Longshore Velocity Research Articles

The shape of fringing tidal flats in engineered estuaries

For the management of estuaries and the preservation of tidal flats it is crucial to understand the tidal flat shape and development. Previous work focused predominantly on the quasi-equilibrium shape of tidal flats along open coasts with a dominant cross-shore flow and wave exposure. This paper evaluates the shape of fringing tidal flats in engineered estuaries, where longshore velocities generally dominate. Using a long-term (20 years) topographic data set of an anthropogenically modified estuary in the Netherlands (the Western Scheldt estuary), we relate key profile shape parameters and changes over time to natural and anthropogenic processes. In an engineered estuary, the tidal flat shape depends on the estuary geometry, hydrodynamic forcings and human interventions. In contrast to open coast tidal flats, the presence of the channel and dominant longshore flow determines the available cross-shore length (accommodation space) of the tidal flat and the shape of the tidal flat. This accommodation space defines the maximum tidal flat height and opportunity for marsh development. We propose the use of the Index of Development, indicating to what extend tidal flats have space to develop. This index is not only influenced by longshore and cross-shore flow, but also (or even more) by hydraulic structures, dike realignments and channel migration. Especially the latter two strongly influence the accommodation space and thereby the maximum tidal flat height and the opportunity for marsh development. For large stretches of the Western Scheldt, the accommodation space is too small, and the majority of the tidal flats do not vertically extent to mean high water. The success of tidal flat and marsh restoration projects depends on the accommodation space.

Investigation of the Spiral Wave Generation and Propagation on a Numerical Circular Wave Tank Model

A two-phase incompressible flow model in three-dimensional cylindrical coordinates is applied for oblique wave generation in a numerical circular wave tank. The governing equations are discretized by a finite volume method, and a mass source function is added to reproduce oblique waves through a spiral wave generator positioned at the center of the tank. The volume of fluid method is implemented to track the free surfaces between the air and fluid, and the zonal embedded grid system is adopted to obtain a grid-independent solution in the cylindrical coordinates. A permeable, circumferentially sloping topography (1:7), similar to a natural beach profile, is set up for investigating wave propagation and other characteristics. The simulation and physical experimental results are compared, which show a good agreement in randomly selected water surface elevation profiles and wave heights under the same wave and sloping conditions. The spiral waves are reproduced and propagated in a phenomenon similar to that observed in the physical experiment. The results also show that the wave-breaking positions differ in different wave conditions and suggest a relationship with cross-shore and longshore velocity distribution in terms of incident wave heights and wave-breaking positions in different wave periods on the same sloping topography. Furthermore, this model can be used to investigate the mechanism of longshore current generation and the influences of beach slope on the generated wave propagation in a sloping topography.

Surf-zone dynamics derived from basin-scale experiments

Surf-zone hydrodynamics forced by oblique wave shoaling and breaking on beach slopes were investigated. The results showed that in wave-basin experiments with incident angles in the range of 15°–30°, wave breaking was initiated at a breaker coefficient of around 0.67, which was significantly less than that predicted from empirical relations based on normally incident waves for a given beach slope and deep-water wave steepness. The measurements also showed that subsequent saturated breaking occurred at a breaker coefficient of around 0.47 that was independent of beach slope in the range of 1:10 to 1:100. This result is likely applicable to both oblique and normally incident waves. It is shown that the measured wave heights and longshore velocity profiles in wave-basin studies can be reasonably well predicted by theory with proper adjustments to the process parameters. Best-match formulations were identified for quantifying bottom friction, eddy viscosity, and energy loss due to surface rollers.

Coastal Trapped Waves: Normal Modes, Evolution Equations, and Topographic Generation

