Longshore Velocity Research Articles

Observations of infragravity wave frequency selection

Field measurements of sea-surface elevation, cross-shore and longshore velocities were obtained during a storm event on the north coast of Zealand, Denmark from a cross-shore array of co-located pressure sensors and current meters. During the major part of this storm, statistically significant spectral peaks at a frequency of f≈0.025 Hz were identified at several cross-shore locations. Based on examinations of cross-shore coherence and phase relationships, it appeared that these motions were due to cross-shore standing wave structures rather than caused by shear wave activity. Comparisons with synthetic edge wave spectra, which were computed assuming a white shoreline surface elevation spectrum, suggested that the identified spectral peaks were not artifacts of a cross-shore standing wave nodal structure but an indication that those frequencies where the peaks occurred were preferentially forced. Analyses of local and spatial phase relationships of velocity and sea-surface elevation, as well as comparisons with numerically calculated theoretical cross-shore wave structures suggested that these waves were standing edge waves with mode number ( n)⩾3. The selected frequency of f≈0.025 Hz corresponds to one of the cut-off frequencies that can theoretically occur on this bathymetric geometry which asymptotes to a constant depth offshore.

Standing edge waves on a pocket beach

Field measurements of edge waves obtained on a narrow pocket beach are described. The beach (named Usgo) is located on the north coast of Spain immediately to the west of the city of Santander and is exposed to the Atlantic Ocean. The edge wave field on this beach is analyzed utilizing data from a longshore array of current meters located in about 3 m of water depth. Frequency spectra of longshore velocities display several significant peaks, suggesting that the edge wave field on this pocket beach consists of several dominant modes. This is in contrast to observations on open coastal beaches [e.g., Oltman‐Shay and Guza, 1987], which show the existence of a broad‐banded edge wave field. We isolate four frequency peaks and examine the longshore variation of the motions at these frequencies and find convincing evidence of a nodal structure. Utilizing the measured bathymetry, we compute the cross‐shore profiles and theoretical wavelengths of edge waves at the dominant frequencies using a numerical solution of the shallow water equations by Howd et al. [1992]. We find that edge waves at these frequencies are theoretically confined between the headlands since their amplitudes decay to insignificant levels within a cross‐shore distance that is less than the offshore extent of the headlands. In addition, assuming perfect sidewall reflections, the theoretical nodal locations closely correspond to those inferred from the measurements. We conclude that discrete standing edge waves resulting from resonances related to the longshore width of the beach are present on Usgo Beach.

Hydrodynamic variability on megatidal beaches, Normandy, France

Several experiments aimed at characterising the hydrodynamics of megatidal beaches outside the surf zone were carried out between 1990 and 1994 on the Cotentin coast of the Cherbourg Peninsula in Normandy. The database was established from the records of several electromagnetic current meters and pressure sensors and from field surveys. The mean spring tidal range on these beaches varies between 9.3 and 11.4 m. The results show the prevalence of strong longshore currents, with velocities up to 0.5 m s −1, on the low- and mid-tidal beach zones. Mostly oriented northward, these currents reflect both a progressive tidal wave and a strong longshore gradient in water level between the Channel Islands embayment and the English Channel. While varying largely during a typical tidal cycle, these longshore velocities are maximum at high tide, reflecting the progressive nature of the tides. This high-tide maximum velocity increases by a factor of 1.5 between the mean tide and mean spring tide, and between the mid- and low-tidal zones due to bed friction effects. Cross-shore velocities are generally weak (<0.1 m s −1), but sometimes stronger in smaller water depths. In the low-tidal zone, they are commonly oriented onshore at the beginning of the rising tide and offshore during the falling tide. This circulation results from a west–east cross-shore gradient in water level that is particularly important around the mean water level. Towards high tides, weak offshore steady flows were observed in the presence of waves. Site-specific relationships were defined in order to characterise the modulation of significant wave height by sea level fluctuations both on the shoreface and in the intertidal zone. The water depth variability during the tidal cycle induces fluctuations in the dissipation by bottom friction, resulting in wave height changes. The influence of tidal currents on the wave height proved to be very small in this context. The tidal fluctuations also influence the instantaneous near-bed currents induced by simultaneous action of non-breaking waves and the tides. During stormy conditions, wave-induced gravity orbital motions dominate the steady flows in the mid-tidal zone, outside the surf zone. At this location, the shallow water friction effect results in weak steady longshore currents, and low water depths explain strong orbital motions. The opposite conditions prevail in the low-tidal zone, where the steady tidal currents are stronger than gravity orbital velocities during a few hours around high tide. Outside this period, with the decrease in water depth and in steady current intensity due to friction effects, the tidal and gravity wave-induced currents have comparable intensities. In both the low- and mid-tidal zones, infragravity motions are weak outside the surf zone. The foregoing results show that outside the surf zone, these megatidal beaches are characterised by wave-dominated mid-tidal zones and tide-dominated low-tidal zones during spring tides. We suggest the term “mixed wave-tide-dominated” for these beaches with very large tidal ranges.

