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The generalized Carrier-Greenspan transform for the shallow water system with arbitrary initial and boundary conditions. (English) Zbl 1477.35177

Summary: We put forward a solution to the initial boundary value (IBV) problem for the nonlinear shallow water system in inclined channels of arbitrary cross section by means of the generalized Carrier-Greenspan hodograph transform [A. Rybkin et al., J. Fluid Mech. 748, 416–432 (2014; Zbl 1416.86007)]. Since the Carrier-Greenspan transform, while linearizing the shallow water system, seriously entangles the IBV in the hodograph plane, all previous solutions required some restrictive assumptions on the IBV conditions, e.g., zero initial velocity, smallness of boundary conditions. For arbitrary non-breaking initial conditions in the physical space, we present an explicit formula for equivalent IBV conditions in the hodograph plane, which can readily be treated by conventional methods. Our procedure, which we call the method of data projection, is based on the Taylor formula and allows us to reduce the transformed IBV data given on curves in the hodograph plane to the equivalent data on lines. Our method works equally well for any inclined bathymetry (not only plane beaches) and, moreover, is fully analytical for U-shaped bays. Numerical simulations show that our method is very robust and can be used to give express forecasting of tsunami wave inundation in narrow bays and fjords.

MSC:

35Q35 PDEs in connection with fluid mechanics
35L50 Initial-boundary value problems for first-order hyperbolic systems
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35Q31 Euler equations

Citations:

Zbl 1416.86007
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References:

