Effects of time dependence in unstratified tidal boundary layers: results from large eddy simulations

A three-dimensional Large Eddy Simulation (LES) model is used to simulate oscillating tidal boundary layers and test previous results obtained from one-dimensional boundary layer models and turbulence measurements in tidal channels. The LES model produces low-order turbulence statistics in agreement with the semi-analytic theory and observations. It shows a logarithmic layer in the mean velocity profile and a linear distribution of Reynolds stress with water depth. However, the eddy viscosity profile predicted by the LES model is not parabolic but better matches a parabolic profile modified by wake effect observed in the outer part of depth-limited steady boundary layers. Low-order turbulence statistics can be scaled by the instantaneous friction velocity at the bottom boundary. Although turbulence intensities in three directions fluctuate over a tidal cycle, their normalized values are in good agreement with those determined from laboratory experiments of steady open-channel flows. The LES model confirms that tidal turbulence is in quasi-equilibrium. However, it also demonstrates the importance of flow acceleration/deceleration term in the depth-integrated momentum balance for the mean flow. Phase differences are found between flows at different heights above the bottom boundary.     

Simulation of nearshore wave processes by a depth-integrated non-hydrostatic finite element model

This paper presents CCHE2D-NHWAVE, a depth-integrated non-hydrostatic finite element model for simulating nearshore wave processes. The governing equations are a depth-integrated vertical momentum equation and the shallow water equations including extra non-hydrostatic pressure terms, which enable the model to simulate relatively short wave motions, where both frequency dispersion and nonlinear effects play important roles. A special type of finite element method, which was previously developed for a well-validated depth-integrated free surface flow model CCHE2D, is used to solve the governing equations on a partially staggered grid using a pressure projection method. To resolve discontinuous flows, involving breaking waves and hydraulic jumps, a momentum conservation advection scheme is developed based on the partially staggered grid. In addition, a simple and efficient wetting and drying algorithm is implemented to deal with the moving shoreline. The model is first verified by analytical solutions, and then validated by a series of laboratory experiments. The comparison shows that the developed wave model without the use of any empirical parameters is capable of accurately simulating a wide range of nearshore wave processes, including propagation, breaking, and run-up of nonlinear dispersive waves and transformation and inundation of tsunami waves.     

Non-hydrostatic finite element model for coastal wave processes

This paper presents the application of the depth-integrated non-hydrostatic finite element model, CCHE2D-NHWAVE (Wei and Jia, 2014), for simulating several types of coastal wave processes. Specifically, the model is applied to (1) predict the swash zone hydrodynamics involving wave bore propagation, (2) resolve wave propagation, breaking, and overtopping in fringing reef environments, (3) study the vegetation effect on wave height reduction through both submerged and emergent vegetation zones using the drag force term technique, and (4) simulate tsunami wave breaking in the nearshore zone and inundation in the coastal area. Satisfactory agreement between numerical results and benchmark data shows that the non-hydrostatic model is capable of modeling a wide range of coastal wave processes. Furthermore, thanks to its simple numerical formulation, the non-hydrostatic model also demonstrates a better computation efficiency when comparing with other numerical models.     

Bottom friction and its effects on periodic long wave propagation

A new set of Boussinesq-type equations describing the free surface evolution and the corresponding depth-integrated horizontal velocity is derived with the bottom boundary layer effects included. Inside the boundary layer the eddy viscosity gradient model is employed to characterize Reynolds stresses and the eddy viscosity is further approximated as a linear function of the distance measured from the seafloor. Boundary-layer velocities are coupled with the irrotational velocity in the core region through boundary conditions. The leading order boundary layer effects on wave propagation appear in the depth-integrated continuity equation to account for the velocity deficit inside the boundary layer. This formulation is different from the conventional approach in which a bottom stress term is inserted in the momentum equation. An iterative scheme is developed to solve the new model equations for the free surface elevation, depth-integrated velocity, the bottom stress, the boundary layer thickness and the magnitude of the turbulent eddy viscosity. A numerical example for the evolution of periodic waves propagating in one-dimensional channel is discussed to illustrate the numerical procedure and physics involved. The differences between the conventional approach and the present formulation are discussed in terms of the bottom frictional stress and the free surface profiles.     

