Long Wave Run-up Research Articles

Efficient Numerical Computations of Long-Wave Run-Up and Their Sensitivity to Grid Nesting

Efficient Numerical Computations of Long-Wave Run-Up and Their Sensitivity to Grid Nesting

Modeling the Evolution and Runup of Breaking Solitary and Solitary-Like Waves on Straight and Composite Slopes

Understanding the runup and inundation of long waves on coasts is of great importance for coastal community as flooding hazards are closely related to safety issues. For many years, solitary and solitary-like waves are frequently considered as a surrogate of extremely long waves for estimating runup and inundation. Since scaling issues are of concern when extending to real-world conditions, large-scale experiments for solitary waves on uniform beaches are reviewed and additional experiments for solitary waves on composite slopes are performed in this study. As such, those experimental data obtained from large-scale physical modeling can be used to validate numerical models and then to extend the range of parameters in terms of wave conditions and slope geometries which cannot be straightforwardly achieved in large-scale experimental works. Considering the computational efficiency, an open-source non-hydrostatic wave-flow model SWASH is used herein. Detailed model-data comparisons in terms of free surface elevation time series and maximum runup heights are carried out for long waves running up and down on beaches with different slope gradients to ensure the accuracy of the SWASH model for such applications. Finally, a simple method for estimating maximum shoreline excursion for solitary waves on a particularly designed composite slope is provided.

Numerical Simulation of Propagation and Run-Up of Long Waves in U-Shaped Bays

Wave propagation and run-up in U-shaped channel bays are studied here in the framework of the quasi-1D Saint-Venant equations. Our approach is numerical, using the momentum conserving staggered-grid (MCS) scheme, as a consistent approximation of the Saint-Venant equations. We carried out simulations regarding wave focusing and run-ups in U-shaped bays. We obtained good agreement with the existing analytical results on several aspects: the moving shoreline, wave shoaling, and run-up heights. Our findings also confirm that the run-up height is significantly higher in the parabolic bay than on a plane beach. This assessment shows the merit of the MCS scheme in describing wave focusing and run-up in U-shaped bays. Moreover, the MCS scheme is also efficient because it is based on the quasi-1D Saint-Venant equations.

Run-up of long waves on background shear currents

Long waves in shallow water propagating over a background shear current towards a sloping beach are investigated, and exact solutions are found using a hodograph transform and separation of variables. Inspired by the work of Carrier and Greenspan on steady waves over a uniform beach profile in the irrotational setting, we study waves which propagate over a background shear current. The shallow-water equations are obtained from the nonlinear Benney equations, and exact solutions are found with help of the hodograph transformation in conjunction with several further changes of variables. The hodograph transformation is effected by finding the Riemann invariants after the equations are written in the standard form of barotropic gas dynamics. In the current work, the background flow features zero mass flux, as would be required by a real flow at a beach. Moreover, in contrast with previous work, the present approach allows separate study of the influence of the strength of the shear current and the slope of the bottom profile. This enables us to provide an estimate of the run-up as a function of the shear flow while keeping the bottom slope constant.

Run-up of long waves generated by bottom-tilting wave maker

ABSTRACTUsing the recently developed bottom-tilting wave maker, run-ups of very long waves are investigated on an adjustable plane beach. Both leading-elevation waves and leading-depression waves are considered, in which the experimental results are compared with the numerical solution of the nonlinear shallow water equations. A surf similarity parameter based on the steepness of the accelerating phase of the wave is suggested. It is shown that both the maximum run-up height and the theoretical breaking criterion can be described in terms of the surf similarity parameter, which is consistent with the results for the solitary waves.

General initial value problem for the nonlinear shallow water equations: Runup of long waves on sloping beaches and bays

We formulate a new approach to solving the initial value problem of the shallow water-wave equations utilizing the famous Carrier–Greenspan transformation (Carrier and Greenspan (1957) [9]). We use a Taylor series approximation to deal with the difficulty associated with the initial conditions given on a curve in the transformed space. This extends earlier solutions to waves with near shore initial conditions, large initial velocities, and in more complex U-shaped bathymetries; and allows verification of tsunami wave inundation models in a more realistic 2-D setting.

