Wavelet-based vortical structure detection and length scale estimate for laboratory spilling waves

Turbulent flow fields under spilling breaking waves are measured by particle image velocimetry and analyzed using the wavelet techniques in a laboratory surf zone. The turbulent vortical structures and corresponding length scales in the flow are detected through the eduction of the most excited mode with local intermittency measure that is found to correlate with the passage of the structure. Distributions and evolution of the educed vortical structures are presented and discussed. Packets of vortical structures with high intermittency is observed to stretch downward below the initially low-intermittency trough level, indicating these structures play a crucial role in turbulent mixing below the trough level. It is found that the probability density functions of the intermittent energy of the educed structures, vorticity and swirl strength display an exponential decay. Ensemble-averaged length scales of the educed vortical structures are found to be about 0.1 to 0.2 times the local water depth, close to the turbulent mixing length reported in the surf zone. The Kolmogorov microscale is evaluated and the turbulent mixing length is estimated using the k − ε relation and mixing length hypothesis. The k − ε relation may overestimate the mixing length scale for energetic descending eddies.     

Laboratory observation of boundary layer flow under spilling breakers in surf zone using particle image velocimetry

A boundary layer flow under spilling breakers in a laboratory surf zone with a smooth bottom is investigated using a high resolution particle image velocimetry (PIV) technique. By cross-correlating the images, oscillatory velocity profiles within a viscous boundary layer of O(1) mm in thickness are resolved over ten points. Using PIV measurements taken for an earlier study and the present study, flow properties in the wave bottom boundary layer (WBBL) over the laboratory surf zone are obtained, including the mean velocities, turbulence intensity, Reynolds stresses, and intermittency of coherent events. The data are then used to estimate the boundary layer thickness, phase variation, and bottom shear stress. It is found that while the time averaged mass transport inside the WBBL is onshore in the outer surf zone, it changes to offshore in the inner surf zone. The zero Eulerian mass transport occurs at h/h_b ≈ 0.92 in the outer surf zone. The maximum overshoot of the streamwise velocity and boundary layer thickness are not constant across the surf zone. The bottom shear stress is mainly contributed by the viscous stress through mean velocity gradient while the Reynolds stress is small and negligible. The turbulence level is higher in the inner surf zone than that in the outer surf zone, although only a slight increase of turbulent intensity is observed inside the WBBL from the outer surf zone to the inner surf zone. The variation of phase inside and outside the WBBL was examined through the spatial velocity distribution. It is found the phase lead is not constant and its value is significantly smaller than previous thought. By analyzing instantaneous velocity and vorticity fields, a remarkable number of intermittent turbulent eddies are observed to penetrate into the WBBL in the inner surf zone. The size of the observed large eddies is about 0.11 to 0.16 times the local water depth. Its energy spectra follow the − 5/3 slope in the inertial subrange and decay exponentially in the dissipation subrange.     

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).     

Large-wave simulation of three-dimensional,cross-shore and oblique,spilling breaking on constant slope beach

In this work, the large-wave simulation (LWS) method is adapted for application in spilling wave breaking over a constant slope beach. According to LWS, large scales of velocities, pressure and free-surface elevation are numerically resolved, while the corresponding unresolved scale effects are taken into consideration by a subgrid scale (SGS) model for wave and eddy stresses. The model may be not fully applicable in very shallow water, close to the shoreline, where the unresolved, turbulent, free-surface oscillation is of the same order with the water depth. Time integration of the Euler equations is achieved by a two-stage fractional scheme, combined with a hybrid scheme for spatial discretization, consisting of finite difference and pseudospectral approximation methods. Model parameters are calibrated by comparison to available experimental data of free-surface elevation and velocities in the surf zone for cross-shore incoming waves. The action of the wave SGS stresses in the outer coastal and surf zones initiates breaking and generates appropriate vorticity, in the form of an eddy structure (surface roller), at the breaking wavefront. At incipient breaking, both advection and gravity contribute to the vorticity flux at the free surface, while only after the full development of the surface roller, the effect of advection becomes stronger. The SGS model is also utilized to simulate propagation, refraction and breaking of oblique incoming waves. The gradual breaking and dissipation of wave crestlines and the surface roller structure along the breaking wavefront are automatically captured without any empirical input, such as data for the roller shape or the wave propagation angle at breaking.     

