Energy properties and shoaling of higher-order stokes waves on a current

The energy density, the energy flux, the set-down, the radiation stress, and some wave energy velocities have been derived correct to fourth order in wave steepness for waves on a vorticity-free current. The energy flux and the set-down have been used for shoaling predictions for finite amplitude waves with and without a net volume flux. The results with a zero volume flux are compared with more accurate shoaling predictions showing rather good accordance, except for large steepnesses. This also applies to the deep water wave energy transport velocity. The results with a net volume flux show that the steepness of the waves reduces the influence of this flux on the wave evolution. Some problems in connection with the orders in Stokes waves are discussed, among others concerning the dispersion relation and the orders of integral properties. Bed shear and accompanying dissipation is neglected.     

Derivation and application of new equations for radiation stress and volume flux

In this paper, new expressions of radiation stress and volume flux for long waves have been analytically derived by inclusion of higher-order surface elevations up to the sixth-order. To quantify these expressions, surface elevations along a beach are first simulated using the fully nonlinear Boussinesq-type model COULWAVE. Then, based on the large amount of numerical data, new equations for radiation stress and volume flux are statistically formulated. The research unveils the essential roles of the Ursell parameter, Irribarren number and wave steepness described by the local wave height, wave length and bottom slope. The study shows the importance of nonlinear wave properties in wave-induced currents and mean water levels (set-up/down). The higher-order formulations produce lower values for radiation stress and volume flux than calculated from the lower-order and linear waves. Case studies suggest that the new formulations produce an accurate estimation for mean water level. However, improvement on the computed current profiles is marginal for some cases. This implies that the accurate prediction of the current profile would require more than just the proposed improvement of the radiation stress and volume flux.     

Scour below marine pipelines in shoaling conditions for random waves

This paper provides an approach by which the scour depth below pipelines in shoaling conditions beneath non-breaking and breaking random waves can be derived. Here the scour depth formula in shoaling conditions for regular non-breaking and breaking waves with normal incidence to the pipeline presented by Cevik and Yüksel [Cevik, E. and Yüksel, Y., (1999). Scour under submarine pipelines in waves in shoaling conditions. ASCE J. Waterw., Port, Coast. Ocean Eng., 125 (1), 9–19.] combined with the wave height distribution including shoaling and breaking waves presented by Mendez et al. [Mendez, F.J., Losada, I.J. and Medina, R., (2004). Transformation model of wave height distribution on planar beaches. Coast. Eng. 50 (3), 97–115.] are used. Moreover, the approach is based on describing the wave motion as a stationary Gaussian narrow-band random process. An example of calculation is also presented.     

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.     

未破碎变浅随机海浪的波面高度概率分布

利用青岛海洋大学物理海洋实验室现代化的大型水槽,设计进行了多种海浪强度下,由深水传入近岸不同坡度水底上的变浅随机海浪的实验.依据实验资料分析结果表明,对变浅非正态海浪过程而言,其波面高度分布取Gram-Charlier级数前3项,所得结果与实验分布符合良好.该分布中σ、λ₃、λ₄3个参量是测点水深和波浪强度的函数,并获得了与无因次参量Hs/d之间的经验关系,为预测变浅随机海浪的波面高度分布提供了可能.     

Modeling wave�Ccurrent bottom boundary layers beneath shoaling and breaking waves

The boundary layer characteristics beneath waves transforming on a natural beach are affected by both waves and wave-induced currents, and their predictability is more difficult and challenging than for those observed over a seabed of uniform depth. In this research, a first-order boundary layer model is developed to investigate the characteristics of bottom boundary layers in a wave–current coexisting environment beneath shoaling and breaking waves. The main difference between the present modeling approach and previous methods is in the mathematical formulation for the mean horizontal pressure gradient term in the governing equations for the cross-shore wave-induced currents. This term is obtained from the wave-averaged momentum equation, and its magnitude depends on the balance between the wave excess momentum flux gradient and the hydrostatic pressure gradient due to spatial variations in the wave field of propagating waves and mean water level fluctuations. A turbulence closure scheme is used with a modified low Reynolds number k-ε model. The model was validated with two published experimental datasets for normally incident shoaling and breaking waves over a sloping seabed. For shoaling waves, model results agree well with data for the instantaneous velocity profiles, oscillatory wave amplitudes, and mean velocity profiles. For breaking waves, a good agreement is obtained between model and data for the vertical distribution of mean shear stress. In particular, the model reproduced the local onshore mean flow near the bottom beneath shoaling waves, and the vertically decreasing pattern of mean shear stress beneath breaking waves. These successful demonstrations for wave–current bottom boundary layers are attributed to a novel formulation of the mean pressure gradient incorporated in the present model. The proposed new formulation plays an important role in modeling the boundary layer characteristics beneath shoaling and breaking waves, and ensuring that the present model is applicable to nearshore sediment transport and morphology evolution.     

