Mathematical notation: help or hindrance?

In this article Geoff Tennant puts forward a range of reasons for using mathematical notation, emphasising the need to allow children learning it time and space to come to terms with it. Examples are given in furthering the argument that the time to introduce notation is after the concept is already fully understood.

Rapid topographic change measured by high-resolution satellite radar at Soufriere Hills Volcano, Montserrat, 2008-2010

High-resolution satellite radar observations of erupting volcanoes can yield valuable information on rapidly changing deposits and geomorphology. Using the TerraSAR-X (TSX) radar with a spatial resolution of about 2 m and a repeat interval of 11-days, we show how a variety of techniques were used to record some of the eruptive history of the Soufriere Hills Volcano, Montserrat between July 2008 and February 2010. After a 15-month pause in lava dome growth, a vulcanian explosion occurred on 28 July 2008 whose vent was hidden by dense cloud. We were able to show the civil authorities using TSX change difference images that this explosion had not disrupted the dome sufficient to warrant continued evacuation. Change difference images also proved to be valuable in mapping new pyroclastic flow deposits: the valley-occupying block-and-ash component tending to increase backscatter and the marginal surge deposits reducing it, with the pattern reversing after the event. By comparing east- and west-looking images acquired 12 hours apart, the deposition of some individual pyroclastic flows can be inferred from change differences. Some of the narrow upper sections of valleys draining the volcano received many tens of metres of rockfall and pyroclastic flow deposits over periods of a few weeks. By measuring the changing shadows cast by these valleys in TSX images the changing depth of infill by deposits could be estimated. In addition to using the amplitude data from the radar images we also used their phase information within the InSAR technique to calculate the topography during a period of no surface activity. This enabled areas of transient topography, crucial for directing future flows, to be captured.

The impact of finer-resolution air-sea coupling on the intraseasonal oscillation of the Indian monsoon

The new HadKPP atmosphere–ocean coupled model is described and then used to determine the effects of sub-daily air–sea coupling and fine near-surface ocean vertical resolution on the representation of the Northern Hemisphere summer intra-seasonal oscillation. HadKPP comprises the Hadley Centre atmospheric model coupled to the K Profile Parameterization ocean-boundary-layer model. Four 30-member ensembles were performed that varied in oceanic vertical resolution between 1 m and 10 m and in coupling frequency between 3 h and 24 h. The 10 m, 24 h ensemble exhibited roughly 60% of the observed 30–50 day variability in sea-surface temperatures and rainfall and very weak northward propagation. Enhancing either only the vertical resolution or only the coupling frequency produced modest improvements in variability and only a standing intra-seasonal oscillation. Only the 1 m, 3 h configuration generated organized, northward-propagating convection similar to observations. Sub-daily surface forcing produced stronger upper-ocean temperature anomalies in quadrature with anomalous convection, which likely affected lower-atmospheric stability ahead of the convection, causing propagation. Well-resolved air–sea coupling did not improve the eastward propagation of the boreal summer intra-seasonal oscillation in this model. Upper-ocean vertical mixing and diurnal variability in coupled models must be improved to accurately resolve and simulate tropical sub-seasonal variability. In HadKPP, the mere presence of air–sea coupling was not sufficient to generate an intra-seasonal oscillation resembling observations.

Integration of chaos theory and mathematical models in building simulation: Part I: Literature review

Current mathematical models in building research have been limited in most studies to linear dynamics systems. A literature review of past studies investigating chaos theory approaches in building simulation models suggests that as a basis chaos model is valid and can handle the increasingly complexity of building systems that have dynamic interactions among all the distributed and hierarchical systems on the one hand, and the environment and occupants on the other. The review also identifies the paucity of literature and the need for a suitable methodology of linking chaos theory to mathematical models in building design and management studies. This study is broadly divided into two parts and presented in two companion papers. Part (I) reviews the current state of the chaos theory models as a starting point for establishing theories that can be effectively applied to building simulation models. Part (II) develops conceptual frameworks that approach current model methodologies from the theoretical perspective provided by chaos theory, with a focus on the key concepts and their potential to help to better understand the nonlinear dynamic nature of built environment systems. Case studies are also presented which demonstrate the potential usefulness of chaos theory driven models in a wide variety of leading areas of building research. This study distills the fundamental properties and the most relevant characteristics of chaos theory essential to building simulation scientists, initiates a dialogue and builds bridges between scientists and engineers, and stimulates future research about a wide range of issues on building environmental systems.

