Constant Bathymetry Research Articles

Nonlinear evolution of water waves and their dispersion relation in coastal waters

Preliminary results of the derivation of a new phase-resolving (deterministic) spatio-temporal nonlinear model of water wave evolution in nondeep waters with constant bathymetry are presented in this paper. The model is the first of its kind to include a nonlinear dispersion relation and cubic (fourwave) interactions. Simulations show the importance of nonlinear dispersion for bound wave components which, if not properly accounted for, result in inaccurate transfer of wave energy. We also investigate the relative importance of cubic nonlinearity, as compared to a quadratic (three-wave interactions) one, and show that it is non-negligible. The model provides the first step before the incorporation of these extensions into a phase-averaged (stochastic) formulation, which can then be used as a more accurate nonlinear source term for wave forecasting models.

Numerical Study of Spurious Inertial Modes in Shallow Water Models for a Variable Bathymetry

In this study we perform a modal analysis of the linear inviscid shallow water equations using a non constant bathymetry, continuous and discontinuous Galerkin approximations. By extracting the discrete eigenvalues of the resulting algebraic linear system written on the form of a generalized eigenvalue / eigenvector problem we first show that the regular variation of the bathymetry does not prevent the presence of spurious inertial modes when centered finite element pairs are used. Secondly, we show that such spurious modes are not present in discontinuous Galerkin discretizations when all variables are approximated in the same descrete space. Such spurious inertial modes have been found very damageable for the quality of inertia-gravity and Rossby modes in ocean modelling.

Effects of hurricane forward speed and approach angle on storm surges: an idealized numerical experiment

The effects of hurricane forward speed (V) and approach angle (θ) on storm surge are important and a systematic investigation covering possible and continuous ranges of these parameters has not been done before. Here we present such a study with a numerical experiment using the Finite Volume Community Ocean Model (FVCOM). The hurricane track is simplified as a straight line, such that V and θ fully define the motion of the hurricane. The maximum surge is contributed by both free waves and a forced storm surge wave moving with the hurricane. Among the free waves, Kelvin-type waves can only propagate in the down-coast direction. Simulations show that those waves can only have a significant positive storm surge when the hurricane velocity has a down-coast component. The optimal values of V and θ that maximize the storm surge in an idealized semi-circular ocean basin are functions of the bathymetry. For a constant bathymetry, the maximum surge occurs when the hurricane approaches the coast from the normal direction when the free wave generation is minimal; for a stepped bathymetry, the maximum surge occurs at a certain acute approach angle which maximizes the duration of persistent wind forcing; a step-like bathymetry with a sloped shelf is similar to the stepped bathymetry, with the added possibility of landfall resonance when the free and forced waves are moving at about the same velocity. For other cases, the storm surge is smaller, given other parameters (hurricane size, maximum wind speed, etc.) unchanged.

Novel Data Assimilation Algorithm for Nearshore Bathymetry

AbstractIt can be expensive and difficult to collect direct bathymetry data for nearshore regions, especially in high-energy locations where there are temporally and spatially varying bathymetric features like sandbars. As a result, there has been increasing interest in remote assessment techniques for estimating bathymetry. Recent efforts have combined Kalman filter–based techniques with indirect video-based observations for bathymetry inversion. Here, we estimate nearshore bathymetry by utilizing observed wave celerity and wave height, which are related to bathymetry through phase-averaged wave dynamics. We present a modified compressed-state Kalman filter (CSKF) method, a fast and scalable Kalman filter method for linear and nonlinear problems with large numbers of unknowns and measurements, and apply it to two nearshore bathymetry estimation problems. To illustrate the robustness and accuracy of our method, we compare its performance with that of two ensemble-based approaches on twin bathymetry estimation problems with profiles based on surveys taken by the U.S. Army Corps of Engineer Field Research Facility (FRF) in Duck, North Carolina. We first consider an estimation problem for a temporally constant bathymetry profile. Then we estimate bathymetry as it evolves in time. Our results indicate that the CSKF method is more accurate and robust than the ensemble-based methods with the same computational cost. The superior performance is due to the optimal low-rank representation of the covariance matrices.

On the existence of solitary waves for Boussinesq type equations and Cauchy problem for a new conservative model

In this paper, we present a long time existence theory for a new enhanced Boussinesq-Type system with constant bathymetry written in a conservative form. We also prove the existence of solitary wave for a large class of asymptotic models, including Beji-Nadaoka, Madsen-Sorensen and Nwogu equations. Furthermore, we give a procedure to calculate numerically these particular solutions and we present some effective computations.

Elliptic Harbor Wave Model with Perfectly Matched Layer and Exterior Bathymetry Effects

AbstractStandard strategies for dealing with the Sommerfeld condition in elliptic mild-slope models require strong assumptions on the wave field in the region exterior to the computational domain. More precisely, constant bathymetry along (and beyond) the open boundary, and parabolic approximations–based boundary conditions are usually imposed. Generally, these restrictions require large computational domains, implying higher costs for the numerical solver. An alternative method for coastal/harbor applications is proposed here. This approach is based on a perfectly matched layer (PML) that incorporates the effects of the exterior bathymetry. The model only requires constant exterior depth in the alongshore direction, a common approach used for idealizing the exterior bathymetry in elliptic models. In opposition to standard open boundary conditions for mild-slope models, the features of the proposed PML approach include (1) completely noncollinear coastlines, (2) better representation of the real unbounded...

