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Energetic particle populations in the solar corona and in the heliosphere appear to have different characteristics even when produced in the same solar flare. It is not clear what causes this difference: properties of the acceleration region, the large-scale magnetic field configuration in the flare, or particle transport effects, such as scattering. We use a combination of non-linear force-free magnetohydrostatic simulations, magnetohydrodynamic and test-particle modelling to investigate magnetic reconnection, particle acceleration and transport in two solar flares events: an M-class flare on June 19th, 2013, and an X-class flare on September 6th, 2011. We show that, although in both events particles are energised at the same locations, the magnetic field structure around the acceleration region results in different characteristics between particle populations precipitating towards the photosphere and those ejected towards the upper corona and the heliosphere. We expect this effect to be ubiquitous when particles are accelerated close to the boundary between open and colsed magnetic fields and, therefore, may be key to solar flares with substantial particle emission into the heliosphere. Furthermore, this analysis elucidates the mechanisms by which escaping particle populations can be created in flares.

We model the time-dependent radio emission from a disc accretion event in a T-Tauri star using 3D, ideal magnetohydrodynamic simulations combined with a gyrosynchrotron emission and radiative transfer model. We predict for the first time, the multifrequency (1–1000 GHz) intensity and circular polarization from a flaring T-Tauri star. A flux tube, connecting the star with its circumstellar disc, is populated with a distribution of non-thermal electrons that is allowed to decay exponentially after a heating event in the disc and the system is allowed to evolve. The energy distribution of the electrons, as well as the non-thermal power-law index and loss rate, are varied to see their effect on the overall flux. Spectra are generated from different lines of sight, giving different views of the flux tube and disc. The peak flux typically occurs around 20–30 GHz and the radio luminosity is consistent with that observed from T-Tauri stars. For all simulations, the peak flux is found to decrease and move to lower frequencies with elapsing time. The frequency-dependent circular polarization can reach 10$-30{{\ \rm per\ cent}}$ but has a complex structure that evolves as the flare evolves. Our models show that observations of the evolution of the spectrum and its polarization can provide important constraints on physical properties of the flaring environment and associated accretion event.

As a frequent and energetic particle accelerator, our Sun provides us with an excellent astrophysical laboratory for understanding the fundamental process of particle acceleration. The exploitation of radiative diagnostics from electrons has shown that acceleration operates on sub-second time scales in a complex magnetic environment, where direct electric fields, wave turbulence, and shock waves all must contribute, although precise details are severely lacking. Ions were assumed to be accelerated in a similar manner to electrons, but γ-ray imaging confirmed that emission sources are spatially separated from X-ray sources, suggesting distinctly different acceleration mechanisms. Current X-ray and γ-ray spectroscopy provides only a basic understanding of accelerated particle spectra and the total energy budgets are therefore poorly constrained. Additionally, the recent detection of relativistic ion signatures lasting many hours, without an electron counterpart, is an enigma. We propose a single platform to directly measure the physical conditions present in the energy release sites and the environment in which the particles propagate and deposit their energy. To address this fundamental issue, we set out a suite of dedicated instruments that will probe both electrons and ions simultaneously to observe; high (seconds) temporal resolution photon spectra (4 keV – 150 MeV) with simultaneous imaging (1 keV – 30 MeV), polarization measurements (5–1000 keV) and high spatial and temporal resolution imaging spectroscopy in the UV/EUV/SXR (soft X-ray) regimes. These instruments will observe the broad range of radiative signatures produced in the solar atmosphere by accelerated particles.

In this study, we inverstigate the acceleration of electrons and ions at current sheets in the flaring solar corona, and their transport into the heliosphere. We consider both generic solar flare models and specific flaring events with a data-driven approach. The aim is to answer two questions: (a) what fraction of particles accelerated in different flares can escape into the heliosphere?; and (b) what are the characteristics of the particle populations propagating towards the chromosphere and into the heliosphere?We use a combination of data-driven 3D magnetohydrodynamics simulations with drift-kinetic particle simulations to model the evolution of the magnetic field and both thermal and non-thermal plasma and to forward-model observable characteristics. Particles are accelerated in current sheets associated with flaring reconnection. When applied to a specific flare, the model successfully predicts observed features such as the location and relative intensity of hard X-ray sources and helioseismic source locations. This confirms the viability of the approach.Using these MHD-particle models, we will show how the magnetic field evolution and particle transport processes affect the characteristics of both energetic electrons and ions in the the inner corona and the heliosphere. The implications for interpretation of in situ measurements of energetic particles by Solar Orbiter and Parker Solar Probe will be discussed.

Decametric radio emission provides a unique insight into the physics of solar and heliospheric plasmas. Along with dynamic spectra, the spatial characteristics of the emission sources observed in solar radio bursts yield important information about the behaviour of high-energy non-thermal electrons, and the state of thermal plasma in the upper solar corona. Recently, it has been shown that sizes and locations of radio sources in the 10-100 MHz range can be used as a diagnostic tool for plasma turbulence in the upper corona and inner heliosphere. However, observations in this spectral range can be strongly affected by limited spatial resolution of the instrument, as well as by the effect of the Earth's ionosphere on radio wave propagation.We describe a new method for correcting radio intensity maps for instrumental and ionospheric effects using observations of a known radio source at an arbitrary location in the sky. Based on this method, we derive sizes and areas of the emission sources in the solar radio bursts observed by the Low-Frequency Array (LOFAR) in 30-45 MHz range. It is shown that the sizes of sources are of the order of ten arcminutes and decrease with increasing frequency. Overall, we find that the sizes and their variation, as well as the shapes of the sources in the considered events are consistent with the theoretical models of turbulent radio-wave scattering in the solar corona developed by Kontar et al. 2019 (Astrophys.J., 884, 122).

