Reconstruction of ancient sea conditions with an example from the Swiss Molasse

Ancient sea conditions can be estimated from the grain size, spacing and steepness of preserved ripple-marks. The element of greatest uncertainty in such reconstructions is the relationship between near-bed orbital diameter of water particles and the ripple spacing. This relationship is simple for vortex ripples of high steepness but is problematical for the low-steepness forms known as post-vortex, rolling-grain or anorbital ripples.

The existence field for wave ripples is between the threshold velocity for sediment movement and the onset of sheet flow, most low-steepness forms occurring close to the bed planation threshold. A range of maximum period of formative waves can be obtained using combinations of orbital diameter and orbital velocity, assuming linear wave theory to be a reasonable approximation.

Probable wave heights, wave lengths and water depths can be investigated using the transformation of wave parameters in shallowing waters and the constraints on wave dimensions provided by the wave-breaking condition. Given reasonable estimates of wave height, crude estimates of wave power allow a comparison of ancient wave-influenced sequences with modern counterparts.

Wave ripple-marks preserved in the Upper Marine Molasse of western Switzerland have been investigated. Results, which are in agreement with regional geology, suggest deposition in a seaway of approximately 100 km width, where moderate period waves (T = 3–6 s) were generated. The depositional facies belts were adjusted to the prevailing waves, tides and fluvial outflows.     

Wave-formed structures and paleoenvironmental reconstruction

Wave-formed sedimentary structures can be powerful interpretive tools because they reflect not only the velocity and direction of the oscillatory currents, but also the length of the horizontal component of orbital motion and the presence of velocity asymmetry within the flow. Several of these aspects can be related through standard wave theories to combinations of wave dimensions and water depth that have definable natural limits. For a particular grain size, threshold of particle movement and that of conversion from a rippled to flat bed indicate flow-velocity limits. The ratio of ripple spacing to grain size provides an estimate of the length of the near-bottom orbital motion. The degree of velocity asymmetry is related to the asymmetry of the bedforms, though it presently cannot be estimated with confidence. A plot of water depth versus wave height (h—H diagram) provides a convenient approach for showing the combination of wave parameters and water depths capable of generating any particular structure in sand of a given grain size. Natural limits on wave height and inferences or assumptions regarding either water depth or wave period based on geologic evidence allow refinement of the paleoenvironmental reconstruction. The assumptions and the degree of approximation involved in the different techniques impose significant constraints. Inferences based on wave-formed structures are most reliable when they are drawn in the context of other evidence such as the association of sedimentary features or progradational sequences.     

Surf zone characterization for integration with remote sensing data

A numerical model that solves the unsteady, incompressible, Reynolds averaged, Navier–Stokes equations has been utilized to simulate 57 cases of monochromatic, breaking waves over a sloping bed. The Volume of Fluid technique is used to track the complex, discontinuous free surface and the Renormalized Group turbulence model is used for closure. The model is validated by comparing predictions with Particle Image Velocimetry data and other empirical results. The model results are used to determine a relationship between the incipient wave breaking height and the maximum orbital velocity as well as a relationship between surf zone width and breaker type. Such expressions may be useful for remote sensing methods like Synthetic Aperture Radar to derive breaker height and classification from image data.     

Wave parameter gradients along the wave ray

Computer methods have been used to track selected deep-water waves along their orthogonals to the surf zone. Three bottom orbital parameters were computed at each step: diameter, maximum velocity and acceleration. These three parameters were then plotted along the several wave rays, thereby providing shoreward gradients.The three gradients have essentially the same geometry, so that any one can be taken as an indicator of the other two. Bottom slope changes are responsible for departure of the gradients from linearity. The bottom orbital gradients are quite different from gradients of wave height and wave length. These results from coasting waves cannot be applied directly to the case of forced, or wind-driven, waves.The non-linear nature of the bottom orbital gradients indicates that the near-shore bottom should exhibit a crude banding, more-or-less parallel with the coast, with different sediment transport and ripple-mark geometry from band to band.     

WAVECALC: an Excel-VBA spreadsheet to model the characteristics of fully developed waves and their influence on bottom sediments in different water depths

The generation and growth of waves in deep water is controlled by winds blowing over the sea surface. In fully developed sea states, where winds and waves are in equilibrium, wave parameters may be calculated directly from the wind velocity. We provide an Excel spreadsheet to compute the wave period, length, height and celerity, as well as horizontal and vertical particle velocities for any water depth, bottom slope, and distance below the reference water level. The wave profile and propagation can also be visualized for any water depth, modeling the sea surface change from sinusoidal to trochoidal and finally cnoidal profiles into shallow water. Bedload entrainment is estimated under both the wave crest and the trough, using the horizontal water particle velocity at the top of the boundary layer. The calculations are programmed in an Excel file called WAVECALC, which is available online to authorized users. Although many of the recently published formulas are based on theoretical arguments, the values agree well with several existing theories and limited field and laboratory observations. WAVECALC is a user-friendly program intended for sedimentologists, coastal engineers and oceanographers, as well as marine ecologists and biologists. It provides a rapid means to calculate many wave characteristics required in coastal and shallow marine studies, and can also serve as an educational tool.     