Quasi-3D modelling of nearshore currents

Existing concepts of wave-induced nearshore current models, in the cross-shore vertical plane (2DV) and depth-integrated (2DH), are combined to a quasi-3D mathematical model. This combination is tested for reproducing correct results in 2DV and 2DH situations. The importance of the various contributions to the wave-induced secondary circulation in the vertical plane is investigated for realistic parameter ranges, which leads to the conclusion that both the non-breaking and the breaking fraction of a random wave field in the surf zone generate important secondary currents.Additional computations show the relevance of a 3D-approach of nearshore currents, even in seemingly simple situations like a plane sloping beach with obliquely incident waves.     

海浪破碎对海洋上混合层中湍能量收支的影响

海浪破碎产生一向下输入的湍动能通量,在近海表处形成一湍流生成明显增加的次层,加强了海洋上混合层中的湍流垂向混合。为了研究海浪破碎对混合层中湍能量收支的影响,文中分析了海浪破碎对海洋上混合层中湍流生成的影响机制,采用垂向一维湍封闭混合模式,通过改变湍动能方程的上边界条件,引入了海浪破碎产生的湍动能通量,并分别对不同风速下海浪破碎的影响进行了数值研究,分析了混合层中湍能量收支的变化。当考虑海浪破碎影响时,近海表次层中的垂直扩散项和耗散项都有显著的增加,该次层中被耗散的湍动能占整个混合层中耗散的总的湍能量的92.0%,比无海浪破碎影响的结果增加了近1倍;由于平均流场切变减小,混合层中的湍流剪切生成减小了3.5%,形成一种存在于湍动能的耗散和垂直扩散之间的局部平衡关系。在该次层以下,局部平衡关系与壁层定律的结论一致,即湍动能的剪切生成与耗散相平衡。研究结果表明,海浪破碎在海表产生的湍动能通量影响了海洋上混合层中的各项湍能量收支间的局部平衡关系。     

Application of a Boussinesq model for the computation of breaking waves: Part 2: Wave-induced setdown and setup on a submerged coral reef

Based on a set of Boussinesq-type equations with improved linear frequency dispersion characteristics in deeper water, the present paper incorporates the simplified effect of spilling wave breaking into the equations. The analysis is restricted to a single horizontal dimension but the method can be extended to include the second horizontal dimension. Inside the surf zone the vertical variation of the horizontal velocity profile is assumed to be composed of an (initially unknown) organised velocity component below the roller and a surface roller travelling with the wave celerity. This leads to a new set of equations which is capable of simulating the transformation of waves before, during and after wave breaking. The model is calibrated and verified by comparison with several wave flume measurements. The results show that the model produces sound physical results.     

Depth-integrated and depth-resolved models of Kara Sea primary production

Primary production (PP) models of the Kara Sea are developed based on data collected on fall expeditions (September–October 1993, 2007, and 2011) and their precision assessment utilizes the dataset collected in September 2013. The algorithms for different model types (depth-integrated and depth-resolved) are compared. The depth-resolved model performs slightly better than the depth-integrated one (the rootmean- square-difference (RMSD) are 0.29 and 0.31, respectively). These algorithms utilize the daily assimilation number (DAN) and photosynthetic efficiency (ψ) as the model coefficients, and surface chlorophyll a (chl a) and photosynthetically active radiation (PAR) as input variables. These algorithms perform better than the models that use chl a alone. Our results suggest that an increase in the performance of the Kara Sea PP models depends on the input of the photophysiological characteristics of phytoplankton (DAN and ψ) and PAR. To a lesser extent, this concerns the advantages of the depth-resolved model over the depth-integrated one. The constructed region-specific Kara Sea PP models combined with satellite-derived chl a and PAR can be used to estimate annual values and long-term variation of PP in hydrologically and hydrochemically similar waters of the Arctic Ocean.     

