Efficient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure

A numerical method for non-hydrostatic, free-surface, irrotational flow governed by the nonlinear shallow water equations including the effects of vertical acceleration is presented at the aim of studying surf zone phenomena. A vertical boundary-fitted grid is used with the water depth divided into a number of layers. A compact finite difference scheme is employed for accurate computation of frequency dispersion requiring a limited vertical resolution and hence, capable of predicting the onset of wave breaking. A novel wet–dry algorithm is applied for a proper handling of moving shoreline. Mass and momentum are strictly conserved at discrete level while the method only dissipates energy in the case of wave breaking. The numerical results are verified with a number of tests and show that the proposed model using two layers without ad-hoc assumptions enables to resolve propagating nonlinear shoaling, breaking waves and wave run-up within the surf and swash zones in an efficient manner.     

A proposed adiabatic formulation of 3‐dimensional global atmospheric models based on potential vorticity

A 2‐time‐level finite difference atmospheric general circulation model based on the semi‐Lagrangian advection of pseudo potential vorticity (which becomes potential vorticity in that part of the domain where the hybrid vertical coordinate becomes isentropic) has been formulated. At low levels, the hybrid vertical coordinate is terrain following. The problem of isentropic potential vorticity possibly becoming ill‐defined in the regions of planetary boundary layer is thus circumvented. The divergence equation is a companion to the (pseudo) potential vorticity equation and the model is thus called a PV‐D model. Many features of a previously developed shallow water PV‐D model are carried over: a modification of the PV equation needed to give computational stability of long Rossby waves; a semi‐Lagrangian semi‐implicit treatment of both the linear and the nonlinear terms; the use of an unstaggered grid in the horizontal; the use of a nonlinear multigrid technique to solve the nonlinear implicit equations. A linear numerical stability analysis of the model's gravity–inertia waves indicates that the potential temperature needs to be separated into horizontal mean and perturbation parts. This allows an implicit treatment of the vertical advection associated with the mean in the thermodynamic equation. Numerical experiments with developing baroclinic waves have been carried out and give realistic results.     

Has wind–wave modeling reached its limit?

This article uses a comparison of four different numerical wave prediction models for hindcast wave conditions in Lake Michigan during a 10-day episode in October 1988 to illustrate that typical wave prediction models based on the concept of a wave energy spectrum may have reached a limit in the accuracy with which they can simulate realistic wave generation and growth conditions. In the hindcast study we compared the model results to observed wave height and period measurements from two deep water NOAA/NDBC weather buoys and from a nearshore Waverider buoy. Hourly wind fields interpolated from a large number of coastal and overlake observations were used to drive the models. The same numerical grid was used for all the models. The results show that while the individual model predictions deviate from the measurements by various amounts, they all tend to reflect the general trend and patterns of the wave measurements. The differences between the model results are often similar in magnitude to differences between model results and observations. Although the four models tested represent a wide range of sophistication in their treatment of wave growth dynamics, they are all based on the assumption that the sea state can be represented by a wave energy spectrum. Because there are more similarities among the model results than significant differences, we believe that this assumption may be the limiting factor for substantial improvements in wave modeling.     

Estimation of Design Wave Heights for Coastal Sea Areas

Based on historical wind fields in the Bohai Sea,a sequence of annual extremal wave heightsis produced with numerical wave models for deep-water and shallow water.The design wave heights withdifferent return periods for the nearest deep-water point and for the shallow water point are estimated onthe basis of P-Ⅲ type,Weibull distribution,and Gumbel distribution;and the corresponding values for theshallow water point are also estimated based on the HISWA model with the input of design wave heightsfor the nearest deep-water point.Comparisons between design wave heights for the shallow water point es-timated on the basis of both distribution functions are HISWA model show that the results from differentdistribution functions scatter considerably,and influenced strongly by return periods;however,the resultsfrom the HISWA model are convergent,that is,the influence of the design wave heights estimated with dif-ferent distribution functions for deep water is weakened,and the estimated values decrease for long     

Wave Numerical Model for Shallow Water

The history of forecasting wind waves by wave energy conservation equation is briefly des-cribed.Several currently used wave numerical models for shallow water based on different wave theoriesare discussed.Wave energy conservation models for the simulation of shallow water waves are introduced,with emphasis placed on the SWAN model,which takes use of the most advanced wave research achieve-ments and has been applied to several theoretical and field conditions.The characteristics and applicabilityof the model,the finite difference numerical scheme of the action balance equation and its source termscomputing methods are described in detail.The model has been verified with the propagation refractionnumerical experiments for waves propagating in following and opposing currents;finally.the model is ap-plied to the Haian Gulf area to simulate the wave height and wave period field there,and the results arecompared with observed data.     

