Variable Bathymetry Research Articles

Propagating of Obliquely Incident, Weakly Viscous Waves over Variable Bathymetry

ABSTRACT Tsai, C.-C.; Lin, Y.-T., and Hsu, T.-W., 2016. Propagating of obliquely incident weakly viscous waves over variable bathymetry. A linear theory for weakly viscous waves propagating obliquely over variable topographies is introduced without considering the boundary layers. Using an eigenfunction-matching method, which involves including the evanescent modes that satisfy the matching conditions, the present problem can be transformed into a system of linear equations. Furthermore, by using a weak-viscosity hypothesis, the proposed method can be degenerated to the traditional eigenfunction-matching method of the potential flow if both the molecular viscosity and the bottom friction are discarded. The applicability of this model was confirmed by comparing it with available theoretical, numerical, and experimental results for both of the normal and oblique incidences.

Circulation and meltwater distribution in the Bellingshausen Sea: From shelf break to coast

AbstractWest Antarctic ice shelves have thinned dramatically over recent decades. Oceanographic measurements that explore connections between offshore warming and transport across a continental shelf with variable bathymetry toward ice shelves are needed to constrain future changes in melt rates. Six years of seal‐acquired observations provide extensive hydrographic coverage in the Bellingshausen Sea, where ship‐based measurements are scarce. Warm but modified Circumpolar Deep Water floods the shelf and establishes a cyclonic circulation within the Belgica Trough with flow extending toward the coast along the eastern boundaries and returning to the shelf break along western boundaries. These boundary currents are the primary water mass pathways that carry heat toward the coast and advect ice shelf meltwater offshore. The modified Circumpolar Deep Water and meltwater mixtures shoal and thin as they approach the continental slope before flowing westward at the shelf break, suggesting the presence of the Antarctic Slope Current. Constraining meltwater pathways is a key step in monitoring the stability of the West Antarctic Ice Sheet.

Time-domain, shallow-water hydroelastic analysis of VLFS elastically connected to the seabed

In order to ensure the safe operation of a VLFS, a combination of mooring, breakwater and other motion reducing systems is employed. In the present work, the transient hydroelastic response of a floating, thin elastic plate, elastically connected to the seabed, is examined. The plate is modelled as an Euler-Bernoulli strip, while the linearized shallow water equations are used for the hydrodynamic modelling. Elastic connectors are approximated by a series of simple spring-dashpot systems positioned along the strip. A higher order finite element scheme is employed for the calculation of the hydroelastic response of the strip-connector configuration over the shallow bathymetry. After the definition of the initial-boundary value problem, its variational form is derived and discussed. Next, on the basis of the aforementioned formulation, an energy balance expression is obtained. The effect of variable bathymetry on the response of a two connector-strip system is examined by means of three seabed profiles featuring a flat bottom, an upslope and a downslope environment. For the flat bottom case, the strip response mitigating effect exerted by the employment of two and three elastic connectors is considered. Finally, by means of the derived energy balance equation, the energy exchange is monitored, providing a valuable insight into the transient phenomena that take place in the studied configurations.

Effects of seabed curvature on the scattered acoustic field

Knowledge of the acoustic properties of the sea bottom provides valuable input into the development and performance of information processing algorithms designed to detect and identify, e.g., man-made objects near to, or embedded in, the seabed. This information is also useful for improving modeling and simulation results in reverberation-limited environments. A critical challenge in this area is in allowing for range dependent bathymetry as it can affect the propagating acoustic field in ways that are difficult to predict. Typical approaches to modeling variable bathymetry assume either piecewise flat or tilted (linear) geometry; however, this only serves as a discrete approximation to reality. This study investigates the effects of mild curvature on the 3D acoustic field in frequency and angle space via finite difference time domain simulations. We present results and discuss the implications on forward modeling of the acoustic interaction with the seabed. [Work supported by the Strategic Environmental Research and Development Program (SERDP).]

