A Eulerian–Lagrangian method for the simulation of wave propagation

A Eulerian–Lagrangian method (ELM) is employed for the simulation of wave propagation in the present research. The wave action conservation equation, instead of the wave energy balance equation, is used. The wave action is conservative and the action flux remains constant along the wave rays. The ELM correctly accounts for this physical characteristic of wave propagation and integrates the wave action spectrum along the wave rays. Thus, the total derivative for wave action spectrum may be introduced into the numerical scheme and the complicated partial differential wave action balance equation is simplified into an ordinary differential equation. A number of test cases on wave propagation are carried out and show that the present method is stable, accurate and efficient. The results are compared with analytical solutions and/or other computed results. It is shown that the ELM is superior to the first-order upwind method in accuracy, stability and efficiency and may better reflect the complicated dynamics due to the complicated bathymetry features in shallow water areas.     

球坐标系下MASNUM海浪数值模式的建立及其应用

为开展海浪对海洋上混合层的搅拌混合作用及其对海气界面通量的影响等研究,在LAGFD WAM区域海浪数值模式基础上建立了球坐标系下的全球海浪数值模式.重点导出了球坐标系下的海浪能量谱平衡方程及其复杂特征线方程,该组方程包含了背景流场对波动传播的调整、波动沿大圆传播的折射等.数值积分则采用复杂特征线嵌入计算格式.初步数值模拟结果表明,该海浪全球数值模式能够较为精确地刻画海浪的动力过程.     

海浪同化预报模型理论基础

将共轭变分同化方法应用于 LAGFD- WAM海浪数值模式 ,导出了海浪谱能量平衡方程的共轭方程以及风输入、破碎、底摩擦、波波非线性相互作用和波流相互作用的相应共轭源函数 ,建立了海浪同化模型 ,数值计算仍采用特征线嵌入计算格式 ,为合成孔径雷达波谱反演资料和卫星高度计有效波高资料同化奠定理论基础     

A note on the derivation of wave action balance equation in frequency space

In this paper the wave action balance equation in terms of frequency-direction spectrum is derived.A theoretical formulation is presented to generate an invariant frequency space to replace the varying wavenumber space through a Jacobian transformation in the wave action balance equation.The physical properties of the Jacobian incorporating the effects of water depths are discussed.The results provide a theoretical basis of wave action balance equations and ensure that the wave balance equations used in the SWAN or other numerical models are correct.It should be noted that the Jacobian is omitted in the wave action balance equations which are identical to a conventional action balance equation.     

LAGFD-WAM numerical wave model——Ⅰ. Basic physical model

The LAGFD-WAM wave model is a third generation wave model. In the present paper the physical aspect of the model was shown in great detail including energy spectrum balance equation, complicated characteristics equations and source functions.     

任意曲线边界条件下缓变水深水域波浪传播的数值模拟

缓坡方程被广泛地应用于描述波浪的传播变形计算,目前一般采用矩形网格求解.将计算域剖分为任意四边形网格,以格林公式为基础,在变量沿单元边界线性变化的假定下,对双曲型的波能守恒方程、波数矢无旋性方程进行离散,同时通过等参单元变换推求节点偏导数值以离散椭圆型光程函数方程,从而建立了任意曲线边界条件下缓变水深水域波浪传播的数值模拟模型.将模型应用于平行直线型等深线地形,并将计算域剖分为不规则四边形网格,对不同入射角、底坡、波高等多种组合情况比较了数值解与解析解,结果表明两者一致.应用于复杂边界的实例,数值模拟结果与物模实验值基本吻合.     

Refraction-Diffraction Mathematical Model for Waves With Lateral Energy Transmission Mechanism and Its Verification

To solve problems concerning wave elements and wave propagation, an effective way is the wave energy balance equation, which is widely applied in oceanography and ocean dynamics for its simple computation. The present papaer advances wave energy balance equations considering lateral energy transmission and energy loss as the governing equation for the study of wave refraction-diffraction. For the mathematical model, numerical simulation is made by means of difference method, and the result is verified with two examples.     

