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Numerical Study for Run-Up of Breaking Waves of Different Polarities on a Sloping Beach

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Extreme Ocean Waves

Abstract

The transformation and run-up of long breaking bell-shaped wave pulses of various polarities is studied numerically in the nonlinear shallow-water theory framework using CLAWPACK software. The considered water basin contains a section of constant depth and a section of a slopping beach. For small-amplitude incident waves regardless their polarity, the results of numerical computations usually coincide with predictions of the nonlinear shallow-water theory for non-breaking waves. Nonlinear effects start to be important when incident wave is located far from the shoreline even for initially small-amplitude waves. With further increase in incident wave amplitude, the wave transforms into the shock wave (bore) before approaching the beach. Run-up characteristics of waves of different polarities are compared. Nonlinear effects and induced energy dissipation caused by wave breaking during its run-up on a beach are more prominent for negative pulses rather than for positive ones.

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References

  • Antuono M, Brocchini M (2007) The boundary value problem for the nonlinear shallow water equation. Stud Appl Math 119:71–91

    Article  Google Scholar 

  • Antuono M, Brocchini M (2008) Maximum run-up, breaking conditions and dynamical forces in the swash zone: a boundary value approach. Coast Eng 55:732–740

    Article  Google Scholar 

  • Antuono M, Brocchini M (2010) Solving the nonlinear shallow-water equations in physical space. J Fluid Mech 643:207–232

    Article  Google Scholar 

  • Bernatskiy A, Nosov M (2012) The role of bottom friction in models of nonbreaking Tsunami wave runup on the shore. Izv Atmos Ocean Phys 48:427–431

    Article  Google Scholar 

  • Carrier GF, Greenspan HP (1958) Water waves of finite amplitude on a sloping beach. J Fluid Mech 4:97–109

    Article  Google Scholar 

  • Carrier GF, Wu TT, Yeh H (2003) Tsunami run-up and draw-down on a plane beach. J Fluid Mech 475:79–99

    Article  Google Scholar 

  • Choi BH, Pelinovsky E, Kim DC, Didenkulova I (2008) Two- and three-dimensional computation of solitary wave runup on non-plane beach. Nonlinear Process Geophys 15:489–502

    Article  Google Scholar 

  • Denissenko P, Didenkulova I, Pelinovsky E, Pearson J (2011) Influence of the nonlinearity on statistical characteristics of long wave runup. Nonlinear Process Geophys 18:967–975

    Article  Google Scholar 

  • Denissenko P, Didenkulova I, Rodin A, Listak M, Pelinovsky E (2013) Experimental statistics of long wave runup on a plane beach. J Coast Res SI65:195–200

    Google Scholar 

  • Didenkulova I (2009) New trends in the analytical theory of long sea wave runup. In: Quak E, Soomere T (eds) Applied wave mathematics: selected topics in solids, fluids, and mathematical methods. Springer, Berlin, pp 265–296

    Google Scholar 

  • Didenkulova I, Pelinovsky EN (2008) Run-up of long waves on a beach: the influence of the incident wave form. Oceanology 48:1–6

    Article  Google Scholar 

  • Didenkulova I, Pelinovsky E (2009) Non-dispersive traveling waves in inclined shallow water channels. Phys Lett A 373(42):3883–3887

    Article  Google Scholar 

  • Didenkulova I, Pelinovsky E (2011a) Nonlinear wave evolution and runup in an inclined channel of a parabolic cross-section. Phys Fluids 23(8):086602

    Article  Google Scholar 

  • Didenkulova I, Pelinovsky E (2011b) Rogue waves in nonlinear hyperbolic systems (shallow-water framework). Nonlinearity 24:R1–R18

    Article  Google Scholar 

  • Didenkulova I, Pelinovsky E (2011c) Runup of tsunami waves in U—shaped bays. Pure Appl Geophys 168:1239–1249

    Article  Google Scholar 

  • Didenkulova I, Zahibo N, Kurkin AA, Levin BV, Pelinovsky EN, Soomere T (2006a) Runup of nonlinearly deformed waves on a coast. Dokl Earth Sci 411:1241–1243

