Numerical prediction of atmospheric blocking - A case study

One of the fundamental questions in dynamical meteorology, and one of the basic objectives of GARP, is to determine the predictability of the atmosphere. In the early planning stage and preparation for GARP a number of theoretical and numerical studies were undertaken, indicating that there existed an inherent unpredictability in the atmosphere which even with the most ideal observing system would limit useful weather forecasting to 2-3 weeks.

Data assimilation with correlated observation errors: experiments with a 1-D shallow water model

Remote sensing observations often have correlated errors, but the correlations are typically ignored in data assimilation for numerical weather prediction. The assumption of zero correlations is often used with data thinning methods, resulting in a loss of information. As operational centres move towards higher-resolution forecasting, there is a requirement to retain data providing detail on appropriate scales. Thus an alternative approach to dealing with observation error correlations is needed. In this article, we consider several approaches to approximating observation error correlation matrices: diagonal approximations, eigendecomposition approximations and Markov matrices. These approximations are applied in incremental variational assimilation experiments with a 1-D shallow water model using synthetic observations. Our experiments quantify analysis accuracy in comparison with a reference or ‘truth’ trajectory, as well as with analyses using the ‘true’ observation error covariance matrix. We show that it is often better to include an approximate correlation structure in the observation error covariance matrix than to incorrectly assume error independence. Furthermore, by choosing a suitable matrix approximation, it is feasible and computationally cheap to include error correlation structure in a variational data assimilation algorithm.

Numerical conformal mapping and mesh generation for polygonal and multiply-connected regions

Details are given of a boundary-fitted mesh generation method for use in modelling free surface flow and water quality. A numerical method has been developed for generating conformal meshes for curvilinear polygonal and multiply-connected regions. The method is based on the Cauchy-Riemann conditions for the analytic function and is able to map a curvilinear polygonal region directly onto a regular polygonal region, with horizontal and vertical sides. A set of equations have been derived for determining the lengths of these sides and the least-squares method has been used in solving the equations. Several numerical examples are presented to illustrate the method.

Influence of shape and absorbing surface: a numerical study of railway noise barriers

Results are presented of a study of a performance of various track-side railway noise barriers, determined by using a two- dimensional numerical boundary element model. The basic model uses monopole sources and has been adapted to allow the sources to exhibit dipole-type radiation characteristics. A comparison of boundary element predictions of the performance of simple barriers and vehicle shapes is made with results obtained by using the standard U.K. prediction method. The results obtained from the numerical model indicate that modifying the source to exhibit dipole characteristics becomes more significant as the height of the barrier increases, and suggest that for any particular shape, absorbent barriers provide much better screening efficiency than the rigid equivalent. The cross-section of the rolling stock significantly affects the performance of rigid barriers. If the position of the upper edge is fixed, the results suggest that simple absorptive barriers provide more effective screening than tilted barriers. The addition of multiple edges to a barrier provides additional insertion loss without any increase in barrier height.

The contributions of domain-general and numerical factors to third-grade arithmetic skills and mathematical learning disability

Explanations of the marked individual differences in elementary school mathematical achievement and mathematical learning disability (MLD or dyscalculia) have involved domain-general factors (working memory, reasoning, processing speed and oral language) and numerical factors that include single-digit processing efficiency and multi-digit skills such as number system knowledge and estimation. This study of third graders (N = 258) finds both domain-general and numerical factors contribute independently to explaining variation in three significant arithmetic skills: basic calculation fluency, written multi-digit computation, and arithmetic word problems. Estimation accuracy and number system knowledge show the strongest associations with every skill and their contributions are both independent of each other and other factors. Different domain-general factors independently account for variation in each skill. Numeral comparison, a single digit processing skill, uniquely accounts for variation in basic calculation. Subsamples of children with MLD (at or below 10th percentile, n = 29) are compared with low achievement (LA, 11th to 25th percentiles, n = 42) and typical achievement (above 25th percentile, n = 187). Examination of these and subsets with persistent difficulties supports a multiple deficits view of number difficulties: most children with number difficulties exhibit deficits in both domain-general and numerical factors. The only factor deficit common to all persistent MLD children is in multi-digit skills. These findings indicate that many factors matter but multi-digit skills matter most in third grade mathematical achievement.

A high frequency $hp$ boundary element method for scattering by convex polygons

In this paper we propose and analyze a hybrid $hp$ boundary element method for the solution of problems of high frequency acoustic scattering by sound-soft convex polygons, in which the approximation space is enriched with oscillatory basis functions which efficiently capture the high frequency asymptotics of the solution. We demonstrate, both theoretically and via numerical examples, exponential convergence with respect to the order of the polynomials, moreover providing rigorous error estimates for our approximations to the solution and to the far field pattern, in which the dependence on the frequency of all constants is explicit. Importantly, these estimates prove that, to achieve any desired accuracy in the computation of these quantities, it is sufficient to increase the number of degrees of freedom in proportion to the logarithm of the frequency as the frequency increases, in contrast to the at least linear growth required by conventional methods.

