Knowing weather in place: the Helm Wind of Cross Fell

The Helm Wind of Cross Fell, North Pennines, is England's only named wind. As a product of the particular landscape found at Cross Fell, the Helm is a true local wind, and a phenomenon that has come to assume great cultural as well as environmental significance in the region and beyond. In this paper we draw on material from county histories, newspaper archives, and documents relating to investigations of the Helm Wind that were conducted by the Royal Meteorological Society between 1884 and 1889, and by British climatologist Gordon Manley (1908–1980), between 1937 and 1939, to document attempts to observe, measure, understand and explain this local wind over a period of 200 years. We show how different ways of knowing the Helm relate to contemporary practices of meteorology, highlighting the shifts that took place in terms of what constituted credible meteorological observation. We also acknowledge the overlapping nature of these ways of knowing and the persistence of multiple testimonies about the Helm and its effects

Land cover and vegetation data from an ecological survey of `key habitat' landscapes in England, 1992-1993

Since 1978, a series of national surveys (Countryside Survey, CS) have been carried out by the Centre for Ecology and Hydrology (CEH) (formerly the Institute of Terrestrial Ecology, ITE) to gather data on the natural environment in Great Britain (GB). As the sampling framework for these surveys is not optimised to yield data on rarer or more localised habitats, a survey was commissioned by the then Department of the Environment (DOE, now the Department for Environment, Food and Rural Affairs, DEFRA) in the 1990s to carry out additional survey work in English landscapes which contained semi-natural habitats that were perceived to be under threat, or which represented areas of concern to the ministry. The landscapes were lowland heath, chalk and limestone (calcareous) grasslands, coasts and uplands. The information recorded allowed an assessment of the extent and quality of a range of habitats defined during the project, which can now be translated into standard UK broad and priority habitat classes. The survey, known as the "Key Habitat Survey", followed a design which was a series of gridded, stratified, randomly selected 1 km squares taken as representative of each of the four landscape types in England, determined from statistical land classification and geological data ("spatial masks"). The definitions of the landscapes are given in the descriptions of the spatial masks, along with definitions of the surveyed habitats. A total of 213 of the 1 km2 square sample sites were surveyed in the summers of 1992 and 1993, with information being collected on vegetation species, land cover, landscape features and land use, applying standardised repeatable methods. The database contributes additional information and value to the long-term monitoring data gathered by the Countryside Survey and provides a valuable baseline against which future ecological changes may be compared, offering the potential for a repeat survey. The data were analysed and described in a series of contract reports and are summarised in the present paper, showing for example that valuable habitats were restricted in all landscapes, with the majority located within protected areas of countryside according to different UK designations. The dataset provides major potential for analyses, beyond those already published, for example in relation to climate change, agri-environment policies and land management. Precise locations of the plots are restricted, largely for reasons of landowner confidentiality. However, the representative nature of the dataset makes it highly valuable for evaluating the status of ecological elements within the associated landscapes surveyed. Both land cover data and vegetation plot data were collected during the surveys in 1992 and 1993 and are available via the following DOI: https://doi.org/10.5285/7aefe6aa-0760-4b6d-9473-fad8b960abd4. The spatial masks are also available from https://doi.org/10.5285/dc583be3-3649-4df6-b67e-b0f40b4ec895

Lyapunov exponents for products of complex Gaussian random matrices

The exact value of the Lyapunov exponents for the random matrix product $P_N = A_N A_{N-1}...A_1$ with each $A_i = \Sigma^{1/2} G_i^{\rm c}$, where $\Sigma$ is a fixed $d \times d$ positive definite matrix and $G_i^{\rm c}$ a $d \times d$ complex Gaussian matrix with entries standard complex normals, are calculated. Also obtained is an exact expression for the sum of the Lyapunov exponents in both the complex and real cases, and the Lyapunov exponents for diffusing complex matrices.Comment: 15 page

A Tale of Two Fractals: The Hofstadter Butterfly and The Integral Apollonian Gaskets

This paper unveils a mapping between a quantum fractal that describes a physical phenomena, and an abstract geometrical fractal. The quantum fractal is the Hofstadter butterfly discovered in 1976 in an iconic condensed matter problem of electrons moving in a two-dimensional lattice in a transverse magnetic field. The geometric fractal is the integer Apollonian gasket characterized in terms of a 300 BC problem of mutually tangent circles. Both of these fractals are made up of integers. In the Hofstadter butterfly, these integers encode the topological quantum numbers of quantum Hall conductivity. In the Apollonian gaskets an infinite number of mutually tangent circles are nested inside each other, where each circle has integer curvature. The mapping between these two fractals reveals a hidden threefold symmetry embedded in the kaleidoscopic images that describe the asymptotic scaling properties of the butterfly. This paper also serves as a mini review of these fractals, emphasizing their hierarchical aspects in terms of Farey fractions

The ecology of British upland landscapes. II. The influence of policy on the current character of the uplands and the potential for change

The paper demonstrates that the British Uplands have been influenced to a great extent by policy - for example, the planting of almost a million hectares of exotic conifers since the Second World War, and the extent of designated areas. Otherwise, climate change transcends policy and is locally important to coastal and high mountain habitats. The different policies affecting the Uplands, such as the Common Agricultural Policy, are described, as are the wide range of designations such as National Parks, which may have a stabilising effect in times of great change. A new trend has started in Scotland in the last 20 years of local initiatives, such as the community ownership of Eigg, however large landowners still dominate. An impact table is presented of the habitats that make up the Uplands and their links to driving forces, with potential changes described that are likely to take place under future policies such as Brexit. Dwarf shrub heath is the habitat affected by many management drivers, whereas habitats such as Inland Rock, are relatively stable but most likely to be affected by climate change