    Article  Google Scholar 

  • Didenkulova I, Zahibo N, Kurkin A, Pelinovsky EN (2006b) Steepness and spectrum of a nonlinearly deformed wave on shallow waters. Izv Atmos Ocean Phys 42:773–776

    Article  Google Scholar 

  • Didenkulova I, Kurkin A, Pelinovsky E (2007a) Run-up of solitary waves on slopes with different profiles. Izv Atmos Ocean Phys 43:384–390

    Article  Google Scholar 

  • Didenkulova I, Pelinovsky E, Soomere T, and Zahibo N (2007b). In: Kundu A (ed) Runup of nonlinear asymmetric waves on a plane beach. Tsunami and nonlinear waves, Springer, Heidelberg, pp 175–190

    Google Scholar 

  • Didenkulova I, Pelinovsky E, Soomere T (2008) Run-up characteristics of tsunami waves of “unknown” shapes. Pure Appl Geophys 165:2249–2264

    Article  Google Scholar 

  • Didenkulova II, Sergeeva AV, Pelinovsky EN, Gurbatov SN (2010) Statistical estimates of characteristics of long-wave run-up on a beach. Izv Atmos Ocean Phys 46:530–532

    Article  Google Scholar 

  • Didenkulova I, Pelinovsky E, Sergeeva A (2011) Statistical characteristics of long waves nearshore. Coast Eng 58:94–202

    Article  Google Scholar 

  • Didenkulova I, Denissenko P, Rodin A, Pelinovsky E (2013) Effects of wave asymmetry on its runup on a beach. J Coast Res SI 65:207–212

    Google Scholar 

  • Didenkulova II, Pelinovsky EN, Didenkulov OI (2014) Run-up of long solitary waves of different polarities on a plane beach. Izv Atmos Ocean Phys 50(5):532–538

    Article  Google Scholar 

  • Dobrokhotov SY, Tirozzi B (2010) Localized solutions of one-dimensional non-linear shallow-water equations with velocity c \(=\) (x)1/2. Russ Math Surv 65:177–179

    Article  Google Scholar 

  • Dutykh D, Labart C, Mitsotakis D (2011) Long wave runup on random beaches. Phys Rev Lett 107:184504

    Article  Google Scholar 

  • Ezersky A, Abcha N, Pelinovsky E (2013a) Physical simulation of resonant wave run-up on a beach. Nonlinear Process Geophys 20:35–40

    Article  Google Scholar 

  • Ezersky A, Tiguercha D, Pelinovsky E (2013b) Resonance phenomena at the long wave run-up on the coast. Nat Hazards Earth Syst Sci 13:2745–2752

    Article  Google Scholar 

  • Kânoğlu U (2004) Nonlinear evolution and runup-drawdown of long waves over a sloping beach. J Fluid Mech 513:363–372

    Article  Google Scholar 

  • Kânoğlu U, Synolakis C (2006) Initial value problem solution of nonlinear shallow water-wave equations. Phys Rev Lett 97:148501

    Article  Google Scholar 

  • LeVeque RJ (2004) Finite volume methods for hyperbolic problems. Cambridge University Press, Cambridge

    Google Scholar 

  • Madsen PA, Fuhrman DR (2008) Run-up of tsunamis and periodic long waves in terms of surf-similarity. Coast Eng 55:209–223

    Article  Google Scholar 

  • Madsen PA, Schaffer HA (2010) Analytical solutions for tsunami run-up on a plane beach: single waves, \(N\)-waves and transient waves. J Fluid Mech 645:27–57

    Article  Google Scholar 

  • Mazova RK, Osipenko NN, Pelinovsky EN (1991) Solitary wave climbing a beach without breaking. Rozpr Hydrotech 54:71–80

    Google Scholar 

  • Pedersen G, Gjevik B (1983) Runup of solitary waves. J Fluid Mech 142:283–299

    Article  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 6:227–249

    Article  Google Scholar 

  • Pelinovsky EN, Rodin AA (2011) Nonlinear deformation of a large-amplitude wave on shallow water. Dokl Phys 56:305–308

    Article  Google Scholar 

  • PelinovskyE N, Rodin AA (2012) Transformation of a strongly nonlinear wave in a shallow-water basin. Izv Atmos Ocean Phys 48:383–390

