Shoaling Internal Solitary Waves and the Formation of Boluses

An internal solitary wave of elevation in a two-layer density stratified system of an incompressible, viscous and homogeneous fluid was studied. The run-up of a wave of elevation encountering different slopes was investigated numerically based on solving the continuity, Navier-Stokes and convective-diffusion equations within the Boussinesq approximation. The commercial software COMSOL Multiphysics was used to conduct the numerical simulations. For gradual shoals, a bolus formed that transported dense fluid up the shoal. The bolus disappeared when it reached its maximum height on the slope due to the draining of the dense fluid. Various shoal angles were simulated to detect the critical angle above which a bolus does not form. An angle of 30 or less resulted in the formation of a bolus. In addition, the simulations demonstrated that the size of the bolus induced by shallower slopes was larger and that the vertical height traveled by the bolus was insensitive to the slope of the shoal.

Microwave backscattering from the sea surface with breaking waves

Based on the effective medium approximation theory of composites, the whitecap-covered sea surface is treated as a medium layer of dense seawater droplets and air. Two electromagnetic scattering models of randomly rough surface are applied to the investigation of microwave backscattering of breaking waves driven by strong wind. The shapes of seawater droplets are considered by calculating the effective dielectric constant of the whitecap layer. The responses of seawater droplets shapes, such as sphere and ellipsoid, to the backscattering coefficient are discussed. Numerical results of the models are in good agreement with the experimental measurements of horizontally and vertically polarized backscattering at microwave frequency 13.9GHz and different incidence angles.

Physical Investigation of Directional Wave Focusing and Breaking Waves in Wave Basin

An experimental scheme for the generation of directional focusing waves in a wave basin is established in this paper. The effects of the directional range, frequency width and center frequency on the wave focusing are studied. The distribution of maximum amplitude and the evolution of time series and spectra during wave packet propagation and the variation of water surface parameters are extensively investigated. The results reveal that the characteristics of focusing waves are significantly influenced by wave directionality and that the breaking criteria for directional waves are distinctly different from those for unidirectional waves.

One-Dimensional Horizontal Boussinesq Model Enhanced for Non-Breaking and Breaking Waves

Based on a set of fully nonlinear Boussinesq equations up to the order of O（μ^2, ε^3μ^2） （where ε is the ratio of wave amplitude to water depth and ,μ is the ratio of water depth to wave length） a numerical wave model is formulated. The model＇s linear dispersion is acceptably accurate to μ ≌ 1.0, which is confirmed by comparisons between the simulat- ed and measured time series of the regular waves propagating on a submerged bar. The moving shoreline is treated numer- ically by replacing the solid beach with a permeable beach. Run-up of nonbreaking waves is verified against the analytical solution for nonlinear shallow water waves. The inclusion of wave breaking is fulfilled by introducing an eddy term in the momentum equation to serve as the breaking wave force term to dissipate wave energy in the surf zone. The model is applied to cross-shore motions of regular waves including various types of breaking on plane sloping beaches. Comparisons of the model test results comprising spatial distribution of wave height and mean water level with experimental data are presented.

A VOF-based numerical model for breaking waves in surf zone

This paper introduces a numerical model for studying the evolution of a periodic wave train, shoaling, and breaking in surf zone. The model can solve the Reynolds averaged Navier-Stokes （RANS） equations for a mean flow, and the k-ε equations for turbulence kinetic energy k and turbulence dissipation rate ε. To track a free surface, the volume of fluid （VOF） function, satisfying the advection equation was introduced. In the numerical treatment, third-order upwind difference scheme was applied to the convection terms of the RANS equations in order to reduce the effect of numerical viscosity. The shoaling and breaking processes of a periodic wave train on gently sloping beaches were modeled. The computed wave heights of a sloping beach and the distribution of breaking wave pressure on a vertical wall were compared with laboratory data.