A Boussinesq-type wave driver for a morphodynamical model to predict short-term morphology

The prediction of near-shore morphology on the time scale of a storm event and the length scale of a few surf zone widths is an active area of research. Intense wave breaking drives offshore-directed currents (undertow) carrying sediment seawards, resulting in offshore bar migration. In contrast, higher order nonlinear properties, such as wave asymmetry (velocity skewness) and velocity asymmetry, are drivers for shoreward transport. These wave processes are included in phase-resolving models such as Boussinesq-type wave models (e.g., TRITON). Short-wave averaging in the wave model yields wave-induced forces (e.g., radiation stress gradients) and a wave asymmetry term. The wave-induced forces are used in a hydrostatic model (e.g., Delft3D flow module) to drive the current and undertow, resulting in a 3D velocity profile. The wave model and hydrostatic model are coupled online with a morphodynamic model (e.g., Delft3D morphology module). The latter computes, based on the 3D flow profile and the wave asymmetry term, the sediment transport and performs the bathymetry updates. The updates are transferred directly back to the hydrodynamic models. The coupling of the wave model TRITON and the Delft3D modules is validated by comparing against extensive laboratory data sets (LIP and Boers) and a field case (Duck94), and show a good performance for the hydrodynamics and a reasonable/fair performance for the bar movements.     

Non-linear propagation of unidirectional wave fields over varying topography

The present study aims at investigating the non-linear triad interaction process affecting shoaling surface gravity wave fields. The triad interaction phenomenon being enhanced towards the shore, the domain of study is extended up to the surf zone. Three 1D non-linear wave models (one phase-resolving and two phase-averaged spectral models) have been implemented and compared to laboratory experiments performed in a wave flume. This set of models includes two existing models and a new one which has been developed in the frame of this work. The models include a breaking dissipation term based on the parametrical model of Battjes and Janssen [Battjes, J.A., Janssen, P.A.E.M., 1978. Energy loss and set-up due to breaking of random waves. Proc. 16th Int. Conf. Coastal Eng. (ASCE), Vol. 1, pp. 569–587.]. The investigations concern the evolution of variance spectra, spectral significant wave height and mean period over a barred bathymetric profile. In addition, the performances of the different models are analysed by computing the spectral source term for triad interactions. We found that all models are able to reproduce the main features of non-linear mechanisms affecting a wave field in the near-shore zone. The phase-resolving model gives the most accurate results for non-breaking situations. It correctly reproduces the non-linear coupling effect in decreasing water depths due to wave–wave interactions, as well as the harmonic release after a bar. However, the model is computationally time-consuming. The CPU time is considerably reduced using phase-averaged models. They give satisfactorily results on harmonic generation. However, they do not reproduce the release of harmonics as water depth increases. In breaking conditions, the variance spectra undergo significant changes under the combined effects of non-linear energy transfers and dissipation. The depth-induced wave breaking model included in the equations provides a good estimate of the energy decay in the surf zone.     

Numerical study of propagation of ship waves on a sloping coast

The aim of this paper is to investigate the propagation of ship waves on a sloping coast on the basis of results simulated by a 2D model. The governing equations used for the present model are the improved Boussinesq-type equations. The wave breaking process is parameterized by adding a dissipation term to the depth-integrated momentum equation. To give the boundary conditions at the ship location, the slender-ship approximation is used. It was verified that, although ship waves are essentially transient, the Snell's law can be applied to predict crest orientation of the wake system on a sloping coast. Based on simulated results, an applicable empirical formula to predict the maximum wave height on the slope is introduced. The maximum wave height estimated by the proposed method agrees well with numerical simulation results.     

Numerical simulation of breaking waves by a multi-scale turbulence model

In this paper, a two-dimensional multi-scale turbulence model is proposed to study breaking waves. The purpose of developing this model is to produce a relatively accurate model with moderate computer requirements. The free surface is tracked by the VOF technique, the log-law profile for the mean velocity is applied at the bottom. Comparing with the Reynolds-Averaged Navier-Stokes models (RANS), the present model shows improving agreement with experimental measurements in terms of surface elevations, particle velocities, wave height distributions and undertow profiles. The subgrid scale (SGS) turbulent transport mechanism is also discussed in the paper. It is found that turbulent production and dissipation are of the same order, but turbulent production is primarily located at the wavefront and above the wave trough, whereas turbulent dissipation is primarily located at the back face of a wave, indicating that in these regions, the assumption of equilibrium is not correct. Below the trough level, the local equilibrium assumption is reasonable. Turbulent convection and diffusion are of the same order at the trough level. Above the trough level, turbulent convection dominates. Under the spilling breaking wave, turbulent kinetic energy is continue to dissipate in the bore region, whereas under the plunging breaking wave, the turbulent kinetic energy is dissipated very rapidly within one wave period.     

