Equivalent Electric Circuit Models of Coronal Magnetic Loops and Related Oscillatory Phenomena on the Sun

Coronal loops, which trace closed magnetic field lines, are the primary structural elements of the solar atmosphere. Complex dynamics of solar coronal magnetic loops, together with action of possible subphotospheric dynamo mechanisms, turn the majority of the coronal loops into current-carrying structures. In that connection none of the loops can be considered as isolated from the surroundings. The current-carrying loops moving relative to each other interact via the magnetic field and currents. One of the ways to take into account this interaction consists in application of the equivalent electric circuit models of coronal loops. According to these models, each loop is considered as an equivalent electric LCR-circuit with variable inductive coefficients L, capacitance C, and resistance R, which depend on shape, scale, position of the loop with respect to neighbouring loops, as well as on the plasma parameters in the magnetic tube. Such an approach enables to describe the process of electric current dynamics in the groups of coronal loops, as well as the related dynamical, energy release and radiation processes. In the present paper we describe the major principles of LCR-circuit models of coronal magnetic loops, and show their application for interpretation of the observed oscillatory phenomena in the loops and in the related radiation.     

Global Non-Potential Magnetic Models of the Solar Corona During the March 2015 Eclipse

Seven different models are applied to the same problem of simulating the Sun’s coronal magnetic field during the solar eclipse on 2015 March 20. All of the models are non-potential, allowing for free magnetic energy, but the associated electric currents are developed in significantly different ways. This is not a direct comparison of the coronal modelling techniques, in that the different models also use different photospheric boundary conditions, reflecting the range of approaches currently used in the community. Despite the significant differences, the results show broad agreement in the overall magnetic topology. Among those models with significant volume currents in much of the corona, there is general agreement that the ratio of total to potential magnetic energy should be approximately 1.4. However, there are significant differences in the electric current distributions; while static extrapolations are best able to reproduce active regions, they are unable to recover sheared magnetic fields in filament channels using currently available vector magnetogram data. By contrast, time-evolving simulations can recover the filament channel fields at the expense of not matching the observed vector magnetic fields within active regions. We suggest that, at present, the best approach may be a hybrid model using static extrapolations but with additional energization informed by simplified evolution models. This is demonstrated by one of the models.     

The Expansion of Coronal Plumes in the Fast Solar Wind

Coronal plumes are believed to be essentially magnetic features: they are rooted in magnetic flux concentrations at the photosphere and are observed to extend nearly radially above coronal holes out to at least 15 solar radii, probably tracing the open field lines. The formation of plumes itself seems to be due to the presence of reconnecting magnetic field lines and this is probably the cause of the observed extremely low values of the Ne/Mg abundance ratio. In the inner corona, where the magnetic force is dominant, steady MHD models of coronal plumes deal essentially with quasi-potential magnetic fields but further out, where the gas pressure starts to be important, total pressure balance across the boundary of these dense structures must be considered. In this paper, the expansion of plumes into the fast polar wind is studied by using a thin flux tube model with two interacting components, plume and interplume. Preliminary results are compared with both remote sensing and solar wind in situ observations and the possible connection between coronal plumes with pressure-balance structures (PBS) and microstreams is discussed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Magnetic configuration of coronal streamers and threads

We give a short account of the most prominent structures of the intermediate corona. Then we propose an axially symmetrical model for coronal streamers, according to which charged particles move along magnetic surfaces whose sources are electrical currents situated in the vicinity of the photosphere. The simplest current configuration is a pair of coaxial, coplanar, circular, and oppositely directed currents parallel to the photosphere. Magnetic surfaces for this current distribution exhibit a helmet-shaped separatrix and a saddle point. The temperature profile along the streamer can be predicted qualitatively if one takes into account the conservation of an adiabatic invariant in the drift theory of the charged particle motion.     

