Thomas Neukirch

We use statistical tools to analyse data from the Solar Dynamics Observatory Helioseismic and Magnetic Imager to determine the distribution of the magnetic flux of photospheric magnetic features and its variation over a full solar cycle. In particular, we use statistical figures of merit to test how well different types of probability density function represent the magnetic flux distribution inferred from the data and how their shape changes over the solar cycle. Our analysis shows that a double power law provides the best representation of the data over the full solar cycle and we present the dependence of the power law exponents on the phase of the solar cycle. The nature of the observed flux distributions at different times during the solar cycle is significant because it could be used to try and infer information about solar magnetic field generation mechanisms. We discuss potential implications of a double power law distribution for solar magnetic field generation.

With current observational methods it is not possible to directly measure the magnetic field in the solar corona with sufficient accuracy. 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. In this paper, we present an extrapolation method based on a family of analytical MHS equilibria that allows for a transition from a non-force-free region to a force-free region. We demonstrate how asymptotic forms of the solutions can help to increase the numerical efficiency of the method. Through both artificial boundary condition testing and a first application to observational data, we validate the method’s effectiveness and practical utility.

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 an extrapolation method based on 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 accuracy of the calculations. We have looked into the asymptotic behaviour of this solution to approximate it by exponential functions which improves the numerical efficiency. We present results with boundary conditions based on artificial magnetograms to test the method, and also the application of the model to observational data.

Current sheets in the collisionless solar wind usually have kinetic spatial scales. In-situ measurements show that these current sheets are often approximately force-free, i.e. the directions of their current density and magnetic field are aligned, despite the fact that the plasma beta is found to be of the order of one. The measurements also often show systematic asymmetric spatial variations of the plasma density and temperature across the current sheets, whilst the plasma pressure is approximately uniform. Neukirch et al. (2020) found exact equilibrium models of force-free collisionless current sheets which allowed for asymmetric plasma density and temperature gradients. These models assumed that the form of the distribution function for electrons and ions is the same. If one assumes that the bulk velocity of the ion population vanishes, the force-free condition is only satisfied approximately. Also, quasi-neutrality requires the presence of a nonvanishing electric potential. We show that an approximate treatment implies that the asymmetries in the density and the temperature only differ by a scale factor from the asymmetries found for the exactly force-free case.T. Neukirch, I. Vasko, A. Artemyev and O. Allanson, ApJ 891, 86 (2020).

Investigating the motion of charged particles in time- and space-dependent electromagnetic fields is central to many areas of space and astrophysical plasmas. Here we present results of studying the energy changes of particle orbits that are trapped in inhomogeneous and time-dependent magnetic fields with rapidly shortening field lines. These so-called collapsing magnetic trap (CMT) models can be useful to better understand the particle energisation processes occurring below the reconnection region in a solar flare. Braking jets may be associated with magnetic reconnection, for example when a sunward flow slows down as it approaches a stronger region of magnetic field. We generalise a 2D CMT model with braking jet (Borissov et al., 2016) to three dimensions and investigate the dynamics of particles in this 3D CMT model. The resulting particle orbits show a sensitive dependence of particle energies on the initial conditions of orbits, with initial pitch angles playing a particularly important role. This sensitive dependence relates to the time evolution of trapping regions that develop in the braking jet region of the CMT, ensuring that some orbits spend a significant time in the loop legs of field lines, whilst others escape these regions for the duration of the simulation. These loop leg trapped particle orbits see significantly lower energy gains than those orbits that repeatedly pass the loop top, with some of these particles even losing energy. This gives us greater insight into the importance of the curvature of collapsing loop tops for the Fermi acceleration mechanism acting on the particles. Borissov A. et al., Solar Physics 291, Issue 5, 1385

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. We therefore 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.

Investigating the motion of charged particles in time- and space-dependent electromagnetic fields is central to many areas of space and astrophysical plasmas. Here we present results of studying the energy changes of particle orbits that are trapped in inhomogeneous magnetic fields with rapidly shortening field lines. These so-called collapsing magnetic trap (CMT) models can be useful for explaining the acceleration of particles below the reconnection region in a solar flare. For both 2D and 3D CMT models (e.g. Giuliani et al. 2005; Grady & Neukirch, 2009), betatron acceleration was considered to be the dominant energisation mechanism. We present new results that have been obtained using an improved version of the 3D CMT model by Grady and Neukirch (2009). Our investigations show that a sizeable portion of particle orbits can gain a significant amount of energy that is not explained by the betatron effect. The other mechanism at play appears to be Fermi acceleration at loop tops, where the particle passes through the region of field that is collapsing the most rapidly. We show that the particles that experience this effect the most have initial positions that are related to specific regions of the magnetic field model and it is these particle orbits whose energy gains are not adequately explained by betatron acceleration alone. In fact, some particle orbits seem to gain energy almost entirely as a result of this Fermi acceleration. One can also show that for suitable initial conditions the same effect can be seen in the 2D CMT model given by Giuliani et al. (2005). This updated understanding of the systems at play for particle acceleration in a CMT can, for example, inform any changes made to future CMT models by accounting for the large number of particles that see energy gains due to Fermi acceleration. Giuliani, P. et al., ApJ 635, 636Grady, K. & Neukirch, T., A&A 508, 1461  

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: 1) 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 modeling); 2) the general nature of the Fokker-Planck equation that can be derived without any prior assumptions regarding its form; 3) the clear dependence of the form of the Fokker-Planck equation and the transport coefficients on given specific timescales.Of particular interest is the ability to derive the governing Fokker-Planck equation, and the required 1st-order (drift) and 2nd-order (diffusion) transport coefficients. Furthermore this method allows for both the inclusion of non-resonant effects, as well as interactions over very short timescales, below a typical de-correlation timescale. All of these nonlinear features have recently been shown to be important for high-energy electron dynamics, and are beyond the reach of the standard resonant diffusion quasilinear theory.This concise derivation and discussion of the weak turbulence and quasilinear theories using the Markovian framework is physically very instructive and may form fundamental groundwork for future studies.Front. Astron. Space Sci. - Research Topic (Plasma Waves in Space Physics: Carrying On the Research Legacies of Peter Gary and Richard Thorne)doi: 10.3389/fspas.2021.805699 (Reviewed by Danny Summers & Jay M. Albert)

Computing the solar coronal magnetic field and plasmaenvironment is an important research topic on it's own rightand also important for space missions like Solar Orbiter toguide the analysis of remote sensing and in-situ instruments.In the inner solar corona plasma forces can be neglected andthe field is modelled under the assumption of a vanishingLorentz-force. Further outwards (above about two solar radii)plasma forces and the solar wind flow has to be considered.Finally in the heliosphere one has to consider that the Sunis rotating and the well known Parker-spiral forms.We have developed codes based on optimization principlesto solve nonlinear force-free, magneto-hydro-static andstationary MHD-equilibria. In the present work we want toextend these methods by taking the solar rotation into account.