Spectral methods for coastal-trapped waves

This note describes efficient and accurate spectral numerical schemes to compute both propagating and evanescent free baroclinic coastal-trapped waves over general depth profiles for arbitrary vertical density profiles in horizontally semi-infinite domains. The general problem is recast into a linear eigenvalue problem for the along-shore wavenumber k, which can be solved directly, without initial guesses or searching, using any standard linear eigenvalue package to find real and complex eigenmodes simultaneously. An equivalent recasting gives the linearised eigenvalue problem for the frequency ω. A novel, nonlinear, boundary condition is derived that is particularly effective for modes whose offshore decay is weak, as in the long-wave limit. The resulting nonlinear eigenvalue problem is solved by a highly efficient Newton-Kantorovich algorithm.     

Bench–shelf system dynamic characteristics and their effects on equipment and contents

Economic losses during past earthquakes are strongly associated with damage and failure to nonstructural equipment and contents. Among the vast types of nonstructural elements, one important category, is scientific equipment in biological or chemical laboratories. These equipment are often mounted on heavy ceramic bench‐tops of bench–shelf systems, which in turn may amplify the dynamic motions imposed. To investigate the seismic response of these types of systems, a series of shake table and field experiments were conducted considering different representative bench and shelf‐mounted equipment and contents. Results from shake table experiments indicate that these equipment are generally sliding‐dominated. In addition, the bench–shelf system is observed to be very stiff and when lightly loaded, has a fundamental frequency between 10 and 16 Hz. An approximate 50% reduction in the first and second fundamental frequencies is observed considering practical loading conditions. Insight into a broader range of system response is provided by conducting eigenvalue and time history analyses. Non‐linear regression through the numerical data indicate acceleration amplification ratios Ω range from 2.6 to 1.4 and from 4.3 to 1.6, for fixed–fixed and pinned–pinned conditions, respectively. Both the experimental and numerical results support the importance of determining the potential dynamic amplification of motion in the context of accurately determining the maximum sliding displacement of support equipment and contents. Copyright © 2006 John Wiley & Sons, Ltd.     

A NEW APPROACH TO THE INVERSION OF SEISMIC DATA*

A new inversion method for seismic reflection data is based on impedance concepts and uses transformations to reduce the essentially non-linear problem to a linear eigenvalue formulation without approximation. A set of reflection data has been inverted using this method. The characteristic impedances of the layers so determined are compared with log data from a reference borehole.     

Tides in the Gulf of Kutch,India

In the Gulf of Kutch on the northwestern shelf of India the semi-diurnal constituents M₂ and S₂ get amplified approximately threefold. In contrast, amplification of the diurnal constituents K₁ and O₁ is small. From analytical and numerical solutions of linear, viscous, cross-sectionally averaged equations for tidal motion in a channel, it is seen that the observed amplification results from a combination of quarter-wavelength resonance, geometric effect due to decrease in width of the channel, and friction.     

Response of non-classically damped structures in the modal subspace

The evaluation of the dynamic response of non-classically damped linear structures requires the solution of an eigenproblem with complex eigenvalues and modal shapes. Since in practice only a small number of complex modes are needed, the complex eigenvalue problem is solved in the modal subspace in which the generalized damping matrix is not uncoupled by classical real modes. It follows that the evaluation of the structural response requires in both cases the determination of complex modes by numerical techniques, which are not as robust as techniques currently used for the solution of the real eigenvalue problem, and the use of complex algebra. In the present paper an unconditionally stable step-by-step procedure is presented for the response of non-classically damped structures in the modal subspace without using complex quantities. The method is based on the evaluation of the fundamental operator in approximated form of the numerical procedure. In addition, the method can be easily modified to incorporate the modal superposition pseudo-static correction terms.     

Analytic expressions for the velocity sensitivity to the elastic moduli for the most general anisotropic media

