One-dimensional numerical models of higher-order Boussinesq equations with high dispersion accuracy

Abstract-Nonlinear water wave propagation passing a submerged shelf is studied experimentally andnumerically. The applicability of the wave propagation model of higher-order Boussinesq equations de-rived by Zou(2000, Ocean Engneering, 27, 557～575) is investigated. Physical experiments areconducted; three different front slopes (1:10, 1:5 and 1:2) of the shelf are set-up in the experimentand their effects on the wave propagation are investigated. Comparisons of the numerical results withtest data are made and the higher-order Boussinesq equations agree well with the measurements since thedispersion of the model is of high accuracy. The numerical results show that the good results can also beobtained for the steep-slope cases although the mild-slope assumption is employed in the derivation of thehigher-order terms in the higher-order Boussinesq equations.     

高阶Boussinesq类方程数值求解及试验验证

基于二阶非线性与色散的Boussinesq类方程,采用改善的Crank-Nicolson方法对不同情况下淹没潜堤上的波浪传播进行数值模拟。高阶方程与传统、改进型的Boussinesq方程计算结果进行比较,高阶方程的计算结果与实验吻合得更好。表明该高阶Boussinesq方程能够精确预测变水深、强非线性的复杂波况,可用于实际近岸海域波浪问题的计算。     

Extended Boussinesq equations for rapidly varying topography

We developed a new Boussinesq-type model which extends the equations of Madsen and Sørensen [1992. A new form of the Boussinesq equations with improved linear dispersion characteristics. Part 2. A slowly varying bathymetry. Coastal Engineering 18, 183-204.] by including both bottom curvature and squared bottom slope terms. Numerical experiments were conducted for wave reflection from the Booij's [1983. A note on the accuracy of the mild-slope equation. Coastal Engineering 7, 191-203] planar slope with different wave frequencies using several types of Boussinesq equations. Madsen and Sørensen's model results are accurate in the whole slopes in shallow waters, but inaccurate in intermediate water depths. Nwogu's [1993. Alternative form of Boussinesq equation for nearshore wave propagation. Journal of Waterway, Port, Coastal and Ocean Engineering 119, 618-638] model results are accurate up to 1:1 (V:H) slope, but significantly inaccurate for steep slopes. The present model results are accurate up to the slope of 1:1, but somewhat inaccurate for very steep slopes. Further, numerical experiments were conducted for wave reflections from a ripple patch and also a Gaussian-shaped trench. For the two cases, the results of Nwogu's model and the present model are accurate, because these models include the bottom curvature term which is important for the cases. However, Madsen and Sørensen's model results are inaccurate, because this model neglects the bottom curvature term.     

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 new form of generalized Boussinesq equations for varying water depth

A new set of equations of motion for wave propagation in water with varying depth is derived in this study. The equations expressed by the velocity potentials and the wave surface elevations include first-order non-linearity of waves and have the same dispersion characteristic to the extended Boussinesq equations. Compared to the extended Boussinesq equations, the equations have only two unknown scalars and do not contain spatial derivatives with an order higher than 2. The wave equations are solved by a finite element method. Fourth-order predictor–corrector method is applied in the time integration and a damping layer is applied at the open boundary for absorbing the outgoing waves. The model is applied to several examples of wave propagation in variable water depth. The computational results are compared with experimental data and other numerical results available in literature. The comparison demonstrates that the new form of the equations is capable of calculating wave transformation from relative deep water to shallow water.     

具有精确色散性的非线性波浪水平二维数学模型

建立具有色散性的水平二维非线性波浪方程,方程的非线性近似到了三阶。方程以波面升高和自由表面速度势表达的微分-积分型数学方程,给出方程的数值求解方法和算例,对方程积分项的处理给出了计算方法。计算结果与Boussinesq方程模型和缓坡方程模型的对应计算结果进行了对比。     

A barotropic model of the linear semidiurnal tide over a continental slope

Semidiurnal tides, and especially the lunar tide M₂, are dominant dynamics in the Bay of Biscay. Strong tidal currents are associated with the presence of a significant continental slope. By combining Newton's gravitation laws and Euler's equations, Laplace's equations contain the astronomical forcing responsible for the observed semidiurnal tides. In shallow waters, this direct forcing is often neglected. We study here its influence on the tidal dynamics over the continental slope through the development of a simple model describing the barotropic semidiurnal dynamics on a transect perpendicular to the slope. This new model results from the combination of two different models, i.e. the one developed by Rosenfeld and Beardsley (1987), which takes into account the tide-generating force, and that of Battisti and Clarke (1982), which neglects it. A first model is developed by neglecting the direct astronomical forcing in equations: it consists in solving a second-order homogeneous propagation equation for the barotropic semidiurnal tide and needs only coastal conditions as well as the knowledge of the along-slope wave number of the solution. For a mean slope typical of the South Brittany area, this non-forced model provides results in accordance with those of Battisti and Clarke and Le Cann (1990): in particular, in the upper part of the slope, it shows a polarization inversion of tidal ellipses characteristic of the tidal dynamics observed in this area. Then, the direct astronomical forcing is kept in equations. The simple model developed without this forcing is fitted in order to solve the resulting forced propagation equation for the barotropic tide. The solution of this second model is the sum of a forced wave responding to the direct astronomical forcing and of a free wave generated at the coastal boundary. Under the same boundary conditions, the results obtained with the influence of the tide-generating force are then compared with those obtained without it. This comparison allows one to apprehend the importance of the direct astronomical forcing on tidal dynamics across the slope: in particular, the main difference appears in deep waters where this forcing induces a phase-lag between the plain and the shelf for the sea-surface slope.     

