Three-dimensional analytical magnetohydrostatic equilibria of rigidly rotating magnetospheres in cylindrical geometry

We present three-dimensional solutions of the magnetohydrostatic equations in the co-rotating frame of reference outside a magnetized rigidly rotating cylinder. We make no symmetry assumption for the magnetic field, but to be able to make analytical progress we neglect outflows and specify a particular form for the current density. The magnetohydrostatic equations can then be reduced to a single linear partial differential equation for a pseudo-potential $U$, from which the magnetic field can be calculated by differentiation. The equation for $U$ can be solved by standard methods. The solutions can also be used to determine the plasma pressure, density and temperature as functions of all three spatial coordinates. Despite the obvious limitations of this approach, it can for example be used as a simple tool to create three-dimensional models for the closed field line regions of rotating magnetospheres without rotational symmetry.Comment: 13 pages, 2 figures, accepted for publication by Geophysical and Astrophysical Fluid Dynamic

On the Relationship between Equilibrium Bifurcations and Ideal MHD Instabilities for Line-Tied Coronal Loops

For axisymmetric models for coronal loops the relationship between the bifurcation points of magnetohydrodynamic (MHD) equilibrium sequences and the points of linear ideal MHD instability is investigated imposing line-tied boundary conditions. Using a well-studied example based on the Gold-Hoyle equilibrium, it is demonstrated that if the equilibrium sequence is calculated using the Grad-Shafranov equation, the instability corresponds to the second bifurcation point and not the first bifurcation point because the equilibrium boundary conditions allow for modes which are excluded from the linear ideal stability analysis. This is shown by calculating the bifurcating equilibrium branches and comparing the spatial structure of the solutions close to the bifurcation point with the spatial structure of the unstable mode. If the equilibrium sequence is calculated using Euler potentials the first bifurcation point of the Grad-Shafranov case is not found, and the first bifurcation point of the Euler potential description coincides with the ideal instability threshold. An explanation of this results in terms of linear bifurcation theory is given and the implications for the use of MHD equilibrium bifurcations to explain eruptive phenomena is briefly discussed.Comment: 22 pages, 6 figures, accepted by Solar Physic

Singular inextensible limit in the vibrations of post-buckled rods: Analytical derivation and role of boundary conditions

In-plane vibrations of an elastic rod clamped at both extremities are studied. The rod is modeled as an extensible planar Kirchhoff elastic rod under large displacements and rotations. Equilibrium configurations and vibrations around these configurations are computed analytically in the incipient post-buckling regime. Of particular interest is the variation of the first mode frequency as the load is increased through the buckling threshold. The loading type is found to have a crucial importance as the first mode frequency is shown to behave singularly in the zero thickness limit in the case of prescribed axial displacement, whereas a regular behavior is found in the case of prescribed axial load.This publication is based in part upon work supported by Award no. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST) (A.G.). A.G. is a Wolfson/Royal Society Merit Award holder. Support from the Royal Society, through the International Exchanges Scheme (Grant IE120203), is also acknowledge

Marktbeherrschung im Tankstellenmarkt Fehlender Binnen- und Außenwettbewerb an der Tankstelle? Deskriptive Evidenz für Marktbeherrschung

Wettbewerbspolitisch stellt sich die Frage, ob im 5er-Oligopol mit Aral, Shell, Esso, Total und Jet Binnenwettbewerb fehlt und ob dieses Oligopol wirksam Außenseiterwettbewerb ausgesetzt ist. Nach neuartigen Daten für vier Metropolen aus 2014, die auf die Marktransparenzstelle zurückgehen, verfolgen Aral und Shell ein Hochpreisstrategie, etwas weniger Esso und Total; Jet und die Nicht- Oligopolisten setzen niedrigere Preise. Morgens starten alle Tankstellen mit hohen Preisen, sukzessive im Laufe des Tages senkend, um in den frühen Abendstunden das niedrigste Niveau zu erreichen; das Vierer-Oligopol ohne Jet erhöht kurz nach 18.00 die Preise, Jet folgt deutlich später, spätestens zu Mitternacht sind alle zurück beim alten Preis. Kontrolliert man für andere Einflussfaktoren auf das Preiserhöhungsausmaß fällt auf, dass Shell gleich stark zu Aral die Preise erhöht, u.U. sogar stärker. Jet und die Nicht-Oligopolisten verhalten sich ebenfalls parallel zu Aral, aber deutlich schwächer. Bei von Shell ausgehenden Preiserhöhungen ist die Reaktionsverbundenheit bei Aral und Jet deutlich geringer. Die stabilen, jedoch markenspezifisch unterschiedlichen Preiserhöhungsmuster sind ein deutliches Zeichen für Parallelverhalten gemäß § 18, V GWB. Eine Marktbeherrschungsvermutung nach § 18, VI GWB darf sich nicht auf die Oligopolzugehörigkeit von Jet stützen