Wave damping over artificial Posidonia oceanica meadow: A large-scale experimental study

An experimental study, conducted in the large wave flume of CIEM in Barcelona, is presented to evaluate the effects of Posidonia oceanica meadows on the wave height damping and on the wave induced velocities. The experiments were performed for irregular waves from intermediate to shallow waters with the dispersion parameter h/λ ranging from 0.09 to 0.29. Various configurations of the artificial P. oceanica meadow were tested for two stem density patterns (360 and 180 stems/m²) and for plant's height ranging from 1/3 to 1/2 of the water depth.The results for wave height attenuation are in good agreement with the analytical expressions found in literature, based on the assumption that the energy loss over the vegetated field is due to the drag forces. Based on this hypothesis, an empirical relationship for the drag coefficient related to the Reynolds number, Re, is proposed. The Reynolds number, calculated using the artificial P. oceanica leaf width as the length scale and the maximum orbital velocity over the meadow edge as the characteristic velocity scale, ranges from 1000 to 3500 and the drag coefficient C_d ranges from 0.75 to 2.0.The calculated wave heights, using the analytical expression from literature and the proposed relationship for the estimation of C_d, are in satisfactory agreement with those measured. Wave orbital velocities are shown to be significantly attenuated inside the meadow and just above the flume bed as indicated by the calculation of an attenuation parameter. Near the meadow edge, energy transfer is found in spectral wave velocities from the longer to the shorter wave period components. From the analysis it is shown that the submerged vegetation attenuates mostly longer waves.     

Parameterization and simulation of near bed orbital velocities under irregular waves in shallow water

A set of empirical formulations is derived that describe important wave properties in shallow water as functions of commonly used parameters such as wave height, wave period, local water depth and local bed slope. These wave properties include time varying near-bed orbital velocities and statistical properties such as the distribution of wave height and wave period. Empirical expressions of characteristic wave parameters are derived on the basis of extensive analysis of field data using recently developed evolutionary algorithms. The field data covered a wide range of wave conditions, though there were few conditions with wave periods greater than 15 s. Comparison with field measurements showed good agreement both on a time scale of a single wave period as well as time averaged velocity moments.     

Wave particle velocities measured with a Doppler current meter

A newly developed three-dimensional Doppler current meter is described and the results of preliminary field experiments are presented where simultaneous measurements of surface elevation and water velocity associated with wave orbital motion were made. The phase difference between the surface elevation and the vertical velocity measured at 1.0 and 0.45 meters below the mean water level is found to be approximately 90, in accord with the theory for surface waves of infinitesimally small amplitudes. The spectral (frequency) density distribution for velocity is also found to agree with that we would expect from the linear theory for the observed frequency distribution of surface elevation. However, the amplitude of velocity is consistently smaller (about 10 %) than that we would expect. This reduction of amplitude is more pronounced in cases where waves are high and the water depth is shallow.     

大糙率珊瑚礁附近规则波非线性特征实验研究

通过波浪水槽实验对大糙率礁面存在下珊瑚礁海岸附近规则波非线性特征参数(偏度、不对称度和厄塞尔数)的变化规律进行了研究。实验采用圆柱体阵列来模拟礁面的粗糙度,测试了一系列规则波工况。结果表明:偏度、不对称度和厄塞尔数的幅值分别在珊瑚礁破碎带结束位置、破碎带内和破碎带开始位置达到最大。3个参数的幅值均随着入射波波高的增大而增大;偏度值随着波浪周期的增大而减小,不对称度幅值和厄塞尔数随着周期的增大而增大;偏度值随着礁坪水深的增大而增大,不对称度幅值和厄塞尔数随着礁坪水深的增大而减小。深水厄塞尔数可以用来描述礁坪上波浪非线性参数的变化,最后给出了用其预测礁坪上3个非线性特征参数的经验关系式。     

The distribution of zero-crossing wave heights and periods in a stationary sea state

Existing theoretical distributions of wave height and period do not reflect measured joint distributions from field data. A simulation methodology is introduced to retain the essential features of the theoretical background in Gaussian random noise but to avoid further compromising assumptions in the interpretation of height and period in the amplitude domain. A joint distribution can be associated directly with an empirical or measured variance spectrum. Spectral shape appears to dominate the detail of predicted joint distributions. There is generally a much sharper decay in probability levels at higher periods than is predicted by theoretical models. For Jonswap spectra, there is a dominant central ridge and a distinct bimodal structure in the joint distribution, features that are not evident in symmetric Gaussian spectral forms. The wave height distributions for Jonswap spectra differ little from the Rayleigh distribution, except at extreme wave heights where Rayleigh overpredicts. The period distributions are strongly sensitive to spectral shape. In the conditional distribution of periods, given the height, the asymptotic median period at extreme wave heights is significantly longer than the mean period for Jonswap spectra, but not for symmetric Gaussian forms.     