The goal of this study is to explore a novel method for the solar photospheric magnetic field diagnostics using Stokes V widths of different magnetosensitive Fe~I spectral lines. We calculate Stokes I and V profiles of several Fe I lines based on a one-dimensional photospheric model VAL C using the NICOLE radiative transfer code. These profiles are used to produce calibration curves linking the intrinsic magnetic field values with the widths of blue peaks of Stokes V profiles. The obtained calibration curves are then tested using the Stokes profiles calculated for more realistic photospheric models based on MHD models of magneto-convection. It is shown that the developed Stokes V widths (SVW) method can be used with various optical and near-infrared lines. Out of six lines considered in this study, FeI 6301 line appears to be the most effective: it is sensitive to fields over ~200G and does not show any saturation up to ~2kG. Other lines considered can also be used for the photospheric field diagnostics with this method, however, only in narrower field value ranges, typically from about 100G to 700-1000G. The developed method can be a useful alternative to the classical magnetic line ratio method, particularly when the choice of lines is limited., Accepted for publication in Astronomy and Astrophysics

The observed properties (i.e., source size, source position, time duration, decay time) of solar radio emission produced through plasma processes near the local plasma frequency, and hence the interpretation of solar radio bursts, are strongly influenced by propagation effects in the inhomogeneous turbulent solar corona. In this work, a 3D stochastic description of the propagation process is presented, based on the Fokker-Planck and Langevin equations of radio-wave transport in a medium containing anisotropic electron density fluctuations. Using a numerical treatment based on this model, we investigate the characteristic source sizes and burst decay times for Type III solar radio bursts. Comparison of the simulations with the observations of solar radio bursts shows that predominantly perpendicular density fluctuations in the solar corona are required, with an anisotropy factor $\sim 0.3$ for sources observed at around 30~MHz. The simulations also demonstrate that the photons are isotropized near the region of primary emission, but the waves are then focused by large-scale refraction, leading to plasma radio emission directivity that is characterized by a half-width-half-maximum of about 40~degrees near 30~MHz. The results are applicable to various solar radio bursts produced via plasma emission., 21 pages, 11 figures, accepted for publication in The Astrophysical Journal

Low-frequency radio observations make it possible to study the solar corona at distances up to 2–3 R ☉. Frequency of plasma emission is a proxy for electron density of the emitting plasma and, therefore, observations of solar radio bursts can be used to probe the density structure of the outer corona. In this study, positions of solar radio sources are investigated using the Low-Frequency Array (LOFAR) spectral imaging in the frequency range 30–50 MHz. We show that there are events where apparent positions of the radio sources cannot be explained using the standard coronal density models. Namely, the apparent heliocentric positions of the sources are 0.1–0.7 R ☉ further from the Sun compared with the positions predicted by the Newkirk model, and these shifts are frequency-dependent. We discuss several possible explanations for this effect, including enhanced plasma density in the flaring corona, as well as scattering and refraction of the radio waves.

The aim of this study is to generate maps of the hard X-ray emission produced by energetic electrons in a solar flare and compare them with observations. The ultimate goal is to test the viability of the combined MHD/test-particle approach for data-driven modelling of active events in the solar corona and their impact on the heliosphere. Based on an MHD model of X-class solar flare observed on the 8th of September 2017, we calculate trajectories of a large number of electrons and protons using the relativistic guiding-centre approach. Using the obtained particle trajectories, we deduce the spatial and energy distributions of energetic electrons and protons, and calculate bremsstrahlung hard X-ray emission using the 'thin target' approximation. Our approach predicts some key characteristics of energetic particles in the considered flare, including the size and location of the acceleration region, energetic particle trajectories and energy spectra. Most importantly, the hard X-ray bremsstrahlung intensity maps predicted by the model are in a good agreement with those observed by RHESSI. Furthermore, the locations of proton and electron precipitation appear to be close to the sources of helioseismic response detected in this flare. Therefore, the adopted approach can be used for observationally-driven modelling of individual solar flares, including manifestations of energetic particles in the corona, as well as inner heliosphere., Comment: Accepted for publication in the Astrophysical Journal (on 19 Aug 2020)

Funding: Carnegie Trust for the Universities of Scotland; Science and Technology Facilities Council (grants ST/N000609/1 and ST/P000428/1). Previous work has confirmed the concept of a magnetohydrodynamic (MHD) avalanche in pre-stressed threads within a coronal loop. We undertook a series of full, three-dimensional MHD simulations in order to create three threads by twisting the magnetic field through boundary motions until an instability ensues. We find that, following the original instability, one unstable thread can disrupt its neighbours with continued driving. A ‘bursty’ heating profile results, with a series of ongoing energy releases, but no evident steady state. For the first time using full MHD, we show that avalanches are a viable mechanism for the storing and release of magnetic energy in the solar corona, as a result of photospheric motions. Publisher PDF

The aim of this work is to investigate and characterise non-thermal particle behaviour in a three-dimensional (3D) magnetohydrodynamical (MHD) model of unstable multi-threaded flaring coronal loops. We have used a numerical scheme which solves the relativistic guiding centre approximation to study the motion of electrons and protons. The scheme uses snapshots from high resolution numerical MHD simulations of coronal loops containing two threads, where a single thread becomes unstable and (in one case) destabilises and merges with an additional thread. The particle responses to the reconnection and fragmentation in MHD simulations of two loop threads are examined in detail. We illustrate the role played by uniform background resistivity and distinguish this from the role of anomalous resistivity using orbits in an MHD simulation where only one thread becomes unstable without destabilising further loop threads. We examine the (scalable) orbit energy gains and final positions recovered at different stages of a second MHD simulation wherein a secondary loop thread is destabilised by (and merges with) the first thread. We compare these results with other theoretical particle acceleration models in the context of observed energetic particle populations during solar flares., 16 pages, 13 figures

Four different approximate approaches used to model the stressing of coronal magnetic fields due to an imposed photospheric motion are compared with each other and the results from a full time-dependent magnetohydrodynamic (MHD) code. The assumptions used for each of the approximate methods are tested by considering large photospheric footpoint displacements. We consider a simple model problem, comparing the full nonlinear magnetohydrodynamic evolution, determined with the Lare2D numerical code, with four approximate approaches. Two of these, magneto-frictional relaxation and a quasi-1D Grad-Shafranov approach, assume sequences of equilibria, whilst the other two methods, a second-order linearisation of the MHD equations and Reduced MHD, are time-dependent. The relaxation method is very accurate compared to full MHD for force-free equilibria for all footpoint displacements but has significant errors when the plasma $\beta_0$ is of order unity. The 1D approach gives an extremely accurate description of the equilibria away from the photospheric boundary layers, and agrees well with Lare2D for all parameter values tested. The linearised MHD equations correctly predict the existence of photospheric boundary layers that are present in the full MHD results. As soon as the footpoint displacement becomes a significant fraction of the loop length, the RMHD method fails to model the sequences of equilibria correctly. The full numerical solution is interesting in its own right, and care must be taken for low $\beta_0$ plasmas if the viscosity is too large., Comment: 15 pages, 9 figures

