scholarly journals Evolution of solitary waves and undular bores in shallow-water flows over a gradual slope with bottom friction

2007 ◽  
Vol 585 ◽  
pp. 213-244 ◽  
Author(s):  
G. A. EL ◽  
R. H. J. GRIMSHAW ◽  
A. M. KAMCHATNOV
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Solitary Waves ◽  
Periodic Wave ◽  
Local Effect ◽  
Bottom Friction ◽  
Undular Bore ◽  
Shallow Water Flows ◽  
Adiabatic Approach ◽  
Variable Topography

This paper considers the propagation of shallow-water solitary and nonlinear periodic waves over a gradual slope with bottom friction in the framework of a variable-coefficient Korteweg–de Vries equation. We use the Whitham averaging method, using a recent development of this theory for perturbed integrable equations. This general approach enables us not only to improve known results on the adiabatic evolution of isolated solitary waves and periodic wave trains in the presence of variable topography and bottom friction, modelled by the Chezy law, but also, importantly, to study the effects of these factors on the propagation of undular bores, which are essentially unsteady in the system under consideration. In particular, it is shown that the combined action of variable topography and bottom friction generally imposes certain global restrictions on the undular bore propagation so that the evolution of the leading solitary wave can be substantially different from that of an isolated solitary wave with the same initial amplitude. This non-local effect is due to nonlinear wave interactions within the undular bore and can lead to an additional solitary wave amplitude growth, which cannot be predicted in the framework of the traditional adiabatic approach to the propagation of solitary waves in slowly varying media.

scholarly journals Combined Effect of Rotation and Topography on Shoaling Oceanic Internal Solitary Waves

2014 ◽  
Vol 44 (4) ◽  
pp. 1116-1132 ◽  
Author(s):  
Roger Grimshaw ◽  
Chuncheng Guo ◽  
Karl Helfrich ◽  
Vasiliy Vlasenko
Keyword(s):  
Solitary Wave ◽  
Wave Packet ◽  
Numerical Simulations ◽  
Solitary Waves ◽  
Combined Effect ◽  
Internal Solitary Wave ◽  
Internal Solitary Waves ◽  
De Vries ◽  
Ostrovsky Equation ◽  
Variable Topography

Abstract Internal solitary waves commonly observed in the coastal ocean are often modeled by a nonlinear evolution equation of the Korteweg–de Vries type. Because these waves often propagate for long distances over several inertial periods, the effect of Earth’s background rotation is potentially significant. The relevant extension of the Kortweg–de Vries is then the Ostrovsky equation, which for internal waves does not support a steady solitary wave solution. Recent studies using a combination of asymptotic theory, numerical simulations, and laboratory experiments have shown that the long time effect of rotation is the destruction of the initial internal solitary wave by the radiation of small-amplitude inertia–gravity waves, and the eventual emergence of a coherent, steadily propagating, nonlinear wave packet. However, in the ocean, internal solitary waves are often propagating over variable topography, and this alone can cause quite dramatic deformation and transformation of an internal solitary wave. Hence, the combined effects of background rotation and variable topography are examined. Then the Ostrovsky equation is replaced by a variable coefficient Ostrovsky equation whose coefficients depend explicitly on the spatial coordinate. Some numerical simulations of this equation, together with analogous simulations using the Massachusetts Institute of Technology General Circulation Model (MITgcm), for a certain cross section of the South China Sea are presented. These demonstrate that the combined effect of shoaling and rotation is to induce a secondary trailing wave packet, induced by enhanced radiation from the leading wave.

scholarly journals Gaussian solitary waves to Boussinesq equation with dual dispersion and logarithmic nonlinearity

2018 ◽  
Vol 23 (6) ◽  
pp. 942-950 ◽  
Author(s):  
Anjan Biswasa ◽  
Mehmet Ekici ◽  
Abdullah Sonmezoglu
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Solitary Waves ◽  
Water Waves ◽  
Trial Function ◽  
Boussinesq Equation ◽  
Solitary Wave Solutions ◽  
Shallow Water Waves ◽  
Wave Solutions ◽  
Extended Trial

This paper discusses shallow water waves that is modeled with Boussinesq equation that comes with dual dispersion and logarithmic nonlinearity. The extended trial function scheme retrieves exact Gaussian solitary wave solutions to the model.

