The catastrophe structure of thermohaline convection in a two-dimensional fluid model and a comparison with low-order box models

Abstract

We impose a surface forcing on the 2D, Boussinesq, thermohaline equations in a rectangular domain, in the form of equatorially symmetric cosine distributions of salinity flux and temperature. This system may be seen as an idealization of the ocean thermohaline circulation on the global scale over intervals of centuries or millenia. Multiple steady states are found numerically. They reflect the competition between the opposite signs of the temperature and salinity-driven equatorially symmetric circulations. There are also pole-to-pole, equatorially asymmetric circulations. In the control space of the temperature and salinity-flux forcing amplitudes, these equilibria form two cusp catastrophes, and transitions between stable equilibria occur through several distinct bifurcations. These catastrophes can be reproduced in simple box models connecting stirred reservoirs through capillary pipes. This steady-state analysis may provide a framework for a better understanding of climatic transitions between different stable regimes of the ocean-atmosphere system.     

Stiffened steel box columns. Part 1: Cyclic behaviour

This paper aims to study the cyclic inelastic behaviour of stiffened steel box columns failed by local and overall interaction instability under a constant compressive axial force and cyclic lateral loading. Such columns find broad application in steel bridge piers. The columns are of box sections with longitudinal stiffeners. In the analysis, a modified two‐surface plasticity model developed at Nagoya University is employed to model material non‐linearity. For comparison, analyses using classical isotropic‐ and kinematic‐hardening models are also carried out. Hysteretic curves and buckling modes obtained from analysis using the two‐surface model and classical models are compared with experimental results. Moreover, the progression of deformation from occurrence of local buckling to structural failure is discussed in detail. The comparisons show that the use of an accurate plasticity model is quite important in the prediction of both the cyclic inelastic behaviour and failure characteristic of steel box columns failed by coupled local and overall instability. It is found that the modified two‐surface model is a satisfactory model in predicting the cyclic hysteretic behaviour of both the thin‐ and thick‐walled steel box columns. Copyright © 2000 John Wiley & Sons, Ltd.     

Stiffened steel box columns. Part 2: Ductility evaluation

The purpose of this study is to evaluate the ultimate strength and ductility capacity of stiffened steel box columns failed by local and overall interaction instability under a constant compressive axial force and cyclic lateral loading. In a companion paper, a finite element formulation accounting for both geometrical and material non‐linearity was developed to obtain cyclic hysteretic behaviour of such columns. In this paper, the effect of loading patterns on the cyclic inelastic behaviour is first studied; then, a parametric study is carried out to investigate the effects of flange plate width–thickness ratio parameter, column slenderness ratio parameter, stiffener's equivalent slenderness ratio parameter, magnitude of axial load, and material type of stiffeners on the strength and ductility of the columns. Last but not least, empirical formulae of both the ultimate strength and ductility capacities are proposed for stiffened steel box columns, and the limit values of various parameters for the required ductility demand are also discussed. Copyright © 2000 John Wiley & Sons, Ltd.     

Visualization and ecohydrologic models: Opening the box

Earth system models synthesize the science of interactions amongst multiple biophysical and, increasingly, human processes across a wide range of scales. Ecohydrologic models are a subset of earth system models that focus particularly on the complex interactions between ecosystem processes and the storage and flux of water. Ecohydrologic models often focus at scales where direct observations occur: plots, hillslopes, streams, and watersheds, as well as where land and resource management decisions are implemented. These models complement field-based and data-driven science by combining theory, empirical relationships derived from observation and new data to create virtual laboratories. Ecohydrologic models are tools that managers can use to ask “what if” questions and domain scientists can use to explore the implications of new theory or measurements. Recent decades have seen substantial advances in ecohydrologic models, building on both new domain science and advances in software engineering and data availability. The increasing sophistication of ecohydrologic models however, presents a barrier to their widespread use and credibility. Their complexity, often encoding 100s of relationships, means that they are effectively “black boxes,” at least for most users, sometimes even to the teams of researchers that contribute to their design. This opacity complicates the interpretation of model results. For models to effectively advance our understanding of how plants and water interact, we must improve how we visualize not only model outputs, but also the underlying theories that are encoded within the models. In this paper, we outline a framework for increasing the usefulness of ecohydrologic models through better visualization. We outline four complementary approaches, ranging from simple best practices that leverage existing technologies, to ideas that would engage novel software engineering and cutting edge human–computer interface design. Our goal is to open the ecohydrologic model black box in ways that will engage multiple audiences, from novices to model developers, and support learning, new discovery, and environmental problem solving.     

