A coupled-mode system with application to nonlinear water waves propagating in finite water depth and in variable bathymetry regions

A non-linear coupled-mode system of horizontal equations is presented, modelling the evolution of nonlinear water waves in finite depth over a general bottom topography. The vertical structure of the wave field is represented by means of a local-mode series expansion of the wave potential. This series contains the usual propagating and evanescent modes, plus two additional terms, the free-surface mode and the sloping-bottom mode, enabling to consistently treat the non-vertical end-conditions at the free-surface and the bottom boundaries. The present coupled-mode system fully accounts for the effects of non-linearity and dispersion, and the local-mode series exhibits fast convergence. Thus, a small number of modes (up to 5–6) are usually enough for precise numerical solution. In the present work, the coupled-mode system is applied to the numerical investigation of families of steady travelling wave solutions in constant depth, corresponding to a wide range of water depths, ranging from intermediate depth to shallow-water wave conditions, and its results are compared vs. Stokes and cnoidal wave theories, as well as with fully nonlinear Fourier methods. Furthermore, numerical results are presented for waves propagating over variable bathymetry regions and compared with nonlinear methods based on boundary integral formulation and experimental data, showing good agreement.     

A coupled-mode model for the refraction–diffraction of linear waves over steep three-dimensional bathymetry

A consistent coupled-mode model recently developed by Athanassoulis and Belibassakis [1], is generalized in 2+1 dimensions and applied to the diffraction of small-amplitude water waves from localized three-dimensional scatterers lying over a parallel-contour bathymetry. The wave field is decomposed into an incident field, carrying out the effects of the background bathymetry, and a diffraction field, with forcing restricted on the surface of the localized scatterer(s). The vertical distribution of the wave potential is represented by a uniformly convergent local-mode series containing, except of the ususal propagating and evanescent modes, an additional mode, accounting for the sloping bottom boundary condition. By applying a variational principle, the problem is reduced to a coupled-mode system of differential equations in the horizontal space. To treat the unbounded domain, the Berenger perfectly matched layer model is optimized and used as an absorbing boundary condition. Computed results are compared with other simpler models and verified against experimental data. The inclusion of the sloping-bottom mode in the representation substantially accelerates its convergence, and thus, a few modes are enough to obtain accurately the wave potential and velocity up to and including the boundaries, even in steep bathymetry regions. The present method provides high-quality information concerning the pressure and the tangential velocity at the bottom, useful for the study of oscillating bottom boundary layer, sea-bed movement and sediment transport studies.     

Experimental and numerical study of the effect of variable bathymetry on the slow-drift wave response of floating bodies

An experimental campaign is reported on the slow-drift motion of a rectangular barge moored at different positions along an inclined beach, at waterdepths ranging from 54 cm to 21 cm, and submitted to irregular beam seas. The beach is achieved by inclining the 24 m long false bottom of the tank at a slope of 5%, from a depth of 1.05 m. The slow-drift component of the measured sway motion is first compared with state-of-the-art calculations based on Newman’s approximation. At 54 cm depth a good agreement is obtained between calculations and measurements. At 21 cm depth the Newman calculations exceed the measured values. When the flat bottom setdown contribution is added up, the calculated values become 2 to 3 times larger than the measured ones. A second-order model is proposed to predict the shoaling of a bichromatic sea-state propagating in varying water-depth. This model is validated through comparisons with an extension of Schäffer’s model for a straight beach [Schäffer HA. Infragravity waves induced by short-wave groups. J Fluid Mech 1993;247:551-88] and with a fully nonlinear Boussinesq model. It appears that the long wave amplitude is much less than predicted by the flat bottom model, and that its phase difference with the short wave envelope also deviates from the flat bottom model prediction. As a result of this phase shift the actual second-order wave loads can be lower than predicted by Newman’s approximation alone. Application of the shoaling model to the barge tests yields a notably better agreement between numerical and experimental values of its slow-drift sway motion.     

