Tsunami evolution and run-up in a large scale experimental facility
Introduction
For modeling of tsunami wave propagation, shoaling and run-up, often solitary or cnoidal waves have been used (Goring, 1979, Synolakis, 1987). However, solitary and cnoidal waves are rarely observed during tsunami propagation. Furthermore, their characteristics are not comparable with most real-field tsunamis as reported by Madsen et al. (2008). Nevertheless, a set of solitary and cnoidal-like waves can be formed in the far field as a result of long wave transformation into an undular bore, as it was observed during the 2004 tsunami in the Indian Ocean near the coast of Thailand (Fig. 1).
More importantly, one cannot generalize a particular tsunami case. In reality, there is a variety of tsunami waves in terms of both periods and wave shapes. Due to the large-distance propagation and complicated bottom and coastal topography, the initial wave changes its shape. Hence, very often even the same tsunami event has very different manifestations in different locations. Therefore, when modeling tsunami, one should also study different wave shapes.
Fig. 2 illustrates the variations in the tsunami profile that occurred along the Japanese coast during the Japan Sea tsunami in 1983. Based on the past tsunami events, tsunamis approaching the shore may broadly be classified as (e.g. Shuto, 1985)
- 1.
non-breaking waves that act as a rapidly rising tide, observed during small and moderate tsunami events after short distance propagation;
- 2.
breaking bore or hydraulic jump (wall of water), observed as a result of wave breaking during large tsunami events after short distance propagation;
- 3.
undular bore, observed after long distance propagation (in terms of wavelength), i.e. the disintegration into series of solitons, see Fig. 1.
The first rigorous solution to the nonlinear shallow water equations was found by Carrier and Greenspan (1958) for non-breaking wave run-up on a plane beach. Based on this approach various shapes of the periodic incident wave trains such as sine wave (Madsen and Fuhrman, 2008), cnoidal wave (Synolakis et al., 1988) and nonlinear deformed periodic wave (Didenkulova et al., 2007a) have been analyzed in the literature. Relevant analysis has also been performed for a variety of solitary waves and single pulses such as soliton (Kânoğlu, 2004, Pedersen and Gjevik, 1983, Synolakis, 1987), sine pulse (Mazova et al., 1991), Lorentz pulse (Pelinovsky and Mazova, 1992), Gaussian pulse (Carrier et al., 2003, Kânoğlu and Synolakis, 2006), N-waves (Tadepalli and Synolakis, 1994), “characterized tsunami waves” (Tinti and Tonini, 2005) and a random set of solitons (Brocchini and Gentile, 2001).
Didenkulova and Pelinovsky (2008) and Didenkulova et al. (2007b) showed that despite the influence of nonlinearity, the differences in shape for all bell-shaped positive pulses, such as solitary wave, Lorentz and sinusoidal pulses are negligible and can be parameterized. This parameterization is especially important for the estimation of tsunami characteristics at the coast, when the shape of the wave approaching the coast is unknown. Also, the asymmetry of a wave approaching the coast and the steepness of its front leads to significant amplification of wave run-up height and shoreline velocity, which was observed during the catastrophic 2004 Indian Ocean tsunami (Didenkulova et al., 2007a). It is also obvious that the bottom and coastal topography has a strong influence on wave run-up characteristics.
It has been demonstrated by Didenkulova et al. (2009) and Didenkulova and Pelinovsky (2010) that some profiles (~ x4/3 from Tinti et al. (2001) and ~x4) mathematically correspond to “nonreflecting” wave propagation and can lead to abnormal wave amplification on the beach. The same effects are also observed in narrow bays (Didenkulova and Pelinovsky, 2009, Didenkulova and Pelinovsky, 2011). The developed theory was successfully applied to explain abnormal run-ups during 2009 Samoa tsunami (Didenkulova, 2013) and 2011 Japan tsunami (Kim et al., 2013).
Breaking bores or hydraulic jumps have been widely studied both theoretically and experimentally. A special case, the so-called dam break problem, became a benchmark for numerical simulations (Liu et al., 2008) and is often used for testing the stability of coastal structures (Haehnel and Daly, 2004, Riggs et al., 2013).
As mentioned above, an undular bore is usually observed after long distance propagation, as it was during the Indian Ocean tsunami (see Fig. 1). Hence, it should be studied in facilities which allow long distance wave propagation, such as the Large Wave Flume (Großer Wellenkanal, GWK), Hannover, Germany. As such facilities are just a few, most undular bore studies are numerical (Brühl and Oumeraci, 2014, Grue et al., 2008; Sriram et al., 2007).
