Upwind solution of singular differential equations arising from steady channel flows

We study ordinary nonlinear singular differential equations which arise from steady conservation laws with source terms. An example of steady conservation laws which leads to those scalar equations is the Saint–Venant equations. The numerical solution of these scalar equations is sought by using the ideas of upwinding and discretisation of source terms. Both the Engquist–Osher scheme and the Roe scheme are used with different strategies for discretising the source terms.

Scale-invariant moving finite elements for nonlinear partial differential equations in two dimensions

A scale-invariant moving finite element method is proposed for the adaptive solution of nonlinear partial differential equations. The mesh movement is based on a finite element discretisation of a scale-invariant conservation principle incorporating a monitor function, while the time discretisation of the resulting system of ordinary differential equations is carried out using a scale-invariant time-stepping which yields uniform local accuracy in time. The accuracy and reliability of the algorithm are successfully tested against exact self-similar solutions where available, and otherwise against a state-of-the-art h-refinement scheme for solutions of a two-dimensional porous medium equation problem with a moving boundary. The monitor functions used are the dependent variable and a monitor related to the surface area of the solution manifold. (c) 2005 IMACS. Published by Elsevier B.V. All rights reserved.

Advances in sequential data assimilation and numerical weather forecasting: an Ensemble Transform Kalman-Bucy Filter, a study on clustering in deterministic Ensemble Square Root Filters, and a test of a new time stepping scheme in an atmospheric model

This dissertation deals with aspects of sequential data assimilation (in particular ensemble Kalman filtering) and numerical weather forecasting. In the first part, the recently formulated Ensemble Kalman-Bucy (EnKBF) filter is revisited. It is shown that the previously used numerical integration scheme fails when the magnitude of the background error covariance grows beyond that of the observational error covariance in the forecast window. Therefore, we present a suitable integration scheme that handles the stiffening of the differential equations involved and doesn’t represent further computational expense. Moreover, a transform-based alternative to the EnKBF is developed: under this scheme, the operations are performed in the ensemble space instead of in the state space. Advantages of this formulation are explained. For the first time, the EnKBF is implemented in an atmospheric model. The second part of this work deals with ensemble clustering, a phenomenon that arises when performing data assimilation using of deterministic ensemble square root filters in highly nonlinear forecast models. Namely, an M-member ensemble detaches into an outlier and a cluster of M-1 members. Previous works may suggest that this issue represents a failure of EnSRFs; this work dispels that notion. It is shown that ensemble clustering can be reverted also due to nonlinear processes, in particular the alternation between nonlinear expansion and compression of the ensemble for different regions of the attractor. Some EnSRFs that use random rotations have been developed to overcome this issue; these formulations are analyzed and their advantages and disadvantages with respect to common EnSRFs are discussed. The third and last part contains the implementation of the Robert-Asselin-Williams (RAW) filter in an atmospheric model. The RAW filter is an improvement to the widely popular Robert-Asselin filter that successfully suppresses spurious computational waves while avoiding any distortion in the mean value of the function. Using statistical significance tests both at the local and field level, it is shown that the climatology of the SPEEDY model is not modified by the changed time stepping scheme; hence, no retuning of the parameterizations is required. It is found the accuracy of the medium-term forecasts is increased by using the RAW filter.

A cell by cell anisotropic adaptive mesh ALE scheme for the numerical solution of the Euler equations

In this paper a cell by cell anisotropic adaptive mesh technique is added to an existing staggered mesh Lagrange plus remap finite element ALE code for the solution of the Euler equations. The quadrilateral finite elements may be subdivided isotropically or anisotropically and a hierarchical data structure is employed. An efficient computational method is proposed, which only solves on the finest level of resolution that exists for each part of the domain with disjoint or hanging nodes being used at resolution transitions. The Lagrangian, equipotential mesh relaxation and advection (solution remapping) steps are generalised so that they may be applied on the dynamic mesh. It is shown that for a radial Sod problem and a two-dimensional Riemann problem the anisotropic adaptive mesh method runs over eight times faster.