Abstract Coastal trapped waves (CTWs) transport energy along coastlines and drive coastal currents and upwelling. CTW modes are nonorthogonal when frequency is treated as the eigenvalue, preventing the separation of modal energy fluxes and quantification of longshore topographic scattering. Here, CTW modes are shown to be orthogonal with respect to energy flux (but not energy) when the longshore wavenumber is the eigenvalue. The modal evolution equation is a simple harmonic oscillator forced by longshore bathymetric variability, where downstream distance is treated like time. The energy equation includes an expression for modal topographic scattering. The eigenvalue problem is carefully discretized to produce numerically orthogonal modes, allowing CTW amplitudes, energy fluxes, and generation to be precisely quantified in numerical simulations. First, a spatially uniform K1 longshore velocity is applied to a continental slope with a Gaussian bump in the coastline. Mode-1 CTW generation increases quadratically with the amplitude of the bump and is maximum when the bump’s length of coastline matches the natural wavelength of the CTW mode, as predicted by theory. Next, a realistic K1 barotropic tide is applied to the Oregon coast. The forcing generates mode-1 and mode-2 CTWs with energy fluxes of 6 and 2 MW, respectively, which are much smaller than the 80 MW of M2 internal-tide generation in this region. CTWs also produce 1-cm sea surface displacements along the coast, potentially complicating the interpretation of future satellite altimetry. Prospects and challenges for quantifying the global geography of CTWs are discussed.

A convolutional neural network based model to predict nearshore waves and hydrodynamics

This study proposes a convolutional neural network (CNN) based model to predict waves and hydrodynamics in the nearshore zone. Specifically, we address the so-called next-frame prediction of wave and hydrodynamic conditions using the previous-step conditions. A laboratory experiment on unsteady rip currents is simulated using the non-hydrostatic model SWASH. The simulated wave and hydrodynamic results are divided into two groups. The first group of data is used to train the CNN model, and the second group of data is used to evaluate the trained model by comparing the simulated water surface elevation, cross-shore velocity, and longshore velocity by SWASH with the corresponding prediction by the trained CNN model. The result shows that the CNN model accurately predicts the wave propagation over the offshore slope, wave breaking over the sandbar, and bore propagation over the shore including detailed wave crest bending and separation. Time-averaged water level comparison further confirms that the CNN model captures the wave setup induced by wave breaking. The CNN model also correctly predicts the cross-shore velocity and longshore velocity driven by wave breaking over sandbars and wave–current interaction. Finally, the low-frequency nearshore circulation pattern predicted by the CNN model is also evaluated by time-averaging the velocity field. The comparison shows that the CNN model correctly predicts the distribution of circulation cells and the number of cells, as simulated by the SWASH model. This work serves as a paradigm to integrate traditional coastal modeling with machine learning to study coastal processes.

Optimization of Bathymetry Estimates for Nearshore Hydrodynamic Models Using Bayesian Methods.

A Bayesian inverse framework is developed to optimize the skill of a predictive numerical model via interpolation of bathymetric measurements to provide the most probable bathymetric surface. The nu cmerical model is a coupled wave flow model and predicts wave and hydrodynamic information (e.g., significant wave height and longshore velocity). The Bayesian method, coupled with Markov chain Monte Carlo (MCMC) optimization, is used to find the bathymetric field, which serves to minimize the residual errors between measured data and the corresponding numerical model results. By using a Bayesian approach, the range of probable model parameters is inferred from the observed data. Monte Carlo simulation is also applied to this numerical model to perform the uncertainty analysis of the model output fields (wave height and flow velocity). This analysis is performed by taking random samples from the probability distribution function (PDF) of inputs and running the model as required until the desired precision (±0.05 m for significant wave height) in output fields is achieved. The case study used in this analysis is the DUCK94 experiment, which was conducted at the US Army Field Research Facility at Duck, North Carolina, in the fall of 1994. The unknown model parameters for the hydrodynamic model involve those controlling bathymetric resolution. Furthermore, the ability of the statistical model to estimate the observed data is tested by running the forward model for two sets of input parameters: the estimated input parameters updated by the previously mentioned statistical model and the prior (noninformative) parameters. Using the model parameters estimated from the Bayesian analysis leads to improved comparisons to data. Using the presented method, the relative errors between the model outputs and the observed data for significant wave height at nearshore gauges is reduced by 30%.