利用波能平衡方程研究波浪折射——绕射的数学模型及数值模拟

波能平衡方程是研究风浪要素及波能传播问题的很有效的方法，计算较为简单，在海洋学及海岸动力学中得到广泛的应用。本文采用考虑波能侧向传递机制和耗损制约的波能平衡方程组作为研究波浪折射——绕射变形的控制方程。对数学模型用差分法进行了数值模拟，并用两个例题进行了验证     

浅海水下地形的SAR遥感仿真研究

结合连续性方程和布拉格后向散射模型,在准一维简化浅海水下地形情况下,建立了浅海水下地形SAR海面相对后向散射强度仿真模型,将浅海水下地形区域的SAR海面后向散射强度的相对变化与大尺度背景流场、海面风场和雷达系统参数等联系起来.海上实验和研究结果表明,浅海水下地形的SAR成像主要由通过受水下地形影响的海表层流场对海表面风引起的微尺度波的水动力调制而获取浅海水下地形信息,其中潮流与水下地形的相互作用过程改变海表层流场,变化的海表层流与海表面微尺度波之间的相互作用改变海表面波的空间分布,雷达波与海表面波之间的相互作用决定雷达海面后向散射强度.因此SAR图像中浅海水下地形或水深信息量的多少不仅与海表层流场和海面风速有关,而且与雷达工作波段、雷达波束入射角和极化方式也密切相关.认为由水下地形变化引起的缓慢变化的表层流场中海表面定常微尺度波谱能量密度的变化满足波作用量谱平衡方程;而在波数空间中,海表面微尺度波谱的成长过程也可以用波数谱平衡方程描述,在此基础上,得出了海表面波高频谱(毛细-重力波)形式的解析表达式.众所周知,浅海水下地形信息是由于水下地形影响下SAR海面后向散射强度与背景海面后向散射强度的相对差异而在SAR图像上的呈现,从而在建立浅海水下地形SAR海面相对后向散射强度仿真模型的基础上,仿真计算了浅海水下地形SAR海面相对后向散射强度相对于海表层流场、海面风场等海况参数和SAR工作波段、SAR波束入射角、极化方式等雷达系统参数的数值仿真结果,分析得到了有关浅海水下地形SAR海面相对后向散射强度的特征和SAR浅海水下地形遥感的最佳海况参数与最佳雷达系统参数,为研究和开展SAR浅海水下地形遥感研究提供了有价值的参考.     

斜坡平台波浪破碎数值模拟

波浪在斜坡地形上破碎,破波后稳定波高多采用物理模型试验方法进行研究,利用近岸波浪传播变形的抛物型缓坡方程和波能流平衡方程,导出了适用于斜坡上波浪破碎的数值模拟方法。首先根据波能流平衡方程和缓坡方程基本型式分析波浪在破波带内的波能变化和衰减率,推导了波浪传播模型中波能衰减因子和破波能量流衰减因子之间的关系;其次,利用陡坡地形上的高阶抛物型缓坡方程建立了波浪传播和波浪破碎数学模型;最后,根据物理模型试验实测数据对数值模拟的效果进行验证。数值计算与试验资料比较表明,该模型可以较好地模拟斜坡地形的波浪传播波高变化。     

Implementation and evaluation of alternative wave breaking formulas in a coastal spectral wave model

This paper describes methods and results of research for incorporating four different parameterized wave breaking and dissipation formulas in a coastal wave prediction model. Two formulations assume the breaking energy dissipation to be limited by the Rayleigh distribution, whereas the other two represent the breaking wave energy by a bore model. These four formulations have been implemented in WABED, a directional spectral wave model based on the wave action balance equation with diffraction, reflection, and wave–current interaction capabilities. Four parameterized wave breaking formulations are evaluated in the present study using two high-quality laboratory data sets. The first data set is from a wave transformation experiment at an idealized inlet entrance, representing four incident irregular waves in a slack tide and two steady-state ebb current conditions. The second data set is from a laboratory study of wave propagation over a complex bathymetry with strong wave-induced currents. Numerical simulation results show that with a proper breaking formulation the wave model can reproduce laboratory data for waves propagating over idealized or complicated bathymetries with ambient currents. The extended Goda wave breaking formulation with a truncated Rayleigh distribution, and the Battjes and Janssen formulation with a bore model produced the best agreement between model and data.     

