Gradient recovery in adaptive finite-element methods for parabolic problems

We derive energy-norm a posteriori error bounds, using gradient recovery (ZZ) estimators to control the spatial error, for fully discrete schemes for the linear heat equation. This appears to be the �rst completely rigorous derivation of ZZ estimators for fully discrete schemes for evolution problems, without any restrictive assumption on the timestep size. An essential tool for the analysis is the elliptic reconstruction technique.Our theoretical results are backed with extensive numerical experimentation aimed at (a) testing the practical sharpness and asymptotic behaviour of the error estimator against the error, and (b) deriving an adaptive method based on our estimators. An extra novelty provided is an implementation of a coarsening error "preindicator", with a complete implementation guide in ALBERTA in the appendix.

A finite element method for nonlinear elliptic problems

We present a Galerkin method with piecewise polynomial continuous elements for fully nonlinear elliptic equations. A key tool is the discretization proposed in Lakkis and Pryer, 2011, allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretization method is that a recovered (finite element) Hessian is a byproduct of the solution process. We build on the linear method and ultimately construct two different methodologies for the solution of second order fully nonlinear PDEs. Benchmark numerical results illustrate the convergence properties of the scheme for some test problems as well as the Monge–Amp`ere equation and the Pucci equation.

A finite element method for second order nonvariational elliptic problems

We propose a numerical method to approximate the solution of second order elliptic problems in nonvariational form. The method is of Galerkin type using conforming finite elements and applied directly to the nonvariational (nondivergence) form of a second order linear elliptic problem. The key tools are an appropriate concept of “finite element Hessian” and a Schur complement approach to solving the resulting linear algebra problem. The method is illustrated with computational experiments on three linear and one quasi-linear PDE, all in nonvariational form.

A finite element-spherical harmonics model for radiative transfer in inhomogeneous clouds. Part II. some applications

EVENT has been used to examine the effects of 3D cloud structure, distribution, and inhomogeneity on the scattering of visible solar radiation and the resulting 3D radiation field. Large eddy simulation and aircraft measurements are used to create realistic cloud fields which are continuous or broken with smooth or uneven tops. The values, patterns and variance in the resulting downwelling and upwelling radiation from incident visible solar radiation at different angles are then examined and compared to measurements. The results from EVENT confirm that 3D cloud structure is important in determining the visible radiation field, and that these results are strongly influenced by the solar zenith angle. The results match those from other models using visible solar radiation, and are supported by aircraft measurements of visible radiation, providing confidence in the new model.

Nature of vegetation and building morphology characteristics across a city: influence on shadow patterns and mean radiant temperatures in London

Vegetation and building morphology characteristics are investigated at 19 sites on a north-south LiDAR transect across the megacity of London. Local maxima of mean building height and building plan area density at the city centre are evident. Surprisingly, the mean vegetation height (zv3) is also found to be highest in the city centre. From the LiDAR data various morphological parameters are derived as well as shadow patterns. Continuous images of the effects of buildings and of buildings plus vegetationon sky view factor (Ψ) are derived. A general reduction of Ψ is found, indicating the importance of including vegetation when deriving Ψ in urban areas. The contribution of vegetation to the shadowing at ground level is higher during summer than in autumn. Using these 3D data the influence on urban climate and mean radiant temperature (T mrt ) is calculated with SOLWEIG. The results from these simulations highlight that vegetation can be most effective at reducing heat stress within dense urban environments in summer. The daytime average T mrt is found to be lowest in the densest urban environments due to shadowing; foremost from buildings but also from trees. It is clearly shown that this method could be used to quantify the influence of vegetation on T mrt within the urban environment. The results presented in this paper highlight a number of possible climate sensitive planning practices for urban areas at the local scale (i.e. 102- 5 × 103 m).

