Wave shoaling and diffraction in current over a mild-slope

The wave relative frequency in the coordinate system moving with current and the angle between the direction of wave propagation and that of current are computed based on the wave dispersion relation. The current field is computed by solving the depth averaged shallow water equations. The wave field is computed by solving the mildslope equation which has taken the current‘s effect into account. A numerical model is established using a finite element method for simulating the wave shoaling and diffraction in current over a mild-slope, and the numerical results are reasonable to compare with the experimental data.     

Wave diffraction in a current over a local shoal

Applying the Green's theorem and a moving oscillating source as the Green's function, an integral equation method is developed for predicting wave height over a shoal in a weak current. The integral equation is discretized by a higher order element method and a numerical code is implemented. To validate the numerical code, comparison is made on wave run-up around a vertical cylinder with McCamy and Fuchs' analytic solution in the still water and the results in currents from numerical wave tanks. Computation has also been carried out for the wave diffraction over a parabolic shoal in a current, and numerical results are given at many sections. The numerical results from the calculation can be used for benchmark test for other methods on wave diffraction in current.     

The Mathematical and Parameterized Expressions for Wave Induced Excess Flow of Momentum

Wave induced excess flow of momentum(WIEFM)is the averaged flow of momentum over a wave period due to wave presence,which may also be called 3-D radiation stress.In this paper,the 3-D current equations with WIEFM are derived from the averaged Navier-Stokes equations over a wave period,in which the velocity is separated into the large-scale background velocity,the wave particle velocity and the turbulent fluctuation velocity.A concept of wave fluctuating layer(WFL)is put forward,which is the vertical column from the wave trough to wave ridge.The mathematical expressions of WIEFM in WFL and below WFL are given separately.The parameterized expressions of WIEFM are set up according to the linear wave theory.The integration of WIEFM in the vertical direction equals the traditional radiation stress(namely 2-D radiation stress)given by Longuet-Higgins and Stewart.     

Modelling of waves and currents around submerged breakwaters

Recent experimental data collected during the DELOS project are used to validate two approaches for simulating waves and currents in the vicinity of submerged breakwaters.The first approach is a phase-averaged method in which a wave model is used to simulate wave transformation and calculate radiation stresses, while a flow model (2-dimensional depth averaged or quasi-3D) is used to calculate the resulting wave driven currents. The second approach is a phase resolving method in which a high order 2DH-Boussinesq-type model is used to calculate the waves and flow.The models predict wave heights that are comparable to measurements if the wave breaking sub-model is properly tuned for dissipation over the submerged breakwater. It is shown that the simulated flow pattern using both approaches is qualitatively similar to that observed in the experiments. Furthermore, the phase-resolving model shows good agreement between measured and simulated instantaneous surface elevations in wave flume tests.     

Energy-momentum tensor and conservation laws for water waves

With the averaged variational principles and the energy-momentum tensor, conservation laws for wave action, mass, momentum and energy for slowly varying water waves are derived systematically. For Stokes waves this enables us to derive energy and momentum conservation laws developed by the concept of radiation stress from general energymomentum tensor, which is as yet not clearly understood. Moreover the change of the wave amplitude of solitary wave over an uneven bottom is obtained from energy conservation equation.     

波浪与抛石潜堤相互作用的MPS法数值模型研究

为了研究波浪与抛石潜堤相互作用过程中大自由表面变形和堤内渗流等强非线性紊流运动问题,利用改进的MPS法,建立了模拟波浪与抛石潜堤相互作用的MPS法数值计算模型。模型将抛石潜堤假定为均质多孔介质,采用Drew的二相流运动方程描述多孔介质内外的流体运动;通过在动量方程中增加非线性阻力项,并引入亚粒子尺度紊流模型,模拟波浪与可渗结构物相互作用过程中的紊流运动。选取“U”型管中多孔介质内渗流过程和孤立波与可渗潜堤相互作用两个典型的渗流问题,通过将数值计算结果与理论解和实测值的对比分析,对所提出的MPS法紊流渗流模型的模拟精度进行验证。结果表明:基于改进的MPS法构建的垂向二维紊流渗流模型可以很好地再现“U”型管中多孔介质内渗流以及波浪作用下可渗潜堤内外的复杂流场,显著缓解流-固界面处的压力震荡与粒子分布不均匀问题,实现了较高的模拟精度。     

Deformation of monochromatic water wave trains propagating over a submerged obstacle in the presence of uniform currents

A nonlinear numerical model based on depth averaged equations and a relevant physical model have been investigated for the deformation of the water wave propagating over a submerged parabolic obstacle in the presence of uniform current. Physical and numerical modeling for wave with both following and opposing currents are done to explore the wave evolutions during passage over the submerged obstacle. A third-order Stokes dispersion relation is utilized in some cases in the computation. Separated flow zone is taken into consideration by two empirical equations obtained from the physical model testing done by the authors. Verification and validation of the numerical model by other published theoretical and experimental data are presented.     

