Showing papers on "Breaking wave published in 1981"

On the detectability of ocean surface waves by real and synthetic aperture radar

Abstract: Real and synthetic aperture radars have been used in recent years to image ocean surface waves. Though wavelike patterns are often discernible on radar images, it is still not fully understood how they relate to the actual wave field. The present paper reviews and extends current models on the imaging mechanism. Linear transfer functions that relate the two-dimensional wave field to the real aperture radar (SLAR) image are calculated by using the two-scale wave model. It is noted that a description of the imaging process by these transfer functions can only be adequate for low to moderate sea states. Possible other mechanisms that contribute to the visibility of waves by real aperture radar at higher sea states, such as Bragg scattering from spontaneously generated short waves at peaked crests or in wave breaking regions, and Rayleigh scattering from air bubbles entrained in the water and from water droplets thrown into the air by breaking waves, are discussed in a qualitative way. The imaging mechanism for synthetic aperture radars (SAR's) is strongly influenced by wave motions (i.e., by the orbital velocity and acceleration associated with the long waves). The phase velocity of the long waves does not enter into the imaging process. Focusing of ocean wave imagery is attributed to orbital acceleration effects. The orbital motions lead to a degradation in resolution which causes image smear as well as a SAR inherent imaging mechanism called velocity bunching. The parameter range for which velocity bunching is a linear mapping process is calculated. It is shown that linearity holds only for a relative small range of ocean wave parameters: The likelihood that the transfer function is linear increases as the direction of wave propagation approaches the range direction, as the wavelength increases, and as the wave height decreases. Linearity is required for applying simple linear system theory for calculating the ocean wave spectrum from the gray level intensity spectrum of the image. Although, in general, the full ocean wave spectrum cannot be recovered from the SAR image by applying simple linear inversion techniques, it is concluded that for many cases in which the ocean wave spectrum is relatively narrow the dominant wavelength and direction can still be retrieved from the image even when the mapping transfer function is nonlinear. Finally, we compare our theoretical models for the imaging mechanisms with existing SLAR and SAR imagery of ocean waves and conclude that our theoretical models are in agreement with experimental data. In particular, our theory predicts that swell traveling in flight (azimuthal) direction is not detectable by SLAR but is detectable by SAR.

An Experimental Investigation of Breaking Waves Produced by a Towed Hydrofoil

Abstract: This paper presents the results of experiments on breaking waves produced by towing a submerged, two-dimensional hydrofoil at constant depth and speed. The wave field consists of a breaker followed by a train of lower, non-breaking waves. The breaker has a small zone of turbulent water riding its forward slope; this zone is called the breaking region. Measurements were made of surface height profiles, the vertical distribution of mean horizontal velocity in the wake of the wave, and the vertical thickness of the wake. The results support the hypothesis that the breaking region imparts a shearing force along the forward slope equal to the component of its weight in that direction. The force produces a turbulent, momentum-deficient wake similar to the wake of a towed, two-dimensional body in an infinite fluid. The vertical thickness of the wake grows in proportion to the square root of distance behind the breaker. The momentum deficit is approximately equal to the maximum momentum flux of a Stokes wave with the same phase speed as the breaker. The surface profile measurements yield several results: the proper independent variables describing the wave are its speed and the slope of its forward face. The relation between breaking wavelength and speed follows the finite-amplitude Stokes wave equation. The amplitude and the vertical extent of the breaking region are both proportional to the phase speed squared; however, they are not functions of the slope of the forward face of the wave. The breaking region has a small oscillation in its length with a regular period of 4.4 the period of a wave with phase speed equal to the hydrofoil speed. The amplitude of the oscillation diminishes with time. It is believed that this oscillation is due to wave components produced when the foil is started from rest.

Gravity waves on water with non-uniform depth and current

Abstract: A mathematical model for the combined refraction-diffraction of linear periodic gravity waves on water is developed, in which the influence of inhomogeneities of depth and current is taken into account. The model is used to compute partial reflection of waves a gully or an undersea slope, with influence of a current. The model is also applied to prismatic wave channels with reflecting side-walls. For a gully bounded by shallows the model predicts the decay of wave height due to radiation of energy in lateral direction. For practical application in regions with arbitrary bottom and current topography a parabolic approximation of the model is derived. This is used as a basis for numerical calculation of waves in a sea region near the coast.