A method to determine the incident wave boundary conditions and its application

When studying the harbor water tranquility, cases are often confronted as that the verification point is not located on the generation line or that the angle between the generation line and the isobath is so large that the differences of the wave climates along the generation line can not be ignored. For these cases, the incident boundary conditions are difficult to evaluate. In order to solve this problem, a combined wave model is developed in the present paper based on the Boussinesq equation and the wave action balance equation. Instead of the one-line wave generation method, a multi-line generation method is proposed for the combined model. Application of this method is given to a case that the harbor is designed with two entrances and the angle between the generation line and the isobath is large and the results are shown reasonable. We suggest that the wave generation method on multi-lines might also be introduced to the wave physical model as the replacement for the one-line generation method.     

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。     

A note on wave action conservation in a dissipative current wave motion

We consider steady, slowly varying water waves propagating on a steady current over a gently sloping bed, so-called current depth refraction. All expressions are correct to second order in wave amplitude. Formulating the energy equation for the fluctuating motion in terms of wave action (wave energy divided by intrinsic angular frequency) results in an expression, where the dissipative term is strikingly similar to wave action itself. It is simply the ‘extra’ dissipation (per unit area) caused by the fluctuating motion (i.e. total dissipation minus the effect of current acting on total mean bed shear stress) divided by the intrinsic angular frequency. We call it ‘wave action dissipation’. An inconsistency in Phillips' (1977) book is pointed out. A new formula for the calculation of wave amplitudes along rays is set forth.     

一般曲线坐标系下波浪传播的数值模拟

在曲线坐标系下,建立了缓变水深水域波浪传播的数值模拟模型.模型适宜于复杂变化的边界形状,克服了各种代数坐标变换的局限性.在建立模型时,将原始的椭圆型缓坡方程的近似型式——依赖时间变化的抛物型方程,作为控制方程,既克服了一般抛物近似方法的缺点,又便利了方程的求解;从开边界条件、不同反射特性的固壁边界条件相统一的表达式出发,对边界条件进行处理;用ADI法数值求解控制方程.对模型的验证表明,数值解与物模实验值吻合良好,模型对于具有复杂边界的工程实际有较强的适应性.     

Wave refraction–diffraction effect in the wind wave model WWM

Based on the extended mild-slope equation, the wind wave model (WWM; Hsu et al., 2005) is modified to account for wave refraction, diffraction and reflection for wind waves propagating over a rapidly varying seabed in the presence of current. The combined effect of the higher-order bottom effect terms is incorporated into the wave action balance equation through the correction of the wavenumber and propagation velocities using a refraction–diffraction correction parameter. The relative importance of additional terms including higher-order bottom components, the wave–bottom interaction source term and wave–current interaction that influence the refraction–diffraction correction parameter is discussed. The applicability of the proposed model to calculate a wave transformation over an elliptic shoal, a series of parallel submerged breakwater induced Bragg scattering and wave–current interaction is evaluated. Numerical results show that the present model provides better predictions of the wave amplitude as compared with the phase-decoupled model of Holthuijsen et al. (2003).     

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.     

特征线计算格式下共轭方程两种导出途径的比较

共轭方程的导出是建立资料同化模型的关键，其导出方式有两种途径：AFD形式与FDA形式。在特征线计算格式基础上针对一类较广泛海洋动力控制方程分析了其两种共轭方程（AFD形式与FDA形式）之间的关系，并将理论结果应用于波谱共轭方程的讨论。     

Wave-Current Propagation over a Frictional Topography

—In this paper the parabolic approximation model based on mild-slope equation is used tostudy wave propagation over a slowly varying and frictional topography under wave-current interaction.A governing equation considering the friction effects is derived by the authors for the first time.A simpli-fied form for the rate of wave energy dissipation is presented on the basis of the wave-current action conser-vation equation and the bottom friction model given by Yoo and O＇connor(1987).Examples reveal thatthe present computational method can be used for the calculation of wave elements for actual engineeringprojects with large water areas.     

多方向不规则波传播变形数值模拟

在推广的缓坡方程数学模型基础上建立了多方向不规则波数学模型,综合考虑了波浪折射、绕射、反射、底摩擦和风能输入等因素。基于线性波浪理论,将波浪方向谱在频率和方向上按等能量分割法离散后,分别计算各组成波的传播变形,再计算合成波要素。缓坡方程数学模型采用改进的ADI法求解,计算效率高,稳定性好。采用椭圆形浅滩不规则波模型试验结果和单突堤不规则波绕射理论解对数学模型进行了验证,数值模拟结果和试验值及理论解符合良好。利用该模型进行了某港港内波浪折射、绕射和反射的联合数值模拟,给出了合理的港内波高分布。