Scattering of elastic waves by a general anisotropic basin. Part 1: a 2D model

Scattering of incident plane harmonic pseudo P‐, SH‐, and SV‐waves by a two‐dimensional basin of arbitrary shape is investigated by using an indirect boundary integral equation approach. The basin and surrounding half‐space are assumed to be generally anisotropic, homogeneous, linearly elastic solids. No material symmetries are assumed. The unknown scattered waves are expressed as linear combinations of full‐space time‐harmonic two‐dimensional Green functions. Using the Radon transform, the Green functions are obtained in the form of finite integrals over a unit circle. An algorithm for the accurate and efficient numerical evaluation of the Green functions is discussed. A detailed convergence and parametric analysis of the problem is presented. Excellent agreement is obtained with isotropic results available in the literature. Steady‐state surface ground motion is presented for semi‐circular basins with generally anisotropic material properties. The results show that surface motion strongly depends upon the material properties of the basin as well as the angle of incidence and frequency of the incident wave. Significant mode conversion can be observed for general triclinic materials which are not present in isotropic models. Comparison with an isotropic basin response demonstrates that anisotropy is very important for assessing the nature of surface motion atop basins. Copyright © 2001 John Wiley & Sons, Ltd.     

Reduction of thermohaline circulation during deglaciation: the effect on atmospheric radiocarbon and CO2

A two-? ocean reservoir model is employed to examine the combined effects of vertical ocean circulation, organic matter extraction and cosmic ray production on the¹⁴C/¹²C in the atmosphere, ocean and sediment. In this model, dissolution of deep-sea calcium carbonate sediment is assumed to respond to the supply of particulate carbonate from the surface ocean and to the dissolved carbonate-ion concentration of the deep-sea.If the vertical ocean circulation decreased by 50% during the maximum rate of deglaciation, the atmospheric¹⁴C/¹²C would have increased concurrently by 10% relative to the pre-bomb present. Further, if cosmic ray production of¹⁴C was 50% greater than present at about 7800 years B.P. as suggested by archeomagnetic measurements, a double maximum of 10% occurs. The first is at about 10.5 ka and the second at 6.5 ka. This result is similar to the variation of¹⁴C/¹²C over the last 10,000 years calculated from the¹⁴C dates measured by Stuiver on the varved Lake of the Clouds. The result is not sensibly altered if 10¹⁷ moles of organic carbon is extracted from the ocean during sea-level rise.Reduction of the thermohaline ocean circulation by 50% over a one- to two-thousand-year interval would not be sensibly detected in the dating of deep-sea sediment. If Broecker's organic nutrient extraction occurs in conjunction with reduction of vertical circulation, a decrease in the atmospheric P_CO2 will precede the eventual rise.     

A box model of circulation and melting in ice shelf caverns

A simple box model of the circulation into and inside the ocean cavern beneath an ice shelf is used to estimate the melt rates of Antarctic glaciers and ice shelves. The model uses simplified cavern geometries and includes a coarse parameterization of the overturning circulation and vertical mixing. The melting/freezing physics at the ice shelf/ocean interface are those usually implemented in high-resolution circulation models of ice shelf caverns. The model is driven by the thermohaline inflow conditions and coupling to the heat and freshwater exchanges at the sea surface in front of the cavern. We tune the model for Pine Island Glacier and then apply it to six other major caverns. The dependence of the melting rate on thermohaline conditions at the ice shelf front is investigated for this set of caverns, including sensitivity studies, alternative parameterizations, and warming scenarios. An analytical relation between the melting rate and the inflow temperature is derived for a particular model version, showing a quadratic dependence of basal melting on small values of the temperature of the inflow, which changes to a linear dependence for larger values. The model predicts melting at all ice shelf bases in agreement with observations, ranging from below a meter per year for Ronne Ice Shelf to about 25 m/year for the Pine Island Glacier. In a warming scenario with a one-degree increase of the inflow temperature, the latter glacier responds with a 1.4-fold increase of the melting rate. Other caverns respond by more than a tenfold increase, as, e.g., Ronne Ice Shelf. The model is suitable for use as a simple fast module izn coarse large-scale ocean models.     

Numerical modeling of general circulation,thermohaline structure,and residence time in Gorgan Bay,Iran

Gorgan Bay is a semi-enclosed basin located in the southeast of the Caspian Sea, Iran. The bay is recognized as a resting place for migratory birds as well as a spawning habitat for native fish. However, apparently, no detailed research on its physical processes has previously been conducted. In this study, a 3D coupled hydrodynamic and solute transport model was used to investigate general circulation, thermohaline structure, and residence time in Gorgan Bay. Model outputs were validated against a set of field observations. Bottom friction and attenuation coefficient of light intensity were tuned in order to achieve optimum agreement with the observations. Results revealed that, due to the interaction between bathymetry and prevailing winds, a barotropic double-gyre circulation, dominating the general circulation, existed during all seasons in Gorgan Bay. Furthermore, temperature and salinity fluctuations in the bay were seasonal, due to the seasonal variability of atmospheric fluxes. Results also indicated that under the prevailing winds, the domain-averaged residence time in Gorgan Bay would be approximately 95 days. The rivers discharging into Gorgan Bay are considered as the main sources of nutrients in the bay. Since their mouths are located in the area with a residence time of over 100 days, Gorgan Bay could be at risk of eutrophication; it is necessary to adopt preventive measures against water quality degradation.     