The effect of jet formation on the velocity distribution in oscillatory flow over flat beds of sand or gravel

It is suggested that the jets or “bursts” of fluid hurled up from flat beds of sand or gravel in oscillatory flow may produce significant modification of the velocity profile. A two-layer model is put forward to investigate this possibility. The calculated velocity profiles are found to reproduce the main features of observed profiles. The experimental results of Kalkanis and Sleath are re-analysed in the light of this model and further measurements of the velocity distribution, designed to extend the experimental range, are reported. The new measurements were made in an oscillating tray rig with a laser doppler anemometer. Finally, the results obtained for the velocity distribution are used to calculate friction and dissipation factors. The calculations show satisfactory agreement with the measurements of Kamphuis.     

An evaluation of wave propagation simulations over a barred beach with a Boussinesq-type model

The FUNWAVE model is used for simulating simulation of monochromatic and irregular wave propagation in a channel with a bar-trough profile. FUNWAVE is based upon the extended Boussinesq equations. The study aims to analyze the model's performance when simulating shoaling, wave breaking and nonlinear interactions that are present in nearshore wave propagation. For that, high-order time domain statistics (root mean-square wave height, skewness, asymmetry and the kurtosis) of the model simulations and of the observations were compared along the whole channel. Also, a frequency domain analysis including standard spectral analysis and the bispectrum was carried out in selected points of the flume. The evaluation included the role of the wave breaking internal model parameters. The main conclusion is that, in general, the one-dimensional version of FUNWAVE simulates quite well the nonlinear transformation of a wave over a bottom with a bar-tough profile, for both regular and irregular wave conditions. The model reproduces the transformation of the wave shape, specially the increasing sharper wave crests and flatter troughs and also the lack of vertical symmetry with crests pitching forward, as it propagates along the domain. However, some differences persist after wave breaking, mainly due to the nature of the wave-breaking module. In this module, the energy dissipation is induced by the increase of viscosity, a rather simple mechanism, without the modification of the wave shape. Also, the energy dissipation develops in a smooth way which is appropriated for spilling breaking waves, but not for plunging breaking waves where the dissipation starts more abruptly.     

海浪破碎对北太平洋海表面温度模拟的影响

基于海浪模式WAVEWATCH Ⅲ模拟北太平洋海浪要素,结合NDBC浮标资料进行验证,发现模拟出的有效波高与浮标测量值具有很好的一致性。基于改进型白冠覆盖率耗散模型,利用海浪模式模拟出的有效波高、有效波周期和摩擦速度等海浪要素计算出单位面积水柱内因海浪破碎产生的湍动能通量。通过改变环流模式sbPOM湍动能方程的上边界条件,引入海浪破碎产生的湍动能通量,并探究海浪破碎对北太平洋海表面温度模拟的影响。研究表明,由于海浪破碎的引入,环流模式sbPOM对北太平洋海表面温度模拟的准确程度得到提升,这为大气模式提供一个准确的北太平洋下边界条件具有重要意义。     

