Runup of nonlinear asymmetric waves on a plane beach

Didenkulova, Ira; Pelinovsky, Efim; Soomere, Tarmo; Zahibo, Narcisse

doi:10.1007/978-3-540-71256-5_8

Cited by 38 publications

(56 citation statements)

References 25 publications

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“…In particular, we especially note in the context of this paper the analyses using the nonlinear shallow water equations by Didenkulova (2009), Didenkulova et al (2006Didenkulova et al ( , 2007, and Pelinovsky (2006) which demonstrate the role of initial nonlinear steepness in increasing the eventual runup height, and that by Didenkulova et al (2014) who found that this nonlinear steepness effect was enhanced when the initial wave was one of depression. However, although these models have proved valuable and insightful, they are non-dispersive and hence do not capture the effects of wavenumber dispersion as the tsunami waves develop shorter length scales in their propagation shoreward.…”

Section: Introductionmentioning

confidence: 73%

Changing forms and sudden smooth transitions of tsunami waves

Grimshaw

Hunt

Chow

2014

J. Ocean Eng. Mar. Energy

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In some tsunami waves travelling over the ocean, such as the one approaching the eastern coast of Japan in 2011, the sea surface of the ocean is depressed by a small metre-scale displacement over a multi-kilometre horizontal length scale, lying in front of a positive elevation of comparable magnitude and length, which together constitute a down-up wave. Shallow water theory shows that the latter travels faster than the former, leading to an interaction, whose description is the issue addressed in this paper, using model equations of the Korteweg-de Vries type. First, we re-examine the undular bore solutions of the Korteweg-de Vries equation which describe how an initial depression wave deforms into a depression rarefaction wave followed by an undular bore of large elevation waves riding on this depression. Then we develop a new extended Korteweg-de Vries equation some of whose solutions can be used to describe the interaction of an elevation wave chasing a depression wave. These show that the two waves coincide at a given position and time producing a maximum elevation. Typically this amplitude is larger than the initial displacement magnitude by a factor which can be as large as two, which may explain anomalous elevations of tsunamis at particular positions along their trajectories. It is physically significant that for these small amplitude waves, no wave breaking occurs and there is no excess dissipation. Then, following the transition, the elevation wave moves ahead of the depression wave and the distance between them increases either linearly or

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Section: Introductionmentioning

confidence: 73%

Changing forms and sudden smooth transitions of tsunami waves

Grimshaw

Hunt

Chow

2014

J. Ocean Eng. Mar. Energy

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“…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.…”

Section: Introductionmentioning

confidence: 70%

“…Second, in the deep ocean a tsunami is essentially a long water wave of relatively small amplitude and hence propagates without significant change of form at a speed ffiffiffiffiffi gh p where h is the ocean depth. In this phase, the tsunami can be either a wave of depression, or a wave of elevation, or a combination of these, see the recent assessments by Didenkulova et al (2007), Arcas and Segur (2012) and Dias et al (2014). In the third phase, the tsunami wave propagates shorewards from the deep ocean over the continental slope and shelf into shallow water.…”

Section: Introductionmentioning

confidence: 98%

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

Grimshaw

Yuan

2016

Nat Hazards

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Although tsunamis in the deep ocean are very long waves of quite small amplitudes, as they propagate shorewards into shallow water, nonlinearity becomes important and the structure of the leading waves depends on the polarity of the incident wave from the deep ocean. In this paper, we use a variable-coefficient Korteweg-de Vries equation to examine this issue, for an initial wave which is either elevation, or depression, or a combination of each. We show that the leading waves can be described by a reduction of the Whitham modulation theory to a solitary wave train. We find that for an initial elevation, the leading waves are elevation solitary waves with an amplitude which varies inversely with the depth, with a pre-factor which is twice the maximum amplitude in the initial wave. By contrast, for an initial depression, the leading wave is a depression rarefaction wave, followed by a solitary wave train whose maximum amplitude of the leading wave is determined by the square root of the mass in the initial wave.

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“…Formally, linear theory is not valid in the vicinity of the shoreline where the wave amplitude becomes comparable with the water depth. In the case of a plane beach of a constant slope it has been demonstrated that the extreme runup characteristics can be calculated rigorously from linear shallow water theory even for the nonlinear problem [Synolakis, 1991;Didenkulova et al, 2006Didenkulova et al, , 2007.…”

Section: Wave Runup Along a Convex Beachmentioning

confidence: 99%

“…The solution of the nonlinear problem strongly depends on the initial waveshape. Various shapes of the periodic incident wave trains such as the sine wave [Kaistrenko et al, 1991;Madsen and Fuhrman, 2007], cnoidal wave [Synolakis, 1991] and nonlinear deformed periodic waves [Didenkulova et al, 2006[Didenkulova et al, , 2007 have been analyzed in the literature. The relevant analysis has been also performed for a variety of solitary waves and single pulses such as soliton [Pedersen and Gjevik, 1983;Synolakis, 1987;Kânoglu, 2004], sine pulse , Lorentz pulse [Pelinovsky and Mazova, 1992], Gaussian pulse [Carrier et al, 2003;Kânoglu and Synolakis, 2006], N waves [Tadepalli and Synolakis, 1994], and ''characterized tsunami waves'' [Tinti and Tonini, 2005].…”

Section: Introductionmentioning

confidence: 99%

Long surface wave dynamics along a convex bottom

Didenkulova¹,

Pelinovsky²,

Soomere³

2009

J. Geophys. Res.

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Long linear wave transformation in a basin of varying depth is studied for a case of a convex bottom profile h(x) ∼ x4/3. This bottom geometry provides the “nonreflecting” wave propagation at least in the framework of the one‐dimensional shallow water equation. In this case, shoaling effects are very strong and wave reflection occurs in the immediate vicinity of the shoreline. The existence of traveling wave solutions (which propagate without reflection) in this geometry is established through construction of a 1:1 transformation of the general 1‐D wave equation to the analogous wave equation with constant coefficients. The general solution of the Cauchy problem consists of two traveling waves propagating in opposite directions and allows a detailed description of the wavefield (vertical displacement and depth‐averaged flow). It is found that generally a zone of weak current is formed between these two waves. Waves are reflected from the coastline so that their profile is inverted with respect to the calm water surface. Long‐wave runup on a beach with this profile is studied for the sine pulse, Korteweg‐de Vries soliton, and N wave. It is shown that in certain cases the runup height along the convex profile is considerably larger than for beaches with a linear slope. The analysis of wave reflection from the border of a shallow coastal area of constant depth and a section with the convex profile shows that a transmitted wave always has a sign‐variable shape. Results of the wave transformation above the convex beach and beaches following a general power law are compared. This simplified model demonstrates the potential importance of the tsunami wave transformation along convex beaches.

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Runup of nonlinear asymmetric waves on a plane beach

Cited by 38 publications

References 25 publications

Changing forms and sudden smooth transitions of tsunami waves

Changing forms and sudden smooth transitions of tsunami waves

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

Long surface wave dynamics along a convex bottom

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