A video-based technique for mapping intertidal beach bathymetry

Measuring the location of the shoreline and monitoring foreshore changes through time are core tasks carried out by coastal engineers for a wide range of research, monitoring and design applications. With the advent of digital imaging technology, shore-based video systems provide continuous and automated data collection, encompassing a much greater range of time and spatial scales than were previously possible using field survey methods.A new video-based technique is presented that utilises full-colour image information, which overcomes problems associated with previous grey-scale methods, which work well at steep (reflective) sites, but are less successful at flatter (dissipative) sites. Identification of the shoreline feature is achieved by the automated clustering of sub-aqueous and sub-aerial pixels in ‘Hue–Saturation–Value’ (HSV) colour space, and applying an objective discriminator function to define their boundary (i.e., ‘shoreline’) within a time-series of consecutive geo-referenced images. The elevation corresponding to the detected shoreline features is calculated on the basis of concurrent tide and wave information, which is incorporated in a model that combines the effects of wave set-up and swash, at both incident and infragravity frequencies.Validation of the technique is achieved by comparison with DGPS survey results, to assess the accuracy of the detection and elevation methods both separately and together. The uncertainties associated with the two sub-components of the model tend to compensate for each other. The mean difference between image-based and surveyed shoreline elevations was less than 15 cm along 85% of the 2-km study region, which corresponded to an horizontal offset of 6 m. The application of the intertidal bathymetry mapping technique in support of CZM objectives is briefly illustrated at two sites in The Netherlands and Australia.     

A new approximation for ocean waves propagating over a beach with variable depth

In this paper, the phenomenon of ocean waves propagating over a beach with variable water depth is re-examined. Based on the assumption of shallow water, a linearised shallow water equation is solved with an arbitrary beach profile. These irregular beach profiles form a set of partial differential equation with variable coefficient as the governing equation, which is the main obstacle in obtaining analytical solutions. In this paper, two families of beach profile are used as examples. A parametric study is conducted to investigate the influence of the beach profiles on the water surface elevation (η) and velocities (u).     

A three-point method for estimating wave reflection of obliquely incident waves over a sloping bottom

A three-point method for estimating wave reflection is proposed to account for monochromatic oblique incident waves propagating over a sloping beach. The amplitudes of reflected wave and incident wave are separated using wave amplitudes measured at three fixed wave gauges with a distance. The applicability of the theory is verified by comparing the simulated results with the available theoretical, numerical and experimental results for the estimation of wave reflection. The sensitivity is also tested to provide a more accurate prediction of the reflection coefficient.     

基于激波捕捉类Boussinesq模型对岛礁地形上波浪传播的数值模拟

为了探究激波捕捉类Boussinesq模型在模拟岛礁地形上规则波和不规则波传播的可行性,采用基于完全非线性Boussinesq方程并具有激波捕捉能力的数值模型Funwave-TVD对规则波和不规则波在岛礁地形上的传播进行了数值模拟,通过与试验数据对比,分析模型中空间步长的影响,验证模型在模拟波高、平均水位分布以及波谱空间演变的能力,结果表明:采用合适的空间步长,模型能较好地模拟规则波和不规则波在岛礁地形上的传播和演化过程。对于规则波,较小的空间步长可改善破碎点处波高峰值的预测,并能更好地预测波浪破碎后波高的空间分布,相比结合经验破碎的Boussinesq模型,Funwave-TVD能更好地模拟规则波在岛礁地形上的破碎,以及破碎以后行进涌波的再生成过程;对于不规则波,Funwave-TVD总体而言能较好地模拟涌浪有效波高、次重力波的生成及空间演化和平均水位,但会低估礁坪上次重力波波高,较粗的空间步长也会低估礁坪上涌浪有效波高。     

基于单景高分辨率光学卫星遥感影像的浅海水下地形提取

本文基于海浪波折射现象和浅水波理论,提出了一种基于单景高分辨率光学遥感影像的浅海地形提取方法。首先,基于浅水波理论推导出适用于浅海区域的水深与海浪波长、频率的定量关系,针对近岸光学遥感图像复杂的海浪特征,讨论了两种海浪波长提取方法,即FFT方法和剖面线法。然后提出了基于长距离波长波动分析的海浪频率计算方法,解决了单景遥感影像的波浪频率计算难题。最后,利用单景QuickBird高分辨率光学遥感影像,以海南岛三亚湾为研究区域进行了应用实验,结果表明,对12m以浅的浅海区域,在不需要任何辅助参数的情况下,反演获得了浅海地形(DEM),经与1:25000比例尺海图的水深对比验证,地形趋势吻合良好,反演水深的均方根误差为1.07m,相对水深误差为16.2%,表明该方法适合于浅海水下地形的提取,且具有无需实测水深数据和环境参数的支持的优点。     

A two-point method for estimating wave reflection over a sloping beach

This work presents a frequency-domain method for estimating incident and reflected waves when normally incident waves’ propagating over a sloping beach in a wave flume is considered. Linear wave shoaling is applied to determine changes of the wave amplitude and phase due to variations of the bathymetry. The wave reflection coefficient is estimated using wave heights measured at two fixed wave gauges with a distance. The present model demonstrates a high capacity of estimating reflection and shoaling coefficients from synthetic wave-amplitude data. Sensitivity tests for the present model due to measurement errors of wave amplitudes and distance of two probes can more accurately predict the reflection coefficients. The measurement error of wave amplitude affects more significantly than measurement error of distance of two probes on calculating reflection coefficient of waves over a sloping bed.     