Resonance of Long Wave Runup on South China Sea’s Piecewise Topographies

Solving the linear shallow water equations on a profile of four segments and matching the solutions on the turning points, runup of periodic long waves is obtained analytically. The resonant phenomena appear when periodic long waves propagate on piecewise topographies. According to the propagation path of tsunami waves, several profiles in the South China Sea are picked up. The topographies can be simplified to a four-segments profile which comprises a horizontal deep seabed, a continental foundation, a continental slope and a continental shelf from the trench to the shore. The runup amplification are calculated both analytically and numerically. Resonant phenomena appear and the values of runup are quite large at the resonant frequencies.

Run-Up of Long Waves in Piecewise Sloping U-Shaped Bays

We present an analytical study of the propagation and run-up of long waves in piecewise sloping, U-shaped bays using the cross-sectionally averaged shallow water equations. The nonlinear equations are transformed into a linear equation by utilizing the generalized Carrier–Greenspan transform (Rybkin et al. J Fluid Mech 748:416–432, 2014). The solution of the linear wave propagation is taken as the boundary condition at the toe of the last sloping segment, as in Synolakis (J Fluid Mech 185:523–545, 1987). We then consider a piecewise sloping bathymetry, and as in Kanoglu and Synolakis (J Fluid Mech 374:1–28, 1998), find the linear solution in the near shore region, which can be used as the boundary condition for the nonlinear problem. Our primary results are an analytical run-up law for narrow channels and breaking criteria for both monochromatic waves and solitary waves. The derived analytical solutions reduce to well-known solutions for parabolic bays and plane beaches. Our analytical predictions are verified in narrow bays via a comparison to direct numerical simulation of the 2-D shallow water equations.

Wave breaking and runup of long waves approaching a cliff over a variable bathymetry

Long waves approaching a cliff are significantly affected by a variable bathymetry. Wave breaking is observed, and determined by both the bathymetry and nonlinear interactions of the waves with the cliff and other waves. Geometric and kinematic measures are applied to determine a breaking criteria, however there are inconsistencies in both when the wave parameters are close to breaking/non-breaking interface values. Wave crest-cliff interaction is discussed in terms of its effect on the phase and fluid velocities. Runup amplification increases with the number of waves in the approaching wavepacket, their amplitude, and the bathymetry slope, but is approximately independent of the wave period. Runup amplifications approaching a factor of 12 are observed for a bathymetry approximating that of the Aran Islands, Ireland.

The run-up of nonlinearly deformed sea waves on the coast of a bay with a parabolic cross-section

The run-up of long waves on the coast of a bay with a parabolic cross-section, where the region of constant depth along the principal axis of the bay is connected with the linearly inclined segment, is considered. The study is carried out analytically in the framework of the nonlinear shallow-water theory under the approximation that the height of the initial wave is small compared to the basin depth, and the reflection from the inflection point of the bottom is negligibly small. Three types of incident waves, viz., a sinusoidal wave and solitary waves of positive and negative polarities, are considered in detail. It is shown that a sinusoidal wave undergoes nonlinear deformation at a segment of constant depth faster than solitary waves of positive and negative polarities. Solitary waves of negative polarity steepen somewhat faster than solitary waves of positive polarity. Waves of positive polarity steepen at wave front, while waves of negative polarity steepen at wave rear. These differences in steepness may become crucial at the wave run-up stage, since the wave run-up height on the coast of a bay with a parabolic cross-section is directly proportional to the steepness of a wave that arrives at the slope and can lead to the anomalous run-up of waves on the coast.

Tsunami evolution and run-up in a large scale experimental facility

In this paper, we study the propagation and run-up of long tsunami-like waves in the 300m long Large Wave Flume (GWK), Hannover, Germany and analyze the feasibility of experiments on tsunami run-up in large facilities. This paper is the continuation of our previous paper (Schimmels et al., 2015, companion paper). The propagation of long period waves over large distances for different shapes is studied experimentally and numerically. Fully nonlinear potential flow theory has been used to model these cases numerically along with Korteweg–de Vries simulations. The theoretical explanation of the observed effects has been discussed. The run-up characteristics of the studied waves with respect to their shape and beach slope were also investigated. Further, a downscaled real tsunami time series is also reproduced experimentally and studied with regard to its possible run-up.