The Evolving Sigmoid: Evidence for Magnetic Flux Ropes in the Corona Before, During, and After CMES

It is generally accepted that the energy that drives coronal mass ejections (CMEs) is magnetic in origin. Sheared and twisted coronal fields can store free magnetic energy which ultimately is released in the CME. We explore the possibility of the specific magnetic configuration of a magnetic flux rope of field lines that twist about an axial field line. The flux rope model predicts coronal observables, including heating along forward or inverse S-shaped, or sigmoid, topological surfaces. Therefore, studying the observed evolution of such sigmoids prior to, during, and after the CME gives us crucial insight into the physics of coronal storage and release of magnetic energy. In particular, we consider (1) soft-X-ray sigmoids, both transient and persistent; (2) The formation of a current sheet and cusp-shaped post-flare loops below the CME; (3) Reappearance of sigmoids after CMEs; (4) Partially erupting filaments; (5) Magnetic cloud observations of filament material.     

Progress, Challenges, and Perspectives of the 3D MHD Numerical Modeling of Oscillations in the Solar Corona

Recent high temporal and spatial resolution satellite observations of the solar corona provide ample evidence of oscillations in coronal structures. The observed waves and oscillations can be used as a diagnostic tool of the poorly known coronal parameters, such as magnetic field, density, and temperature. The emerging field of coronal seismology relies on the interpretation of the various coronal oscillations in terms of theoretically known wave modes, and the comparison of observed and theoretical wave mode properties for the determination of the coronal parameters. However, due to complexity of coronal structures the various modes are coupled, and the application of linear theory of idealized structures to coronal loops and active regions limits the usefulness of such methods. Improved coronal seismology can be achieved by the development of full 3D MHD dynamical model of relevant coronal structures and the oscillation phenomena. In addition to improved accuracy compared to linear analysis, 3D MHD models allow the diagnostic method to include nonlinearity, compressibility, and dissipation. The current progress made with 3D MHD models of waves in the corona is reviewed, and the challenges facing further development of this method are discussed in the perspective of future improvement that will be driven by new high resolution and high cadence satellite data, such as received from Hinode and STEREO, and expected from SDO.     

Modeling of dynamic evolution of coronal loops

Parameters of expanding magnetic loops and arches and of mass flows generated by them in the corona have been computed in a 1D two-fluid approximation. Two possible trigger mechanisms of the coronal transients have been considered: (i) sudden increase of the background magnetic field strength, and (ii) heating and compression plasma inside these magnetic structures. We discuss the formation of shock waves and their dependence on dynamics and geometry of the magnetic structures.     

Observations of inverted-V electron precipitation

The characteristics of inverted-V electron precipitation fluxes deduced predominantly from observations by the Atmosphere Explorer satellites are reviewed. The energy and pitch angle distributions are presented and shown to be generally in agreement with acceleration by a parallel electrostatic potential. Characteristics of secondary electrons are examined, and effects of beam plasma instabilities on these electrons are discussed. The properties of the monoenergetic component are compared with theoretical models of creating parallel DC electric fields, and found to favor the anomalous resistivity model. The article also discusses relations of inverted-V events with other auroral phenomena including auroras, electrostatic shocks, convective electric field reversals, field-aligned currents and wave emissions. The principal conclusions are: (1) plasma sheet electrons are continuously accelerated to form inverted-V structures in the pre-midnight hemisphere independent of substorm phase, (2) the acceleration processes are probably related to large scale electrostatic wave turbulence observed at altitudes of a few thousand kilometers, (3) narrow bursts of intense electron precipitation fluxes are found to be imbedded within some inverted-V's. It is argued that the narrow bursts of intense electron precipitation have the proper characteristics to cause discrete auroral arcs in the atmosphere. We suggest that these narrow bursts are accelerated by an electrostatic shock at higher altitude and capable of producing discrete auroral arcs below the observing satellite.     

