Shallow Waves Research Articles

From the paddle to the beach – A Boussinesq shallow water numerical wave tank based on Madsen and Sørensen’s equations

This article describes a one-dimensional numerical model of a shallow-water flume with an in-built piston paddle moving boundary wavemaker. The model is based on a set of enhanced Boussinesq equations and the nonlinear shallow water equations. Wave breaking is described approximately, by locally switching to the nonlinear shallow water equations when a critical wave steepness is reached. The moving shoreline is calculated as part of the solution. The piston paddle wavemaker operates on a movable grid, which is Lagrangian on the paddle face and Eulerian away from the paddle. The governing equations are, however, evolved on a fixed mapped grid, and the newly calculated solution is transformed back onto the moving grid via a domain mapping technique. Validation test results are compared against analytical solutions, confirming correct discretisation of the governing equations, wave generation via the numerical paddle, and movement of the wet/dry front. Simulations are presented that reproduce laboratory experiments of wave runup on a plane beach and wave overtopping of a laboratory seawall, involving solitary waves and compact wave groups. In practice, the numerical model is suitable for simulating the propagation of weakly dispersive waves and can additionally model any associated inundation, overtopping or inland flooding within the same simulation.

Horizon effects for surface waves in wave channels and circular jumps

Surface waves in classical fluids experience a rich array of black/white hole horizon effects. The dispersion relation depends on the characteristics of the fluid (in our case, water and silicon oil) as well as on the fluid depth and the wavelength regime. In some cases, it can be tuned to obtain a relativistic regime plus high-frequency dispersive effects. We discuss two types of ongoing analogue white-hole experiments: deep water waves propagating against a counter-current in a wave channel and shallow waves on a circular hydraulic jump.

Rupture process of the 2001 May 7 Mw 4.3 Ekofisk induced earthquake

SUMMARY On 2001 May 7, following unintentional water injection, a moderate size induced earthquake struck the Ekofisk oil field, North Sea. Despite of its relatively moderate magnitude, clear low-frequency waveforms could be recorded up to more than 2000 km epicentral distance, suggesting a slow rupture at very shallow depth and wave propagation through low-velocity shallow structures. The event poses a rare opportunity to constrain rupture velocity, duration and rise time of a superficial M > 4 event occurring on a horizontal plane in soft, water-saturated sediments. Two previous studies discussed the earthquake point source finding vertical dip-slip focal mechanisms with opposite senses of P and T axes. A further investigation was thus required to provide a basis for a deeper discussion of the failure dynamics. We significantly improve the used data set, test different earth models and derive a point source as well as a kinematic rupture model. We carefully discuss parameter uncertainties and effects related to shallow sources and wave propagation through different crustal structures to resolve the previous controversy. We additionally provide a kinematic rupture model, based on apparent source times derived from Rayleigh and Love waves. The waveforms resolve a predominant unilateral rupture along a horizontal plane at about 2 km depth. We derive an unusually slow rupture, consequence of a slow rupture velocity of about 500 m s –1 and a long rise time of about 7 s. An independent modelling of GPS- based static displacements allows to confirm the focal mechanism polarity and to locate the centroid at the eastern side of the field, resulting in a much larger seismic moment in comparison with dynamic seismic moment. The rupture directivity is confirmed by the relative location of the centroid with respect to the epicentre, which is set at the site of water injection.

SU‐E‐T‐287: The Clinical Evaluation of a Novel Amplitude‐Based Binning Algorithm for 4D CT Reconstruction