HOS-ocean: Open-source solver for nonlinear waves in open ocean based on High-Order Spectral method

HOS-ocean is an efficient High-Order Spectral code developed to solve the deterministic propagation of nonlinear wavefields in open ocean. HOS-ocean is released as open-source, developed and distributed under the terms of GNU General Public License (GPLv3). Along with the source code, a documentation under wiki format is available which makes easy the compilation and execution of the source files. The code has been shown to be accurate and efficient. Program summaryProgram title: HOS-oceanCatalogue identifier: AEZS_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEZS_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 30332No. of bytes in distributed program, including test data, etc.: 2656194Distribution format: tar.gzProgramming language: Fortran.Computer: Tested on Intel Xeon E5504 and Intel Core i7.Operating system: Any system with a Fortran compiler: tested on Linux, OS X and Windows 7.RAM: From several MB up to several GB, depending on problem (512×512, M=3: 385 MB and 256×256, M=3: 99 MB)Classification: 4.12.External routines: FFTW 3.3.4 [1], LAPACK [2] and makedepf90 (linux.die.net/man/1/makedepf90)Nature of problem:HOS-ocean has been developed to study the propagation of highly nonlinear sea-states over large domains and long duration.Solution method:HOS-ocean is an implementation of the High-Order Spectral method, which solves the problem formulated on the free surface by means of a pseudo-spectral method.Restrictions:HOS-ocean is dedicated to the propagation of wave fields in infinite and finite constant depth, the evolution over variable bathymetry is not treated. Furthermore, simulations are restricted to non-breaking waves.Running time:2D simulation of irregular wavefield with Nx=1024 modes and an HOS order M=5: t≅2.010−1s per wave period3D simulation of irregular wavefield with Nx=256, Ny=256 modes, an HOS order M=3: t≅10s per wave period.

Mechanical balance laws for fully nonlinear and weakly dispersive water waves

The Serre–Green–Naghdi system is a coupled, fully nonlinear system of dispersive evolution equations which approximates the full water wave problem. The system is known to describe accurately the wave motion at the surface of an incompressible inviscid fluid in the case when the fluid flow is irrotational and two-dimensional. The system is an extension of the well known shallow-water system to the situation where the waves are long, but not so long that dispersive effects can be neglected. In the current work, the focus is on deriving mass, momentum and energy densities and fluxes associated with the Serre–Green–Naghdi system. These quantities arise from imposing balance equations of the same asymptotic order as the evolution equations. In the case of an even bed, the conservation equations are satisfied exactly by the solutions of the Serre–Green–Naghdi system. The case of variable bathymetry is more complicated, with mass and momentum conservation satisfied exactly, and energy conservation satisfied only in a global sense. In all cases, the quantities found here reduce correctly to the corresponding counterparts in both the Boussinesq and the shallow-water scaling. One consequence of the present analysis is that the energy loss appearing in the shallow-water theory of undular bores is fully compensated by the emergence of oscillations behind the bore front. The situation is analyzed numerically by approximating solutions of the Serre–Green–Naghdi equations using a finite-element discretization coupled with an adaptive Runge–Kutta time integration scheme, and it is found that the energy is indeed conserved nearly to machine precision. As a second application, the shoaling of solitary waves on a plane beach is analyzed. It appears that the Serre–Green–Naghdi equations are capable of predicting both the shape of the free surface and the evolution of kinetic and potential energy with good accuracy in the early stages of shoaling.

Meteo-marine parameters for highly variable environment in coastal regions from satellite radar images

The German Bight of the North Sea is the area with highly variable sea state conditions, intensive ship traffic and with a high density of offshore installations, e.g. wind farms in use and under construction. Ship navigation and the docking on offshore constructions is impeded by significant wave heights HS>1.3m. For these reasons, improvements are required in recognition and forecasting of sea state HS in the range 0–3m. Thus, this necessitates the development of new methods to determine the distribution of meteo-marine parameters from remote sensing data with an accuracy of decimetres for HS. The operationalization of these methods then allows the robust automatic processing in near real time (NRT) to support forecast agencies by providing validations for model results.A new empirical algorithm XWAVE_C (C=coastal) for estimation of significant wave height from X-band satellite-borne Synthetic Aperture Radar (SAR) data has been developed, adopted for coastal applications using TerraSAR-X (TS-X) and Tandem-X (TD-X) satellites in the German Bight and implemented into the Sea Sate Processor (SSP) for fully automatic processing for NRT services.The algorithm is based on the spectral analysis of subscenes and the model function uses integrated image spectra parameters as well as local wind information from the analyzed subscene. The algorithm is able to recognize and remove the influence of non-sea state produced signals in the Wadden Sea areas such as dry sandbars as well as nonlinear SAR image distortions produced by e.g. short wind waves and breaking waves. Also parameters of very short waves, which are not visible in SAR images and produce only unsystematic clutter, can be accurately estimated. The SSP includes XWAVE_C, a pre-filtering procedure for removing artefacts such as ships, seamarks, buoys, offshore constructions and slicks, and an additional procedure performing a check of results based on the statistics of the whole scene.The SSP allows an automatic processing of TS-X images with an error RMSE=25cm and Scatter Index SI=20% for total significant wave height HS from sequences of TS-X StripMap images with a coverage of ∼30km×300km across the German Bight. The SSP was tuned spatially with model data of the German Weather Service’s (DWD) CWAM (Coastal WAve Model) with 900m horizontal resolution and tuned in situ with 6 buoys located in DWD model domain in the German Bight. The collected, processed and analyzed data base for the German Bight consists of more than 60 TS-X StripMap scenes/overflights with more than 200 images since 2013 with sea state acquired in the domain HS=0–7m with a mean value of 1.25m over all available scenes at buoy locations.The paper addresses the development and implementation of XWAVE_C, and presents the possibilities of SSP delivering sea state fields by reproducing local HS variations connected with local wind gusts, variable bathymetry and moving wind fronts under different weather conditions.