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

By assuming the flow is uniform along the narrow long bays, the 2-D nonlinear shallow-water equations are reduced to a linear semi-axis variable-coefficient 1-D wave equation via the generalized Carrier–Greenspan transformation. The run-up of long waves in constantly sloping U-shaped and V-shaped bays is studied both analytically and numerically within the framework of the 1-D nonlinear shallow-water theory. An analytic solution, in the form of a double integral, to the resulting linear wave equation is obtained by utilizing the Hankel transform, and consequently the solution to the tsunami run-up problem is developed by applying the inverse generalized Carrier–Greenspan transform. The presented solution is a generalization of the solutions found by Carrier et al. (2003) and Didenkulova and Pelinovsky (2011) for the case of a plane beach and a parabolic bay, respectively. The shoreline dynamics in U-shaped and V-shaped bays are computed via a double integral through standard integration techniques.

Run-up of nonlinear long waves in U-shaped bays of finite length: analytical theory and numerical computations

Run-up of long waves in sloping U-shaped bays is studied analytically in the framework of the 1-D nonlinear shallow water theory. By assuming that the wave flow is uniform along the cross section, the 2-D nonlinear shallow water equations are reduced to a linear semi-axis variable-coefficient 1-D wave equation via the generalized Carrier-Greenspan transformation (Rybkin et al. in J Fluid Mech 748:416–432, 2014). A spectral solution is developed by solving the linear semi-axis variable-coefficient 1-D equation via separation of variables and then applying the inverse Carrier-Greenspan transform. To compute the run-up of a given long wave, a numerical method is developed to find the eigenfunction decomposition required for the spectral solution in the linearized system. The run-up of a long wave in a bathymetry characteristic of a narrow canyon is then examined.

Runup of Nonlinear Long Waves in Trapezoidal Bays: 1-D Analytical Theory and 2-D Numerical Computations

Long nonlinear wave runup on the coasts of trapezoidal bays is studied analytically in the framework of one-dimensional (1-D) nonlinear shallow-water theory with cross-section averaging, and is also studied numerically within a two-dimensional (2-D) nonlinear shallow water theory. In the 1-D theory, it is assumed that the trapezoidal cross-section channel is inclined linearly to the horizon, and that the wave flow is uniform in the cross-section. As a result, 1-D nonlinear shallow-water equations are reduced to a linear, semi-axis variable-coefficient 1-D wave equation by using the generalized Carrier–Greenspan transformation [Carrier and Greenspan (J Fluid Mech 1:97–109, 1958)] recently developed for arbitrary cross-section channels [Rybkinet al. (Ocean Model 43–44:36–51, 2014)], and all characteristics of the wave field can be expressed by implicit formulas. For detailed computations of the long wave runup process, a robust and effective finite difference scheme is applied. The numerical method is verified on a known analytical solution for wave runup on the coasts of an inclined parabolic bay. The predictions of the 1-D model are compared with results of direct numerical simulations of inundations caused by tsunamis in narrow bays with real bathymetries.

Run-up of long solitary waves of different polarities on a plane beach

We study the run-up of long solitary waves of different polarities on a beach in the case of composite bottom topography: a plane sloping beach transforms into a region of constant depth. We confirm that nonlinear wave deformation of positive polarity (wave crest) resulting in an increase in the wave steepness leads to a significant increase in the run-up height. It is shown that nonlinear effects are most strongly pronounced for the run-up of a wave with negative polarity (wave trough). In the latter case, the run-up height of such waves increases with their steepness and can exceed the amplitude of the incident wave.

Resonance phenomena at the long wave run-up on the coast

Abstract. Run-up of long waves on a beach consisting of three pieces of constant but different slopes is studied. Linear shallow-water theory is used for incoming impulse evolution, and nonlinear corrections are obtained for the run-up stage. It is demonstrated that bottom profile influences the run-up characteristics and can lead to resonance effects: increase of wave height, particle velocity, and number of oscillations. Simple parameterization of tsunami source through an earthquake magnitude is used to calculate the run-up height versus earthquake magnitude. It is shown that resonance effects lead to the sufficient increase of run-up heights for the weakest earthquakes, and a tsunami wave does not break on chosen bottom relief if the earthquake magnitude does not exceed 7.8.

