The domain derivative in rough-surface scattering and rigorous estimates for first-order perturbation theory

We investigate Fréchet differentiability of the scattered field with respect to variation in the boundary in the case of time–harmonic acoustic scattering by an unbounded, sound–soft, one–dimensional rough surface. We rigorously prove the differentiability of the scattered field and derive a characterization of the Fréchet derivative as the solution to a Dirichlet boundary value problem. As an application of these results we give rigorous error estimates for first–order perturbation theory, justifying small perturbation methods that have a long history in the engineering literature. As an application of our rigorous estimates we show that a plane acoustic wave incident on a sound–soft rough surface can produce an unbounded scattered field.

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.

First-order turbulence closure for modelling complex canopy flows

Simple first-order closure remains an attractive way of formulating equations for complex canopy flows when the aim is to find analytic or simple numerical solutions to illustrate fundamental physical processes. Nevertheless, the limitations of such closures must be understood if the resulting models are to illuminate rather than mislead. We propose five conditions that first-order closures must satisfy then test two widely used closures against them. The first is the eddy diffusivity based on a mixing length. We discuss the origins of this approach, its use in simple canopy flows and extensions to more complex flows. We find that it satisfies most of the conditions and, because the reasons for its failures are well understood, it is a reliable methodology. The second is the velocity-squared closure that relates shear stress to the square of mean velocity. Again we discuss the origins of this closure and show that it is based on incorrect physical principles and fails to satisfy any of the five conditions in complex canopy flows; consequently its use can lead to actively misleading conclusions.

Radiative-dynamical climatology of the first-generation Canadian middle atmosphere model

The Canadian Middle Atmosphere Modelling (MAM) project is a collaboration between thé Atmospheric Environment Service (AES) of Environment Canada and several Canadian universities. Its goal is thé development of a comprehensive General Circulation Model of the troposphere-stratosphere-mesosphere System, starting from the AES/CCCma third-generation atmospheric General Circulation Model. This paper describes the basic features of the first-generation Canadian MAM and some aspects of its radiative-dynamical climatology. Standard first-order mean diagnostics are presented for monthly means and for the annual cycle of zonal-mean winds and temperatures. The mean meridional circulation is examined, and comparison is made between thé steady diabatic, downward controlled, and residual stream functions. It is found that downward control holds quite well in the monthly mean through most of the middle atmosphere, even during equinoctal periods. The relative roles of different drag processes in determining the mean downwelling over the wintertime polar middle stratosphere is examined, and the vertical structure of the drag is quantified.

Transreal arithmetic as a consistent basis for paraconsistent logics

Paraconsistent logics are non-classical logics which allow non-trivial and consistent reasoning about inconsistent axioms. They have been pro- posed as a formal basis for handling inconsistent data, as commonly arise in human enterprises, and as methods for fuzzy reasoning, with applica- tions in Artificial Intelligence and the control of complex systems. Formalisations of paraconsistent logics usually require heroic mathe- matical efforts to provide a consistent axiomatisation of an inconsistent system. Here we use transreal arithmetic, which is known to be consis- tent, to arithmetise a paraconsistent logic. This is theoretically simple and should lead to efficient computer implementations. We introduce the metalogical principle of monotonicity which is a very simple way of making logics paraconsistent. Our logic has dialetheaic truth values which are both False and True. It allows contradictory propositions, allows variable contradictions, but blocks literal contradictions. Thus literal reasoning, in this logic, forms an on-the- y, syntactic partition of the propositions into internally consistent sets. We show how the set of all paraconsistent, possible worlds can be represented in a transreal space. During the development of our logic we discuss how other paraconsistent logics could be arithmetised in transreal arithmetic.

Can the frequency of blocking be described by a red noise process?

The frequency of persistent atmospheric blocking events in the 40-yr ECMWF Re-Analysis (ERA-40) is compared with the blocking frequency produced by a simple first-order Markov model designed to predict the time evolution of a blocking index [defined by the meridional contrast of potential temperature on the 2-PVU surface (1 PVU ≡ 1 × 10−6 K m2 kg−1 s−1)]. With the observed spatial coherence built into the model, it is able to reproduce the main regions of blocking occurrence and the frequencies of sector blocking very well. This underlines the importance of the climatological background flow in determining the locations of high blocking occurrence as being the regions where the mean midlatitude meridional potential vorticity (PV) gradient is weak. However, when only persistent blocking episodes are considered, the model is unable to simulate the observed frequencies. It is proposed that this persistence beyond that given by a red noise model is due to the self-sustaining nature of the blocking phenomenon.