Improved representation of breaking wave energy dissipation in parametric wave transformation models

An improved formulation to describe breaking wave energy dissipation is presented and incorporated into a previous parametric cross-shore wave transformation model [Baldock, T.E., Holmes, P., Bunker, S., Van Weert, P., 1998. Cross-shore hydrodynamics within an unsaturated surf zone. Coastal Engineering 34, 173–196]. The new formulation accounts for a term in the bore dissipation equation neglected in some previous modelling, but which is shown to be important in the inner surf zone. The only free model parameter remains the choice of γ, the ratio of wave height to water depth at initial breaking, and a well-established standard parameter is used for all model runs. The proposed model is compared to three sets of experimental data and a previous version of the model which was extensively calibrated against field and laboratory data. The model is also compared to the widely used model presented by Thornton and Guza (1983) [Thornton, E.B., Guza, R.T., 1983. Transformation of wave height distribution. Journal of Geophysical Research 88 (No.C10), 5925–5938].     

A note on wave energy dissipation over steep beaches

This paper revisits the derivation of the parametric surf zone model proposed by Baldock et al. [Baldock, T. E., Holmes, P., Bunker, S. & Van Weert, P. 1998 Cross-shore hydrodynamics within an unsaturated surf zone. Coast. Eng. 34, 173–196.]. We show that a consistent use of the proposed Rayleigh distribution for surf zone wave heights results in modification of the expressions for the bulk dissipation rate and enhanced dissipation levels on steep beaches and over-saturated surf zone conditions. As a consequence, the modification proposed herein renders the model robust even on steep beaches where it could otherwise develop a shoreline singularity.     

Calibration and modification of energy dissipation models for irregular wave breaking

This study is undertaken to recalibrate eight existing energy dissipation models and find out the suitable models, which can be used to compute H_rms for a wide range of experimental conditions. The examination shows that the coefficients in the existing models are not the optimal values for a wide range of experimental conditions. Using the new calibrated coefficients, all existing models can be used for computing H_rms and the model of Battjes, J.A., Stive, M.J.F. [1985. Calibration and verification of a dissipation model for random breaking waves. Journal of Geophysical Research 90 (C5), 9159–9167] gives the best predictions. The existing models are also modified by changing the breaker height formulas in the dissipation models. The accuracy of most existing models is improved significantly by using the suitable breaker height formula.     

The effect of groundwater on topographic changes in a gravel beach

In recent years, several attempts to stabilize the beach by control of the percolation of water have been proposed. However, morphodynamics in the surf zone is still not clear because of the complexity of wave actions and sediment transport. Especially, there is a little research on gravel beach morphodynamics including wave breaking in the surf zone. The present study investigates experimentally how groundwater level influences topographic changes in a gravel beach and simulates numerically the wave fields and flow patterns in the surf zone, considering the porosity of the media and the presence of groundwater. In experiments, water-level control tank was designed to control the simulated groundwater elevation and the wave flume was divided into two parts to maintain a constant mean water level. The experimental results show that the berm formed in the upper portion of the shoreline moves up the beach as the groundwater level falls and the lower the groundwater level, the steeper the beach surface. The numerical model was developed to clarify these features capable of simulating the difference of groundwater and mean water level. Numerical results showed different flow patterns due to the groundwater elevation; wave run-up weakens and wave run-down strengthens by the seaward currents caused by elevated groundwater. These deformations of the flow pattern explain well how the beach profile is affected by the groundwater elevation.     

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.     

Calculations of wave transformation across the surf zone

The accuracy of predicting wave transformation in the nearshore is very important to wave hydrodynamics, sediment transport and design of coastal structures. An efficient numerical model based on the time-dependent mild-slope equation is presented in this paper for the estimation of wave deformation across the surf zone. This model incorporates an approximate nonlinear shoaling formula and an energy dissipation factor due to wave breaking to improve the accuracy of the calculation of wave height deformation prior to wave breaking and also in the surf zone. The model also computes the location of first wave breaking, wave recovery and second wave breaking, if physical condition permits. Good agreement is found upon comparison with experimental data over several one-dimensional beach profiles, including uniform slope, bar and step profiles.     

Laboratory measurements of large-scale near-bed turbulent flow structures under spilling regular waves

The instantaneous turbulent velocity field created by the breaking of spilling regular waves on a plane slope was measured in a plane running parallel to the slope using particle image velocimetry. The measurement plane was located at a height of about 1 mm above the bed. The measurement area encompassed the region where the large eddies generated at incipient wave breaking impinged on the bottom inside the surf zone. A total of 30 trials were conducted under identical experimental conditions. In each trial, six consecutive wave cycles were recorded. The measured velocity fields were separated into a mean flow and a turbulence component by ensemble averaging. The instantaneous turbulent velocity fields were analyzed to determine the occurrence frequency, location, geometry and evolution of the large eddies, and their contributions to instantaneous shear stresses, turbulent kinetic energy and turbulence energy fluxes. The motion of single glass spheres along the bed was also investigated. The two-phase flow measurements showed that the velocity and displacement of large solid particles on a smooth bed were significantly affected by the magnitude and direction of turbulence velocities. Overall, this study has examined the kinematic and dynamic properties of large eddies impinging on the bed and the interaction of these large-scale turbulent flow structures with the mean flow. The study has also highlighted the important role of large eddies in sediment transport.     

