Abstract
A run-up of irregular long sea waves on a beach with a constant slope is studied within the framework of the nonlinear shallow-water theory. This problem was solved earlier for deterministic waves, both periodic and pulse ones, using the approach based on the Legendre transform. Within this approach, it is possible to get an exact solution for the displacement of a moving shoreline in the case of irregular-wave run-up as well. It is used to determine statistical moments of run-up characteristics. It is shown that nonlinearity in a run-up wave does not affect the velocity moments of the shoreline motion but influences the moments of mobile shoreline displacement. In particular, the randomness of a wave field yields an increase in the average water level on the shore and decrease in standard deviation. The asymmetry calculated through the third moment is positive and increases with the amplitude growth. The kurtosis calculated through the fourth moment turns out to be positive at small amplitudes and negative at large ones. All this points to the advantage of the wave run-up on the shore as compared to a backwash at least for small-amplitude waves, even if an incident wave is a Gaussian stationary process with a zero mean. The probability of wave breaking during run-up and the applicability limits for the derived equations are discussed.
This is a preview of subscription content, log in via an institution to check access.
References
N. E. Vol’tsinger, K. A. Klevannyi, and E. N. Pelinovskii, Long-Wave Dynamics of Coastal Zone (Gidrometeoizdat, Leningrad, 1989) [in Russian].
H. Yeh and P. L.-F. Liu, C. Synolakis, Long-Wave Runup (World Sci, Singapore, 1996).
Ph. L.-F. Liu, H. Yeh, C. Synolakis, Advanced Numerical Models for Simulating Tsunami Waves and Runup (World Sci, Singapore, 2008).
B. V. Levin and M. A. Nosov, Tsunami Physics (Yanus-K, Moscow, 2005) [in Russian].
R. G. Dean and T. L. Walton, Wave Setup. A Handbook of Coastal and Ocean Engineering, Ed. by Y.C. Kim (World Sci., Singapore, 2009), pp. 1–23.
D. A. Huntley, R. T. Guza, and A. J. Bowen, “A Universal Form for Shoreline Run-Up Spectra,” J. Geophys. Res. 82(18), 2577–2581 (1977).
I. Didenkulova, E. Pelinovsky, and T. Soomere, “Run-Up Characteristics of Tsunami Waves of “Unknown” Shapes,” Pure Appl. Geophys., 165(11/12), 2249–2264 (2008).
S. N. Gurbatov, A. N. Malakhov, and A. I. Saichev, Nonlinear Random Waves in Nondispersion Media (Nauka, Moscow, 1990).
V. I. Klyatskin, Stochastic Equations and Waves in Randomly-Inhomogeneous Media (Nauka, Moscow, 1980) [in Russian].
I. I. Didenkulova and E. N. Pelinovsky, “Run-up of Long Waves on a Beach: The Influence of the Incident Wave Form,” Okeanologiya 48(1), 5–10 (2008) [Oceanology 48 (1), 1–6 (2008)].
Ch. Kharif, E. Pelinovsky, and A. Slunyaev, Rogue Waves in the Ocean (Springer, Berlin, Heidelberg, 2009).
Author information
Authors and Affiliations
Additional information
Original Russian Text © I.I. Didenkulova, A.V. Sergeeva, E.N. Pelinovsky, S.N. Gurbatov, 2010, published in Izvestiya AN. Fizika Atmosfery i Okeana, 2010, Vol. 46, No. 4, pp. 571–574.
Rights and permissions
About this article
Cite this article
Didenkulova, I.I., Sergeeva, A.V., Pelinovsky, E.N. et al. Statistical estimates of characteristics of long-wave run-up on a beach. Izv. Atmos. Ocean. Phys. 46, 530–532 (2010). https://doi.org/10.1134/S0001433810040122
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001433810040122