An analytic investigation of oscillations within a harbor of constant slope

Longitudinal and transverse oscillations within a harbor of constant slope are analyzed. Based on the linear shallow water approximation, longitudinal oscillations are described with Bessel equations. Ignoring friction, oscillations are forced using the period of the incident perpendicular wave field by the method of matched asymptotics. The analytic results show that the varying depth shifts the resonant wave numbers to lower values than those for the same geometric harbor with constant depth. Furthermore, we extend the shallow water equations to a linear, weakly dispersive, Boussinesq-type equation by modifying the offshore velocity component, and then use it to investigate possible existing transverse oscillations in the harbor of constant slope. These oscillations are types of standing edge waves. Their character is quite sensitive to the boundary condition at the backwall of the harbor.     

Hydraulic Modeling of A Curtain-Walled Dissipater by the Coupling of RANS and Boussinesq Equations

A hybrid numerical method for the hydraulic modeling of a curtain-walled dissipater of reflected waves from breakwa-ters is presented. In this method, a zonal approach that combines a nonlinear weakly dispersive wave (Boussinesq-type equation) method and a Reynolds-Averaged Navier-Stokes (RANS) method is used. The Boussinesq-type equation is solved in the far field to describe wave transformation in shallow water. The RANS method is used in the near field to re-solve the turbulent boundary layer and vortex flows around the structure. Suitable matching conditions are enforced at the interface between the viscous and the Boussinesq region. The Coupled RANS and Boussinesq method successfully resolves the vortex characteristics of flow in the vicinity of the structure, while unexpected phenomena like wave re-reflection are effectively controlled by lengthening the Boussinesq region. Extensive results on hydraulic performance of a curtain-walled dissipater and the mechanism of dissipation of reflected waves     

An explicit finite difference model for simulating weakly nonlinear and weakly dispersive waves over slowly varying water depth

In this paper, a modified leap-frog finite difference (FD) scheme is developed to solve Non linear Shallow Water Equations (NSWE). By adjusting the FD mesh system and modifying the leap-frog algorithm, numerical dispersion is manipulated to mimic physical frequency dispersion for water wave propagation. The resulting numerical scheme is suitable for weakly nonlinear and weakly dispersive waves propagating over a slowly varying water depth. Numerical studies demonstrate that the results of the new numerical scheme agree well with those obtained by directly solving Boussinesq-type models for both long distance propagation, shoaling and re-fraction over a slowly varying bathymetry. Most importantly, the new algorithm is much more computationally efficient than existing Boussinesq-type models, making it an excellent alternative tool for simulating tsunami waves when the frequency dispersion needs to be considered.     

关于Boussinesq型水波方程理论和应用研究的综述

Boussinesq型方程是研究水波传播与演化问题的重要工具之一,本文就1967-2018年常用的Boussinesq型水波方程从理论推导和数值应用两个方面进行了回顾,以期推动该类方程在海岸(海洋)工程波浪水动力方向的深入研究和应用。此类方程推导主要从欧拉方程或Laplace方程出发。在一定的非线性和缓坡假设等条件下,国内外学者建立了多个Boussinesq型水波方程,并以Stokes波的相关理论为依据,考察了这些方程在相速度、群速度、线性变浅梯度、二阶非线性、三阶非线性、波幅离散、速度沿水深分布以及和(差)频等多方面性能的精度。将Boussinesq型水波方程分为水平二维和三维两大类,并对主要Boussinesq型水波方程的特性进行了评述。进而又对适合渗透地形和存在流体分层情况下的Boussinesq型水波方程进行了简述与评论。最后对这些方程的应用进行了总结与分析。     

Applicability of numerical models to nonlinear dispersive waves

The applicability of three different wave-propagation models in nonlinear dispersive wave fields has been investigated. The numerical models tested here are based on three different wave theories: a fully nonlinear potential theory, a Stokes second-order theory, and a Boussinesq-type theory with an improved dispersion relation. Physical experiments and computations were conducted for wave evolutions during passage over a submerged shelf under various wave conditions. As expected, the fully nonlinear solutions agree better with the measurements than do the other solutions. Although the second-order solution has sufficient accuracy for smaller-amplitude wave cases, the truncation after the third harmonics causes significant discrepancies in wave form for larger waves. In addition, the second-order model markedly overestimates the first- and second-harmonic amplitudes in transmitted waves. The Boussinesq model provides excellent predictions of wave profile over the shelf even in larger wave cases. However, this model also overestimates the magnitudes of several higher harmonics in transmitted waves. These facts may indicate that energy transfer from bound components into free waves in these higher harmonics cannot be accurately evaluated by the Boussinesq-type equations.     

