Distribution of bottom stress and tidal energy dissipation in a well-mixed estuary

Estimates of area-averaged tidal bottom stress are made for four channel segments of the Great Bay Estuary, N.H. Current and sealevel measurements are used to estimate acceleration and pressure gradient terms in the equation of motion, while the equation of motion itself is used to infer the remaining stress term. Dynamic terms, bottom stress values, friction coefficients and energy dissipation rates are estimated for each site. The analysis shows that while throughout the estuary the principal force balance is between the frictional stress and the pressure gradient forcing, RMS values of total bottom stress range from 2·67 to 10·38 Nm^?2 and friction coefficients vary from 0·015 to 0·054. Both stress and energy dissipation are largest in the seaward portion of the estuary with an order of magnitude decrease in dissipation at the most inland site.These distributions of stress and energy dissipation are consistent with cotidal charts of the principal semi-diurnal tidal constituent (M₂) which indicate that the estuary is composed of a highly dissipative more progressive tidal wave regime seaward and a less dissipative standing wave regime landward.     

Propagation of shallow water waves in an open parabolic channel using the WKB perturbation technique

A singular perturbation analysis based on the WKB technique to study the hydrodynamic performance of periodic ocean waves that are incident on an open parabolic channel of constant depth is proposed. We derive a linear model to predict the propagation of the long ocean waves into the channel. In this manner, the spatial distribution for the surface elevation of the ocean waves inside the channel as a function of two dimensionless parameters, namely, a kinematical parameter, κ and a geometrical parameter ε, is governed by a second-order ordinary differential equation. The kinematical parameter κ denotes the ratio of the potential head, due to gravity, to the kinetic head of the ocean waves along the longitudinal axis of the parabolic channel. Meanwhile, ε is a dimensionless geometrical parameter that represents a characteristic ratio of the parabolic channel. Using matching conditions, simple expressions for the reflection and transmission coefficients are obtained.     

Wave modelling in the vicinity of submerged breakwaters

This paper describes a simple method for modelling wave breaking over submerged structures, with the view of using such modelling approach in a coastal area morphodynamic modelling system.A dominant mechanism for dissipating wave energy over a submerged breakwater is depth-limited wave breaking. Available models for energy dissipation due to wave breaking are developed for beaches (gentle slopes) and require further modifications to model wave breaking over submerged breakwaters.In this paper, wave breaking is split into two parts, namely: 1) depth-limited breaking modelled using Battjes and Janssen's (1978) theory [Battjes, J.A. and Jannsen, J.P.F.M. (1978). Energy loss and setup due to breaking of random waves. Proceedings of the 16th Int. Conf. Coast. Eng., Hamburg, Germany, pp. 569-587.] and 2) steepness limited breaking modelled using an integrated form of the Hasselmann's whitecapping dissipation term, commonly used in fully spectral wind–wave models. The parameter γ₂, governing the maximum wave height at incipient breaking (H_max = γ₂d) is used as calibration factor to tune numerical model results to selected laboratory measurements. It is found that γ₂ varies mainly with the relative submergence depth (ratio of submergence depth at breakwater crest to significant wave height), and a simple relationship is proposed. It is shown that the transmission coefficients obtained using this approach compare favourably with those calculated using published empirical expressions.     

Interaction mechanisms among waves,currents and a submerged plate

Wave-induced loads on a submerged plate, representative of submerged breakwater, coastal-bridge deck and a certain type of wave energy converter, in a uniform current are investigated in this study using fully nonlinear numerical wave tanks (NWTs) based on potential flow theory. The coupling effect of wave and current is explored, and the underlying interaction mechanisms of the hydrodynamic forces are described. The presence of a background current modifies the frequency dispersion. It produces changes of the water-surface elevation, and also has an effect on wave-induced loads. Depending on the nonlinearity, higher harmonic wave components are generated above the submerged plate. These contribute to the wave forces. It is found that the horizontal and the vertical force, hence the moment, are affected in the opposite way by the currents. The Doppler shifted effect dominates the vertical force and the moment on the plate. Whereas, the Doppler shifted effect and the generation of higher wave harmonics play opposite roles on the horizontal forces. The contribution of 2^nd order harmonics is found to be up to 30% of the linear component. The current-induced drag force, represented by the advection term ρU∂φ/∂x in the pressure equation, is found to lead to a decrease in the moment for the most range of wavelengths considered, and an increase in the moment for a small range of longer waves.     