3D hydroelastic analysis of very large floating bodies over variable bathymetry regions

The coupled-mode model developed by Belibassakis and Athanassoulis (J Fluid Mech 531:221–249, 2005) is extended and applied to the hydroelastic analysis of three-dimensional large floating bodies of shallow draft lying over variable bathymetry regions. The method is also applicable to the problem of wave interaction with ice sheets of small thickness. A general bathymetry is assumed, characterized by a continuous depth function, joining two regions of constant, but possibly different, depth. We consider the scattering problem of harmonic incident surface waves, under the combined effects of variable bathymetry and a floating elastic plate of orthogonal planform shape. Under the assumption of small-amplitude waves and plate deflections, the hydroelastic problem is formulated within the context of linearized water-wave and thin elastic-plate theory. To consistently treat the wave field beneath the elastic floating plate, down to the sloping bottom boundary, a complete, local, hydroelastic-mode series expansion of the wave field is used, enhanced by an appropriate sloping-bottom mode. The latter enables the consistent satisfaction of the Neumann bottom-boundary condition on a general topography. Numerical results concerning floating structures over flat and inhomogeneous seabed are presented, and the effects of wave direction, bottom slope and bottom corrugations on the hydroelastic responses are discussed.

Development and validation of a non-linear spectral model for water waves over variable depth

In the present paper two numerical schemes for propagating waves over a variable bathymetry in an existing High-Order Spectral (HOS) model are introduced. The first scheme was first developed by Liu and Yue (1998), and the second one is an improved scheme which considers two independent orders of non-linearity: one for the bottom and one for the free-surface elevation. We investigate the numerical properties (accuracy, convergence rate, efficiency) of both schemes with respect to the numerical parameters on a simple configuration. To validate the proposed schemes, we first consider Bragg reflection from a sinusoidal bottom patch — as an example of a small bottom variation around a mean water depth. The second validation case focuses on a larger bottom variation with the study of the shoaling of linear waves. Finally, an application is performed to demonstrate the applicability of the proposed model to non-linear cases where the bottom variation is important. In this concern, the very well documented test case of a 2D underwater bar is studied in detail. Comparisons are provided with both experimental and numerical results.

Comparison between Consistent Coupled-Mode System and Eigenfunction Matching Method for Solving Water Wave Scattering

Both the consistent coupled-mode system (CCMS) and the eigenfunction matching method (EMM) are well-known models for simulating the propagation of small-amplitude water waves over variable bathymetry. In this study, a thorough comparison is performed through numerical experiments. For the CCMS, a bottom-sloping mode is coupled in the mildslope equation with evanescent modes, then the CCMS are discretized by the finite-element method with high-order shape functions. For the EMM, the bottom profile is approximated in terms of successive flat shelves separated by abrupt steps, and then eigensolutions on the shelves are matched by the conservation of mass and momentum. To perform error analysis, numerical solutions are compared with Roseau's analytical solution and the semi-analytical solutions of the integral equation method. Numerical results indicate that the CCMS and EMM are accurate up to six and four decimal places, respectively. On the other hand, the EMM is more efficient for short waves because multiple waves can be approximated by few shelves. In addition, improvements in their accuracy over the mild-slope system without bottomsloping mode are shown to be significant.