Comparison of time-dependent extended mild-slope equations for random waves

The extended mild-slope equations of Suh et al. [Suh, K.D., Lee, C., Park, W.S., 1997. Time-dependent equations for wave propagation on rapidly varying topography. Coastal Eng., 32, 91–117] and Lee et al. [Lee, C., Kim, G., Suh, K.D., 2003. Extended mild-slope equation for random waves. Coastal Eng., 48, 277–287] are compared analytically and numerically to determine their applicability to random wave transformation. The geometric optics approach is used to compare the two models analytically. In the model of Suh et al., the wave number of the component wave with a local angular frequency ω is approximated with an accuracy of O(ω − ω¯) at a constant water depth, where ω¯ is the carrier frequency of random waves. In the model of Suh et al., however, the diffraction effects and higher-order bottom effects are considered only for monochromatic waves, and the shoaling coefficient of random waves is not accurately approximated. This inaccuracy arises because the model of Suh et al. was derived for regular waves. In the model of Lee et al., all the parameters of random waves such as wave number, shoaling coefficient, diffraction effects, and higher-order bottom effects are approximated with an accuracy of O(ω − ω¯). This approximation is because the model of Lee et al. was developed using the Taylor series expansion technique for random waves. The result of dispersion relation analysis suggests the use of the peak and weighted-average frequencies as a carrier frequency for Suh et al. and Lee et al. models, respectively. All the analytical results are verified by numerical experiments of shoaling of random waves over a slightly inclined bed and diffraction of random waves through a breakwater gap on a flat bottom.     

Seasonal Variability of Near-Surface Heat Budget of Selected Oceanic Areas in the North Tropical Indian Ocean

The results obtained from an Ocean General Circulation Model (OGCM), the Modular Ocean Model 2.2, forced with the National Center for Environmental Prediction/National Center for Atmospheric Research reanalysis data, and observational data have been utilized to document the climatological seasonal cycle of the upper ocean response in the Tropical Indian Ocean. We address the various roles played by the net surface heat flux and the local and remote ocean dynamics for the seasonal variation of near-surface heat budget in the Tropical Indian Ocean. The investigation is based in seven selected boxes in the Arabian Sea, Bay of Bengal and the Equatorial Indian Ocean. The changes of basin-wide heat budget of ocean process in the Arabian Sea and the Western Equatorial Indian Ocean show an annual cycle, whereas those in the Bay of Bengal and the Eastern Equatorial Indian Ocean show a semi-annual cycle. The time tendency of heat budget in the Arabian Sea depends on both the net surface heat flux and ocean dynamics while on the other hand, that in the Bay of Bengal depends mainly on the net surface flux. However, it has been found that the changes of heat budget are very different between western and eastern regional sea areas in the Arabian Sea and the Bay of Bengal, respectively. This difference depends on seasonal variations of the different local wind forcing and the different ocean dynamics associated with ocean eddies and Kelvin and Rossby waves in each regional sea areas. We also discuss the comparison and the connection for the seasonal variation of near-surface heat budget among their regional sea areas. This revised version was published online in July 2006 with corrections to the Cover Date.     

A formal derivation and numerical modelling of the improved Boussinesq equations for varying depth

A formal derivation of the improved Boussinesq equations of Madsen and Sørens (1992) is presented to provide the correct forms of the depth-gradient related terms. Linear shoaling characteristics of the new equations are investigated by the method of Madsen and Sørensen (1992) and by the energy flux concept separately and found to agree perfectly, whereas these approaches give conflicting results for the equations derived by Madsen and Sørensen (1992). Furthermore, Nwogu's (1993) modified Boussinesq model is found to produce a linear shoaling-gradient identical with the present work. Numerical modelling of the derived equations for directional waves is carried out by three-time-level finite-difference approximations. A higher-order radiation condition is implemented for effective absorption of the outgoing waves. Several test cases are included to demonstrate the performance of the model.     