Local balance in the air-sea boundary processes

A new growth equation for wind waves of simple spectrum is presented upon three basic concepts. The period and the wave height of significant waves in dimensionless forms, which are considered to correspond to the peak frequency and the energy level, respectively, are used as representative quantities of wind waves. One of the three basic concepts is the concept of local balance, and the other two concern the acquisition of wave energy and the dissipation of wave energy, respectively. It is shown from some actual data that the equation, together with two universal constants concerning the acquisition and the dissipation of wave energy (B=6.2×10^?2 andK=2.16×10^?5, respectively), is applied universally to wide ranges of wind waves from those in a wind-wave tunnel to fully developed sea in the open ocean. “The three-second power law for wind waves of simple spectrum”, and a few relations as the lemmas, are derived, such that the mean surface transport due to the orbital motion of wind waves is always proportional to the friction velocity in wind, and that the steepness is inversely proportional to the root of the wave age. It is also derived that the portion of wind stress which directly enters the wind waves decreases exponentially with increasing wave age and is 7.5 % of the total wind stress for very young waves. Also, equations are presented as to the increase of momentum of drift current, and as to the supply of turbulent energy by wind waves into the upper ocean.     

Numerical simulation of wave transformation and runup incorporating porous media wave absorber and turbulence models

A numerical wave tank is first established using the Navier–Stokes equations and the VOF method assuming laminar flow. The standard k–ε, realizable k–ε and RNG k–ε turbulent models are then incorporated to the numerical tank. An effective numerical method for wave absorption utilizing the energy-dissipating property of porous media is also included. To validate the accuracy of the proposed models, the propagation of a solitary wave, where analytical solution is available for comparison, is first simulated. This is followed by the simulation of irregular wave runup on a composite seawall, wave propagation over submerged bars and wave refraction and diffraction over an elliptic shoal, where experimental data are available for comparison. All computed results agree well with either the analytical solution or the experimental data.     

The continuity equation for longshore current velocity with breaker angle adjusted for a wave-current interaction

The continuity equation for mean longshore current velocity, V = gmT sin 2θ_b, agrees with selected field and laboratory data covering a wide range of conditions. Agreement between continuity equation and data is improved by eliminating those laboratory data which imply deep-water wave crests at angles near or greater than 90 degrees to the shoreline. Agreement between continuity equation and data is further improved by adjusting breaker angles to account for convection of the breaker point by the longshore current. Breaker point convection increases breaker angle by an amount predictable from the analysis developed here. This increase in angle is significant in those laboratory experiments with breaking wave crests at high angles to the shoreline.In the continuity equation, m is bottom slope, T is wave period, and θ_b is breaker angle, but breaker height does not appear. According to radiation stress theory, mean velocity does depend on breaker height, but only weakly. Consistency between the two approaches would require a dimensionless velocity, C_b/gT, to be relatively constant, which it is. (The same dimensionless velocity appears in the analyses of breaker point convection.) The continuity equation is functionally independent of friction and mixing, in keeping with its derivation from simple conservation of mass considerations. The equation has no adjustable coefficients. The degress of agreement with data and the internal consistency of the analysis suggests that it is a good predictor of mean velocity in ordinary longshore currents.     

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.     

Improved representation of breaking wave energy dissipation in parametric wave transformation models

An improved formulation to describe breaking wave energy dissipation is presented and incorporated into a previous parametric cross-shore wave transformation model [Baldock, T.E., Holmes, P., Bunker, S., Van Weert, P., 1998. Cross-shore hydrodynamics within an unsaturated surf zone. Coastal Engineering 34, 173–196]. The new formulation accounts for a term in the bore dissipation equation neglected in some previous modelling, but which is shown to be important in the inner surf zone. The only free model parameter remains the choice of γ, the ratio of wave height to water depth at initial breaking, and a well-established standard parameter is used for all model runs. The proposed model is compared to three sets of experimental data and a previous version of the model which was extensively calibrated against field and laboratory data. The model is also compared to the widely used model presented by Thornton and Guza (1983) [Thornton, E.B., Guza, R.T., 1983. Transformation of wave height distribution. Journal of Geophysical Research 88 (No.C10), 5925–5938].     