On the convergence of the no response test

The no response test is a new scheme in inverse problems for partial differential equations which was recently proposed in [D. R. Luke and R. Potthast, SIAM J. Appl. Math., 63 (2003), pp. 1292–1312] in the framework of inverse acoustic scattering problems. The main idea of the scheme is to construct special probing waves which are small on some test domain. Then the response for these waves is constructed. If the response is small, the unknown object is assumed to be a subset of the test domain. The response is constructed from one, several, or many particular solutions of the problem under consideration. In this paper, we investigate the convergence of the no response test for the reconstruction information about inclusions D from the Cauchy values of solutions to the Helmholtz equation on an outer surface $\partial\Omega$ with $\overline{D} \subset \Omega$. We show that the one‐wave no response test provides a criterion to test the analytic extensibility of a field. In particular, we investigate the construction of approximations for the set of singular points $N(u)$ of the total fields u from one given pair of Cauchy data. Thus, the no response test solves a particular version of the classical Cauchy problem. Also, if an infinite number of fields is given, we prove that a multifield version of the no response test reconstructs the unknown inclusion D. This is the first convergence analysis which could be achieved for the no response test.

Quantifying the effects of perturbing the physics of an interactive sulfur scheme using an ensemble of GCMs on the climateprediction.net platform

Aerosols from anthropogenic and natural sources have been recognized as having an important impact on the climate system. However, the small size of aerosol particles (ranging from 0.01 to more than 10 μm in diameter) and their influence on solar and terrestrial radiation makes them difficult to represent within the coarse resolution of general circulation models (GCMs) such that small-scale processes, for example, sulfate formation and conversion, need parameterizing. It is the parameterization of emissions, conversion, and deposition and the radiative effects of aerosol particles that causes uncertainty in their representation within GCMs. The aim of this study was to perturb aspects of a sulfur cycle scheme used within a GCM to represent the climatological impacts of sulfate aerosol derived from natural and anthropogenic sulfur sources. It was found that perturbing volcanic SO2 emissions and the scavenging rate of SO2 by precipitation had the largest influence on the sulfate burden. When these parameters were perturbed the sulfate burden ranged from 0.73 to 1.17 TgS for 2050 sulfur emissions (A2 Special Report on Emissions Scenarios (SRES)), comparable with the range in sulfate burden across all the Intergovernmental Panel on Climate Change SRESs. Thus, the results here suggest that the range in sulfate burden due to model uncertainty is comparable with scenario uncertainty. Despite the large range in sulfate burden there was little influence on the climate sensitivity, which had a range of less than 0.5 K across the ensemble. We hypothesize that this small effect was partly associated with high sulfate loadings in the control phase of the experiment.

QUAGMIRE v1.3: a quasi-geostrophic model for investigating rotating fluids experiments

QUAGMIRE is a quasi-geostrophic numerical model for performing fast, high-resolution simulations of multi-layer rotating annulus laboratory experiments on a desktop personal computer. The model uses a hybrid finite-difference/spectral approach to numerically integrate the coupled nonlinear partial differential equations of motion in cylindrical geometry in each layer. Version 1.3 implements the special case of two fluid layers of equal resting depths. The flow is forced either by a differentially rotating lid, or by relaxation to specified streamfunction or potential vorticity fields, or both. Dissipation is achieved through Ekman layer pumping and suction at the horizontal boundaries, including the internal interface. The effects of weak interfacial tension are included, as well as the linear topographic beta-effect and the quadratic centripetal beta-effect. Stochastic forcing may optionally be activated, to represent approximately the effects of random unresolved features. A leapfrog time stepping scheme is used, with a Robert filter. Flows simulated by the model agree well with those observed in the corresponding laboratory experiments.

The Representative Soil Sampling Scheme of England and Wales: the spatial variation of topsoil nutrient status and pH between 1971 and 2001

The Representative Soil Sampling Scheme (RSSS) has monitored the soil of agricultural land in England and Wales since 1969. Here we describe the first spatial analysis of the data from these surveys using geostatistics. Four years of data (1971, 1981, 1991 and 2001) were chosen to examine the nutrient (available K, Mg and P) and pH status of the soil. At each farm, four fields were sampled; however, for the earlier years, coordinates were available for the farm only and not for each field. The averaged data for each farm were used for spatial analysis and the variograms showed spatial structure even with the smaller sample size. These variograms provide a reasonable summary of the larger scale of variation identified from the data of the more intensively sampled National Soil Inventory. Maps of kriged predictions of K generally show larger values in the central and southeastern areas (above 200 mg L-1) and an increase in values in the west over time, whereas Mg is fairly stable over time. The kriged predictions of P show a decline over time, particularly in the east, and those of pH show an increase in the east over time. Disjunctive kriging was used to examine temporal changes in available P using probabilities less than given thresholds of this element. The RSSS was not designed for spatial analysis, but the results show that the data from these surveys are suitable for this purpose. The results of the spatial analysis, together with those of the statistical analyses, provide a comprehensive view of the RSSS database as a basis for monitoring the soil. These data should be taken into account when future national soil monitoring schemes are designed.