For non‐linear kinematic inversion of elastic anisotropy parameters and related investigations of the sensitivity of seismic data, the derivatives of the wavespeed (phase velocity and group velocity) with respect to the individual elastic moduli are required. This paper presents two analytic methods, called the eigenvalue and eigenvector methods, to compute the derivatives of the wavespeeds for wave propagation in a general anisotropic medium, which may be defined by up to 21 density‐normalized elastic moduli. The first method employs a simple and compact form of the eigenvalue (phase velocity) and a general form of the group velocity, and directly yields general expressions of the derivatives for the three wave modes (qP, qS₁, qS₂). The second method applies simple eigenvector solutions of the three wave modes and leads to other general forms of the derivatives. These analytic formulae show that the derivatives are, in general, functions of the 21 elastic moduli as well as the wave propagation direction, and they reflect the sensitivity of the wavespeeds to the individual elastic moduli. Meanwhile, we give results of numerical investigations with some examples for particular simplified forms of anisotropy. They show that the eigenvalue method is suitable for the qP‐, qS₁‐ and qS₂‐wave computations and mitigates the singularity problem for the two quasi‐shear waves. The eigenvector method is preferable to the eigenvalue method for the group velocity and the derivative of the phase velocity because it involves simpler expressions and independent computations, but for the derivative of the group velocity the derivative of the eigenvector is required. Both methods tackle the singularity problem and are applicable to any degree of seismic anisotropy for all three wave modes.     

Effect of the mesoscale hydrographic features on larval fish distribution across the shelf break of East China Sea

Surveys of fish larvae and oceanographic conditions were conducted along transects across the shelf break region in the East China Sea (ECS) in May 2001. The objective of this study was to investigate the distribution and assemblages of fish larvae across the shelf break region and their relationships to mesoscale hydrographic features. There was a warm surface streamer from the Kuroshio which extended toward the shelf region of the ECS; concurrently, we observed the intrusion of the less saline shelf water into the subsurface layer towards the offshore. In all 66 taxa (65 families and 1 order) of larvae were collected by oblique net tows using a bongo net, the larval fish density and number of families sampled were lower in the low-temperature area. Based on cluster analysis and environmental factor, two larval fish assemblages were identified: off-shelf/Kuroshio and shelf break. In shelf break assemblage, Auxis spp. and Diaphus spp. were abundant in the warm surface streamer. In contrast, Maurolicus japonicus, Synagrops spp., Bregmaceros sp. and Champsodon spp. were found in the off-shelf region where the offshore intrusion of less-saline water occurred. This different pattern of the horizontal distribution would reflect vertical distribution of the larvae in the water column. Moreover, copepod nauplii density was higher in the upper layer of the shelf region. Thus, dynamic interactions between the Kuroshio and shelf waters at the frontal boundary are concluded to potentially affect larval fish transport and prey availability.     

The seasonal occurrence of Neocalanus cristatus,Neocalanus plumchrus,and Eucalanus bungii over the shelf of the northern Gulf of Alaska

Field collections of zooplankton from the northern Gulf of Alaska demonstrate that the large oceanic copepods Neocalanus cristatus, N. plumchrus, and Eucalanus bungii are seasonal members of shelf and coastal zooplankton communities. They account for more than 25% of the net zooplankton biomass (0.333 mm) during the spring and summer months.The presence, absence, and subsequent reappearance of these copepods over the shelf is apparently related to their seasonal shifts in vertical distributions and to a persistent pattern of onshore Ekman transport. During periods when these copepods occur in the surface water of the bordering deep ocean (late autumn to summer) they are swept shoreward over the shelf. During times when they occur below the Ekman layer, they are generally absent from the shelf.The ecological significance of this seasonally introduced biomass is discussed in terms of what is understood about these species from studies elsewhere in the North Pacific Ocean.     

Celtic Sea and Armorican current structure and the vertical distributions of temperature and chlorophyll

Vertical sections of temperature and chlorophyll a across the slopes and shelf of the Celtic Sea in the summer show the characteristic regimes; oceanic, slope, shelf, frontal, and mixed. Increases of surface chlorophyll a are commonly observed along the shelf tidal fronts where the thermocline outcrops at the surface, and also at the shelf-break. The variations in phytoplankton biomass are most readily interpreted in terms of the effects of physical mixing processes due to wind and tide on the availability of inorganic nutrients and light energy. On the shelf, mixing processes, both due to internal waves, inertial currents, and to boundary induced turbulence caused by tidal shear associated with the sea floor, play an important role in determining the observed vertical structures. A numerical model is used to define regions where tidal mixing processes are likely to be relatively important and provides the physical framework for interpreting the temperature and chlorophyll a profiles.     