Solar flares are explosive events in the solar corona, representing fast conversion of magnetic energy into thermal and kinetic energy, and hence radiation, due to magnetic reconnection. Modelling is essential for understanding and predicting these events. However, self-consistent modelling is extremely difficult due to the vast spatial and temporal scale separation between processes involving thermal plasma (normally considered using magnetohydrodynamic (MHD) approach) and non-thermal plasma (requiring a kinetic approach). In this mini-review we consider different approaches aimed at bridging the gap between fluid and kinetic modelling of solar flares. Two types of approaches are discussed: combined MHD/test-particle (MHDTP) models, which can be used for modelling the flaring corona with relatively small numbers of energetic particles, and hybrid fluid-kinetic methods, which can be used for modelling stronger events with higher numbers of energetic particles. Two specific examples are discussed in more detail: MHDTP models of magnetic reconnection and particle acceleration in kink-unstable twisted coronal loops, and a novel reduced-kinetic model of particle transport., Comment: Accepted as a mini-review in Advances in Space Research

Early magnetographic observations indicated that magnetic field in the solar photosphere has unresolved small-scale structure. Near-infrared and optical data with extremely high spatial resolution show that these structures have scales of few tens of kilometres, which are not resolved in the majority of solar observations. The goal of this study is to establish the effect of unresolved photospheric magnetic field structure on Stokes profiles observed with relatively low spatial resolution. Ultimately, we aim to develop methods for fast estimation of the photospheric magnetic filling factor and line-of-sight gradient of the photospheric magnetic field, which can be applied to large observational data sets. We exploit 3D MHD models of magneto-convection developed using MURAM code. Corresponding profiles of Fe I 6301.5 and 6302.5 $\mathrm{\AA}$ spectral lines are calculated using NICOLE radiative transfer code. The resulting I and V Stokes [x,y,$\lambda$] cubes with reduced spatial resolution of 150 km are used to calculate magnetic field values as they would be obtained in observations with Hinode/SOT or SDO/HMI. Three different methods of the magnetic filling factor estimation are considered: the magnetic line ratio method, Stokes V width method and a simple statistical method. We find that the statistical method and the Stokes V width method are sufficiently reliable for fast filling factor estimations. Furthermore, we find that Stokes $I\pm V$ bisector splitting gradient can be used for fast estimation of line-of-sight gradient of the photospheric magnetic field., Comment: Submitted to Astronomy & Astrophysics on 14/05/2018

Magnetic reconnection is believed to be the primary mechanism by which non-potential energy stored in coronal magnetic fields is rapidly released during solar eruptive events. Unfortunately, owing to the small spatial scales on which reconnection is thought to occur, it is not directly observable in the solar corona. However, larger scale processes, such as associated inflow and outflow, and signatures of accelerated particles have been put forward as evidence of reconnection. We explore the origin of a persistent Type I radio source that accompanies a coronal X--shaped structure during its passage across the disk. Of particular interest is the time range around a partial collapse of the structure that is associated with inflow, outflow, and signatures of particle acceleration. Imaging radio observations from the NRH were used to localise the radio source. SDO AIA EUV observations from the same time period were analysed, looking for evidence of inflows and outflows. Further \texttt{mpole} magnetic reconstructions using SDO HMI observations allowed the magnetic connectivity associated with the radio source to be determined. The Type I radio source was well aligned with a magnetic separator identified in the extrapolations. During the partial collapse, gradual (1 km/s) and fast (5 km/s) inflow phases and fast (30 km/s) and rapid (80-100 km/s) outflow phases were observed, resulting in an estimated reconnection rate of ~ 0.06. The radio source brightening and dimming was found to be co-temporal with increased soft x-ray emission in both RHESSI and GOES. We interpret the brightening and dimming of the radio emission as evidence for accelerated electrons in the reconnection region responding to a gradual fall and rapid rise in electric drift velocity, in response to the inflowing and outflowing field lines. These results present a comprehensive example of 3D null-point reconnection., Comment: 7 pages, 4 figure

The mega amp spherical tokamak (MAST) was a low aspect ratio device (R/a = 0.85/0.65 ~ 1.3) with similar poloidal cross-section to other medium-size tokamaks. The physics programme concentrates on addressing key physics issues for the operation of ITER, design of DEMO and future spherical tokamaks by utilising high resolution diagnostic measurements closely coupled with theory and modelling to significantly advance our understanding. An empirical scaling of the energy confinement time that favours higher power, lower collisionality devices is consistent with gyrokinetic modelling of electron scale turbulence. Measurements of ion scale turbulence with beam emission spectroscopy and gyrokinetic modelling in up-down symmetric plasmas find that the symmetry of the turbulence is broken by flow shear. Near the non-linear stability threshold, flow shear tilts the density fluctuation correlation function and skews the fluctuation amplitude distribution. Results from fast particle physics studies include the observation that sawteeth are found to redistribute passing and trapped fast particles injected from neutral beam injectors in equal measure, suggesting that resonances between the m = 1 perturbation and the fast ion orbits may be playing a dominant role in the fast ion transport. Measured D–D fusion products from a neutron camera and a charged fusion product detector are 40% lower than predictions from TRANSP/NUBEAM, highlighting possible deficiencies in the guiding centre approximation. Modelling of fast ion losses in the presence of resonant magnetic perturbations (RMPs) can reproduce trends observed in experiments when the plasma response and charge-exchange losses are accounted for. Measurements with a neutral particle analyser during merging-compression start-up indicate the acceleration of ions and electrons. Transport at the plasma edge has been improved through reciprocating probe measurements that have characterised a geodesic acoustic mode at the edge of an ohmic L-mode plasma and particle-in-cell modelling has improved the interpretation of plasma potential estimates from ball-pen probes. The application of RMPs leads to a reduction in particle confinement in L-mode and H-mode and an increase in the core ionization source. The ejection of secondary filaments following type-I ELMs correlates with interactions with surfaces near the X-point. Simulations of the interaction between pairs of filaments in the scrape-off layer suggest this results in modest changes to their velocity, and in most cases can be treated as moving independently. A stochastic model of scrape-off layer profile formation based on the superposition of non-interacting filaments is in good agreement with measured time-average profiles. Transport in the divertor has been improved through fast camera imaging, indicating the presence of a quiescent region devoid of filament near the X-point, extending from the separatrix to ψ n ~ 1.02. Simulations of turbulent transport in the divertor show that the angle between the divertor leg on the curvature vector strongly influences transport into the private flux region via the interchange mechanism. Coherence imaging measurements show counter-streaming flows of impurities due to gas puffing increasing the pressure on field lines where the gas is ionised. MAST Upgrade is based on the original MAST device, with substantially improved capabilities to operate with a Super-X divertor to test extended divertor leg concepts. SOLPS-ITER modelling predicts the detachment threshold will be reduced by more than a factor of 2, in terms of upstream density, in the Super-X compared with a conventional configuration and that the radiation front movement is passively stabilised before it reaches the X-point. 1D fluid modelling reveals the key role of momentum and power loss mechanisms in governing detachment onset and evolution. Analytic modelling indicates that long legs placed at large major radius, or equivalently low at the target compared with the X-point are more amenable to external control. With MAST Upgrade experiments expected in 2019, a thorough characterisation of the sources of the intrinsic error field has been carried out and a mitigation strategy developed.