Further improvements to the higher-order Boussinesq equations: Bragg reflection

Enhancements for the Bragg reflection are introduced for three sets of 2D higher order Boussinesq equations to improve the prediction of the Bragg reflection. The extension of the approach to other sets of Boussinesq equations is discussed. The analytical solutions for the Bragg reflection over an infinite number of sinusoidal bars are derived for these Boussinesq models and compared to the exact theoretical solution in order to determine the optimized values of the parameters in the new enhancement terms. Numerical simulations are also carried out for the Bragg reflection over a finite number of sand bars and compared with corresponding measurements to validate the enhancements. Comparisons with other forms of Boussinesq models are made to discuss the applicability of different forms of Boussinesq models to rapidly varying topography with sand bars. The effects of the mild slope assumption on the prediction of Bragg reflection and of wave reflection on a plane self are also discussed.     

完全非线性孤立波的直墙反射

报道了应用边界积分方法模拟完全非线性孤立波的传播与直墙反射,给出了波形演变过程。结果表明,本模型对计算孤立波的传播与直墙反射是有效的。三阶Boussinesq方程的孤立波解比低阶方程的孤立波解更接近完全非线性的数值解.当来波波高增大时,孤立波直墙反射的相位滞后变小。若考虑大波高孤立波的直墙反射或波——波相互作用,一阶理论预报的相位滞后往往低估实际情况。     

Coastal trapped waves in a two-layer ocean: Wave properties when the density interface intersects a sloping bottom

Properties of coastal trapped waves when the pycnocline intersects a sloping bottom are studied using a two-layer model which has slopes in both layers. In this system there is an infinite discrete sequence of modes, and four different sorts of waves exist: the barotropic Kelvin wave, the upper shelf wave, the lower shelf wave and the internal Kelvin-type wave. They all propagate with the coast to their right in the Northern Hemisphere. The upper and lower shelf waves are due to the topographic-effect on the upper-layer and lower-layer slopes, respectively. Their motions are dominant in the respective layers being accompanied by significant interface elevations. The properties of the upper (lower) shelf wave are almost unaffected by the existence of a lower-layer (upper-layer) slope. The motion of the internal Kelvin-type wave is confined to the region around the line where the density interface intersects the bottom slope.The modes, except that with the fastest phase speed (the barotropic Kelvin wave), are assigned mode numbers in order of descending frequency. Characteristics of Mode 1 change with wavenumber; the upper shelf wave for small wavenumbers and the internal Kelvin-type wave for large wavenumbers (high frequencies). The higher modes of Mode 2 and above can be classified into the upper and lower shelf waves.     

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.     

A phase-averaged Boussinesq model with effective description of carrier wave group and associated long wave evolution

From the phase-resolving improved Boussinesq equations (Beji and Nadaoka, Ocean Engineering 23 (1996) 691), a phase-averaged Boussinesq model for water waves is derived by more effectively describing carrier wave groups and accompanying long wave evolution with less CPU time. Linear shoaling characteristics of carrier wave equations are investigated and found to agree exactly with the analytical expression obtained from the constancy of energy flux for the improved Boussinesq equations themselves, showing that the present model equations are the results of a consistent derivation procedure regarding energy considerations. Numerical simulations of the derived equations for the single wave group and narrow-banded random waves show the validity of the present model and its high performance, especially on the CPU time.     

非线性波传播的新型数值模拟模型及其实验验证

以一种新型的Boussinesq型方程为控制方程组,采用五阶Runge-Kutta-England格式离散时间积分,采用七点差分格式离散空间导数,并通过采用恰当的出流边界条件,从而建立了非线性波传播的新型数值模拟模型.通过对均匀水深水域内波浪传播的数值模拟说明,模型能较好地模拟大水深水域和强非线性波的传播.通过设置不同的入射波参数来进行潜堤地形上波浪传播的物理模型实验,并将数值解与物理模型实验结果进行了比较.     