Purpose: To evaluate the performance of the first commercial amplitude‐binning algorithm for 4DCT reconstruction, to compare its reconstructed image quality to that of the phase‐binning algorithm, and to investigate possible clinical effects from changing algorithms. Methods: A department customized 4DCT motion phantom was utilized to evaluate the geometry accuracy (correct phase sequencing and motion) of the amplitude‐binning algorithm. 64 4DCT patient cases, in which 60(4) cases were acquired with 4D chest (abdomen) protocol, were collected and evaluated on the respiratory motion artifact severity. The MIP images generated from both algorithms for 10 SBRT cases were further evaluated by two radiation oncologists regarding the tumor sizes and locations. Results: The phantom experiments illustrated that, as expected, maximum inhalation occurred between 90%–10% phases, while maximum exhalation occurred between 40%–60% phases. The amplitude‐binning algorithm provides reasonable phase sequencing and motion within the 4D images. The amplitude‐binning algorithm yielded fewer respiratory motion artifacts on 24/64 cases (37.5%), comparable artifacts to the phase‐binning algorithm on 35/64 cases (54.7%), and slightly greater artifacts on 5/64 cases (7.8%). The 10 SBRT cases indicated that the reconstructed tumor sizes and locations were comparable on 9 MIP image pairs generated from amplitude‐binning and phase‐binning, respectively. However, amplitude‐binning yielded smaller tumor size on one case probably due to the very shallow respiratory amplitude occurring over several breathing cycles. Conclusions: Overall, amplitude‐binning algorithm yielded fewer or equivalent artifacts on 92.2% of cases (59/64 cases), and no dramatic difference on reconstructed tumor sizes and locations from the MIP images based on amplitude and phase binning algorithms. We suggest utilizing the amplitude‐binning 4DCT images for clinical practice, especially for SBRT cases. If very shallow respiratory waves or severe motion artifacts were found on specific cases, physicists should work with the physicians to provide the best achievable 4D images.

Dark Solitons for a Generalized Korteweg-de Vries Equation with Time-Dependent Coefficients

We consider the evolution of long shallow waves in a convecting fluid when the critical Rayleigh number slightly exceeds its critical value within the framework of a perturbed Korteweg-de Vries (KdV) equation. In order to study the wave dynamics of nonlinear pulse propagation in an inhomogeneous KdV media, a generalized form of the considered model with time-dependent coefficients is presented. By means of the solitary wave ansatz method, exact dark soliton solutions are derived under certain parametric conditions. The results show that the soliton parameters (amplitude, inverse width, and velocity) are influenced by the time variation of the dependent model coefficients. The existence of such a soliton solution is the result of the exact balance among nonlinearity, third-order and fourth-order nonlinear dispersions, diffusion, dissipation, and reaction.

Cloud formation along mountain ridges on Titan

Cassini radar passes have shown a number of mountain ranges on Titan. Radar data covering approximately one quarter of Titan's surface places mountains in primarily equatorial regions with the mean height of about 900 m. The flow of air over topographic features can both trigger and enhance cloud formation. Orographically induced clouds near terrestrial mountain ranges include shallow wave clouds produced from upslope flow as well as precipitating stratus and cumulus type clouds; mountains can provide the perturbations needed to trigger convective clouds. The Titan regional atmospheric modeling system (TRAMS) has been used to explore a number of convective cloud properties and is now used to report on clouds formed when a mountain peak is placed within the model domain. Using a range of heights and surface winds compatible with Cassini/Huygens data, constraints can be placed on the scenarios in which clouds can be expected to form. Given sufficiently humid conditions (at least 50% humidity), convection is triggered. For drier environments similar to the Huygens landing site, short-lived, optically thin clouds form from air rising upslope. Precipitation is also seen in the cases of the convective clouds, which could have implications for the eroded appearance of Titan's mountains.

Observations of a Super Cloud Cluster Accompanied by Synoptic-Scale Eastward-Propagating Precipitating Systems over the Indian Ocean

Abstract The multiscale structure of a super cloud cluster (SCC) over the equatorial Indian Ocean, observed in November and December 2006, was investigated using data from satellite microwave sensors and surface-based radars. The smaller-scale structure of this SCC was marked by a complicated relationship between rainfall systems and upper-tropospheric cloud shields, which moved eastward and westward, respectively, with a cycle of 2–4 days. In the analyses, attention was given to the structure of slow eastward-propagating (5–11 m s−1) precipitating systems and related synoptic-scale (∼2000 km) disturbances. A case study of one of the systems revealed that it consisted of several lines of convective cells with a depth that was usually shallower than 10 km unless the cells encountered the westward-moving cloud shields. The environment of the convective lines was characterized by persistent unstable conditions with an increase of the westerly flow in the lower troposphere, suggesting the existence of a synoptic-scale upward motion. Composite analyses revealed that each rainfall system formed in a region of zonal flow convergence near the surface and divergence near 300 hPa. The vertical temperature structure tilted westward with height below this pressure level and eastward aloft, similar to that of a convectively coupled Kelvin wave. These results suggest that a SCC involves a group of synoptic-scale shallow waves propagating eastward. An additional analysis over the western Pacific also showed the predominance of eastward propagation in a SCC, demonstrating the advantage of satellite microwave sensors over infrared ones in monitoring the multiscale structure of tropical convection.

