Shoaling Surface Gravity Waves Cause a Force and a Torque on the Bottom

Freely propagating surface gravity waves are observed to slow down and to stop at a beach when the bottom has a relatively gentle upward slope toward the shore and the frequency range of the waves covers the most energetic wind waves (sea and swell). Essentially no wave reflection can be seen and the measured reflected energy is very small compared to that transmitted shoreward. One consequence of this is that the flux of the wave’s linear momentum decreases in the direction of wave propagation, which is equivalent to a time rate of change of the momentum. It takes a force to cause the time rate of change of the momentum. Therefore, the bottom exerts a force on the waves in order to decrease the momentum flux. By Newton’s third law (action equals reaction) the waves then impart an equal but opposite force to the bottom. In shallow (but finite) water depths the wave force per unit bottom area is calculated, for normal angle of incidence to the beach, to be directly proportional to the square of the wave amplitude and to the bottom slope and inversely proportional to the mean depth; it is independent of the wave frequency. Constants of proportionality are: 1/4, the fluid density and the acceleration of gravity. Swell attenuation near coasts and some characteristics of sand movement in the near-shore region are not inconsistent with the algebraic structure of the wave force formula. Since the force has a depth variation which is significantly faster than that of the dimensions of the particle orbits in the vertical direction, the bottom induces a torque on the fluid particles that decreases the angular momentum flux of the waves. By an extension of Newton’s third law, the waves also exert an equal but opposite torque on the bottom. And because the bottom force on the waves exists over a horizontal distance, it does work on the waves and decreases their energy flux. Thus, theoretically, the fluxes of energy, angular and linear momentum are not conserved for shoaling surface gravity waves. Mass flux, associated with the Stokes drift, is assumed to be conserved, and the wave frequency is constant for a steady medium.     

Determination of wave energy dissipation factor and numerical simulation of wave height in the surf zone

Based on the time-dependent mild slope equation including the effect of wave energy dissipation, an expression for the energy dissipation factor is derived in conjunction with the wave energy balance equation. The wave height of regular and irregular waves is numerically simulated by use of the parabolic mild slope equation considering the energy dissipation due to wave breaking. Comparison of numerical results with experimental data shows that the expression for the energy dissipation factor is reasonable. The effects of the wave breaking coefficient on the breaking point and the distribution of wave height after breaking are discussed through the study of a specific experimental topography.     

of regular and irregular waves from a partially perforated caisson breakwater with a rock-filled core

The reflection of regular and irregular waves from a partially perforated caisson breakwater with a rock-filled core is examined. The present mathematical model is developed by means of the matched eigenfunction method. Numerical results of the present model are compared with the experimental data of different researchers. Numerical examples are given to examine the effect of rock fill on the reflection coefficient. The differences between regular and irregular waves are also investigated by means of theoretical and experimental results. It is found that the minimum reflection coefficient of irregular waves is larger than that of corresponding regular waves, but the contrary is the case for the maximum reflection coefficient.     

确定震源位置,发震时刻及区域平均波速的最优化方法

本文详细讨论了利用井下和地面观测直达波到时资料联合确定震源位置,发震时刻及区域平均波速的最优化方法,该方法对震源深度及波速的确定给予特别的重视与处理。理论检验和实际算例表明,用该方法编制的程序定位精度高,计算速度快,可以在地层差异小的地区推广应用。     

Instability of planetary waves in a high vertical resolution model

Abstract

A high vertical resolution model is used to examine the instability of a baroclinic zonal flow and a finite amplitude topographically forced wave. Two families of unstable modes are found, consisting of zonally propagating most unstable modes, and stationary unstable modes. The former have time scale and spatial structure similar to baroclinic synoptic disturbances, but are localized in space due to interaction with the zonally asymmetric forcing. These modes transport heat efficiently in both the zonal and meridional directions. The second family of stationary unstable modes has characteristics of modes of low frequency variability of the atmosphere. They have time scales of 10 days and longer, and are of planetary scale with an equivalent barotropic vertical structure. The horizontal structure resembles blocking flows. They are maintained by available potential energy of the basic wave, and have large zonal heat fluxes. The results for both families of modes are interpreted in terms of an interaction between forcing and baroclinic instability to create favoured regions for eddy development. Applications to baroclinic planetary waves are also considered.     

Open-ocean response and normal mode excitation in an eddy-resolving general circulation model

Abstract

Analysis of a two-layer, flat-bottom, steady-wind driven, eddy-resolving general circulation model reveals a distinct separation in frequency of baroclinic and barotropic motion in the region distant from the model Gulf Stream. The far-field motions at periods less (greater) than about 100 days are predominantly barotropic (baroclinic), unlike the near-field, eddy-generating, free-jet region which contains barotropic and baroclinic energy throughout the modei frequency range. The far-field barotropic energy produces a peak in the model sea-level spectra between 25 and 50 days with a magnitude comparable to energy levels observed in spectra of sea level from oceanic island tide gauges. The far-field barotropic motion is clearly composed of large-scale, resonant, barotropic normal modes drive by mesoscale activity of the turbulent, free-jet region. Oceanic mesoscale turbulence may therefore provide for planetary normal modes an excitation mechanism distinct from atmospheric forcing. The open-ocean, barotropic, model response is very similar to that of a fluctuating-wind driven model, which suggests that atmospheric and intrinsic forcing of mid-ocean eddies may be of comparable importance.     