Propagation of solitary waves through significantly curved shallow water channels

1998 ◽  
Vol 362 ◽  
pp. 157-176 ◽  
Author(s):  
AIMIN SHI ◽  
MICHELLE H. TENG ◽  
THEODORE Y. WU
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Solitary Waves ◽  
Wave Amplitude ◽  
Water Channels ◽  
Numerical Results ◽  
Channel Width ◽  
Boussinesq Model ◽  
Reflected Wave ◽  
Transmission And Reflection

Propagation of solitary waves in curved shallow water channels of constant depth and width is investigated by carrying out numerical simulations based on the generalized weakly nonlinear and weakly dispersive Boussinesq model. The objective is to investigate the effects of channel width and bending sharpness on the transmission and reflection of long waves propagating through significantly curved channels. Our numerical results show that, when travelling through narrow channel bends including both smooth and sharp-cornered 90°-bends, a solitary wave is transmitted almost completely with little reflection and scattering. For wide channel bends, we find that, if the bend is rounded and smooth, a solitary wave is still fully transmitted with little backward reflection, but the transmitted wave will no longer preserve the shape of the original solitary wave but will disintegrate into several smaller waves. For solitary waves travelling through wide sharp-cornered 90°-bends, wave reflection is seen to be very significant, and the wider the channel bend, the stronger the reflected wave amplitude. Our numerical results for waves in sharp-cornered 90°-bends revealed a similarity relationship which indicates that the ratios of the transmitted and reflected wave amplitude, excess mass and energy to the original wave amplitude, mass and energy all depend on one single dimensionless parameter, namely the ratio of the channel width b to the effective wavelength λe. Quantitative results for predicting wave transmission and reflection based on b/λe are presented.

Observations of fine structure changes in shoaling internal solitary waves based on seismic oceanography method

2021 ◽  
Author(s):  
Haibin Song ◽  
Yi Gong ◽  
Yongxian Guan ◽  
Wenhao Fan ◽  
Yunyan Kuang
Keyword(s):  
Fine Structure ◽  
Solitary Wave ◽  
Phase Velocity ◽  
Shallow Water ◽  
Solitary Waves ◽  
Internal Solitary Wave ◽  
Internal Solitary Waves ◽  
Seismic Events ◽  
Velocity Amplitude ◽  
Seismic Oceanography

<p>In the study of shoaling internal solitary waves, the observation and research on the internal fine structure and the effect of the topography are still insufficient. We try to make up for such insufficiency by seismic oceanography method. A first-mode depression internal solitary wave was observed propagating on the continental slope in the northeast South China Sea near Dongsha Atoll. We used common offset gathers (COGs) to obtain a series of images of this internal solitary wave that evolved over time, and studied the changes in internal fine structure by analyzing the seismic events in COG migrated sections. We found that the seismic events were broken during the shoaling, which was caused by the instability induced by internal solitary wave. We picked six events which represent six waveform and analyzed their evolution. It was found that the change in shape of waveform at different depths is different. The waveform in deep water deforms before that in shallow water, and the waveform in shallow water deforms to a greater degree. In addition, we also counted four parameters of phase velocity, amplitude, wavelength, and slopes of front and rear during the shoaling. The results show that the phase velocity and amplitude of waveform in shallow water increases, the wavelength decreases, and the slope of rear gradually becomes larger than that of the front. We have compared the observed changes with previous study made by numerical simulation.</p>

The Atmospheric Solitary Wave

1955 ◽  
Vol 36 (10) ◽  
pp. 511-518 ◽  
Author(s):  
Abdul Jabbar Abdullah
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Solitary Waves ◽  
Local Weather

Solitary waves have been observed in shallow water. In the present paper some evidence is presented for the existence of similar phenomena in the atmosphere. A possible mechanism for the formation of atmospheric solitary waves is described, and a case study is discussed in support of this mechanism. Some speculations are made about some possible effects of these disturbances on the local weather.

scholarly journals Modelling turbulence generation in solitary waves on shear shallow water flows

2015 ◽  
Vol 773 ◽  
pp. 49-74 ◽  
Author(s):  
G. L. Richard ◽  
S. L. Gavrilyuk
Keyword(s):  
Shallow Water ◽  
Solitary Waves ◽  
Velocity Fluctuation ◽  
Small Region ◽  
Vertical Coordinate ◽  
Water Flows ◽  
Shallow Water Flows ◽  
Dispersive Model ◽  
Wave Profiles ◽  
Turbulence Generation

We derive a dispersive model of shear shallow water flows which takes into account a non-uniform horizontal velocity. This model generalizes the Green–Naghdi model to the case of shear flows. Besides the classical dispersion term in the Green–Naghdi model related to the acceleration of the free surface, it also contains a new dispersion parameter related to the flow structure. This parameter is related to the second moment of the velocity fluctuation with respect to the vertical coordinate. The distinction between shearing and turbulence based on the scale of variation of the velocity fluctuation is proposed. In particular, an equation for the turbulence generation is derived. Solitary waves for this model are obtained in explicit form. Comparison of solitary wave profiles with experimental ones is also performed. The agreement is very good apart from the small region near the top of the wave.

