Even–odd analysis on a complex shoreline

An even–odd signal decomposition is performed on a complex shoreline having a longshore sediment transport gradient. The expected impact of erosion due to a navigation channel and structures is discussed and implications of the transport gradient on the decomposed shoreline signal are noted.     

Application of one-line model to the prediction of shoreline change

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Calculation of beach change under interacting cross-shore and longshore processes

This paper presents a mathematical approach and numerical model that simulates beach and dune change in response to cross-shore processes of dune growth by wind and dune erosion by storms, and by gradients in longshore sand transport that will alter shoreline position. Sub-aerial transport processes are represented, whereas sub-aqueous transport is neglected. The system is tightly coupled morphologically, with the berm playing a central role. For example, the potential for sand to be transported to the dune by wind depends on berm width, and sand lost in erosion of the dune during storms can widen the berm. Morphologic equilibrium considerations are introduced to improve reliability of predictions and stability of the non-linear model. An analytical solution is given under simplification to illustrate properties of the model. Sensitivity tests with the numerical solution of the coupled equations demonstrate model performance, with one test exploring beach and dune response to potential increase in storm-wave height with global warming. Finally, the numerical model is applied to examine the consequences of groin shortening at Westhampton Beach, Long Island, New York, as an alternative for providing a sand supply to the down-drift beach. Results indicate that the sand will be released over several decades as the shoreline and dune move landward in adjustment to the new equilibrium condition with the shortened groins.     

Modelling analysis of the sensitivity of shoreline change to a wave farm

Change of shoreline wave climate caused by the installation of a wave farm is assessed using the SWAN wave model. The 30 MW-rated wave farm is called the ‘Wave Hub’ and will be located 20 km off the north coast of Cornwall, UK. Changes in significant wave height and mean wave period due to the presence of the Wave Hub are presented. The results suggest that the shoreline wave climate will be affected, although the magnitude of effects decreases linearly as wave energy transmitted increases. At probable wave energy transmission levels, the predicted change in shoreline wave climate is small.     

A numerical model of beach morphological evolution due to waves and currents in the vicinity of coastal structures

A numerical model was developed of beach morphological evolution in the vicinity of coastal structures. The model includes five sub-models for random wave transformation, surface roller development, nearshore wave-induced currents, sediment transport, and morphological evolution. The model was validated using high-quality data sets obtained during experiments with a T-head groin and a detached breakwater in the basin of the Large-scale Sediment Transport Facility at the Coastal and Hydraulics Laboratory in Vicksburg, Miss, USA. The simulations showed that the model reproduced well the wave conditions, wave-induced currents, and beach morphological evolution in the vicinity of coastal structures. Both salient and tombolo formation behind a T-head groin and a detached breakwater were simulated with good agreement compared to the measurements.     

Alongshore fluid motions in the swash zone of a sandy and gravel beach

Field experiments were conducted on a low-gradient, high-energy sandy beach (Truc Vert, France) and a steep, low-energy gravel beach (Slapton, UK) to examine alongshore-directed currents within the swash zone. At Truc Vert, data were collected over 33 tidal cycles with offshore significant wave heights of 1–4 m and periods of 5–12 s. At Slapton data were collected during 12 tides with wave heights of 0.3–1 m and periods of 4–9 s. The swash motion was predominantly at infragravity frequencies at Truc Vert and incident frequencies at Slapton.     

Surf zone dynamics simulated by a Boussinesq type model. Part I. Model description and cross-shore motion of regular waves

This is the first of three papers on the modelling of various types of surf zone phenomena. In this first paper, part I, the model is presented and its basic features are studied for the case of regular waves. The model is based on two-dimensional equations of the Boussinesq type and it features improved linear dispersion characteristics, possibility of wave breaking, and a moving boundary at the shoreline. The moving shoreline is treated numerically by replacing the solid beach by a permeable beach characterized by an extremely small porosity. Run-up of nonbreaking waves is verified against the analytical solution for nonlinear shallow water waves. The inclusion of wave breaking is based on the surface roller concept for spilling breakers using a geometrical determination of the instantaneous roller thickness at each point and modelling the effect of wave breaking by an additional convective momentum term. This is a function of the local wave celerity, which is determined interactively. The model is applied to cross-shore motions of regular waves including various types of breaking on plane sloping beaches and over submerged bars. Model results comprise time series of surface elevations and the spatial variation of phase-averaged quantities such as the wave height, the crest and trough elevations, the mean water level, and the depth-averaged undertow. Comparisons with physical experiments are presented. The phaseaveraged balance of the individual terms in the momentum and energy equation is determined by time-integration and quantities such as the cross-sectional roller area, the radiation stress, the energy flux and the energy dissipation are studied and discussed with reference to conventional phase-averaged wave models. The companion papers present cross-shore motions of breaking irregular waves, swash oscillations and surf beats (part II) and nearshore circulations induced by breaking of unidirectional and multidirectional waves (part III).     

