An extended analytic solution for combined refraction and diffraction of long waves over circular shoals

A new analytic solution of the mild-slope long wave equation is derived for studying the effects of bottom topography on combined refraction and diffraction. The solution is essentially of a series form involving the Bessel functions of real orders but is found to be singular as the bottom tends to be parabolic. Numerical evaluation of the solution nearby the singularity requires some special considerations. The particular solution under the singular condition is also given. Study on combined refraction and diffraction for waves around a circular island on the top of a shoal, which is radially described by a power function with two independent parameters, indicates that there exist an extremely high wave zone near the top of a hyperparabolic shoal. It is also found that the intensity of wave ray focusing increases significantly as the mean slope decreases. A direct consequence of the wave ray focusing is the concentration of wave energy and an increase of the maximal wave runup height around the island.     

An energy-controlling boundary condition for partial wave reflections in the mild slope equation

An energy-controlling technique to actively manage the reflective property of waves from solid boundary is presented. As linear waves propagate through an energy-controlling area, a reduction in wave heights occurs due to energy dissipation, which can be placed under direct control through the imaginary part of the wavenumber and phase velocity. Based on this relationship, the present study investigates a new method to control reflected waves with desired heights in the mild slope equation model. The method is validated through numerical tests for various reflection coefficients and the results confirm the promising use of energy-controlling boundary condition for partial wave reflections.     

修正型缓坡方程的有限元模型

与缓坡方程相比,修正型缓坡方程增加了地形曲率项和坡度平方项,从而提高了数值求解的复杂性。本文将计算域划分为内域和外域,内域为水深变化区域,使用修正型缓坡方程,其中的地形曲率项和坡度平方项可用有限单元各节点的水深信息和单元插值函数表示,外域为水深恒定区,速度势满足Helmholtz方程,通过内外域的边界匹配建立有限元方程,并用高斯消去法求解。进而分别模拟了波浪传过Homma岛和圆形浅滩的变形,其结果与相关的解析解和实验数据吻合良好,证明了本文有限元模型的正确性。同时,通过与实验数据的对比也明显看出,在地形坡度较陡的情况下,修正型缓坡方程较缓坡方程具有更高的计算精度。     

Modeling nonlinear wave–wave interactions with the elliptic mild slope equation

An approach is developed to simulate wave–wave interactions using nonlinear elliptic mild-slope equation in domains where wave reflection, refraction, diffraction and breaking effects must also be considered. This involves the construction of an efficient solution procedure including effective boundary treatment, modification of the nonlinear equation to resolve convergence issues, and validation of the overall approach. For solving the second-order boundary-value problem, the Alternating Direction Implicit (ADI) scheme is employed, and the use of approximate boundary conditions is supplemented, for improved accuracy, with internal wave generation method and dissipative sponge layers. The performance of the nonlinear model is investigated for a range of practical wave conditions involving reflection, diffraction and shoaling in the presence of nonlinear wave–wave interactions. In addition, the transformation of a wave spectrum due to nonlinear shoaling and breaking, and nonlinear resonance inside a rectangular harbor are simulated. Numerical calculations are compared with the results from other relevant nonlinear models and experimental data available in literature. Results show that the approach developed here performs reasonably well, and has thus improved the applicability of this class of wave transformation models.     

Mild-slope equation for water waves propagating over non-uniform currents and uneven bottoms

I~IOXThe interaction between surface waves and ambient currents and nearshore topography lies atone of the heat of morphological medelling. Accurate predictions of how wave propagates overcurrents and topography, and of the consequent erosion and dePOSition of sand on a beach or tidalflat are vital when assessing how a coastline may be affected by changing conditions.The mild-slope equation was introduced by Berkhoff (1972) as a way of approximating therefraction-diffraction of linearized s…     

An analytic solution to the wave bottom boundary layer governing equation under arbitrary wave forcing

Existing models of the wave bottom boundary layer have focused on the vertical and temporal dynamics associated with monochromatic forcing. While these models have made significant advances, they do not address the more complicated dynamics of random wave forcing, commonly found in natural environments such as the surf zone. In the closed form solution presented here, the eddy viscosity is assumed to vary temporally with the bed shear velocity and linearly with depth, however, the solution technique is valid for any eddy viscosity which is separable in time and space. A transformation of the cross-shore velocity to a distorted spatial domain leads to time-independent boundary conditions, allowing for the derivation of an analytic expression for the temporal and vertical structure of the cross-shore velocity under an arbitrary wave field. The model is compared with two independent laboratory observations. Model calculations of the bed shear velocity are in good agreement with laboratory measurements made by Jonsson and Carlsen (1976, J. Hydraul. Res., 14, 45–60). A variety of monochromatic, skewed, and asymmetric wave forcing conditions, characteristic of those found in the surf zone, are used to evaluate the relative effects on the bed shear. Because the temporal variation of the eddy viscosity is assumed proportional to the bottom shear, a weakly nonlinear interaction is created, and a fraction of the input monochromatic wave energy is transferred to the odd harmonics. For a monochromatic input wave, the ratio of the third harmonic of velocity at the bed to the first is <10%. However, for a skewed and asymmetric input wave, this ratio can be as large as 30% and is shown to increase with increasing root-mean-square input wave acceleration. The work done by the fluid on the bed is shown to be a maximum under purely skewed waves and is directed onshore. Under purely asymmetric waves, the work done is significantly smaller and directed offshore.     