Experimental investigation of vanes as a means of beach protection

Vanes are essentially flat vertical flow guiding structures. In the beach protection application vanes are placed, inshore of the breaking zone, in the upper part of the vertical water column and with a predefined angle of attack to the design wave orthogonal. They are thus subjected to a generally inshore-directed flow, which induces a lift, and drag force on the vane. These forces, in turn, induce a flow that counteracts the wave-induced longshore current, thus decreasing the longshore current velocity, which facilitates sediment deposition. Two experiments were conducted in a small-scale wave basin with the purpose of investigating the vane function. Both experiments were conducted with a movable bed that formed a crenulate-shaped bay between two headland structures. The first experiment showed that the vanes created a salient upstream of the vanes. This result is also supported by initial prototype results. The second experiment showed that the longshore velocity was reduced upstream of the vane and that the bulk of the velocity reduction took place in the nearshore part of the breaking zone. The main flow feature is identified as a large horizontal vortex inshore of the vane.

The run-up of weakly-two-dimensional solitary pulses

Abstract. The run-up of solitary-type pulses propagating at a small angle with respect to the shore normal is analysed by means of a weakly-two-dimensional extension of a solution of the nonlinear shallow water equations for a non-breaking, solitary pulse incident and reflecting on an inclined plane beach similar to that of Synolakis (1987). A simple analytic expression for the longshore velocity of the solitarytype pulse is given along with examples of computations. The proposed solution can be employed in modelling run-up flow properties of solitary-type pulses (e.g. tsunamis, primary waves of wave groups propagating in shallow waters, ...). The hodograph transformation that is used and the flow properties are illustrated in terms of contour plots. A limiting pulse amplitude is defined such that breakdown of the solution occurs. A solution for the run-up of multiplesolitary-pulses in shallow waters is also described. Some of the salient characteristics are illustrated and discussed. Breakdown conditions are analytically defined also for the multiple-solitary-pulses solution. A strong condition is given which couples information on both pulses amplitudes and distances. An easier (but weaker) version of the criterion is given in terms of a pair of decoupled formulae one for the Pulses amplitudes and the second for their initial positions. Very large run-up is achieved because of the merging of two or more solitary pulses which are smaller than the limiting Pulse. The role of pulse separation within a group of solitary Pulses is also analysed in terms of both a 'nonlinearity parameter' N and a 'groupiness parameter' G. It is found that a critical distance exists between two pulses which minimizes the back-wash velocity and, as a consequence, the nonlinearity parameter N.

On deep western boundary currents

The Stommel and Arons [1960a, b] theory of the deep circulation of the world ocean does not directly address the physics of deep boundary currents. In the present paper a maximally simple steady state physics for deep western boundary currents is developed and integrated into the Stommel‐Arons model. This is accomplished by modifying the assumptions of the latter to include effective horizontal eddy viscous terms everywhere in the deep water, including in the interior. The resulting solution for the meridional velocity in a rectangular ocean on a beta plane is mathematically analogous to that of Munk [1950] for the wind‐driven circulation. In the interior the meridional velocity reduces identically to that of Stommel and Arons. Among other benefits, the solution eliminates the theoretical difficulties associated with the crossing of the equator by the deep western boundary current. As an initial test of the theory's consistency with observation, theoretical plots of pressure and meridional velocity as a function of longitude are qualitatively compared with hydrographic sections and inferred meridional velocity fields in the eastern and western basins of the abyssal North Atlantic. Also, the theoretical profile for the meridional velocity as a function of longitude is qualitatively compared with observations of longshore velocity as a function of offshore distance in the deep western boundary current of the North Atlantic Deep Water.