Chromospheric and coronal heating mechanisms II

We review the mechanisms which are thought to provide steady heating of chromospheres and coronae. It appears now fairly well established that nonmagnetic chromospheric regions of latetype stars are heated by shock dissipation of acoustic waves which are generated in the stellar surface convection zones. In the case of late-type giants there is additional heating by shocks from pulsational waves. For slowly rotating stars, which have weak or no magnetic fields, these two are the dominant chromospheric heating mechanisms.Except for F-stars, the chromospheric heating of rapidly rotating late-type stars is dominated by magnetic heating either through MHD wave dissipation (AC mechanisms) or through magnetic field dissipation (DC mechanisms). The MHD wave and magnetic field energy comes from fluid motions in the stellar convection zones. Waves are also generated by reconnective events at chromospheric and coronal heights. The high-frequency part of the motion spectrum leads to AC heating, the low frequency part to DC heating. The coronae are almost exclusively heated by magnetic mechanisms. It is not possible to say at the moment whether AC or DC mechanisms are dominant, although presently the DC mechanisms (e.g., nanoflares) appear to be the more important. Only a more detailed study of the formation of and the dissipation in small-scale structures can answer this question.The X-ray emission in early-type stars shows the presence of coronal structures which are very different from those in late-type stars. This emission apparently arises in the hot post-shock regions of gas blobs which are accelerated in the stellar wind by the intense radiation field of these stars.     

Mass flows in coronal loops

Although static loop models are often used to describe the structure of coronal loops, it is evident on both observational and theoretical grounds that mass motions play a crucial role in the physics of the corona and transition region. First we review the observations of emission-line broadening and wavelength shifts, which imply the presence of random motions and systematic downflows in coronal loops. Some discrepancies in the observations are discussed. It is argued that velocities due to gas pressure gradients are the most likely explanation for the observed flows. A number of models that have been proposed for these motions are reviewed. The implications of the various models on observations of the corona and transition region by SOHO are discussed.     

Modeling the out-of-ecliptic interplanetary magnetic field in the declining phase of sunspot Cycle 22

We have developed a new model of the coronal and interplanetary magnetic field. The model includes the effects of large-scale horizontal electric currents flowing in the inner corona, of the warped heliospheric current sheet, and of heliospheric volume currents in the super-Alfvenic solar wind. The model determines the interplanetary magnetic field (IMF) strength as well as its polarity from measurements of the photospheric magnetic field. A detailed comparison between the observed and calculated in-ecliptic IMF Bx in Cycles 22, confirms the fitness of the optimal set of free parameters inferred using data in Cycle 21. We can predict the latitudinal gradient of Bx in the declining phase of Cycle 22 and the temporal variation of the amplitude of the radial component of the IMF at various latitudes. The calculated IMF polarity and Bx strength agree best with the in-ecliptic observations when the photospheric field (measured with a 5250Å magnetograph) is scaled up by a factor of two. Ulysses may provide the critical data to improve the model and check these inferences.     

Current Fragmentation and Particle Acceleration in Solar Flares

Particle acceleration in solar flares remains an outstanding problem in plasma physics and space science. While the observed particle energies and timescales can perhaps be understood in terms of acceleration at a simple current sheet or turbulence site, the vast number of accelerated particles, and the fraction of flare energy in them, defies any simple explanation. The nature of energy storage and dissipation in the global coronal magnetic field is essential for understanding flare acceleration. Scenarios where the coronal field is stressed by complex photospheric motions lead to the formation of multiple current sheets, rather than the single monolithic current sheet proposed by some. The currents sheets in turn can fragment into multiple, smaller dissipation sites. MHD, kinetic and cellular automata models are used to demonstrate this feature. Particle acceleration in this environment thus involves interaction with many distributed accelerators. A series of examples demonstrate how acceleration works in such an environment. As required, acceleration is fast, and relativistic energies are readily attained. It is also shown that accelerated particles do indeed interact with multiple acceleration sites. Test particle models also demonstrate that a large number of particles can be accelerated, with a significant fraction of the flare energy associated with them. However, in the absence of feedback, and with limited numerical resolution, these results need to be viewed with caution. Particle in cell models can incorporate feedback and in one scenario suggest that acceleration can be limited by the energetic particles reaching the condition for firehose marginal stability. Contemporary issues such as footpoint particle acceleration are also discussed. It is also noted that the idea of a “standard flare model” is ill-conceived when the entire distribution of flare energies is considered.     

