On the Shoaling of Solitary Waves in the Presence of Short Random Waves

Miao Tian Department of Civil and Coastal Engineering, University of Florida, Gainesville, Florida

Search for other papers by Miao Tian in
Current site
Google Scholar
PubMed
Close
,
Alex Sheremet Department of Civil and Coastal Engineering, University of Florida, Gainesville, Florida

Search for other papers by Alex Sheremet in
Current site
Google Scholar
PubMed
Close
,
James M. Kaihatu Zachry Department of Civil Engineering, Texas A&M University, College Station, Texas

Search for other papers by James M. Kaihatu in
Current site
Google Scholar
PubMed
Close
, and
Gangfeng Ma Department of Civil and Environmental Engineering, Old Dominion University, Norfolk, Virginia

Search for other papers by Gangfeng Ma in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

Overhead video from a small number of laboratory tests conducted by Kaihatu et al. at the Tsunami Wave Basin at Oregon State University shows that the breaking point of a shoaling solitary wave shifts to deeper water if random waves are present. The analysis of the laboratory data collected confirms that solitary waves indeed tend to break earlier in the presence of random wave field, and suggests that the effect is the result of the radiation stresses gradient induced by the random wave fields. A theoretical approach based on the forced KdV equation is shown to successfully predict the shoaling process of the solitary wave. An ensemble of tests simulated using a state-of-the-art nonhydrostatic model is used to test the statistical significance of the process. The results of this study point to a potentially significant oceanographic process that has so far been ignored and suggest that systematic research into the interaction between tsunami waves and the swell background could increase the accuracy of tsunami forecasting.

Corresponding author address: Miao Tian, Department of Civil and Coastal Engineering, University of Florida, 365 Weil Hall, Gainesville, FL 32611. E-mail: mtian04.18@ufl.edu

Abstract

Overhead video from a small number of laboratory tests conducted by Kaihatu et al. at the Tsunami Wave Basin at Oregon State University shows that the breaking point of a shoaling solitary wave shifts to deeper water if random waves are present. The analysis of the laboratory data collected confirms that solitary waves indeed tend to break earlier in the presence of random wave field, and suggests that the effect is the result of the radiation stresses gradient induced by the random wave fields. A theoretical approach based on the forced KdV equation is shown to successfully predict the shoaling process of the solitary wave. An ensemble of tests simulated using a state-of-the-art nonhydrostatic model is used to test the statistical significance of the process. The results of this study point to a potentially significant oceanographic process that has so far been ignored and suggest that systematic research into the interaction between tsunami waves and the swell background could increase the accuracy of tsunami forecasting.

Corresponding author address: Miao Tian, Department of Civil and Coastal Engineering, University of Florida, 365 Weil Hall, Gainesville, FL 32611. E-mail: mtian04.18@ufl.edu
Save
  • Aida, I., K. Kajiura, T. Hatori, and T. Momoi, 1964: A tsunami accompanying the Niigata earthquake of June 16, 1964 (in Japanese). Bull. Earthquake Res. Inst. Univ. Tokyo,42, 741–780.

    • Search Google Scholar
    • Export Citation
  • Bouws, E., H. Gunther, W. Rosenthal, and C. L. Vincent, 1985: Similarity of the wind spectrum in finite depth water: 1. Spectral form. J. Geophys. Res., 90, 975986, doi:10.1029/JC090iC01p00975.

    • Search Google Scholar
    • Export Citation
  • Chui, C. K., 1992: An Introduction to Wavelet: Wavelet Analysis and Its Applications. Academic Press, 264 pp.

  • Constantin, A., and R. Johnson, 2008: Propagation of very long water waves with vorticity over variable depth with applications to tsunamis. Fluid Dyn. Res., 40, 175211, doi:10.1016/j.fluiddyn.2007.06.004.

    • Search Google Scholar
    • Export Citation
  • El, G., R. Grimshaw, and A. Kamchatnov, 2007: Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction. J. Fluid Mech., 585, 213244, doi:10.1017/S0022112007006817.