The ecology of British upland landscapes. I. Composition of landscapes, habitats, vegetation and species

A primary requirement for policy objectives is reliable figures on the composition of any region. Currently there is no comprehensive, definitive set of statistics for the British Uplands, hence the present paper. An overview of the background to the region is first provided, together with some examples of the available figures and a discussion of their limitations. The paper uses a formal structure, with landscapes at the highest level followed by habitats, then vegetation, and finally species, with exact definitions of the categories applied at all levels. The figures are produced from a survey of stratified, random one kilometre squares. The tables give comprehensive figures for Great Britain (GB) as a whole, and also England, Wales and Scotland. The Uplands are shown to cover 38 % of the country. In terms of UK Broad Habitats, Bog is the most common overall (2062 k ha). It is estimated that 41 % of upland vegetation in Britain is grazed by sheep, and Cervus elephus (red deer) are particularly evident in Scotland. Walls (mainly drystone) are the most important linear feature (84 k km) but hedgerows (30 k km) are also widespread. The major vegetation classes are those linked to moorlands and bogs (about 25 %) but those associated with fertile soils are also common (10 %). In terms of species, Potentilla erecta (tormentil) is the most frequent species with four other acid grassland species in the top ten. Calluna vulgaris (ling heather) has the highest cover in Great Britain (14.8 %)

Geometric phases for non-degenerate and degenerate mixed states

This paper focuses on the geometric phase of general mixed states under unitary evolution. Here we analyze both non-degenerate as well as degenerate states. Starting with the non-degenerate case, we show that the usual procedure of subtracting the dynamical phase from the total phase to yield the geometric phase for pure states, does not hold for mixed states. To this end, we furnish an expression for the geometric phase that is gauge invariant. The parallelity conditions are shown to be easily derivable from this expression. We also extend our formalism to states that exhibit degeneracies. Here with the holonomy taking on a non-abelian character, we provide an expression for the geometric phase that is manifestly gauge invariant. As in the case of the non-degenerate case, the form also displays the parallelity conditions clearly. Finally, we furnish explicit examples of the geometric phases for both the non-degenerate as well as degenerate mixed states.Comment: 23 page

The landscape ecological impact of afforestation on the British uplands and some initiatives to restore native woodland cover

The majority of forest cover in the British Uplands had been lost by the beginning of the Nineteenth Century, because of felling followed by overgrazing by sheep and deer. The situation remained unchanged until a government policy of afforestation, mainly by exotic conifers, after the First World War up to the present day. This paper analyses the distribution of these predominantly coniferous plantations, and shows how they occupy specific parts of upland landscapes in different zones throughout Britain. Whilst some landscapes are dominated by these new forests, elsewhere the blocks of trees are more localised. Although these forests virtually eliminate native ground vegetation, except in rides and unplanted land, the major negative impacts are at the landscape level. For example, drainage systems are altered and ancient cultural landscape patterns are destroyed. These impacts are summarised and possible ways of amelioration are discussed. By contrast, in recent years, a series of projects have been set up to restore native forest cover, as opposed to the extensive plantations of exotic species. Accordingly, the paper then provides three examples of such initiatives designed to restore native forests to otherwise bare landscapes, as well as setting them into a policy context. Whilst such projects cover a limited proportion of the British Uplands they nevertheless restore forest to landscapes at a local level

Quantifying the impact of an extreme climate event on species diversity in fragmented temperate forests: the effect of the October 1987 storm on British broadleaved woodlands

1. We report the impact of an extreme weather event, the October 1987 severe storm, on fragmented woodlands in southern Britain. We analysed ecological changes between 1971 and 2002 in 143 200-m2 plots in 10 woodland sites exposed to the storm with an ecologically equivalent sample of 150 plots in 16 non-exposed sites. In both years, understorey species-richness, species composition, soil pH and woody basal area of the tree and shrub canopy were measured. 2. We tested the hypothesis that the storm had deflected sites from the wider national trajectory of an increase in woody basal area and reduced understorey species-richness associated with ageing canopies and declining woodland management. We also expected storm disturbance to amplify the background trend of increasing soil pH, a UK-wide response to reduced atmospheric sulphur deposition. Path analysis was used to quantify indirect effects of storm exposure on understorey species richness via changes in woody basal area and soil pH. 3. By 2002, storm exposure was estimated to have increased mean species richness per 200 m2 by 32%. Woody basal area changes were highly variable and did not significantly differ with storm exposure. 4. Increasing soil pH was associated with a 7% increase in richness. There was no evidence that soil pH increased more as a function of storm exposure. Changes in species richness and basal area were negatively correlated: a 3.4% decrease in richness occurred for every 0.1-m2 increase in woody basal area per plot. 5. Despite all sites substantially exceeding the empirical critical load for nitrogen deposition, there was no evidence that in the 15 years since the storm, disturbance had triggered a eutrophication effect associated with dominance of gaps by nitrophilous species. 6. Synthesis: Although the impacts of the 1987 storm were spatially variable in terms of impacts on woody basal area, the storm had a positive effect on understorey species richness. There was no evidence that disturbance had increased dominance of gaps by invasive species. This could change if recovery from acidification results in a soil pH regime associated with greater macronutrient availability

AR and MA representation of partial autocorrelation functions, with applications

We prove a representation of the partial autocorrelation function (PACF), or the Verblunsky coefficients, of a stationary process in terms of the AR and MA coefficients. We apply it to show the asymptotic behaviour of the PACF. We also propose a new definition of short and long memory in terms of the PACF.Comment: Published in Probability Theory and Related Field