Context. Forced magnetic reconnection, a reconnection event triggered by external perturbation, should be ubiquitous in the solar corona. Energy released during such cases can be much greater than that which was introduced by the perturbation. The exact dynamics of magnetic reconnection events are determined by the structure and complexity of the reconnection region: the thickness of reconnecting layers, the field curvature; the presence, shapes and sizes of magnetic islands. It is unclear how the properties of the external perturbation and the initial current sheet affect the reconnection region properties, and thereby the reconnection dynamics and energy release profile.Aims. We investigate the effect of the form of the external perturbation and initial current sheet on the evolution of the reconnection region and the energy release process. Chiefly we explore the non-linear interactions between multiple, simultaneous perturbations, which represent more realistic scenarios. Future work will use these results in test particle simulations to investigate particle acceleration over multiple reconnection events.Methods. Simulations are performed using Lare2d, a 2.5D Lagrangian-remap solver for the visco-resistive MHD equations. The model of forced reconnection is extended to include superpositions of sinusoidal driving disturbances, including localised Gaussian perturbations. A transient perturbation is applied to the boundaries of a region containing a force-free current sheet. The simulation domain is sufficiently wide to allow multiple magnetic islands to form and coalesce.Results. Island coalescence contributes significantly to energy release and involves rapid reconnection. Long wavelength modes in perturbations dominate the evolution, without the presence of which reconnection is either slow, as in the case of short wavelength modes, or the initial current sheet remains stable, as in the case of noise perturbations. Multiple perturbations combine in a highly non-linear manner: reconnection is typically faster than when either disturbance is applied individually, with multiple low-energy events contributing to the same total energy release.

Astrophysical fluids have very large Reynolds numbers and therefore turbulence is their natural state. Magnetic reconnection is an important process in many astrophysical plasmas, which allows restructuring of magnetic fields and conversion of stored magnetic energy into heat and kinetic energy. Turbulence is known to dramatically change different transport processes and therefore it is not unexpected that turbulence can alter the dynamics of magnetic field lines within the reconnection process. We shall review the interaction between turbulence and reconnection at different scales, showing how a state of turbulent reconnection is natural in astrophysical plasmas, with implications for a range of phenomena across astrophysics. We consider the process of magnetic reconnection that is fast in magnetohydrodynamic (MHD) limit and discuss how turbulence—both externally driven and generated in the reconnecting system—can make reconnection independent on the microphysical properties of plasmas. We will also show how relaxation theory can be used to calculate the energy dissipated in turbulent reconnecting fields. As well as heating the plasma, the energy dissipated by turbulent reconnection may cause acceleration of non-thermal particles, which is briefly discussed here.

In this article, we discuss the idea of a hierarchy of instabilities that can rapidly couple the disparate scales of a turbulent plasma system. First, at the largest scale of the system, L, current carrying flux ropes can undergo a kink instability. Second, a kink instability in adjacent flux ropes can rapidly bring together bundles of magnetic flux and drive reconnection, introducing a new scale of the current sheet width, l, perhaps several ion inertial lengths (δ i ) across. Finally, intense current sheets driven by reconnection electric fields can destabilize kinetic waves such as ion cyclotron waves as long as the drift speed of the electrons is large compared to the ion thermal speed, v D ≫v i . Instabilities such as these can couple MHD scales to kinetic scales, as small as the proton Larmor radius, ρ i .

New results from MAST are presented that focus on validating models in order to extrapolate to future devices. Measurements during start-up experiments have shown how the bulk ion temperature rise scales with the square of the reconnecting field. During the current ramp up models are not able to correctly predict the current diffusion. Experiments have been performed looking at edge and core turbulence. At the edge detailed studies have revealed how filament characteristic are responsible for determining the near and far SOL density profiles. In the core the intrinsic rotation and electron scale turbulence have been measured. The role that the fast ion gradient has on redistributing fast ions through fishbone modes has led to a redesign of the neutral beam injector on MAST Upgrade. In H-mode the turbulence at the pedestal top has been shown to be consistent with being due to electron temperature gradient modes. A reconnection process appears to occur during ELMs and the number of filaments released determines the power profile at the divertor. Resonant magnetic perturbations can mitigate ELMs provided the edge peeling response is maximised and the core kink response minimised. The mitigation of intrinsic error fields with toroidal mode number n>1 has been shown to be important for plasma performance., 34 pages, 10 figures. This is an author-created, un-copyedited version of an article submitted for publication in Nuclear Fusion. IoP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it

We acknowledge the financial support of STFC through the Consolidated grants to the University of St Andrews and the University of Manchester. For the first time, we demonstrate how an MHD avalanche might occur in a multithreaded coronal loop. Considering 23 non-potential magnetic threads within a loop, we use 3D MHD simulations to show that only one thread needs to be unstable in order to start an avalanche even when the others are below marginal stability. This has significant implications for coronal heating in that it provides for energy dissipation with a trigger mechanism. The instability of the unstable thread follows the evolution determined in many earlier investigations. However, once one stable thread is disrupted, it coalesces with a neighboring thread and this process disrupts other nearby threads. Coalescence with these disrupted threads then occurs leading to the disruption of yet more threads as the avalanche develops. Magnetic energy is released in discrete bursts as the surrounding stable threads are disrupted. The volume integrated heating, as a function of time, shows short spikes suggesting that the temporal form of the heating is more like that of nanoflares than of constant heating. Publisher PDF

In the present work we aim to study particle acceleration in twisted coronal loops. For this purpose, an MHD model of magnetic reconnection in a linearly unstable twisted magnetic fluxtube is considered. Further, the electric and magnetic fields obtained in the MHD simulations are used to calculate proton and electron trajectories in the guiding-centre approximation. It is shown that particle acceleration in such a model is distributed rather uniformly along the coronal loop and the high-energy population remains generally neutral. It also follows from the model that the horizontal cross-section of the volume occupied by high-energy particles near the loop footpoints increases with time, which can be used as an observational proxy.