Eulerian and Lagrangian aspects of the longshore drift in the surf and swash zones

A theoretical study of the mean longshore mass flux and longshore drift velocities due to waves obliquely incident on a uniform sloping beach is presented. Analysis is performed using both Eulerian and Lagrangian representations of the flow. A number of results based on inviscid solutions are obtained concerning the mean longshore mass flux 〈My〉 inside the swash zone. Thus 〈My〉 depends on the parameters (wave amplitude, wave frequency, and wave number) of waves incident at a shoreline in a similar fashion to the dependence of the mass flux occurring between the crests and troughs of propagating waves on the same wave parameters. The mass flux depends on the square of the local wave amplitude, even for very steep waves which break before reaching the shoreline. A Lagrangian approach shows that particle paths are not closed even offshore of the breaking point and that “zigzag transport” is characteristic of Lagrangian particle paths in the two‐dimensional horizontal flow seaward of the swash zone. The cross‐shore profile of the longshore drift velocity is analyzed. Very weak longshore drift velocity characterizes the nonbreaking waves up to the swash zone. Onshore of the breaking point we find that the longshore drift velocity has a quasi‐linear profile up to the maximum velocity reached near the shoreline. Effects of seabed friction are included in the computation of cross‐shore profiles of the longshore drift velocity. A sensitivity analysis reveals that even for relatively small friction parameters (ƒ≈0.1), velocities inside the swash zone are greatly reduced. The reduction in longshore drift velocity is greater for the Lagrangian profile than for the Eulerian one, and for larger friction parameters (ƒ≥0.1) the maximum longshore velocity moves toward the breaking point, where the friction effects are smaller. Finally, steady state profiles of the longshore drift velocities (Eulerian and Lagrangian) are computed.

Particle Tracking Simulation of Pollutant Discharges

A two-dimensional particle tracking simulation of pollutant discharges in a vertically well-mixed tidal current is performed for an idealized straight coast with parametrically defined parallel bathymetry and realistic spatially and temporally varying longshore velocity and diffusivity fields. Two methods of random-walk simulation in varying diffusivity media are compared: one where the random walk is performed in a transformed (spatially) uniform diffusivity space and the other where the random walk takes place in the untransformed space. For the special case of a steady coastal current—and subject to a suitable choice of a random-number generator, time step, and particle flux—the stochastic solutions in either case are found to be in reasonable agreement with the analytic solution of a previous researcher. Further simulations of steady and of “optimal” discharges in tidal flows with and without a mean longshore current are presented. The major conclusion is that in the absence of shorelines, discharging optimally—in the sense of maintaining a constant global maximum concentration in the presence of a varying current—allows a much greater mass of pollutant to be discharged over the tidal cycle. However, in the presence of shorelines discharging optimally leads to much higher shoreline concentrations and thus has little effect in reducing global maximum concentrations. This suggests that there is little advantage to be gained by varying the discharge in such cases.

One-dimensional modelling of individual waves and wave-induced longshore currents in the surf zone

A probabilistic model ( Wavis-model) was developed to describe the propagation and transformation of individual waves (wave by wave approach). The individual waves shoal until an empirical criterion for breaking is satisfied. Wave height decay after breaking is modelled by using an energy dissipation method. Wave-induced set-up and set-down and breaking-associated longshore currents are also modelled. Laboratory and field data were used to calibrate and verify the model. The model was calibrated by adjusting the wave breaking coefficient (as a function of local wave steepness and bottom slope) to obtain optimum agreement between measured and computed wave height. Four tests carried out in the large Delta flume of Delft Hydraulics were considered. Generally, the measured H 1 3 -wave heights are reasonably well represented by the model in all zones from deep water to the shallow surf zone. The fraction of breaking waves was reasonably well represented by the model in the upsloping zones of the bottom profile. Verification of the model results with respect to wave-induced longshore current velocities was not extensive, because of a lack of data. In case of a barred profile the measured longshore velocities showed a relatively uniform distribution in the (trough) zone between the bar crest and the shoreline, which could to some extent be modelled by including space-averaging of the radiation force gradient, horizontal mixing and longshore water surface gradients related to variations in set-up. In case of a monotonically upsloping profile the cross-shore distribution of the longshore current velocities is reasonably well represented.

