Models and observations have demonstrated that twisted flux ropes (TFRs) play a significant role in the structure and eruptive dynamics of active regions. Their role in the dynamics of the quiet Sun atmosphere has remained elusive, their fundamental relevance emerging mainly from theoretical models showing that they form and erupt as a result of flux cancellation. Here, Hinode high-resolution photospheric vector magnetic field measurements are integrated with advanced environment reconstruction models: TFRs develop on various scales and are associated with the appearance of mesospots. The developing TFRs contain sufficient free magnetic energy to match the requirements of the recently observed “campfires” discovered by Solar Orbiter in the quiet Sun. The free magnetic energy is found to be large enough to trigger eruptions, while the magnetic twist is large enough to trigger confined eruptions, heating the chromosphere. TFRs are also connected to larger-scale magnetic fields such as supergranulation loops, allowing the generation of Alfvén waves at the top of the chromosphere that can propagate along them. High-resolution magnetohydrodynamic simulations, incorporating subsurface dynamo activity at an unprecedented 30 km spatial resolution, confirm that TFRs are ubiquitous products of the permanent small-scale dynamo engine that feeds their formation, destabilization, eruption via flux emergence, submergence, and cancellation of their chromospheric feet, similar to the dynamics driving large-scale eruptive events. Future investigations, especially with the Daniel K. Inouye Solar Telescope and Solar Orbiter, will deepen our understanding of TFRs in the context of atmospheric heating.

Measurements and modelling of a large confined eruption on the Sun show that its evolution is controlled by a multilayer magnetic cage containing a twisted flux rope, which can sometimes be ejective.

We consider a magnetic field B occupying the simply connected domain D and having all its field lines tied to the boundary S of D. We assume here that B has a simple topology, i.e., the mapping M from positive to negative polarity areas of S associating to each other the two footpoints of any magnetic line, is continuous. We first present new formulae for the helicity H of B relative to a reference field B r having the same normal component B n on S, and for its field line helicity h relative to a reference vector potential C r of B r . These formulae make immediately apparent the well known invariance of these quantities under all the ideal MHD deformations that preserve the positions of the footpoints on S. They express indeed h and H either in terms of M and B n , or in terms of the values on S of a pair of Euler potentials of B . We next show that, for a specific choice of C r , the field line helicity h of B fully characterizes the magnetic mapping M and then the topology of the lines. Finally, we give a formula that describes the rate of change of h in a situation where the plasma moves on the perfectly conducting boundary S without changing B n and/or non-ideal processes, described by an unspecified term N in Ohm’s law, are at work in some parts of D.

model of the heating of the quiet Sun, in which magnetic fields are generated by a subphotospheric fluid dynamo intrinsically connected to granulation, shows fields expanding into the chromosphere, where plasma is heated at the rate required to match observations by small-scale eruptions that release magnetic energy and drive sonic motions, while the corona is heated by the dissipation of Alfvén waves.

It is now becoming necessary to access the global magnetic structure of the solar low corona at a large scale in order to understand its physics and more particularly the conditions of energization of the magnetic fields and the multiple connections between distant active regions (ARs) which may trigger eruptive events in an almost coordinated way. Various vector magnetographs, either on board spacecraft or ground-based, currently allow to obtain vector synoptic maps, composite magnetograms made of multiple interactive ARs, and full disk magnetograms. We present a method recently developed for reconstructing the global solar coronal magnetic field as a nonlinear force-free magnetic field in spherical geometry, generalizing our previous results in Cartesian geometry. This method is implemented in the new code XTRAPOLS, which thus appears as an extension of our active region scale code XTRAPOL. We apply our method by performing a reconstruction at a specific time for which we dispose of a set of composite data constituted of a vector magnetogram provided by SDO/HMI, embedded in a larger full disk vector magnetogram provided by the same instrument, finally embedded in a synoptic map provided by SOLIS. It turns out to be possible to access the large scale structure of the corona and its energetic contents, and also the AR scale, at which we recover the presence of a twisted flux rope in equilibrium.

Let ℬ be a class of topologically equivalent regular magnetic fields occupying a cylindrical domain of axis Oz and produced by ideal MHD deformations of the uniform field , and let ℬff ⊂ ℬ be constituted of the fields of ℬ which are force-free. We derive topological lower and upper bounds on the energy of the fields in ℬ and in ℬff, respectively. We also establish new formulae for the field line helicity and the relative magnetic helicity of the fields in ℬ, and give a sufficient criterion of ideal MHD linear stability for the fields in ℬff.

Modelling the solar magnetic field using observations of the photospheric field in the four-day period preceding a coronal mass ejection shows that the formation and later ejection of a twisted rope of magnetic flux provides the physical mechanism responsible for the ejection.

