Boundary integral methods in high frequency scattering

In this article we review recent progress on the design, analysis and implementation of numerical-asymptotic boundary integral methods for the computation of frequency-domain acoustic scattering in a homogeneous unbounded medium by a bounded obstacle. The main aim of the methods is to allow computation of scattering at arbitrarily high frequency with finite computational resources.

Boundary integral equations on unbounded rough surfaces: Fredholmness and the finite section method

We consider a class of boundary integral equations that arise in the study of strongly elliptic BVPs in unbounded domains of the form $D = \{(x, z)\in \mathbb{R}^{n+1} : x\in \mathbb{R}^n, z > f(x)\}$ where $f : \mathbb{R}^n \to\mathbb{R}$ is a sufficiently smooth bounded and continuous function. A number of specific problems of this type, for example acoustic scattering problems, problems involving elastic waves, and problems in potential theory, have been reformulated as second kind integral equations $u+Ku = v$ in the space $BC$ of bounded, continuous functions. Having recourse to the so-called limit operator method, we address two questions for the operator $A = I + K$ under consideration, with an emphasis on the function space setting $BC$. Firstly, under which conditions is $A$ a Fredholm operator, and, secondly, when is the finite section method applicable to $A$?

Condition number estimates for combined potential boundary integral operators in acoustic scattering

We study the classical combined field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle, namely the indirect formulation due to Brakhage-Werner/Leis/Panic, and the direct formulation associated with the names of Burton and Miller. We obtain lower and upper bounds on the condition numbers for these formulations, emphasising dependence on the frequency, the geometry of the scatterer, and the coupling parameter. Of independent interest we also obtain upper and lower bounds on the norms of two oscillatory integral operators, namely the classical acoustic single- and double-layer potential operators.

The influence of the atmospheric boundary layer on nocturnal layers of noctuids and other moths migrating over southern Britain

Insects migrating at high altitude over southern Britain have been continuously monitored by automatically-operating, vertical-looking radars over a period of several years. During some occasions in the summer months, the migrants were observed to form well-defined layer concentrations, typically at heights of 200-400 m, in the stable night-time atmosphere. Under these conditions, insects are likely to have control over their vertical movements and are selecting flight heights which are favourable for long-range migration. We therefore investigated the factors influencing the formation of these insect layers by comparing radar measurements of the vertical distribution of insect density with meteorological profiles generated by the UK Met. Office’s Unified Model (UM). Radar-derived measurements of mass and displacement speed, along with data from Rothamsted Insect Survey light traps provided information on the identity of the migrants. We present here three case studies where noctuid and pyralid moths contributed substantially to the observed layers. The major meteorological factors influencing the layer concentrations appeared to be: (a) the altitude of the warmest air, (b) heights corresponding to temperature preferences or thresholds for sustained migration and (c), on nights when air temperatures are relatively high, wind-speed maxima associated with the nocturnal jet. Back-trajectories indicated that layer duration may have been determined by the distance to the coast. Overall, the unique combination of meteorological data from the UM and insect data from entomological radar described here show considerable promise for systematic studies of high-altitude insect layering.

The moist boundary layer under a mid-latitude weather system

Mid-latitude weather systems are key contributors to the transport of atmospheric water vapour, but less is known about the role of the boundary layer in this transport. We expand a conceptual model of dry boundary-layer structure under synoptic systems to include moist processes, using idealised simulations of cyclone waves to investigate the three-way interaction between the boundary layer, atmospheric moisture and large-scale dynamics. Forced by large-scale thermal advection, boundary-layer structures develop over large areas, analogous to the daytime convective boundary layer, the nocturnal stable boundary layer and transitional regimes between these extremes.