Integration of chaos theory and mathematical models in building simulation: Part II: Conceptual frameworks

Current mathematical models in building research have been limited in most studies to linear dynamics systems. A literature review of past studies investigating chaos theory approaches in building simulation models suggests that as a basis chaos model is valid and can handle the increasing complexity of building systems that have dynamic interactions among all the distributed and hierarchical systems on the one hand, and the environment and occupants on the other. The review also identifies the paucity of literature and the need for a suitable methodology of linking chaos theory to mathematical models in building design and management studies. This study is broadly divided into two parts and presented in two companion papers. Part (I), published in the previous issue, reviews the current state of the chaos theory models as a starting point for establishing theories that can be effectively applied to building simulation models. Part (II) develop conceptual frameworks that approach current model methodologies from the theoretical perspective provided by chaos theory, with a focus on the key concepts and their potential to help to better understand the nonlinear dynamic nature of built environment systems. Case studies are also presented which demonstrate the potential usefulness of chaos theory driven models in a wide variety of leading areas of building research. This study distills the fundamental properties and the most relevant characteristics of chaos theory essential to (1) building simulation scientists and designers (2) initiating a dialogue between scientists and engineers, and (3) stimulating future research on a wide range of issues involved in designing and managing building environmental systems.

A simplified mathematical model for urban microclimate simulation

Techniques for modelling urban microclimates and urban block surfaces temperatures are desired by urban planners and architects for strategic urban designs at the early design stages. This paper introduces a simplified mathematical model for urban simulations (UMsim) including urban surfaces temperatures and microclimates. The nodal network model has been developed by integrating coupled thermal and airflow model. Direct solar radiation, diffuse radiation, reflected radiation, long-wave radiation, heat convection in air and heat transfer in the exterior walls and ground within the complex have been taken into account. The relevant equations have been solved using the finite difference method under the Matlab platform. Comparisons have been conducted between the data produced from the simulation and that from an urban experimental study carried out in a real architectural complex on the campus of Chongqing University, China in July 2005 and January 2006. The results show a satisfactory agreement between the two sets of data. The UMsim can be used to simulate the microclimates, in particular the surface temperatures of urban blocks, therefore it can be used to assess the impact of urban surfaces properties on urban microclimates. The UMsim will be able to produce robust data and images of urban environments for sustainable urban design.

Four-dimensional variational data assimilation for high resolution nested models

Four-dimensional variational data assimilation (4D-Var) is used in environmental prediction to estimate the state of a system from measurements. When 4D-Var is applied in the context of high resolution nested models, problems may arise in the representation of spatial scales longer than the domain of the model. In this paper we study how well 4D-Var is able to estimate the whole range of spatial scales present in one-way nested models. Using a model of the one-dimensional advection–diffusion equation we show that small spatial scales that are observed can be captured by a 4D-Var assimilation, but that information in the larger scales may be degraded. We propose a modification to 4D-Var which allows a better representation of these larger scales.

Near real-time flood detection in urban and rural areas using high resolution Synthetic Aperture Radar images

A near real-time flood detection algorithm giving a synoptic overview of the extent of flooding in both urban and rural areas, and capable of working during night-time and day-time even if cloud was present, could be a useful tool for operational flood relief management. The paper describes an automatic algorithm using high resolution Synthetic Aperture Radar (SAR) satellite data that builds on existing approaches, including the use of image segmentation techniques prior to object classification to cope with the very large number of pixels in these scenes. Flood detection in urban areas is guided by the flood extent derived in adjacent rural areas. The algorithm assumes that high resolution topographic height data are available for at least the urban areas of the scene, in order that a SAR simulator may be used to estimate areas of radar shadow and layover. The algorithm proved capable of detecting flooding in rural areas using TerraSAR-X with good accuracy, and in urban areas with reasonable accuracy. The accuracy was reduced in urban areas partly because of TerraSAR-X’s restricted visibility of the ground surface due to radar shadow and layover.