    • Search Google Scholar
    • Export Citation
  • Farge, M., 1992: Wavelet transforms and their applications to turbulence. Annu. Rev. Fluid Mech., 24, 395457, doi:10.1146/annurev.fl.24.010192.002143.

    • Search Google Scholar
    • Export Citation
  • Fritz, H. M., and Coauthors, 2012: The 2011 Japan tsunami current velocity measurements from survivor videos at Kesennuma Bay using LiDAR. Geophys. Res. Lett., 39, L00G23, doi:10.1029/2011GL050686.

    • Search Google Scholar
    • Export Citation
  • Geist, E., V. Titov, and C. Synolakis, 2006: Tsunami: Wave of change. Sci. Amer., 294, 5663, doi:10.1038/scientificamerican0106-56.

  • Goring, D. G., 1979: Tsunamis—The propagation of long waves onto a shelf. W. M. Keck Laboratory of Hydraulics and Water Resources, California Institute of Technology, Rep. KH-R-38, 356 pp.

  • Goupillaud, P., P. A. Grossman, and J. Morlet, 1984: Cycle-octave and related transforms in seismic signal analysis. Geoexploration, 23, 85102, doi:10.1016/0016-7142(84)90025-5.

    • Search Google Scholar
    • Export Citation
  • Grilli, S. T., S. Vogelmann, and P. Watts, 2002: Development of a 3D numerical wave tank for modeling tsunami generation by underwater landslides. Eng. Anal. Boundary Elem., 26, 301313, doi:10.1016/S0955-7997(01)00113-8.

    • Search Google Scholar
    • Export Citation
  • Grimshaw, R., 1971: The solitary wave in water of variable depth. Part 2. J. Fluid Mech., 46, 611622, doi:10.1017/S0022112071000739.

  • Grimshaw, R., 1979: Slowly varying solitary waves. I. Korteweg-de Vries equation. Proc. Roy. Soc. London,A368, 359–375, doi:10.1098/rspa.1979.0135.

  • Grimshaw, R., E. Pelinovsky, T. Talipova, and O. Kurkina, 2010: Internal solitary waves: Propagation, deformation and disintegration. Nonlinear Processes Geophys.,17, 633–649, doi:10.5194/npg-17-633-2010.

  • Grimshaw, R., C. Guo, K. Helfrich, and V. Vlasenko, 2014: Combined effect of rotation and topography on shoaling oceanic internal solitary waves. J. Phys. Oceanogr., 44, 1116–1132, doi:10.1175/JPO-D-13-0194.1.

    • Search Google Scholar
    • Export Citation
  • Grue, J., E. N. Pelinovsky, D. Fructus, T. Talipova, and C. Kharif, 2008: Formation of undular bores and solitary waves in the Strait of Malacca caused by the 26 December 2004 Indian Ocean tsunami. J. Geophys. Res.,113, C05008, doi:10.1029/2007JC004343.

  • Ippen, A. T., and G. Kulin, 1954: The shoaling and breaking of the solitary waves. Proc. Fifth Int. Conf. on Coastal Engineering, New York, NY, ASCE, 27–47.

  • Johnson, R. S., 1973a: On the development of a solitary wave moving over an uneven bottom. Math. Proc. Cambridge Philos. Soc., 73, 183203, doi:10.1017/S0305004100047605.

    • Search Google Scholar
    • Export Citation
  • Johnson, R. S., 1973b: On an asymptotic solution of the Korteweg–de Vries equation with slowly varying coefficients. J. Fluid Mech., 60, 813824, doi:10.1017/S0022112073000492.

    • Search Google Scholar
    • Export Citation
  • Kaihatu, J. M., and H. M. El Safty, 2011: The interaction of tsunamis with ocean swell: An experimental study. Proc. 30th Int. Conf. on Ocean, Offshore and Arctic Engineering, Rotterdam, Netherlands, ASME, 747–754, doi:10.1115/OMAE2011-49936.

  • Kaihatu, J. M., D. Devery, R. J. Erwin, and J. T. Goertz, 2012: The interaction between short ocean swell and transient long waves: Dissipative and nonlinear effects. Proc. 33rd Conf. on Coastal Engineering, Santander, Spain, ASCE, wave.20. [Available online at https://journals.tdl.org/icce/index.php/icce/article/view/7012.]