Reactive Halogens in the Marine Boundary Layer (RHaMBLe): the tropical North Atlantic experiments

The NERC UK SOLAS-funded Reactive Halogens in the Marine Boundary Layer (RHaMBLe) programme comprised three field experiments. This manuscript presents an overview of the measurements made within the two simultaneous remote experiments conducted in the tropical North Atlantic in May and June 2007. Measurements were made from two mobile and one ground-based platforms. The heavily instrumented cruise D319 on the RRS Discovery from Lisbon, Portugal to São Vicente, Cape Verde and back to Falmouth, UK was used to characterise the spatial distribution of boundary layer components likely to play a role in reactive halogen chemistry. Measurements onboard the ARSF Dornier aircraft were used to allow the observations to be interpreted in the context of their vertical distribution and to confirm the interpretation of atmospheric structure in the vicinity of the Cape Verde islands. Long-term ground-based measurements at the Cape Verde Atmospheric Observatory (CVAO) on São Vicente were supplemented by long-term measurements of reactive halogen species and characterisation of additional trace gas and aerosol species during the intensive experimental period. This paper presents a summary of the measurements made within the RHaMBLe remote experiments and discusses them in their meteorological and chemical context as determined from these three platforms and from additional meteorological analyses. Air always arrived at the CVAO from the North East with a range of air mass origins (European, Atlantic and North American continental). Trace gases were present at stable and fairly low concentrations with the exception of a slight increase in some anthropogenic components in air of North American origin, though NOx mixing ratios during this period remained below 20 pptv. Consistency with these air mass classifications is observed in the time series of soluble gas and aerosol composition measurements, with additional identification of periods of slightly elevated dust concentrations consistent with the trajectories passing over the African continent. The CVAO is shown to be broadly representative of the wider North Atlantic marine boundary layer; measurements of NO, O3 and black carbon from the ship are consistent with a clean Northern Hemisphere marine background. Aerosol composition measurements do not indicate elevated organic material associated with clean marine air. Closer to the African coast, black carbon and NO levels start to increase, indicating greater anthropogenic influence. Lower ozone in this region is possibly associated with the increased levels of measured halocarbons, associated with the nutrient rich waters of the Mauritanian upwelling. Bromide and chloride deficits in coarse mode aerosol at both the CVAO and on D319 and the continuous abundance of inorganic gaseous halogen species at CVAO indicate significant reactive cycling of halogens. Aircraft measurements of O3 and CO show that surface measurements are representative of the entire boundary layer in the vicinity both in diurnal variability and absolute levels. Above the inversion layer similar diurnal behaviour in O3 and CO is observed at lower mixing ratios in the air that had originated from south of Cape Verde, possibly from within the ITCZ. ECMWF calculations on two days indicate very different boundary layer depths and aircraft flights over the ship replicate this, giving confidence in the calculated boundary layer depth.

On the dependence of boundary-layer ventilation on frontal type

Case study simulations with idealized tracers have been used to determine the relationship between the dynamics and conceptual representations of different midlatitude frontal systems and the amount, distribution, and time scale of boundary layer ventilation by these systems. The key features of ventilation by a kata– and ana–cold frontal system are found to be quantitatively and also often qualitatively similar to the main ventilation pathways, which are the conveyor belts, cloud head, and other convection. The conveyor belts and cloud head occur within cloud, implying that they can be identified using satellite imagery. Differences in the transport by the two systems can be related to their conceptual representations and include a sensitive dependence on the diurnal cycle for the kata- but not the ana-cold frontal case.

Observations of changes in marine boundary layer clouds

Recent research outlined by the Intergovernmental Panel on Climate Change (IPCC) highlights the response of marine boundary layer (MBL) clouds to warming associated with increasing greenhouse gases as a major contributor to uncertainties in model projections of climate change. Understanding how MBL clouds respond to increasing temperatures is hampered by the relative scarcity of marine surface observations and the difficulty of retrieving accurate parameters remotely from satellites. In this study we combine data from surface observations with that from the International Satellite Cloud Climatology Project (ISCCP), CloudSat and CALIPSO, with a view to investigating the spatial distribution and variations in MBL cloud fraction and cloud liquid water path (LWP). These results are then compared with the treatment of MBL clouds in the UK Met Office HadGEM models. Future work will assess how variations in LWP impact the top of atmosphere radiative energy balance using data from the Geostationary Earth Radiation Budget (GERB), in order to quantify the response of MBL clouds on interannual timescales to a changing climate

Distribution and mobility of trace elements in soils and vegetation around the mining and smelting areas of Tharsis, Riotinto and Huelva, Iberian Pyrite Belt, SW Spain