Collusive upward gasoline price movements in medium-sized German cities

Do we have effective competition between the gasoline's big five oligopolists (Aral, Shell, Esso, Total and Jet) and fringe gasoline stations? Using 2014 Market Transparency price data from 66 cities with populations between 60,000 and 100,000, we analyze which brands lead price increases, the first average price mark-up in the evening, and the trend on price increases until midnight. Furthermore, we measure the response time it takes for competitors to react to these price increases, and how much prices change from the beginning to the end of a day. By watching local activities of the big brands, it is possible to measure how smaller businesses, such as Jet or independent retailers, react to Aral's and Shell's price changes. Multivariate estimations allows to control for gasoline type (regular or diesel), school holidays, weekends, weekdays, location -such as East or West Germany-, wholesale and starting prices. Descriptive results show the typical patterns. Aral (or Shell) will start a price increase round, and then Shell (or Aral) will more or less immediately follow. Total, Esso and Non-Oligopolists react within one or two hours. Jet behaves more as an "outsider" with later reaction times and lower price mark-ups. Multivariate estimation indicates that the single cause "price change by competitors" is less important and nearly irrelevant for Jet

Three-dimensional solar active region magnetohydrostatic models and their stability using Euler potentials

Active regions (ARs) are typical magnetic structures found in the solar atmosphere. We calculate several magnetohydrostatic (MHS) equilibrium models that include the effect of a finite plasma-$\beta$ and gravity and that are representative of these structures in three dimensions. The construction of the models is based on the use of two Euler potentials, $\alpha$ and $\beta$, that represent the magnetic field as ${\bf B}=\nabla \alpha \times \nabla \beta$. The ideal MHS nonlinear partial differential equations are solved numerically using finite elements in a fixed 3D rectangular domain. The boundary conditions are initially chosen to correspond to a potential magnetic field (current-free) with known analytical expressions for the corresponding Euler potentials. The distinctive feature is that we incorporate the effect of shear by progressively deforming the initial potential magnetic field. This procedure is quite generic and allows us to generate a vast variety of MHS models. The thermal structure of the ARs is incorporated through the dependence of gas pressure and temperature on the Euler potentials. Using this method we achieve the characteristic hot and over-dense plasma found in ARs, but we demonstrate that the method can also be applied to study configurations with open magnetic field lines. Furthermore, we investigate basic topologies that include neutral lines. Our focus is on the force balance of the structures and we do not consider the energy balance in the constructed models. In addition, we address the difficult question of the stability of the calculated 3D models. We find that if the plasma is convectively stable, then the system is not prone in general to develop magnetic Rayleigh-Taylor instabilities.Comment: Accepted for publication in A&

Coronal magnetic field extrapolation using a specific family of analytical 3D MHS equilibria

Funding: LN acknowledges financial support by the University of St Andrews, and TN acknowledges support by the United Kingdom’s Science and Research Council (STFC) via Consolidated Grant ST/W001195/1.With current observational methods it is not possible to determine the magnetic field in the solar corona accurately. Therefore, coronal magnetic field models have to rely on extrapolation methods using photospheric magnetograms as boundary conditions. In recent years, due to the increased resolution of observations and the need to resolve non-force-free lower regions of the solar atmosphere, there have been increased efforts to use magnetohydrostatic (MHS) field models instead of force-free extrapolation methods. Although numerical methods to calculate MHS solutions can deal with non-linear problems and hence provide more accurate models, analytical three-dimensional MHS equilibria can also be used as a numerically relatively “cheap” complementary method. We discuss a family of analytical MHS equilibria that allows for a transition from a non-force-free region to a force-free region. The solution involves hypergeometric functions and while routines for the calculation of these are available, this can affect both the speed and the numerical accuracy of the calculations. Therefore, we look into the asymptotic behaviour of this solution in order to numerically approximate it through exponential functions aiming to improve the numerical efficiency. We present an illustrative example by comparing field line profiles, density and pressure differences between the exact solutions, the asymptotic solution and a hybrid model where the use of the hypergeometric function is restricted to an area around the transitional region between the non-force-free and the force-free domain.Publisher PDFNon peer reviewe