Seabed shear stress and bedload transport due to asymmetric and skewed waves

A simple conceptual formulation to compute seabed shear stress due to asymmetric and skewed waves is presented. This formulation generalizes the sinusoidal wave case and uses a variable friction factor to describe the physics of the boundary layer and to parameterize the effects of wave shape. Predictions of bed shear stresses agree with numerical computations using a standard boundary layer model with a k–ε turbulence closure. The bed shear stress formulation is combined with a Meyer-Peter and Müller-type formula to predict sheet flow bedload transport under asymmetric and skewed waves for a horizontal or sloping bed. The predictions agree with oscillatory water tunnel measurements from the literature.     

A coupled numerical model for simulation of wave breaking and hydraulic performances of a composite seawall

A numerical model is developed by combining a porous flow model and a two-phase flow model to simulate wave transformation in porous structure and hydraulic performances of a composite type low-crest seawall. The structure consists of a wide submerged reef, a porous terrace at the top and an impermeable rear wall. The porous flow model is based on the extended Navier-Stokes equations for wave motion in porous media and k−ε turbulence equations. The two-phase flow model combines the water domain with the air zone of finite thickness above water surface. A unique solution domain is established by satisfying kinematic boundary condition at the interface of air and water. The free surface advection of water wave is modeled by the volume of fluid method with newly developed fluid advection algorithm. Comparison of computed and measured wave properties shows reasonably good agreement. The influence of terrace width and structure porosity is investigated based on numerical results. It is concluded that there exist optimum value of terrace width and porosity that can maximize hydraulic performances. The velocity distributions inside and in front of the structure are also investigated.     

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.     

Internal characteristics and numerical analysis of plunging breaker on a slope

Many investigations about the direct measurements of velocities to clarify the internal mechanism of the breaker have been carried out as a result of recent progress in the measuring techniques.This research attempts to clarify the breaking wave transformation system on a slope by an experiment and numerical analysis. In an experiment, the velocities in the surf zone were measured directly using an electromagnetic current meter, and the space distribution characteristic of the vorticity ω = (∂u/∂y − ∂u/∂x) and the skewness γ = (∂u/∂y + ∂u/∂x) were examined. Also, occurrence situations of the vortices at the time of water mass inrush were measured by video tape recorder (VTR) image processing. However, because the breaker is a violent phenomenon that is entrained with plentiful bubbles, the extent to which we can clarify breaker transformation in experiments is limited. Numerical simulations are substituted for experiments as a method to clarify breaker transformation.In numerical analysis, finite amplitude wave analysis based on the potential theory (non-viscous fluid) is possible before wave breaking; however, the analysis must take into account the viscous fluid after breaking. So, we use the Reynolds equations to develop a numerical simulation system of the breaker transformation on a sloping bottom. The numerical energy dissipation model of the breaker was compared to the experimental results, and a modified Simplified Marker and Cell (SMAC) method is presented. The internal characteristics of the breaker transformation are described using application examples.     

