A high-order arbitrarily unstructured finite-volume model of the global atmosphere: tests solving the shallow-water equations

Simulations of the global atmosphere for weather and climate forecasting require fast and accurate solutions and so operational models use high-order finite differences on regular structured grids. This precludes the use of local refinement; techniques allowing local refinement are either expensive (eg. high-order finite element techniques) or have reduced accuracy at changes in resolution (eg. unstructured finite-volume with linear differencing). We present solutions of the shallow-water equations for westerly flow over a mid-latitude mountain from a finite-volume model written using OpenFOAM. A second/third-order accurate differencing scheme is applied on arbitrarily unstructured meshes made up of various shapes and refinement patterns. The results are as accurate as equivalent resolution spectral methods. Using lower order differencing reduces accuracy at a refinement pattern which allows errors from refinement of the mountain to accumulate and reduces the global accuracy over a 15 day simulation. We have therefore introduced a scheme which fits a 2D cubic polynomial approximately on a stencil around each cell. Using this scheme means that refinement of the mountain improves the accuracy after a 15 day simulation. This is a more severe test of local mesh refinement for global simulations than has been presented but a realistic test if these techniques are to be used operationally. These efficient, high-order schemes may make it possible for local mesh refinement to be used by weather and climate forecast models.

Sparse space-time boundary element methods for the heat equation

The goal of this work is the efficient solution of the heat equation with Dirichlet or Neumann boundary conditions using the Boundary Elements Method (BEM). Efficiently solving the heat equation is useful, as it is a simple model problem for other types of parabolic problems. In complicated spatial domains as often found in engineering, BEM can be beneficial since only the boundary of the domain has to be discretised. This makes BEM easier than domain methods such as finite elements and finite differences, conventionally combined with time-stepping schemes to solve this problem. The contribution of this work is to further decrease the complexity of solving the heat equation, leading both to speed gains (in CPU time) as well as requiring smaller amounts of memory to solve the same problem. To do this we will combine the complexity gains of boundary reduction by integral equation formulations with a discretisation using wavelet bases. This reduces the total work to O(h

Finite sample weighting of recursive forecast errors

This paper proposes and tests a new framework for weighting recursive out-of-sample prediction errors according to their corresponding levels of in-sample estimation uncertainty. In essence, we show how to use the maximum possible amount of information from the sample in the evaluation of the prediction accuracy, by commencing the forecasts at the earliest opportunity and weighting the prediction errors. Via a Monte Carlo study, we demonstrate that the proposed framework selects the correct model from a set of candidate models considerably more often than the existing standard approach when only a small sample is available. We also show that the proposed weighting approaches result in tests of equal predictive accuracy that have much better sizes than the standard approach. An application to an exchange rate dataset highlights relevant differences in the results of tests of predictive accuracy based on the standard approach versus the framework proposed in this paper.

Finite-stretching corrections to the Milner-Witten-Cates theory for polymer brushes

This paper investigates finite-stretching corrections to the classical Milner-Witten-Cates theory for semi-dilute polymer brushes in a good solvent. The dominant correction to the free energy originates from an entropic repulsion caused by the impenetrability of the grafting surface, which produces a depletion of segments extending a distance $\mu \propto L^{-1}$ from the substrate, where $L$ is the classical brush height. The next most important correction is associated with the translational entropy of the chain ends, which creates the well-known tail where a small population of chains extend beyond the classical brush height by a distance $\xi \propto L^{-1/3}$. The validity of these corrections is confirmed by quantitative comparison with numerical self-consistent field theory.

Water wave scattering by finite arrays of circular structures

The scattering of small amplitude water waves by a finite array of locally axisymmetric structures is considered. Regions of varying quiescent depth are included and their axisymmetric nature, together with a mild-slope approximation, permits an adaptation of well-known interaction theory which ultimately reduces the problem to a simple numerical calculation. Numerical results are given and effects due to regions of varying depth on wave loading and free-surface elevation are presented.

Impact of mass lumping on gravity and Rossby waves in 2D finite-element shallow-water models

The goal of this study is to evaluate the effect of mass lumping on the dispersion properties of four finite-element velocity/surface-elevation pairs that are used to approximate the linear shallow-water equations. For each pair, the dispersion relation, obtained using the mass lumping technique, is computed and analysed for both gravity and Rossby waves. The dispersion relations are compared with those obtained for the consistent schemes (without lumping) and the continuous case. The P0-P1, RT0 and P-P1 pairs are shown to preserve good dispersive properties when the mass matrix is lumped. Test problems to simulate fast gravity and slow Rossby waves are in good agreement with the analytical results.

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 $A$?

