Detection of nonlinear waves and their contribution to ocean wave spectra Part II: Observation

This is a Part II of a paper of nonlinearities of wind waves in the deep open ocean. As shown in Part I, bound waves in deep sea are detectable by extracting secondorder Doppler spectra from the Doppler spectra of HF (high-frequency) radio waves scattered from the sea surface. There is a remarkable agreement between the calculated and measured Doppler spectra, considering the noise levels in measured Doppler spectra and the uncertainties in directional properties. The theoretical expression for bound waves is thus verified. Furthermore, the upper limit in calculating the Doppler spectra for the second-order approximation is presented from field observations, although we cannot conclude that it is equivalent to the limitation of the second-order bound wave theory. It is shown that analysis of radio wave scattering by the sea surface is one useful means of understanding the nonlinear properties of ocean waves.     

Detection of nonlinear waves and their contribution to ocean wave spectra Part I: Theoretical consideration

This is a Part I of a paper of nonlinearities of wind waves in the deep open ocean. First, considerations are given in order to verify the theoretical expression for bound waves from observed data. We compare the contribution of bound waves and double Bragg scattering to the second-order scattering, and we show that the contribution of bound waves is larger, and that bound waves can be detected by measuring the Doppler spectra of HF (high-frequency) radio wave scattering from the sea surface. Moreover, if the theory of the HF radio wave scattering from the sea surface is verified, so is the second-order perturbation theory for bound waves. Then, the contributions of bound waves to ocean wave spectra are investigated on the basis of the nonlinear theory. The bound waves are shown to modify frequency spectra and wave directional distributions at higher frequencies, and it is shown that although the modifications of frequency spectra are smaller for a two-dimensional field case than for a one-dimensional field case, they are not negligible at higher frequencies. On the other hand, the modifications of wave directional distributions are shown to be significant at higher frequencies. These discussions become significant only when bound wave predictions are verified in the open ocean. Consequently, it is shown that nonlinearities of water waves are important in considering both radio wave scattering from the sea surface and the detailed structures of ocean wave spectra at high frequencies.     

Tidal Current Measurements Using VHF Radar and ADCP in the Normand Breton Gulf: Comparison of Observations and Numerical Model

Performance and operational feasibility of very high-frequency (VHF) Doppler radar have been demonstrated in a region dominated by strong tidal currents. An analysis of remote measurements of sea surface currents acquired by Courants de Surface MEsureacutes par Radar (COSMER)-pulsed Doppler radar during Evaluation et Preacutevision de l'Environnement Littoral (EPEL) experiment (supported by the French Navy) is presented in this paper. The VHF COSMER radar was deployed to provide continuous sea surface current measurements within an area of about 25 km times 25 km in the Normand Breton Gulf, France. This paper presents VHF measurement comparisons with observations such as acoustic Doppler current profiler (ADCP), as well as comparisons with numerical model TELEMAC 2-D. Results of tidal waves extraction, using harmonic analysis and residual currents, are shown in this paper. We also present a case where radar method is limited, due to the presence of additional peaks in the Doppler spectrum     

Random wave-current interactions in water of varying depth

Refraction of incoherent random gravity waves with currents and bottom topography results in spatial variations in the spectral characteristics of the free surface. Prediction of such variations based on the radiation transfer equation is in a simple analytic form for the case of one dimensional inhomogeneities in currents and topography. This analytic form is examined in terms of two-dimensional wave number- and polar frequency-direction spectra along the associated dynamic and kinematic constraints relevant to wave breaking and reflection. Results are specialized to the simplest case of horizontal shear currents in deep and shallow water with explicit examples to illustrate the relative and combined effects of currents and topography on free surface spectra.     

Higher Harmonics Induced by Waves Propagating over A Submerged Obstacle in the Presence of Uniform Current

To investigate higher harmonics induced by a submerged obstacle in the presence of uniform current, a 2D fully nonlinear numerical wave flume（NWF） is developed by use of a time-domain higher-order boundary element method（HOBEM） based on potential flow theory. A four-point method is developed to decompose higher bound and free harmonic waves propagating upstream and downstream around the obstacle. The model predictions are in good agreement with the experimental data for free harmonics induced by a submerged horizontal cylinder in the absence of currents. This serves as a benchmark to reveal the current effects on higher harmonic waves. The peak value of non-dimensional second free harmonic amplitude is shifted upstream for the opposing current relative to that for zero current with the variation of current-free incident wave amplitude, and it is vice versa for the following current. The second-order analysis shows a resonant behavior which is related to the ratio of the cylinder diameter to the second bound mode wavelength over the cylinder. The second-order resonant position slightly downshifted for the opposing current and upshifted for the following current.     