A numerical experiment on nearshore circulation in standing edge waves

The unsteady shallow-water vorticity equation dominating nearshore flow on a gently sloping plane beach has been solved by using the implicit finite difference technique under the assumption of constant viscosity over the flow field. The result of computation showed that pairs of nearshore circulation cells are generated through the nonlinear effect of flow in the boundary layer formed by the run-up movement of a standing edge wave along a shoreline and the paired cell has the spacing of half a wavelength of the edge wave. When the leaky-mode standing wave of Lamb with the same wave period as the edge wave and the wave crest parallel to a shoreline was superposed on the edge wave field, the alongshore spacing of circulation cell doubled and seaward flow in the cell concentrated in the narrow zone like a ‘rip current’. Although no effect of breaking waves is considered in the computation, such a mechanism may also generate some kinds of nearshore circulation systems observed in a sea.     

强非线性和色散性Boussinesq方程数值模型检验

采用同位网格有限差分法,建立了强非线性和色散性Boussinesq方程数值计算模型。以稳恒波Fourier近似解给定入射波边界条件,对均匀水深深水和浅水域不同非线性的行进波、缓坡地形上深水至浅水域的浅水变形波、以及缓坡和陡坡地形上的波浪水槽实验进行了数值计算,并将计算结果与解析解、解析数值解以及实验值进行了较为详细的比较,从而检验了模型的色散性、非线性以及不同底坡下非线性波的浅水变形性能。     

Wash Waves Generated by High Speed Displacement Ships

It is difficult to compute far-field waves in a relative large area by using one wave generation model when a large calculation domain is needed because of large dimensions of the waterway and long distance of the required computing points. Variation of waterway bathymetry and nonlinearity in the far field cannot be included in a ship fixed process either. A coupled method combining a wave generation model and wave propagation model is then used in this paper to simulate the wash waves generated by the passing ship. A NURBS-based higher order panel method is adopted as the stationary wave generation model; a wave spectrum method and Boussinesq-type equation wave model are used as the wave propagation model for the constant water depth condition and variable water depth condition, respectively. The waves calculated by the NURBS-based higher order panel method in the near field are used as the input for the wave spectrum method and the Boussinesq-type equation wave model to obtain the far-field waves. With this approach it is possible to simulate the ship wash waves including the effects of water depth and waterway bathymetry. Parts of the calculated results are validated experimentally, and the agreement is demonstrated. The effects of ship wash waves on the moored ship are discussed by using a diffraction theory method. The results indicate that the prediction of the ship induced waves by coupling models is feasible.     

不规则波作用下沿岸流流速分布规律分析

基于MIKE21 PMS,对1/10、1/20、1/40、1/100四种不同坡度地形下不规则波浪产生的沿岸流进行数值模拟。将模拟得到的流速随着离开静水岸线距离的变化曲线进行无量纲化处理,并选用高斯分布函数对流速曲线进行拟合,进而得到高斯分布函数中流速最大值与其出现位置以及流速分布宽度等参数随着入射波浪要素和地形坡度的变化关系。研究结果表明,文中给出的高斯分布函数适用于描述不规则波浪作用生成的沿岸流流速分布规律。     

Generation of waves in Boussinesq models using a source function method

A method for generating waves in Boussinesq-type wave models is described. The method employs a source term added to the governing equations, either in the form of a mass source in the continuity equation or an applied pressure forcing in the momentum equations. Assuming linearity, we derive a transfer function which relates source amplitude to surface wave characteristics. We then test the model for generation of desired incident waves, including regular and random waves, for both one and two dimensions. We also compare some model results with analytical solution and available experiment data.     

A theory of semigeostrophic gravity waves and its application to the intrusion of a density current along a coast

Semigeostrophic gravity waves associated with a coastal boundary current, which has finite and uniform potential vorticity and is bounded away from the coastline by a density front on the ocean surface, are investigated. It is shown that the semigeostrophic coastal current has two waves which are named here the Semigeostrophic Coastal Wave (SCW) and the Semigeostrophic Frontal Wave (SFW). The SCW becomes an elementary Kelvin wave at some limit while the SFW is caused by the existence of the surface density front. The SCW appears mainly as variations in the upper layer depth at the coast and as alongshore velocity at the density front. On the other hand, the SFW appears mainly as variations in the width of the current. When the weak nonlinearity and ageostrophic effect are included, these semigeostrophic gravity waves satisfy the Kortweg- de Vries equation, which suggests that the local changes in the width and/or velocity of the semigeostrophic coastal current propagate as wave-like disturbances.     