Matched mode geoacoustic inversion of broadband signals in shallow water

A matched mode geoacoustic inversion is presented for broadband acoustic signals recorded in experiments at a shallow water region of the New England Bight. First the modal behavior of the waveguide in the presence of 3D effects due to the water column and bottom bathymetry is examined. Spatial measurements of the temperature and salinity profiles and bathymetry provide data for calculating 3D propagation of the broadband acoustic wave field. Frequency shift of the broadband signal along one of the source-receiver paths in the experiment is attributed to internal waves propagating in the water column. Geoacoustic model parameters of the seabed along a source-receiver track with constant bathymetry are estimated by matched mode processing through matching the phase of the mode signals recorded at a Vertical Line Array. A warping transform was applied to identify and isolate modes, and the filtered modes were inverse transformed back to the time domain to reconstruct the mode signals. The inversion results are compared with the results from other inversion methods to assess the performance of the inversion scheme for estimating reliable values of the geoacoustic parameters in the experiment region.

A GIS-Based Tool for Representing Larval Dispersal for Marine Reserve Selection*

Increasing efforts to implement marine protected areas (MPAs) as a means of managing marine ecosystems have created a need for evaluating potential spatial management plans. Almost all marine populations are metapopulations, connected reproductively by the dispersal of pelagic larvae. Models of marine population dynamics must account for larval connectivity, but despite recent advances connectivity patterns are still poorly understood. To allow more informed decision making when complete information on dispersal is lacking, we have developed a method based on geographic information systems (GIS) for representing larval dispersal distributions based on bathymetry and typical flows in the coastal ocean. These distributions reflect (1) generally greater flow in directions along, rather than across, lines of constant bathymetry and (2) lesser flow in shallow near-shore waters. We demonstrate how to parameterize this two-dimensional method for depicting larval dispersal based on comparisons to local oceanographic data. We then compare the predictions of the two-dimensional method to those of a simpler one-dimensional alternative in a population model used to evaluate proposed MPAs along the coast of central California. The method produces reasonable larval dispersal patterns and appears to include the effects of bathymetry on population dynamics better than commonly used one-dimensional methods and without requiring the significantly greater investment of developing particle-tracking circulation models. An important advantage of a two-dimensional approach is more realistic portrayal of the dependence of population persistence on the cross-shelf dimension of available habitat.

A numerical analysis of a class of generalized Boussinesq‐type equations using continuous/discontinuous FEM

SUMMARYAn improved class of Boussinesq systems of an arbitrary order using a wave surface elevation and velocity potential formulation is derived. Dissipative effects and wave generation due to a time‐dependent varying seabed are included. Thus, high‐order source functions are considered. For the reduction of the system order and maintenance of some dispersive characteristics of the higher‐order models, an extra O(μ2n+2) term (n ∈ ℕ) is included in the velocity potential expansion. We introduce a nonlocal continuous/discontinuous Galerkin FEM with inner penalty terms to calculate the numerical solutions of the improved fourth‐order models. The discretization of the spatial variables is made using continuous P2 Lagrange elements. A predictor‐corrector scheme with an initialization given by an explicit Runge–Kutta method is also used for the time‐variable integration. Moreover, a CFL‐type condition is deduced for the linear problem with a constant bathymetry. To demonstrate the applicability of the model, we considered several test cases. Improved stability is achieved. Copyright © 2011 John Wiley & Sons, Ltd.

Extraction of acoustic normal mode depth functions using vertical line array data.

A method for extracting the normal modes of acoustic propagation in the shallow ocean from sound recorded on a vertical line array (VLA) of hydrophones as a source travels nearby is presented. The mode extraction is accomplished by performing a singular value decomposition (SVD) of individual frequency components of the signal's temporally averaged, spatial cross-spectral density matrix. The SVD produces a matrix containing a mutually orthogonal set of basis functions, which are proportional to the depth-dependent normal modes, and a diagonal matrix containing the singular values, which are proportional to the modal source excitations and mode eigenvalues. The conditions under which the method is expected to work are found to be (1) sufficient depth sampling of the propagating modes by the VLA receivers; (2) sufficient source-VLA range sampling, and (3) sufficient range interval traversed by the source. The mode extraction method is applied to data from the Area Characterization Test II, conducted in September 1993 in the Hudson Canyon Area off the New Jersey coast. Modes are successfully extracted from cw tones recorded while (1) the source traveled along a range-independent track with constant bathymetry and (2) the source traveled up-slope with gradual changes in bathymetry. In addition, modes are successfully extracted at multiple frequencies from ambient noise.

Circulation Induced by Tidal Jets

Steady potential motion induced by a tidal jet in an ebbing ocean is analyzed by representing the jet as a distribution of sink singularities and relating the sink strength to jet solutions for steady bottom‐frictional turbulent jets over constant bathymetry. Circulation patterns are determined for various inlet configurations of practical interest and for different values of the bottom friction parameter. Particular attention is given to the jet induced alongshore current which is shown to be sensitive to the presence of jetties and also to the bottom friction experienced by the jet. The present theory is applied to Fort Pierce Inlet, Florida, where severe erosion is known to exist on both sides of the inlet. It is suggested that the alongshore current, albeit small in magnitude, is a potential contributor to the movement of suspended beach material toward the inlet due to its continual occurrence during every ebb tide.