    Google Scholar 

  • Pritchard D, Dickinson L (2007) The near-shore behaviour of shallow-water waves with localized initial conditions. J Fluid Mech 591:413–436

    Article  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

    Article  Google Scholar 

  • Shermeneva M, Shugan I (2006) The computations of the wave runup on gentle beach. Lett J Tech Phys 32:33–38

    Article  Google Scholar 

  • Soloviev SL, Mazova RK (1994) On the influence of sign of leading tsunami wave on run-up height on the coast. Sci Tsunami Hazards 12:2531

    Google Scholar 

  • Spielvogel LO (1975) Runup of single waves on a sloping beach. J Fluid Mech 74:685–694

    Article  Google Scholar 

  • Stefanakis TS, Dias F, Dutykh D (2011) Local run-up amplification by resonant wave interactions. Phys Rev Lett 107:124502

    Article  Google Scholar 

  • Stoker JJ (1957) Water waves: mathematical theory with applications. Interscience Publishers, New York

    Google Scholar 

  • Synolakis CE (1991) Tsunami runup on steep slopes: how good linear theory really is? Nat Hazards 4:221–234

    Article  Google Scholar 

  • Synolakis CE, Deb MK, Skjelbreia JE (1988) The anomalous behavior of the run-up of cnoidal waves. Phys Fluids 31:3–5

    Article  Google Scholar 

  • Synolakis CE (1987) The runup of solitary waves. J Fluid Mech 185:523–545

    Article  Google Scholar 

  • Tadepalli S, Synolakis CE (1994) The runup of \(N\)-waves. Proc R Soc Lond A445:99–112

    Article  Google Scholar 

  • Tinti S, Tonini R (2005) Analytical evolution of Tsunamis induced by near-shore earthquakes on a constant-slope ocean. J Fluid Mech 535:33–64

    Article  Google Scholar 

  • Zahibo N, Pelinovsky E, Golinko V, Osipenko N (2006) Tsunami wave runup on coasts of narrow bays. Int J Fluid Mech Res 33:106–118

    Article  Google Scholar 

  • Zahibo N, Didenkulova I, Kurkin A, Pelinovsky E (2008) Steepness and spectrum of nonlinear deformed shallow water wave. Ocean Eng 35:47–52

    Article  Google Scholar 

Download references

Acknowledgments

The results presented in this paper are obtained with the support of State Contract No 2014/133 and grants RFBR (14-02-00983, 14-05-00092, 15-35-20563, 15-55-45053), MK-1146.2014.5 and SF0140007s11. Authors also acknowledge the support from CENS through the European Regional Development Fund (ERDF). Some aspects of the appearance of the extreme run-up characteristics are considered in the framework of a Volkswagen grant.

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Authors and Affiliations

  1. Nizhny Novgorod State Technical University n. a. R.E. Alekseev, Nizhny Novgorod, Russia

    Artem Rodin, Ira Didenkulova & Efim Pelinovsky

  2. Institute of Cybernetics at Tallinn University of Technology, Tallinn, Estonia

    Artem Rodin

  3. Marine Systems Institute, Tallinn University of Technology, Tallinn, Estonia

    Ira Didenkulova

  4. Institute of Applied Physics, Nizhny Novgorod, Russia

    Efim Pelinovsky

  5. Special Research Bureau for Automation of Marine Researches, Yuzhno-Sakhalinsk, Russia

    Efim Pelinovsky

Authors
  1. Artem Rodin
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  2. Ira Didenkulova
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  3. Efim Pelinovsky
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Correspondence to Artem Rodin .

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Editors and Affiliations

  1. Institute of Applied Physics and Nizhny Novgorod State Technical University, Nizhny Novgorod, Russia

    Efim Pelinovsky

  2. Institut de Recherche sur les Phénomènes Hors Equilibre (IRPHE), Marseille, France

    Christian Kharif

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Rodin, A., Didenkulova, I., Pelinovsky, E. (2016). Numerical Study for Run-Up of Breaking Waves of Different Polarities on a Sloping Beach. In: Pelinovsky, E., Kharif, C. (eds) Extreme Ocean Waves. Springer, Cham. https://doi.org/10.1007/978-3-319-21575-4_9

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