Energetic particles are critical components of plasma populations found throughout the universe. In many cases particles are accelerated to relativistic energies and represent a substantial fraction of the total energy of the system, thus requiring extremely efficient acceleration processes. The production of accelerated particles also appears coupled to magnetic field evolution in astrophysical plasmas through the turbulent magnetic fields produced by diffusive shock acceleration. Particle acceleration is thus a key component in helping to understand the origin and evolution of magnetic structures in, e.g. galaxies. The proximity of the Sun and the range of high-resolution diagnostics available within the solar atmosphere offers unique opportunities to study the processes involved in particle acceleration through the use of a combination of remote sensing observations of the radiative signatures of accelerated particles, and of their plasma and magnetic environment. The SPARK concept targets the broad range of energy, spatial and temporal scales over which particle acceleration occurs in the solar atmosphere, in order to determine how and where energetic particles are accelerated. SPARK combines highly complementary imaging and spectroscopic observations of radiation from energetic electrons, protons and ions set in their plasma and magnetic context. The payload comprises focusing-optics X-ray imaging covering the range from 1 to 60 keV; indirect HXR imaging and spectroscopy from 5 to 200 keV, γ-ray spectroscopic imaging with high-resolution LaBr3 scintillators, and photometry and source localisation at far-infrared wavelengths. The plasma environment of the regions of acceleration and interaction will be probed using soft X-ray imaging of the corona and vector magnetography of the photosphere and chromosphere. SPARK is designed for solar research. However, in addition it will be able to provide exciting new insights into the origin of particle acceleration in other regimes, including terrestrial gamma-ray flashes (TGF), the origin of γ-ray bursts, and the possible existence of axions.

It has been proposed that the million degree temperature of the corona is due to the combined effect of barely-detectable energy releases, so called nanoflares, that occur throughout the solar atmosphere. Alas, the nanoflare density and brightness implied by this hypothesis means that conclusive verification is beyond present observational abilities. Nevertheless, we investigate the plausibility of the nanoflare hypothesis by constructing a magnetohydrodynamic (MHD) model that can derive the energy of a nanoflare from the nature of an ideal kink instability. The set of energy-releasing instabilities is captured by an instability threshold for linear kink modes. Each point on the threshold is associated with a unique energy release and so we can predict a distribution of nanoflare energies. When the linear instability threshold is crossed, the instability enters a nonlinear phase as it is driven by current sheet reconnection. As the ensuing flare erupts and declines, the field transitions to a lower energy state, which is modelled by relaxation theory, i.e., helicity is conserved and the ratio of current to field becomes invariant within the loop. We apply the model so that all the loops within an ensemble achieve instability followed by energy-releasing relaxation. The result is a nanoflare energy distribution. Furthermore, we produce different distributions by varying the loop aspect ratio, the nature of the path to instability taken by each loop and also the level of radial expansion that may accompany loop relaxation. The heating rate obtained is just sufficient for coronal heating. In addition, we also show that kink instability cannot be associated with a critical magnetic twist value for every point along the instability threshold.

We review basic theoretical concepts in particle acceleration, with particular emphasis on processes likely to occur in regions of magnetic reconnection. Several new developments are discussed, including detailed studies of reconnection in three-dimensional magnetic field configurations (e.g., current sheets, collapsing traps, separatrix regions) and stochastic acceleration in a turbulent environment. Fluid, test-particle, and particle-in-cell approaches are used and results compared. While these studies show considerable promise in accounting for the various observational manifestations of solar flares, they are limited by a number of factors, mostly relating to available computational power. Not the least of these issues is the need to explicitly incorporate the electrodynamic feedback of the accelerated particles themselves on the environment in which they are accelerated. A brief prognosis for future advancement is offered., This is a chapter in a monograph on the physics of solar flares, inspired by RHESSI observations. The individual articles are to appear in Space Science Reviews (2011)

Proton and electron acceleration in a fragmenting periodic current sheet (CS) is investigated, based on the forced magnetic reconnection scenario. The aim is to understand the role of CS fragmentation in high-energy beam generation in solar flares. We combine magnetohydrodynamics and test-particle models to consider particle trajectories consistent with a time-dependent reconnection model. It is shown that accelerated particles in such a model form two distinct populations. Protons and electrons moving in open magnetic field have energy spectra that are a combination of the initial Maxwellian distribution and a power-law high-energy (E > 20 keV) part. The second population contains particles moving in a closed magnetic field around O-points. These particles move predominantly along the guiding field and their energies fall within quite a narrow range between similar to 1 MeV and similar to 10 MeV. It is also found that particles moving in an open magnetic field have a considerably wider pitch-angle distribution.

A 1-D fluid code simulation of the region above the racetrack in a magnetron discharge has been developed to yield predictions for the temporal evolution of the axial plasma parameters during pulsed DC operation. To take into account anomalous cross-field diffusion transport rates, an effective magnetic field profile has been calculated based on the real experimental values, to yield ratios in the effective electron gyro and collision frequencies comparable with Bohm diffusion. Pulse frequencies from 100 to 500 kHz are considered, with gas pressures ranging from 0.25 to 0.65 Pa. The model results for the plasma potential, electron and ion densities, during the pulse cycle agree well with experimental measurements made in the Liverpool magnetron using time-resolved Langmuir and emissive probes. In the bulk plasma, the predictions for electron temperature are close to the measured values; however, in the sheath region they are artificially high due to the model assumption that only Maxwellian distributed electrons exist in all regions of the discharge. This inadequacy in the model and the methods to improve has been discussed. Results for DC magnetrons are also shown.