欧拉模型模拟波浪的抑制数值耗散效应的一种简单方法

数值波浪水槽是研究波浪及波浪与结构相互作用的常用工具,可在真实尺度下产生波浪,并提供流场的详细数据。然而,大部分数值波浪水槽都存在数值耗散和数值色散问题,数值耗散使波能缓慢消散,数值弥散在波传播过程中使波频移。本文在有限差分法（FDM）求解欧拉方程的基础上,提出了一种抑制数值耗散效应的简单方法,考虑阻尼项的影响,对波的传播解进行了解析求解。该方法的主要思想是在动量方程中附加一个源项,其强度由数值阻尼效应的强度决定。本文通过对规则线性波、Stokes波和不规则波的数值模拟,验证了该方法的有效性。结果表明,本文方法可有效减小数值波浪水槽中存在的数值耗散现象。     

The role of sea water viscidity in modeling the vertical movement of internal tides

A numerical closure scheme has been developed to introduce dissipation processes in particular for the vertical movement of internal tides. This scheme is based on the assumption that a vertically oscillating water mass disturbs the pressure field and feels the viscosity from its neighborhood at the same time. The horizontal viscosity term, referred in this paper as the internal-tide viscosity (ITV) term, is retained in the vertical movement equation, which introduces a quasi-hydrostatic assumption. Therefore, a new expression of the total perturbation pressure has been derived. By applying this expression in a 5′ × 5′ z-coordinate regional ocean model, the results show great improvements. With consideration of the ITV-term, the numerically enhanced vertical movement locally near a ridge has been damped in a z-coordinate system, and the propagation of internal tides away from the ridge has been converted into a more reasonable dissipative mode. With the tunable parameter C_w equals to 0.2, the values of the simulated vertical velocity have been reduced to approximately 50%. And the simulated thermocline structure has been preserved, as well.     

Profiles of fully developed (Airy) waves in different water depths

Airy waves have a sinusoidal profile in deep water that can be modeled by a time series at any point x and time t, given by η(x,t) = (H_o/2) cos[2πx/L_o − 2πt/T_w], where H_o is the deepwater height, L_o is the deepwater wavelength, and T_w is the wave period. However, as these waves approach the shore they change in form and dimension so that this equation becomes invalid. A method is presented to reconstruct the wave profile showing the correct wavelength, wave height, wave shape, and displacement of the water surface with respect to the still water level for any water depth.     

A numerical model for wave motions and turbulence flows in front of a composite breakwater

A mathematical model based on the Volume-Averaged/Reynolds Averaged Navier-Stokes (VARANS) equations is developed to describe surface wave motions in the vicinity of a coastal structure, which could be either a rigid solid structure or a permeable structure or a combination of both. In the VARANS equations, the volume-averaged Reynolds stress is modeled by adopting the nonlinear eddy viscosity assumption. The model equations for the volume-averaged turbulent kinetic energy and its dissipation rate are derived by taking the volume-average of the standard k−ϵ equations. Because of the volume-averaging process, the effects of the small-scale turbulence in porous media are introduced. The performance of the model is checked by comparing numerical solutions with the experimental data related to a composite breakwater reported by Sakakiyama and Liu [Coast. Eng. 121 (2001) 117].     

近岸波浪破碎区不规则波浪的数值模拟

基于近岸不规则波浪传播的抛物型缓坡方程和两类波浪破碎能量损耗因子,对近岸波浪破碎区不规则波浪的波高分布进行了数值模拟,并结合实验结果对数值模拟结果进行了验证分析,结果表明采用两类波浪破碎能量损耗因子所模拟的破碎区波高与实测值均吻合良好,波浪破碎能量损耗因子及波浪破碎指标对破碎区波浪波高分布影响较明显。     

Real scalar field scattering in the nearly extremal Schwarzschildben de Sitter space

Reasonable approximations are introduced to investigate the real scalar field scattering in the nearly extremal Schwarzschild--de Sitter (SdS) space. The approximations naturally lead to the invertible x(r) and the global replacement of the true potential by a P?shl--Teller one. Meanwhile, the Schr?dinger-like wave equation is transformed into a solvable form. Our numerical solutions to the wave equation show that the wave is characteristically similar to the harmonic under the tortoise coordinate x, while the wave piles up near the two horizons and the wavelength tends to its maximum as the potential approaches to the peak under the radial coordinate r.     

