Non-breaking and breaking solitary wave run-up

Abstract

The run-up of non-breaking and breaking solitary waves on a uniform plane beach connected to a constant-depth wave tank was investigated experimentally and numerically. If only the general characteristics of the run-up process and the maximum run-up are of interest, for the case of a breaking wave the post-breaking condition can be simplified and represented as a propagating bore. A numerical model using this bore structure to treat the process of wave breaking and subsequent shoreward propagation was developed. The nonlinear shallow water equations (NLSW) were solved using the weighted essentially non-oscillatory (WENO) shock capturing scheme employed in gas dynamics. Wave breaking and post-breaking propagation are handled automatically by this scheme and ad hoc terms are not required. A computational domain mapping technique was used to model the shoreline movement. This numerical scheme was found to provide a relatively simple and reasonably good prediction of various aspects of the run-up process. The energy dissipation associated with wave breaking of solitary wave run-up (excluding the effects of bottom friction) was also estimated using the results from the numerical model.

Runup of Tsunami Waves in U-Shaped Bays

“…In the nonlinear theory the same effects can be found for a particular case of the linearly inclined bay with a parabolic cross-section m = 2 ). The solution of the nonlinear problem in this case can be obtained with the use of the Legendre (hodograph) transformation, which has been very popular for long wave runup on a plane beach (Carrier and Greenspan, 1958;Pedersen and Gjevik, 1983;Synolakis, 1987;Tadepalli and Synolakis, 1996;Li, 2000;Li and Raichlen, 2001;Carrier et al, 2003;Kânoğlu, 2004;Tinti and Tonini, 2005;Kânoğlu and Synolakis, 2006;Didenkulova et al, 2006;2008a;Antuono and Brocchini, 2007;Pritchard and Dickinson, 2007) and is valid for non-breaking waves. In this case the nonlinear system (3) can be reduced to the linear equation (Choi et al, 2008; …”

On the solitary wave paradigm for tsunamis

“…Since the early 1970s, it has been frequently assumed that solitary (or cnoidal) waves can be used to model some of the important features of tsunamis approaching the beach and shoreline, and that these theories, originating from the KdV equation, can define the proper input waves for physical or mathematical models of tsunamis. Examples from the literature are numerous see, e.g., Goring [1978], Synolakis [1986Synolakis [ , 1987, , , Tadepalli and Synolakis [1994], Yeh et al [1994], , Liu et al [1995], Li [2000], Li and Raichlen [2001, Tonkin et al [2003], Jensen et al [2003], Kobayashi and Lawrence [2004], Craig [2006], Synolakis and Bernard [2006] and Lakshmanan [2007]. Another example of the popularity of this concept is the recent NOAA Technical Memorandum [Synolakis et al, 2007] discussing necessary analytical and experimental benchmarking for numerical tsunami models.…”

Experimental Investigation on Reflection and Energy Loss of Tsunami Over the Permeable Beach

“…These variables are incident wave height (Hi), reflecting wave height (Hr), water depth in the constant depth region (d), run-up height (Ru), water density (w), sand density (s), sand diameter (D), the angle of the slope (), the acceleration of gravity (g), total energy of incident wave (Ei), and total energy of reflecting wave (Er). This expression can be written as a functional form: 8) Choosing the variables d, s and g as the independent physical variables, the following dimensionless parameters are obtained via  theory: 9) where Hi/d is dimensionless incident wave height, Hr/d is dimensionless reflecting wave height, Ru/d is dimensionless run-up height, Gsb = s / w is specific gravity of sand, D/d is dimensionless diameter of sand, cot is dimensionless slope angle, Ei /wd 3 is dimensionless total energy of incident wave, and Er /wd 3 is dimensionless total energy of reflecting wave, E/wd 3 is the dimensionless wave energy loss.…”

“…Then, during the run-down process, the potential energy begins to transform to kinetic energy with the kinetic energy being equal to the potential energy. During this process, the total energy is constant where the energy is conserved [9].…”