. Cree, W. C., & Swaters, G. E. (1991). On the topographic dephasing and amplitude modulation of nonlinear Rossby wave interactions. Geophysical and Astrophysical Fluid Dynamics, 61, 75–99. Scholar
  14. Dubinina, V. A., Kurkin, A. A., Pelinovsky, E. N., & Poloukhina, O. E. (2006). Resonance three-wave interactions of Stokes edge waves. Izvestiya, Atmospheric and Oceanic Physics, 42, 254–261.CrossRefGoogle Scholar
  15. Dubinina, V. A., Kurkin, A. A., & Polukhina, O. E. (2008). On the nonlinear interactions in triads of edge waves on the sea shelf. Physical Oceanography, 18, 117–132.CrossRefGoogle Scholar
  16. Fujima, K., Dozono, R., & Shigemura, T. (2000). Generation and propagation of tsunami accompanying edge waves on a uniform shelf. Coastal Engineering Journal, 42, 211–236.CrossRefGoogle Scholar
  17. Geist, E. L. (2009). Phenomenology of tsunamis: Statistical properties from generation to runup. Advances in Geophysics, 51, 107–169.CrossRefGoogle Scholar
  18. Geist, E. L. (2012). Near-field tsunami edge waves and complex earthquake rupture. Pure and Applied Geophysics. Scholar
  19. Geist, E. L. (2016). Non-linear resonant coupling of tsunami edge waves using stochastic earthquake source models. Geophysical Journal International, 204, 878–891. Scholar
  20. Golovachev, E. V., Kochergin, I. E., & Pelinovsky, E. N. (1992). The effect of the Airy phase during propagation of edge waves. Soviet Journal of Physical Oceanography, 3, 1–7.CrossRefGoogle Scholar
  21. Greenspan, H. P. (1956). The generation of edge waves by moving pressure distributions. Journal of Fluid Mechanics, 1, 574–592.CrossRefGoogle Scholar
  22. Hasselmann, K. (1966). Feynamn diagrams and interaction rules of wave–wave scattering processes. Reviews of Geophysics, 4, 1–32.CrossRefGoogle Scholar
  23. Herrero, A., & Bernard, P. (1994). A kinematic self-similar rupture process for earthquakes. Bulletin of the Seismological Society of America, 84, 1216–1228.Google Scholar
  24. Hindmarsh, A. C. (1983). ODEPACK, a systematized collection of ODE solvers. In R. S. Stepleman (Ed.), Scientific computing. Amsterdam: North-Holland.Google Scholar
  25. Hsieh, W. W., & Mysak, L. A. (1980). Resonant interactions between shelf waves, with applications to the Oregon Shelf. Journal of Physical Oceanography, 10, 1729–1741.CrossRefGoogle Scholar
  26. Ishii, H., & Abe, K. (1980). Propagation of tsunami on a linear slope between two flat regions. Part I edge wave. Journal of Physics of the Earth, 28, 531–541.CrossRefGoogle Scholar
  27. 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.CrossRefGoogle Scholar
  28. Kartashova, E. (2011). Nonlinear resonance analysis: Theory, computation, applications. Cambridge: Cambridge University Press.Google Scholar
  29. Kartashova, E., & L’vov, V. S. (2007). Model of intraseasonal oscillations in Earth’s atmosphere. Physical Review Letters, 98, 198501.CrossRefGoogle Scholar
  30. Kartashova, E., & L’vov, V. S. (2008). Cluster dynamics of planetary waves. EPL (Europhysics Letters), 83, 50012.CrossRefGoogle Scholar
  31. Kartashova, E., Raab, C., Feurer, C., Mayrhofer, G., & Schreiner, W. (2008). Symbolic computation for nonlinear wave resonances. In E. Pelinovsky & C. Kharif (Eds.), Extreme ocean waves (pp. 95–126). Dordrecht: Springer Netherlands.CrossRefGoogle Scholar
  32. Kaup, D. J., Reiman, A., & Bers, A. (1979). Space–time evolution of nonlinear three-wave interactions. I. Interaction in a homogeneous medium. Reviews of Modern Physics, 51, 275–309.CrossRefGoogle Scholar
  33. Kenyon, K. E. (1970). A note on conservative edge wave interactions. Deep-Sea Research, 17, 197–201.Google Scholar
  34. Kirby J. T., Putrevu U., & Özkan-Haller H. T. (1998). Evolution equations for edge waves and shear waves on longshore uniform beaches. In: Proceedings of 26th International Conference on Coastal Engineering. ASCE, Copenhagen, Denmark, pp. 203–216.Google Scholar
  35. Kurkin, A., & Pelinovsky, E. (2002). Focusing of edge waves above a sloping beach. European Journal of Mechanics B/Fluids, 21, 561–577.CrossRefGoogle Scholar
  36. Lavallée, D., Liu, P., & Archuleta, R. J. (2006). Stochastic model of heterogeneity in earthquake slip spatial distributions. Geophysical Journal International, 165, 622–640.CrossRefGoogle Scholar
  37. Lavallée, D., Miyake, H., & Koketsu, K. (2011). Stochastic model of a subduction-zone earthquake: Sources and ground motions for the 2003 Tokachi-oki, Japan, earthquake. Bulletin of the Seismological Society of America, 101, 1807–1821.CrossRefGoogle Scholar
  38. Leblond, P. H., & Mysak, L. A. (1978). Waves in the ocean. Amseterdam: Elsevier.Google Scholar