The time evolution of scour around offshore structures

This paper describes the development of an engineering model to predict the development of scour evolution through time around an offshore structure under current, wave and combined wave–current flows. The model has been tested against a range of data as well as being run for idealised tests. From the results of these tests issues with respect to scour prediction have been highlighted. In particular, the initial growth rate of scour appears to be too rapid based on the large-scale laboratory tests and tidal field data against which the model has been compared. Application of the model to a field-scale study has been completed which shows that in shallow water depths storm waves dominate the scour process. In deeper water depths, currents generally dominate even under storm conditions, although this will be dependent on the wave height and period and the actual water depth. The scour time evolution predictor (Step) model can be applied to scour assessment studies where there is a requirement to have an indication of the time-history of scour. The model clearly has a role in offshore wind turbine studies, but it is not limited to this sector and can be applied to bridge scour situations or other situations where scouring around monopile structures is an issue. Furthermore, if the scour at other types of structures or objects needs to be evaluated then provided the equilibrium scour depth and time-scale functions are available these can be implemented within the framework of the Step model.

内部波と転回音線の位置の関係が音波伝搬に与える影響

In the sound propagation in shallow water, the influence of an internal wave is worthy of attention. We investigated the influence of the shallow water internal wave on the sound propagation on the continental shelf. The internal wave was assumed to be 100 m wide, with an amplitude of 22.5 m. The relation between the position of the internal wave and the turning ray was examined at 50 m in sound source depth in the thermocline. Because the turning position and the internal wave overlap, a strong scattering is caused at the rear and underside of the internal wave. Moreover, the width of the influence to the turning position is ±400 m regardless of the position of the internal wave. We also determined that the influence increases according to the extent of the internal wave width.

Nonlinear shallow water tropical instability waves on the equatorial β‐plane: Genesis of two distinct types of waves

The nonlinearity of Tropical Instability Waves (TIWs) is studied using shallow‐water equations with various mean states from the equatorial Pacific Ocean. In the early linear stage, unstable TIWs, centered near 5°N with a wavelength about 1000 km and a period about one month, dominate the whole domain. However neutral Yanai waves with periods about 15–22 days emerge near the equator when the unstable TIWs grow into fully nonlinear vortices and begin to rotate, which stabilizes the mean states substantially. The strength of these Yanai waves are sensitive to the instability of the initial mean flow. Meanwhile the TIWs centered near 5°N are slowed down and weakened. The external forcing terms are found to be important for them to retain their dominance from 3°N to 7°N and also be able to suppress the late emerging Yanai waves if strong enough.

Effect of ocean acidity on the mid-frequency ambient noise field in shallow water.

Due to increased acidification, Hester et al. [Geophys. Res. Lett. 35, L19601] estimate that by midcentury the sound absorption of seawater will decrease by almost 40%. In the present study, the expected relative increase in the mid-frequency ambient noise field is estimated. Emphasis is placed on shallow water scenarios with typical summertime sound speed profiles and frequencies from 1 to 5 kHz. A model is developed that assumes shallow water internal waves will redistribute energy from high-order acoustic modes excited by noise sources near the sea surface into ducted, low-order modes that propagate with low loss. Calculations are made as a function of pH level, the strength of the internal waves, and the bottom composition. The modeled noise field is beamformed and the effect on the so-called ambient noise notch is estimated.

Exact Periodic Solitary Solutions to the Shallow Water Wave Equation

Exact solutions to the shallow wave equation are studied based on the idea of the extended homoclinic test and bilinear method. Some explicit solutions, such as the one soliton solution, the doubly-periodic wave solution and the periodic solitary wave solutions, are obtained. In addition, the properties of the solutions are investigated.

Acoustic wave scattering in turbulence and in internal waves.