Physical Processes and Fine-grained Sediment Dynamics, Coast of Surinam, South America

Abstract: The prograding Holocene mud wedge between the Amazon and Orinoco Rivers in the trade wind belt of northeastern South America provides a modern-day example of muds accumulating under moderate wave-energy conditions. Gigantic shore-attached mudbanks (10 km 20 km), composed partly of thixotropic fluid-mud gel, front this coast every 30-60 km to form a buffer to wave attack and a temporary storage for fine-grained sediments. This mesotidal coast (tide range 2.0 m) with gentle offshore slope (0.0006) allows the exposure twice a day of extensive tidal flat deposits, which are backed by mangrove swamps on a well-developed chenier-plain complex. Field experiments were conducted in Surinam du ing 1975 and 1977 to provide new information on process-form relationships in this interesting but unusual muddy environment. Simultaneous measurements of waves, currents, tide elevation, suspended-sediment concentration, and variations in mud density show that soft intertidal and subtidal muds are suspended at both tide and wave frequency. Suspended-sediment concentrations typically exceed 1,000 mg/l at the surface as incoming solitary-like waves partially disperse fluid mud into overlying water on a falling or rising tide. Redeposition of mud may occur near time of high tide. The strong attenuation of shallow-water waves by these muds provides conditions that are favorable for further sedimentation. High concentrations of suspended fluid mud, together with solitary-like waves from the northeast throughout the year, can lead to extraordinarily high net sediment transport rates in the nearshore zone. Calculations based on solitary-wave theory and on data obtained from this study indicate that 15-65 106 m3 of mud can move along shore each year without involving breaking waves, the concept of radiation stress and a nearshore circulation cell, or bedload transport. Farther offshore, outside the zone of wave dominance, wind-driven currents and the Guiana Current combine to transport muds to the northwest, consistent with the observed direction of mudflat migration.

Near-limiting gravity waves in water of finite depth

Abstract: Progressive, irrotational gravity waves of constant form exist as a two-parameter family. The first parameter, the ratio of mean depth to wavelength, varies from zero (the solitary wave) to infinity (the deep-water wave). The second parameter, the wave height or amplitude, varies from zero (the infinitesimal wave) to a limiting value dependent on the first parameter. For limiting waves the wave crest ceases to be rounded and becomes angled, with an included angle of 120°. Most methods of calculating finite-amplitude waves use either a form of series expansion or the solution of an integral equation. For waves nearing the limiting amplitude many terms (or nodal points) are needed to describe the wave form accurately. Consequently the accuracy even of recent solutions on modern computers can be improved upon, except at the deep-water end of the range. The present work extends an integral equation technique used previously in which the angled crest of the limiting wave is included as a specific term, derived from the well known Stokes corner flow. This term is now supplemented by a second term, proposed by Grant in a study of the flow near the crest. Solutions comprising 80 terms at the shallow-water end of the range, reducing to 20 at the deep-water end, have defined many field and integral properties of the flow to within 1 to 2 parts in 106. It is shown that without the new crest term this level of accuracy would have demanded some hundreds of terms while without either crest term many thousands of terms would have been needed. The practical limits of the computing range are shown to correspond, to working accuracy, with the theoretical extremes of the solitary wave and the deep-water wave. In each case the results agree well with several previous accurate solutions and it is considered that the accuracy has been improved. For example, the height: depth ratio of the solitary wave is now estimated to be 0.833 197 and the height: wavelength ratio of the deep-water wave to be 0.141063. The results are presented in detail to facilitate further theoretical study and early practical application. The coefficients defining the wave motion are given for 22 cases, five of which, including the two extremes, are fully documented with tables of displacement, velocity, acceleration, pressure and time. Examples of particle orbits and drift profiles are presented graphically and are shown for the extreme waves to agree very closely with simplified calculations by Longuet-Higgins. Finally, the opportunity has been taken to calculate to greater accuracy the long-term Lagrangian-mean angular momentum of the maximum deep-water wave, according to the recent method proposed by Longuet-Higgins, with the conclusion that the level of action is slightly above the crest.