Trace elements may present an environmental hazard in the vicinity of mining and smelting activities. However, the factors controlling their distribution and transfer within the soil and vegetation systems are not always well defined. Total concentrations of up to 15,195 mg center dot kg (-1) As, 6,690 mg center dot kg(-1) Cu, 24,820 mg center dot kg(-1) Pb and 9,810 mg center dot kg(-1) Zn in soils, and 62 mg center dot kg(-1) As, 1,765 mg center dot kg(-1) Cu, 280 mg center dot kg(-1) Pb and 3,460 mg center dot kg (-1) Zn in vegetation were measured. However, unusually for smelters and mines of a similar size, the elevated trace element concentrations in soils were found to be restricted to the immediate vicinity of the mines and smelters (maximum 2-3 km). Parent material, prevailing wind direction, and soil physical and chemical characteristics were found to correlate poorly with the restricted trace element distributions in soils. Hypotheses are given for this unusual distribution: (1) the contaminated soils were removed by erosion or (2) mines and smelters released large heavy particles that could not have been transported long distances. Analyses of the accumulation of trace elements in vegetation (median ratios: As 0.06, Cu 0.19, Pb 0.54 and Zn 1.07) and the percentage of total trace elements being DTPA extractable in soils (median percentages: As 0.06%, Cu 15%, Pb 7% and Zn 4%) indicated higher relative trace element mobility in soils with low total concentrations than in soils with elevated concentrations.

Factors affecting distribution and mobility of trace elements (Cu, Pb, Zn) in a perennial grapevine (Vitis vinifera L.) in the Champagne region of France

Soil and Vitis vinifera L (coarse and fine roots, leaves, berries) concentration and geochemical partitioning of Cu, Pb and Zn were determined in a contaminated calcareous Champagne plot to assess their mobility and transfer. Accumulation ratios in roots remained low (0.1-0.4 for Cu and Zn, <0.05 for Pb). Differences between elements resulted from vegetation uptake strategy and soil partitioning. Copper, significantly associated with the oxidisable fraction (27.8%), and Zn with the acid soluble fraction (33.3%), could be mobilised by rhizosphere acidification and oxidisation, unlike Pb, essentially contained in the reducible fraction (72.4%). Roots should not be considered as a whole since the more reactive fine roots showed higher accumulation ratios than coarse ones. More sensitive response of fine roots, lack of correlation between chemical extraction results and vegetation concentrations, and very limited translocation to aerial parts showed that fine root concentrations should be used when assessing bioavailability. (C) 2008 Elsevier Ltd. All rights reserved.

Unification of the probe and singular sources methods for the inverse boundary value problem by the no-response test

In this article, we use the no-response test idea, introduced in Luke and Potthast (2003) and Potthast (Preprint) and the inverse obstacle problem, to identify the interface of the discontinuity of the coefficient gamma of the equation del (.) gamma(x)del + c(x) with piecewise regular gamma and bounded function c(x). We use infinitely many Cauchy data as measurement and give a reconstructive method to localize the interface. We will base this multiwave version of the no-response test on two different proofs. The first one contains a pointwise estimate as used by the singular sources method. The second one is built on an energy (or an integral) estimate which is the basis of the probe method. As a conclusion of this, the probe and the singular sources methods are equivalent regarding their convergence and the no-response test can be seen as a unified framework for these methods. As a further contribution, we provide a formula to reconstruct the values of the jump of gamma(x), x is an element of partial derivative D at the boundary. A second consequence of this formula is that the blow-up rate of the indicator functions of the probe and singular sources methods at the interface is given by the order of the singularity of the fundamental solution.

How to save a bad element with weak boundary conditions

The P-1-P-1 finite element pair is known to allow the existence of spurious pressure (surface elevation) modes for the shallow water equations and to be unstable for mixed formulations. We show that this behavior is strongly influenced by the strong or the weak enforcement of the impermeability boundary conditions. A numerical analysis of the Stommel model is performed for both P-1-P-1 and P-1(NC)-P-1 mixed formulations. Steady and transient test cases are considered. We observe that the P-1-P-1 element exhibits stable discrete solutions with weak boundary conditions or with fully unstructured meshes. (c) 2005 Elsevier Ltd. All rights reserved.