Non-invasive imaging of roots with high resolution X-ray micro-tomography

ray micro-tomography is a well-established technique for non-invasive imaging and evaluation of heterogeneous materials. An inexpensive X-ray micro-tomography system has been designed and built for the specific purposes of examining root growth and root/soil interactions. The system uses a silver target X-ray source with a focal spot diameter of 80 mum, an X-ray image intensifier with a sampling aperture of about 100 mum, and a sample with a diameter of 25 mm. Pre-germinated wheat and rape seeds were grown for up to 8-10 days in plastic containers in a sandy loam soil sieved to < 250 μm, and imaged with the X-ray system at regular intervals. The quality of 3 D image obtained was good allowing the development and growth of both root axes and some first-order laterals to be observed. The satisfactory discrimination between soil and roots enabled measurements of root diameter (wheat values were 0.48-1.22 mm) in individual tomographic slices and, by tracking from slice to slice, root lengths were also measured. The measurements obtained were generally within 10% of those obtained from destructive samples measured manually and with a flat-bed scanner. Further developments of the system will allow more detailed examination of the root: soil interface.

The acoustic signature of fluid flow in complex porous media

Effective medium approximations for the frequency-dependent and complex-valued effective stiffness tensors of cracked/ porous rocks with multiple solid constituents are developed on the basis of the T-matrix approach (based on integral equation methods for quasi-static composites), the elastic - viscoelastic correspondence principle, and a unified treatment of the local and global flow mechanisms, which is consistent with the principle of fluid mass conservation. The main advantage of using the T-matrix approach, rather than the first-order approach of Eshelby or the second-order approach of Hudson, is that it produces physically plausible results even when the volume concentrations of inclusions or cavities are no longer small. The new formulae, which operates with an arbitrary homogeneous (anisotropic) reference medium and contains terms of all order in the volume concentrations of solid particles and communicating cavities, take explicitly account of inclusion shape and spatial distribution independently. We show analytically that an expansion of the T-matrix formulae to first order in the volume concentration of cavities (in agreement with the dilute estimate of Eshelby) has the correct dependence on the properties of the saturating fluid, in the sense that it is consistent with the Brown-Korringa relation, when the frequency is sufficiently low. We present numerical results for the (anisotropic) effective viscoelastic properties of a cracked permeable medium with finite storage porosity, indicating that the complete T-matrix formulae (including the higher-order terms) are generally consistent with the Brown-Korringa relation, at least if we assume the spatial distribution of cavities to be the same for all cavity pairs. We have found an efficient way to treat statistical correlations in the shapes and orientations of the communicating cavities, and also obtained a reasonable match between theoretical predictions (based on a dual porosity model for quartz-clay mixtures, involving relatively flat clay-related pores and more rounded quartz-related pores) and laboratory results for the ultrasonic velocity and attenuation spectra of a suite of typical reservoir rocks. (C) 2003 Elsevier B.V. All rights reserved.

Modelling impacts of pollution in river systems: a new dispersion model and a case study of mine discharges in the Abrud, Aries and Mures river system in Transylvania, Romania

A new model of dispersion has been developed to simulate the impact of pollutant discharges on river systems. The model accounts for the main dispersion processes operating in rivers as well as the dilution from incoming tributaries and first-order kinetic decay processes. The model is dynamic and simulates the hourly behaviour of river flow and pollutants along river systems. The model has been applied to the Aries and Mures River System in Romania and has been used to assess the impacts of potential dam releases from the Roia Montan Mine in Transylvania, Romania. The question of mine water release is investigated under a range of scenarios. The impacts on pollution levels downstream at key sites and at the border with Hungary are investigated.

Monte Carlo phase diagram for diblock copolymer melts

A partial phase diagram is constructed for diblock copolymer melts using lattice-based Monte Carlo simulations. This is done by locating the order-disorder transition (ODT) with the aid of a recently proposed order parameter and identifying the ordered phase over a wide range of copolymer compositions (0.2 <= f <= 0.8). Consistent with experiments, the disordered phase is found to exhibit direct first-order transitions to each of the ordered morphologies. This includes the spontaneous formation of a perforated-lamellar phase, which presumably forms in place of the gyroid morphology due to finite-size and/or nonequilibrium effects. Also included in our study is a detailed examination of disordered cylinder-forming (f=0.3) diblock copolymers, revealing a substantial degree of pretransitional chain stretching and short-range order that set in well before the ODT, as observed previously in analogous studies on lamellar-forming (f=0.5) molecules. (c) 2006 American Institute of Physics.