Analysis of some key parametrizations in a beach profile morphodynamical model

A numerical process-based model to forecast beach profile morphodynamics has been developed. In the present paper, an analysis of various modelling approaches and key parametrizations involved in the estimation of the wave-driven current and the suspended sediment concentration is carried out.Several resolution techniques for the 1DV horizontal (i.e., in the x-direction perpendicular to coastline) momentum equation governing the Mean Horizontal Velocity (MHV) are analysed. In the first kind of techniques, the mean horizontal velocity is computed from the momentum equation, whereas the Mean Water Level (MWL) is computed using a parametrization of the depth-averaged momentum equation. Two boundary or integral conditions are thus needed. In the second kind, both mean horizontal velocity and mean water level gradient in the x-direction are the unknowns of the momentum equation, thus, three boundary or integral conditions are needed. Various additional conditions are discussed. We show that using a technique of the first kind is equivalent to imposing the difference between the surface and the bottom shear stresses in the 1D vertical equation. Both techniques lead to results that are in good agreement with the Delta Flume experimental data, provided the Stokes drift flow discharge is imposed as an additional condition. The influence of the breaking roller model and of the turbulent viscosity parametrization are also analysed.Suspended sediment transport by the mean current and wave-induced bedload transport are taken into account in the sediment flux. Three turbulent diffusivity parametrizations are compared for suspended sediment concentration estimations. A linear profile for the turbulent diffusivity taking into account the wave bottom shear stress and the surface wave breaking turbulence production is shown to give the best results. Using experimental data, we put forward the poor estimation of the bottom sediment concentration given by the three implemented parametrizations. We thus propose a new parametrization relying on a Shields parameter based on the breaking roller induced surface shear stress. Using this new parametrization, the bottom profile used in the tests keeps its two bars which disappear otherwise. However, the morphodynamical model still overestimates the bars offshore motion, a bias already observed in other models.     

Equilibrium beach profiles under breaking and non-breaking waves

Simple theoretical models to determine the equilibrium profile shape under breaking and non-breaking waves are presented. For the case of breaking waves, it is assumed that the seaward transport in the undertow is locally balanced by a net vertical sedimentation, so that no bottom changes occur at equilibrium. The parameterization of the water and sediment flux in the surf zone yields a power curve for the equilibrium profile with a power of 2/3, which is in agreement with previous field investigations on surf zone profile shapes. Three different models were developed to derive the profile shape under non-breaking waves, namely (1) a variational formulation where the wave energy dissipation in the bottom boundary layer is minimized over the part of the profile affected by non-breaking waves, (2) an integration of a small-scale sediment transport formula over a wave period where the slope conditions that yield zero net transport determine equilibrium, and (3) a conceptual formulation of mechanisms for onshore and offshore sediment transport where a balance between the mechanisms defines equilibrium conditions. All three models produced equilibrium profile shapes of power-type with the power typically in the range 0.15–0.30. Comparison with field data supported the results obtained indicating different powers for the equilibrium profile shape under breaking and non-breaking waves.     

A Hybrid Finite-Volume and Finite Difference Scheme for Depth-Integrated Non-Hydrostatic Model

A depth-integrated, non-hydrostatic model with hybrid finite difference and finite volume numerical algorithm is proposed in this paper. By utilizing a fraction step method, the governing equations are decomposed into hydrostatic and non-hydrostatic parts. The first part is solved by using the finite volume conservative discretization method, whilst the latter is considered by solving discretized Poisson-type equations with the finite difference method. The second-order accuracy, both in time and space, of the finite volume scheme is achieved by using an explicit predictor-correction step and linear construction of variable state in cells. The fluxes across the cell faces are computed in a Godunov-based manner by using MUSTA scheme. Slope and flux limiting technique is used to equip the algorithm with total variation dimensioning property for shock capturing purpose. Wave breaking is treated as a shock by switching off the non-hydrostatic pressure in the steep wave front locally. The model deals with moving wet/dry front in a simple way. Numerical experiments are conducted to verify the proposed model.     

A morphological stability analysis for a long straight barred coast

A morphological stability analysis is carried out for a long straight coast with a longshore bar. The situation with oblique wave incidence and a wave-driven longshore current is considered. The flow and sediment transport are described by a numerical modelling system. The models comprise: (i) a wave model with depth refraction, shoaling and wave breaking, (ii) a depth integrated model for wave driven currents and (iii) a sediment transport model for the bed load transport and the suspended load transport in combined waves and current. The direction of the sediment transport is taken to be parallel to the depth integrated mean current velocity, neglecting the effects of a bed slope and secondary currents. An instability is found to develop around the bar crest. The instability is periodic in the alongshore direction, and tends to form rip channels and to steepen the offshore face of the bar between the rip channels. The alongshore wave length of the most unstable perturbation is determined for different combinations of the wave conditions and the geometry of the profile.