Efficient shock capturing for isentropic flows using arithmetic averaging

A shock capturing scheme is presented for the equations of isentropic flow based on upwind differencing applied to a locally linearized set of Riemann problems. This includes the two-dimensional shallow water equations using the familiar gas dynamics analogy. An average of the flow variables across the interface between cells is required, and this average is chosen to be the arithmetic mean for computational efficiency, leading to arithmetic averaging. This is in contrast to usual ‘square root’ averages found in this type of Riemann solver where the computational expense can be prohibitive. The scheme is applied to a two-dimensional dam-break problem and the approximate solution compares well with those given by other authors.

An efficient shock capturing scheme for the pseudo-unsteady equations representing steady inviscid flow

An efficient algorithm is presented for the solution of the steady Euler equations of gas dynamics. The scheme is based on solving linearised Riemann problems approximately and in more than one dimension incorporates operator splitting. The scheme is applied to a standard test problem of flow down a channel containing a circular arc bump for three different mesh sizes.

An efficient shock capturing algorithm for compressible flows in a duct of variable cross-section

A numerical scheme is presented for the solution of the Euler equations of compressible flow of a gas in a single spatial co-ordinate. This includes flow in a duct of variable cross-section as well as flow with slab, cylindrical or spherical symmetry and can prove useful when testing codes for the two-dimensional equations governing compressible flow of a gas. The resulting scheme requires an average of the flow variables across the interface between cells and for computational efficiency this average is chosen to be the arithmetic mean, which is in contrast to the usual ‘square root’ averages found in this type of scheme. The scheme is applied with success to five problems with either slab or cylindrical symmetry and a comparison is made in the cylindrical case with results from a two-dimensional problem with no sources.

A shock capturing scheme for steady, supercritical, free-surface flow

Abstract A finite difference scheme is presented for the solution of the two-dimensional shallow water equations in steady, supercritical flow. The scheme incorporates numerical characteristic decomposition, is shock capturing by design and incorporates space-marching as a result of the assumption that the flow is wholly supercritical in at least one space dimension. Results are shown for problems involving oblique hydraulic jumps and reflection from a wall.

Shock capturing scheme for steady, supersonic, isentropic flow

A finite difference scheme is presented for the solution of the two-dimensional equations of steady, supersonic, isentropic flow. The scheme incorporates numerical characteristic decomposition, is shock-capturing by design and incorporates space marching as a result of the assumption that the flow is wholly supersonic in at least one space dimension. Results are shown for problems involving oblique hydraulic jumps and reflection from a wall.

Reconstruction of historical pollination rates reveals linked declines of pollinators and plants

Widespread reports of low pollination rates suggest a recent anthropogenic decline in pollination that could threaten natural and agricultural ecosystems. Nevertheless, unequivocal evidence for a decline in pollination over time has remained elusive because it was not possible to determine historical pollination rates. Here we demonstrate a widely applicable method for reconstructing historical pollination rates, thus allowing comparison with contemporary rates from the same sites. We focused on the relationship between the oil-collecting bee Rediviva peringueyi (Melittidae) and the guild of oil-secreting orchid species (Coryciinae) that depends on it for pollination. The guild is distributed across the highly transformed and fragmented lowlands of the Cape Region of South Africa. We show that rehydrated herbarium specimens of Pterygodium catholicum, the most abundant member of the guild, contain a record of past pollinator activity in the form of pollinarium removal rates. Analysis of a pollination time series showed a recent decline in pollination on Signal Hill, a small urban conservation area. The same herbaria contain historical species occurrence data. We analyzed this data and found that there has been a contemporaneous shift in orchid guild composition in urban areas due to the local extirpation of the non-clonal species, consistent with their greater dependence on seeds and pollination for population persistence.

Historicizing counterurbanization: in-migration and the reconstruction of rural space in Berkshire (UK), 1901-51

This article uses census data for Berkshire to argue that large-scale counterurbanization began much earlier than is generally recognized in some parts of southern England. This was not just movement down the urban hierarchy, which as Pooley and Turnbull have demonstrated was a long-term feature of England’s settlement system, but in some cases at least amenity-driven migration to rural areas of the kind increasingly recognized as a core component of recent counterurbanization. Despite a reduction of acreage Berkshire’s rural districts saw a 54% rise in population between 1901 and 1951. The sub-regional pattern of growth is assessed to gauge whether ‘clean break’ migration to the remote west of the county (which remained effectively out of commuting range from London throughout the period) was taking place, or whether counterurbanization was confined to the more accessible eastern districts. However, whilst population did increase in both west and east, it was in fact the central districts that grew most impressively. Three case study parishes are investigated in order to gauge the nature and consequences of counterurbanization at a local level. Professional and business migrants figure prominently, seeking to preserve and promote the rural attributes of their new communities, without however cutting their ties to urban centres. It is argued that migration to rural Berkshire in the first half of the twentieth century cannot adequately be described either as a form of extended suburbanization or an anti-metropolitan ‘clean break’. Rather, early counterurbanization marks the first stage on the long road to a post-productivist countryside, in which countryside becomes detached from agriculture, there is socio-economic convergence between town and country, and the ‘rural’ increasingly becomes defined by landscape and identity rather than economic function.

Implications of non-cylindrical flux ropes for magnetic cloud reconstruction techniques and the interpretation of double flux rope events

Magnetic clouds (MCs) are a subset of interplanetary coronal mass ejections (ICMEs) which exhibit signatures consistent with a magnetic flux rope structure. Techniques for reconstructing flux rope orientation from single-point in situ observations typically assume the flux rope is locally cylindrical, e.g., minimum variance analysis (MVA) and force-free flux rope (FFFR) fitting. In this study, we outline a non-cylindrical magnetic flux rope model, in which the flux rope radius and axial curvature can both vary along the length of the axis. This model is not necessarily intended to represent the global structure of MCs, but it can be used to quantify the error in MC reconstruction resulting from the cylindrical approximation. When the local flux rope axis is approximately perpendicular to the heliocentric radial direction, which is also the effective spacecraft trajectory through a magnetic cloud, the error in using cylindrical reconstruction methods is relatively small (≈ 10∘). However, as the local axis orientation becomes increasingly aligned with the radial direction, the spacecraft trajectory may pass close to the axis at two separate locations. This results in a magnetic field time series which deviates significantly from encounters with a force-free flux rope, and consequently the error in the axis orientation derived from cylindrical reconstructions can be as much as 90∘. Such two-axis encounters can result in an apparent ‘double flux rope’ signature in the magnetic field time series, sometimes observed in spacecraft data. Analysing each axis encounter independently produces reasonably accurate axis orientations with MVA, but larger errors with FFFR fitting.