A process based shape equation for a static equilibrium beach planform

Coastal erosion is a major problem for shorelines everywhere in the world. In order to mitigate beach stability problems it is necessary to determine the equilibrium shape of the beach planform. Current equilibrium planform shape equations are only valid under certain simplifications; for instance, they are unable to predict the effect of nearshore islands and/or rocky bottoms, as well as the effect of several diffraction points. The aim of this paper is to present a new process-based shape equation that is able to overcome those limitations and estimate the static equilibrium shoreline of complex bathymetry beaches. The equation is based on the hypothesis that a pocket beach gets its static equilibrium planform when the mean surf-zone longshore velocity averaged over a period of time is null (V¯=0) in every point along the beach. Based on this hypothesis, the direction of the shoreline that nulls the mean surf zone longshore velocity along the beach is evaluated. Therefore, the shape equation is based on the longshore current velocity formula. The new equilibrium shape equation is applied to two Spanish beaches, namely Milagro beach and Cala Millor beach. Milagro beach has a very smooth bathymetry and the parabolic shape equation developed by Hsu and Evans (1989) is able to predict the equilibrium shoreline. Cala Millor is more complex: it has some rock outcrops along the beach and the parabolic shape equation does not work. The equilibrium shape equation presented in this paper simulates the equilibrium shoreline of both beaches successfully with a high R2 (around 0.96) between the modelled shoreline and the real shoreline in both case studies.

Modeling of the meso- and submesoscale dynamic processes in the Black sea coastal zones

The article presents an analysis of simulation results for the Black Sea circulation. The calculations were carried out with a horizontal resolution of 1.6~km and taking into account the real atmospheric forcing for 2006 and 2011. Eddies with characteristic mesoscale and submesoscale spatio-temporal parameters were reconstructed in the velocity fields. It is shown that the simulated hydrophysical fields correspond to observed data. The most intensive generation of eddies with the scale of less than 10~km and orbital velocities of 20 -- 30~cm/s was observed in the neighborhood of the Crimean Peninsula, in the north-eastern and south-eastern parts of the sea. Vortex structures in the shelf zones were formed in all seasons with weak winds of directions. Their size depended on the depth and length of the shelf, as well as on the dimensions of coastal capes. The analysis of spatial and temporal variability of salinity fields and the longshore velocity component in the vertical cross-section passing through eddy centers showed that two main mechanisms of coastal eddies formation dominated in the Black Sea. The first mechanism is baroclinic instability, which arises with increasing horizontal density gradient in the periphery of large eddies or currents due to water rising/lowering. The second mechanism is the streamlining of shore inhomogeneities by the coastal current with a velocity higher than 40~cm/s.

Hydrodynamic Modeling of Sedimentation in the Navigation Channel of Damietta Harbor in Egypt

In this study, a simplified multicomponent technique for modeling the 2D sediment transport in the vicinity of navigation channels to harbors is formulated. The main simplification assumes that longshore current is the only factor transporting sediments to the offshore navigation channel. The technique requires the application of three numerical models: NMLONG model (ID depth averaged finite difference wave model), RMA2 (2D depth averaged finite element hydrodynamic model) and SED2D (2D depth averaged finite element sediment transport model). RMA2 is forced at the inflow boundary using longshore velocity profiles generated by the incoming waves via the application of NMLONG model. Output of NMLONG provides also the boundary condition of SED2D which takes a velocity field input from RMA2. A variable manning coefficient is used inside RMA2 domain to account for wave roughness. Data collected in the period from 1989 to 1997 in the vicinity of the navigation channel of Damietta Harbor (Egypt) is used to calibrate the multicomponent technique. The results show that the technique explains well the observed spatial variation of bed change in the navigation channel.The calibrated technique is then applied to study different solutions to reduce sedimentation in Damietta Harbor navigation channel. The results showed that the optimum solution found can significantly reduce the amount of sedimentation. For sedimentation caused by west-to-east longshore current, the reduction amount range is 62–74% (i.e. only 26–38% of the current sedimentation shall occur). For sedimentation caused by east-to-west longshore current, the reduction percentage reaches 60–70%.