Application of the data filtration algorithm in reconstructing the structure of trapped waves over the Crimean shelf

The data collected in the course of a local dynamic experiment near the Crimean shore during the summer of 1991 are interpreted, using the filtration algorithm based on variation principles. As a result, the ratio between the mode amplitudes of the long-shore velocity structure has been obtained, which is consistent with the physical views about shelf water dynamics.

Suspended-sediment transport in the surf zone: response to breaking waves

A field experiment designed to investigate the influence of wave breaking on suspended-sediment transport was conducted at Duck, NC, from 6 to 9 September 1985. Arrays of optical backscatter sensors, electromagnetic current meters and pressure sensors were deployed at five positions on a shore-normal transect that spanned the surf zone. At each position measurements were made of cross-shore and longshore velocity, sea-surface fluctuations, and suspended sediment at five levels above the bed. Experimental data runs were conducted when incident swell waves ( H s = 0.5m, T= 10–12s) broke (primarily plunging) within the experimental transect. This paper describes the spatial characteristics of the plunge-to-bore tranformation region and describes (1) the cross-shore variability of sediment resuspension, including the mean concentrations and mean suspended load; (2) the net longshore and cross-shore flux across the surf zone; (3) mean suspended-sediment profiles as a function of wave type, e.g. plunging, spilling and bore, and unbroken at four positions across the surf zone; and (4) discusses the relative contribution of each wave type to the net longshore and cross-shore sediment flux.

Numerical simulation of coastal upwelling and inerfacial instability of a rotaion and stratified fluid

The evolution of the coastal upwelling and interfacial instability of a stratified and rotating fluid is studied numerically by using large-eddy simulation. Upwelling is generated near the sidewall of a rotating annulus by the shear at the top. The fluid initially consists of a stably stratified ‘two-layer’ structure with a narrow interface separating the two layers. The large-scale motion of the flow is simulated by solving the time-dependent non-hydrostaic incompressible Navier-Stockes and scalar transport equations while the small-scale motion is represented by a dynamic subgrid-scale model. The upwelling process contains both stable and unstable stratification. The vertical structure of upwelling consists of a persistent primary front, a trailing mixing zone on te shore side of the front, and a temporary secondary front which leads a top inversion layer. The longshore velocity profile has two maxima which occur at the edge of the sidewall boundary layer and at the density front. The upwelled density front is unstable to azimuthal perturbations and baroclinic waves develop and grow to large amplitude. Pairs of cyclonic and anticyclonic waves appear at the front which form ‘jet-streams’. The secondary front is unstable to azimuthal perturbations. Its instability, and the associated drop of the top inversion layer, take the form of radial bands which subsequently break up into isolated patches and eventually sink. The computed values of various upwelling time and length scales are compared to and are in good agreement with past experimental data.

Three Dimensional Structure of Nearshore Currents

Three dimensional structure of nearshore currents is obtained by detailed measurements of velocity by Fiber-LDV using a plane wave tank. Vertical distribution of cross-shore velocity is similar to sum of uniform flow and velocity distribution of two dimensional wave flume. Vertical distribution of longshore velocity is nearly uniform flow. Velocity measurements of the upper reaches of trough level of wave are important, because vertical distribution of velocity changes rapidly at the nearly trough level.