Effect of asymmetry of incident wave on the maximum runup height

Didenkulova, I., Denissenko, P., Rodin, A. and Pelinovsky, E., 2013. Effect of asymmetry of incident wave on the maximum runup height.Shoaling and runup of long waves on a beach form a classical task for coastal oceanography and engineering. Though many empirical and theoretical formulae have been developed in this field, most of them are targeted to typical waves and situations, while the greatest hazard is caused by extreme events, such as, for example, extreme storms and catastrophic tsunamis. From this point of view it was shown theoretically that one of the most important parameters which influence the wave runup height is the steepness of the incident wave front and the asymmetry of the incident wave. It helped to provide a simple explanation to the extreme runup observed during the catastrophic 2004 Indonesian tsunami event. However, the theoretical results were obtained under many assumptions (ideal fluid, no wave breaking, no bottom friction) and have not been validated. Here we present an experimental study performed in the Large Wave Flume (GWK), Hannover, Germany, which is focused on the influence of the asymmetry caused by the non-linear deformation of incident waves on their runup on a plane beach. The series of experiments are aimed to validate the theoretical formulae for runup height of asymmetric waves. Obtained results are in a good agreement with theoretical predictions and corresponding formulae are recommended to be considered in wave forecasts.

DUNE EROSION ABOVE REVETMENTS

In a situation with a narrow dune, the dune base can be protected with a revetment to reduce dune erosion during extreme events. To quantify the effects of a revetment on storm impact, the functionality of the numerical storm impact model XBeach (Roelvink et al., 2009) is extended to account for the complex morphodynamics around revetments. Here the focus is on dune erosion above revetments, which is simulated with a simple avalanching algorithm that is triggered by the combined runup of short waves and long waves. The simulated runup statistics depend on the incident wave groupiness and associated long wave variance.

A depth-averaged 2D shallow water model for breaking and non-breaking long waves affected by rigid vegetation

This paper presents a depth-averaged two-dimensional shallow water model for simulating long waves in vegetated water bodies under breaking and non-breaking conditions. The effects of rigid vegetation are modelled in the form of drag and inertia forces as sink terms in the momentum equations. The drag coefficient is treated as a calibrated bulk constant and also determined using two empirical formulas as functions of stem Reynolds number, Froude number, and vegetation volume fraction. The governing equations are solved using an explicit finite-volume method based on rectangular mesh with the Harten, Lax, and van Leer approximate Riemann solver with second-order piecewise linear reconstruction for the streamwise convection fluxes, a second-order upwind scheme for the lateral convection fluxes, and a stable centred difference scheme for the water surface gradient terms. The model was tested using five laboratory experiments, including steady flow in a flume with alternate vegetation zones, solitary wave in a vegetated flatbed flume, long-wave runup on a partially-vegetated sloping beach, the dam-break wave overtopping an obstacle, and breaking the solitary wave on a sloping beach. The computed water levels, flow velocities, wave heights, and runups are in generally good agreement with experimental observations. The model was then applied to assess the hydrodynamic effectiveness and limitations of vegetation in coastal and river protection. It is shown that vegetation along the coastal shoreline has a positive benefit in reducing wave runup on sloping beaches, whereas vegetation in open channels causes conflicting impacts: reducing inundation in the downstream areas, but increasing flood risk in a certain distance upstream.

On the runup of long waves on a plane beach

Using the records of free surface fluctuations at several locations during the 2011 Japan Tohoku tsunami, we first show that the leading tsunami waves in both near‐field and far‐field regions are small amplitude long waves. These leading waves are very different from solitary waves. We then focus on investigating the evolution and runup of non‐breaking long waves on a plane beach, which is connected to a constant depth region. For this purpose, we develop a Lagrangian numerical model to solve the nonlinear shallow water equations. The Lagrangian approach tracks the moving shoreline directly without invoking any additional approximation. We also adopt and extend the analytical solutions by Synolakis (1987) and Madsen and Schäffer (2010) for runup and rundown of cnoidal waves and a train of multiple solitary waves. The analytical solutions for cnoidal waves compare well with the existing experimental data and the direct numerical results when wave amplitudes are small. However, large discrepancies appear when the incident amplitudes are finite. We also examine the relationship between the maximum runup height and the leading wave form. It is concluded that for a single wave the accelerating phase of the incident wave controls the maximum runup height. Finally, using the analytical solutions for the approximated wave forms of the leading tsunamis recorded at Iwate South station from the 2011 Tohoku Japan tsunami, we estimate the runup height.