Estimation of sea surface temperature from the Spinning Enhanced Visible and Infra Red Imager, improved using numerical weather prediction

Most of the operational Sea Surface Temperature (SST) products derived from satellite infrared radiometry use multi-spectral algorithms. They show, in general, reasonable performances with root mean square (RMS) residuals around 0.5 K when validated against buoy measurements, but have limitations, particularly a component of the retrieval error that relates to such algorithms' limited ability to cope with the full variability of atmospheric absorption and emission. We propose to use forecast atmospheric profiles and a radiative transfer model to simulate the algorithmic errors of multi-spectral algorithms. In the practical case of SST derived from the Spinning Enhanced Visible and Infrared Imager (SEVIRI) onboard Meteosat Second Generation (MSG), we demonstrate that simulated algorithmic errors do explain a significant component of the actual errors observed for the non linear (NL) split window algorithm in operational use at the Centre de Météorologie Spatiale (CMS). The simulated errors, used as correction terms, reduce significantly the regional biases of the NL algorithm as well as the standard deviation of the differences with drifting buoy measurements. The availability of atmospheric profiles associated with observed satellite-buoy differences allows us to analyze the origins of the main algorithmic errors observed in the SEVIRI field of view: a negative bias in the inter-tropical zone, and a mid-latitude positive bias. We demonstrate how these errors are explained by the sensitivity of observed brightness temperatures to the vertical distribution of water vapour, propagated through the SST retrieval algorithm.

Parameterised diabatic processes in numerical simulations of an extratropical cyclone

The parameterisation of diabatic processes in numerical models is critical for the accuracy of weather forecasts and for climate projections. A novel approach to the evaluation of these processes in models is introduced in this contribution. The approach combines a suite of on-line tracer diagnostics with off-line trajectory calculations. Each tracer tracks accumulative changes in potential temperature associated with a particular parameterised diabatic process in the model. A comparison of tracers therefore allows the identification of the most active diabatic processes and their downstream impacts. The tracers are combined with trajectories computed using model-resolved winds, allowing the various diabatic contributions to be tracked back to their time and location of occurrence. We have used this approach to investigate diabatic processes within a simulated extratropical cyclone. We focus on the warm conveyor belt, in which the dominant diabatic contributions come from large-scale latent heating and parameterised convection. By contrasting two simulations, one with standard convection parameterisation settings and another with reduced parameterised convection, the effects of parameterised convection on the structure of the cyclone have been determined. Under reduced parameterised convection conditions, the large-scale latent heating is forced to release convective instability that would otherwise have been released by the convection parameterisation. Although the spatial distribution of precipitation depends on the details of the split between parameterised convection and large-scale latent heating, the total precipitation amount associated with the cyclone remains largely unchanged. For reduced parameterised convection, a more rapid and stronger latent heating episode takes place as air ascends within the warm conveyor belt.

Analytical and numerical solutions describing the inward solidification of a binary melt

We present a mathematical model describing the inward solidification of a slab, a circular cylinder and a sphere of binary melt kept below its equilibrium freezing temperature. The thermal and physical properties of the melt and solid are assumed to be identical. An asymptotic method, valid in the limit of large Stefan number is used to decompose the moving boundary problem for a pure substance into a hierarchy of fixed-domain diffusion problems. Approximate, analytical solutions are derived for the inward solidification of a slab and a sphere of a binary melt which are compared with numerical solutions of the unapproximated system. The solutions are found to agree within the appropriate asymptotic regime of large Stefan number and small time. Numerical solutions are used to demonstrate the dependence of the solidification process upon the level of impurity and other parameters. We conclude with a discussion of the solutions obtained, their stability and possible extensions and refinements of our study.

Communicating probabilistic information from climate model ensembles—lessons from numerical weather prediction

Climate model ensembles are widely heralded for their potential to quantify uncertainties and generate probabilistic climate projections. However, such technical improvements to modeling science will do little to deliver on their ultimate promise of improving climate policymaking and adaptation unless the insights they generate can be effectively communicated to decision makers. While some of these communicative challenges are unique to climate ensembles, others are common to hydrometeorological modeling more generally, and to the tensions arising between the imperatives for saliency, robustness, and richness in risk communication. The paper reviews emerging approaches to visualizing and communicating climate ensembles and compares them to the more established and thoroughly evaluated communication methods used in the numerical weather prediction domains of day-to-day weather forecasting (in particular probabilities of precipitation), hurricane and flood warning, and seasonal forecasting. This comparative analysis informs recommendations on best practice for climate modelers, as well as prompting some further thoughts on key research challenges to improve the future communication of climate change uncertainties.