Context. The heating of solar coronal plasma to millions of degrees is likely to be due to the superposition of many small energy-releasing events, known as nanoflares. Nanoflares dissipate magnetic energy through magnetic reconnection. Aims. A model has been recently proposed in which nanoflare-like heating naturally arises, with a sequence of dissipation events of various magnitudes. It is proposed that heating is triggered by the onset of ideal instability, with energy release occurring in the nonlinear phase due to fast magnetic reconnection. The aim is to use numerical simulations to investigate this heating process. Methods. Three-dimensional magnetohydrodynamic numerical simulations of energy release are presented for a cylindrical coronal loop model. Initial equilibrium magnetic-field profiles are chosen to be linearly unstable, with a two-layer parameterisation of the current profile. The results are compared with calculations of linear instability, with line-tying, which are extended to account for a potential field layer surrounding the loop. The energy release is also compared with predictions that the field relaxes to a state of minimum magnetic energy with conserved magnetic helicity (a constant a force-free field). Results. The loop initially develops a helical kink, whose structure and growth rate are generally in accordance with linear stability theory, and subsequently a current sheet forms. During this phase, there is a burst of kinetic energy while the magnetic energy decays. A new relaxed equilibrium is established that corresponds quite closely to a constant a field. The fraction of stored magnetic energy released depends strongly on the initial current profile, which agrees with the predictions of relaxation theory. Conclusions. Energy dissipation events in a coronal loop are triggered by the onset of ideal kink instability. Magnetic energy is dissipated, leading to large or small heating events according to the initial current profile.

A linked set of Monte Carlo applications has been developed in order to investigate the sputtering, deposition, and ionization processes in a circular direct current unbalanced magnetron discharge. Particles respond to prescribed electric and magnetic fields, the former taken from experimental measurements, and self-consistent plasma behavior resulting in changes in the fields is not accounted for. The motion of energetic electrons emitted from the target surface by ion impacts is followed in the gas phase in order to characterize ionization and excitation collisions and elastic scattering with argon filling gas. The inhomogeneous erosion track profile is computed and compared with experimental measurements. The transport of titanium sputtered neutrals between the target and substrate surfaces is then analyzed using both a rigid sphere collision model and an interatomic potential model to describe collisions between sputtered neutrals and background gas atoms. The radial emission distribution of sputtered atoms is taken from the electron transport model. The mean arrival energy and the angular distribution of titanium neutrals impinging on the substrate surface, and the metal density profile between target and substrate are calculated. Finally, the electron impact ionization of titanium neutrals in a plasma formed by a mixture of titanium (10% of argon density) and argon atoms is simulated, motivated by the promising possibility of controlling the deposition process by influencing the direction of the ion flux.

In the present study we investigate magnetic reconnection in twisted magnetic fluxtubes with different initial configurations. In all considered cases, energy release is triggered by the ideal kink instability, which is itself the result of applying footpoint rotation to an initially potential field. The main goal of this work is to establish the influence of the field topology and various thermodynamic effects on the energy release process. Specifically, we investigate convergence of the magnetic field at the loop footpoints, atmospheric stratification, as well as thermal conduction. In all cases, the application of vortical driving at the footpoints of an initally potential field leads to an internal kink instability. With the exception of the curved loop with high footpoint convergence, the global geometry of the loop change little during the simulation. Footpoint convergence, curvature and atmospheric structure clearly influences the rapidity with which a loop achieves instability as well as the size of the subsequent energy release. Footpoint convergence has a stabilising influence and thus the loop requires more energy for instability, which means that the subsequent relaxation has a larger heating effect. Large-scale curvature has the opposite result: less energy is needed for instability and so the amount of energy released from the field is reduced. Introducing a stratified atmosphere gives rise to decaying wave phenomena during the driving phase, and also results in a loop that is less stable., Submitted to Solar Physics

We study a model of particle acceleration coupled with an MHD model of magnetic reconnection in unstable twisted coronal loops. The kink instability leads to the formation of helical currents with strong parallel electric fields resulting in electron acceleration. The motion of electrons in the electric and magnetic fields of the reconnecting loop is investigated using a test-particle approach taking into account collisional scattering. We discuss the effects of Coulomb collisions and magnetic convergence near loop footpoints on the spatial distribution and energy spectra of high-energy electron populations and possible implications on the hard X-ray emission in solar flares.

The solar corona, the tenuous outer atmosphere of the Sun, is orders of magnitude hotter than the solar surface. This 'coronal heating problem' requires the identification of a heat source to balance losses due to thermal conduction, radiation and (in some locations) convection. The review papers in this Theo Murphy meeting issue present an overview of recent observational findings, large- and small-scale numerical modelling of physical processes occurring in the solar atmosphere and other aspects which may affect our understanding of the proposed heating mechanisms. At the same time, they also set out the directions and challenges which must be tackled by future research. In this brief introduction, we summarize some of the issues and themes which reoccur throughout this issue. Postprint

We acknowledge the financial support of STFC through the Consolidated grant to the University of St Andrews. Context. Heating the solar corona to several million degrees requires the conversion of magnetic energy into thermal energy. In this paper, we investigate whether an unstable magnetic thread within a coronal loop can destabilise a neighbouring magnetic thread. Aims. By running a series of simulations, we aim to understand under what conditions the destabilisation of a single magnetic thread can also trigger a release of energy in a nearby thread. Methods. The 3D magnetohydrodynamics code, Lare3d, is used to simulate the temporal evolution of coronal magnetic fields during a kink instability and the subsequent relaxation process. We assume that a coronal magnetic loop consists of non-potential magnetic threads that are initially in an equilibrium state. Results. The non-linear kink instability in one magnetic thread forms a helical current sheet and initiates magnetic reconnection. The current sheet fragments, and magnetic energy is released throughout that thread. We find that, under certain conditions, this event can destabilise a nearby thread, which is a necessary requirement for starting an avalanche of energy release in magnetic threads. Conclusions. It is possible to initiate an energy release in a nearby, non-potential magnetic thread, because the energy released from one unstable magnetic thread can trigger energy release in nearby threads, provided that the nearby structures are close to marginal stability. Publisher PDF

Twisted coronal loops should be ubiquitous in the solar corona. Twisted magnetic fields contain excess magnetic energy, which can be released during magnetic reconnection, causing solar flares. The aim of this work is to investigate magnetic reconnection, and particle acceleration and transport in kink-unstable twisted coronal loops, with a focus on the effects of resistivity, loop geometry and atmospheric stratification. Another aim is to perform forward-modelling of bremsstrahlung emission and determine the structure of hard X-ray sources. We use a combination of magnetohydrodynamic (MHD) and test-particle methods. First, the evolution of the kinking coronal loop is considered using resistive MHD model, incorporating atmospheric stratification and loop curvature. Then, the obtained electric and magnetic fields and density distributions are used to calculate electron and proton trajectories using a guiding-centre approximation, taking into account Coulomb collisions. It is shown that electric fields in twisted coronal loops can effectively accelerate protons and electrons to energies up to 10 MeV. High-energy particles have hard, nearly power-law energy spectra. The volume occupied by high-energy particles demonstrates radial expansion, which results in the expansion of the visible hard X-ray loop and a gradual increase in hard X-ray footpoint area. Synthesised hard X-ray emission reveals strong footpoint sources and the extended coronal source, whose intensity strongly depends on the coronal loop density.