江苏海岸中部近岸冬季潮汐和潮流特征

南黄海西侧的江苏海岸近岸区域,素以地形复杂、潮流强劲、悬沙输运剧烈著称,但是较长期的同步潮位和潮流观测数据仍然缺乏,尤其是在近岸(20 km)浅水(20 m)区域。2014年1月在大丰港附近开展了连续潮位和潮流观测,获得的数据揭示了一系列特征。此地潮汐潮流为正规半日潮,浅水分潮显著。平均潮差为3.05 m,最显著的两个分潮为M2和S2分潮,振幅分别为1.45 m和0.52 m。潮流最显著的半日分潮M2分潮和最显著的浅水分潮M4分潮在沿岸方向上振幅分别为0.84m/s和0.12m/s,在跨岸方向上振幅分别为0.24 m/s和0.01 m/s,沿岸方向占绝对优势。潮波的沿岸传播介于前进波和驻波之间,驻波的特征稍强。M2分潮潮流椭圆最大流(长轴)方向为南偏东7.4°。存在冬季沿岸向北的余流,垂向平均值的大小为2.2 cm/s。以上潮汐潮流特征为该区域海洋物质输运研究提供了基础资料。     

Shock-capturing Boussinesq-type model for nearshore wave processes

This paper describes the formulation and validation of a nearshore wave model for tropical coastal environment. The governing Boussinesq-type equations include the conservative form of the nonlinear shallow-water equations for shock capturing. A Riemann solver supplies the inter-cell flux and bathymetry source term, while a Godunov-type scheme integrates the evolution variables in time. The model handles wave breaking through momentum conservation with energy dissipation based on an eddy viscosity concept. The computed results show very good agreement with laboratory data for wave propagation over a submerged bar, wave breaking and runup on plane beaches as well as wave transformation over fringing reefs. The model accurately describes transition between supercritical and subcritical flows as well as development of dispersive waves in the processes.     

Experimental study of the instabilities of alongshore currents on plane beaches

A laboratory experiment on alongshore currents was conducted for two plane beaches, with gradients 1:40 and 1:100, to investigate the instability of alongshore currents. Complicated and strongly unstable alongshore current motions were observed. In order to clearly examine the spatial and temporal variations of the shear instability of the currents, digital images from a charge-coupled device (CCD) recorded the deformations of dye batches released in the surf zone. Some essential characteristics of the shear instability were obtained from analyses of images showing the temporal variation of the dye patches.A high-resolution spectral analysis technique (the maximum entropy method, or MEM) was used to analyze the dominant frequency of the observed oscillation, along with the trigonometric regression method for determining the variations of the oscillation strength in the cross-shore direction. The propagation speed of the dye patch was obtained by tracking the movement over time of fixed locations in the dye patch, such as its peak, in the longshore direction. This data was then fitted linearly.Alongshore and cross-shore velocity time series acquired from sensors showed clearly that large-amplitude, long-period (about 50 s or 100 s) oscillations were present for all sensors deployed in the cross-shore direction under regular and irregular wave conditions. The analysis found that the maximum shear wave amplitude was approximately one-sixth of the maximum for the mean alongshore current, and occurred approximately at the position of the maximum of the mean alongshore current for irregular waves. The spatial structure of the shear waves was studied by analyzing collected images of the dye patches. The phase velocity of the meandering movements was obtained by measuring the magnitude of the oscillations of the dye patches in the alongshore direction with respect to time. The results suggest that the propagation speed of the shear instability was approximately one-half to three-quarters of the maximum mean longshore current for both regular and irregular waves.Linear instability analysis theory was applied to the characteristics of alongshore current instability, which suggested that there are two instability modes related to the observed oscillations: the frontshear mode observed for the 1:100 slope, and the backshear mode observed for the 1:40 slope. Theoretical analyses agreed with the experimental results in both cases. The velocity profile of the mean longshore current was found to affect the instability mode significantly, leading to further investigations on the influence of the velocity profiles and to provide support for the above conclusions.     