Energy dissipation through wind-generated breaking waves

Wave breaking is an important process that controls turbulence properties and fluxes of heat and mass in the upper oceanic layer.A model is described for energy dissipation per unit area at the ocean surface attributed to wind-generated breaking waves,in terms of ratio of energy dissipation to energy input,windgenerated wave spectrum,and wave growth rate.Also advanced is a vertical distribution model of turbulent kinetic energy,based on an exponential distribution method.The result shows that energy dissipation rate depends heavily on wind speed and sea state.Our results agree well with predictions of previous works.

A second order volume of fluid (VOF) scheme for numerical simulation of 2-D breaking waves

Among all environmental forces acting on ocean structures and marine vessels, those resulting from wave impacts are likely to yield the highest loads. Being highly nonlinear, transient and complex, a theoretical analysis of their impact would be impossible without numerical simulations. In this paper, a pressure-split two-stage numerical algorithm is proposed based on Volume Of Fluid （VOF） methodology. The algorithm is characterized by introduction of two pressures at each half and full cycle time step, and thus it is a second-order accurate algorithm in time. A simplified second-order Godunov-type solver is used for the continuity equations. The method is applied to simulation of breaking waves in a 2-D water tank, and a qualitative comparison with experimental photo observations is made. Quite consistent results are observed between simulations and experiments. Commercially available software and Boundary Integral Method （BIM） have also been used to simulate the same problem. The results from present code and BIM are in good agreement with respect to breaking location and timing, while the results obtained from the comrnercial software which is only first-order accurate in time has clearly showed a temporal and spatial lag, verifying the need to use a higher order numerical scheme.

Experimental Study of Pore Water Pressure and Bed Profile Change Under Regular Breaking Waves

There lies a close relationship between the seabed destruction and the distribution of pore water pressure under the action of breaking wave. The experiments were carried out in a wave flume with a 1：30 sloping sandy seabed to study regular breaking wave induced pore water pressure. A wide range of measurements from the regular wave runs were reported, including time series of wave heights, pore pressures. The video records were analysed to measure the time development of the seabed form and the characteristics of the orbital motion of the sand in the wave breaking region. The pore water pressure in the breaker zone showed the time variation depending on the wave phases including wave breaking and bore propagation. The time-averaged pore water pressure was higher near the seabed surface. The peak values of pore water pressure increase significantly at the breaking point. The direction of pore water pressure difference forces in the breaker zone is of fundamental importance for a correct description of the sediment dynamics. The upwards- directed pressure differences may increase sand transport by reducing the effective weight of the sediment, thereby increasing the bed form evolution. The seabed configuration changed greatly at the wave breaking zone and a sand bar was generated remarkably. The amplitude of the pore water pressure changed with the seabed surface. The results are to improve the understanding of sand transport mechanisms and seabed responses due to breaking regular waves over a sloping sandy bed.

A method for detecting the breaking of wind-generated waves in deep water

The breaking of wind-generated waves is an important phenomenon in the ocean, having close relation to many aspects of the ocean, such as air-sea interaction, ocean wave dynamics, oceanic remote sensing and ocean engineering. The first problem encountered in both its theoretical study and practical measurement is how to detect the breaking of waves.

The study on the bottom friction and the breaking coefficient for typhoon waves in radial sand ridges—the Lanshayang Channel as an example

Owing to the interactions among the complex terrain, bottom materials, and the complicate hydrodynam-ics, typhoon waves show special characteristics as big waves appeared at the high water level （HWL） and small waves emerged at low and middle water levels （LWL and MWL） in radial sand ridges （RSR）. It is as-sumed that the mud damping, sandy bed friction and wave breaking effects have a great influence on the typhoon wave propagation in this area. Under the low wave energy, a mud layer will form and transport into the shallow area, thus the mud damping effects dominate at the LWL and the MWL. And high Collins coef-ficient （c around 1） can be applied to computing the damping effects at the LWL and the MWL. But under the high wave energy, the bottom sediment will be stirred and suspended, and then the damping effects disappear at the HWL. Thus the varying Collins coefficient with the water level method （VCWL） is imple-mented into the SWAN to model the typhoon wave process in the Lanshayang Channel （LSYC） of the RSR, the observed wave data under “Winnie” （“9711”） typhoon was used as validation. The results show that the typhoon wave in the RSR area is able to be simulated by the VCWL method concisely, and a constant wave breaking coefficient （γ） equaling 0.78 is better for the RSR where wide tidal flats and gentle bed slopes exist.