Modeling coronal streamers and their eruption

Numerical solutions of the time-dependent MHD equations are used to generate ambient coronal streamer structures in a corona characteristic of that near solar minimum. The streamers are then disrupted by slow photospheric shear motion at the base of magnetic field lines within the closed field region, which is currently believed to be responsible for producing at least some CMEs. In contrast to several other simulations of this phenomena, the polytropic index is maintained at a value of 5/3 through the addition of coronal heating. Observations are used as a guide in determining the thermodynamic structure and plasma beta in the ambient corona. For a shear speed of 2.5 km/sec, the streamer configuration evolves slowly for about 65 hours before erupting outward with the formation of a CME. The bright CME leading edge travels outward at a speed of about 240 km/sec, and the sheared field lines follow at a somewhat slower speed. A closed magnetic field region is ejected as the magnetic field lines that were opened by the CME reconnect and reform the streamer.     

Transverse Oscillations of Coronal Loops

On 14 July 1998 TRACE observed transverse oscillations of a coronal loop generated by an external disturbance most probably caused by a solar flare. These oscillations were interpreted as standing fast kink waves in a magnetic flux tube. Firstly, in this review we embark on the discussion of the theory of waves and oscillations in a homogeneous straight magnetic cylinder with the particular emphasis on fast kink waves. Next, we consider the effects of stratification, loop expansion, loop curvature, non-circular cross-section, loop shape and magnetic twist. An important property of observed transverse coronal loop oscillations is their fast damping. We briefly review the different mechanisms suggested for explaining the rapid damping phenomenon. After that we concentrate on damping due to resonant absorption. We describe the latest analytical results obtained with the use of thin transition layer approximation, and then compare these results with numerical findings obtained for arbitrary density variation inside the flux tube. Very often collective oscillations of an array of coronal magnetic loops are observed. It is natural to start studying this phenomenon from the system of two coronal loops. We describe very recent analytical and numerical results of studying collective oscillations of two parallel homogeneous coronal loops. The implication of the theoretical results for coronal seismology is briefly discussed. We describe the estimates of magnetic field magnitude obtained from the observed fundamental frequency of oscillations, and the estimates of the coronal scale height obtained using the simultaneous observations of the fundamental frequency and the frequency of the first overtone of kink oscillations. In the last part of the review we summarise the most outstanding and acute problems in the theory of the coronal loop transverse oscillations.     

CME Theory and Models

This chapter provides an overview of current efforts in the theory and modeling of CMEs. Five key areas are discussed: (1) CME initiation; (2) CME evolution and propagation; (3) the structure of interplanetary CMEs derived from flux rope modeling; (4) CME shock formation in the inner corona; and (5) particle acceleration and transport at CME driven shocks. In the section on CME initiation three contemporary models are highlighted. Two of these focus on how energy stored in the coronal magnetic field can be released violently to drive CMEs. The third model assumes that CMEs can be directly driven by currents from below the photosphere. CMEs evolve considerably as they expand from the magnetically dominated lower corona into the advectively dominated solar wind. The section on evolution and propagation presents two approaches to the problem. One is primarily analytical and focuses on the key physical processes involved. The other is primarily numerical and illustrates the complexity of possible interactions between the CME and the ambient medium. The section on flux rope fitting reviews the accuracy and reliability of various methods. The section on shock formation considers the effect of the rapid decrease in the magnetic field and plasma density with height. Finally, in the section on particle acceleration and transport, some recent developments in the theory of diffusive particle acceleration at CME shocks are discussed. These include efforts to combine self-consistently the process of particle acceleration in the vicinity of the shock with the subsequent escape and transport of particles to distant regions.     

Theory of fragmented energy release in the Sun

The magnetic energy released inside an active region is closely related to its formation and evolution. Following the evolution of a collection of flux tubes inside the convection zone and above the photosphere we can show that many nonlinear structures (current sheets, shock waves, double layers etc.) are formed. We propose in this review that coronal heating, flares and particle acceleration are due to the interaction of the plasma with these nonlinear structures. Approaching active regions as a driven complex dynamical system we can show that several coherent ensembles of the nonlinear structures will appear spontaneously. The statistical analysis of these structures is a major problem in solar physics. We can also show that many observed large scale structures are the result of the convolution of non-observable fragmentation in the energy release process.     