Suspended sediment transport in the surf zone: Response to incident wave and longshore current interaction

This paper reports the results of a field experiment carried out in the inner surf zone of a dissipative beach at San Marine, Oregon, in 1984. Observations were made at a location 70 m seaward of the mean high water line (mean depth 1.3 m) during a period when significant offshore wave heights were 3–5 m and the surf zone was 500 m wide. Instruments were deployed in a closely spaced array and measurements included cross-shore and longshore velocity, sea surface fluctuations, and sediment concentration profiles. The results reported in this paper are from a particular 44-min data run characterized by high waves and strong longshore currents. In the first section of this paper the field measurements are summarized to provide insight into the resuspension process and observed sediment concentrations and fluxes on time scales including incident and infragravity bands, and mean values for the entire data set. In the second section a numerical model is presented that incorporates the nonlinear interaction of waves and currents and calculates velocity profiles, boundary shear stress, suspended sediment profiles, and longshore and cross-shore particle flux over 0.5 s intervals throughout the time series. Finally, model calculations of mean suspended load and net longshore and cross-shore flux of sediment are compared to observations. The results show that the model calculates mean suspended load and net longshore and cross-shore flux of sediment within a factor of two of the observations. The model suggests that, for this data set, incident waves and longshore currents contribute equally to the suspended load. For the longshore particle flux, the current contribution is approximately 1.7 times the wave contribution (63% vs 37%), primarily because currents carry suspended sediment higher in the water column where it is advected by stronger longshore currents. Additionally, model results suggest that wave/current interactions enhance suspended sediment load and longshore sand transport by approximately 50–60% over the transport forced by waves alone.

The longshore flow component in low-energy rip channels: The Mediterranean, Israel

We examined in the field the near-bed longshore ( v) flow component in rip channels at the Herzliyya beach, Israel, on the Southeastern Mediterranean. Field experiments, conducted under relatively calm wave conditions ( T sig = 3–6 s, H sig = 0.3–1.1 m) included current and wave monitoring and morphological mapping. Time series of instantaneous and smoothed shore-parallel current velocities and of the orbital components, were analysed, including statistical and spectral processing. The peak instantaneous velocities of waves and currents combined were in excess of the threshold motion of the local sand, taken as 0.3 m/s. Some velocities even exceeded the threshold for sheet flow transport ∼0.9 m/s. The feeder and the neck demonstrated the highest longshore flow components, whereas the rip head, typically showed the lowest longshore velocities. The unsteadiness of the net longshore current was very high. Feeders typically showed a mean unidirectional longshore current. However, directional fluctuations gradually increased towards the rip head. The data suggest a clear spectral structure; most energy (46–67%) was in the gravity range with 13–35% subharmonic and minor (13–16%) in the infragravity band. Systematic cross-shore spatial trends of the energy spectra components were linked to bar morphology and to exposure to incident waves.

Frictionally Modified Continental Shelf Waves and the Subinertial Response to Wind and Deep-0cean Forcing

Abstract Analytic solutions am obtained for the barotropic shelf circulation caused by wind and deep-ocean forcing at subinertial frequencies. An Inclined beach model of the continental shelf is used and only situations in which bottom friction is important are considered. Three different alongshore forces are considered: pressure gradients and currents (maintained by the deep ocean) at the shelf break and wind stress, over the shelf. In each case the model is formulated as a boundary value problem in which the boundary conditions are determined by the forcing mechanism. In general, a damped resonant response occurs when the forcing function has the same longshore velocity as an unforced continental shelf wave and is most significant for the fim mode. In the case of forcing by an alongshore pressure gradient at the edge of the shelf, this leads to the amplification of the pressure signal toward the coast. The model frequencies and structures are determined for various frictional values. When friction is s...

Velocities in combined undertow and longshore currents

The flow composed of a periodic wave motion with turbulent fluctuations caused by breaking and a (slowly varying) mean current is considered. The mean current consists of the combined cross-shore and longshore current velocities generated by the waves which means the profile of the total velocity is highly three dimensional. The Reynolds equations for the currents are derived and the influence of cross-shore and longshore currents on one another is neglected. For the special case of a cylindrical coast and turbulent shear stresses modelled by an eddy viscosity, the equations for the longshore velocity variation V( x, z) over depth and distance from the shore are found to satisfy a Poisson equation. A perturbation solution is derived to a second order for V( x, z) and the classical solutions for V( x) from the literature are found to be logically related to the new solution, which (briefly described) always yields somewhat larger velocities. Measurements in the literature show the same qualitative variation of the longshore velocity over depth but a detailed comparison is not made.