Transformation and Breaking of Irregular Waves on Very Gentle Slopes

Based on theoretical analysis, numerical calculation, and experimental study. this paper discusses breaker indices of irregular waves, transformation of wave spectrum, characteristics and computation of breaking waves, as well as the critical beach slope under which waves will not break. Computed results are in good agreement with laboratory physical model test data and ocean wave field measurements.

Detection of Breaking Waves Using X-Band Pulse Radar in the Nearshore Region

An X-band pulsed Doppler microwave radar has been used to determine the characteristics of breaking waves. Field experiments were conducted at the Shuang-Si estuary in the north of Taiwan in the winter of 2005. Analyses on maxima radar cross section and Doppler frequency shift are done to characterize wave breaking zones. Based on observations of breaking waves, the wave breaking zones are shown to be located at water depths of 1.8 to 2.2 m in the experimental site. In general, the results indicate that a radar system has the potential to delineate the spatial variation of breaking waves clearly and that this is sufficient to achieve a measurement operation for near-shore air-sea interaction events.

A novel synthetic aperture radar scattering model for sea surface with breaking waves

In this study a novel synthetic aperture radar(SAR)scattering model for sea surface with breaking waves is proposed.Compared with existing models,the proposed model considers an empirical relationship between wind speed and wave breaking scattering to present the contribution of wave breaking.Moreover,the scattering weight factor p,and wave breaking rate q,are performed to present the contribution of the quasi-specular scattering term,Bragg scattering term,and wave breaking scattering term to the total scattering from the sea surface.To explore the modeling accuracy of sea-surface scattering,a simulated normalized radar cross-section(NRCS)and measured NRCS are compared.The proposed model generated the simulated NRCS and a matching GF-3 dataset was used for the measured NRCS.It was revealed that the performance of the VV polarization of our model was much better than that of HH polarization,with a correlation of 0.91,bias of-0.14 dB,root mean square error(RMSE)of 1.26 dB,and scattering index(SI)of-0.11.In addition,the novel model is explored and compared with the geophysical model of CMODs and satellite-measured NRCS from GF-3 SAR wave mode imagery.For an incidence angle 40°–41°,the relationship between the NRCS and wind speed,relative wind direction is proposed.As with the SAR-measured NRCS,the performance of VV polarization was much better than HH polarization,with a correlation of 0.99,bias of-0.25 dB,RMSE of 0.64 dB,and SI of-0.04.

Study on the Set-up and Set-down Induced by Breaking Waves Over A Reef

This paper proposes an equation to calculate breaking wave induced wave set-up and set-down along reef flat. The mathematical equation was derived based on the theory of radiation stress and the conservation of wave energy. The equation is primarily determined by several physical variables including the breaking wave index, the stable wave index, the attenuation coefficient of wave energy flux, and the flow velocity in the re-stabilization zone. A series of laboratory experiments were carried out to calibrate the theoretical equations. Specifically, the breaking wave index,the stable wave index, and the velocity over the reef flat were measured in the laboratory. The attenuation coefficient of wave energy flux in our theoretical equation was determined by calibration by comparing with the laboratory measured wave height. Furthermore, it has been put forward that the velocity based on cnoidal wave theory could be used to determine the velocity over the reef flat if there is no velocity measurement available. Overall, the proposed equation can provide satisfactory prediction of wave set-up and set-down along the reef flat.

Breaking of Solitary Waves on Uniform Slopes

The propagation, shoaling and breaking of solitary waves on mild slopes are simulated by boundary element method. In this paper, the criterion of breaking solitary waves on mild slopes is discussed. The criterion is that the ratio of horizontal velocity of water particles on the wave crest to wave celerity equals one. However, the case that the ratio of horizontal velocity of water particles on the wave crest to wave celerity is below one but the front face of wave profile becomes vertical is also considered as a breaking criterion. According to the above criteria, the breaking index for slopes 1:10 to 1:25 is studied. The result is compared to other researchers'. The deformation of solitary waves on slopes is discussed and the distribution of fluid velocities at breaking is shown.