Ultimate Lateral Capacity of Rigid Pile in <Emphasis Type="Italic">c</Emphasis>–<Emphasis Type="Italic">φ</Emphasis> Soil

To date no analytical solution of the pile ultimate lateral capacity for the general c–φ soil has been obtained. In the present study, a new dimensionless embedded ratio was proposed and the analytical solutions of ultimate lateral capacity and rotation center of rigid pile in c–φ soils were obtained. The results showed that both the dimensionless ultimate lateral capacity and dimensionless rotation center were the univariate functions of the embedded ratio. Also, the ultimate lateral capacity in the c–φ soil was the combination of the ultimate lateral capacity (f _c ) in the clay, and the ultimate lateral capacity (f _φ ) in the sand. Therefore, the Broms chart for clay, solution for clay (φ=0) put forward by Poulos and Davis, solution for sand (c=0) obtained by Petrasovits and Awad, and Kondner’s ultimate bending moment were all proven to be the special cases of the general solution in the present study. A comparison of the field and laboratory tests in 93 cases showed that the average ratios of the theoretical values to the experimental value ranged from 0.85 to 1.15. Also, the theoretical values displayed a good agreement with the test values.     

Linear wave reflection by trench with various shapes

Two types of analytical solutions for waves propagating over an asymmetric trench are derived. One is a long-wave solution and the other is a mild-slope solution, which is applicable to deeper water. The water depth inside the trench varies in proportion to a power of the distance from the center of the trench (which is the deepest water depth point and the origin of x-coordinate in this study). The mild-slope equation is transformed into a second-order ordinary differential equation with variable coefficients based on the longwave assumption [Hunt's, 1979. Direct solution of wave dispersion equation. Journal of Waterway, Port, Coast. and Ocean Engineering 105, 457–459] as approximate solution for wave dispersion. The analytical solutions are then obtained by using the power series technique. The analytical solutions are compared with the numerical solution of the hyperbolic mild-slope equations. After obtaining the analytical solutions under various conditions, the results are analyzed.     

Effect of under connected plates on the hydrodynamic efficiency of the floating breakwater

In this paper, the hydrodynamic efficiency of a floating breakwater system is experimentally studied by use of physical models. Regular waves with wide ranges of wave heights and periods are tested. The efficiency of the breakwater is presented as a function of the wave transmission, reflection, and energy dissipation coefficients. Different parameters affecting the breakwater efficiency are investigated, e.g. the number of the under connected vertical plates, the length of the mooring wire, and the wave length. It is found that, the transmission coefficient kt decreases with the increase of the relative breakwater width B/L, the number of plates n and the relative wire length l/h, while the reflection coefficient kr takes the opposite trend. Therefore, it is possible to achieve kt values smaller than 0.25 and kr values larger than 0.80 when B/L is larger than 0.25 for the case of l/h-1.5 and n=4. In addition, empirical equations used for estimating the transmission and reflection coefficients are developed by using the dimensionless analysis, regression analysis and measured data and verified by different theoretical and experimental results.     

Improvement and application of a saturation based wave dissipation function in SWAN model

Wave dissipation characteristics in SWAN (Simulating Waves Nearshore) model are investigated through numerical experiments. It is found that neither the fully developed integral parameters of wind waves (significant wave height and peak frequency) nor the high frequency spectral tail can be well reproduced by the default wave dissipation source terms. A new spectral dissipation source term is proposed, which comprises saturation based dissipation above two times of peak frequency and improved whitecapping dissipation at lower frequency spectrum. The reciprocal wave age (u /c p ) is involved into the whitecapping model to adjust dissipation rate at different wind speed. The Phillips higher frequency saturation parameter in the saturation-based dissipation is no longer taken as a constant, but varies with wave age. Numerical validations demonstrate that both the wind wave generation process and higher frequency spectrum of wind waves can be well simulated by the new wave dissipation term.