The wind field over wind-waves

The velocity fluctuations of wind over wind-waves in a wind tunnel are measured with a X-type hot-wire anemometer at some heights over the water surface.The observed vertical profiles of the wave-induced velocity fluctuations and the wave-induced Reynolds stress at the wave spectral peak frequency are different from those expected from the inviscid quasi-laminar model;i.e., the observed vertical profiles of the power spectral density of the wave-induced horizontal or vertical velocity fluctuations of wind have the minimum value at the height much heigher than the critical layer, and the value of the wave-induced Reynolds stress is negative at several heights over the water surface. From the comparison between the experimental results and the numerical solutions of a linear model of the turbulent shear flow over the wavy boundary, it is shown that the discrepancy described above can be attributed to the atmospheric turbulence.     

An analytic solution to the wave bottom boundary layer governing equation under arbitrary wave forcing

Existing models of the wave bottom boundary layer have focused on the vertical and temporal dynamics associated with monochromatic forcing. While these models have made significant advances, they do not address the more complicated dynamics of random wave forcing, commonly found in natural environments such as the surf zone. In the closed form solution presented here, the eddy viscosity is assumed to vary temporally with the bed shear velocity and linearly with depth, however, the solution technique is valid for any eddy viscosity which is separable in time and space. A transformation of the cross-shore velocity to a distorted spatial domain leads to time-independent boundary conditions, allowing for the derivation of an analytic expression for the temporal and vertical structure of the cross-shore velocity under an arbitrary wave field. The model is compared with two independent laboratory observations. Model calculations of the bed shear velocity are in good agreement with laboratory measurements made by Jonsson and Carlsen (1976, J. Hydraul. Res., 14, 45–60). A variety of monochromatic, skewed, and asymmetric wave forcing conditions, characteristic of those found in the surf zone, are used to evaluate the relative effects on the bed shear. Because the temporal variation of the eddy viscosity is assumed proportional to the bottom shear, a weakly nonlinear interaction is created, and a fraction of the input monochromatic wave energy is transferred to the odd harmonics. For a monochromatic input wave, the ratio of the third harmonic of velocity at the bed to the first is <10%. However, for a skewed and asymmetric input wave, this ratio can be as large as 30% and is shown to increase with increasing root-mean-square input wave acceleration. The work done by the fluid on the bed is shown to be a maximum under purely skewed waves and is directed onshore. Under purely asymmetric waves, the work done is significantly smaller and directed offshore.     

On the determination of modal attenuation coefficients andcompressional wave attenuation profiles in a range-dependent environmentin Nantucket Sound

Techniques for determining modal attenuation coefficients and the compressional wave attenuation profile of the bottom in shallow water are presented. The input data consists of spatially well-sampled measurements of the pressure field versus range due to a monochromatic point source, which can be provided by either real or synthetic aperture horizontal arrays. Several methods are described for obtaining modal attenuation coefficients from the pressure field or its Hankel transform. The bottom attenuation profile is related to the modal attenuation coefficients through an integral equation that is solved using linear inverse theory. The methods are demonstrated using both noise-free and noisy synthetic data. The results of inverting experimental data from Nantucket Sound at 140 Hz and 220 Hz are presented. including the resolution and variance estimates of the inferred bottom attenuation profiles. Separation of the contributions of other attenuating mechanisms that can be confused with compressional wave attenuation (shear, rough surface scattering) is also discussed     

The statistical distribution of nearbed wave orbital velocity in intermediate coastal water depth

The statistical distribution of wave orbital velocity in intermediate coastal water depth has been quantitatively determined from the comprehensive field velocity data collected near the seabed in this study. Two ocean ADV current meters, which were mounted at 0.5 m above the seabed on two separate stainless steel tripods sitting on the seabed, were used to measure instantaneous water particle velocities at a 2 Hz sampling rate for 17.07 min every hour in two coastal water depths of 11 m and 23 m in nine field deployments over a period of 2 years. The zero-crossing method is applied to analyse the field velocity data collected in each field deployment to obtain a large sample of wave orbital velocity amplitudes of individual waves. Based on the collected field velocity data, it is found that the histogram of instantaneous wave orbital velocities perfectly follows the Gaussian distribution as commonly assumed, while the histogram of wave orbital velocity amplitudes is less accurately described by the Rayleigh distribution than the modified Rayleigh and the Weibull distribution. It is also found that large orbital velocity amplitudes are generally overestimated by the Rayleigh distribution, but well predicted by the modified Rayleigh and the Weibull distribution. The expected value of maximum orbital velocity in a velocity record of finite size is also derived from the three distributions and found to agree well with the present field data.     