The Heliospheric Magnetic Field at Solar Maximum: Ulysses Observations

At solar maximum, the large-scale structure of the heliospheric magnetic field (HMF) reflects the complexity of the Sun's coronal magnetic fields. The corona is characterised by mostly closed magnetic structures and short-lived, small coronal holes. The axis of the Sun's dipole field is close to the solar equator; there are also important contributions from the higher order terms. This complex and variable coronal magnetic configuration leads to a much increased variability in the HMF on all time scales, at all latitudes. The transition from solar minimum to solar maximum conditions, as reflected in the HMF, is described, as observed by Ulysses during its passage to high southern heliolatitudes. The magnetic signatures associated with the interaction regions generated by short-lived fast solar wind streams are presented, together with the highly disordered period in mid-1999 when there was a considerable reorganisation in coronal structures. The magnetic sector structure at high heliolatitudes shows, from mid-1999, a recognisable two-sector structure, corresponding to a highly inclined Heliospheric Current Sheet. A preliminary investigation of the radial component of the magnetic field indicates that it remains, on average, constant as a function of heliolatitude. Intervals of highly Alfvénic fluctuations in the rarefaction regions trailing the interaction regions have been, even if intermittently, identified even close to solar maximum. This revised version was published online in August 2006 with corrections to the Cover Date.     

Electrodynamic coupling of the magnetosphere and ionosphere

The auroral zone ionosphere is coupled to the outer magnetosphere by means of field-aligned currents. Parallel electric fields associated with these currents are now widely accepted to be responsible for the acceleration of auroral particles. This paper will review the theoretical concepts and models describing this coupling. The dynamics of auroral zone particles will be described, beginning with the adiabatic motions of particles in the converging geomagnetic field in the presence of parallel potential drops and then considering the modifications to these adiabatic trajectories due to wave-particle interactions. The formation of parallel electric fields can be viewed both from microscopic and macroscopic viewpoints. The presence of a current carrying plasma can give rise to plasma instabilities which in a weakly turbulent situation can affect the particle motions, giving rise to an effective resistivity in the plasma. Recent satellite observations, however, indicate that the parallel electric field is organized into discrete potential jumps, known as double layers. From a macroscopic viewpoint, the response of the particles to a parallel potential drop leads to an approximately linear relationship between the current density and the potential drop.The currents flowing in the auroral circuit must close in the ionosphere. To a first approximation, the ionospheric conductivity can be considered to be constant, and in this case combining the ionospheric Ohm's Law with the linear current-voltage relation for parallel currents leads to an outer scale length, above which electric fields can map down to the ionosphere and below which parallel electric fields become important. The effects of particle precipitation make the picture more complex, leading to enhanced ionization in upward current regions and to the possibility of feedback interactions with the magnetosphere.Determining adiabatic particle orbits in steady-state electric and magnetic fields can be used to determine the self-consistent particle and field distributions on auroral field lines. However, it is difficult to pursue this approach when the fields are varying with time. Magnetohydrodynamic (MHD) models deal with these time-dependent situations by treating the particles as a fluid. This class of model, however, cannot treat kinetic effects in detail. Such effects can in some cases be modeled by effective transport coefficients inserted into the MHD equations. Intrinsically time-dependent processes such as the development of magnetic micropulsations and the response of the magnetosphere to ionospheric fluctuations can be readily treated in this framework.The response of the lower altitude auroral zone depends in part on how the system is driven. Currents are generated in the outer parts of the magnetosphere as a result of the plasma convection. The dynamics of this region is in turn affected by the coupling to the ionosphere. Since dissipation rates are very low in the outer magnetosphere, the convection may become turbulent, implying that nonlinear effects such as spectral transfer of energy to different scales become important. MHD turbulence theory, modified by the ionospheric coupling, can describe the dynamics of the boundary-layer region. Turbulent MHD fluids can give rise to the generation of field-aligned currents through the so-called -effect, which is utilized in the theory of the generation of the Earth's magnetic field. It is suggested that similar processes acting in the boundary-layer plasma may be ultimately responsible for the generation of auroral currents.     

Small scale structures in the solar corona

The observational characteristics of the small scale magnetic structures are summarized. The temperature structure and temporal variability of the emission from coronal bright points, that pervade the source region of the solar wind in coronal holes and the quiet sun, and from active regions are shown to be remarkably similar. Particular emphasis is given to observations, potentially feasible with SOHO, that could resolve some of the outstanding issues regarding the role of the small scale magnetic structures in the energy balance and properties of the solar wind.