Plane Beach Research Articles

Flow kinematics of focused wave groups on a plane beach in the U.K. Coastal Research Facility

Measurements are presented of the water particle kinematics of focused wave groups generated in the U.K. Coastal Research Facility. Single and repeated wave groups are considered at normal and 20° incidence to a 1:20 plane beach. The single focused wave groups model extreme transient events without the complication of reflections during the data acquisition process. A symmetry-based separation of harmonics method is used to interpret the water particle kinematics at the point of focus. Although the largest component is linear, there are also considerable second order kinematics terms (both low frequency and high frequency). Away from the free surface, the 2nd order difference contribution to the kinematics is a return current opposed to the direction of wave advance. For repeated wave groups, the measured kinematics confirms the presence of a low frequency free wave, followed by higher frequency waves of the main group and trailing higher order harmonic waves. In the breaker and surf zones, there is also evidence of the saw-tooth behaviour of broken waves, followed by scatter due to breaker-induced turbulence. Pulsatile wave breaking of repeated wave groups at oblique incidence is found to drive a longshore current.

Physics of nearshore bed pattern formation under regular or random waves

We present an investigation into the growth of nearshore, rhythmic patterns. A comprehensive linear stability model of the surf and shoaling zones is used to examine which type of pattern, transverse or crescentic bar, is likely to form under different wave conditions. In contrast to earlier studies we examine normal and near‐normal incidence on a plane beach. In doing so we reproduce results of earlier, more restricted studies and thereby identify the physical mechanisms leading to the growth of different patterns. This paper also focuses on the role of random wave height distribution compared with regular waves and identifies conditions likely to lead to pattern growth. To this end, an amended wave height dissipation function is presented, which allows us to move between random and regular regimes. It is found that a sharply defined surf breakpoint leads to larger growth rates and crescentic‐bar‐type features. In contrast, a large spread in breaking gives rise to transverse bar patterns with reduced growth rates. Transverse bar alongshore spacing is typically about 1/4 to 1/2 the width of the surf zone, while crescentic bar spacing is larger, up to twice this width. It is also shown that pattern types are influenced by the wave height to depth ratio in the surf zone. This indicates that sites with substantial inner surf zone wave energy and thus greater energy available to move sediment will give rise to transverse bar patterns. A new, propagating mode is identified in such cases, which exists for normal wave incidence. Finally, the role of wave shoaling and wave refraction, either on the bed or on the currents is examined. Crescentic bars seem to be a very robust feature as they stem from the model even if those three effects are ignored. Thus the only essential feedback for their formation is the coupling between depth‐controlled breaking and the evolving bathymetry. In contrast, transverse bar formation is very sensitive to wave refraction being enhanced by refraction over the bed and weakened by refraction over the current.

Boussinesq modelling of transient rip currents

The flow on a plane beach with a random, directionally spread wave field was simulated with a Boussinesq model. The random wave spectra were directionally symmetric with their central direction perpendicular to the beach, so no constant longshore current was generated. Variable wave-averaged currents were generated because of the spatially variable wave field, and sometimes formed offshore directed rip currents that appear in variable longshore locations. The rip currents are associated with a vortex pair which is generated within the surfzone and subsequently propagates offshore. Analysis of the vorticity balance show that the main vorticity input occurs within the inner surfzone. Three different beach slopes and four different wave spectra are simulated. The frequency, duration, and intensity of the transient rips depend on both the beach slope and the incident wave spectra. The results have important engineering implications for the transport of material in the nearshore zone, in particular on longshore uniform beaches.

Tsunami Wave Runup on Coasts of Narrow Bays

The runup of tsunami waves on the coasts of the barrow bays, channels and straits is studied in the framework of the nonlinear shallow water theory. Using the narrowness of the water channel, the onedimensional equations are applied; they include the variable cross-section of channel. It is shown that the analytical solutions can be obtained with use of the hodograph (Legendre) transformation similar to the wave runup on the plane beach. As a result, the linear wave equation is derived and all physical variables (water displacement, fluid velocity, coordinate and time) can be determined. The dynamics of the moving shoreline (boundary of the flooding zone) is investigated in details. It is shown that all analytical formulas for the moving shoreline can be obtained explicitly. Two examples of the incident wave shapes are analysed: sine wave and parabolic pulse. The last example demonstrates that even for approaching of the crest only, the flooding can appear very quickly; then water will recede relatively slowly, and then again quickly return to the initial state. Correspondence: Professor Efim Pelinovsky Laboratory of Hydrophysics and Nonlinear Acoustics, Institute of Applied Physics, 46 Uljanov Street, Nizhny Novgorod, 603950 Russia Email: Pelinovsky@hydro.appl.sci-nnov.ru Phone: 007-8312-164839 Fax: 007-8312-365976