Three-Dimensional Instability of Finite-Amplitude Water Waves

Abstract: Computations based on the full water-wave equations reveal that there are two distinct types of instabilities for gravity waves of finite amplitude on deep water. One is predominantly two dimensional and is related to all the known results for special cases. The other is predominantly three dimensional and becomes dominant when the wave steepness is sufficiently large.

Dynamical cooling induced by dissipating internal gravity waves

Abstract: In this analysis we show that vertically propagating internal gravity waves induce a downward transfer of sensible heat from regions of wave dissipation, and that this transfer of heat may result in a net cooling of regions of the upper atmosphere.

On the structure of turbulent flow over a progressive water wave: theory and experiment in a transformed, wave-following co-ordinate system

Abstract: An investigation of the turbulent flow structure over a progressive water wave, as well as the structure of the wave-induced flow field in a transformed wave-following frame, is reported. Experimental results are given for a free-stream velocity of 2·4 m s−1 over a 1 Hz mechanically generated deep-water wave. The velocity components were measured with a cross hot-film probe oscillating in a transformed wave-following frame. The amplitude and phase of the wave-induced velocity components are deduced by correlation to the generated water wave. The mean flow tends to follow the wave form so that the water wave should not be regarded as surface roughness. The mean velocity profile is basically log-linear and is similar to that over a smooth plate, because ripples riding on the waves do not produce sufficient roughness to interfere with the wind field. The wave-induced motion in the free stream is irrotational; but, in the boundary layer, it has strong shear behaviour related to the wave-associated Reynolds stress. The shear stress production as well as the energy production from the mean flow is concentrated near the interface. A phase jump of 180° in the wave-induced turbulent Reynolds stresses in the middle of the boundary layer was observed. The relationships between the induced turbulent Reynolds stresses and the induced velocities are of an eddy-viscosity type.

An experimental study of critical layers

Abstract: Laboratory experiments have been made to investigate the development of internal gravity waves as they approach a critical layer where their phase speed is equal to that of the mean flow. The waves are produced in the accelerating flow of a stratified fluid in a long tilted tube in which the lower boundary has sinusoidal corrugations. As found in earlier experiments, the waves are not observed to propagate beyond the critical layer. Near the layer their amplitude increases, with the development of regions in which the fluid is gravitationally unstable. Kelvin-Helmholtz instability is not observed, perhaps because of viscous effects.A model is devised which describes the weakly nonlinear development of the waves. This is solved numerically. The results compare favourably with the experiments until gravitational instability is imminent. The numerical model is used to estimate both the second order Eulerian ‘jet’, which develops below the critical layer, and the Stokes drift. In the cases examined, the maximum drift is stronger than the jet and opposite in direction. The numerical model predicts the regions of wave breaking quite well.Internal gravity waves in the ocean may be modified by transient critical layers, for example those caused by vertically-propagating near-inertial oscillations.

Internal gravity waves in the solar atmosphere. I - Adiabatic waves in the chromosphere

Abstract: The properties of adiabatic and linear internal gravity waves propagating in a solar wind model are discussed, using nonlinearity criteria unique to gravity waves to estimate wave-breaking heights. The results are used to deduce information on the possible role of gravity waves in the chromospheric energy balance. Maximum vertical velocity amplitudes for gravity waves are estimated to be on the order of 2 km/sec or less, and maximum horizontal velocity amplitudes are less than 6 km/sec, with temperature perturbations as large as 1000-2000 K. It is also estimated that gravity waves with an incident energy flux of one million ergs/sq cm-sec can propagate upward to a maximum height of 900-1000 km above the visible surface before nonlinearities lead to wave breaking, while those with an energy flux of 100,000 ergs/sq cm-sec can reach maximum heights of 1400-1600 km.