A near real-time algorithm for flood detection in urban and rural areas using high resolution Synthetic Aperture Radar images

A near real-time flood detection algorithm giving a synoptic overview of the extent of flooding in both urban and rural areas, and capable of working during night-time and day-time even if cloud was present, could be a useful tool for operational flood relief management and flood forecasting. The paper describes an automatic algorithm using high resolution Synthetic Aperture Radar (SAR) satellite data that assumes that high resolution topographic height data are available for at least the urban areas of the scene, in order that a SAR simulator may be used to estimate areas of radar shadow and layover. The algorithm proved capable of detecting flooding in rural areas using TerraSAR-X with good accuracy, and in urban areas with reasonable accuracy.

A method for merging flow-dependent forecast error statistics from an ensemble with static statistics for use in high resolution variational data assimilation

The background error covariance matrix, B, is often used in variational data assimilation for numerical weather prediction as a static and hence poor approximation to the fully dynamic forecast error covariance matrix, Pf. In this paper the concept of an Ensemble Reduced Rank Kalman Filter (EnRRKF) is outlined. In the EnRRKF the forecast error statistics in a subspace defined by an ensemble of states forecast by the dynamic model are found. These statistics are merged in a formal way with the static statistics, which apply in the remainder of the space. The combined statistics may then be used in a variational data assimilation setting. It is hoped that the nonlinear error growth of small-scale weather systems will be accurately captured by the EnRRKF, to produce accurate analyses and ultimately improved forecasts of extreme events.

Northern Hemisphere extratropical cyclones in a warming climate in the HiGEM high-resolution climate model

Changes to the Northern Hemisphere winter (December, January and February) extratropical storm tracks and cyclones in a warming climate are investigated. Two idealised climate change experiments with HiGEM1.1, a doubled CO2 and a quadrupled CO2 experiment, are compared against a present day control run. An objective feature tracking method is used and a focus given to regional changes. The climatology of extratropical storm tracks from the control run is shown to be in good agreement with ERA-40, while the frequency distribution of cyclone intensity also compares well. In both simulations the mean climate changes are generally consistent with the simulations of the IPCC AR4 models, with a strongly enhanced surface warming at the winter pole and the reduced lower tropospheric warming over the North Atlantic Ocean associated with the slowdown of the Meridional Overturning Circulation. The circulation changes in the North Atlantic are different between the two idealised simulations with different CO2 forcings. In the North Atlantic the storm tracks are influenced by the slowdown of the MOC, the enhanced surface polar warming, and the enhanced upper tropical troposphere warming, giving a north eastward shift of the storm tracks in the 2XCO2 experiment, but no shift in the 4XCO2 experiment. Over the Pacific, in the 2XCO2 experiment, changes in the mean climate are associated with local temperature changes, while in the 4XCO2 experiment the changes in the Pacific are impacted by the weakened tropical circulation. The storm track changes are consistent with the shifts in the zonal wind. Total cyclone numbers are found to decrease over the Northern Hemisphere with increasing CO2 forcing. Changes in cyclone intensity are found using 850hPa vorticity, mean sea level pressure, and 850hPa winds. The intensity of the Northern Hemisphere cyclones is found to decrease relative to the control.

Impact of increasing resolution and a warmer climate on extreme weather from NH extra-tropical cyclones

The effect of a warmer climate on the properties of extra-tropical cyclones is investigated using simulations of the ECHAM5 global climate model at resolutions of T213 (60 km) and T319 (40 km). Two periods representative of the end of the 20th and 21st centuries are investigated using the IPCC A1B scenario. The focus of the paper is on precipitation for the NH summer and winter seasons, however results from vorticity and winds are also presented. Similar number of events are identified at both resolutions. There are, however, a greater number of extreme precipitation events in the higher reso- lution run. The difference between maximum intensity distributions are shown to be statistically significant using a Kolmogorov-Smirnov test. A Generalised Pareto Distribution is used to analyse changes in extreme precipitation and wind events. In both resolutions, there is an increase in the number of ex- treme precipitation events in a warmer climate for all seasons, together with a reduction in return period. This is not associated with any increased verti- cal velocity, or with any increase in wind intensity in the winter and spring. However, there is an increase in wind extremes in the summer and autumn associated with tropical cyclones migrating into the extra-tropics.