Reply to discussion of “On the transport of suspended sediment by a swash event on a plane beach” [ Coastal Engineering 52 (2005) 1–23

Baldock and Alsina (2005) (hereafter BA05) identified two separate errors in PH05: both errors were introduced when the paper was being written rather than when the computations were carried out, and we are confident that they do not affect the other results which we present. Firstly, in the plots of the concentration field c(x,t) in Figs. 5, 6 and 9, the Fsnapshots_ are incorrectly labelled: in all these figures, snapshots are taken at intervals of 0.36, with the first snapshot being taken at t=0.2. Correctly labelled versions of these figures are given below. Secondly, there is an error in Eq. (29) of PH05. In the first line cL (t;n) should read simply cL(t;n): the expansion given is for the full solution rather than just for the component representing pre-suspended sediment. This error is responsible for a slight confusion: BA05 point out correctly that c should be independent of qe, since the concentration of presuspended sediment evolves solely by advection and deposition. For

An alternative dispersion equation for water waves over an inclined bed

A classical approach to extending the validity of Airy's dispersion relation for surface gravity waves by Friedrichs (1948) to gentle slopes (of special inclinations) is here re-examined with extended small-slope asymptotics using the full linear harmonic function theory combined with the method of steepest descent. A new dispersion relation emerges that appears to give significantly increased accuracy over sloping beds when tested on the plane beach problem with various forms of the mild-slope equation (MSE) and global error reductions of the order 50% are noted in some 'from deep to shore' computations. Unlike the classical formula, the new formula predicts a discontinuous wavenumber at a place where the bottom slope is discontinuous. Preliminary tests examining the reflection coefficient with the basic (early version) MSE over ramp-type profiles indicate that this is not a major problem and numerical results using wavenumber calculated by the new dispersion relation are qualitatively similar to those of the modified MSE (MMSE) developed in Chamberlain & Porter (1995). When the new formula is applied (with mass conservation) to the MMSE on the ramp, results are almost identical to those of a full linear model for inclines having a gradient up to 8:1. It is also shown that the dominant asymptotic analysis, responsible for the new formula, is valid for all slope angles a < n/2 and not just the special angles considered by Friedrichs.

Computing the Continuous-spectrum Linearised Bounded Standing Wave on a Plane Bed of Arbitrary Slope

The problem is revisited of continuous-spectrum small-amplitude bounded standing waves over a plane beach of arbitrary incline. Three documented approaches are compared and discussed vis-??-vis computational difficulties and this reveals, in particular, that the computation in established models, for particular beach angles, is optimised by seeking solutions to a functional integro-difference equation in a L2 space. This simplification is found to be impossible for general beach angles and developed instead is an alternative method initially based on Cauchy Principal value convergence of the inversion integral but improved (by linear combination of solutions) and developed into a comprehensive computational package which is validated by three fundamental tests. A single f77 routine is provided for the user through a web-link. Comparisons with established but hitherto un-computed methods are found numerically to favour the present development. Examples presented include both very steep and very shallow beaches and both potentials and velocities are calculated and displayed throughout the water column.

On the transport of suspended sediment by a swash event on a plane beach, by D. Pritchard and A.J. Hogg

On the transport of suspended sediment by a swash event on a plane beach, by D. Pritchard and A.J. Hogg

Two-dimensional wave-current interaction with a locally enriched quadtree grid system

Wave-induced currents may drive nearshore mixing and transport processes, including coastal pollutant dispersion, littoral drift, and long-term morphological changes through beach erosion and accretion. In this study, a numerical model is newly developed to simulate wave climate and localized currents in complicated coastal environments. The model developed is based on a quadtree grid system. The two-dimensional hydrodynamic governing equations are solved by using an explicit Adams-Bashforth finite difference scheme. Effects of wave breaking, shoaling, refraction, diffraction, wave-current interaction, set-up and set-down, turbulent mixing, bed friction, and shoreline movement are incorporated in the model. Results are presented for set-up at a beach in a flume due to normally incident waves, and longshore currents generated by oblique waves on a plane beach.