Spatial and temporal variations in spectra of storm waves across a barred nearshore

Wave staffs and electromagnetic current meters were deployed on a profile across a two-bar system at Wendake Beach, southern Georgian Bay. This paper examines spatial and temporal changes in the characteristics of wave form, and the spectra of surface elevation and on-offshore current motion, during one storm. Non-linear effects of wave shoaling and breaking across the bars result in the appearance of secondary waves and both the wave and on-offshore current spectra have significant harmonic peaks during most of the storm. Significant low-frequency energy occurs only during the peak of the storm. While the peak frequency remains constant across the bar system, the proportion of energy in the primary peak is greatest in the troughs and lowest over the bar crests and there are similar changes in the proportion of energy in the first harmonic. However, in both surface elevation spectra and on-offshore current spectra, the greatest proportion of energy is found in frequencies related to the incident wind waves.     

Short-wave measurements by a fixed tower-based and a drifting buoysystem

The small-scale roughness of the sea surface acts as an important link in air-sea interaction processes. Radar and sonar waves are scattered by short surface waves providing the basis for remote sensing methods of the sea surface. At high wind speeds, breaking waves occur. Bubbles penetrate into the water and drastically increase acoustical reverberation, transmission loss and ambient noise. Thus, the development of short waves and wave breaking have to be known to apply radar remote sensing to the surface and to deduce from radar backscatter which sonar conditions prevail. To measure the wind dependence of short waves an experimental device was constructed for use from stationary platforms. It is nearly all-weather capable and can easily be handled by a crane. On the other hand, frequencies of short waves measured in a fixed position are extremely frequency shifted by currents. This limits the usefulness of tower-based measurements, e.g., the short wave modulation by wind and waves or currents can only be estimated in a rough approximation. Consequently, a buoy was developed to reduce the frequency shifts. The principle of the buoy is to drift in the local surface current and to follow the amplitudes of long waves. Therefore, short waves are measured in facets of long waves and the Doppler shifts are minimized. The wind is measured at a constant height above the long wave profile and relative to the moving facets. The paper describes the conventional measuring device and points out the necessity of the drifting buoy system. Examples of wind and wave spectra are presented and short wave modulations by long waves are depicted, too. From these measurements, new insights in short wave behaviour have to be expected     

Propagation of shallow water waves in an open parabolic channel using the WKB perturbation technique

A singular perturbation analysis based on the WKB technique to study the hydrodynamic performance of periodic ocean waves that are incident on an open parabolic channel of constant depth is proposed. We derive a linear model to predict the propagation of the long ocean waves into the channel. In this manner, the spatial distribution for the surface elevation of the ocean waves inside the channel as a function of two dimensionless parameters, namely, a kinematical parameter, κ and a geometrical parameter ε, is governed by a second-order ordinary differential equation. The kinematical parameter κ denotes the ratio of the potential head, due to gravity, to the kinetic head of the ocean waves along the longitudinal axis of the parabolic channel. Meanwhile, ε is a dimensionless geometrical parameter that represents a characteristic ratio of the parabolic channel. Using matching conditions, simple expressions for the reflection and transmission coefficients are obtained.     

X-band low-grazing-angle ocean backscatter obtained during LOGAN1993

This paper reviews X-band ocean microwave backscatter data from the LOGAN (LOw Grazing ANgle) experiment conducted on the Chesapeake Light Tower by the Naval Air Warfare Center. The data were collected under varied wind, sea, and swell conditions that provide some new insights into low-grazing-angle backscatter phenomena. Transient backscatter peaks called “sea spikes” have long been associated with deep-water breaking waves; however, they have yet to be fully reconciled with backscatter and hydrodynamic theories. New analysis techniques have been applied to the LOGAN data that take advantage of the unique characteristics of sea spikes and their dynamics. High-resolution Doppler spectra are organized relative to the space-time centroids of the sea-spike clusters and conditionally averaged by RCS strength. The mean Doppler variation of the strongest sea spikes then map the breaking-wave structure just as Doppler histories measured at moderate grazing angles map the dynamics of the dominant linear surface-wave components. While breaking waves are manifest to some degree in backscatter data at all grazing angles, a non-Bragg-scatter mechanism accentuates the crest scattering at low grazing angles. The phenomena potentially can be exploited for remote ocean sensing and imaging     