A Review About SAR Technique for Shallow Water Bathymetry Surveys

Synthetic Aperture Radar(SAR)has become one of the important tools for shallow water bathymetry surveys.This has significant economic efficiency compared with the traditional bathymetry surveys.Numerical models have been developed to simulate shallow water bathymetry SAR images.Inversion of these models makes it possible to assess the water depths from SAR images.In this paper,these numerical models of SAR technique are reviewed,and examples are illustrated including in the coastal areas of China.Some issues about SAR technique available and the research orientation in future are also discussed.     

南黄海辐射沙脊群海域的水深遥感

南黄海的辐射沙脊群海域地形地貌复杂多变,应用遥感技术来测量其水下地形不仅有可能,而且势在必行。在对沿海及长江口门内外遥感反演水下地形研究的基础上,选择沙脊群海域中有代表性的一块水域为研究区,同步测量其水下地形和水流、悬沙含量,用实测地形和MODIS影像数据建立水深反演模型,同时分析该区域的悬沙光谱特征,选择悬沙遥感参数,以MODIS数据的水深反演模型和沙脊群水域的悬沙遥感参数为依据,建立削弱悬沙影响的水深遥感模型,从而反演出1.5×104km2的辐射沙脊群水下地形。从反演结果看,所建模型的相关系数二次方的值为0.664,F-值为18662,平均绝对误差为1.52m,平均相对误差为24%,5~15m水深段的平均相对误差只有18%,反演效果较好。     

Investigation of the generation and propagation of low frequency internal waves: A case study for the east coast of India

SAR (Synthetic Aperture Radar) images in the northwest Bay of Bengal indicate the existence of internal waves and their occurrence and intensity is topography dependent as indicated by in-situ data and satellite information. To complement and comprehend the observations, a three-dimensional Princeton Ocean Model is applied to study the generation and propagation of internal waves. The model domain is configured with a variable curvilinear grid and the input fields comprise bathymetry, initial temperature and salinity, wind stress, air-sea heat flux and tidal information. The numerical investigation indicated a predominant activity of internal waves in the north, and the rationale is three-fold. The first one could be the stable stratification due to fresh water discharge from head-bay major river system, secondly, the significant magnitude/range of the tides and finally, the bathymetry in the coastal waters off Paradip is about 12% shallower compared to that of Visakhapatnam and further south. The cumulative effect of these causes the predominance of internal waves in the north. The core of the energy is essentially in the low-frequency range and the model is able to simulate semi-diurnal and diurnal components reasonably well up to 6 h frequency (0.162 cph).     

Effects of tropical cyclone paths and shelf bathymetry on inducement of severe storm surges in the Gulf of Thailand

The influences of tropical cyclone paths and shelf bathymetry on the inducement of extreme sea levels in a regional bay are investigated. A finite volume coastal ocean model(FVCOM) has been configured for the Gulf of Thailand-Sunda Shelf. A parametric wind model is used to drive the FVCOM. The contributions of the tropical cyclone characteristics are determined through a scenario-based study. Validation based on a historical extreme sea level event shows that the model can resolve the oscillation mechanism well. The intensification of severe storm surges in the region highly depends on four factors including phase propagation of the storm surge wave determined by the landfall position, funnel effect caused by locality of the coastline, and shelf bathymetry determined by the state of mean sea level and coastline crossing angle of the storm path. The coexistence of these factors can cause particular regions e.g. the Surat Thani Bay, inner Gulf of Thailand and Ca Mau Peninsular to experience a larger surge magnitude. These areas are found to be highly related to monsoon troughs that develop during the onset and early northeastern monsoon season(October–November).     

Laboratory study of the nonlinear transformation of irregular waves over a mild slope

This paper considers the nonlinear transformation of irregular waves propagating over a mild slope （1：40）. Two cases of irregular waves, which are mechanically generated based on JONSWAP spectra, are used for this purpose. The results indicate that the wave heights obey the Rayleigh distribution at the offshore location; however, in the shoaling region, the heights of the largest waves are underestimated by the theoretical distributions. In the surf zone, the wave heights can be approximated by the composite Weibull distribution. In addition, the nonlinear phase coupling within the irregular waves is investigated by the wavelet-based bicoherence. The bicoherence spectra reflect that the number of frequency modes participating in the phase coupling increases with the decreasing water depth, as does the degree of phase coupling. After the incipient breaking, even though the degree of phase coupling decreases, a great number of higher harmonic wave modes are also involved in nonlinear interactions. Moreover, the summed bicoherence indicates that the frequency mode related to the strongest local nonlinear interactions shifts to higher harmonics with the decreasing water depth.     