Analytical solutions of acoustic wave scattering have been obtained for turbulent currents and for shallow water internal waves. The scattering characteristics of the two are quit different. The most striking difference is that the maximum scattering is in only one direction by turbulence. Whereas there are two directions of maximum scattering in internal waves. In turbulence, acoustic waves scatter quite strongly while internal waves acoustic waves scatter very weakly. Therefore, turbulence changes acoustic wave (coherent) energy rapidly into (incoherent) scattered acoustic energy. The scattering of acoustic energy by internal waves is weak, on the order of 0.001 of that of turbulence. The theoretical results are verified with transmission loss measured in Kanmon Strait and at New Jersey Shelf with good agreements. [Work supported by ONR Code 322 OA.]

Preliminary Investigations into Macroscopic Attenuated Total Reflection—Fourier Transform Infrared Imaging of Intact Spherical Domains: Spatial Resolution and Image Distortion

This paper describes preliminary investigations into the spatial resolution of macro attenuated total reflection (ATR) Fourier transform infrared (FT-IR) imaging and the distortions that arise when imaging intact, convex domains, using spheres as an extreme example. The competing effects of shallow evanescent wave penetration and blurring due to finite spatial resolution meant that spheres within the range 20-140 microm all appeared to be approximately the same size ( approximately 30-35 microm) when imaged with a numerical aperture (NA) of approximately 0.2. A very simple model was developed that predicted this extreme insensitivity to particle size. On the basis of these studies, it is anticipated that ATR imaging at this NA will be insensitive to the size of intact highly convex objects. A higher numerical aperture device should give a better estimate of the size of small spheres, owing to superior spatial resolution, but large spheres should still appear undersized due to the shallow sampling depth. An estimate of the point spread function (PSF) was required in order to develop and apply the model. The PSF was measured by imaging a sharp interface; assuming an Airy profile, the PSF width (distance from central maximum to first minimum) was estimated to be approximately 20 and 30 microm for IR bands at 1600 and 1000 cm(-1), respectively. This work has two significant limitations. First, underestimation of domain size only arises when imaging intact convex objects; if surfaces are prepared that randomly and representatively section through domains, the images can be analyzed to calculate parameters such as domain size, area, and volume. Second, the model ignores reflection and refraction and assumes weak absorption; hence, the predicted intensity profiles are not expected to be accurate; they merely give a rough estimate of the apparent sphere size. Much further work is required to place the field of quantitative ATR-FT-IR imaging on a sound basis.

Wave effects on submarine groundwater seepage measurement

Variation of sea-surface and water pressure above the sea bed induces temporal variation of submarine groundwater discharge over time scales ranging from seconds to years. Hydrodynamic theory and measurements suggest that wave-induced exchange between a permeable sediment bed and overlying water column is significant but conventional seepage-meter studies have focussed mainly on tidal dynamics and ignored waves. At Cockburn Sound in Western Australia we measured wave-induced flow reversals through a seepage meter of amplitude 60 cm d - 1 ; however, it was unclear whether the measured flows were real seepage or, partly or wholly, an artifact of wave action on the seepage meter. A numerical model of seepage patterns beneath a vented benthic chamber demonstrated an observer effect introduced by the chamber and not previously identified. Placing a chamber on the sediment bed disturbed the pressure field and changed both the pattern and magnitude of the wave-induced flow. A separate analysis of benthic-chamber movements under the action of shallow surface waves established that micron-scale movements of the chamber at the wave frequency were sufficient to produce apparent seepage amplitudes of O(1–100) cm d −1. We concluded that wave action is a key control on bed seepage and should not be neglected without justification in direct-measurement studies of marine bed discharge. A systematic error during each wave cycle can accumulate to a significant measurement error if the wave cycle error is large or if wave-induced flow is the dominant component of the seepage. In the latter case, the error could potentially be misinterpreted as a steady seepage component.