Regional Differences in the Prediction of Extratropical Cyclones by the ECMWF Ensemble Prediction System

A regional study of the prediction of extratropical cyclones by the European Centre for Medium-Range Weather Forecasts (ECMWF) Ensemble Prediction System (EPS) has been performed. An objective feature-tracking method has been used to identify and track the cyclones along the forecast trajectories. Forecast error statistics have then been produced for the position, intensity, and propagation speed of the storms. In previous work, data limitations meant it was only possible to present the diagnostics for the entire Northern Hemisphere (NH) or Southern Hemisphere. A larger data sample has allowed the diagnostics to be computed separately for smaller regions around the globe and has made it possible to explore the regional differences in the prediction of storms by the EPS. Results show that in the NH there is a larger ensemble mean error in the position of storms over the Atlantic Ocean. Further analysis revealed that this is mainly due to errors in the prediction of storm propagation speed rather than in direction. Forecast storms propagate too slowly in all regions, but the bias is about 2 times as large in the NH Atlantic region. The results show that storm intensity is generally overpredicted over the ocean and underpredicted over the land and that the absolute error in intensity is larger over the ocean than over the land. In the NH, large errors occur in the prediction of the intensity of storms that originate as tropical cyclones but then move into the extratropics. The ensemble is underdispersive for the intensity of cyclones (i.e., the spread is smaller than the mean error) in all regions. The spatial patterns of the ensemble mean error and ensemble spread are very different for the intensity of cyclones. Spatial distributions of the ensemble mean error suggest that large errors occur during the growth phase of storm development, but this is not indicated by the spatial distributions of the ensemble spread. In the NH there are further differences. First, the large errors in the prediction of the intensity of cyclones that originate in the tropics are not indicated by the spread. Second, the ensemble mean error is larger over the Pacific Ocean than over the Atlantic, whereas the opposite is true for the spread. The use of a storm-tracking approach, to both weather forecasters and developers of forecast systems, is also discussed.

Differences in phosphorus retention and release in soils amended with animal manures and sewage sludge

Excessive levels of P in agricultural soils pose a threat to local water quality. This study evaluated (i) time-dependent changes in soil P sorption (expressed as a phosphorus sorption index, PSI) and P availability (as resin P) during incubation (100 d) with poultry litter, cattle slurry, sewage sludge, or KH2PO4, added on a P-equivalent basis (100 mg P kg(-1)), and (ii) the subsequent kinetics of P release, measured by repeated extractions with a mixed cation-anion exchange resin. Soil exchangeable Ca and ammonium oxalate-extractable Fe and Al were also determined at 100 d of incubation. The small decrease in P sorption in the litter and sludge treatments (25%), compared with that in the slurry and KH2PO4 treatments (52%) between 20 and 100 d of incubation was attributed partly to the formation of new adsorption sites for P. Subsequent P release was described by a power equation: Resin P = a(extraction number)(b), where the constants a and b represent resin P obtained with a single extraction and the rate of P release per resin extraction, respectively. On average, the rate of P release decreased in the order: KH2PO4 and slurry > litter > sludge, and was inversely related to exchangeable Ca content of the incubated soils (R-2 = 0.57). The slower rates of P release in the litter and sludge treatments (P < 0.001) are attributed to the large values for exchangeable Ca (1050-2640 and 1070-2710 mg kg(-1), respectively) in these amended soils. Future research concerned with short-term declines in environmentally harmful levels of P in recently amended soils should consider the differential effects of the amendments on soil P dynamics.

Individual differences in the effects of perceived controllability on pain perception: critical role of the prefrontal cortex

The degree to which perceived controllability alters the way a stressor is experienced varies greatly among individuals. We used functional magnetic resonance imaging to examine the neural activation associated with individual differences in the impact of perceived controllability on self-reported pain perception. Subjects with greater activation in response to uncontrollable (UC) rather than controllable (C) pain in the pregenual anterior cingulate cortex (pACC), periaqueductal gray (PAG), and posterior insula/SII reported higher levels of pain during the UC versus C conditions. Conversely, subjects with greater activation in the ventral lateral prefrontal cortex (VLPFC) in anticipation of pain in the UC versus C conditions reported less pain in response to UC versus C pain. Activation in the VLPFC was significantly correlated with the acceptance and denial subscales of the COPE inventory [Carver, C. S., Scheier, M. F., & Weintraub, J. K. Assessing coping strategies: A theoretically based approach. Journal of Personality and Social Psychology, 56, 267–283, 1989], supporting the interpretation that this anticipatory activation was associated with an attempt to cope with the emotional impact of uncontrollable pain. A regression model containing the two prefrontal clusters (VLPFC and pACC) predicted 64% of the variance in pain rating difference, with activation in the two additional regions (PAG and insula/SII) predicting almost no additional variance. In addition to supporting the conclusion that the impact of perceived controllability on pain perception varies highly between individuals, these findings suggest that these effects are primarily top-down, driven by processes in regions of the prefrontal cortex previously associated with cognitive modulation of pain and emotion regulation.

Individual Differences in Amygdala and Ventromedial Prefrontal Cortex Activity are Associated with Evaluation Speed and Psychological Well-being

Using functional magnetic resonance imaging, we examined whether individual differences in amygdala activation in response to negative relative to neutral information are related to differences in the speed with which such information is evaluated, the extent to which such differences are associated with medial prefrontal cortex function, and their relationship with measures of trait anxiety and psychological well-being (PWB). Results indicated that faster judgments of negative relative to neutral information were associated with increased left and right amygdala activation. In the prefrontal cortex, faster judgment time was associated with relative decreased activation in a cluster in the ventral anterior cingulate cortex (ACC, BA 24). Furthermore, people who were slower to evaluate negative versus neutral information reported higher PWB. Importantly, higher PWB was strongly associated with increased activation in the ventral ACC for negative relative to neutral information. Individual differences in trait anxiety did not predict variation in judgment time or in amygdala or ventral ACC activity. These findings suggest that people high in PWB effectively recruit the ventral ACC when confronted with potentially aversive stimuli, manifest reduced activity in subcortical regions such as the amygdala, and appraise such information as less salient as reflected in slower evaluative speed.