The influence of vegetation and building morphology on shadow patterns and mean radiant temperatures in urban areas: model development and evaluation

The solar and longwave environmental irradiance geometry (SOLWEIG) model simulates spatial variations of 3-D radiation fluxes and mean radiant temperature (T mrt) as well as shadow patterns in complex urban settings. In this paper, a new vegetation scheme is included in SOLWEIG and evaluated. The new shadow casting algorithm for complex vegetation structures makes it possible to obtain continuous images of shadow patterns and sky view factors taking both buildings and vegetation into account. For the calculation of 3-D radiation fluxes and T mrt, SOLWEIG only requires a limited number of inputs, such as global shortwave radiation, air temperature, relative humidity, geographical information (latitude, longitude and elevation) and urban geometry represented by high-resolution ground and building digital elevation models (DEM). Trees and bushes are represented by separate DEMs. The model is evaluated using 5 days of integral radiation measurements at two sites within a square surrounded by low-rise buildings and vegetation in Göteborg, Sweden (57°N). There is good agreement between modelled and observed values of T mrt, with an overall correspondence of R 2 = 0.91 (p < 0.01, RMSE = 3.1 K). A small overestimation of T mrt is found at locations shadowed by vegetation. Given this good performance a number of suggestions for future development are identified for applications which include for human comfort, building design, planning and evaluation of instrument exposure.

Contribution of land use changes to near-surface air temperatures during recent summer extreme heat events in the Phoenix Metropolitan Area

The impact of 1973–2005 land use–land cover (LULC) changes on near-surface air temperatures during four recent summer extreme heat events (EHEs) are investigated for the arid Phoenix, Arizona, metropolitan area using the Weather Research and Forecasting Model (WRF) in conjunction with the Noah Urban Canopy Model. WRF simulations were carried out for each EHE using LULC for the years 1973, 1985, 1998, and 2005. Comparison of measured near-surface air temperatures and wind speeds for 18 surface stations in the region show a good agreement between observed and simulated data for all simulation periods. The results indicate consistent significant contributions of urban development and accompanying LULC changes to extreme temperatures for the four EHEs. Simulations suggest new urban developments caused an intensification and expansion of the area experiencing extreme temperatures but mainly influenced nighttime temperatures with an increase of up to 10 K. Nighttime temperatures in the existing urban core showed changes of up to 2 K with the ongoing LULC changes. Daytime temperatures were not significantly affected where urban development replaced desert land (increase by 1 K); however, maximum temperatures increased by 2–4 K when irrigated agricultural land was converted to suburban development. According to the model simulations, urban landscaping irrigation contributed to cooling by 0.5–1 K in maximum daytime as well as minimum nighttime 2-m air temperatures in most parts of the urban region. Furthermore, urban development led to a reduction of the already relatively weak nighttime winds and therefore a reduction in advection of cooler air into the city.

Simulated changes in the relationship between tropical ocean temperatures and the western African monsoon during the mid-Holocene

Results from nine coupled ocean-atmosphere simulations have been used to investigate changes in the relationship between the variability of monsoon precipitation over western Africa and tropical sea surface temperatures (SSTs) between the mid-Holocene and the present day. Although the influence of tropical SSTs on the African monsoon is generally overestimated in the control simulations, the models reproduce aspects of the observed modes of variability. Thus, most models reproduce the observed negative correlation between western Sahelian precipitation and SST anomalies in the eastern tropical Pacific, and many of them capture the positive correlation between SST anomalies in the eastern tropical Atlantic and precipitation over the Guinea coastal region. Although the response of individual model to the change in orbital forcing between 6 ka and present differs somewhat, eight of the models show that the strength of the teleconnection between SSTs in the eastern tropical Pacific and Sahelian precipitation is weaker in the mid-Holocene. Some of the models imply that this weakening was associated with a shift towards longer time periods (from 3–5 years in the control simulations toward 4–10 years in the mid-Holocene simulations). The simulated reduction in the teleconnection between eastern tropical Pacific SSTs and Sahelian precipitation appears to be primarily related to a reduction in the atmospheric circulation bridge between the Pacific and West Africa but, depending on the model, other mechanisms such as increased importance of other modes of tropical ocean variability or increased local recycling of monsoonal precipitation can also play a role.