Bifurcations of Solitary Waves of a Simple Equation

2020 ◽  
Vol 30 (09) ◽  
pp. 2050138 ◽  
Author(s):  
Jiaopeng Yang ◽  
Rui Liu ◽  
Yiren Chen
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Traveling Wave ◽  
Wave Speed ◽  
Blow Up ◽  
Periodic Wave ◽  
Simple Equation ◽  
Singular Line ◽  
Wave System ◽  
Bifurcation Method

In this paper, we consider a simple equation which involves a parameter [Formula: see text], and its traveling wave system has a singular line. Firstly, using the qualitative theory of differential equations and the bifurcation method for dynamical systems, we show the existence and bifurcations of peak-solitary waves and valley-solitary waves. Specially, we discover the following novel properties: (i) In the traveling wave system, there exist infinitely many periodic orbits intersecting at a point, or two points and passing through the singular line, and there is no singular point inside a homoclinic orbit. (ii) When [Formula: see text], in the equation there exist three types of bifurcations of valley-solitary waves including periodic wave, blow-up wave and double solitary wave. (iii) When [Formula: see text], in the equation there exist two types of bifurcations of valley-solitary wave including periodic wave and blow-up wave, but there is no double solitary wave bifurcation. Secondly, we perform numerical simulations to visualize the above properties. Finally, when [Formula: see text] and the constant wave speed equals [Formula: see text], we give exact expressions to the above phenomena.

scholarly journals Staggered Conservative Scheme for 2-Dimensional Shallow Water Flows

Fluids ◽  
2020 ◽  
Vol 5 (3) ◽  
pp. 149
Author(s):  
Novry Erwina ◽  
Didit Adytia ◽  
Sri Redjeki Pudjaprasetya ◽  
Toni Nuryaman
Keyword(s):  
Wave Propagation ◽  
Numerical Model ◽  
Solitary Wave ◽  
Shallow Water ◽  
Threshold Value ◽  
Staggered Grid ◽  
Conservative Scheme ◽  
Shallow Water Flows ◽  
Run Up ◽  
N Wave

Simulating discontinuous phenomena such as shock waves and wave breaking during wave propagation and run-up has been a challenging task for wave modeller. This requires a robust, accurate, and efficient numerical implementation. In this paper, we propose a two-dimensional numerical model for simulating wave propagation and run-up in shallow areas. We implemented numerically the 2-dimensional Shallow Water Equations (SWE) on a staggered grid by applying the momentum conserving approximation in the advection terms. The numerical model is named MCS-2d. For simulations of wet–dry phenomena and wave run-up, a method called thin layer is used, which is essentially a calculation of the momentum deactivated in dry areas, i.e., locations where the water thickness is less than the specified threshold value. Efficiency and robustness of the scheme are demonstrated by simulations of various benchmark shallow flow tests, including those with complex bathymetry and wave run-up. The accuracy of the scheme in the calculation of the moving shoreline was validated using the analytical solutions of Thacker 1981, N-wave by Carrier et al., 2003, and solitary wave in a sloping bay by Zelt 1986. Laboratory benchmarking was performed by simulation of a solitary wave run-up on a conical island, as well as a simulation of the Monai Valley case. Here, the embedded-influxing method is used to generate an appropriate wave influx for these simulations. Simulation results were compared favorably to the analytical and experimental data. Good agreement was reached with regard to wave signals and the calculation of moving shoreline. These observations suggest that the MCS method is appropriate for simulations of varying shallow water flow.

scholarly journals Stability of solitary waves for a generalized higher-order Shallow water equation

Filomat ◽  
2014 ◽  
Vol 28 (5) ◽  
pp. 1007-1017 ◽  
Author(s):  
Nurhan Dündar ◽  
Necat Polat
Keyword(s):  
Solitary Wave ◽  
Shallow Water ◽  
Solitary Waves ◽  
Shallow Water Equation ◽  
Higher Order ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Existence And Stability ◽  
Stability Of Solitary Waves

In this work, we consider solitary wave solutions of a generalized higher-order shallow water equation. We investigate the existence and stability of solitary waves of the equation.

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