# Run-Up of Long Waves in Piecewise Sloping U-Shaped Bays

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## Abstract

We present an analytical study of the propagation and run-up of long waves in piecewise sloping, U-shaped bays using the cross-sectionally averaged shallow water equations. The nonlinear equations are transformed into a linear equation by utilizing the generalized Carrier–Greenspan transform (Rybkin et al. J Fluid Mech 748:416–432, 2014). The solution of the linear wave propagation is taken as the boundary condition at the toe of the last sloping segment, as in Synolakis (J Fluid Mech 185:523–545, 1987). We then consider a piecewise sloping bathymetry, and as in Kanoglu and Synolakis (J Fluid Mech 374:1–28, 1998), find the linear solution in the near shore region, which can be used as the boundary condition for the nonlinear problem. Our primary results are an analytical run-up law for narrow channels and breaking criteria for both monochromatic waves and solitary waves. The derived analytical solutions reduce to well-known solutions for parabolic bays and plane beaches. Our analytical predictions are verified in narrow bays via a comparison to direct numerical simulation of the 2-D shallow water equations.

## Keywords

Long-wave run-up shallow water wave equations carrier–greenspan transformation piecewise sloping bays## Notes

### Acknowledgements

D. Anderson, H. Hartle, and A. Raz were supported by the National Science Foundation Research Experience for Undergraduate program (Grant #1411560) and the Geophysical Institute, University of Alaska Fairbanks. D. Nicolsky acknowledges support from the Geophysical Institute, University of Alaska Fairbanks. E. Pelinovsky acknowledges support from the Grant of the RF President for state support of leading scientific schools (NSH-6637.2016.5) and RFBR grant (15-45-02061). A. Rybkin acknowledges support from the National Science Foundation Grant DMS-1411560. We would also like to thank Research Computing Systems at the Geophysical Institute for providing supercomputing resources required for the FUNWAVE trials.