It has been demonstrated in the companion paper (Schimmels et al., 2015) that existing testing facilities with piston type wave makers (having a large stroke) can effectively be used to generate long waves, including very long tsunami-like waves. In this paper, we will focus on the propagation and run-up characteristics of these long waves and will consider waves of various shapes including real (scaled down) tsunami records.
The paper is arranged in the following manner. After a brief description of the applied numerical models based on fully nonlinear potential flow theory (FNPT) and Korteweg–de Vries (KdV), the experimental data of wave propagation are compared with the results of numerical simulations. Then, the conditions for undular bore formation are also verified theoretically and finally the same waves/wave trains propagate onshore and climb “hypothetical” beaches of different slopes, confirming that by constructing a mild slope in a large scale experimental facility, one can study the characteristics of tsunami waves of various types.
Section snippets
Nonlinear potential flow theory (FNPT)
The fluid is assumed to be incompressible. The flow is irrotational and viscous forces are neglected in this model. This simplifies the flow problem to be defined by Laplace's equation involving a velocity potential Φ (x,z) given by,
A potential flow in a sub-domain with a wave maker at one end and nonlinear free surface boundary conditions is considered. Prescribed Neumann and Dirichlet boundary conditions are applied on the boundaries. Considering the flume bottom as flat with no flow
Experimental test cases
The large scale experimental study was carried out in the Large Wave Flume (Großer Wellenkanal, GWK) of Forschungszentrum Küste (FZK). A detailed description of the experimental setup and the generation methodology has been given in Schimmels et al. (2015) and will not be repeated here. However, for the completeness of the study, the tested cases are reported in Table 1. For the generation of the waves, Schimmels et al. (2015) used a generic approach by combining several solitons (sech2-waves)
Long wave propagation
In this section, the results of the previous test cases are reported and discussed along with the numerical modeling based on FNPT-FEM. Since, the generated waves are long, in addition to FNPT-FEM simulations, the Korteweg–de Vries model is also used to compare it with the experimental data. This will give us the range of validity of used numerical models.
The test cases are classified into five different categories based on their shape, namely, regular waves, solitary waves of positive
Comparison between experiments and numerical simulations
The length of the tank for the FNPT-FEM simulations was 300 m, with a numerical beach at the far end. The numbers of nodes used in the simulation are 800 and 16 in the x and y directions, respectively with a time step of 0.05 s. This has been kept constant throughout this paper, as it produced acceptable results. The details about the numerical convergence of this model can be referred in Sriram et al., 2006, Sriram et al., 2010. The simulations were carried out on a Dell Laptop with 1.28 GB RAM
Wave breaking and undular bore formation
The evolution of long waves in shallow water depends on the wave amplitude (Stoker, 1957, Teles Da Silva and Peregrine, 1990). Large amplitude waves (ηmax/d ≥ 0.5) propagate following a non-dispersive scenario, getting steeper and steeper during propagation and eventually form a breaking bore. Waves of smaller amplitude are influenced by dispersive effects and evolve into an undular bore. All waves considered in this paper are of small amplitude (ηmax/d < 0.5) and, therefore, follow the second
Run-up of long waves in GWK
As it has been mentioned before, one of the main questions in tsunami related studies are run-up and its impact on the coast. From this perspective, it is extremely important to make sure that the generated waves can also be used for this kind of studies. Based on the current data, one can easily question the height of the generated waves, which was rather small, and might not lead to sufficient run-up.
The natural slope in GWK is quite steep 1:6. Considering our water depth of 1 m the length of
Conclusions
The potential of large scale facilities and in particular the 300 meter long Large Wave Flume (GWK) for tsunami wave propagation and run-up has been examined. Three types of long waves: non-breaking surging waves, breaking bore (or hydraulic jump) and undular bore (or split-up of waves) have been considered in the feasibility analysis. The generation of non-breaking waves has been demonstrated by the example of a downscaled tsunami time-series and small-amplitude elongated solitary waves. For
Acknowledgment
V. Sriram and I. Didenkulova would like to thank the Alexander von Humboldt (AvH) foundation for their grants and network meetings and RFBR-DST program for grants RFBR 15-55-45053 and INT/RUS/RFBR/P-203. I. Didenkulova further acknowledges the basic part of the State Contract No 2016/133 and grant MD-6373.2016.5. A. Sergeeva would like to acknowledge support from the Volkswagen Foundation and RFBR grants 14-02-00983, 14-05-00092 and 15-35-20563. Authors are very grateful to Dr. Vasily Titov
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