THEORETICAL MODEL FOR NEARSHORE CIRCULATIONS

The dynamics of nearshore circulations is investigated using mass, momentum and energy conservation equations with bottom friction, lateral mixing and wave-current interaction. By means of introducing a perturbation expansion for the mean variables, the first-order solutions are found in the surf and offshore zones according to the boundary conditions at the coast. It is found that:(l) The rip velocity attains a maximum value in the offshore region near the break point as Yr* becomes larger. (2) The longshore velocities become higher in the surf zone and lower in the offshore region with increasing Yr*. (3) The rip and longshore velocities in the surf zone become relatively smaller due to the effects of the bottom friction, and that the rip and longshore velocities in the surf zone become smaller due to the wave-current interaction.

THREE DIMENSIONAL FLOW PROFILES ON LITTORAL BEACHES

The problem of combined cross-shore and longshore currents generated by waves in and around a surf zone is considered in its full three-dimensional formulation. The equations for the two current components are decoupled and it is found that for a cylindrical coast with no longshore variations the longshore current variation with depth and distance from the shoreline satisfies a Poisson equation. This equation is solved by a perturbation method and it is shown that the longshore velocities are always larger than the velocities found by classical theory. In the simple uncoupled case, the full 3-D current profile is constructed by combining the results with cross - shore velocities determined in previous publications. Also, the total velocities are larger than velocities found from simple depth averaged models.

Momentum and heat balances in steady coastal currents

The momentum balance for coastal currents forced by a longshore wind stress and a longshore density gradient is examined for coastlines of various orientations. We isolate the coastal circulation by requiring that the longshore wind stress and density gradient become uniform in the adjacent deep ocean in which the longshore bottom pressure gradient is assumed to be zero, and derive a vorticity equation for the longshore current for a prescribed cross-shelf variation of wind stress and density structure. Solutions of this equation are obtained for a uniform longshore wind stress along a coastline with an exponential shelf in two situations: (A) For a uniform cross-shelf density on the inner shelf and a zero longshore bottom pressure gradient on the outer shelf, the division between the two regions occurring approximately at the depth of the main thermocline, and (B) for a thermohaline structure which is independent of the cross-shelf co-ordinate such that the coastal current is barotropic. It is shown that on a zonal coast the coastal transports due to the wind stress forcing are of O(σ τ Wl ϱ 0β ) , and due to the density gradient forcing of O(− σ ϱ 0 gh T 2/ β∂δ 0/ ∂l) where τ Wl and − ϱ 0 ∂δ 0/ ∂l are respectively, the longshore wind stress and surface density gradient, ϱ 0 is a reference density for seawater, g is the acceleration of gravity, h T is the scale depth for the main thermocline, β= df dy where f=2Ω sinϕ in which Ω is the angular speed of rotation of Earth, and ϕ is the latitude, and σ= W T W is a shelf parameter in which W T is the exponential shelf scale depth and W is an eddy scale. On eastern boundaries the transports are less [especially in (A)] and are identically zero on western boundaries on which the transport is determined by external (non-local) processes. The longshore velocity structure on the western boundary depends strongly on σ and incorporates a system of currents and counter-currents. The heat balance for the coastal circulation is investigated by considering the solution of the heat conservation equation for the surface temperature with the cross-shelf boundary conditions of zero heat flux through the coast and into deep ocean. It is found that the sea surface temperature anomaly is zero for wind stress forcing and positive for density gradient forcing (at constant salinity). Mixed forcing however may lead to upwelling which is characterized by a negative sea surface anomaly. Upwelling occurs on non-western boundaries on which there is a longshore wind stress of sufficient intensity or on western boundaries where there is an external transport of sufficient magnitude to generate a net transport up the longshore surface density gradient. This occurs on eastern coasts where there is poleward advection of warm water and sufficiently strong equatorward meridional winds, and on western coasts in a polar gyre where there is a strong equatorward external transport and a poleward temperature gradient. The Ekman circulation maintains the upwelling anomalies.