A Numerical Model for Nonlinear Wave Propagation on Non-uniform Current

On the basis of the new type Boussinesq equations (Madsen et al.,2002),a set of equations explicitly including the effects of currents on waves are derived.A numerical implementation of the present equations in one dimension is described.The numerical model is tested for wave propagation in a wave flume of uniform depth with current present.The present numerical results are compared with those of other researchers.It is validated that the present numerical model can reasonably reflect the nonlinear influences of currents on waves.Moreover,the effects of inputting different incident boundary conditions on the calculated results are studied.     

A 3-D time-domain coupled model for nonlinear waves acting on a box-shaped ship fixed in a harbor

A 3-D time-domain numerical coupled model is developed to obtain an efficient method for nonlinear waves acting on a box-shaped ship fixed in a harbor. The domain is divided into the inner domain and the outer domain. The inner domain is the area beneath the ship and the flow is described by the simplified Euler equations. The remaining area is the outer domain and the flow is defined by the higher-order Boussinesq equations in order to consider the nonlinearity of the wave motions. Along the interface boundaries between the inner domain and the outer domain, the volume flux is assumed to be continuous and the wave pressures are equal. Relevant physical experiment is conducted to validate the present model and it is shown that the numerical results agree with the experimental data. Compared the coupled model with the flow in the inner domain governed by the Laplace equation, the present coupled model is more efficient and its solution procedure is simpler, which is particularly useful for the study on the effect of the nonlinear waves acting on a fixed box-shaped ship in a large harbor.     

Bragg resonant reflection of carrier waves composing wave groups

Numerical analyses for the Bragg resonant reflection of carrier waves associated long waves due to sinusoidally varying seabeds are performed by using a set of coupled ordinary differential equations derived from the Boussinesq equations. The Boussinesq equations are firstly approximated with the Fourier decomposition. The coupled governing equations are then derived and used to simulate evolution of both short and long wave components. It is also found that wave groups are generated by two carrier waves with slightly different frequencies. The wave energy of the initial wave components is transferred to other harmonic components during propagation over a long distance. Evolution and reflection of both short and long waves were largely affected by nonlinearity.     

波浪水槽中非线性浅水波传播特性与模拟

通过建立解析解、进行数值模拟和物理实验，研究了波浪水槽中非线性浅水波浪传播特性，给出了数值模拟中对应造波板做正弦运动的二阶入射边界条件。数值模拟采用高阶Boussinesq方程。实验结果、数值结果和解析解进行对比，并讨论了解析解的适用范围、高次谐波的产生及三波相互作用问题。     

Numerical simulation for the two-dimensional nonlinear shallow water waves

This study deals with the general numerical model to simulate the two-dimensional tidal flow, flooding wave (long wave) and shallow water waves (short wave). The foundational model is based on nonlinear Boussinesq equations. Numerical method for modelling the short waves is investigated in detail. The forces, such as Coriolis forces, wind stress, atmosphere and bottom friction, are considered. A two-dimensional implicit difference scheme of Boussinesq equations is proposed. The low-reflection outflow open boundary is suggested. By means of this model,both velocity fields of circulation current in a channel with step expansion and the wave diffraction behind a semi-infinite breakwater are computed, and the results are satisfactory.     

Alternative forms of the higher-order Boussinesq equations: Derivations and validations

An alternative form of the Boussinesq equations is developed, creating a model which is fully nonlinear up to O(μ⁴) (μ is the ratio of water depth to wavelength) and has dispersion accurate to the Padé [4,4] approximation. No limitation is imposed on the bottom slope; the variable distance between free surface and sea bottom is accounted for by a σ-transformation. Two reduced forms of the model are also presented, which simplify O(μ⁴) terms using the assumption ε = O(μ^2/3) (ε is the ratio of wave height to water depth). These can be seen as extensions of Serre's equations, with dispersions given by the Padé [2,2] and Padé [4,4] approximations. The third-order nonlinear characteristics of these three models are discussed using Fourier analysis, and compared to other high-order formulations of the Boussinesq equations. The models are validated against experimental measurements of wave propagation over a submerged breakwater. Finally, the nonlinear evolution of wave groups along a horizontal flume is simulated and compared to experimental data in order to investigate the effects of the amplitude dispersion and the four-wave resonant interaction.     

Generalized boussinesq model for periodic non-linear shallow-water waves

This paper presents the development of a generalized Boussinesq (gB) model for the periodic non-linear shallow-water waves. An incident cnoidal wave solution for the gB model is derived and applied to the wave simulation. A set of radiation boundary conditions is also established to transmit effectively the cnoidal waves out of the computational domain. The classical solutions of the second-order cnoidal waves are discussed within the content of the KdV equation and the generalized Boussinesq equations. An Euler's predictor-corrector finite-difference algorithm is used for numerical computation. The propagation of normally incident cnoidal waves in a channel is studied. The simulated wave profiles agree well with the analytical results. The temporal and spatial evolution of an obliquely incident cnoidal wave is also modelled. The phenomenon of Mach reflection is discussed.