Effect of vertically logarithmic steady currents on shallow surface waves

The combined wave-current flow has been solved by researchers by assuming wave over either depthwise constant or linear current profile. Some complicated nonlinear current profiles have also been considered to simulate various shear currents. We consider a nonlinear current vertically logarithmic in nature and examine its interaction with a periodic surface wave. The Navier-Stokes equations for incompressible flow are solved for the current part and by using periodic boundary conditions. The effect of logarithmic current on wave components is assessed. The corresponding celerity and dispersion equation yields a close-form solution for the shallow-wave approximation. Several comparative trends between wave-only, wave with log current, and wave with constant current for the wave following/opposing these currents have been discussed. The flow properties of the first order are presented which can be applicable to the real inland and coastal flows, where progressive waves are ubiquitous over a depthwise logarithmic current. The work is further extended to the second-order semiempirical wave component by using past experimental data on the wave spectrum of combined flow. Published in Morskoi Gidrofizicheskii Zhurnal, No. 3, pp. 20–40, May–June, 2008.     

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.     

A General Linear Wave Theory for Water Waves Propagating over Uneven Porous Bottoms

Starting from the widespread phenomena of porous bottoms in the near shore region, considering fully the diversity of bottom topography and wave number variation, and including the effect of evanescent modes, a general linear wave theory for water waves propagating over uneven porous bottoms in the near shore region is established by use of Green‘s scond identity. This theory can be reduced to a number of the most typical mild-slope equations curreutly in use and provide a reliable research basis for follow-up development of nonlinear water wave theory involving porous bottoms.     

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.     

Non-hydrostatic modelling of regular wave transformation and current circulation in an idealized reef-lagoon-channel system

The wave-induced setup and circulation in a two dimensional horizontal (2DH) reef-lagoon-channel system is investigated by a non-hydrostatic model. The simulated results agree well with observations from the laboratory experiments, revealing that the model is valid in simulating wave transformation and currents over reefs. The effects of incident wave height, period, and reef flat water depth on the mean sea level and wave-driven currents are examined. Results show that the distributions of mean sea level and current velocities on the reef flat adjacent to the channel vary significantly from those in the area close to the side walls. From the wave averaged current field, an obvious alongshore flux flowing from the reef flat to the channel is captured. The flux from the reef flat composes the second source of the offshore rip current, while the first source is from the lagoon. A detailed momentum balance analysis shows that the alongshore current is mainly induced by the pressure gradient between the reef flat and the channel. In the lagoon, the momentum balances are between the pressure and radiation stress gradient, which drives flow towards the channel. Along the channel, the offshore current is mainly driven by the pressure gradient.     

A new efficient 3D non-hydrostatic free-surface flow model for simulating water wave motions

A new three-dimensional, non-hydrostatic free surface flow model is presented. For simulating water wave motions over uneven bottoms, the model employs an explicit project method on a Cartesian the staggered gird system to solve the complete three-dimensional Navier–Stokes equations. A bi-conjugated gradient method with a pre-conditioning procedure is used to solve the resulting matrix system. The model is capable of resolving non-hydrostatic pressure by incorporating the integral method of the top-layer pressure treatment, and predicting wave propagation and interaction over irregular bottom by including a partial bottom-cell treatment. Four examples of surface wave propagation are used to demonstrate the capability of the model. Using a small of vertical layers (e.g. 2–3 layers), it is shown that the model could effectively and accurately resolve wave shoaling, non-linearity, dispersion, fission, refraction, and diffraction phenomena.     

Wave Currents Generated by a Moving Load in a Liquid Covered with Ice

We study horizontal wave currents generated in a liquid of finite depth by a load of constant intensity moving over the floating ice cover and analyze the dependences of the space structure of the field of wave velocities on the characteristics of the ice cover and the velocity of motion of the load. It is shown that the velocity of wave currents caused by flexural waves can increase with the velocity of motion of the load, whereas the wave currents caused by the gravity waves decay monotonically. The ice compression increases the velocity of horizontal wave currents.     