Multigrain sedimentation/erosion model based on cross-shore equilibrium sediment distribution: Application to nourishment design

In the light of global warming and sea level rise there are many coastal beaches that suffer from erosion. Beach nourishment has become a common practice to maintain the sediment balance on a shore-face. In this paper, a three-dimensional numerical model for evaluating long-term impact of beach nourishment projects has been developed. The model addresses the longstanding complex issue of coastal morphology and sediment grain size distribution from an unconventional angle, which exploits the strong links between grain size distribution and the prevailing transport direction of each sediment constituent under ‘average’ wave and storm action. The present model predicts the redistribution of nourished sediment according to the subtle clues implied by equilibrium distribution curves and latest coastal wave transformation theories. After verification against recent field observations in Terschelling, The Netherlands, the model was used to predict long-term effects of different beach nourishment strategies. It was found that: (a) given the source sediment available in Terschelling the tactics of large volume and less frequent implementation are better than otherwise; and (b) from a pure engineering point of view, waterline nourishment outperforms offshore trough nourishment.The model offers an additional tool for coastal engineers to evaluate the feasibility, effectiveness and the optimization of dumping locations for beach nourishment projects. It is also a useful tool for stratigraphic modelling of shallow-marine sedimentation in conjunction with sea level changes.     

Wave pressures on a vertical wall due to short-crested waves: fifth-order approximation

Non-linear wave pressure induced by short-crested waves on a vertical wall is an important factor to be considered in the design of coastal structures. The existing models to estimate the wave pressure in engineering design are limited to the third-order solution ([Hsu et al., 1979]). In this paper, an analytical solution up to the fifth-order is derived through perturbation approximation. This analytical closed-form solution is used to investigate the contributions of the higher-order components in short-crested waves. It is found that fifth-order components significantly affect the change of pressure, especially in shallow water and larger waves.     

Mud mass transport due to waves based on an empirical rheology model featured by hysteresis loop

A vertical 2-D water–mud numerical model is developed for estimating the rate of mud mass transport under wave action. A nonlinear semi-empirical rheology model featured by remarkable hysteresis loops in the relationships of the shear stress versus both the shear strain and the rate of shear strain of mud is applied to this water–mud model. A logarithmic grid in the vertical direction is employed for numerical treatment, which increases the resolution of the flow in the neighborhood of both sides of the interface. Model verifications are given through comparisons between the calculated and the measured mud mass transport velocities as well as wave height changes.     

Surf zone dynamics simulated by a Boussinesq type model. Part II: surf beat and swash oscillations for wave groups and irregular waves

This is the second of three papers on the modelling of various types of surf zone phenomena. In the first paper the general model was described and it was applied to study cross-shore motion of regular waves in the surf zone. In this paper, part II, we consider the cross-shore motion of wave groups and irregular waves with emphasis on shoaling, breaking and runup as well as the generation of surf beats. These phenomena are investigated numerically by using a time-domain Boussinesq type model, which resolves the primary wave motion as well as the long waves. As compared with the classical Boussinesq equations, the equations adopted here allow for improved linear dispersion characteristics and wave breaking is modelled by using a roller concept for spilling breakers. The swash zone is included by incorporating a moving shoreline boundary condition and radiation of short and long period waves from the offshore boundary is allowed by the use of absorbing sponge layers. Mutual interaction between short waves and long waves is inherent in the model. This allows, for example, for a general exchange of energy between triads rather than a simple one-way forcing of bound waves and for a substantial modification of bore celerities in the swash zone due to the presence of long waves. The model study is based mainly on incident bichromatic wave groups considering a range of mean frequencies, group frequencies, modulation rates, sea bed slopes and surf similarity parameters. Additionally, two cases of incident irregular waves are studied. The model results presented include transformation of surface elevations during shoaling, breaking and runup and the resulting shoreline oscillations. The low frequency motion induced by the primary-wave groups is determined at the shoreline and outside the surf zone by low-pass filtering and subsequent division into incident bound and free components and reflected free components. The model results are compared with laboratory experiments from the literature and the agreement is generally found to be very good. Finally the paper includes special details from the breaker model: time and space trajectories of surface rollers revealing the breakpoint oscillation and the speed of bores; envelopes of low-pass filtered radiation stress and surface elevation; sensitivity of surf beat to group frequency, modulation rate and bottom slope is investigated. Part III of this work (Sørensen et al., 1998) presents nearshore circulations induced by the breaking of unidirectional and multi-directional waves.     