Modeling Loads Caused by Breaking Waves Striking Vertical Walls

Breaking waves can have tremendous destructive impact on vertical walls, yet they are poorly understood. By using particle imaging velocimetry （PIV） technology and high-precision pressure transducers, actual breaking wave loads on vertical walls were studied. By simultaneously comparing the flow field structure and wave pressure, the mechanisms of breaking wave pressure could be analyzed. The probability distribution of the peak value of the first impact of a breaking wave was investigated. The results showed that the impact pressure p is mainly distributed in the range of 0.25-2.75 pv2, with the greatest possible probability at p/pv2 = 0.75.

Effects of Breaking Waves on Composite Backscattering from Ship-Ocean Scene

The existence of the sea surface is bound to affect the electromagnetic （EM） scattering from marine targets. When dealing with the composite scattering from targets over a sea surface by applying high-frequency EM theories, the total scattering field can be decomposed into three parts in low sea states, namely, the direct scattering from the sea surface, the direct scattering from targets and the coupling scattering between the sea surface and targets. With regard to high sea states, breaking waves make the direct scattering from the sea surface and the coupling scattering more complicated. To solve this issue, a scattering model is proposed to analyze the composite scattering from a ship over a rough sea surface under high sea states. To consider the effect of breaking waves, a three dimensional geometric model is adopted together with Ufimtsev＇s theory of edge waves for the scattering from a breaker. In addition, the coupling scattering between targets and breaking waves is taken into account by considering all possible scattering paths. The simulated results indicate that the influence of breaking waves on the scattering field from the sea surface and on the coupling field is non-negligible, and the numerical results also show the effectiveness of the proposed scattering model.

Flow characteristics and bubble statistics during the fragmentation process of the ingested main cavity in plunging breaking waves

Plunging breaking waves play an important role in the exchange of heat,momentum,and mass between the atmosphere and ocean.In this paper,a series of direct numerical simulations is conducted to investigate the fragmentation process of the ingested main cavity in plunging breaking waves.The two-phase Navier-Stokes equations are solved using the finite-volume method based on adaptive refinement meshes.The free surface is captured using a geometrical volume of fluid method.Both 2-D,3-D simulations are conducted.Instantaneous flow fields at different stages of wave breaking are presented and quantitative analysis for bubbles is performed.The 2-D instantaneous vorticity field and local velocity field are visualized to discuss the general flow characteristics during the fragmentation process.Then a 2-D parametric study is conducted to investigate the differences in the flow characteristics during the fragmentation process under different wave parameters including initial wave steepness(ε),Bond number(Bo),and Reynolds number(Re).3-D vortex structures are shown to further investigate the mechanisms behind the differences in the flow characteristics.The bubble size distributions under two different initial wave steepness are also discussed with their relationship to the fragmentation process of the ingested main cavity.This research offers a significant understanding of the distinct procedures and fundamental dynamics involved in wave breaking,enhancing our comprehension of this intricate event.

WAVE TRANSFORMATION AND BREAKING OVER A RECTANGULAR STEP

An idealized numerical wave flume has been established by finite element method on the bases of Navier Stokes equations through prescribing the appropriate boundary conditions for the open boundary,incident boundary,free surface and solid boundary in this paper.The characteristics of waves propagating over a step have been investigated by this numerical model.The breaker wave height is determined depending on the kinetic criterion.The numerical model is verified by laboratory experiments,and the empirical formula for the damping of wave height due to breaking is also given by experiments.

Theoretical and Experimental Study of Breaking Wave on Sloping Bottoms

This paper studies the continuous evolution of breaking wave for the surface water waves propagating on a sloping beach. A Lagrangian asymptotic solution is derived. According to the solution coupled with the wave breaking criteria and the equations of water particles motion, the wave deformation and the continuous wave breaking processes for the progressive water waves propagating on a sloping bottom can be derived. A series of experiments are also conducted to compare with the theoretical solution. The results show that the present solution can reasonably describe the plunging or spilling wave breaking phenomenon.