Implementation of the vortex force formalism in the coupled ocean-atmosphere-wave-sediment transport (COAWST) modeling system for inner shelf and surf zone applications

The coupled ocean-atmosphere-wave-sediment transport modeling system (COAWST) enables simulations that integrate oceanic, atmospheric, wave and morphological processes in the coastal ocean. Within the modeling system, the three-dimensional ocean circulation module (ROMS) is coupled with the wave generation and propagation model (SWAN) to allow full integration of the effect of waves on circulation and vice versa. The existing wave-current coupling component utilizes a depth dependent radiation stress approach. In here we present a new approach that uses the vortex force formalism. The formulation adopted and the various parameterizations used in the model as well as their numerical implementation are presented in detail. The performance of the new system is examined through the presentation of four test cases. These include obliquely incident waves on a synthetic planar beach and a natural barred beach (DUCK’ 94); normal incident waves on a nearshore barred morphology with rip channels; and wave-induced mean flows outside the surf zone at the Martha’s Vineyard Coastal Observatory (MVCO).Model results from the planar beach case show good agreement with depth-averaged analytical solutions and with theoretical flow structures. Simulation results for the DUCK’ 94 experiment agree closely with measured profiles of cross-shore and longshore velocity data from Garcez Faria et al. (1998, 2000). Diagnostic simulations showed that the nonlinear processes of wave roller generation and wave-induced mixing are important for the accurate simulation of surf zone flows. It is further recommended that a more realistic approach for determining the contribution of wave rollers and breaking induced turbulent mixing can be formulated using non-dimensional parameters which are functions of local wave parameters and the beach slope. Dominant terms in the cross-shore momentum balance are found to be the quasi-static pressure gradient and breaking acceleration. In the alongshore direction, bottom stress, breaking acceleration, horizontal advection and horizontal vortex forces dominate the momentum balance. The simulation results for the bar/rip channel morphology case clearly show the ability of the modeling system to reproduce horizontal and vertical circulation patterns similar to those found in laboratory studies and to numerical simulations using the radiation stress representation. The vortex force term is found to be more important at locations where strong flow vorticity interacts with the wave-induced Stokes flow field. Outside the surf zone, the three-dimensional model simulations of wave-induced flows for non-breaking waves closely agree with flow observations from MVCO, with the vertical structure of the simulated flow varying as a function of the vertical viscosity as demonstrated by Lentz et al. (2008).

The morphological response of a nearshore double sandbar system to constant wave forcing

Results of 2DH morphodynamic computations are presented to quantify the temporal evolution of the crescentic patterns emerging in a double nearshore bar system in response to constant wave boundary forcing. Sixteen different conditions varying both offshore wave height and angle of wave incidence were applied. The mean length scales of the emerging irregular crescentic patterns are linearly proportional to the local longshore velocity over the inner and outer bars. For similar longshore velocities, the length scales of the outer bar are larger than of the inner bar. This is explained by accounting for the difference in water depth above the bar crest. The variable morphological response times can be explained by including additional bathymetrical parameters. The active volume of the bar, defined by the breaker index, plays an important role in this response time. With larger active volumes the bar responds more rapidly to identical boundary conditions. Also, bars with a smaller total volume respond more quickly. This faster response is due to the steeper active volume of the bars. Different initial perturbations resulted in different locations of the emerging features, showing that their location is sensitive to the initial bathymetry. However, the range in length scales and response times due to the different perturbations was significantly smaller than those obtained for the different hydrodynamic conditions. Based on the present findings we hypothesize that morphological length scales in the field are rarely in equilibrium with the concurrent offshore wave height and angle of incidence owing to the slow response of the sandbars under constant conditions relative to the stochastic nature of natural wave forcing.

Rip Currents with Varying Gap Widths

The behavior of rip currents with changing bar and gap lengths was tested in a series of laboratory experiments using video-tracked floating drifters. Qualitatively, currents showed strong differences in overall character with changing topography. As narrow gaps became wider, the circulation tended to first travel parallel to the shoreline instead of immediately offshore, although strong offshore-directed flows were found for all topographies. Quantitatively, the width of the rip neck was found to increase slowly when compared to the increasing width of the rip channel. Maximum offshore and longshore velocities showed little dependence on bar or gap lengths, and were found to scale well with the predicted rate of generation of circulation. Finally, volumetric offshore flow rates through the rip channel showed no proportionality with the bar length as predicted by the mass transport hypothesis of rip current strength.