On resonant Rossby–Haurwitz triads

The dynamics of non-divergent flow on a rotating sphere are described by the conservation of absolute vorticity. The analytical study of the non-linear barotropic vorticity equation is greatly facilitated by the expansion of the solution in spherical harmonics and truncation at low order. The normal modes are the well-known Rossby–Haurwitz (RH) waves, which represent the natural oscillations of the system. Triads of RH waves, which satisfy conditions for resonance, are of critical importance for the distribution of energy in the atmosphere.
We show how non-linear interactions of resonant RH triads may result in dynamic instability of large-scale components. We also demonstrate a mathematical equivalence between the equations for an orographically forced triad and a simple mechanical system, the forced-damped swinging spring. This equivalence yields insight concerning the bounded response to a constant forcing in the absence of damping. An examination of triad interactions in atmospheric reanalysis data would be of great interest.     

Intercomparison of wave-driven current models

Seven numerical models which simulate waves and currents in the surf-zone are tested for the case of a reduced-scale detached breakwater subjected to the action of regular waves with normal incidence. The computed wave heights, water levels and velocities are compared with measurements collected in an experimental wave basin. The wave height decay in the surf-zone is predicted reasonably well. Set-up and currents appear to be less well predicted. This intercomparison exercise shows that radiation stresses are systematically overestimated by formulations used in the models, mean bottom shear stresses are not always co-linear with the mean bottom velocity vector in shallow water, and turbulence modelling in the surf-zone requires a sophisticated     

Validation of the three-wave quasi-kinetic approximation for the spectral evolution in shallow water

This paper aims at validating the three-wave quasi-kinetic approximation for the spectral evolution of weakly nonlinear gravity waves in shallow water. The problem is investigated using a one-dimensional numerical wave propagation model, formulated in the spectral representation. This model includes both a nonlinear triad interactions term and a wave breaking dissipation term. Some numerical tests were carried out in order to show the importance of using the triad nonlinear term in wave propagation spectral models, particularly to describe both behavior of the spectral integral parameters and of the spectral shape evolution in shallow water depth. Furthermore; a comparison against different set of experimental observations was carried out. Comparing the numerical results with the experimental observations made it possible to show the modeling efficiency of the three-wave quasi-kinetic approximation.     

A time‐split, forward‐backward numerical model for solving a nonhydrostatic and compressible system of equations

In this paper, we present a numerical procedure for solving a 2‐dimensional, compressible, and nonhydrostatic system of equations. A forward‐backward integration scheme is applied to treat high‐frequency and internal gravity waves explicitly. The numerical procedure is shown to be neutral in time as long as a Courant–Friedrichs–Lewy criterion is met. Compared to the leap‐frog‐scheme most models use, this method involves only two time steps, which requires less memory and is also free from unstable computational modes. Hence, a time‐filter is not needed. Advection and diffusion terms are calculated with a time step longer than sound‐wave related terms, so that extensive computer time can be saved. In addition, a new numerical procedure for the free‐slip bottom boundary condition is developed to avoid using inaccurate one‐sided finite difference of pressure in the surface horizontal momentum equation when the terrain effect is considered. We have demonstrated the accuracy and stability of this new model in both linear and nonlinear situations. In linear mountain wave simulations, the model results match the corresponding analytical solution very closely for all three cases presented in this paper. The analytical streamlines for uniform flow over a narrow mountain range were obtained through numerical integration of Queney's mathematical solution. It was found Queney's original diagram is not very accurate. The diagram had to be redrawn before it was used to verify our model results. For nonlinear tests, we simulated the famous 1972 Boulder windstorm and a bubble convection in an isentropic enviroment. Although there are no analytical solutions for the two nonlinear tests, the model results are shown to be very robust in terms of spatial resolution, lateral boundary conditions, and the use of the time-split scheme.     