Boundary element formulation of the Mild-Slope equation for harmonic water waves propagating over unidirectional variable bathymetries

This paper presents a boundary element formulation for the solution of the Mild-Slope equation in wave propagation problems with variable water depth in one direction. Based on Green׳s function approximation proposed by Belibassakis [1], a complete fundamental-solution kernel is developed and combined with a boundary element scheme for the solution of water wave propagation problems in closed and open domains where the bathymetry changes arbitrarily and smoothly in a preferential direction. The ability of the proposed formulation to accurately represent wave phenomena like refraction, reflection, diffraction and shoaling, is demonstrated with the solution of some example problems, in which arbitrary geometries and variable seabed profiles with slopes up to 1:3 are considered. The obtained results are also compared with theoretical solutions, showing an excellent agreement that demonstrates its potential.

A computational method to predict and study underwater noise due to pile driving.

A hybrid modeling approach that uses the parabolic equation (PE) with an empirical source model is presented to study and predict the underwater noise due to pile driving in shallow, inhomogeneous environments over long propagation ranges. The empirical source model uses a phased point source array to simulate the time-dependent pile source. The pile source is coupled with a broadband application of a PE wave propagation model that includes range dependent geoacoustic properties and bathymetry. Simulation results are shown to be in good agreement with several acoustic observations of pile driving in the Columbia River between Portland, OR and Vancouver, WA. The model is further applied to predict sound levels in the Columbia River and study the effects of variable bathymetry and sediment configurations on underwater sound levels.

WAVE PROPAGATION ON A FLUME: NUMERICAL SIMULATION

This paper presents the numerical simulations done by using the waves2Foam, an OpenFOAM® library, to simulate the propagation of regular waves without breaking in a three-dimensional flume. The numerical code solves the unsteady Navier-Stokes equations and uses a Volume-of-Fluid (VoF) method to identify the free-surface. A regular incident wave with a 1.5s period and 0.1m wave height was considered. This is one of the conditions, from the wide range of wave flume tests conducted at the National Laboratory for Civil Engineering (LNEC), whose objective was to analyze the hydrodynamics of wave transformation and wave breaking for different incident conditions over a variable bathymetry. Comparisons are made between the numerical and the experimental results. These comparisons include time-series of wave-gauges records at several locations along the flume and the corresponding amplitude spectra; significant wave height and average period evolution along the flume; time-series of the velocity components at one section of the flume, measured at the middle of the water column; and hodograph representation of the velocity components, in the middle of the water column, in the xy, xz, and yz planes, along the flume. It was found that the numerical results obtained are close to the experimental data. The observed differences are attributable to numerical inaccuracies as well as the differences between the wave generation method in the numerical and experimental tests.

Numerical Simulation of Non-linear Water Waves over Variable Bathymetry

Abstract We present a numerical method for the computation of surface gravity waves in the presence of variable bathymetry. The evolution is described by an alternative form of the usual Hamiltonian system where the substrate boundary value problem for the potential gives its place to an infinite system of second order partial differential equations on horizontal fields with appropriate boundary conditions and a linear algebraic constraint. We employ the finite difference method to solve this time-independent problem and the classic Runge-Kutta method for the time-integration of the evolution equations. Special care is taken for the consistent wave generation and absorption at the lateral boundaries of a reasonably sized computational domain. Simulations of non-linear waves in the presence of undulated bottoms (Bragg reflection) are presented and validated against experimental measurements.

A DEPTH-INTEGRATED EQUATION FOR LARGE SCALE MODELING OF TSUNAMI IN WEAKLY COMPRESSIBLE FLUID

Pressure waves generated by fast seabed movement in weakly compressible sea water, namely hydro-acoustic waves travel at the sound celerity in water (about 1500 m/s). These waves are precursors of the counterpart long free-surface gravity waves and contain significant information on the tsunamigenic source. Measurement of hydro-acoustic waves can therefore anticipate the tsunami arrival and significantly enhance the promptness and accuracy of tsunami early warning systems. In this paper derivation of a novel depth-integrated numerical model for reproduction of hydroacoustic waves is presented and the application of this computationally ecient model on two devastating historical tsunamis of Mediterranean Sea in real bathymetry analyzed to reveal the eect of variable bathymetry. On the basis of the model results, some hints for deep sea observatory are given.

A normal mode inner product to account for acoustic propagation over horizontally variable bathymetry

This talk will consider the conversion of normal mode functions over local variations in bathymetry. Mode conversions are accomplished through an inner product, which enables the modes compromising the field at each range-dependent step to be written as a function of those in the preceding step. The efficiency of the method results from maintaining a stable number of modes throughout the calculation of the acoustic field. A verification of the inner product is presented by comparing results from its implementation in a simple mode model to that of a closed-form solution for the acoustic wedge environment. A solution to the more general problem of variable bottom slope, which involves a decomposition of bathymetric profiles into a sequence of wedge environments, will also be discussed. The overall goal of this research is the development and implementation of a rigorous shallow water acoustic propagation solution which executes in a time window to support tactical applications.