Fixed-frequency Stokes wave expansion

The study describes a new fixed-frequency Stokes wave theory that differs from previous Stokes wave theories that fix the wave number. The present wave expansion analytically reveals that the wavelength increases with wave height and exceeds than the wavelength obtained by linear wave theory. A method proposed to comparably transform the wave celerity of Fenton's [Fenton, J.D., 1985. A fifth-order Stokes theory for steady waves. Journal of Waterway, Port, Coastal and Ocean Engineering 111, 216–234.] wave theory to the present one. A direct calculation of the wavelength is introduced for practical solutions, avoiding the need to solve a nonlinear equation using an iterative numerical method.     

Internal solitary waves shoaling onto a shelf: Comparisons of weakly-nonlinear and fully nonlinear models for hyperbolic-tangent stratifications

In this study the evolution of internal solitary waves shoaling onto a shelf is considered. The results of high resolution two-dimensional numerical simulations of the incompressible Euler equations are compared with the predictions of several weakly-nonlinear shoaling models of the Korteweg–de Vries family including the Gardner equation and the cubic regularized long wave (or Benjamin–Bona–Mahoney) equation. Wave models in both physical x–t space and in s–x space are considered where s is a commonly used characteristic time variable. The effects of rotation, background currents and damping are ignored. The Boussinesq and rigid lid approximations are also used. The shoaling internal solitary waves generally fission into several waves. Reflected waves are negligible in the cases considered here. Several hyperbolic tangent stratifications are considered with and without a critical point. Among the equations in x–t space the cubic regularized long wave equation gives the best predictions. The Gardner equation in s–x space gives the best predictions of the shape of the leading waves on the shelf, but for many stratifications it predicts a propagation speed that is too large.     

Shoaling waves: A discussion

Surface gravity waves are commonly observed to slow down and to stop at a beach without any noticeable reflection taking place. We assume that as a consequence the waves are continuously giving up their linear and angular momenta, which they carry with them, along with energy, as they propagate into gradually decreasing mean depths of water. It takes a force to cause a time rate of decrease in the linear momentum and a torque to produce a time rate of decrease in the angular momentum. Both a force and a torque operate on the shoaling waves, due to the presence of the sloping bottom, to cause the diminution of their linear and angular momenta. By Newton’s third law, action equals reaction, an equal but opposite force and torque are exerted on the bottom. No other mechanisms for transferring linear and angular momenta are included in the model. Since the force on the waves acts over a horizontal distance during shoaling, work is done on the waves and energy flux is not conserved. Bottom friction, wave interaction with a mean flow, scattering from small-scale bottom irregularities and set-up are neglected. Mass flux is conserved, which leads to a shoreward monotonic decrease in amplitude consistent with available swell data. The formula for the time-independent force on the bottom agrees qualitatively with observations in seven different ways: four for swell attenuation and three for sediment transport on beaches. Ardhuin (2006) argues against a mean force on the bottom that is not hydrostatic, mainly by using conservation of energy flux. He also applies the action balance equation to shoaling waves. Action is a difficult concept to grasp for motion in a continuum; it cannot be easily visualized, and it is not really necessary for solving the shoaling wave problem. We prefer angular momentum because it is clearly related to the observed orbital motion of the fluid particles in progressive surface waves. The physical significance of wave action for surface waves has been described recently by showing that in deep water action is equivalent to the magnitude of the wave’s orbital angular momentum (Kenyon and Sheres, 1996). Finally, Ardhuin requires that there be a significant exchange of linear momentum between shoaling waves and an unspecified mean flow, although the magnitude and direction of the exchange are not predicted. No mention is made of what happens to the orbital angular momentum during shoaling. Mass flux conservation is not stated.     

基于二阶斯托克斯波理论的辐射应力垂向分布

基于二阶斯托克斯波理论推导了辐射应力的垂向分布表达式,通过算例讨论了辐射应力在深水和有限水深条件下的垂向分布规律,并与基于微幅波理论的辐射应力进行了比较.结果表明,在波浪非线性不强时,基于二阶斯托克斯波理论的辐射应力与基于微幅波理论的辐射应力表达式计算结果接近;而当水深较浅波浪非线性较强时,基于二阶斯托克斯波理论的辐射应力在近表面处明显大于基于微幅波理论的辐射应力.采用二阶斯托克斯波理论推导的波浪辐射应力更为合理地反映了波浪非线性效应.     

