Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-24T14:54:07.689Z Has data issue: false hasContentIssue false
  • English
  • Français

Long wave propagation and run-up in converging bays

Published online by Cambridge University Press:  03 June 2016

Takenori Shimozono
Takenori Shimozono*
Affiliation:
Department of Civil Engineering, The University of Tokyo, 7-3-1, Hongo, Bunkyo-ku, Tokyo, 113-8656, Japan
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Analytical solutions are derived to describe two-dimensional wave evolution in converging bays. Three bay types of different cross-sections are studied: U-shaped, V-shaped and cusped bays. For these bays, the two-dimensional linear shallow water equations can be reduced to one-dimensional linear dispersive wave equations if the transverse flow acceleration inside them is assumed to be small. The derived solutions are characterized as the leading-order plane-wave solutions with higher-order corrections for two-dimensionality due to wave refraction. Wave amplitude longitudinally increases with different rates for the three bay types, whereas it exhibits weak parabolic variations in the transverse direction. Wave refraction significantly affects relatively short waves, contributing to wave energy transfer to the inner bay in a different manner depending on the bay type. The perturbation analysis of very high-order wave celerity suggests that the solutions are valid only when the ratio of the bay width to the wavelength is smaller than a certain limit that differs with bay type. Beyond the limit, the higher-order effect is no longer a minor correction, implying that wave behaviours become highly two-dimensional and possibly cause total reflection. The higher-order effect on the run-up height at the bay head is found to be small within the applicable range of the solution, and thus, the run-up formula neglecting the transverse flows has a wide validity. We also discuss the limitation of run-up height by wave breaking on the basis of a breaking criterion from previous studies.