Quantification of changes in current intensities induced by wave overtopping around low-crested structures

The phenomenon of overtopping is traditionally studied for well-emerged harbour structures and often focuses on safety and stability. In this paper laboratory tests are presented and analysed to sharpen the hypothesis that overtopping is capable of changing the horizontal circulation pattern around low-crested structures. A unique data set from laboratory experiments was acquired in the wave basin at Delft University of Technology. The experiments were performed using an emerged impermeable low-crested structure (three freeboards and three different wave conditions for each freeboard) and yielded nine different combinations of set-up and overtopping driving forces. Using this information it was possible to quantify the changes in cross-shore and longshore velocity induced by the overtopping and the set-up changes under the different freeboard and wave conditions described. It is found that overtopping enhances the outgoing flows (longshore velocities parallel to the structure) away from the lee side of the structure and dampens the water level gradient driven flow towards the structure.

Longshore Current and Sediment Transport on Beaches

A numerical model based on the time-averaged continuity, cross-shore momentum, longshore momentum, and energy equations is developed to predict the cross-shore variations of the mean and standard deviation of the free surface elevation and depth-averaged cross-shore and longshore velocities under obliquely incident irregular breaking waves. The suspended sediment volume per unit horizontal area is estimated using the computed energy dissipation rates due to wave breaking and bottom friction. The longshore suspended sediment transport rate is estimated as the product of the longshore current and suspended sediment volume. The developed model is compared with limited field and laboratory data. The calibrated model is in fair agreement with the data. The longshore suspended sediment transport rate is shown to be approximately proportional to the square of the longshore current. The developed model appears promising but will need to be evaluated using extensive data sets.

Integral properties of the swash zone and averaging. Part 3. Longshore shoreline boundary conditions for wave-averaged nearshore circulation models

The aim of the present work, final of a three-part series, is to analyse in detail flow motions within the swash zone and define suitable shoreline boundary conditions for the longshore flow for wave-averaged circulation models. The analyses of Parts 1 and 2 are extended to cover horizontally two-dimensional flows. An analytical solution for the longshore motion representing the drift velocity of the whole swash zone water mass is found. This is seen to be approximated well by the ratio between the time integral of the longshore momentum flux crossing the swash lower boundary and the swash zone net water volume. Further, a complete set of shoreline boundary conditions, taking into account wave–wave interactions, is obtained on the basis of fully numerical solutions of the nonlinear shallow-water equations. The main focus of the work is to clarify the structure of the shoreline boundary conditions for the longshore flow, but attention has also been paid to their derivation and assessment from the numerical solutions. The latter have been obtained on the basis of a fairly broad range of input wave conditions which, though biased towards those typical of reflective beaches, are believed to cover conditions also typical of moderate dissipative beaches. Two main terms are found to contribute to the longshore drift velocity: (i) a term, proportional to the shallow-water velocity, accounting for short-wave interactions, frictional swash zone forces and continuous forcing due to non-breaking wave nonlinearities and (ii) a drift-type term representing the momentum transfer due to wave breaking.

Fluctuating circulation forced by unsteady multidirectional breaking waves

We consider nearshore rotational currents directly forced by unsteady multidirectional wave breaking. Scaling relationships, simplified analytical solutions, and asymptotic limits are developed for the maximum forced cross-shore and longshore velocities. In all cases, forced longshore velocities are considerably larger than cross-shore velocities. On longshore-uniform beaches, strong fluctuating velocities are found for (i) large waves; (ii) strong directional spreading; and (iii) short peak wave periods. When topographic inhomogeneities control longshore scales of wave breaking, overall scaling changes and the largest fluctuating velocities are found for (a) large waves; (b) long wave periods; and (c) topographic features that vary quickly in the longshore direction. The ratio of fluctuating rotational velocities to mean longshore current does not depend on the wave height or period, but instead on the bottom friction, slope, deep water wave angle, and details of the wave spectrum.

Longshore current forcing by irregular waves

The terms in the wave‐averaged depth‐integrated alongshore momentum balance are populated with detailed measurements from a laboratory basin experiment. Obliquely incident irregular waves were generated in a large wave basin equipped with instruments to collect free surface elevation and velocity measurements over a sand beach. Pumps matched the wave‐generated currents, and experimental evidence indicates that the equations can justifiably be simplified by assuming alongshore uniformity. While the use of data measured below the trough level is straightforward, the velocity field in the region near the surface is extrapolated from the measured positions and constrained by conservation of cross‐shore mass flux. The role of the dispersive mixing mechanism due to depth‐dependent currents is shown to be small but acts to shift the peak of the longshore velocity shoreward slightly. Radiation stresses are the primary forcing mechanism, and computed stresses based on measured wave‐induced hydrodynamics are considerably smaller than the widely used linear representation, indicating that the small‐amplitude assumption of linear theory is not appropriate in the surf zone. The time‐averaged bottom shear stress remains as the primary balance to the gradient in alongshore momentum, and the inferred shear stress compares favorably with the values computed using the quadratic drag law and a spatially constant friction coefficient.