Weak turbulence and quasilinear diffusion for relativistic wave-particle interactions via a Markov approach

Funding: OA acknowledges financial support from the University of Exeter and from the United Kingdom Natural Environment Research Council (NERC) Independent Research Fellowship NE/V013963/1. OA and CW acknowledge financial support from the NERC Highlight Topic Grant NE/P017274/1 (Rad-Sat), and from United Kingdom Science and Technology Facilities Council (STFC) via Consolidated Grant ST/W000369/1. TE acknowledges financial support from an Early Career Fellowship, split jointly by the Leverhulme Trust (ECF2019-155) and the University of Leicester in the first instance (2019-21), but presently the University of Glasgow (2021-). TN acknowledges financial support from the STFC via Consolidated Grant ST/S000402/1. The University of Exeter cover the Open Access Publication Fee via a UKRI block grant.We derive weak turbulence and quasilinear models for relativistic charged particle dynamics in pitch-angle and energy space, due to interactions with electromagnetic waves propagating (anti-)parallel to a uniform background magnetic field. We use a Markovian approach that starts from the consideration of single particle motion in a prescribed electromagnetic field. This Markovian approach has a number of benefits, including: (i) the evident self-consistent relationship between a more general weak turbulence theory and the standard resonant diffusion quasilinear theory (as is commonly used in e.g. radiation belt and solar wind modelling); (ii) the general nature of the Fokker-Planck equation that can be derived without any prior assumptions regarding its form; (iii) the clear dependence of the form of the Fokker-Planck equation and the transport coefficients on given specific timescales. The quasilinear diffusion coefficients that we derive are not new in and of themselves, but this concise derivation and discussion of the weak turbulence and quasilinear theories using the Markovian framework is physically very instructive. The results presented herein form fundamental groundwork for future studies that consider phenomena for which some of the assumptions made in this manuscript may be relaxed.Publisher PDFPeer reviewe

Particle acceleration in collapsing magnetic traps with a braking plasma jet

Collapsing magnetic traps (CMTs) are one proposed mechanism for generating non-thermal particle populations in solar flares. CMTs occur if an initially stretched magnetic field structure relaxes rapidly into a lower-energy configuration, which is believed to happen as a by-product of magnetic reconnection. A similar mechanism for energising particles has also been found to operate in the Earth's magnetotail. One particular feature proposed to be of importance for particle acceleration in the magnetotail is that of a braking plasma jet, i.e. a localised region of strong flow encountering stronger magnetic field which causes the jet to slow down and stop. Such a feature has not been included in previously proposed analytical models of CMTs for solar flares. In this work we incorporate a braking plasma jet into a well studied CMT model for the first time. We present results of test particle calculations in this new CMT model. We observe and characterise new types of particle behaviour caused by the magnetic structure of the jet braking region, which allows electrons to be trapped both in the braking jet region and the loop legs. We compare and contrast the behaviour of particle orbits for various parameter regimes of the underlying trap by examining particle trajectories, energy gains and the frequency with which different types of particle orbit are found for each parameter regime.PostprintPeer reviewe

Analyzing AIA flare observations using convolutional neural networks

TL acknowledges support by the UK's Science and Technology Facilities Council (STFC) Doctoral Training Centre Grant ST/P006809/1 (ScotDIST). TN and CP both acknowledge support by the STFC Consolidated Grant ST/S000402/1.In order to efficiently analyse the vast amount of data generated by solar space missions and ground-based instruments, modern machine learning techniques such as decision trees, support vector machines (SVMs) and neural networks can be very useful. In this paper we present initial results from using a convolutional neural network (CNN) to analyse observations from the Atmospheric Imaging Assembly (AIA) in the 1,600Å wavelength. The data is pre-processed to locate flaring regions where flare ribbons are visible in the observations. The CNN is created and trained to automatically analyse the shape and position of the flare ribbons, by identifying whether each image belongs into one of four classes: two-ribbon flare, compact/circular ribbon flare, limb flare, or quiet Sun, with the final class acting as a control for any data included in the training or test sets where flaring regions are not present. The network created can classify flare ribbon observations into any of the four classes with a final accuracy of 94%. Initial results show that most of the images are correctly classified with the compact flare class being the only class where accuracy drops below 90% and some observations are wrongly classified as belonging to the limb class.Publisher PDFPeer reviewe