Modeling of the Turbulence in the Water Column under Breaking Wind Waves

Past studies have shown that there is a wave-enhanced, near-surface mixed-layer in which the dissipation rate is greater than that derived from the “law of the wall”. In this study, turbulence in water columns under wind breaking waves is investigated numerically and analytically. Improved estimations of dissipation rate are parameterized as surface source of turbulent kinetic energy (TKE) for a more accurate modelling of vertical profile of velocity and TKE in the water column. The simulation results have been compared with the experimental results obtained by Cheung and Street (1988) and Kitaigorodskii et al. (1983), with good agreement. The results show that the numerical full model can well simulate the near-surface wave-enhanced layer and suggest that the vertical diffusive coefficients are highly empirical and related to the TKE diffusion, the shear production and the dissipation. Analytical solutions of TKE are also derived for near surface layer and in deep water respectively. Near the surface layer, the dissipation rate is assumed to be balanced by the TKE diffusion to obtain the analytical solution; however, the balance between the dissipation and the shear production is applied at the deep layer. The analytical results in various layers are compared with that of the full numerical model, which confirms that the wave-enhanced layer near the surface is a diffusion-dominated region. The influence of the wave energy factor is also examined, which increases the surface TKE flux with the wave development. Under this region, the water behavior transits to satisfy the classic law of the wall. Below the transition depth, the shear production dominantly balances the dissipation. This revised version was published online in August 2006 with corrections to the Cover Date.     

Ocean wave transmission by submerged reef—A physical model study

Ocean waves can be destructive as steeper waves due to their high energy eroding the sandy beaches. During storm surge or high tide, the water level rises and if large waves occur, they will break closer to the beach, releasing enormous amount of energy resulting in strong currents. This causes heavy loss of beach material due to large-scale erosion. If these waves are made to break prematurely and away from the beach, they can be attenuated so as to reduce beach erosion. The reef, which is a homogeneous pile of armour units without a core, breaks the steeper ocean waves, dissipates a major portion of their energy and transmits attenuated waves. This paper experimentally investigates the armour stone stability of the submerged reef and the influence of its varying distance from shore and crest width on ocean wave transmission.     

Cross-shore variations of wave groupiness by wavelet transform

This paper proposes a new definition of groupiness factor (GF) based on the local wavelet energy density of the wave time series to describe the groupiness degree of waves. The main advantage of this new GF is that the effect of the operational definition on it is smaller than that on SIWEH-based GF or envelop-based GF. Then, the new GF is used to study the groupiness variations of mechanically generated irregular waves in a wave flume propagating on a slope of 1:45. The results of present study show that the decrease of groupiness in the coast is triggered by breaking. And energy distribution along the record time for the first harmonics of waves in the surf zone, which becomes more uniform than that out the surf zone, is the main reason causing the decrease of groupiness.     

Large-scale experiments on beach profile evolution and surf and swash zone sediment transport induced by long waves,wave groups and random waves

New large-scale laboratory data are presented on the influence of long waves, bichromatic wave groups and random waves on sediment transport in the surf and swash zones. Physical model testing was performed in the large-scale CIEM wave flume at UPC, Barcelona, as part of the SUSCO (swash zone response under grouping storm conditions) experiment in the Hydralab III program (Vicinanza et al., 2010). Fourteen different wave conditions were used, encompassing monochromatic waves, bichromatic wave groups and random waves. The experiments were designed specifically to compare variations in beach profile evolution between monochromatic waves and unsteady waves with the same mean energy flux. Each test commenced with approximately the same initial profile. The monochromatic conditions were perturbed with free long waves, and then subsequently substituted with bichromatic wave groups with different bandwidth and with random waves with varying groupiness. Beach profile measurements were made at half-hourly and hourly intervals, from which net cross-shore transport rates were calculated for the different wave conditions. Pairs of experiments with slightly different bandwidth or wave grouping show very similar net cross-shore sediment transport patterns, giving high confidence to the data set. Consistent with recent small-scale experiments, the data clearly show that in comparison to monochromatic conditions the bichromatic wave groups reduce onshore transport during accretive conditions and increase offshore transport during erosive conditions. The random waves have a similar influence to the bichromatic wave groups, promoting offshore transport, in comparison to the monochromatic conditions. The data also indicate that the free long waves promote onshore transport, but the conclusions are more tentative as a result of a few errors in the test schedule and modifications to the setup which reduced testing time. The experiments suggest that the inclusion of long wave and wave group sediment transport is important for improved near-shore morphological modeling of cross-shore beach profile evolution, and they provide a very comprehensive and controlled series of tests for evaluating numerical models. It is suggested that the large change in the beach response between monochromatic conditions and wave group conditions is a result of the increased significant and maximum wave heights in the wave groups, as much as the presence of the forced and free long waves induced by the groupiness. The equilibrium state model concept can provide a heuristic explanation of the influence of the wave groups on the bulk beach profile response if their effective relative fall velocity is larger than that of monochromatic waves with the same incident energy flux.     

