A multiple shooting approach for the numerical treatment of stellar structure and evolution

We present a new numerical method for solving the system of partial differential equations describing the structure and evolution of a spherically symmetric star. As usual, we employ the transversal method of lines in order to split the equations into a coupled spatial and temporal part. The novel features of the algorithm are the following: (a) Instead of using the Lagrangian picture we formulate the system of partial differential equations in the Eulerian picture. (b) We reformulate the equations of stellar structure as a multipoint boundary-value problem. By means of this reformulation the rather clumsy iterative matching procedure of stellar atmosphere and interior is avoided. (c) The multipoint boundary-value problem is solved by the multiple shooting method. This approach not only ensures a high accuracy of the stellar models calculated at each time step but also allows the free boundaries inside the star due to different energy transport mechanisms to be located exactly. (d) The time derivatives involved in the stellar-structure equations are discretized implicitly to second order accuracy. Moreover, at each time step, the chemical abundances are determined by using a sophisticated update procedure. In this way, a high accuracy is achieved with respect to the integration in time. The algorithm has turned out to be exceedingly reliable and numerically accurate. This is shown by the evolution of a 1 M_⊙ star up to the hydrogen-shell burning phase. In this example, the virial theorem, the law of mass conservation, and the law of energy conservation is fulfilled to a hitherto unattainable degree of accuracy. Since the multiple shooting method, which is at the heart of our approach, is a perfect example of a parallel algorithm, the computational speed of the algorithm might be substantially improved provided easy-to-program, high-performance parallel computers with sufficiently many processors become available in the near future.     

Readout Techniques for Drift and Low Frequency Noise Rejection in Infrared Arrays

Three different methods are presented to subtract thermal drifts and low-frequency noise from the signal of infrared array. The first is dead pixels with open Indium bumps, the second is reference output as implemented on the Hawaii2 multiplexer, and the third is dark pixels to emulate reference cells having a capacity connected to the gate of the unit cell field-effect transistor (FET). The third method is the most effective and yields a reduction in readout noise from15.4–9.4 erms. A novel method will be described to extend this readout technique to the Aladdin 1 K × 1 K InSb array. This revised version was published online in July 2006 with corrections to the Cover Date.     

Die Steuerung des vertikalen Austauschkoeffizienten durch die Temperaturstruktur in der unteren Troposphäre

Zusammenfassung Zwei unabhängige Verfahren zur direkten Bestimmung des vertikalen Austauschkoeffizienten auf dem Wege über Aerosolmessungen werden über lange Zeiträume hinweg angewandt um gesicherte Beziehungen zwischen Austauschkoeffizient und meteorologischen Zustandsgrössen, vorallem des vertikalen Temperaturgrdienten, in der unteren Troposphäre zwischen 700 und 3000 m NN abzuleiten. Das eine Verfahren beruht auf der Messung der RaB-Konzentration in der Luft in 700, 1800 und 3000 m NN, das andere in der aerologischen Registrierung der totalen Luftleitfähigkeit zwischen 700 und 1800 m. Im ersten Falle können aus den mittleren Konzentrations gradienten die mittleren Austauschkoeffizienten direkt berechnet werden, im zweiten werden die Luftleitfähigkeitswerte zunächst in Partikelkonzentrationen umgerechnet. Diese gestatten die Be rechnung von differentiellen Austauschkoeffizienten über beliebig kleine Höhenintervalle. Die wichtigsten Ergebnisse sind: a) Der mittlere Austauschkoeffizient zwischen 700–800 und 1800–300 m lässt sich bei geringer statischer Streuung als Funktion des mittleren Temperaturgradienten in der jeweiligen Schicht angeben. Nebeneinflüsse: Luftkörperwechsel, Kondensationswärme, Windscherung. b) Der mittlere Austauschkoeffizient lässt sich als Funktion der Stabilitäenergie in der betrachteten Schicht darstellen, Nebeneinflüsse wie bei a). c) Der Austauschkoeffizient durch eine Inversionsschicht hindruch ist eine Funktion des grössten positiven Temperaturgradienten in der Inversion. d) Der vertikale Austausch durch eine beliebig dicke Schicht hindurch wird im wesentlichen durch den kleinsten differentiellen Austauschkoeffizienten innerhald dieser Schicht bestimmt. — Schliesslich werden Ergebnisse über Feinstrukturuntersuchungen im Bereich von Unstetigkeitsschichten mitgeteilt.