Variability of the subtropical shelf front off eastern South America: Winter 2003 and summer 2004

Hydrographic data collected during surveys carried out in austral winter 2003 and summer 2004 are used to analyze the distributions of temperature (T) and salinity (S) over the continental shelf and slope of eastern South America between 27°S and 39°S. The water mass structure and the characteristics of the transition between subantarctic and subtropical shelf water (STSW), referred to as the subtropical shelf front (STSF), as revealed by the vertical structure of temperature and salinity are discussed. During both surveys, the front intensifies downward and extends southwestward from the near coastal zone at 33°S to the shelf break at 36°S. In austral winter subantarctic shelf water (SASW), derived from the northern Patagonia shelf, forms a vertically coherent cold wedge of low salinity waters that locally separate the outer shelf STSW from the fresher inner shelf Plata Plume Water (PPW) derived from the Río de la Plata. Winter T–S diagrams and cross-shelf T and S distributions indicate that mixtures of PPW and tropical water only occur beyond the northernmost extent of pure SASW, and form STSW and an inverted thermocline characteristic of this region. In summer 2004, dilution of Tropical water (TW) occurs at two distinct levels: a warm near surface layer, associated to PPW–TW mixtures, similar to but significantly warmer than winter STSW, and a colder (T∼16 °C) salinity minimum layer at 40–50 m depth, created by SASW–STSW mixtures across the STSF. In winter, the salinity distribution controls the density structure creating a cross-shore density gradient, which prevents isopycnal mixing across the STSF. Temperature stratification in summer induces a sharp pycnocline providing cross-shelf isopycnal connections across the STSF. Cooling and freshening of the upper layer observed at stations collected along the western edge of the Brazil Current suggest offshore export of shelf waters. Low T and S filaments, evident along the shelf break in the winter data, suggest that submesoscale eddies may enhance the property exchange across the shelf break. These observations suggest that as the subsurface shelf waters converge at the STSF, they flow southward along the front and are expelled offshore, primarily along the front axis.     

Free and steady state vibration of non-linear structures using a finite element–non-linear eigenvalue technique

The problem of free vibration of non-linear structures is considered initially. It is shown that this problem can be represented as a non-linear eigenvalue problem. Variational principles for non-linear eigenvalue problems are defined. These variational principles are implemented with finite element models to define numerical approximations for the free vibration problem. The solution of these approximate equations provides a set of non-linear modal vectors and natural frequencies which vary with the amplitude of the solution. The non-linear eigenvalue parameters can be used in modal expansion approximations for the non-linear transient or steady state response of structural systems. To demonstrate the proposed techniques the free vibration and steady state vibration characteristics of a geometrically non-linear circular plate are determined.     

A computationally efficient algorithm for the solution of eigenproblems for large structures with non-proportional damping using Lanczos method

In this paper, a solution method is presented to solve the eigenproblem arising in the dynamic analysis of non-proportional damping systems with symmetric matrices. The method is based on the Lanczos method to generate one pair of Krylov subspaces consisting of trial vectors, which is then used to reduce a large eigenvalue problem into a much smaller one. The method retains the n order quadratic eigenproblem, without employing the method of matrix augmentation traditionally used to cast the problem as a linear eigenproblem of order 2n. In this process, the method preserves the sparseness and symmetry of the system matrices and does not invoke complex arithmetic; thus making it very economical for use in solving large problems. Numerical results are presented to demonstrate the efficiency and accuracy of the proposed method. Copyright © 1999 John Wiley & Sons, Ltd.     

Dimethylsulfide and Phaeocystis poucheti in the southeastern Bering Sea

Dimethylsulfide (DMS), a volatile excretion product of marine phytoplankton, was determined in the water column during the spring phytoplankton bloom on the southeastern Bering Sea shelf. In the same samples, a broad range of variables which characterize the biological processes in this region were measured. DMS was correlated with phytoplankton chlorophyll in the outer shelf and oceanic domains, but not in the middle shelf domain. A very strong correlation between the cell density of the haptophyte Phaeocystis poucheti and the DMS concentration in seawater was found, which suggests that this species accounts for most of the DMS present in the study region. We propose that in P. poucheti and certain other phytoplankton species the excretion of DMS is incidental to the release of acrylic acid which serves to inhibit bacterial attack upon the algae.     

Efficient conceptual model for simulating the effect of aquifer heterogeneity on natural groundwater discharge to rivers

When linearity can be assumed (linear response of heads to stresses), stream–aquifer flow exchange can be simulated as the drainage of a number of independent linear reservoirs. This conceptual model, which can be mathematically deduced in a univocal way from an eigenvalue solution of the linear groundwater flow problem, facilitates the understanding of the physical phenomenon and the analysis of influencing factors. The number of reservoirs required to simulate stream depletion in some ideal homogeneous cases of stream–aquifer connection was analyzed in detail in a previous investigation using analytical eigenvalue solutions [16]. However, most aquifers are heterogeneous in nature and numerical solutions must be employed to analyze whether they could also be simulated using few reservoirs. This paper presents a stochastic analysis of the influence of heterogeneity on the simulation of natural groundwater discharges in aquifers connected to rivers, as a series of linear reservoirs. A Monte-Carlo approach was employed to perform this study. The results show that, on a monthly time scale, many cases (even heterogeneous aquifers) can be simulated using just a few reservoirs with sufficient accuracy and at minimum computational cost. Therefore, this modeling technique can be useful to efficiently simulate the integrated management of complex water resources systems at the basin scale (with many aquifers, reservoirs, demands, etc.) that need to simultaneously consider surface and groundwater flow and stream–aquifer interaction.     