Twisted magnetic flux ropes are ubiquitous in laboratory and astrophysical plasmas, and the merging of such flux ropes through magnetic reconnection is an important mechanism for restructuring magnetic fields and releasing free magnetic energy. The merging-compression scenario is one possible start-up scheme for spherical tokamaks, which has been used on the Mega Amp Spherical Tokamak (MAST). Two current-carrying plasma rings or flux ropes approach each due to mutual attraction, forming a current sheet and subsequently merge through magnetic reconnection into a single plasma torus, with substantial plasma heating. Two-dimensional resistive and Hall-magnetohydrodynamic simulations of this process are reported, including a strong guide field. A model of the merging based on helicity-conserving relaxation to a minimum energy state is also presented, extending previous work to tight-aspect-ratio toroidal geometry. This model leads to a prediction of the final state of the merging, in good agreement with simulations and experiment, as well as the average temperature rise. A relaxation model of reconnection between two or more flux ropes in the solar corona is also described, allowing for different senses of twist, and the implications for heating of the solar corona are discussed.

The fields of solar radiophysics and solar system radio physics, or radio heliophysics, will benefit immensely from an instrument with the capabilities projected for SKA. Potential applications include interplanetary scintillation (IPS), radio-burst tracking, and solar spectral radio imaging with a superior sensitivity. These will provide breakthrough new insights and results in topics of fundamental importance, such as the physics of impulsive energy releases, magnetohydrodynamic oscillations and turbulence, the dynamics of post-eruptive processes, energetic particle acceleration, the structure of the solar wind and the development and evolution of solar wind transients at distances up to and beyond the orbit of the Earth. The combination of the high spectral, time and spatial resolution and the unprecedented sensitivity of the SKA will radically advance our understanding of basic physical processes operating in solar and heliospheric plasmas and provide a solid foundation for the forecasting of space weather events., Comment: 15 pages, Proceedings of Advancing Astrophysics with the Square Kilometre Array (AASKA14). 9 -13 June, 2014. Giardini Naxos, Italy. Online at http://pos.sissa.it/cgi-bin/reader/conf.cgi?confid=215, id.169

We study test panicle trajectories in the vicinity of a three-dimensional (3D) magnetic null point during spine reconnection. Particles are injected into the steady-state non-uniform magnetic and electric fields derived by Priest & Titov (1996), and the equations of motion numerically integrated. We use input parameters typical of the solar corona, for which reconnection has been suggested as the fundamental mechanism responsible for particle acceleration in flare events. We show that substantial acceleration is possible in the 3D spine reconnection configuration, in the strong electric field regime. The energy gain is strongly dependent on the location of injection into the simulation box, as was the case in 2D X-point configurations. In our 3D geometry, we first vary the location of injection within a plane through the spine, and derive an analytical value for the injection angle for which maximum energy gain is achieved. Secondly we vary the azimuthal location of particle injection and show that as one moves away from the plane with maximum electric field magnitude, higher final energies can be achieved, though this requires substantially longer times.

A coronal heating model is proposed which predicts heating by a series of discrete events of various energies, analogous to the observed range of events from large scale flares through various transient brightening phenomena down to the often discussed "nanoflares". We suggest that an energy release event occurs when a field becomes linearly unstable to ideal MHD modes, with dissipation during the nonlinear phase of such an instability due to reconnection in fine-scale structures such as current sheets. The energy release during this complex dynamic period can be evaluated by assuming the field relaxes to a minimum energy state subject to the constraint of helicity conservation. A model problem is studied: a cylindrical coronal loop, with a current profile generated by slow twisting of the photospheric footpoints parameterised by two values of α (the ratio of current density to field strength). Different initial α profiles, corresponding to different footpoint twisting profiles, lead to energy release events of a wide range of magnitudes, but our model predicts an observationally realistic minimum size for these events.

Models of the ideal magnetohydrodynamic (MHD) stability of spheromaks and spherical tokamaks are presented, including the effects of current on the open flux which plays a key role in helicity-injected current drive. The stability of spheromak equilibria with both open and closed flux and realistic current profiles representative of helicity-injected state is investigated, where a kink instability in the open flux is shown to dominate a tilt mode in the closed flux as the open flux current density is increased. A previous one-dimensional model is extended to more realistic two-dimensional equilibria which properly incorporate a region of closed magnetic flux as well as open flux penetrating the boundaries at electrodes. A new stability code SCOTS has been developed and benchmarked which can determine the growth rates of ideal MHD modes in this geometry. The coordinate system for this code has been developed such that it extends smoothly across the separatrix between closed and open flux, thus not imposing...

Magnetic reconnection and particle acceleration due to the kink instability in twisted coronal loops can be a viable scenario for confined solar flares. Detailed investigation of this phenomenon requires reliable methods for observational detection of magnetic twist in solar flares, which may not be possible solely through extreme UV and soft X-ray thermal emission. Polarisation of microwave emission in flaring loops can be used as one of the detection criteria. The aim of this study is to investigate the effect of magnetic twist in flaring coronal loops on the polarisation of gyro-synchrotron microwave (GSMW) emission, and determine whether it could provide a means for magnetic twist detection. We use time-dependent magnetohydrodynamic and test-particle models developed using LARE3D and GCA codes to investigate twisted coronal loops relaxing following the kink-instability. Synthetic GSMW emission maps (I and V Stokes components) are calculated using GX simulator. It is found that flaring twisted coronal loops produce GSMW radiation with a gradient of circular polarisation across the loop. However, these patterns may be visible only for a relatively short period of time due to fast magnetic reconfiguration after the instability. Their visibility also depends on the orientation and position of the loop on solar disk. Typically, it would be difficult to see these characteristic polarisation pattern in a twisted loop seen from the top (close to the centre of the solar disk), but easier in a twisted loop seen from the side (i.e. observed very close to the limb)., Accepted for publication in Astronomy & Astrophysics

Merging-compression start-up in the Mega Ampere Spherical Tokamak provides an opportunity to investigate the merging of flux ropes through magnetic reconnection, and the self-organization into a single flux rope, in a low-plasma-β, high-Lundquist-number plasma. We present an overview of simulations of this process using the compressible Hall-MHD equations in two dimensions. Preliminary results of an analytical model of the self-organization, assuming a helicity-conserving relaxation to a minimum energy state, are also presented. The relevance of these models to solar plasmas is discussed.