Rip currents, mega-cusps, and eroding dunes

Dune erosion is shown to occur at the embayment of beach mega-cusps O(200 m alongshore) that are associated with rip currents. The beach is the narrowest at the embayment of the mega-cusps allowing the swash of large storm waves coincident with high tides to reach the toe of the dune, to undercut the dune and to cause dune erosion. Field measurements of dune, beach, and rip current morphology are acquired along an 18 km shoreline in southern Monterey Bay, California. This section of the bay consists of a sandy shoreline backed by extensive dunes, rising to heights exceeding 40 m. There is a large increase in wave height going from small wave heights in the shadow of a headland, to the center of the bay where convergence of waves owing to refraction over the Monterey Bay submarine canyon results in larger wave heights. The large alongshore gradient in wave height results in a concomitant alongshore gradient in morphodynamic scale. The strongly refracted waves and narrow bay aperture result in near normal wave incidence, resulting in well-developed, persistent rip currents along the entire shoreline.

The alongshore variations of the cuspate shoreline are found significantly correlated with the alongshore variations in rip spacing at 95% confidence. The alongshore variations of the volume of dune erosion are found significantly correlated with alongshore variations of the cuspate shoreline at 95% confidence. Therefore, it is concluded the mega-cusps are associated with rip currents and that the location of dune erosion is associated with the embayment of the mega-cusp.     

Amplitude variations of edge waves on a shelf slowly varying in the alongshore direction

Within the framework of the linear shallow-water theory, the dynamics of edge waves over a shelf characterized by a cylindrical bottom relief is investigated under the assumption that shelf parameters vary slowly in the alongshore direction. An asymptotic theory and an energy approach are used to calculate the amplitude of the edge wave. In the analytic form, the results are obtained for shelves of three different profiles with parameters varying along the shore: an infinite linear profile, a concave exponential profile, and a stepwise profile.     

A higher-order non-hydrostatic σ model for simulating non-linear refraction–diffraction of water waves

A higher-order non-hydrostatic σ model is developed to simulate non-linear refraction–diffraction of water waves. To capture non-linear (or steep) waves, a 4th-order spatial discretization is utilized to approximate the large horizontal pressure gradient. A higher-order top-layer pressure treatment is further implemented to resolve wave propagation. The model's characteristics including linear wave dispersion and non-linearity are carefully examined. The accuracy of the present model using only two vertical layers is validated by laboratory data and the available results predicted by the non-linear Schrödinger equation, Boussinesq-type equations, the non-linear mild slope equation, and the Laplace equation. Features of harmonic generation as well as the influences of dispersion and non-linearity on wave energy transfer processes are discussed.     

Wavenumber–frequency analysis of the landslide-generated tsunamis at a conical island

In this paper we investigate the propagation of tsunamis generated by landslides around the coast of an island to understand the propagation and trapping mechanisms of the waves. Records of shoreline displacement have been processed using the wavenumber–frequency analysis (k–f). We identify the dispersion relation followed by the waves that propagate alongshore. It appears that the 0th-order edge wave mode is the only one relevant for shoreline run-up. Furthermore, it is shown that the edge wave dispersion relation is a quantitative tool to estimate the phase and group wave celerities. A very good agreement is found when comparing the wave celerities as calculated from the experimental records, against those predicted by the edge wave theory. Furthermore the analysis of the tsunami around the island has been carried out by means of the two-dimensional k–f. Using as input data a high-space resolution dataset of free surface elevations it is found that other modes, both trapped (1st-order edge waves) and non-trapped (free radiating waves), occur during the propagation.     

An Analytical Solution for Nearshore Circulation Driven by Focused/Defocused Waves

An offshore shoal or bar refracts ocean surface waves and causes wave focusing/defocusing on the adjacent beach.Wave focal patterns characterized by alongshore variations in wave height, wave angle, and breaking location induce alongshore non-uniformities of wave setup and nearshore circulation, e.g., rip currents and alongshore currents, in the surfzone. A simplified analytic model for nearshore circulation generated by focused/defocused waves on a planar beach is developed and theoretical solutions are obtained using transport stream function and perturbations in alongshore distributions of wave height and wave angle at the breaker line. The analytic model suggests that alongshore currents are strongly affected by a pair of counter-rotating vortices generated shoreward of the wave focal zone. The vortices are persistent, and their strengths depend on the amplitudes of alongshore variations in wave height and wave angle. The alongshore gradient in wave height tends to intensify the vortices while the convergence of wave angle tends to weaken the vortices. Divergent flows associated with the vortices in the surfzone are intense,strengthening alongshore currents in the downstream of the wave focal zone and weakening alongshore currents or causing flows reversal in the upstream. Alongshore currents are modulated by rip currents associated with the wave focusing/defocusing patterns.