Pure and Applied Geophysics

, Volume 172, Issue 3–4, pp 885–899 | Cite as

Runup of Nonlinear Long Waves in Trapezoidal Bays: 1-D Analytical Theory and 2-D Numerical Computations

  • M. W. Harris
  • D. J. Nicolsky
  • E. N. Pelinovsky
  • A. V. Rybkin


Long nonlinear wave runup on the coasts of trapezoidal bays is studied analytically in the framework of one-dimensional (1-D) nonlinear shallow-water theory with cross-section averaging, and is also studied numerically within a two-dimensional (2-D) nonlinear shallow water theory. In the 1-D theory, it is assumed that the trapezoidal cross-section channel is inclined linearly to the horizon, and that the wave flow is uniform in the cross-section. As a result, 1-D nonlinear shallow-water equations are reduced to a linear, semi-axis variable-coefficient 1-D wave equation by using the generalized Carrier–Greenspan transformation [Carrier and Greenspan (J Fluid Mech 1:97–109, 1958)] recently developed for arbitrary cross-section channels [Rybkin et al. (Ocean Model 43–44:36–51, 2014)], and all characteristics of the wave field can be expressed by implicit formulas. For detailed computations of the long wave runup process, a robust and effective finite difference scheme is applied. The numerical method is verified on a known analytical solution for wave runup on the coasts of an inclined parabolic bay. The predictions of the 1-D model are compared with results of direct numerical simulations of inundations caused by tsunamis in narrow bays with real bathymetries.


Wave run-up shallow water wave equations Carrier–Greenspan transformation numerical simulation 



This work was done as part of the REU program run by the fourth author in the summer of 2012, and was supported by NSF grant DMS 1009673. The government support is highly appreciated. We are grateful to the following other participants who also contributed to this project: Jeremiah Harrington, Lander Ver Hoef, and Viacheslav Garayshin. DJ acknowledges support from the Cooperative Institute for Alaska Research with funds from the National Oceanic and Atmospheric Administration under cooperative agreement NA08OAR4320751 with the University of Alaska. EP acknowledges support from State Contract 2014/133, RFBR grant (14-05-00092), and VolkswagenStiftung.


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • M. W. Harris
    • 1
  • D. J. Nicolsky
    • 2
  • E. N. Pelinovsky
    • 3
    • 4
  • A. V. Rybkin
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of Alaska FairbanksFairbanksUSA
  2. 2.Geophysical InstituteUniversity of Alaska FairbanksFairbanksUSA
  3. 3.Institute of Applied PhysicsNizhny NovgorodRussia
  4. 4.Nizhny Novgorod State Technical UniversityNizhny NovgorodRussia

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