## References

- Abramowitz, M., & Stegun, I. (1965).
*“Chapter 17”, handbook of mathematical functions with formulas, graphs, and mathematical tables*. New York: Dover.Google Scholar - Antuono, M., & Brocchini, M. (2007). The boundary value problem for the nonlinear shallow water equations.
*Studies in Applied Mathematics*,*119*, 73–93.CrossRefGoogle Scholar - Arfken, G. B., Weber, H. J., & Harris, F. E. (2012).
*Mathematical methods for physicists: A comprehensive guide*. Waltham: Academic Press.Google Scholar - Bowman, F. (1958).
*Introduction to Bessel functions*. North Chelmsford, Massachusetts: Courier Corporation.Google Scholar - Brown, J. W., & Churchill, R. V. (1993).
*Fourier series and boundary value problems*. New York: McGraw-Hill.Google Scholar - Carrier, G. (1966). Run-up of nonlinear long waves in u-shaped bays of finite length: Analytical theory and numerical computations.
*Journal of Fluid Mechanics*,*24*, 641–659.CrossRefGoogle Scholar - Carrier, G., & Greenspan, H. (1957). Water waves of finite amplitude on a sloping beach.
*Journal of Fluid Mechanics*,*01*, 97–109.Google Scholar - Carrier, G., Wu, T., & Yeh, H. (2003). Tsunami run-up and draw-down on a plane beach.
*Journal of Fluid Mechanics*,*475*, 79–99. doi: 10.1017/S0022112002002653.CrossRefGoogle Scholar - Choi, B., Pelinovsky, E., Kim, D., Didenkulova, I., & Woo, S. (2008). Two- and three-dimensional computation of solitary wave runup on non-plane beach.
*Nonlinear Process in Geophysics*,*15*, 489–502.CrossRefGoogle Scholar - Didenkulova, I., & Pelinovsky, E. (2009). Non-dispersive traveling waves in inclined shallow water channels.
*Physics Letters A*,*373*, 3883–3887.CrossRefGoogle Scholar - Didenkulova, I., & Pelinovsky, E. (2011). Runup of tsunami waves in U-shaped bays.
*Pure and Applied Geophysics*,*168*, 1239–1249.CrossRefGoogle Scholar - Dunbar, P., & Weaver, C. (2015). U.S. states and territories. national tsunami hazard assessment: Historical record and sources for waves—Update. Tech. rep., National Oceanic and Atmospheric Administration (NOAA).Google Scholar
- Fuentes, M., Ruiz, J., & Riquelme, S. (2015). The runup on a multilinear sloping beach model.
*Geophysical Journal International*,*201*, 915928. doi: 10.1093/gji/ggv056.CrossRefGoogle Scholar - Garashin, V., Harris, M., Nicolsky, D., Pelinovsky, E., & Rybkin, A. (2016). An analytical and numerical study of long wave run-up in U-shaped and V-shaped bays.
*Applied Mathematics and Computation*,*297*, 187–197.CrossRefGoogle Scholar - Harris, M., Nicolsky, D., Pelinovsky, E., Pender, J., & Rybkin, A. (2015). Run-up of nonlinear long waves in U-shaped bays of finite length: Analytical theory and numerical computations.
*Journal of Ocean Engineering and Marine Energy*,*2*, 113–127.CrossRefGoogle Scholar - Kanoglu, U. (2004). Nonlinear evolution and runup-drawdown of long waves over a sloping beach.
*Journal of Fluid Mechanics*,*513*, 363–372.CrossRefGoogle Scholar - Kanoglu, U., & Synolakis, C. (2006). Initial value problem solution of nonlinear shallow water-wave equations.
*Physical Review Letters*,*148501*, 97. doi: 10.1103/PhysRevLett.97.148501.Google Scholar - Kanoglu, U., & Synolakis, C. E. (1998). Long wave runup on piecewise linear topographies.
*Journal of Fluid Mechanics*,*374*, 1–28.CrossRefGoogle Scholar - Kanoglu, U., Titov, V., Bernard, E., & Synolakis, C. (2015). Tsunamis: bridging science, engineering and society.
*Philosophical Transactions of the Royal Society A*,*373*(2053), 20140369. doi: 10.1098/rsta.2014.0369.CrossRefGoogle Scholar - Keller, J., & Keller, H. (1964). Water wave run-up on a beach. ONR Research Report Contract No NONR-3828(00)Google Scholar
- Kirby, J., Wei, G., Chen, Q., Kennedy, A., & Dalrymple, R. (1998). FUNWAVE 1.0, Fully nonlinear Boussinesq wave model. documentation and user’s manual. Research Report CACR-98-06, Center for Applied Coastal Research, Department of Civil and Environmental Engineering, University of DelawareGoogle Scholar
- Li, Y., & Raichlen, F. (2001). Solitary wave runup on plane slopes.
*Journal of Waterway, Port, Coastal & Ocean Engineering*,*127*, 33–44.CrossRefGoogle Scholar - Madsen, P., Fuhrman, D., & Schäffer, H. (2008). On the solitary wave paradigm for tsunamis.
*Journal of Geophysical Research*,*113*, C12012.CrossRefGoogle Scholar - MATLAB. (2014).
*Matlab and statistics toolbox release 2012b*. Natick: The Mathworks, Inc.Google Scholar - NTHMP (ed) (2012) Proceedings and results of the 2011 NTHMP Model Benchmarking Workshop, NOAA Special Report, U.S. Department of Commerce/NOAA/NTHMP, National Tsunami Hazard Mapping Program [NTHMP], Boulder, CO.Google Scholar
- Pelinovsky, E., & Mazova, R. (1992). Exact analytical solutions of nonlinear problems of tsunami wave run-up on slopes with different profiles.
*Natural Hazards*,*6*, 227–249.CrossRefGoogle Scholar - Rybkin, A., Pelinovsky, E., & Didenkulova, I. (2014). Non-linear wave run-up in bays of arbitrary cross-section:generalization of the Carrier–Greenspan approach.
*Journal of Fluid Mechanics*,*748*, 416–432.CrossRefGoogle Scholar - Shi, F., Kirby, J., Harris, J., Geiman, J., & Grilli, S. (2012a). A high-order adaptive time-stepping tvd solver for boussinesq modeling of breaking waves and coastal inundation.
*Ocean Modeling*,*43–44*, 36–51.CrossRefGoogle Scholar - Shi, F., Kirby, J. T., Harris, J. C., Geiman, J. D., & Grilli, S. T. (2012b). A high-order adaptive time-stepping TVD solver for boussinesq modeling of breaking waves and coastal inundation.
*Ocean Modelling*,*43*, 36–51.CrossRefGoogle Scholar - Shimozono, T. (2016). Long wave propagation and run-up in converging bays.
*Journal of Fluid Mechanics*,*798*, 457–484. doi: 10.1017/jfm.2016.327.CrossRefGoogle Scholar - Stoker, J. (1957).
*Water waves: The mathematical theory with applications*. New York: Interscience Publishers.Google Scholar - Synolakis, C. (1987). The runup of solitary waves.
*Journal of Fluid Mechanics*,*185*, 523–545.CrossRefGoogle Scholar - Synolakis, C. (1991). Tsunami runup on steep slopes: How good linear theory really is?
*Natural Hazards*,*4*, 221–234.CrossRefGoogle Scholar - Synolakis, C., & Bernard, E. (2006). Tsunami science before and beyond Boxing Day 2004.
*Philosophical Transactions of the Royal Society A*,*364*, 2231–2265.CrossRefGoogle Scholar - Synolakis, C., Bernard, E., Titov, V., Kanoglu, U., & Gonzalez, F. (2008). Validation and verification of tsunami numerical models.
*Pure Applied Geophysics*,*165*, 2197–2228.CrossRefGoogle Scholar - Tadepalli, S., & Synolakis, C. (1994). The runup of N-waves.
*Proceedings of the Royal Socity of London*,*A445*, 99–112.CrossRefGoogle Scholar - Tadepalli, S., & Synolakis, C. E. (1988). Roots of \(f(z) = j_{\gamma }(z)-ij_{\gamma +1}(z)\) and the evaluation of integrals with cylindrical function kernels.
*Quarterly of Applied Mathematics*,*XLVI*, 105–108.Google Scholar - Tehranirad, B., Kirby, J., Ma, G., & Shi, F. (2012a). Tsunami benchmark results for nonhydrostatic wave model NHWAVE (Version 1.1). Research report no. cacr-12-03, Center for Applied Coastal Research, University of Delaware, NewarkGoogle Scholar
- Tehranirad, B., Shi, F., Kirby, J., Harris, J., & Grilli, S. (2012b). Tsunami benchmark results for fully nonlinear boussinesq wave model funwave-tvd, version 1.0. In: Proceedings and Results of the 2011 NTHMP Model Benchmarking Workshop, U.S. Department of Commerce/NOAA/NTHMP, NOAA Special Report, Boulder, CO, pp. 1–81. Available at http://nthmp.tsunami.gov
- Zahibo, N., Pelinovsky, E., Golinko, V., & Osipenko, N. (2006). Tsunami wave runup on coasts of narrow bays.
*International Journal of Fluid Mechanics Research*,*331*, 106–118.Google Scholar