Linear stability analysis for the differentially heated rotating annulus

We use linear stability analysis to approximate the axisymmetric to nonaxisymmetric transition in the differentially heated rotating annulus. We study an accurate mathematical model that uses the Navier–Stokes equations in the Boussinesq approximation. The steady axisymmetric solution satisfies a two-dimensional partial differential boundary value problem. It is not possible to compute the solution analytically, and thus, numerical methods are used. The eigenvalues are also given by a two-dimensional partial differential problem, and are approximated using the matrix eigenvalue problem that results from discretizing the linear part of the appropriate equations.

A comparison is made with experimental results. It is shown that the predictions using linear stability analysis accurately reproduce many of the experimental observations. Of particular interest is that the analysis predicts cusping of the axisymmetric to nonaxisymmetric transition curve at wave number transitions, and the wave number maximum along the lower part of the axisymmetric to nonaxisymmetric transition curve is accurately determined. The correspondence between theoretical and experimental results validates the numerical approximations as well as the application of linear stability analysis.

A linear stability analysis is also performed with the effects of centrifugal buoyancy neglected. Along the lower part of the transition curve, the results are significantly qualitatively and quantitatively different than when the centrifugal effects are considered. In particular, the results indicate that the centrifugal buoyancy is the cause of the observation of a wave number maximum along the transition curve, and is the cause of a change in concavity of the transition curve.     

A subspace modal superposition method for non-classically damped systems

For structures with non-proportional damping, complex eigenvectors or mode shapes must be used in order to decoe the equations of motion. The resulting equations can then be solved in a systematic way. The necessity of solvie complex eigenvalue problem of a large system remains an obstacle for the practical application of the method. This stres utilizes the fact that in practice only a small number of the complex modes are needed. Therefore, these complex modes be approximated by a linear combination of a small number of the undamped modes, which can be obtained by established methods with less cost. An additional eigenvalue problem is then solved in a subspace with a much sm dimension to provide the best combination coefficient for each complex mode. The method of solution for the decoue equations is then carried over, using the approximate complex modes expressed in undamped mode shapes, to resue simple formulas for the time- and frequency-domain solution. Thus, an efficient modal superposition method is develoe for non-proportionally damped systems. The accuracy of this approximate method is studied through an example. Comparing the frequency response result using the approximate method with that using the exact complex modes, found that the error is negligible.     

New approaches for non‐classically damped system eigenanalysis

This paper presents three new approaches for solving eigenvalue problems of non‐classically damped linear dynamics systems with fewer calculations than the conventional state vector approach. In the latter, the second‐order differential equation of motion is converted into a first‐order system by doubling the size of the matrices. The new approaches simplify the approach and reduce the number of calculations. The mathematical formulations for the proposed approaches are presented and the numerical results compared with the existing method by solving a sample problem with different damping properties. Of the three proposed approaches, the expansion approach was found to be the simplest and fastest to compute. Copyright © 2005 John Wiley & Sons, Ltd.     

Partial eigensolution of damped structural systems by Arnoldi's method

An efficient numerical algorithm is developed to solve the quadratic eigenvalue problems arising in the dynamic analysis of damped structural systems. The algorithm can even be applied to structural systems with non-symmetric matrices. The algorithm is based on the use of Arnoldi's method to generate a Krylov subspace of trial vectors, which is then used to reduce a large eigenvalue problem to a much smaller one. The reduced eigenvalue problem is solved and the solutions are used to construct approximate solutions to the original large system. In the process, the algorithm takes full advantage of the sparseness and symmetry of the system matrices and requires no complex arithmetic, therefore, making it very economical for use in solving large problems. The numerical results from test examples are presented to demonstrate that a large fraction of the approximate solutions calculated are very accurate, indicating that the algorithm is highly effective for extracting a number of vibration modes for a large dynamic system, whether it is lightly or heavily damped.