The effect of (mis-specified) GARCH filters on the finite sample distribution of the BDS test

This paper considers the effect of using a GARCH filter on the properties of the BDS test statistic as well as a number of other issues relating to the application of the test. It is found that, for certain values of the user-adjustable parameters, the finite sample distribution of the test is far-removed from asymptotic normality. In particular, when data generated from some completely different model class are filtered through a GARCH model, the frequency of rejection of iid falls, often substantially. The implication of this result is that it might be inappropriate to use non-rejection of iid of the standardised residuals of a GARCH model as evidence that the GARCH model ‘fits’ the data.

Technical note: a simple procedure for removing temporal discontinuities in ERA-Interim upper stratospheric temperatures for use in nudged chemistry-climate model simulations

This note describes a simple procedure for removing unphysical temporal discontinuities in ERA-Interim upper stratospheric global mean temperatures in March 1985 and August 1998 that have arisen due to changes in satellite radiance data used in the assimilation. The derived temperature adjustments (offsets) are suitable for use in stratosphere-resolving chemistry-climate models that are nudged (relaxed) to ERA-Interim winds and temperatures. Simulations using a nudged version of the Canadian Middle Atmosphere Model (CMAM) show that the inclusion of the temperature adjustments produces temperature time series that are devoid of the large jumps in 1985 and 1998. Due to its strong temperature dependence, the simulated upper stratospheric ozone is also shown to vary smoothly in time, unlike in a nudged simulation without the adjustments where abrupt changes in ozone occur at the times of the temperature jumps. While the adjustments to the ERA-Interim temperatures remove significant artefacts in the nudged CMAM simulation, spurious transient effects that arise due to water vapour and persist for about 5 yr after the 1979 switch to ERA-Interim data are identified, underlining the need for caution when analysing trends in runs nudged to reanalyses.

Global snow mass measurements and the effect of stratigraphic detail on inversion of microwave brightness temperatures

Snow provides large seasonal storage of freshwater, and information about the distribution of snow mass as Snow Water Equivalent (SWE) is important for hydrological planning and detecting climate change impacts. Large regional disagreements remain between estimates from reanalyses, remote sensing and modelling. Assimilating passive microwave information improves SWE estimates in many regions but the assimilation must account for how microwave scattering depends on snow stratigraphy. Physical snow models can estimate snow stratigraphy, but users must consider the computational expense of model complexity versus acceptable errors. Using data from the National Aeronautics and Space Administration Cold Land Processes Experiment (NASA CLPX) and the Helsinki University of Technology (HUT) microwave emission model of layered snowpacks, it is shown that simulations of the brightness temperature difference between 19 GHz and 37 GHz vertically polarised microwaves are consistent with Advanced Microwave Scanning Radiometer-Earth Observing System (AMSR-E) and Special Sensor Microwave Imager (SSM/I) retrievals once known stratigraphic information is used. Simulated brightness temperature differences for an individual snow profile depend on the provided stratigraphic detail. Relative to a profile defined at the 10 cm resolution of density and temperature measurements, the error introduced by simplification to a single layer of average properties increases approximately linearly with snow mass. If this brightness temperature error is converted into SWE using a traditional retrieval method then it is equivalent to ±13 mm SWE (7% of total) at a depth of 100 cm. This error is reduced to ±5.6 mm SWE (3 % of total) for a two-layer model.

Dynamical and observational constraints on tropical Pacific sea surface temperatures at the last glacial maximum

Asynchronously coupled atmosphere and ocean general circulation model simulations are used to examine the consequences of changes in the west/east sea-surface temperature (SST) gradient across the equatorial Pacific at the last glacial maximum (LGM). Simulations forced by the CLIMAP SST for the LGM, where the west/east SST gradient across the Pacific is reduced compared to present, produce a reduction in the strength of the trade winds and a decrease in the west/east slope of the equatorial thermocline that is incompatible with thermocline depths newly inferred from foraminiferal assemblages. Stronger-than-present trade winds, and a more realistic simulation of the thermocline slope, are produced when eastern Pacific SSTs are 2°C cooler than western Pacific SSTs. Our study highlights the importance of spatial heterogeneity in tropical SSTs in determining key features of the glacial climate.