Direct production of droplets from breaking wind-waves —its observation by a multi-colored overlapping exposure photographing technique

Abstract: The micro-scale configurations of the surface of breaking wind-waves and their processes of change were investigated through wind-wave tank experiments, with use of a simple, new photographic technique of multi-colored overlapping exposures. The most conspicuous micro-scale phenomenon was the appearance of small projections, mainly on the crest of the wave and the succeeding stretching and breaking of these projections into small droplets. This sequence of events shows the process of the direct production of droplets by breaking waves. The movements of such directly produced droplets were also measured and their characteristics in relation to the phase of the wave are discussed. DOI: 10.1111/j.2153-3490.1981.tb01781.x

A preliminary investigation of the interaction of internal gravity waves with a steady shearing motion

Abstract: Preliminary experimental results are presented which describe the interaction of an internal-wave field with a steady shearing motion. The results are primarily qualitative and presented in the form of photographs of shadowgraph images. Several internal-wave sources are used, and both critical- and non-critical-layer flows are examined. The results of these observations are interpreted in terms of several existing theories. For critical-layer flows the primary result is that virtually none of the internal-wave momentum flux penetrates the critical-level region, and under certain conditions a critical-layer instability develops resulting in the generation of turbulence. Such wave-induced turbulence is also observed for certain non-critical-layer flows and is believed to be the result of a convective instability.

Elevation and velocity measurements of laboratory shoaling waves

Abstract: Measurements of wave elevation and orbital velocity in the shoaling, breaking, and bore regime of single-frequency laboratory waves show that third-order Stokes theory, when energy flux is conserved, predicts the wave height change and harmonic growth in the regime where the Ursell number Ur = (H/ h)/(kh)2 is 0(1) or less. Shoreward of the Stokes region and up to the breakpoint, harmonic amplitudes are well described by the cnoidal theory. It is shown theoretically that a smooth transition regime exists between Stokes and cnoidal regions for waves which eventually break by plunging. The wave profile asymmetry about the vertical plane observed in near-breaking waves and bores is due to slow changes of phase of the harmonics relative to the primary wave as the wave train shoals. By contrast, only asymmetry about the horizontal plane is possible in the Stokes and cnoidal wave theories, since these classical solutions allow no relative phase shifts between harmonics. Velocity measurements made with hot-film anemometers show that ‘unorganized’ fluctuations at the bottom under breaking waves are of the order of half the rms amplitude of the wave-induced ‘organized’ flow. The correlation between surface elevation and bottom velocity under breakers and bores suggests that turbulence contributes more strongly to the unorganized flow at the bottom under plunging than under spilling waves.

On the theory of nonlinear wave-wave interactions among geophysical waves

Abstract: The problem of nonlinear wave-wave interactions is reformulated, in a Eulerian framework, for two classical geophysical systems: barotropic Rossby waves and internal gravity waves on a vertical plane. The departure of the dynamical fields from the equilibrium state is expanded in the linear-problem eigenfunctions, using their properties of orthogonality and completeness. The system is then completely described by the expansion amplitudes, whose evolution is controlled by a system of equations (with quadratic nonlinearity) which is an exact representation of the original model equations. There is no a priori need for the usual multiple-time-scale analysis, or any other perturbation expansion, to develop the theory. These or other approximations (like truncation of the expansion basis or the Boltzmann equation for a stochastic description) can, if desired, be performed afterwards.The evolution of the system is constrained mainly by the conservation of energy E and pseudo-momentum P, properties related to time and space homogeneity of the model equations. Conservation of E and P has, in turn, some interesting consequences: (a) a generalization of Fjortoft's theorem, (b) a class of exact nonlinear solutions (which includes the system of one single wave), and (c) conservation of E and P in an arbitrarily truncated system (which is useful in the development of approximations of the problem).The properties of all possible resonant triads are shown and used to estimate the order of magnitude of off-resonant coupling coefficients.The results are used in two different problems: the stability of a single wave (maximum growth rates are evaluated in both the strong- and weak-interactions limits) and the three-wave system. The general solution (for any initial condition and for both the resonant and off-resonant cases) of the latter is presented.