  • Kowalik, Z., T. Proshutinsky, and A. Proshutinsky, 2006: Tide-tsunami interactions. Sci. Tsunami Hazards,24, 242–256. [Available online at http://tsunamisociety.org/244kowalik.pdf.]

  • Longuet-Higgins, S., 1987: The propagation of short surface waves on longer gravity waves. J. Fluid Mech., 177, 293306, doi:10.1017/S002211208700096X.

    • Search Google Scholar
    • Export Citation
  • Longuet-Higgins, S., and R. Stewart, 1962: Radiation stress and mass transport in gravity waves with application to surf beats. J. Fluid Mech., 13, 481504, doi:10.1017/S0022112062000877.

    • Search Google Scholar
    • Export Citation
  • Ma, G.-F., F. Shi, and J. T. Kirby, 2012: Shock-capturing non-hydrostatic model for fully dispersive surface wave processes. Ocean Modell., 43–44, 2235, doi:10.1016/j.ocemod.2011.12.002.

    • Search Google Scholar
    • Export Citation
  • Madsen, O. S., and C. C. Mei, 1969: The transformation of a solitary wave over an uneven bottom. J. Fluid Mech., 39, 781791, doi:10.1017/S0022112069002461.

    • Search Google Scholar
    • Export Citation
  • Madsen, P. A., D. R. Fuhrman, and H. A. Schäffer, 2008: On the solitary wave paradigm for tsunamis. J. Geophys. Res.,113, C12012, doi:10.1029/2008JC004932.

  • Matsuyama, M., M. Ikeno, T. Sakakiyama, and T. Takeda, 2007: A study of tsunami wave fission in an undistorted experiment. Pure Appl. Geophys., 164, 617631, doi:10.1007/s00024-006-0177-0.

    • Search Google Scholar
    • Export Citation
  • Mei, C. C., M. Stiassnie, and D.-P. Yue, 2005: Theory and Applications of Ocean Surface Waves. Advanced Series on Ocean Engineering, Vol. 23, World Scientific, 1136 pp.

  • Mellor, G. L., 1996: Introduction to Physical Oceanography. American Institute of Physics, 260 pp.

  • Nazarenko, S., 2011: Wave Turbulence. Lecture Notes in Physics, Vol. 825, Springer, 279 pp.

  • Osborne, A. R., 2010: Nonlinear Ocean Waves and the Inverse Scattering Transform. International Geophysics Series, Vol. 97, Academic Press, 944 pp.

  • Peregrine, D. H., 1983: Breaking waves on beaches. Annu. Rev. Fluid Mech., 15, 149178, doi:10.1146/annurev.fl.15.010183.001053.

  • Synolakis, C. E., 1991: Green’s law and the evolution of solitary waves. Phys. Fluids, 3A, 490491, doi:10.1063/1.858107.

  • Synolakis, C. E., and J. E. Skjelbreia, 1993: Evolution of maximum amplitude of solitary waves on plane beaches. J. Waterw. Port Coastal Ocean Eng., 119, 323342, doi:10.1061/(ASCE)0733-950X(1993)119:3(323).

    • Search Google Scholar
    • Export Citation
  • Tadepalli, S., and C. Synolakis, 1994: The run-up of N-waves on sloping beaches. Proc. Roy. Soc. London, A445, 99–112, doi:10.1098/rspa.1994.0050.

    • Search Google Scholar
    • Export Citation
  • Torrence, C., and G. P. Compo, 1998: A practical guide to wavelet analysis. Bull. Amer. Meteor. Soc., 79, 6178, doi:10.1175/1520-0477(1998)079<0061:APGTWA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Whitham, G. B., 1974: Linear and Nonlinear Waves. Pure and Applied Mathematics Series, Wiley, 636 pp.

  • Zhang, J., and W. K. Melville, 1990: Evolution of weakly nonlinear short waves riding on long gravity waves. J. Fluid Mech., 214, 321346, doi:10.1017/S0022112090000155.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 190 62 10
PDF Downloads 140 45 8