基于可见光遥感测深技术的长江口南支河段河势演变规律研究

以长江口南支河段为研究区,利用LandsatETM+遥感影像反演的1999年水下地形和该河段2002年实测水深数据,结合长江口来水来沙现状和河段特征,分析了河段的冲淤变化和沙体迁移规律,提出了需采取的治理措施。研究结果表明:(1)从1999到2002年南支河段总体发生冲刷,年平均冲刷量为0.1926亿m3,河道内白茆沙、新浏河沙和新浏河沙包的沙头年平均后退467,374和421m;(2)上、下扁担沙之间由于串沟发育,两者有分离的趋势,沙体5m等深线的变动格局显示两沙体沙尾向下移动导致新桥水道上段水深加深、下段南侧水深变浅;(3)固定白茆沙和上、下扁担沙沙体,对维持南支河段航道的航运、稳定南北槽河势具有重要作用。     

A short cut numerical method for determination of periods of free oscillations for basins with irregular geometry and bathymetry

A short cut numerical method for evaluation of the modes of free oscillations of the basins which have irregular geometry and bathymetry is presented in this paper. In the method, a single wave is inputted to the basin as an initial impulse. The respective agitation in the basin is computed by using the numerical method solving the nonlinear form of long wave equations. The time histories of water surface fluctuations at different locations due to propagation of the waves in relation to the initial impulse are stored and analyzed by the fast Fourier transform technique (FFT) and energy spectrum curves for each location are obtained. The frequencies of each mode of free oscillations are determined from the peaks of the spectrum curves. The method is tested by using regular shaped flat bottom basins with different depths. The computed periods of free oscillations are compared with the theoretical values. The accuracy and performance of the method are discussed. As a case study for the application to the basins of irregular shape and bathymetry, the periods of free oscillations of the sea of Marmara is determined and discussed.     

Model for the Processing and Estimation of Dual Frequency Echo Sounder Observations in Detailed Bathymetries

The dataset acquired by an echo sounder generally comprises outliers and erroneous measurements which must be automatically processed. The main aim of this research is the development and validation of an algorithm which permits detecting outliers, estimating their real value from data acquired with a multifrequency echo sounder, and verifying the measure according to the desired accuracy. The process considers the nature of the seabed and improves the density and number of lectures without globally compromising the precision, but enhancing it in areas with anomalous measurements. Twenty-nine percent of this type of measurements, which would have been otherwise rejected, have been estimated.     

A new bathymetric model for the central Fram Strait

Based on data from R/V Polarstern multibeam sonar surveys between 1984 and 1997 high resolution bathymetry has been generated for the central Fram Strait. The area insonified covers approx. 36,500 km² between 78–80°N and 0–7.5°E allowing the creation of a Digital Terrain Model (DTM) with 100 m grid spacing. The DTM was utilized for contouring and generation of a new series of bathymetric charts (AWI Bathymetric Charts of the Fram Strait, AWI BCFS) at a scale of 1:100,000. The paper starts with a brief introduction to the regional setting of the study area comprising information on the local links between bathymetry, sea ice transport and water mass exchange. The bathymetric feature names used in this article and how they were chosen is outlined. Next, the input data and processing applied are described. Thereafter the newly created grid and contour data are put into context with existing data sets. Finally the main bathymetric features of the area are characterized and the generated data products available for public disposal are specified.     

A theory for waves interacting with porous structures with multiple regions

This study presents an analytical solution for the problem of waves passing a submerged porous structure, using a multi-region method in the solution scheme considering the characteristics of geometry and composing materials of the porous structure. Using the flux and pressure conditions on horizontal boundaries and interfaces, the orthogonal property of wave motion within the porous layers through water depth is derived, and applied in the solution process. The flux and pressure conditions on vertical boundaries and interfaces are integrated to give a set of linear matrix equations, through which the unknown coefficients are solved. Comparisons of the present method with previous studies are preceded in verification, which suggests the validity and practicability of the present study, with a further expectation of extending our work to build a mild-slope equation over multiple-layer porous medium in the future.