  39. Longuet-Higgins, M. S., & Gill, A. E. (1967). Resonant interactions between planetary waves. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 299, 120–144. Scholar
  40. Lynch, P. (2003). Resonant Rossby wave triads and the swinging spring. Bulletin of the American Meteorological Society, 84, 605–616. Scholar
  41. Lynch, P. (2009). On resonant Rossby–Haurwitz triads. Tellus A, 61, 438–445. Scholar
  42. Lynch, P., & Houghton, C. (2004). Pulsation and precession of the resonant swinging spring. Physica D: Nonlinear Phenomena, 190, 38–62. Scholar
  43. Mai P. M., & Beroza G. C. (2002). A spatial random field model to characterize complexity in earthquake slip. Journal of Geophysical Research, 107.
  44. Mei, C. C., Stiassnie, M., & Yue, D. K.-P. (2005). Theory and applications of ocean surface waves. Part 2: Nonlinear aspects. Singpore: World Scientific.Google Scholar
  45. Miller, G. R., Munk, W. H., & Snodgrass, F. E. (1962). Long-period waves over California’s continental borderland Part II. Tsunamis. Journal of Marine Research, 20, 31–41.Google Scholar
  46. Monserrat, S., Vilibic, I., & Rabinovich, A. B. (2006). Meteotsunamis: Atmospherically induced destructive ocean waves in the tsunami frequency band. Natural Hazards and Earth System Sciences, 6, 1035–1051.CrossRefGoogle Scholar
  47. Munk, W., Snodgrass, F. E., & Carrier, G. F. (1956). Edge waves on the continental shelf. Science, 123, 127–132.CrossRefGoogle Scholar
  48. Munk, W., Snodgrass, F. E., & Gilbert, F. (1964). Long waves on the continental shelf: An experiment to separate trapped and leaky modes. Journal of Fluid Mechanics, 20, 529–554.CrossRefGoogle Scholar
  49. 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
  50. Pelinovsky, E., Polukhina, O., & Kurkin, A. (2010). Rogue edge waves in the ocean. European Physical Journal Special Topics, 185, 34–44.CrossRefGoogle Scholar
  51. Rabinovich, A. B., Candella, R. N., & Thomson, R. E. (2011). Energy decay of the 2004 Sumatra tsunami in the World Ocean. Pure and Applied Geophysics, 168, 1919–1950.CrossRefGoogle Scholar
  52. Reznik, G. M., Piterbarg, L. I., & Kartashova, E. A. (1993). Nonlinear interactions of spherical Rossby modes. Dynamics of Atmospheres and Oceans, 18, 235–252. Scholar
  53. Saito, T., Inazu, D., Tanaka, S., & Miyoshi, T. (2013). Tsunami coda across the Pacific Ocean following the 2011 Tohoku-Oki earthquake. Bulletin of the Seismological Society of America, 103, 1429–1443. Scholar
  54. Slunyaev, A., Didenkulova, I. I., & Pelinovsky, E. (2011). Rogue waters. Contemporary Physics, 52, 571–590.CrossRefGoogle Scholar
  55. Snodgrass, F. E., Munk, W. H., & Miller, G. R. (1962). Long-period waves over California’s continental borderland Part I. Background spectra. Journal of Marine Research, 20, 3–30.Google Scholar
  56. Sobarzo, M., Garcés-Vargas, J., Bravo, L., Tassara, A., & Quiñones, R. A. (2012). Observing sea level and current anomalies driven by a megathrust slope-shelf tsunami: The event on February 27, 2010 in central Chile. Continental Shelf Research, 49, 44–55. Scholar
  57. Toledo, B. A., Chian, A. C. L., Rempel, E. L., Miranda, R. A., Muñoz, P. R., & Valdivia, J. A. (2013). Wavelet-based multifractal analysis of nonlinear time series: The earthquake-driven tsunami of 27 February 2010 in Chile. Physical Review E, 87, 022821.CrossRefGoogle Scholar
  58. Vela, J., Pérez, B., González, M., Otero, L., Olabarrieta, M., Canals, M., et al. (2014). Tsunami resonance in Palma Bay and Harbor, Majorca Island, as induced by the 2003 western Mediterranean earthquake. The Journal of Geology, 122, 165–182. Scholar
  59. Vennell, R. (2010). Resonance and trapping of topographic transient ocean waves generated by a moving atmospheric disturbance. Journal of Fluid Mechanics, 650, 427–443.CrossRefGoogle Scholar
  60. Wilhelmsson, H., Stenflo, L., & Engelmann, F. (1970). Explosive instabilities in the well-defined phase description. Journal of Mathematical Physics, 11, 1738–1742. Scholar
  61. Yamazaki Y., & Cheung K. F. (2011). Shelf resonance and impact of near-field tsunami generated by the 2010 Chile earthquake. Geophysical Research Letters, 38.
  62. Yankovsky A. E. (2009). Large-scale edge waves generated by hurricane landfall. Journal of Geophysical Research, 114.

Copyright information

© This is a U.S. government work and its text is not subject to copyright protection in the United States; however, its text may be subject to foreign copyright protection 2018

Authors and Affiliations

  1. 1.US Geological SurveyMenlo ParkUSA

Personalised recommendations