High resolution downscaled ocean waves (DOW) reanalysis in coastal areas

Large-scale wave reanalysis databases (0.1°–1° spatial resolution) provide valuable information for wave climate research and ocean applications which require long-term time series (> 20 years) of hourly sea state parameters. However, coastal studies need a more detailed spatial resolution (50–500 m) including wave transformation processes in shallow waters. This specific problem, called downscaling, is usually solved applying a dynamical approach by means of numerical wave propagation models requiring a high computational time effort. Besides, the use of atmospheric reanalysis and wave generation and propagation numerical models introduce some uncertainties and errors that must be dealt with. In this work, we present a global framework to downscale wave reanalysis to coastal areas, taking into account the correction of open sea significant wave height (directional calibration) and drastically reducing the CPU time effort (about 1000 ×) by using a hybrid methodology which combines numerical models (dynamical downscaling) and mathematical tools (statistical downscaling). The spatial wave variability along the boundaries of the propagation domain and the simultaneous wind fields are taking into account in the numerical propagations to performance similarly to the dynamical downscaling approach. The principal component analysis is applied to the model forcings to reduce the data dimension simplifying the selection of a subset of numerical simulations and the definition of the wave transfer function which incorporates the dependency of the wave spatial variability and the non-uniform wind forcings. The methodology has been tested in a case study on the northern coast of Spain and validated using shallow water buoys, confirming a good reproduction of the hourly time series structure and the different statistical parameters.     

Eliminating finite‐amplitude non‐physical oscillations in the time evolution of adjoint model solutions introduced by the leapfrog time‐integration scheme

This note provides a detailed theoretical derivation for the removal of non‐physical finite‐amplitude computational oscillations from the solution of the adjoint of a discretized model using the leapfrog finite‐difference scheme. Numerical results are shown using a 1‐dimensional shallow water equation model.     

Numerical modeling of dynamic responses and mooring forces of submerged floating breakwater

Using the volume of fluid (VOF) method, a numerical model is developed to estimate the nonlinear dynamics of a pontoon type moored submerged breakwater under wave action and the forces acting on the mooring lines, for both the vertical and inclined mooring alignments. The model is developed for a two-dimensional wave field in a vertical plane. The finite displacements of the breakwater such as sway, heave and roll in a very small time step are considered and the numerical grid cells intersected by the breakwater surfaces for changing its position due to wave action are treated using the concept of porous body model. Also, two-dimensional experimental studies are carried out to investigate the performance of the proposed model. The comparison of the computed and measured results reveals that the developed numerical model can reproduce well the dynamics of the floating body and the mooring line forces.     

The applicability of the shallow water equations for modelling violent wave overtopping

The shallow water equations (SWE) have been used to model a series of experiments examining violent wave overtopping of a near-vertical sloping structure with impacting wave conditions. A finite volume scheme was used to solve the shallow water equations. A monotonic reconstruction method was applied to eliminate spurious oscillations and ensure proper treatment of bed slope terms. Both the numerical results and physical observations of the water surface closely followed the relevant Rayleigh probability distributions. However, the numerical model overestimated the wave heights and suffered from the lack of dispersion within the shallow water equations. Comparisons made on dimensionless parameters for the overtopping discharge and percentage of waves overtopping between the numerical model and the experimental observations indicated that for the lesser impacting waves, the shallow water equations perform satisfactorily and provide a good alternative to computationally more expensive methods.     

Unsteady ship waves in shallow water of varying depth based on Green–Naghdi equation

Based on Green–Naghdi equation this work studies unsteady ship waves in shallow water of varying depth. A moving ship is regarded as a moving pressure disturbance on free surface. The moving pressure is incorporated into the Green–Naghdi equation to formulate forcing of ship waves in shallow water. The frequency dispersion term of the Green–Naghdi equation accounts for the effects of finite water depth on ship waves. A wave equation model and the finite element method (WE/FEM) are adopted to solve the Green–Naghdi equation. The numerical examples of a Series 60 (C_B=0.6) ship moving in shallow water are presented. Three-dimensional ship wave profiles and wave resistance are given when the ship moves in shallow water with a bed bump (or a trench). The numerical results indicate that the wave resistance increases first, then decreases, and finally returns to normal value as the ship passes a bed bump. A comparison between the numerical results predicted by the Green–Naghdi equation and the shallow water equations is made. It is found that the wave resistance predicted by the Green–Naghdi equation is larger than that predicted by the shallow water equations in subcritical flow , and the Green–Naghdi equation and the shallow water equations predict almost the same wave resistance when , the frequency dispersion can be neglected in supercritical flows.     