The surface temperatures of Earth: steps towards integrated understanding of variability and change

Surface temperature is a key aspect of weather and climate, but the term may refer to different quantities that play interconnected roles and are observed by different means. In a community-based activity in June 2012, the EarthTemp Network brought together 55 researchers from five continents to improve the interaction between scientific communities who focus on surface temperature in particular domains, to exploit the strengths of different observing systems and to better meet the needs of different communities. The workshop identified key needs for progress towards meeting scientific and societal requirements for surface temperature understanding and information, which are presented in this community paper. A "whole-Earth" perspective is required with more integrated, collaborative approaches to observing and understanding Earth's various surface temperatures. It is necessary to build understanding of the relationships between different surface temperatures, where presently inadequate, and undertake large-scale systematic intercomparisons. Datasets need to be easier to obtain and exploit for a wide constituency of users, with the differences and complementarities communicated in readily understood terms, and realistic and consistent uncertainty information provided. Steps were also recommended to curate and make available data that are presently inaccessible, develop new observing systems and build capacities to accelerate progress in the accuracy and usability of surface temperature datasets.

The potential impact of changes in lower stratospheric water vapour on stratospheric temperatures over the past 30 years

This study investigates the potential contribution of observed changes in lower stratospheric water vapour to stratospheric temperature variations over the past three decades using a comprehensive global climate model (GCM). Three case studies are considered. In the first, the net increase in stratospheric water vapour (SWV) from 1980–2010 (derived from the Boulder frost-point hygrometer record using the gross assumption that this is globally representative) is estimated to have cooled the lower stratosphere by up to ∼0.2 K decade−1 in the global and annual mean; this is ∼40% of the observed cooling trend over this period. In the Arctic winter stratosphere there is a dynamical response to the increase in SWV, with enhanced polar cooling of 0.6 K decade−1 at 50 hPa and warming of 0.5 K decade−1 at 1 hPa. In the second case study, the observed decrease in tropical lower stratospheric water vapour after the year 2000 (imposed in the GCM as a simplified representation of the observed changes derived from satellite data) is estimated to have caused a relative increase in tropical lower stratospheric temperatures by ∼0.3 K at 50 hPa. In the third case study, the wintertime dehydration in the Antarctic stratospheric polar vortex (again using a simplified representation of the changes seen in a satellite dataset) is estimated to cause a relative warming of the Southern Hemisphere polar stratosphere by up to 1 K at 100 hPa from July–October. This is accompanied by a weakening of the westerly winds on the poleward flank of the stratospheric jet by up to 1.5 m s−1 in the GCM. The results show that, if the measurements are representative of global variations, SWV should be considered as important a driver of transient and long-term variations in lower stratospheric temperature over the past 30 years as increases in long-lived greenhouse gases and stratospheric ozone depletion.

Measuring ion temperatures and studying the ion energy balance in the high-latitude ionosphere

Data are presented for a nighttime ion heating event observed by the EISCAT radar on 16 December 1988. In the experiment, the aspect angle between the radar beam and the geomagnetic field was fixed at 54.7°, which avoids any ambiguity in derived ion temperature caused by anisotropy in the ion velocity distribution function. The data were analyzed with an algorithm which takes account of the non-Maxwellian line-of-sight ion velocity distribution. During the heating event, the derived spectral distortion parameter (D∗) indicated that the distribution function was highly distorted from a Maxwellian form when the ion drift increased to 4 km s−1. The true three-dimensional ion temperature was used in the simplified ion balance equation to compute the ion mass during the heating event. The ion composition was found to change from predominantly O4 to mainly molecular ions. A theoretical analysis of the ion composition, using the MSIS86 model and published values of the chemical rate coefficients, accounts for the order-of-magnitude increase in the atomic/molecular ion ratio during the event, but does not successfully explain the very high proportion of molecular ions that was observed.