Forces and moment on a horizontal plate due to regular and irregular waves in the presence of current

This work presents an experimental study of a submerged plate used as a breakwater for coastal areas protection. Questions addressed concern the influence of current on the reflective power of the plate, and its influence on the hydrodynamic loads exerted on it. Results concern both monochromatic and irregular waves. Generally speaking, an influence of the current is found, changing the reflecting power of the structure up to 50%. A homogenized behavior of the loads and moments is found in the presence of currents, meaning that the load values become less sensitive to the frequency. Furthermore, the influence of waves reflected by the wave absorber, representing partially reflective conditions at the shore, is found to be of same order in the absence of current. In any case, the linear behavior of the breakwater is emphasized through the irregular waves approach.     

The SWAN model used to study wave evolution in a flume

The SWAN numerical model is used to model the evolution of JONSWAP wave spectra and hence the significant wave height of waves in a tank. Comparison with experiment has shown that modelling triad interactions in the numerical model leads to too low predictions of spectra and significant wave height and should therefore be excluded. The modelling of the breaking constant was also investigated, by looking at the use of a constant breaking constant, Nelson formula, and Goda formula (added into SWAN for this study). Using a constant value of 0.78 within SWAN gave the best comparison between theory and experiment.     

A mathematical model of spit growth and barrier elongation: Application to Fire Island Inlet (USA) and Badreveln Spit (Sweden)

A mathematical model of spit growth and barrier elongation adjacent to an inlet (of arbitrary width), supplied by sediment coming from longshore sediment transport, was developed based on the spit growth model proposed by Kraus (1999). The fundamental governing equation is the conservation equation for sand, where the width of the spit is assumed constant during growth. The portion of the longshore sediment transport feeding the spit has been estimated based on the ratio between the depth of the inlet channel and the depth of active longshore transport. Sediment transport from the channel due to the inlet flow, as well as other sinks of sand (e.g., dredging), are taken into account. Measured data on spit elongation at Fire Island Inlet, United States, and at Badreveln Spit, Sweden, were used to validate the model. The simulated results agree well with the measured data at both study sites, where spit growth at Fire Island was restricted by the inlet flow and the growth at Badreveln Spit was unrestricted. The model calculation for Fire Island Inlet indicates that the dredging to maintain channel navigation is the major reason for the stable period observed from 1954 to 1994 at the Fire Island barrier. The average annual net longshore transport rate at the eastern side of the Fire Island inlet obtained in this study was about 220,000 m³/yr, of which approximately 165,000 m³/yr (75% of the net longshore transport) is deposited in the inlet feeding the spit growth, whereas the remaining portion (25%) is bypassed downdrift through the ebb shoal complex.     

Circulation and heat budget of the northern Adriatic Sea (Italy) due to a Bora event in January 2001: a numerical model study

The Princeton Ocean Model was implemented to investigate the response of northern Adriatic Sea during the Bora event in January 2001 when strong wind and surface cooling was reported. The model has been run with realistic wind stress, surface heat flux and river runoffs forcings continuously from 1 January 1999 to 31 January 2001. The wind stress and surface heat flux was computed by the bulk parameterization, using the European Centre for Medium Range Weather Forecast analysis fields and the Comprehensive Ocean Atmosphere Data Set cloud data. All the freshwater sources along the Adriatic coastlines were represented by point or line source functions. Open boundary conditions in the Ionian Sea along a latitudinal boundary were nested within a large scale model of the Mediterranean Sea. The numerical study found that, before the Bora event of 13–17 January 2001, the water column of the northern Adriatic Sea was stratified by salinity, and the temperature was already cooler at the surface and over the shallower shelf region. The pre-Bora circulation of the northern Adriatic Sea was relatively weak and baroclinic with maximum surface currents occurred near the Italian coast. During the Bora event, the water column was well mixed in the most of coastal region of the northern Adriatic Sea. The atmospheric cooling produced colder water over the northern and western Adriatic Coast. The circulation of the northern Adriatic Sea was barotropic and dominantly wind driven, with maximum current speed of about 1 m s⁻¹. The numerical study also demonstrated that the Bora event decreased the heat content of the water column with an area averaged value of 205 W m⁻² over the shallow northern shelf. It was concluded that the heat budget of the northern Adriatic Sea during the Bora event was a balance between the surface heat loss, horizontal net heat inflow and resulting heat content decrease. The horizontal advection played a particularly important role in controlling the water temperature change over the shallower northern shelf.