Experimental study of the transformation of bound long waves over a mild slope with ambient currents

The effect of currents on the variation of cross-shore bound long waves forced by bichromatic waves over a plane slope was investigated in the laboratory. In still water the growth rate of the shoaling bound long waves over the slope is proportional to h^– 5/2 (h is still-water depth). It was found that the opposing current makes the amplitudes of the bound long waves greater than those of still water for all cases. However, the amplitudes of bound long waves in a following current are reduced in the weakly modulated cases but are enhanced in the fully modulated case.     

采掘坑位置对珊瑚礁海岸波浪传播变形影响试验

为研究珊瑚礁坪上采掘坑位置变化对珊瑚礁海岸波浪传播变形的影响, 本文通过物理模型试验测试了采掘坑在不同位置和无坑情况下一系列不规则波工况的波浪特征。结果表明, 随着采掘坑位置朝岸线附近移动直至无坑时, 岸线附近的短波波高逐渐减小; 采掘坑的存在减弱了岸线附近的低频长波波高, 当采掘坑位于岸线附近时, 长波波高还受到局部水深增加的影响而进一步减弱。采掘坑从礁缘移动至岸线附近直到无坑时, 岸线附近的增水逐渐增大, 这种趋势在礁坪水深较大时更为明显。通过相干函数分析, 证明了礁坪上低频长波是由于短波群破碎点的移动而产生, 采掘坑位置的变化对低频长波的产生无明显影响; 通过传递函数分析, 验证了礁坪上的低频长波存在一阶共振放大效应, 采掘坑的存在减弱了这种放大效应, 当坑位于礁坪中间和岸线附近时, 这种减弱效应更为显著。     

Experimental investigation of the load exerted by nonstationary internal solitary waves on a submerged slender body over a slope

Laboratory experiments are performed in a large stratified fluid flume to examine the characteristics of the load on a submerged slender body that is exerted by a nonstationary internal solitary wave (ISW) from its interaction with a gentle slope. The nonstationary ISW over the slope and its load on the body are measured by using multi-channel conductivity-probe arrays and a specially designed force measurement device, respectively, and the body’s vertical and horizontal positions on the load are determined by analyzing the effects of the incident ISW’s amplitude. The experimental results show that the load on the slender body increases as the incident ISW’s amplitude increases; additionally, the effect of oscillations is enhanced because of the ISW’s distortion, breaking and fission. The oscillating action from fission waves becomes dominant as the amplitude reaches a certain value. Additionally, the load is correlated with body’s vertical position relative to the pycnocline. The magnitudes of the vertical and horizontal forces reach a maximum and minimum in the pycnocline, respectively, and the horizontal force in this direction is the opposite above and below the pycnocline. Compared to a case without a slope, the load on the slender body increases because of the nonstationary ISW, and its effect on the maximum force is transferred to the pycnocline. When the body’s horizontal position is located close to the top of the slope, the direction of the horizontal and vertical forces remains consistent, but its acting time becomes longer. In addition, high-frequency actions on the slender body are impacted by nonstationary ISWs near the slope’s top.     

An analytical solution for the local suspended sediment concentration profile in tidal sea region

The time-averaged and oscillatory solutions of the one-dimensional vertical (1DV) advection–diffusion equation for the suspended sediment have been derived analytically in a tidal sea region of finite water depth. The basic equation assumes constant eddy diffusivity and settling velocity. No net flux condition is set at the sea surface, while a boundary condition with the erosion rate and depositional velocity is prescribed at the sea bottom. The time-averaged solution has been derived in a straightforward manner, while the advection–diffusion equation governing the oscillatory concentration has been first transformed to a simple diffusion equation and then solved using the Galerkin-eigenfunction method. The former is given in a closed form, while the latter is presented in a series solution.A set of calculations has been performed to examine the change in the vertical structure as well as magnitude of the concentration response function. A possible use of the solution to make an estimate of the erosion rate at the sea bottom based on the concentration information at the sea surface is discussed.     

An approximated method for the solution of elliptic problems in thin domains: Application to nonlinear internal waves

Realistic numerical simulations of nonlinear internal waves (NLIWs) have been hampered by the need to use computationally expensive nonhydrostatic models. In this paper, we show that the solution to the elliptic problem arising from the incompressibility condition can be successfully approximated by a few terms (three at most) of an expansion in powers of the ratio (horizontal grid spacing)/(total depth). For an n dimensional problem, each term in the expansion is the sum of a function that satisfies a one-dimensional second-order ODE in the vertical direction plus, depending on the surface boundary condition, the solution to an n-1 dimension elliptic problem, an evident saving over having to solve the original n-dimensional elliptic problem. This approximation provides the physically correct amount of dispersion necessary to counteract the nonlinear steepening tendency of NLIWs. Experiments with different types of NLIWs validate the approach. Unlike other methods, no ad hoc artificial dispersion needs to be introduced.