The Representative Soil Sampling Scheme of England and Wales: a statistical analysis of topsoil nutrient status and pH between 1971 and 2001

The Representative Soil Sampling Scheme of England and Wales has recorded information on the soil of agricultural land in England and Wales since 1969. It is a valuable source of information about the soil in the context of monitoring for sustainable agricultural development. Changes in soil nutrient status and pH were examined over the period 1971-2001. Several methods of statistical analysis were applied to data from the surveys during this period. The main focus here is on the data for 1971, 1981, 1991 and 2001. The results of examining change over time in general show that levels of potassium in the soil have increased, those of magnesium have remained fairly constant, those of phosphorus have declined and pH has changed little. Future sampling needs have been assessed in the context of monitoring, to determine the mean at a given level of confidence and tolerable error and to detect change in the mean over time at these same levels over periods of 5 and 10 years. The results of a non-hierarchical multivariate classification suggest that England and Wales could be stratified to optimize future sampling and analysis. To monitor soil quality and health more generally than for agriculture, more of the country should be sampled and a wider range of properties recorded.

Atmosphere feedbacks during ENSO in a coupled GCM with a modified atmospheric convection scheme

The too diverse representation of ENSO in a coupled GCM limits one’s ability to describe future change of its properties. Several studies pointed to the key role of atmosphere feedbacks in contributing to this diversity. These feedbacks are analyzed here in two simulations of a coupled GCM that differ only by the parameterization of deep atmospheric convection and the associated clouds. Using the Kerry–Emanuel (KE) scheme in the L’Institut Pierre-Simon Laplace Coupled Model, version 4 (IPSL CM4; KE simulation), ENSO has about the right amplitude, whereas it is almost suppressed when using the Tiedke (TI) scheme. Quantifying both the dynamical Bjerknes feedback and the heat flux feedback in KE, TI, and the corresponding Atmospheric Model Intercomparison Project (AMIP) atmosphere-only simulations, it is shown that the suppression of ENSO in TI is due to a doubling of the damping via heat flux feedback. Because the Bjerknes positive feedback is weak in both simulations, the KE simulation exhibits the right ENSO amplitude owing to an error compensation between a too weak heat flux feedback and a too weak Bjerknes feedback. In TI, the heat flux feedback strength is closer to estimates from observations and reanalysis, leading to ENSO suppression. The shortwave heat flux and, to a lesser extent, the latent heat flux feedbacks are the dominant contributors to the change between TI and KE. The shortwave heat flux feedback differences are traced back to a modified distribution of the large-scale regimes of deep convection (negative feedback) and subsidence (positive feedback) in the east Pacific. These are further associated with the model systematic errors. It is argued that a systematic and detailed evaluation of atmosphere feedbacks during ENSO is a necessary step to fully understand its simulation in coupled GCMs.

Errors associated with the preparation of aseptic products in UK hospital pharmacies: lessons from the national aseptic error reporting scheme

Background Pharmacy aseptic units prepare and supply injectables to minimise risks. The UK National Aseptic Error Reporting Scheme has been collecting data on pharmacy compounding errors, including near-misses, since 2003. Objectives The cumulative reports from January 2004 to December 2007, inclusive, were analysed. Methods The different variables of product types, error types, staff making and detecting errors, stage errors detected, perceived contributory factors, and potential or actual outcomes were presented by cross-tabulation of data. Results A total of 4691 reports were submitted against an estimated 958 532 items made, returning 0.49% as the overall error rate. Most of the errors were detected before reaching patients, with only 24 detected during or after administration. The highest number of reports related to adult cytotoxic preparations (40%) and the most frequently recorded error was a labelling error (34.2%). Errors were mostly detected at first check in assembly area (46.6%). Individual staff error contributed most (78.1%) to overall errors, while errors with paediatric parenteral nutrition appeared to be blamed on low staff levels more than other products were. The majority of errors (68.6%) had no potential patient outcomes attached, while it appeared that paediatric cytotoxic products and paediatric parenteral nutrition were associated with greater levels of perceived patient harm. Conclusions The majority of reports were related to near-misses, and this study highlights scope for examining current arrangements for checking and releasing products, certainly for paediatric cytotoxic and paediatric parenteral nutrition preparations within aseptic units, but in the context of resource and capacity constraints.