A free rossby wave in the troposphere and stratosphere during January 1979

Abstract: A large-scale (zonal wave number 1), westward-propagating disturbance present in the troposphere and the stratosphere in January 1979 is shown to be similar to a free Rossby wave. Its vertical and horizontal structures are reasonably consistent with those of a theoretically predicted wave mode of the second class. It is similar to the so-called 16-day wave described earlier, and, therefore, not an unusual occurrence. It is argued that this traveling wave may be preferentially excited by random forcing and that it can interact with quasistationary waves forced by mountains and by heat sources. It is shown, for example, that this interaction can cause fluctuations in eddy heat transport throughout the atmosphere.

Visualization of airflow separation over wind-wave crests under moderate wind

Abstract: An intermittently-smoking smoke-wire was devised to visualize the airflow structure over individual crests of actual wind waves. The device was used under a moderate wind 6 m s-1 (maximum speed in the vertical cross-section) at a fetch 3.8 m in a wind-wave tunnel. Airflow patterns with separation were clearly visualized over wind-wave crests which were not accompanied by wave breaking characterized by air entrainment. A classification of 41 samples of airflow structures showed that two distinct patterns (with and without separation) exist, with significant frequency of occurrence for each.

A numerical model of flow over sand waves in water of finite depth

Abstract: Summary. A model of turbulent flow above sand waves in water of finite depth is described. Closure assumptions are based on an eddy viscosity proportional to the square root of the local value of turbulent kinetic energy and mixing length dependent upon distance from the lower boundary. Results are presented for some idealized cases to investigate the effects of wave slope, water depth, Froude number and wave shape. The implications of the model for the transport of sediment are discussed and the development of the wave investigated. It is found that the crest of the wave will become sharper for lower flow rates, as has been observed in the sea. Comparisons are made with recent measurements made over sand waves in the Columbia River.

Solitary wave solutions for a model of the two-way propagation of water waves in a channel

Abstract: Bona and Smith (6) have suggested that the coupled system of equations has the same formal justification as other Boussinesq-type models for the two-way propagation of one-dimensional water waves of small but finite amplitude in a channel with a flat bottom. The variables u and η represent the velocity and elevation of the free surface, respectively. Using the energy invariant they show that for a restricted, but nevertheless physically relevant, class of initial data, the system (1·1) has solutions which exist for all time, and that in such circumstances the wave height is bounded solely in terms of the initial data.

Breaking-Limited Wave Heights

Abstract: Empirical distributions of wave heights derived from field data differ from the theoretical distribution of wave heights as given by the Rayleigh law. A principal cause of this discrepancy is suggested to be that the conventional theory does not in any way account for wave breaking. Based on the assumption that the sea surface is characterized with a narrow-band spectrum, the probability distribution of wave heights limited by breaking, due to a Stokes-type limiting steepness in deep or finite water depths, is derived. The breaking-limited distribution provides a probability discription which compares with two well documented field observations more favorably than the Rayleigh law.

Finite-amplitude surface waves in electrohydrodynamics

Abstract: The stability of weakly nonlinear waves on the surface of a fluid layer in the presence of an applied electric field is investigated by using the derivative expansion method. A nonlinear Schrodinger equation for the complex amplitude of quasi-monochromatic traveling wave is derived. The wave train of constant amplitude is unstable against modulation. The equation governing the amplitude modulation of the standing wave is also obtained which yields the nonlinear cut-off wave number.

Model of gravity wave growth due to fluctuations in the air‐sea coupling parameter

Abstract: The linear Miles-Phillips model of the coupling between water waves and a turbulent wind field is generalized to include fluctuations in the air-sea coupling parameter in the dynamic equations The model is solved exactly, assuming the fluctuations to be delta correlated in time The energy spectrum of the water wave field is shown to grow exponentially at rates higher then those predicted by the linear Miles-Phillips model These increased initial growth rates are consistent with those observed in open ocean measurements

