An analytical and numerical study of long wave run-up in U-shaped and V-shaped bays

Garayshin, V.V.; Harris, M. W.; Nicolsky, Dmitry; Pelinovsky, Efim; Rybkin, Alexei

doi:10.1016/j.amc.2016.01.005

Cited by 12 publications

(18 citation statements)

References 19 publications

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“…Similar to the previous numerical experiment, we consider a zero-velocity initial condition (η 0 , 0) and run it by the standard CG to compute the maximum run-up (t = t r ), rundown (t = t d ) and the secondary run-up t = t s . The secondary run-up of a Gaussian wave is a feature of the V-shaped bay as it was noted by Garayshin et al [20], Nicolsky et al [30]. Now, while modeling the wave dynamics, we save the time history of η (x, t) and u (x, t) at some point x = x * near the shore (e.g.…”

Section: Verification Of the Boundary Value Problemmentioning

confidence: 99%

“…(12) is the standard (constant coefficient) 1 + 1 wave equation and hence can be solved by the d'Alembert formula [15]. For an arbitrary power bay ( f (y) ∼ |y| m , m > 0, which corresponds to S(H ) ∼ H (m+1)/m ) there is no d'Alembert solution but a similar to (12) equation takes place, which can be solved by the very same techniques [20] as for the plane beach. Note that the plane beach corresponds to m = ∞.…”

Section: Introductionmentioning

confidence: 99%

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The Generalized Carrier–Greenspan Transform for the Shallow Water System with Arbitrary Initial and Boundary Conditions

et al. 2020

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We put forward a solution to the initial boundary value (IBV) problem for the nonlinear shallow water system in inclined channels of arbitrary cross section by means of the generalized Carrier-Greenspan hodograph transform (Rybkin et al. in J Fluid Mech, 748:416-432, 2014). Since the Carrier-Greenspan transform, while linearizing the shallow water system, seriously entangles the IBV in the hodograph plane, all previous solutions required some restrictive assumptions on the IBV conditions, e.g., zero initial velocity, smallness of boundary conditions. For arbitrary non-breaking initial conditions in the physical space, we present an explicit formula for equivalent IBV conditions in the hodograph plane, which can readily be treated by conventional methods. Our procedure, which we call the method of data projection, is based on the Taylor formula and allows us to reduce the transformed IBV data given on curves in the hodograph plane to the equivalent data on lines. Our method works equally well for any inclined bathymetry (not only plane beaches) and, moreover, is fully analytical for U-shaped bays. Numerical simulations show that our method is very robust and can be used to give express forecasting of tsunami wave inundation in narrow bays and fjords.

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Section: Verification Of the Boundary Value Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The Generalized Carrier–Greenspan Transform for the Shallow Water System with Arbitrary Initial and Boundary Conditions

et al. 2020

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“…Analytical solutions to the wave profile runup have also been found in more complex bathymetries using the CSA SWEs, for parabolic bays (Didenkulova & Pelinovsky, ) and more generally in U‐shaped and V‐shaped bays (Anderson et al, ; Didenkulova & Pelinovsky, ; Garayshin et al, ; Rybkin et al, ). Preliminary results by Harris et al (), Garayshin et al (), and Anderson et al () show that the runup at the head of the U‐shaped bay computed by the 2‐D SWEs equations and that computed by the CSA SWEs are comparable in height and timing. A limitation of the previously mentioned results for the U‐shaped bays is the symmetric and simple nature of the cross sections of the bathymetries under consideration.…”

Section: Introductionmentioning

confidence: 99%

“…We recall that the vast majority of results that have been derived by use of Carrier-Greenspan (CG) transform in the context of plane beaches, for example, Carrier and Greenspan (1957), Synolakis (1991Synolakis ( , 1987, Carrier et al (2003), and Kanoglu and Synolakis (2006). Analytical solutions to the wave profile runup have also been found in more complex bathymetries using the CSA SWEs, for parabolic bays (Didenkulova & Pelinovsky, 2011b) and more generally in U-shaped and V-shaped bays (Anderson et al, 2017;Didenkulova & Pelinovsky, 2011a;Garayshin et al, 2016;Rybkin et al, 2014). Preliminary results by Harris et al (2015), Garayshin et al (2016), and Anderson et al (2017) show that the runup at the head of the U-shaped bay computed by the 2-D SWEs equations and that computed by the CSA SWEs are comparable in height and timing.…”

Section: Introductionmentioning

confidence: 99%

Long Wave Runup in Asymmetric Bays and in Fjords With Two Separate Heads

et al. 2018

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Modeling of tsunamis in glacial fjords prompts us to evaluate applicability of the cross‐sectionally averaged nonlinear shallow water equations to model propagation and runup of long waves in asymmetrical bays and also in fjords with two heads. We utilize the Tuck‐Hwang transformation, initially introduced for the plane beaches and currently generalized for bays with arbitrary cross section, to transform the nonlinear governing equations into a linear equation. The solution of the linearized equation describing the runup at the shore line is computed by taking into account the incident wave at the toe of the last sloping segment. We verify our predictions against direct numerical simulation of the 2‐D shallow water equations and show that our solution is valid both for bays with an asymmetric L‐shaped cross section, and for fjords with two heads—bays with a W‐shaped cross section.

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“…Mathematically, the validity of channel theory application for solving such problems was proved in [14]. From the practical point of view, its accuracy is assessed by a comparison with direct numerical solution of shallow water equations for waves in the bay of Alaska [15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Wave Dynamics in the Channels of Variable Cross-Section

Pelinovsky¹,

Didenkulova²,

Shurgalina³

2017

PhO

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Dynamics of long sea waves in the channels of variable depth and variable rectangular cross-section is discussed within various approximations -from the shallow water equations to those of nonlinear dispersion theory. General approach permitting to find traveling (non-reflective) waves in inhomogeneous channels is demonstrated within the framework of the shallow water linear theory. The appropriate conditions are determined by solving a system of ordinary differential equations. The so-called self-consistent channel in which the width is connected with its depth in a specific way is studied in detail. Within the linear theory of shallow water, a wave does not reflect from the bottom irregularities. The wave shape remains unchanged on the records of the wave gauges (mareographs) fixed along the channel axis, but it varies in space. Nonlinearity and dispersion lead to the wave transformation in such a channel. Within the framework of the shallow water weakly nonlinear theory, the wave shape is described by the Riemann solution, and the wave breaks (gradient catastrophe) quicker in the zones of decreasing depth. The modified Korteweg -de Vries equation describing evolution of a solitary wave of weak but finite amplitude in a self-consistent channel, the depth of which can vary arbitrary, is derived. Some examples of a solitary wave transformation in such a channel are analyzed (particularly, a soliton adiabatic transformation in the channel with the slowly varying parameters, and a solitary wave fission into the group of solitons after it has passed the zone where the depth changes abruptly. The obtained solutions extend the class of those represented earlier by S.F. Dotsenko and his colleagues.

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An analytical and numerical study of long wave run-up in U-shaped and V-shaped bays

Cited by 12 publications

References 19 publications

The Generalized Carrier–Greenspan Transform for the Shallow Water System with Arbitrary Initial and Boundary Conditions

The Generalized Carrier–Greenspan Transform for the Shallow Water System with Arbitrary Initial and Boundary Conditions

Long Wave Runup in Asymmetric Bays and in Fjords With Two Separate Heads

Wave Dynamics in the Channels of Variable Cross-Section

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