On the transport of suspended sediment by a swash event on a plane beach

We develop solutions for the transport of suspended sediment by a single swash event following the collapse of a bore on a plane beach, and we investigate the morphodynamical role that such transport may play. Although the intrinsic asymmetry between uprush and backwash velocities tends to encourage the export of sediment, we find that swash events may be effective in distributing across the swash zone much or all of the sediment mobilised by bore collapse; additionally, settling lag effects may promote a weak onshore movement of sediment. We quantify both effects in terms of the properties of the sediment and of the swash event, and comment on the relationship between our findings and recent field studies of swash zone sediment transport.

Rip channel development on nonbarred beaches: The importance of a lag in suspended‐sediment transport

On approximately planar beaches, rip channel development is often preceded by a period in which jet‐like rip currents develop in apparently random locations, and dissipate after minutes to tens of minutes. The subsequent development of sharp‐edged, trough‐like channels extending across the surf zone suggests that rip currents on planar beaches cause local erosion. Conversely, channels are known to cause localized offshore‐directed flow, and once channels have formed on approximately planar beaches, rip currents no longer occur in apparently random locations, but are restricted to the locations of the channels. Apparently, the excavation of channels by rip currents on planar beaches triggers a positive feedback between the morphological development and the flow. However, theoretical analysis indicates that, when depth increases with distance from shore, and sediment transport is treated as a function only of local flow conditions, channel development in the vicinity of a rip current may not occur. In numerical simulations (using a “cellular” model of nonbathymetrically driven rip currents) in which sediment flux on a planar beach is driven by approximately realistic rip current velocity patterns, deposition occurs under parts of the rip currents (especially in the seaward half of the surf zone), and these “rip ridges” cause a negative feedback. In these simulations, as in most models treating surf zone sediment transport, sediment flux is assumed to be strictly a function of local hydrodynamic conditions. However, Observations of sediment‐laden rip‐current plumes extending well beyond the surf zone suggest that suspended sediment transport is not always in equilibrium with local conditions. Other simulations employ a treatment of suspended‐sediment transport that allows for a lag in the adjustment of fluxes to the changes in local hydrodynamic conditions that the sediment is advected through. With this nonlocal sediment‐transport, the flux of sediment out of the surf zone in a rip current is increased, ridges do not develop under any part of the rip currents, and the morphodynamic feedback becomes positive.

Long wave forcing on a barred beach

We present new laboratory data on long wave forcing over a barred beach profile under random wave breaking conditions. The data include incident and radiated wave amplitudes, wave set-up, and detailed measurements of the cross-shore variation in long wave amplitude, including shoreline (swash) amplitudes. The total surf zone width was varied via changes in both wave height and the water level over the bar crest. The data obtained from the barred beach are also compared with previous data obtained from a plane beach under essentially identical short wave forcing conditions. The presence of the bar induces a frequency downshift in the spectral peak of the radiated long waves, a consequence of the increased surf zone width on the barred beach and a clear signature of long wave forcing by a time-varying breakpoint. Further comparisons of the two data sets suggest that the bar leads to resonant trapping and amplification (or suppression) of the shoreline motion at discrete long wave frequencies. Well-defined standing long wave motion occurs at discrete frequencies inside the bar and the resonant response is consistent with a simple seiche between the bar crest and shoreline, in agreement with previous numerical model studies. The long wave structure offshore of the breakpoint depends on the relative positions of the bar, shoreline and breakpoint, and is inconsistent with a numerical solution for a free standing long wave over the barred beach profile.

Inclined wavemaker inlinear water wave theory

A model for a paddle-type wavemaker coupled to 2D fluid flow over a plane beach isstudied. For purely outgoing wave motion, we find that the wavemaker has positive and negative added moment of inertias. When the wavemaker is used to extract energy, the efficiency is found to be at most 100% which is achieved when there is no reflected wave.

A grid-free, nonlinear shallow-water model with moving boundary

A meshless method based on the radial basis function is derived to solve the nonlinear, nondispersive shallow-water equations. The one-dimensional initial boundary value problem has a fixed boundary at one end and a free boundary at the other. The formulation employs a Lagrangian–Eulerian scheme to track the movement of the free boundary and transform the problem to a time-independent domain. The radial basis function evaluates the spatial derivatives in the numerical solution, while the Wilson- θ method integrates the development of the flow in time. The model is applied to calculate the flow of floodwater resulting from dam collapse and the run-up of waves on a plane beach. Comparisons of the computed results with analytical and finite difference solutions demonstrate the accuracy and capability of this meshless model in engineering applications.