Energy transfer among internal gravity modes: Weak and strong interactions

Abstract: The general characteristics of the energy spectrum for internal gravity waves in the ocean are well known from the large body of recent experimental observations. The theoretical understanding has not developed at the same rate, perhaps due to the limitation of linear or quasi-linear theories, which can cope only with weak interaction processes and are inadequate for representing the more violent and sporadic wave breaking processes present in nature. A detailed study of energy transfer among two-dimensional internal gravity modes in a fully nonlinear regime was performed. Wave-wave interactions and overturning were included in the solutions of a two-dimensional numerical model, and the results are presented here. A background spectrum of finite amplitude, random internal gravity wave field was generated by a long time integration of a two-dimensional model with random body forcing. Over this background field, two sets of experiments were performed: spike-random, where energy at low, medium, and high wave numbers were introduced and integrated in time, and band-random, where energy was introduced over a band of wave numbers instead of introducing only discrete modes. The results can be summarized as follows. Multiple triad interactions will result in a filling of the energy spectrum when energy is introduced in a particular band of wave numbers. For bands where the energy level is high enough to result in nonlinear time scales of only a few intrinsic periods, wave-wave interactions (resonant and nonresonant) provide the mechanism for filling the spectrum. The energy transfer becomes more and more rapid with increasing energy, and no universal spectrum appears to result from these processes. As the energy input increases, energy will accumulate in high wave numbers until localized instabilities (over-turning) occur. From that point on, these high wave numbers will remain at a saturation such that any additional energy input at the saturated band, either directly or via wave-wave interactions, will result in localized mixing. On the other hand, additional energy input at bands other than the saturated band will result in an increase of low and medium wave band energy (via wave-wave interactions) until an equilibrium level is achieved. The equilibrium level of any particular band will depend on the high wave number bands being saturated. For instance, any energy above the equilibrium at low wave numbers will produce localized mixing in physical space almost instantaneously. This does not mean that the low wave numbers are saturated, as their energy levels can be much lower than a saturation level. What takes place at or near an equilibrium level is that the contributions from high and low wave numbers result in localized regions in physical space where the criterion for instability is almost met. In fact, this superposition effect means that low and medium wave numbers are far from meeting any breaking criterion when taken individually, yet cannot tolerate any additional input energy when in the presence of a saturated band of high wave numbers. It was found also that the dissipation is approximately constant over the wave numbers and small compared with the large transfer of energy between neighboring waves. However, if bands of waves are considered, very little energy is transferred between neighboring bands above the equilibrium level. Rather, a direct cascade of energy from low to high wave numbers occurs due to localized instabilities which result in overturning, and it is this amount of energy flux which is dissipated by the high wave numbers.

Wave-energy extraction by a submerged cylindrical resonant duct

Abstract: A cylindrical duct absorbing energy from incident surface waves is considered. The asymptotic properties of the scattering and radiation potentials are determined, to yield the hydrodynamic quantities on which the energy absorption characteristics of the duct can be shown to depend. It is shown that it is possible to tune the resonant response of the duct to absorb the maximum theoretical energy at a given frequency. Curves are presented showing the variation of energy absorption and the amplitude of the duct response with frequency for various depths of submergence and various tuning frequencies.

Resonant type apparatus for absorbing wave energy arranged at wave-breaking facilities

Abstract: A resonant type apparatus for absorbing wave energy arranged at wave-breaking facilities, which comprises, a caisson having a bottom plate, side plates, a backside plate and a top plate at least a part of which is omitted. A water chamber in the caisson has a length in a directional parallel to the side walls which is larger than 1/4 of the wave length Lc of a stationary wave within the water chamber, and a node of the stationary wave in the water chamber is formed at a distance Lc/4 from the backside plate. A pendulum is arranged at the position of the node of the stationary wave for swinging with a natural period Tp which is substantially the same value as the natural period Tw of the stationary wave, whereby the pendulum is swung by the stationary wave to absorb and convert wave energy to useful available energy at high efficiency.

Further comments on the behavior of acceleration waves of arbitrary shape

Abstract: Converging waves with nonzero initial critical amplitude are completely characterized. It is shown that for a converging wave a necessary and sufficient condition for the initial critical amplitude to be zero is that the converging wave is spherical.