Radar scattering on gravity-capillary waves: Laboratory investigation

The mechanism of the resonance (Bragg) scattering of microwaves by gravity-capillary waves (GCWs) is checked experimentally. Resonant regularities of backscattered signal intensity are obtained as functions of frequency and GCW propagation direction. It is shown that the width of the resonance curves is determined by the width of the directivity pattern of the scatterometer’s antenna. The excitation of the GCW second harmonic and the spatial structure of the wave field at the GCW doubled frequency are investigated. The ratio of the amplitudes of the free wave and forced harmonic that originated during excitation of the primary wave is determined. The resonance curve is obtained for the second-order scattering of radio waves (on forced harmonics and free GCWs). The correction to the backscattering cross section is investigated in the second order of smallness relative to the Bragg term.     

Microwave scattering by surface waves on water

The electromagnetic backscattering from small-amplitude water waves has been studied using a microwave model (2.5 GHz). The first- and second-order backscatter Doppler spectra have been measured and compared with the theoretical models, and good agreement has been found. In this paper, the specific case of two water waves propagating at right angles to each other will be presented, and the level of contribution of both the hydrodynamic and the electromagnetic effects on the backscatter radar Doppler spectrum will be considered.     

A Calculation Method for Incidence and Reflection of Irregular Waves Along Seawalls

Some researches have been made in this aspect.In the method by Walton Jr.(1992),incidentwaves are supposed to be the overlapping result of M component waves with different frequencies whichmay take different directions,the direction of incident waves should be available in advance,but in fact thedirection of incident waves is not available.In our study,incident waves are supposed to be composed ofM overlapping component waves with different frequencies,and different frequencies have different direc-tions.Based on the irregular wave reflection theory,the calculation formulas of wave direction,complexamplitude of incident waves,and complex amplitude of reflected waves in surface which are composed ofcomponent waves are derived by means of discrete Fourier transform.Then,the frequency spectra of inci-dent waves and reflected waves and the reflection coefficient of waves with corresponding frequencies areobtained.Verification of the method and the calculation results from in-situ measured data indicate thatt     

Singularity and Breaking of the Third Order Stokes Waves

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The effects of reflection coefficient of the harbour sidewall on the performance of floating breakwaters

This paper studies the interaction of linear water waves with a moored floating breakwater with a leeward boundary composed from a vertical wall. This describes a real modelling for the case of ports in contrary to the problems of unbounded domains. It involves the application of a partial reflection boundary condition for this sidewall. In fact, the partial reflection problem in practical application is of great importance in the design of a harbour or breakwater and mainly for short waves. The reflection coefficient of the harbour boundary (sidewall) plays an important role in modifying the performance of the floating breakwater. Moreover, it reduces the resonant peaks appearing inside the ports due to the energy accumulation in an enclosed domain. The effects of the variation of structural parameters of the breakwater on the transmitted wave height are discussed in details for various values of partial reflection.     

The structure of ocean-surface currents measured by Doppler radar

The spatial structure of surface currents collected using a Doppler Radar system off the Florida Keys has been investigated. Surface current vectors were acquired within a rectilinear grid containing 700 nodes, where each node was spaced 1 km apart. Principal component analyses indicate that at least 63% of the variance of the surface currents at the 700 nodal locations can be accounted for by only three principal components. The principal modes of circulation during two separate experiments were extracted and found to be basically the same, although the first two modes were reversed between the two experiments. Coherence spectra were determined for nodal spacings between 1.76 km and 10.48 km, and the results of these analyses indicate that over most of the experimental area, approximately 60% of the variance is coherent over separations of 10.48 km. Application of a synoptic time-series model indicates that accurate prediction of the mean currents over each of the 65-h subrecord lengths was the dominant factor in controlling model performance, and that on the order of 10% error could be expected in using the time-series model to predict the low-frequency fluctuations. This indicates that time-series modeling of surface currents may be feasible and useful in estimating the long-term mixing characteristics of contaminants transported in the surface layer     