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Summary The vertical-mass-exchange coefficient was determined from aerosol measurements by two independent methods; the measurements were kept up for a considerable period of time. Thus, well-substantiated relationships between the exchange coefficient on the one hand and meteorological parameters — especially the lapse rate — on the other were found to exist in the lower troposphere between 700 and 1800 m.The first of the two methods is based on measurements of RaB concentration in air at 700, 1800 and 3000 m, the second on recordings of soundings of overall air conductivity between 700 and 1800 m a.s.l. In the first case, mean exchange coefficients were computed from RaB concentration gradients given as averages from RaB measurements at the above-specified levels; in the second, the air conductivity data were used to compute particle concentration profiles, from which incremental exchange coefficients can be computed for thin-layer increments. The latter may be made as thin as is consistent with the vertical resolution of the soundings. The major results obtained were the following: a) The mean exchange coefficient for the 700–1800 and the 1800–3000 m layer is found to be a function of the average temperature lapse rate in the respective layer; this relationship is subject only to some slight statistic alscatter, and is affected in some measure by air-mass changes, wind shear and heat of condensation. b) The mean exchange coefficient is a function of the stability energy of the layer concerned, subject to the influences listed in a). c) The exchange coefficient of an inversion layer is a function of the greatest inverse lapse rate existing in any part of the inversion. d) Vertical mass exchange in or through a layer of any thickness is primarily determined by the lowest incremental mass-exchange coefficient of any thinner layer contained therein. — The study is supplemented by results of fine-structure analyses of discontinuity layers and their surroundings.

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Die Beschaffung der Einrichtungen, die Durchführung und Auswertung der Messungen wurde durch Mittel der US Army, European Research Office (Verträge DA-91-591-EUC-3936, DAJA-37-67C-0254 und DAJA-37-68C-0331) und des Bundesministeriums der Verteidigung der Deutschen Bundesrepublik ermöglicht. Für die grosszügige Förderung der Arbeiten haben wir sehr zu danken.     

The atmospheric heat budget over the western part of the Tibetan plateau during Monex, 1979

The atmospheric heat source strength over western Tibet has been computed for the period beginning with the last ten days in May, 1979 and extending through August, 1979. Our results show a significantly smaller heat source than that obtained by other authors. The discrepancy is mainly due to adjustments in the dray, coefficient suggested by observations and numerical modeling experiments. We subdivided western Tibet into northern and southern parts. In the north sensible heating, SH, provides the dominant input into the atmospheric heat source, whereas in the southern part latent heat, LP, offers a significant contribution after the start of the rainy season.Detailed heat budget calculations were also carried out over limited regions of southwestern Tibet which hau good station coverage. During periods with area-averaged rainfall ≤1 mm/day an atmospheric heat source maximum was located over southwestern Tibet near the 500 hPa level, while a heat sink dominated the upper troposphere in a layer of subsidence. When rainfall exceeded 4 mm/day, ascending motions and heal sources prevailed throughout the troposphere with maxima near 400 hPa. Time series analyses of the heat sourcs components show that the total atmospheric heat source is strongly modulated by the release of latent heat. Atmospheric radiational cooling reveals a phase shift in its relation with precipitation. During the first part of the observation period a correlation of that cooling exists mainly with the net radiation at the top of the atmosphere, during the last part with the net radiation at the ground.     

The propagation effect of commuting to work in the spatial transmission of COVID-19

Journal of Geographical Systems - This work is concerned with the spatiotemporal dynamics of the coronavirus disease 2019 (COVID-19) in Germany. Our goal is twofold: first, we propose a novel...     