System identification of linear structures based on Hilbert–Huang spectral analysis. Part 1: normal modes

Based on the Hilbert–Huang spectral analysis, a method is proposed to identify multi‐degree‐of‐freedom (MDOF) linear systems using measured free vibration time histories. For MDOF systems, the normal modes have been assumed to exist. In this method, the measured response data, which are polluted by noises, are first decomposed into modal responses using the empirical mode decomposition (EMD) approach with intermittency criteria. Then, the Hilbert transform is applied to each modal response to obtain the instantaneous amplitude and phase angle time histories. A linear least‐square fit procedure is proposed to identify the natural frequency and damping ratio from the instantaneous amplitude and phase angle for each modal response. Based on a single measurement of the free vibration time history at one appropriate location, natural frequencies and damping ratios can be identified. When the responses at all degrees of freedom are measured, the mode shapes and the physical mass, damping and stiffness matrices of the structure can be determined. The applications of the proposed method are illustrated using three linear systems with different dynamic characteristics. Numerical simulation results demonstrate that the proposed system identification method yields quite accurate results, and it offers a new and effective tool for the system identification of linear structures in which normal modes exist. Copyright © 2003 John Wiley & Sons, Ltd.     

Site effect assessment using KiK-net data: Part 1. A simple correction procedure for surface/downhole spectral ratios

When data is available, the estimation of site effects is usually performed using the “standard spectral ratio” (SSR) technique with respect to an outcropping, reference rock site. This study uses the Japanese KiK-net network, which has more than 600 pairs of surface-downhole stations allowing the computation of empirical borehole transfer functions, consisting of mean spectral ratios of surface over downhole recordings. The borehole transfer function deviates from the SSR in two respects: the reference is located at depth, and the downhole velocity varies from one site to another. These differences bias the estimation of the transfer function with reference to a standard outcrop rock site. The goal of this paper is to develop a simple and robust methodology to correct for such bias. The proposed correction procedure consists of two steps: a depth correction designed to account, in a simplified and physically acceptable way, for the existence at depth of destructive interferences and the absence of free-surface effects in the high-frequency range; and an impedance correction designed to normalize the shear wave velocity at depth. The depth correction involves a simple, frequency-dependent curve to be adapted for each site as a function of the first destructive interference frequency at depth. The impedance normalization combines the use of “generic” rock velocity profiles and a quarter-wavelength approach, resulting in a smooth frequency-dependent amplitude correction. The proposed methodology is applied on a large subset of KiK-net data in view of analysing the correlation between site amplification factors and site parameters in a companion paper.     

Response spectral amplification ratios from 1- and 2-dimensional nonlinear soil site models

In order to determine the effect of geometry on the ground response of 2-dimensional (2-D) basins filled with soils that can develop nonlinear response, we use three basin models with width/depth ratios 3, 6 and 10. The three basins are subjected to a suite of rock site records with various magnitudes and source distances. We compute response spectral amplification ratios at four locations on the surface of the 2-D basins, and determine the average variation of the amplification ratios with respect to excitation spectra, for peak ground acceleration (PGA) and 3 spectral periods of 0.2, 0.5, 1 s. Similarly, we compute the average response spectral amplification ratios for two 1-dimensional (1-D) nonlinear models, one having the soil profile at the basin centre and the other having a soil profile at half the depth of the basin. From the relationship between the average amplification ratios and excitation spectra, we determine the cross-over point in terms of excitation spectral values that separate the amplification range from the deamplification range. Our results show that the cross-over point varies significantly from one location to another on the ground surface and from one basin to another, in a range of 0.3–1.1g for PGA. The effects of basin geometry are very strong at weak and moderate excitation, but decrease with increasing excitation spectra in a significant portion around the basin centre. Our results provide some justification for using 1-D models for 2-D basins with a width/depth ratio ?6 if the soil site is subjected to strong ground shaking.     

Local amplification of deep mining induced vibrations Part 1: Experimental evidence for site effects in a coal basin

The work presented in this paper lies under the scope of a research program aiming to assess the impact of deep coal mining induced vibrations on the surface constructions. The concerned section of the program is dedicated to the study of geological site effects and their influence on the mining induced vibrations for which the experimental investigations have been carried out and developed in this paper (Part 1). The empirical methods based on H/V spectral ratios have been applied on data sets provided from mining induced vibrations recorded within private residences above the deep coalmine as well as complementary measurements of ambient noise. The results evidence an amplified zone in the southern part of the Gardanne basin where drilling data confirmed the presence of particularly fractured and soft stratigraphic units. This joint analysis of induced seismicity and ambient noise enabled to validate the method based on H/V ratios applied to the mining context.     

Resolution issues in numerical models of oceanic and coastal circulation

The baroclinic and barotropic properties of ocean processes vary on many scales. These scales are determined by various factors such as the variations in coastline and bottom topography, the forcing meteorology, the latitudinal dependence of the Coriolis force, and the Rossby radius of deformation among others. In this paper we attempt to qualify and quantify scales of these processes, with particular attention to the horizontal resolution necessary to accurately reproduce physical processes in numerical ocean models. We also discuss approaches taken in nesting or down-scaling from global/basin-scale models to regional-scale or shelf-scale models. Finally we offer comments on how vertical resolution affects the representation of stratification in these numerical models.