Vorticity and intermittency within the pre-breaking region of spilling breakers

This paper presents measurements and analysis of fluid velocity within the context of spilling waves. The data have been collected using 2-D Laser Doppler Velocimetry in pre-breaking monochromatic waves generated in a wave tank. The analysis is performed using orthogonal wavelets and, in addition to the classical criterion adopted in applying Taylor's hypothesis, a new algorithm is proposed for the eduction of eddies at different length scales. The contribution of different scale vortices is computed, and phase is resolved. Microvortices (smaller than the breaker height but larger than the dissipative vortices) and mid-size vortices (with length ranging from the breaker height to the wave length) carry out most turbulence energy under wave crest. The phase average vorticity and strain rate is computed at different wave lengths, with the analysis of intermittence. The intermittency factor shows spikes in the wave crest, especially for turbulence in small vortices.     

Laboratory study of wave and turbulence velocities in a broad-banded irregular wave surf zone

The characteristics of wave and turbulence velocities created by a broad-banded irregular wave train breaking on a 1:35 slope were studied in a laboratory wave flume. Water particle velocities were measured simultaneously with wave elevations at three cross-shore locations inside the surf zone. The measured data were separated into low-frequency and high-frequency time series using a Fourier filter. The measured velocities were further separated into organized wave-induced velocities and turbulent velocity fluctuations by ensemble averaging. The broad-banded irregular waves created a wide surf zone that was dominated by spilling type breakers. A wave-by-wave analysis was carried out to obtain the probability distributions of individual wave heights, wave periods, peak wave velocities, and wave-averaged turbulent kinetic energies and Reynolds stresses. The results showed that there was a consistent increase in the kurtosis of the vertical velocity distribution from the surface to the bottom. The abnormally large downward velocities were produced by plunging breakers that occurred from time to time. It was found that the mean of the highest one-third wave-averaged turbulent kinetic energy values in the irregular waves was about the same as the time-averaged turbulent kinetic energy in a regular wave with similar deep-water wave height to wavelength ratio. It was also found that the correlation coefficient of the Reynolds stress varied strongly with turbulence intensity. Good correlation between u′ and w′ was obtained when the turbulence intensity was high; the correlation coefficient was about 0.3–0.5. The Reynolds stress correlation coefficient decreased over a wave cycle, and with distance from the water surface. Under the irregular breaking waves, turbulent kinetic energy was transported downward and landward by turbulent velocity fluctuations and wave velocities, and upward and seaward by the undertow. The undertow in the irregular waves was similar in vertical structure but lower in magnitude than in regular waves, and the horizontal velocity profiles under the low-frequency waves were approximately uniform.     

Laboratory study of wave and turbulence characteristics in narrow-band irregular breaking waves

Wave elevations and water particle velocities were measured in a laboratory surf zone created by the breaking of a narrow-band irregular wave train on a 1/35 plane slope. The incident waves form wave groups that are strongly modulated. It is found that the waves that break close to the shoreline generally have larger wave-height-to-water-depth ratios before breaking than the waves that break farther offshore. After breaking, the wave-height-to-water-depth ratio for the individual waves approaches a constant value in the inner surf zone, while the standard deviation of the wave period increases as the still water depth decreases. In the outer surf zone, the distribution of the period-averaged turbulent kinetic energy is closely correlated to the initial wave heights, and has a wider variation for narrow-band waves than for broad-band waves. In the inner surf zone, the distribution of the period-averaged turbulent kinetic energy is similar for narrow-band waves and broad-band waves. It is found that the wave elevation and turbulent kinetic energy time histories for the individual waves in a wave group are qualitatively similar to those found in a spilling regular wave. The time-averaged transport of turbulent kinetic energy by the ensemble-averaged velocity and turbulence velocity under the irregular breaking waves are also consistent with the measurements obtained in regular breaking waves. The experimental results indicate that the shape of the incident wave spectrum has a significant effect on the temporal and spatial variability of wave breaking and the distribution of turbulent kinetic energy in the outer surf zone. In the inner surf zone, however, the distribution of turbulent kinetic energy is relatively insensitive to the shape of the incident wave spectrum, and the important parameters are the significant wave height and period of the incident waves, and the beach slope.