Two-dimensional numerical simulations are used to investigate nonlinear aspects of forced magnetic reconnection in a zero β highly conducting plasma. This is representative of the solar corona, where reconnection may be induced by external perturbations, e.g., at the photospheric boundary of the corona. The aim is to investigate the energy dissipation by the reconnection, which may provide a mechanism for heating the coronal plasma. The field is taken to be initially a sheared force-free equilibrium in a slab, and the effects of applying a slow deformation to the boundaries are investigated. Previous analytical studies assuming small departures from the initial equilibrium have found that a current sheet forms during an initial ideal phase of evolution, which subsequently relaxes to a reconnected equilibrium, releasing some magnetic energy. The linear theory predicts that the energy release has a singularity when the field is marginally stable to the tearing mode. The nonlinear evolution of the field is c...

We report on further studies of current drive in the SPHEX spheromak device when operated as a spherical tokamak. Previous work has demonstrated that the current amplification factor, defined as the ratio of the toroidal plasma current to the driving gun current, increased linearly with the applied toroidal field. Results are presented from a more detailed analysis of this scaling for a fixed gun current of 60 kA, utilizing direct internal measurements of current density and equilibrium modelling of internal magnetic fields. The data indicate that the increase in the current amplification factor is largely the result of changes in the wavelength of the helical distortion of the open flux (identified with the global n = 1 mode) connected to the helicity injector: the increase in toroidal current on closed flux surfaces is much more modest. We discuss changes in current profiles at differing applied toroidal fields and demonstrate the operating regime over which helicity-injected current drive is effective in SPHEX. The implications of this work for proposed helicity injection schemes in the next generation of spherical tokamaks are discussed.

We describe an experiment, the UMIST Linear System (ULS), in which a hydrogen plasma stream, guided by a longitudinal magnetic field, is injected through a diaphragm containing an orifice into a separately-pumped target chamber in which the neutral hydrogen pressure can be raised to a maximum of 8 mTorr. The stream is about 6 mm in diameter, has an electron temperature of up to 15 eV and an ion flux of 3×1018 s-1; it is supersonic with Mach number up to M≈3. We have studied both the passage of the stream through the orifice and the interaction of the supersonic plasma with neutral hydrogen in the target chamber. We find that transmission is incomplete even when the orifice diameter is five times that of the plasma; we attribute this to the presence of ion trajectories which extend well outside the visible plasma and are intercepted by the diaphragm. In the target chamber, the stream does not broaden, but the ion flux decreases approximately exponentially with distance, with a scale length of the order of the mean free path for momentum transfer in ion-neutral collisions, and much less than that expected for other processes, such as charge exchange or electron-ion recombination. Elastic collisions alone cannot decrease the flux, but would lead to a large accumulation of slow ions in thermal equilibrium with the neutral gas, which must be limited by some other loss process: collisional diffusion and electron-ion recombination are too slow, leading to a density approaching 1020 m-3. The observed density is of the order of 1018 m-3, requiring a process with a rate of 10-100 times faster. Calculated rates for molecular-activated recombination (MAR) of the slow ions are of this order, and the predicted density agrees with our observations to order of magnitude.

A model is developed to describe a coronal loop, which may originate from a photospheric source of smaller size than the coronal radius of the loop. The energy and relative helicity of the loop are evaluated, as are two alternative estimates of the energy available for coronal heating. Both of these estimates are strongly dependent on the size of the photospheric footprint of the loop. A coronal heating rate is then deduced, based on a nanoflare-type scenario, where slowly accumulated energy is rapidly released as heat. An explicit calculation is carried out for one particular choice of loop length and coronal radius, with dissipation timescale and photospheric radius as parameters. Two main conclusions are reached. Firstly, the proposed mechanism can make a significant contribution to coronal heating. Secondly, the mechanism is more effective for a more concentrated photospheric flux source.

Oscillations with toroidal mode number n=1 are ubiquitous in helicity injected spheromaks and spherical tokamaks, and play a crucial role in current drive. It has been proposed that these arise from a current driven instability of the open flux tube. Stability calculations are presented to confirm this, and they are compared with experimental data from the Spheromak Experiment (SPHEX) [M. Rusbridge et al., Plasma Phys. Control. Fusion 39, 683 (1997)]. The equilibria are modelled as piece-wise constant μ profile force-free plasmas with different values for the μ in the open (μc) and closed (μa) flux regions. A stability map in μc,μa space is then calculated. The SPHEX experimental data is also reduced to the same space both as a culmination of direct single point measurements of μ and as a time history of the reconstructed equilibrium from a particular shot. The results show a favorable comparison of the stability map with experiment, both in magnitude and shape. The effect of inserting a central current-c...

A novel technique is presented that aids the reconstruction of solar coronal magnetic fields by augmenting the visible solar surface boundary conditions with an estimate of the simultaneous conditions on the far side of the Sun. In converting from line-of-sight measurements, two alternative assumptions are considered: (1) ignore the plasma beta differences between corona and photosphere and so find a radial field component that is consistent with an overlying potential (or force free) field corona; and (2) characterize the change in beta by a boundary layer, matching the coronal field to the photosphere where the field, seen in projection, is taken to be wholly radial. The radial magnetic field (or radial field component) over the unseen hemisphere of the Sun is deduced from information held within a time series of the Solar and Heliospheric Observatory Michelson Doppler imager photospheric magnetograms centered on the time of interest and combined with a full disc of visible data for that time. Comparison is made with the more usual synoptic map boundary conditions to test the time sensitivity of the problem. Several methods of extrapolation to the far side are assessed using data centered on an observation at the end of August 1996, when a large equatorial coronal hole was present on one side of the Sun. The corresponding magnetic field equilibrium in the corona is found, assuming a simple potential approximation, which employs an outer “source surface” in addition to the derived inner boundary condition. Together with the inherent assumption that the evolution of the global field is slow, the validity of the technique is confirmed by the self-consistency of the results.