Static and dynamic modeling of a SAR imaged ocean scene

A number of models exist that attempt to explain wave imagery obtained with a synthetic aperture radar (SAR). These models are of two types; static models that depend on instantaneous surface features and dynamic models that employ surface velocities. Radar backscatter values (sigma_{0}) were calculated from 1.3- and 9.4-GHz SAR data collected off Marineland, FL. Thesigma_{0}data (averaged over many wave trains) collected at Marineland can best be modeled by the Bragg-Rice-Phillips model which is based on roughness variation and the complex dielectric constant of oceans. This result suggests that capillaries on the surface of oceanic waves are the primary cause for the surface return observed by a SAR. Salinity and temperature of the sea at small and medium incidence angles produce little effect upon sea-surface reflection coefficients atX-band, for either of the linear polarizations. The authors' observation of moving ocean, imaged by the SAR and studied in the SAR optical correlator, support a theory that the ocean surface appears relatively stationary in the absence of currents. The reflecting surface is most likely moving slowly (i.e., capillaries) relative to the phase velocity of the large gravity waves.     

Effect of the Planetary Boundary Layer Horizontal Inhomogeneity in the Two-Dimensional Fluid Motion Model

The generalized two-dimensional vortex equation is derived for an incompressible viscous fluid in a rotating system for a vertically averaged flow taking into account the variability of the boundary layer characteristics. The resulting equation contains parameters and their spatial derivatives determined by the second moments of functions describing the vertical profiles of the flow components. Numerical experiments demonstrate the influence of the boundary-layer horizontal inhomogeneity on the evolution of the vorticity field of a pair of atmospheric vortices.     

Numerical Simulation of the Three-Dimensional Wave-Induced Currents on Unstructured Grid

By coupling the three-dimensional hydrodynamic model with the wave model, numerical simulations of the three-dimensional wave-induced current are carried out in this study. The wave model is based on the numerical solution of the modified wave action equation and eikonal equation, which can describe the wave refraction and diffraction. The hydrodynamic model is driven by the wave-induced radiation stresses and affected by the wave turbulence. The numerical implementation of the module has used the finite-volume schemes on unstructured grid, which provides great flexibility for modeling the waves and currents in the complex actual nearshore, and ensures the conservation of energy propagation. The applicability of the proposed model is evaluated in calculating the cases of wave set-up, longshore currents, undertow on a sloping beach, rip currents and meandering longshore currents on a tri-cuspate beach. The results indicate that it is necessary to introduce the depth-dependent radiation stresses into the numerical simulation of wave-induced currents, and comparisons show that the present model makes better prediction on the wave procedure as well as both horizontal and vertical structures in the wave-induced current field.     

Modelling the influence of currents on wave propagation at the entrance of the Tagus estuary

A study of the wave propagation and of the consequences of the influence of currents on waves in the Tagus estuary is performed in the present work. For this purpose a high-resolution SWAN domain was coupled to a wave prediction system based on the two state of the art phase averaged wave models, WAM for wave generation and SWAN for nearshore wave transformation. The most important factors affecting the incoming waves are the local currents and the wind. These influences were evaluated by performing SWAN simulations in the target area with and respectively without considering the tide level and tide induced currents. The model results were compared with wave measurements, validating in this way the results of the wave prediction system developed herewith.     

磨刀门河口枯季波生流场数值模拟研究

本文建立曲线坐标系下的双曲型缓坡方程波浪模型和考虑波浪辐射应力影响的深度平均2D潮流数学模型,首次研究了磨刀门河口2011年地形条件下的枯季波生流场。受波浪作用影响,落潮阶段,波浪作用方向与流向相反,在波浪顶托效应下拦门沙沙脊及外坡处流速普遍减小,形成两个主要回流区,口门外东西两侧浅滩处流速也减小,东西两汊及横洲深槽流速增大;涨潮阶段,波浪作用方向与流向相同,拦门沙沙脊及外坡处流速增大,沙脊处出现冲越流,口门两侧浅滩处流速增大,横洲深槽流速减小。     

Zonal versus meridional velocity variance in satellite observations and realistic and idealized ocean circulation models

Global, high-quality, satellite-based observation of oceanic currents over the past 13 years has revealed ubiquitous quasi-horizontal eddies in the mesoscale (tens to hundreds of kilometers), confirming the view of a highly turbulent ocean suggested by observational programs in the 1970s. Idealized quasigeostrophic turbulence models suggest mesoscale turbulent flow can vary between isotropic, and highly anisotropic zonal jets. Here we compare the zonal and meridional velocity variance from satellite altimetry. We find that, for an unexplained reason and despite the chaotic nature of turbulence, the surface flow is organized into mesoscale patches where either zonal or meridional velocity variance dominates. The patches persist over 13 years, much longer than the turbulent timescale of a few months. Implications include potentially highly anisotropic redistribution of tracers by the mesoscale flow. Zonally averaged velocity variances reveal a slight preference for meridional over zonal velocity variance. Realistic primitive equation models succeed in reproducing both the patchy structure in local preference for either zonal or meridional velocity variance, and the zonally averaged preference for meridional variance. Idealized models of fully developed, quasigeostrophic turbulence fail in both regards.