Modeling process asymmetries with Laplace moving average

Many records in environmental science exhibit asymmetries: for example in shallow water and with variable bathymetry, the sea wave time series shows front–back asymmetries and different shapes for crests and troughs. In such situation, numerical models are available but their computational cost and complexity are high. A stochastic process aimed at modeling such asymmetries has recently been proposed, the Laplace moving average process, which consists in applying a linear filter on a non-Gaussian noise built using the generalized Laplace distribution. The objective is to propose a new non-parametric estimator for the kernel involved in the definition of this process. Results based on a comprehensive numerical study will be shown in order to evaluate the performances of the proposed method.

Cloaking of a vertical cylinder in waves using variable bathymetry

AbstractThe paper describes a process which allows a vertical circular cylinder subject to plane monochromatic surface gravity waves to appear invisible to the far-field observer. This is achieved by surrounding the cylinder with an annular region of variable bathymetry. Two approaches are taken to investigate this effect. First a mild-slope approximation is applied to the governing linearised three-dimensional water wave equations to formulate a depth-averaged two-dimensional wave equation with varying wavenumber over the variable bathmetry. This is then solved by formulating a domain integral equation, solved numerically by discretisation. For a given set of geometrical and wave parameters, the bathymetry is selected by a numerical optimisation process and it is shown that the scattering cross-section is reduced towards zero with increasing refinement of the bathymetry. A fully three-dimensional boundary-element method, based on the WAMIT solver (see www.wamit.com) but adapted here to allow for depressions in the bed, is used to assess the accuracy of the mild-slope results and then further numerically optimise the bathymetry towards a cloaking structure. Numerical results provide strong evidence that perfect cloaking is possible for the fully three-dimensional problem. One practical application of the results is that cloaking implies a reduced mean drift force on the cylinder.

Wind-driven waves in a shallow estuarine lake with muddy substrates: St Lucia, South Africa

Zikhali, V., Tirok, K., Stretch, D., 2014. Wind-driven waves in a shallow estuarine lake with muddy substrates: St Lucia, South Africa. In: Green, A.N. and Cooper, J.A.G. (eds.), Proceedings 13th International Coastal Symposium (Durban, South Africa), Journal of Coastal Research, Special Issue No. 70, pp. 729-735, ISSN 0749-0208.Wind-waves in shallow lakes or estuaries with muddy substrates can drive sediment re-suspension and cause high turbidity levels that can negatively impact the productivity of photosynthetic organisms. This investigation evaluates the efficacy of a simple semi-empirical model (Young and Verhagen, 1996 : Coastal Engineering, 29, 47–78) for predicting the wave characteristics in these systems in order to include their effects in ecosystem models. The southern basin of the St Lucia estuarine lake in South Africa was used for a case study. Average depths are about 1 m with fetches up to approximately 10 km. Substrate materials vary from sandy to muddy with deeper locations predominantly the latter. An array of pressure sensing wave poles was deployed to measure significant wave heights and periods to compare with model predictions. The influence of the wind speed, fetch, fetch-averaged depth, and substrate composition were evaluated. Most of the observed waves were fetch limited during the conditions that prevailed during the two field trips. The results indicate that the model adequately captures the high energy wave events for persistent wind speeds and directions, but that there is considerable variability in its performance generally. Some of this variability can be attributed to difficulties in estimating appropriate fetch and depth parameters for variable winds and in the context of a lake with compound shape and variable bathymetry. There was no clear evidence of significant wave attenuation due to the muddy substrates.

Bearing estimation from vector sensor intensity processing for autonomous underwater gliders

Data have been collected on acoustic vector sensors mounted on autonomous underwater gliders in the Monterey Bay during 2012–2013. In this work, we show results of intensity processing to estimate bearing to impulsive sources of interest. These sources included small explosive shots deployed by local fisherman, and humpback whale vocalizations. While the highly impulsive shot data produced unambiguous bearing estimations, the longer duration whale vocalizations showed a fairly wide spread in bearing. The causes of the ambiguity in bearing estimation are investigated in the context of the highly variable bathymetry of the Monterey Bay Canyon, as well as the coherent multipath interference in the longer duration calls.