[1] Alekseenko, S.; Dontsova, M.; Pelinovsky, D., Global solutions to the shallow water system with a method of an additional argument, Appl. Anal., 69, 9, 1444-1465 (2017) · Zbl 1375.35354
[2] Anderson, D.; Harris, M.; Hartle, H.; Nicolsky, D.; Pelinovsky, E.; Raz, A.; Rybkin, A., Run-up of long waves in piecewise sloping u-shaped bays, J. Pure Appl. Geophys., 174, 3185-3207 (2017)
[3] Antuono, M.; Brocchini, M., The boundary value problem for the nonlinear shallow water equations, Stud. Appl. Math., 119, 73-93 (2007)
[4] Antuono, M.; Brocchini, M., Solving the nonlinear shallow-water equations in physical space, J. Fluid Mech., 643, 207-232 (2010) · Zbl 1189.76083
[5] Carrier, G.; Greenspan, H., Water waves of finite amplitude on a sloping beach, J. Fluid Mech., 01, 97-109 (1958) · Zbl 0080.19504
[6] Carrier, G.; Wu, T.; Yeh, H., Tsunami run-up and draw-down on a plane beach, J. Fluid Mech., 475, 79-99 (2003) · Zbl 1051.86003
[7] Chugunov, V.; Fomin, S.; Noland, W.; Sagdiev, B., Tsunami runup on a sloping beach, Comput. Math. Methods, 2, e1081 (2020)
[8] Chugunov, V.; Fomin, S.; Shankar, R., Influence of underwater barriers on the distribution of tsunami waves, J. Geophys. Res. Oceans, 119, 7568-7591 (2014)
[9] Craig, W., Surface water waves and tsunamis, J. Dyn. Differ. Equ., 18, 3, 525-549 (2006) · Zbl 1207.76016
[10] Craig, W.; Groves, M., Hamiltonian long-wave approximations to the water-wave problem, Wave Motion, 19, 367-389 (1994) · Zbl 0929.76015
[11] Craig, W.; Guyenne, P.; Kalisch, H., A new model for large amplitude long internal waves, C. R. Mec., 332, 525-530 (2004) · Zbl 1223.35256
[12] Craig, W.; Guyenne, P.; Kalisch, H., Hamiltonian long wave expansions for free surfaces and interfaces, Commun. Pure Appl. Math., 58, 12, 1587-1641 (2005) · Zbl 1151.76385
[13] Craig, W.; Guyenne, P.; Nicholls, D.; Sulem, C., Hamiltonian long-wave expansions for water waves over a rough bottom, Proc. R. Soc. Lond. Ser. A, 461, 839-873 (2005) · Zbl 1145.76325
[14] Craig, W.; Wayne, C., Mathematical aspects of surface water waves, Russ. Math. Surv., 62, 3, 453-473 (2007) · Zbl 1203.76023
[15] Didenkulova, I.; Pelinovsky, E., Non-linear wave evolution and run-up in an inclined channel of a parabolic cross-section, Phys. Fluids, 23, 086602 (2011)
[16] Didenkulova, I.; Pelinovsky, E., Rogue waves in nonlinear hyperbolic systems (shallow-water framework), Nonlinearity, 24, R1-R18 (2011) · Zbl 1213.35297
[17] Dobrokhotov, S.; Medvedev, S.; Minenkov, D., On transforms reducing one-dimensional systems of shallow-water to the wave equation with sound speed \(c^2 = x\), Math. Notes, 93, 704-714 (2013) · Zbl 1307.76010
[18] Dobrokhotov, S.; Nazaikinskii, V.; Tirozzi, B., Asymptotic solution of the one-dimensional wave equation with localized initial data and with degenerating velocity: I, Russ. J. Math. Phys., 17, 4, 434-450 (2010) · Zbl 1387.35404
[19] Dobrokhotov, S.; Tirozzi, B., Localized solutions of one-dimensional non-linear shallow-water equations with velocity \(c=\sqrt{x} \), Russ. Math. Surv., 65, 1, 177-179 (2010) · Zbl 1196.35161
[20] Garayshin, V.; Harris, M.; Nicolsky, D.; Pelinovsky, E.; Rybkin, A., An analytical and numerical study of long wave run-up in u-shaped and v-shaped bays, Appl. Math. Comput., 297, 187-197 (2016) · Zbl 1410.86008
[21] Harris, M.; Nicolsky, D.; Pelinovsky, E.; Pender, J.; Rybkin, A., Run-up of nonlinear long waves in u-shaped bays of finite length: analytical theory and numerical computations, J. Ocean Eng. Mar. Energy, 2, 113-127 (2016)
[22] Harris, M.; Nicolsky, D.; Pelinovsky, E.; Rybkin, A., Runup of nonlinear long waves in trapezoidal bays: 1-D analytical theory and 2-D numerical computations, Pure Appl. Geophys., 172, 885-899 (2015)
[23] Johnson, RS, A Modern Introduction to the Mathematical Theory of Water Waves (1997), Cambridge: Cambridge University Press, Cambridge · Zbl 0892.76001
[24] Kanoglu, U., Nonlinear evolution and runup-drawdown of long waves over a sloping beach, J. Fluid Mech., 513, 363-372 (2004) · Zbl 1107.76016
[25] Kanoglu, U.; Synolakis, C., Initial value problem solution of nonlinear shallow water-wave equations, Phys. Rev. Lett., 148501, 97 (2006)
[26] Kanoglu, U., Synolakis, C. E.: Tsunami dynamics, forecasting, and mitigation. In: Shroder, J. F., Ellis, J. T., Sherman, D. (Eds.) Chapter 2 Hazards and Disasters Series: Coastal and Marine Hazards, Risks, and Disasters. Elsevier, pp. 15-57 (2015). doi:10.1016/B978-0-12-396483-0.00002-9
[27] Kanoglu, U.; Titov, V.; Bernard, E.; Synolakis, C., Tsunamis: bridging science, engineering and society, Philos. Trans. R. Soc. A, 373, 2053, 20140369 (2015)
[28] Lannes, D.: The water waves problem: mathematical analysis and asymptotics. In: Mathematical Surveys and Monographs. American Mathematical Society, Providence vol. 188, p 321 (2013) ISBN 978-0-8218-9470-5 · Zbl 1410.35003
[29] Madsen, P., Fuhrman, D., Schäffer, H.: On the solitary wave paradigm for tsunamis. J. Geophys. Res. Oceans 113, C12012 (2008). doi:10.1029/2008JC004932
[30] Nicolsky, D.; Pelinovsky, E.; Raza, A.; Rybkin, A., General initial value problem for the nonlinear shallow water equations: runup of long waves on sloping beaches and bays, Phys. Lett. A, 382, 38, 2738-2743 (2018)
[31] NTHMP (ed.): Proceedings and results of the 2011 NTHMP Model Benchmarking Workshop, NOAA Special Report, Boulder, CO. U.S. Department of Commerce/NOAA/NTHMP, National Tsunami Hazard Mapping Program [NTHMP], pp. 436 (2012)
[32] Pelinovsky, E.: Waves in geophysical fluids. In: Grue, J., Trulsen, K. (Eds.) Hydrodynamics of Tsunami Waves, pp. 1-48. CISM Courses and Lectures, No. 489. Springer, Berlin (2006) · Zbl 1241.76092
[33] Raz, A.; Nicolsky, D.; Rybkin, A.; Pelinovsky, E., Long wave run-up in asymmetric bays and in fjords with two separate heads, J. Geophys. Res. Oceans, 123, 3, 2066-2080 (2018)
[34] Rybkin, A.; Pelinovsky, E.; Didenkulova, I., Non-linear wave run-up in bays of arbitrary cross-section:generalization of the Carrier-Greenspan approach, J. Fluid Mech., 748, 416-432 (2014) · Zbl 1416.86007
[35] Stoker, J., Water Waves: The Mathematical Theory with Applications (1957), New York: Interscience Publishers, New York · Zbl 0078.40805
[36] Synolakis, C., The runup of solitary waves, J. Fluid Mech., 185, 523-545 (1987) · Zbl 0633.76021
[37] Synolakis, C., Tsunami runup on steep slopes: how good linear theory really is?, Nat. Hazards, 4, 221-234 (1991)
[38] Synolakis, C.; Bernard, E., Tsunami science before and beyond Boxing Day 2004, Philos. Trans. R. Soc. A, 364, 2231-2265 (2006)
[39] Synolakis, C.; Bernard, E.; Titov, V.; Kanoglu, U.; Gonzalez, F., Validation and verification of tsunami numerical models, Pure Appl. Geophys., 165, 2197-2228 (2008)
[40] Tuck, E.; Hwang, L., Long wave generation on a sloping beach, J. Fluid Mech., 51, 449-461 (1972) · Zbl 0234.76012
[41] Zahibo, N.; Pelinovsky, E.; Golinko, V.; Osipenko, N., Tsunami wave runup on coasts of narrow bays, Int. J. Fluid Mech. Res., 33, 106-118 (2006)
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