Transformation model of wave height distribution on planar beaches

A new probability density function (pdf) for the transformation of depth-limited wave height distributions is presented. Assuming the bore approach for modeling the energy dissipation in the inner surf zone to be valid, an analytical expression for the transformation of wave height distribution including shoaling and breaking on a planar beach is obtained. The resulting expression for the pdf is formulated with a single function and only one shape parameter, which is calibrated as a function of the local root-mean-square (rms) wave height-to-water depth ratio and the local Iribarren number. The transformed pdf is able to reproduce the shape of field and laboratory measured wave height histograms and the sharp change in the shape of the wave height distribution in depth-limited breaking conditions for low exceedance probability. Results show that the theory is appropriate to represent wave height distribution transformation over shallow foreshores or in the surf zone. Alternatively, a combination of the new model with existing state-of-the-art wave energy propagation models allows the complete definition of the wave height distribution transformation on a planar beach.

Practical spreading of driving forces for wave-driven currents

Profiles of wave-induced current on plane beaches are generally smooth. Previously, the smooth nature was often explained by involving lateral mixing concept. This paper describes a new spreading approach instead of the previous lateral mixing approach. The spreading of the driving forces leads to similar results on the velocity profiles of the previous approach. The results of the new method show good agreement with some laboratory measurements for a plane beach. The present method was then extended for two-dimensional plan problems. The present approach was tested against a laboratory experiment, and the computed flow field agrees reasonably well with the measured flow field. A merit of the present method is that the computational time increment is not restricted by the lateral mixing.

Un modèle morphologique côtier pour la création de barres rythmiques

ABSTRACT A hydrodynamic and morphodynamic model for sandy beaches is presented. One interesting point is that the effects of the undertow and the wave asymetry are computed, which are the main factors in the creation of the bar. A study of the position and the initial vertical growth of the bar has been made with varying wave conditions and beach slope. This model is integrated in a commercial 2DH software. And it has also been coupled with a model of linear stability which gives the initial bottom, perturbations be added to an initial plane beach. The complete model allows the creation of a rythmic bar.

Bounded capillary–gravity waves obliquely incident on a plane beach

Surface tension effects upon bounded standing capillary–gravity waves over a flat beach are investigated in a linearised framework of a classical non-hydrostatic model of a perfect fluid. Existing theory for oblique pure gravity waves is extended to capillary–gravity waves. The solutions are presented in a way that is easy to compute using a combination of finite sums and Laplace integrals. Examples are given for oblique incidence. Solutions are determined, subject to boundedness at the contact line. It is found that surface tension lowers the amplitude, generally lengthens the wavelength and reduces the refractive effect of shoaling. However, very near the shore-line it is found that the refractive effect is reversed by the combined action of surface tension and shoaling in very shallow water. The Hocking edge condition is briefly examined.

On fine sediment transport by long waves in the swash zone of a plane beach

We calculate suspended sediment transport, erosion and deposition under reflected long waves on a plane beach. Our method employs the shallow-water equations, and is based on the calculation of the concentration field in Lagrangian coordinates: this allows results to be obtained readily throughout the whole domain, including the often troublesome shoreline region. These results suggest that the tendency of wave-dominated muddy shores to export sediment and erode over long time scales is not due to wave forcing at infragravity frequencies, since the sediment transport under such waves is localized close to the shoreline, and is principally directed shorewards. They also provide easily reproduced test cases against which to validate the numerical methods used in more detailed studies of coastal sediment transport and morphodynamics.

Effects of cross-shore boundary condition errors in nearshore circulation modeling

The paper analyzes the effect which prescribed errors in the cross-shore boundary conditions for a computational domain along a beach have on the flow field predicted inside the domain. This problem is relevant because errors in boundary conditions are unavoidable when modeling limited domains of a nearshore region. For simplicity, we consider a longshore uniform plane beach with monochromatic, obliquely incident waves, and assume depth uniform currents. It is then studied analytically and numerically how small perturbations of the boundary conditions along both upstream and downstream cross-shore boundaries spread inside the computational domain. It is found that the errors at the upstream cross-shore boundary tend to spread over a long distance downstream of the boundary, while the influence of the errors in the downstream boundary condition is limited to the adjacent upstream area of the computational domain. Both the numerical and analytical solutions show that the errors introduced at the upstream boundary decay exponentially in the surf zone at a rate proportional to the bottom friction. A simple formula is developed to estimate the influence distance of the upstream errors. If we consider the mismatch in the volume flux at the upstream boundary, the error merely redistributes in the cross-shore direction to conserve volume. In the case of excessive flux or velocity specified at the cross-shore boundaries, a circulation cell tends to appear in the offshore region where the errors caused by the boundary mismatch increase with the cross-shore width of the model domain.