A model for Doppler peak spectral shift for low grazing angle sea scatter

A model is formulated for Doppler spectral characteristics of radar sea scatter for low grazing angles, and is compared with previous radar measurements reported in the literature. The Doppler model is based upon the two-scale model for radar scatter, with scatterer motions hypothesized as due to the orbital wave velocity of the large-scale waves, Stokes and wind drift currents, and the phase velocity of the small-scale Bragg scatterers. Expressions for Doppler shifts due to these motions are derived, and are given as a function of wave height, wave period, and wind speed. Although this model appears to account for the peak Doppler shift of the sea-scatter Doppler spectrum for vertical polarization, it is insufficient to describe horizontally and cross-polarized data, which have larger mean Doppler shifts. However, these two cases are found to scale very closely with the nearly simultaneous vertically polarized data for the variety of environmental conditions reported. Implications of the extension of these results to higher-angle remote-sensing applications are discussed.     

Estimating coastal design extremes for given risks: A new approach

For many reasons, e.g., port operations, coastal construction planning, undersea structure survival, and underwater transport, man wishes to know the extreme values that are likely to occur in coastal oceanographic variables. This paper presents a hybrid statistics/ computer simulation method that uses archived oceanographic observations to estimate confidence levels on the most extreme values likely to occur over a given period in the future. The difference from previously developed methods is the ability to estimate the most extreme value over a time period for a given probability (as opposed to estimating the probability of exceeding a given value) and the ability to obtain results from empirical data without a great deal of theoretical oceanography. The method is applied to the California coast for a period of 100 years on the following variables: bottom surge particle velocity by water depth, wave height by water depth, wavelength by water depth, wave period, current velocity, regions of high density, regions of low density, and earthquake magnitude. Values are given for the 99- and 99.9-percent probability levels.     

Intercomparison of wave-driven current models

Seven numerical models which simulate waves and currents in the surf-zone are tested for the case of a reduced-scale detached breakwater subjected to the action of regular waves with normal incidence. The computed wave heights, water levels and velocities are compared with measurements collected in an experimental wave basin. The wave height decay in the surf-zone is predicted reasonably well. Set-up and currents appear to be less well predicted. This intercomparison exercise shows that radiation stresses are systematically overestimated by formulations used in the models, mean bottom shear stresses are not always co-linear with the mean bottom velocity vector in shallow water, and turbulence modelling in the surf-zone requires a sophisticated     

A New Approach to High-Order Boussinesq-Type Equations with Ambient Currents

A new approach to high-order Boussinesq-type equations with ambient currents is presented. The current velocity is assumed to be uniform over depth and of the same magnitude as the shallow water wave celerity. The wave velocity field is expressed in terms of the horizontal and vertical wave velocity components at an arbitrary water depth level. Linear operators are introduced to improve the accuracy of the kinematic condition at the sea bottom. The dynamic and kinematic conditions at the free surface are expressed in terms of wave velocity variables defined directly on the free surface. The new equations provide high accuracy of linear properties as well as nonlinear properties from shallow to deep water, and extend the applicable range of relative water depth in the case of opposing currents.     

A GIS-based WFD oriented typology of shallow micro-tidal soft bottom using wave exposure and turbidity mapping

Our GIS based project aims at producing a classification scheme to develop a typology of the bottom of the Bay of Gdansk in the southern Baltic. The typology was based on the abiotic factors which are used to define water body types by the European Water Framework Directive (WFD). Significance analysis of particular factors has shown that within the discussed area wave exposure seems to play the most important role. All other factors are to a greater or lesser degree correlated with these two. Taking into consideration the shallows and the varied coastline of the investigated area it was decided to make use of the SWAN numerical wave model to determine the influence of wave impact upon the bottom. The model was used to produce raster maps of orbital velocity near the bottom for each wind scenario. With the help of the GIS analysis the maps were turned into layers: the mean velocity and the maximum velocity at the bottom. To produce the layer of yearly amount of solar radiation a GIS model was built which main parameters were the layer of depth and three layers of turbidity for three seasons. The layers of the maximum orbital velocity and of the solar radiation at the bottom were then used in a classificatory procedure consisted in an iterative sequence of the three following steps: cluster formation, dendrogram analysis and classification using the maximum likelihood method. Ecological importance of the classification has been obtained by means of the aggregation of a part of classes based upon the statistics calculated for them within the GIS system with the help of the zonal function out of the following parameters: salinity, depth, mean and maximum orbital velocity at the bottom, temperature differences between warm and cold seasons, solar radiation, and type of sediments. The method proposed here makes it possible to produce high resolution thematic maps of the bottom even with incomplete data cover of the investigated area.