The influence of hilly terrain on canopy-atmosphere carbon dioxide exchange

Topography influences many aspects of forest-atmosphere carbon exchange; yet only a small number of studies have considered the role of topography on the structure of turbulence within and above vegetation and its effect on canopy photosynthesis and the measurement of net ecosystem exchange of CO2 (N-ee) using flux towers. Here, we focus on the interplay between radiative transfer, flow dynamics for neutral stratification, and ecophysiological controls on CO2 sources and sinks within a canopy on a gentle cosine hill. We examine how topography alters the forest-atmosphere CO2 exchange rate when compared to uniform flat terrain using a newly developed first-order closure model that explicitly accounts for the flow dynamics, radiative transfer, and nonlinear eco physiological processes within a plant canopy. We show that variation in radiation and airflow due to topography causes only a minor departure in horizontally averaged and vertically integrated photosynthesis from their flat terrain values. However, topography perturbs the airflow and concentration fields in and above plant canopies, leading to significant horizontal and vertical advection of CO2. Advection terms in the conservation equation may be neglected in flow over homogeneous, flat terrain, and then N-ee = F-c, the vertical turbulent flux of CO2. Model results suggest that vertical and horizontal advection terms are generally of opposite sign and of the same order as the biological sources and sinks. We show that, close to the hilltop, F-c departs by a factor of three compared to its flat terrain counterpart and that the horizontally averaged F-c-at canopy top differs by more than 20% compared to the flat-terrain case.

LiDAR mapping of tidal marshes for ecogeomorphological modelling in the TIDE project

The European research project TIDE (Tidal Inlets Dynamics and Environment) is developing and validating coupled models describing the morphological, biological and ecological evolution of tidal environments. The interactions between the physical and biological processes occurring in these regions requires that the system be studied as a whole rather than as separate parts. Extensive use of remote sensing including LiDAR is being made to provide validation data for the modelling. This paper describes the different uses of LiDAR within the project and their relevance to the TIDE science objectives. LiDAR data have been acquired from three different environments, the Venice Lagoon in Italy, Morecambe Bay in England, and the Eden estuary in Scotland. LiDAR accuracy at each site has been evaluated using ground reference data acquired with differential GPS. A semi-automatic technique has been developed to extract tidal channel networks from LiDAR data either used alone or fused with aerial photography. While the resulting networks may require some correction, the procedure does allow network extraction over large areas using objective criteria and reduces fieldwork requirements. The networks extracted may subsequently be used in geomorphological analyses, for example to describe the drainage patterns induced by networks and to examine the rate of change of networks. Estimation of the heights of the low and sparse vegetation on marshes is being investigated by analysis of the statistical distribution of the measured LiDAR heights. Species having different mean heights may be separated using the first-order moments of the height distribution.

The propagation mechanism of a vortex on the β plane

The propagation velocity and propagation mechanism for vortices on a β plane are determined for a reduced-gravity model by integrating the momentum equations over the β plane. Isolated vortices, vortices in a background current, and initial vortex propagation from rest are studied. The propagation mechanism for isolated anticyclones as well as cyclones, which has been lacking up to now, is presented. It is shown that, to first order, the vortex moves to generate a Coriolis force on the mass anomaly of the vortex to compensate for the force on the vortex due to the variation of the Coriolis parameter. Only the mass anomaly of the vortex is of importance, because the Coriolis force due to the motion of the bulk of the layer moving with the vortex is almost fully compensated by the Coriolis force on the motion of the exterior flow. Because the mass anomaly of a cyclone is negative the force and acceleration have opposite sign. The role of dipolar structures in steadily moving vortices is discussed, and it is shown that their overall structure is fixed by the steady westward motion of the mass anomaly. Furthermore, it is shown that reduced-gravity vortices are not advected with a background flow. The reason for this behavior is that the background flow changes the ambient vorticity gradient such that the vortex obtains an extra self-propagation term that exactly cancels the advection by the background flow. Last, it is shown that a vortex initially at rest will accelerate equatorward first, after which a westward motion is generated. This result is independent of the sign of the vortex.