Role of fish herbivory in structuring the vertical distribution of canopy algae Cystoseira spp. in the Mediterranean Sea

Biological interactions can strongly influ- ence the distribution of organisms. In the Mediter- ranean Sea, macroherbivore densities vary with depth in a highly predictable manner, but the direct effects of grazing on the vertical distribution of macrophytes remain largely unknown. Species of the genus Cysto- seira provide a unique system in which to test the effects of herbivores on algal distribution in the Mediterranean Sea, since they form highly prominent forest-like assemblages along the entire sublittoral zone, with different species dominating the canopy at different depths. We measured the algal cover of Cys- toseira spp. in habitats characterised by dominance of these algae at different depths, and quantified the den- sity of herbivorous fish at each habitat. The shallow sublittoral (ca. 5 m) was the only habitat where abun- dant Cystoseira forests coexisted with high densities of herbivorous fish. Reciprocal in situ feeding experi- ments showed that Cystoseira spp. from wave exposed shores (0 m) and deep habitats (40 m) are highly sus- ceptible to being eaten when placed within reach of herbivorous fish and are, thus, restricted to spatial refuges with locally reduced herbivory. In contrast, Cystoseira spp. from shallow rocky bottoms (5 m) and wave sheltered shores (0.2 m) are resistant to con- sumers and are not affected by transplantation to areas of high fish densities. Our findings challenge the widely accepted view that herbivorous fish have only a modest effect on structuring algal assemblages in tem- perate waters and show that these consumers can have a particularly important effect on the vertical distribu- tion of algal assemblages in the Mediterranean Sea.

Single-path acoustic scintillation results from the Shallow Water 2006 Experiment.

In “sound transmission through a fluctuating ocean,” Flattxe et al. described saturation of a single acoustic path as that path becoming a number of interfering uncorrelated micropaths due to refraction by internal waves. The probability density function of intensity becomes exponential with a scintillation index of 1.0. In deep water, however, full saturation is not achieved due to weak scattering and absorption. Mid‐frequency (1–10 kHz) data from the Shallow Water 2006 Experiment are used to determine single‐path intensity statistics. At a range of 1 km in water 80 m deep, an acoustic path is isolated that went through two upper turning points separated by a single bottom reflection. The data were collected during a period when large nonlinear internal waves were absent. The scintillation index calculated from the data increases with frequency until reaching a maximum of 1.2 around 6 kHz. It then decreases to 1.0, suggesting that single‐path saturation can be achieved at mid‐frequencies in shallow water. The probability density functions of intensity at various frequencies show a trend toward exponential. Because shallow water internal waves are dominated by the first mode, uncorrelated micropaths are an unlikely mechanism for producing the observed saturation. [Work supported by ONR.]

Propagation of water wave over a periodically perforated bottom and the band structure

The band structures of shallow water surface wave propagating over a periodically perforated bottom are calculated by the plane wave expansion method，and complete band gaps appear. Through considering three drilling patterns （square holes arrayed in square lattice，circular holes arrayed in square and hexagonal lattices），we find that the number，width，frequency and location of band gaps are related to the area，shape，orientation of drilled holes and the lattice. These rules are similar to those in photonic/ phononic crystals.

A generalized expansion method for nonlinear wave equations

A generalized Jacobian/exponential expansion method for finding the exact traveling wave solutions of a nonlinear partial differential equation is discussed. We use this method to construct many new, previously undiscovered exact solutions for the Boussinesq and modified KdV equations. We also apply it to the shallow long wave approximate equations. New solutions are deduced for this system of partial differential equations.

Non-linear output feedback tracking control for AUVs in shallow wave disturbance condition

This article presents a non-linear output feedback tracking controller deisgn for autonomous underwater vehicles (AUVs) operating in shallow water area. In a shallow water environment, significant disturbances due to shallow water waves affect the motion of marine vehicles greatly. Since it is not energy efficient to counteract the oscillatory disturbances due to waves, it is critical to obtain the wave information or wave induced disturbance information and design an energy efficient controller to reduce the action of actuators to counteract wave disturbances to avoid wear and tear on actuators. In this article, a non-linear observer is first designed to estimate the low frequency (LF) motion of AUVs and to filter out wave-frequency (WF) motion of AUVs due to shallow water wave by using position and attitude measurements. Based on the designed observer, a non-linear output feedback controller is subsequently derived by using the observer backstepping technique. By using this approach, the AUV achieves global exponential tracking without excessive energy consumption to counteract the wave disturbance and also avoids excessive wear and tear on thrusters. Global exponential stability (GES) of overall observer-controller system is proved through Lyapunov stability theory. A set of simulations is carried out by using the KAMBARA (Silpa-Anan 2001) AUV model to demonstrate the performance of the proposed observer and output feedback controller.