Test of edge wave forcing during formation of rhythmic beach morphology

This paper reports on concurrent measurements of evolving rhythmic beach morphology and the associated water motion on a macrotidal, coarse sandy beach. Beach cusps with a spacing of 19 m developed over a 3‐day period of low‐energy swell after which they were destroyed by energetic wind waves. The beach cusps were clearly accretionary features and developed mainly during the rising tide. Detailed hydrodynamic measurements were conducted at three locations in the swash and inner surf zone in the vicinity of a developing cusp horn and revealed that energetic standing, mode‐zero, subharmonic edge waves were not present during the initiation of the beach cusps. Specifically, analysis of the hydrodynamic data indicated that: (1) the amount of subharmonic energy was limited (<10% of the total energy); (2) none of the computed energy spectra exhibited a significant peak at the subharmonic frequency; (3) the cross‐shore velocity amplitude at the subharmonic frequency was 2–3 times larger than the longshore velocity amplitude; (4) the longshore current motion at the subharmonic frequency was barely coherent between adjacent locations in the surf zone; and (5) the time series of pressure and cross‐shore current were in phase at either side of the developing cusp horn. It is concluded that, at least during this event of beach cusp formation, energetic standing edge waves are not a prerequisite to initiate the development of rhythmic beach morphology.

不規則波による沿岸流速に及ぼす砕波モデル選択の影響

Various factors affecting the velocities of longshore currents induced by obliquely incident random waves are examined through numerical calculation and literature review. Several models of the random breaking process are used for mutual comparison. These models produce some differences in the estimate of wave height decay in the surf zone, and the differences cause large changes in the longshore velocities. Efficiency of horizontal mixing also depends on the wave breaking model employed. The eddy viscosity formulation by Larson and Kraus (1991) is found better suited to random waves than those by Longuet-Higgins (1970) and Battjes (1975). Introduction of surface rollers in numerical models is necessary in shifting the longshore current profile toward the shoreline and brings forth appreciable strength of longshore currents at the trough of a barred beach. Compared to significant consequence of the selection of wave breaking models, eddy viscosity, and surface roller strength, a questionof linear or nonlinear formulation of shear stress along the seabed seems trivial.

Cross-shore distribution of longshore sediment transport rates on a barred non-tidal beach

This study was conducted to relate the cross-shore distribution of longshore sediment transport and grain size characteristics to cross-shore and longshore current velocities on a sandy low-energy beach in a non-tidal embayment of the Baltic Sea. Simultaneous measurements of current velocities and amount of sand caught in streamer traps were made on 31 sampling runs on 6 d in April 1999 at three fixed sites including the swash zone on the upper foreshore, the lower foreshore, and the crest of the most landward of four bars. Spilling waves broke frequently on the bar but rarely on the lower foreshore, even during onshore wind speeds up to 11.0 m s−1. Waves always broke as plunging waves at the step at the base of the upper foreshore and were converted directly into swash. The greatest longshore current velocities in the swash occurred when wind speeds and water levels were greatest, but wind direction was nearly directly onshore. Longshore velocities were greater in the swash zone than at other sites except when relatively strong winds blew nearly parallel to the shoreline, causing a pronounced wind-induced current at the other two sites. Calculated longshore shear stress and rate of sediment trapped were highly correlated on the bar (r=0.90), less highly correlated in the swash zone, and least highly correlated (r=0.66) on the lower foreshore. Mean trapping rates in the swash were 14.6 times greater than on the lower foreshore and 7.2 times greater than on the bar. Greater trapping rates in the swash are attributed to the greater turbulence mobilizing sediments in the uprush and backwash. Little of the finer-grained sediment on the offshore sites was reworked under low energy conditions. The study reveals the dominance of swash transport on steep, reflective, low-energy beaches where wave energy dissipation takes place over small distances on the upper foreshore.