Determination of the concentration of chemical main and trace elements (chemical matrix) in the aerosol from 1972 to 1982 at a North-alpine pure air station at 1780 m a.s.l.

Summary Aerosol particles have been separated by filter at a mountain station at 1780 m a.s.l. continuously since 1972. The station (a WMO-BAPMoN station with extended program) is freely exposed to all sides and in no way influenced by local aerosol sources. In the cold seasons it lies almost exclusively above the convection layer (very often over inversions) so that real background data can be acquired in these cases. By means of analytical methods the chemical components are determined on the exposed filters. This includes (apart from crust elements) also cations and anions from man-made and maritime sources as well as some important trace elements (in all, 15 components). Filters are exposed over 1–4 days. Filter exchange is determined by the respective weather situation with due regard to all changes so that the requirements for a reasonable correlation of the measured values with the meteorological conditions are met. Additionally, all meteorological parameters are determined, as well as intensity of the vertical exchange, number density of Aitken nuclei, concentration of natural radioactivity (RaB), and other variables. In the present paper the components of the aerosol chemical matrix are related to the behavior of the parameters below and the results are discussed in detail: Dependence of the individual components on total aerosol mass per volume, type of air mass; representation of two single cases of long-range transport of crust elements (source areas: Sahara, Colorado); vertical exchange coefficient, concentration of natural radioactivity, temperature gradient between valley and mountain station, temperature at the sampling site, relative humidity, precipitation rate, wind direction. occurrence of fog.Zusammenfassung An einer allseitig frei exponierten Bergstation in 1780 m NN werden seit 1982 die Aerosolpartikel auf Filtern gesammelt. Die Station (eine WMO-BAPMoN-Station mit erweitertem Programm) wird nicht durch lokale Aerosolquellen in irgendeiner Weise beeinflußt. Sie liegt in den kalten Jahreszeiten fast ausschließlich über der Konvektionsschicht (sehr oft oberhalb von Inversionen), so daß in diesen Fällen echte Background-Daten erfaßt werden. Mittels analytischer Verfahren werden an den exponierten Filtern alle chemischen Hauptkomponenten (von den Krustenelementen abgesehen auch die Kationen und Anionen sowohl anthropogenen als auch maritimen Ursprungs), sowie einige gewichtige Spurenelemente (insgesamt 15 Komponenten) bestimmt. Die Filter werden 1–4 Tage lang exponiert. Der Filterwechsel wird durch den jeweiligen Wetterzustand bzw. seiner Veränderung gesteuert, so daß eine gute Voraussetzung für eine Zuordnung der Meßwerte zu den meteorologischen Bedingungen gegeben ist. Zusätzlich werden alle meteorologischen Parameter, sowie die Intensität des Vertikalaustausches, Zahl der Aitken-Kerne, Konzentration der natürlichen Radioaktivität (RaB) und andere Größen bestimmt. In der vorliegenden Arbeit werden die Komponenten der chemischen Aerosolmatrix zum Verhalten der folgenden Parameter in Beziehung gesetzt und die Ergebnisse eingehend diskutiert:Abhängigkeit der Einzelkomponenten von: Gesamt-Aerosohnasse pro Volumen; Typ der Luftmasse und Darstellung von zwei Einzelfällen eines Ferntransportes von Krustenelementen (Quellgebiete: Sahara, Colorado); vertikaler Austauschkoeffizient; Konzentration der natürlichen Radioaktivität; Temperaturgradient zwischen Tal- und Bergstation; Temperatur an der Probennahmestelle; relative Feuchte; Niederschlagsintensität; Windrichtung; Auftreten von Nebel.

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Bestimmung der Konzentration chemischer Haupt- und Spurenstoffe (chemische Matrix) im Aerosol von 1972 bis 1982 an einer nordalpinen Reinluftstation in 1780 m NN.Teil II: Parametrische Korrelationsanalyse der chemischen Matrix mit Berücksichtigung meteorologischer Größen