Wave characteristics past a flexible fishnet

The scattering of surface waves by a flexible fishnet is studied analytically. The fishnet is modelled as a porous flexible barrier displaced solely by hydrodynamic force like a catenary. The objective is to investigate how a flexible permeable barrier affects the passing waves in the way they are transmitted and reflected, as observed by the fact that the water inside a fishfarm surrounded by fishnets is significantly calmer than that outside. The boundary value problems are solved by defining the reflection coefficient in terms of velocity potential and then the full solutions are obtained by suitable application of the eigenfunction expansion method and the least squares approximation method. The variations of the reflection coefficient, hydrodynamic pressure, barrier deformation and surface wave elevation are determined with respect to the barrier length, porosity and stiffness. It is observed that as the fishnet gets more flexible, its deformation increases and the reflection coefficient decreases, whereas as the fishnet gets more porous, more water can pass through it and thus the reflection coefficient, barrier deformation and the hydrodynamic force are reduced. The flexibility of the barrier behaves like its porosity by allowing more wave energy to act on it through its deformation and hence reduce the reflection and hydrodynamic force of the incident waves acting on the barrier.     

The statistics of HF sea-echo Doppler spectra

Several important statistical properties of the HF sea echo and its Doppler power spectrum, which are useful in optimizing the design of radar oceanographic experiments, are established. First- and second-order theories show that the echo signal (e.g., the voltage) should be Gaussian; this is confirmed with experimental surface-wave data i) by comparison of the normalized standard deviation of the power spectrum at a given frequency with its predicted value of unity, and ii) by cumulative distribution plots of measured spectral amplitudes on Rayleigh probability charts. The normalized standard deviation of the dominant absolute peak amplitudes of the power spectrum (which wander slightly in frequency) are shown from experimental data to besim 0.7for the first-order peaks andsim 0.5for the second-order peaks. The autocorrelation coefficient of the power spectra is derived from measured data and interpreted in terms of the spectral peak widths; from this information, the correlation time (or time between independent power spectrum samples) iS shown to besim 25-50s for radar frequencies above 7 MHz. All of these statistical quantities are observed to be independent of sea state, scattering cell size, and relatively independent of radar operating frequency. These quantities are then used to establish the statistical error (and confidence interval) for radar remote sensing of sea state, and it is shown, for example, that 14 power spectral samples result in a sample average whose rms error about the true mean is 1.0 dB.     

An experimental study of wave scattering by a vertical slotted barrier in the presence of a current

In coastal waters, tidal currents and surface waves co-exist. In this study, the influence of a steady current on the scattering of waves by a vertical slotted barrier is investigated experimentally in a wave flume. The separation of the incident and reflected waves is carried out by a two-point method that takes into account the effects of the current. Results show that currents significantly increase the wave-energy loss by the barrier and remarkably reduce the wave transmission through the barrier. It is suggested that the tidal currents should be taken into consideration in an economical design of slotted breakwaters.     

An analysis of the second-order Doppler return from the ocean surface

A theoretical analysis of the scattering of high-frequency (HF) electromagnetic waves from rough surfaces is proposed. The analysis, when applied to a model for the ocean surface, leads to new predictions of its second-order backscattered radar cross section in addition to those provided by existing theories. Some of the predictions of the theory have already been verified experimentally.     

Peculiarities of backscattering in multifrequency nadir probing of the sea surface

Backscattering of centimeter electromagnetic waves during nadir probing of the sea surface is considered in an approximation of the Kirchoff method and a two-scale model of the scattering surface. A model of the effective reflection coefficient is developed under the assumption of the specified roughness spectrum model, and the coefficients for Ku (0.021 m), C (0.055 m), and S (0.09 m) bands are calculated. The dependence of the backscattering cross section on the wind speed, wind fetch, and the height of the swell are numerically investigated. The proximity of the numerical estimates and the experimental data is demonstrated. The dependence of the difference of two cross sections on the wind speed or on one of the backscattering cross sections is analyzed in dual-frequency measurements. It is shown that the numerical model made it possible to describe the nonmonotonic behavior of the difference cross section for the Ku-C and Ku-S pairs for the first time. The developed effective reflection coefficient model made it possible to predict in the numerical experiment the behavior of the difference cross section of a new frequency pair (C and S) for which in situ measurements have not yet been performed.