Simplified higher-order Boussinesq equations: I. Linear simplifications

In this paper, we derive and test simplified higher-order Boussinesq equations, i.e., higher-order Boussinesq equations which only show lower-order terms. Simplifications are performed linearly for flat beds and slopes of O(∇h). With proper coefficient choice, dispersion and shoaling properties are found to be good, while interior fluid velocities show relatively greater error at high wavenumbers.The resulting sets of equations are found to be variants of already-existing equations, which may be easily modified to improve performance. The new equations have dispersion identical to previous results but significantly improved shoaling.     

Acoustic measurements of the velocity field beneath shoaling and breaking waves

An acoustic current meter attached to a servo-hydraulic surface-following device was used to obtain near-surface velocity measurements beneath breaking and near breaking surface gravity waves shoaling on a 1:40 beach. The data are compared to velocities predicted by two adaptations of linear theory: superposition and stretching. For unbroken and near breaking waves, the predictions are in close agreement with the measurements. For breaking and broken waves, near surface crest velocity measurements are influenced by air entrainment; trough velocities are relatively well predicted. The problems associated with the acoustic measurement of near-surface velocities are highlighted.     

强非线性和色散性Boussinesq方程数值模型检验

采用同位网格有限差分法,建立了强非线性和色散性Boussinesq方程数值计算模型。以稳恒波Fourier近似解给定入射波边界条件,对均匀水深深水和浅水域不同非线性的行进波、缓坡地形上深水至浅水域的浅水变形波、以及缓坡和陡坡地形上的波浪水槽实验进行了数值计算,并将计算结果与解析解、解析数值解以及实验值进行了较为详细的比较,从而检验了模型的色散性、非线性以及不同底坡下非线性波的浅水变形性能。     

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).     

Three-dimensional structure of wave-induced momentum flux in irrotational waves in combined shoaling-refraction conditions

In the paper, the three-dimensional structure of the wave-induced momentum flux in irrotational waves propagating over a two-dimensional, irregular bathymetry is analyzed. The expansion method developed by de Vriend and Kitou [de Vriend, H.J., Kitou, N., 1990a. Incorporation of wave effects in a 3D hydrostatic mean current model. Delft Hydraulics Report H-1295. de Vriend, H.J., Kitou, N., 1990b. Incorporation of wave effects in a 3D hydrostatic mean current model. Proc. 22nd Int. Coast. Eng. Conf. ASCE, 1005–1018.] for unidirectional waves has been extended to derive expressions for velocity components in three-dimensional waves over sloping bottom. The vertical wave-induced momentum flux resulting from this solution has been shown to be vertically-varying (contrary to the 2D-V case) and to act as a counterbalance for the vertical variability of the other wave forcing terms in the momentum equations. Thus, the total wave forcing remains depth-invariant, but—contrary to the ‘traditional’ solution based on the radiation stress concept—it does not depend explicitly on the direction of wave propagation and is a simple function of gradients of wave energy and water depth only. One of the most important consequences of this fact is the lack of the longshore-current-generating force in the case of non-dissipative waves approaching a shore with a bottom profile uniform in the along-shore direction. To illustrate the meaning of the new solution, the wave forcing due to waves approaching a barred beach has been analysed in detail. Also, the present solution has been shown to give the same results as the one obtained by extending of the approach by Rivero and Arcilla [Rivero, F.J., Arcilla, A.S., 1995. On the vertical distribution of 〈ũw˜〉. Coast. Eng. 25, 137–152.] to three dimensions.     

基于四阶完全非线性Boussinesq水波方程二维波浪传播数值模型

建立基于四阶完全非线性Boussinesq水波方程的二维波浪传播数值模型。采用Kennedy等提出的涡粘方法模拟波浪破碎。在矩形网格上对控制方程进行离散,采用高精度的数值格式对离散方程进行数值求解。对规则波在具有三维特征地形上的传播过程进行了数值模拟,通过数值模拟结果与实验结果的对比,对所建立的波浪传播模型进行了验证。同时,为了考察非线性对波浪传播的影响,给出和上述模型具有同阶色散性、变浅作用性能但仅具有二阶完全非线性特征的波浪模型的数值结果。通过对比两个模型的数值结果以及实验数据,讨论非线性在波浪传播过程中的作用。研究结果表明,所建立的Boussinesq水波方程在深水范围内不但具有较精确的色散性和变浅作用性能,而且具有四阶完全非线性特征,适合模拟波浪在近岸水域的非线性运动。