Tsunami run-up and draw-down on a plane beach

Abstract

Tsunami run-up and draw-down motions on a uniformly sloping beach are evaluated based on fully nonlinear shallow-water wave theory. The nonlinear equations of mass conservation and linear momentum are first transformed to a single linear hyperbolic equation. To solve the problem with arbitrary initial conditions, we apply the Fourier–Bessel transform, and inversion of the transform leads to the Green function representation. The solutions in the physical time and space domains are then obtained by numerical integration. With this semi-analytic solution technique, several examples of tsunami run-up and draw-down motions are presented. In particular, detailed shoreline motion, velocity field, and inundation depth on the shore are closely examined. It was found that the maximum flow velocity occurs at the moving shoreline and the maximum momentum flux occurs in the vicinity of the extreme draw-down location. The direction of both the maximum flow velocity and the maximum momentum flux depend on the initial waveform: it is in the inshore direction when the initial waveform is predominantly depression and in the offshore direction when the initial waves have a dominant elevation characteristic.

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; …”

Validation of the GPU-Accelerated CFD Solver ELBE for Free Surface Flow Problems in Civil and Environmental Engineering

“…The benchmark reference data for this 2D runup problem was obtained from the analytical solution of NSW equations of Carrier et al [40], for the initial tsunami wave profile shown in Figure 5 (without initial velocity) and a given beach slope (1:10). The benchmark task was to compute the free surface elevation in the runup region at three points in time (t = 160, 175, 220 s) and to compare those to the reference solution.…”

“…The viscosity for the given LB parameters in Figure 5a still is several orders of magnitude higher than the viscosity of water (ν ≈ 25 m 2 /s), but a further decrease does not significantly change the results. [40] (alternating dots and dashes) for three points in time.…”

Depression and elevation tsunami waves in the framework of the Korteweg–de Vries equation

“…In this paper, we are concerned with the third phase, where the increasing effect of nonlinearity may lead to quite different outcomes, depending on the wave polarity as the wave transverses the deep ocean, see Carrier et al (2003) and Fernando et al (2008) for instance. Although initial depression waves are potentially just as damaging as initial elevation waves, they have not received the same attention, although we note the theoretical studies by Synolakis (1994, 1996), and most recently by Grimshaw et al (2015), the analysis of field data by Soloviev and Mazova (1994), and the experiments of Kobayashi and Lawrence (2004), Klettner et al (2012), Rossetto et al (2011) and Charvet et al (2013).…”

“…Many studies of the development of a tsunami as it approaches the shore have used the nonlinear shallow water equations to examine the connection between the incident wave mass, amplitude and polarity, on the shoreline impact, see for instance Synolakis (1994, 1996), Carrier et al (2003), Madsen and Schaffer (2010) and Didenkulova and Pelinovsky (2011). In particular, studies by Didenkulova (2009), Didenkulova et al (2006Didenkulova et al ( , 2007 and Pelinovsky (2006) using the nonlinear shallow water equations have elucidated the role of initial steepness in increasing the eventual run-up height, and we especially note that Didenkulova et al (2014) found that this nonlinear steepness effect was enhanced when the initial wave was one of depression.…”