JFM classification

Geophysical and Geological Flows: Coastal engineering Waves/Free-surface Flows: Surface gravity waves Geophysical and Geological Flows: Topographic effects
Type
Papers
Information
Journal of Fluid Mechanics , Volume 798 , 10 July 2016 , pp. 457 - 484
Check for updates
Copyright
© 2016 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Antuono, M. & Brocchini, M. 2007 The boundary value problem for the nonlinear shallow water equations. Stud. Appl. Maths 119 (1), 73–93.Google Scholar
Carrier, G. F. & Greenspan, H. P. 1958 Water waves of finite amplitude on a sloping beach. J. Fluid Mech. 4 (01), 97–109.Google Scholar
Didenkulova, I. 2013 Tsunami runup in narrow bays: the case of samoa 2009 tsunami. Nat. Hazards 65 (3), 1629–1636.Google Scholar
Didenkulova, I., Didenkulov, O. & Pelinovsky, E. 2015 A note on the uncertainty in tsunami shape for estimation of its run-up heights. J. Ocean Engng Mar. Energy 1 (2), 199–205.Google Scholar
Didenkulova, I. & Pelinovsky, E. 2011 Runup of tsunami waves in u-shaped bays. Pure Appl. Geophys. 168 (6), 1239–1249.Google Scholar
Didenkulova, I., Pelinovsky, E. & Soomere, T. 2008 Runup characteristics of symmetrical solitary tsunami waves of ‘unknown’ shapes. Pure Appl. Geophys. 165 (11), 2249–2264.Google Scholar
Friedrichs, C. T. & Aubrey, D. G. 1994 Tidal propagation in strongly convergent channels. J. Geophys. Res. 99 (C2), 3321–3336.Google Scholar
Jay, D. A. 1991 Green’s law revisited: tidal long-wave propagation in channels with strong topography. J. Geophys. Res. 96 (C11), 20585–20598.Google Scholar
Kawai, H., Satoh, M., Kawaguchi, K. & Seki, K. 2013 Characteristics of the 2011 tohoku tsunami waveform acquired around japan by nowphas equipment. Coast. Engng J. 55 (03), 1350008.Google Scholar
Keller, J. B. & Keller, H. B.1964 Water wave run-up on a beach. Tech. Rep. ONR Research Report Contract No. NONR-3828(00). Department of the Navy, Washington, D.C.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Liu, H., Shimozono, T., Takagawa, T., Okayasu, A., Fritz, H. M., Sato, S. & Tajima, Y. 2013 The 11 March 2011 tohoku tsunami survey in rikuzentakata and comparison with historical events. Pure Appl. Geophys. 170 (6), 1033–1046.CrossRefGoogle Scholar
Madsen, P. A. & Schaeffer, H. A. 2010 Analytical solutions for tsunami runup on a plane beach: single waves, n-waves and transient waves. J. Fluid Mech. 645, 27–57.Google Scholar
Mori, N. & Takahashi, T. 2012 Nationwide post event survey and analysis of the 2011 Tohoku earthquake tsunami. Coast. Engng J. 54 (01), 1250001.Google Scholar
Pelinovsky, E. & Mazova, R. 1992 Exact analytical solutions of nonlinear problems of tsunami wave run-up on slopes with different profiles. Nat. Hazards 6 (3), 227–249.CrossRefGoogle Scholar
Peregrine, D. H. 1968 Long waves in a uniform channel of arbitrary cross-section. J. Fluid Mech. 32 (02), 353–365.Google Scholar
Peregrine, D. H. 1969 Solitary waves in trapezoidal channels. J. Fluid Mech. 35 (01), 1–6.Google Scholar
Prandle, D. 2003 Relationships between tidal dynamics and bathymetry in strongly convergent estuaries. J. Phys. Oceanogr. 33 (12), 2738–2750.Google Scholar
Rayleigh, Lord 1879 On reflection of vibrations at the confines of two media between which the transition is gradual. Proc. Lond. Math. Soc. s1-11 (1), 51–56.Google Scholar
Rybkin, A., Pelinovsky, E. & Didenkulova, I. 2014 Nonlinear wave run-up in bays of arbitrary cross-section: generalization of the Carrier–Greenspan approach. J. Fluid Mech. 748, 416–432.Google Scholar
Savenije, H. H. G., Toffolon, M., Haas, J. & Veling, E. J. M. 2008 Analytical description of tidal dynamics in convergent estuaries. J. Geophys. Res. 113, C10025.Google Scholar
Shimozono, T., Cui, H., Pietrzak, J. D., Fritz, H. M., Okayasu, A. & Hooper, A. J. 2014 Short wave amplification and extreme runup by the 2011 Tohoku tsunami. Pure Appl. Geophys. 171 (12), 3217–3228.CrossRefGoogle Scholar
Shimozono, T., Sato, S., Okayasu, A., Tajima, Y., Fritz, H. M., Liu, H. & Takagawa, T. 2012 Propagation and inundation characteristics of the 2011 Tohoku tsunami on the central sanriku coast. Coast. Engng J. 54 (01), 1250004.Google Scholar
Shuto, N. 1972 Standing waves in front of a sloping dike. Coast. Engng Japan 15, 13–23.Google Scholar
Stoker, J. J. 1957 Water Waves: The Mathematical Theory with Applications. Wiley.Google Scholar
Synolakis, C. E. 1987 The runup of solitary waves. J. Fluid Mech. 185, 523–545.Google Scholar
Synolakis, C. E. 1991 Tsunami runup on steep slopes: How good linear theory really is. Nat. Hazards 4 (2), 221–234.CrossRefGoogle Scholar
Tadepalli, S. & Synolakis, C. E. 1994 The run-up of N-waves on sloping beaches. Proc. R. Soc. Lond. A 445 (1923), 99–112.Google Scholar
Teng, M. H. & Wu, T. Y. 1992 Nonlinear water waves in channels of arbitrary shape. J. Fluid Mech. 242, 211–233.Google Scholar
Teng, M. H. & Wu, T. Y. 1994 Evolution of long water waves in variable channels. J. Fluid Mech. 266, 303–317.Google Scholar
Zahibo, N., Pelinovsky, E., Golinko, V. & Osipenko, N. 2006 Tsunami wave runup on coasts of narrow bays. Intl J. Fluid Mech. Res. 33, 106–118.Google Scholar
Supplementary material: File

Shimozono supplementary material

Shimozono supplementary material 1

Download Shimozono supplementary material(File)
File 741.1 KB