复杂地形下海浪数值模式的特征线计算格式

LAGFD-WAM海浪数值模式是一种第三代海浪数值模式，通过求解波数谱平衡方程，并考虑风输入、波浪破碎耗散、底摩擦耗散、波波非线性相互作用和波流相互作用等源函数，模拟波数空间下的海浪方向谱，并依此获得海浪的波高、周期和平均波向。该模式的一个显著特点是采用特征线嵌入格式求解海浪的传播。在进行浅水区域的海浪模拟时，特征线嵌入格式的数值计算方案是否合理对海浪数值模拟结果产生直接的影响。为此LAGFD-WAM海浪数值模式提出了一种新的特征线混合数值计算格式，并应用于浅水海浪数值模拟。结果表明，采用该计算方法，能够使数值模拟结果与实测结果很好符合。     

Deterministic combination of numerical and physical coastal wave models

A deterministic combination of numerical and physical models for coastal waves is developed. In the combined model, a Boussinesq model MIKE 21 BW is applied for the numerical wave computations. A piston-type 2D or 3D wavemaker and the associated control system with active wave absorption provides the interface between the numerical and physical models. The link between numerical and physical models is given by an ad hoc unified wave generation theory which is devised in the study. This wave generation theory accounts for linear dispersion and shallow water non-linearity. Local wave phenomena (evanescent modes) near the wavemaker are taken into account. With this approach, the data transfer between the two models is thus on a deterministic level with detailed wave information transmitted along the wavemaker.     

非线性效应对浅水水波变形的影响

本文采用波数矢量无旋和波能守恒方程建立了一个考虑非线性作用的浅水水波变形数值模型,模型中采用Ｂａｔｔｊｅｓ关系与波数矢量无旋,波能守恒方程一起来求解波浪在浅水中变形的波浪要素,在波能守恒方程中考虑了底摩擦的影响。利用本文提出的数值模型对一个斜坡浅滩水域波浪折射绕射现象进行了验证,验证计算中用一个非线性经验弥散关系近似浅水水波变形的非线性效应并与用线性弥散关系的计算结果进行了比较,结果说明使用非线性     

四叉树网格下的椭圆型缓坡方程数值模型研究

波浪是近岸海域关键的水动力因素之一。考虑到近岸地形复杂、波浪演化显著的特点,建立了四叉树网格体系下的椭圆型缓坡方程数值模型,采用有限体积法对模型进行数值离散,应用GPBiCG(m, n)算法求解离散后的控制方程。模型中根据波浪波长布局计算网格,生成多层次四叉树网格,对复杂计算域有较好的适应性,并且在离散和方程求解中无需引入形函数、不产生复杂的交叉项,节约了存储空间和计算时间。将模型成功应用于物理模型实验及Acapulco海湾的波浪场数值模拟,结果表明该模型能够准确、高效地模拟近岸波浪场,可为近岸波浪场的模拟提供一定的理论和技术支持。     

一个改进的二阶Boussinesq方程模型在线性波浪反射问题中的适用性研究

基于改进型的二阶Boussinesq方程,在交错网络下建立数值模型.利用模型模拟波浪在常水深情况下的传播,波浪反射系数均低于2％.利用该模型模拟波浪在平斜坡前的反射,并将数值结果与解析解进行对比.结果表明,对于相对水深较大情况,坡度较陡时模拟结果明显偏大;对 于相对水深较小情况,坡度超过1:1时,数值结果仍与解析解有....     

A two dimensional hydrodynamic and sediment transport model for dam break based on finite volume method with quadtree grid

This study aims to develop a robust, accurate and computationally efficient hydrodynamic and sediment transport model for dam break flows. The two dimensional shallow water equations are resolved based on the finite volume method with an unstructured quadtree mesh. The sediment transport and bed evolution modules are coupled with hydrodynamic module to predict simultaneously the hydrodynamics, sediment concentrations and morphological changes. The interface flux is computed by the HLL approximate Riemann solver with second order accuracy. The effects of pressure and gravity are included in source term in this model, which can simplify the computation and eliminate numerical imbalance between source and flux terms. For dam break flows occurring in complicated geometries, the quadtree rectangular mesh is used to refine the interesting area and important part. The model is first verified against results from laboratory experiments, existing numerical models and real life case. It is then used to simulate dam break flows over a mobile bed to investigate the bed evolution. The results are compared with experimental data and field data with good agreement. The method is simple, efficient, and conservative. It shows promise for handling hydrodynamic simulation and sediment transport for a wide range of dam break flows.