Magnetic Rossby waves in a stably stratified layer near the surface of the Earth's outer core

Abstract

The stratification profile of the Earth's magnetofluid outer core is unknown, but there have been suggestions that its upper part may be stably stratified. Braginsky (1984) suggested that the magnetic analog of Rossby (planetary) waves in this stable layer (the ‘H’ layer) may be responsible for a portion of the short-period secular variation. In this study, we adopt a thin shell model to examine the dynamics of the H layer. The stable stratification justifies the thin-layer approximations, which greatly simplify the analysis. The governing equations are then the Laplace's tidal equations modified by the Lorentz force terms, and the magnetic induction equation. We linearize the Lorentz force in the Laplace's tidal equations and the advection term in the magnetic induction equation, assuming a zeroth order dipole field as representative of the magnetic field near the insulating core-mantle boundary. An analytical β-plane solution shows that a magnetic field can release the equatorial trapping that non-magnetic Rossby waves exhibit. A numerical solution to the full spherical equations confirms that a sufficiently strong magnetic field can break the equatorial waveguide. Both solutions are highly dissipative, which is a consequence of our necessary neglect of the induction term in comparison with the advection and diffusion terms in the magnetic induction equation in the thin-layer limit. However, were one to relax the thin-layer approximations and allow a radial dependence of the solutions, one would find magnetic Rossby waves less damped (through the inclusion of the induction term). For the magnetic field strength appropriate for the H layer, the real parts of the eigenfrequencies do not change appreciably from their non-magnetic values. We estimate a phase velocity of the lowest modes that is rather rapid compared with the core fluid speed typically presumed from the secular variation.     

Nonlinear wave propagation on a sphere: Interaction between Rossby waves and gravity waves; stability of the Rossby waves

Abstract

An analytical spectral model of the barotropic divergent equations on a sphere is developed using the potential-stream function formulation and the normal modes as basic functions. Explicit expressions of the coefficients of nonlinear interaction are obtained in the asymptotic case of a slowly rotating sphere, i.e. when the normal modes can be expressed as single spherical harmonics.     

Shallow Water Dissipation Processes for Wind Waves off the Mackenzie Delta

This study focuses on two physical processes for waves in shallow waters off the Mackenzie Delta: bottom friction and depth-induced breaking terms. We use field observations of winds and waves, the state-of-the-art Simulating Waves Nearshore (SWAN) model, and reanalysis wind and wave data. The two field observation periods are an August 2008 field experiment, during which in situ field data were collected, and an Arctic storm when data were recorded by buoy measurements from 4 to 6 August 1991. Wind and wave development processes are analyzed during these two periods with comparisons to observed winds and waves. Our analyses show that bottom friction is the main shallow water physical process during the August 2008 field experiment, whereas depth-induced breaking is the dominant shallow water physical process during the 4–6 August 1991 storm, in conjunction with the effects of bottom friction. The SWAN wave model is used to investigate the shallow water physical processes during these two observation periods. Simulation results indicate that the model can give reasonable results, with an appropriate Collins coefficient of 0.006 and a wave breaking parameter of 0.55 to represent bottom friction and depth-induced breaking physics, respectively.

RÉSUMÉ?[Traduit par la rédaction] Cette étude porte sur deux processus physiques concernant les vagues dans les eaux peu profondes au large du delta du Mackenzie : les termes du frottement contre le fond et du déferlement lié à la profondeur. Nous utilisons des observations du vent et des vagues, le modèle d'avant-garde SWAN (Simulating Waves Nearshore) et des données de vent et de vagues réanalysées. Les deux périodes d'observations sont une expérience sur le terrain réalisée en août 2008, au cours de laquelle des données de terrain ont été recueillies, et une tempête arctique lors de laquelle des mesures faites par bouée du 4 au 6 août 1991 ont été enregistrées. Nous analysons les processus dévolution du vent et des vagues durant ces deux périodes, et comparons avec le vent et les vagues observées. Nos analyses montrent que le frottement contre le fond est le processus physique en eaux peu profondes le plus important durant l'expérience sur le terrain d'août 2008, alors que le déferlement lié à la profondeur est le processus physique en eaux peu profondes dominant pendant la tempête arctique du 4 au 6 août 1991, en combinaison avec les effets du frottement contre le fond. Nous nous servons du modèle de vagues SWAN pour étudier les processus physiques en eaux peu profondes durant ces deux périodes d'observations. Les résultats des simulations indiquent que le modèle peut donner des résultats raisonnables, avec un coefficient de Collins approprié de 0,006 et un paramètre de déferlement de 0,55 pour représenter la physique du frottement contre le fond et du déferlement lié à la profondeur, respectivement.     

The formation and evolution of an isolated submarine valley in the North Channel,Irish Sea: an investigation of Beaufort's Dyke

Beaufort's Dyke is a submarine depression located in the North Channel of the Irish Sea. With a maximum depth of 312 m, the dyke is one of the deepest areas within the European continental shelf. Integration and interpretation of 450 km of sparker seismic data and full‐coverage bathymetric data derived from multi‐beam echo sounder surveys allow for the investigation of the formation processes of Beaufort's Dyke and the evolution of geomorphological features within it. The dyke, formed by composite subglacial processes dominated by subglacial meltwater discharge, is interpreted as a tunnel valley. The regional isolation of Beaufort's Dyke may be explained by the bounding of the North Channel by the bedrock masses of Ireland and Scotland, coupled with the exploitation of structural weakness along a fault plane and presence of halite in the eroded substrate enhancing the erosive potential of the overlying glacier. Beaufort's Dyke has probably been maintained as an open feature by strong rectilinear tidal currents. The morphology of lunate sediment waves and a large parabolic bedform towards the south of the dyke contradict the observed dominant S–N mean hydrodynamic flow recorded within the North Channel, suggesting an alternative hydrodynamic regime either within the dyke or during bedform creation. Copyright © 2011 John Wiley & Sons, Ltd.