This chapter contains sections titled: Introduction The Model Problem : Shearing of a 2D Cartesian Arcade Three Dimensionnal MHD of Twisted Flux Ropes Conclusion

Context. High-resolution vector magnetographs either onboard spacecrafts or satellites (HMI/SDO, etc.) or ground based (SOLIS, etc.) now gives access to vector synoptic maps, composite magnetograms made of multiple interactive active regions, and full disk magnetograms. It thus become possible to reconstruct the coronal magnetic field on the full Sun scale.Aims. We present a method for reconstructing the global solar coronal magnetic field as a nonlinear force-free field. It is based on a well-posed Grad-Rubin iterative scheme adapted to spherical coordinatesMethods. This method is a natural extension to spherical geometry of the one we previously developed in Cartesian geometry. It is implemented in the code XTRAPOLS, which is a massively parallel code. It allows dealing with the strong constraints put on the computational methods by having to handle the very large amounts of data contained in high-resolution large-scale magnetograms. The method exploits the mixed elliptic-hyperbolic nature of the Grad-Rubin boundary value problem. It uses a finite-difference method for the elliptic part and a method of characteristics for the hyperbolic part. The computed field guarantees to be divergence free up to round-off errors, by introducing a representation in terms of a vector potential satisfying specific gauge conditions. The construction of the latter - called here the restricted DeVore gauge - is described in detail in an appendix.Results. We show that XTRAPOLS performs well by applying it to the reconstruction of a particular semi-analytic force-free field that has already been considered by various authors.

The low solar corona is dominated by the magnetic field which is created inside the sun by a dynamo process and then emerges into the atmosphere. This magnetic field plays an important role in most structures and phenomena observed at various wavelengths such as prominences, small and large scale eruptive events, and continuous heating of the plasma, and therefore it is important to understand its threedimensional properties in order to elaborate e cient theoretical models. Unfortunately, the magnetic field is di cult to measure locally in the hot and tenuous corona. But this can be done at the level of the cooler and denser photosphere, and several instruments with high resolution vector magnetographs are currently available (THEMIS, Imaging Vector Magnetograph (IVM), the Advanced Stokes Polarimeter (ASP), SOLIS, HINODE, Solar Dynamics Observatory (SDO), or will be shortly available by future telescopes such as EST and solar missions as SOLAR-ORBITER. This has lead solar physicists to develop an approach which consists in ” reconstructing” the coronal magnetic field from boundary data given on the photosphere. We will discuss some of the issues encountered in solving this problem as well our recent progress and results at the scale of active region scales or the larger one such as full sun scale.

We present exact analytical examples of three types of axisymmetric toroidal magnetostatic equilibria with nested flux surfaces: (1) current reversal equilibria, for which the net toroidal current switches from a negative to a positive value when moving away from the magnetic axis; these equilibria have a non-monotonic pressure profile, in accordance with Hammett et al.’s theorem stating that the pressure on the current reversal surface has to exceed the volume-averaged pressure within that surface; (2) zero core current equilibria, in which the toroidal current density vanishes inside some flux surface; and (3) surface current equilibria, constituted of an arbitrary number of nested layers inside which the plasma pressure is constant and the magnetic field force-free, with two adjacent layers being separated by a current sheet. All these configurations are obtained by shaping in an adequate way the arbitrary function which intervenes in the class of generalized isodynamic equilibria first constructed by Palumbo and recovered later on by Bishop and Taylor. A derivation of these equilibria by a method slightly different from Palumbo’s is given in an Appendix.

We consider a particular class of three-dimensional magnetostatic equilibria in which the plasma is submitted to a vertical gravitational field and the gradient of the total (thermal+magnetic) pressure vanishes. We show analytically that an equilibrium in that class makes the energy an absolute minimum in the set of all the configurations accessible from it by an arbitrary finite deformation constrained by ideal MHD and imposed to vanish on a rigid conducting wall (line-tying condition). Along with energy conservation, this implies the nonlinear ideal stability of that equilibrium in the following sense. Suppose that a perturbation of energy w(0) is applied at time t = 0 and thus evolves by obeying the nonlinear MHD equations. Then some measure of the sizes of the plasma velocity and the deformation of the structure can be made to stay at any t >= 0 below an arbitrarily prescribed value by choosing w(0) small enough. Nonlinear stability also holds true for a configuration obtained by superposing an equilibrium of the previous type and a nonmagnetic equilibrium which is also an energy minimizer-for instance an equilibrium with uniform specific entropy, which is shown to have that property. Our result applies to a subset of a family of equilibria, computed by B. C. Low, which includes in particular the standard Kippenhahn-Schluter model describing the magnetic support of solar corona prominences.

We construct exact analytical solutions of the equations describing the equilibrium of a self-gravitating magnetized fluid body, possibly rigidly rotating, by superposing two solutions of finite energy defined in the whole space, one describing a non-magnetized gravitating equilibrium (ST1) and the other describing a magnetized non-gravitating equilibrium (ST2). A large number of ST1s can be found in the literature and directly used for our constructions, and we thus concentrate on ST2s, which are difficult to obtain. We derive some of their general properties and exhibit two explicit classes of axisymmetric "elementary" such equilibria. The first one is extracted from the stellar models proposed by Prendergast and by Kutvitskii & Solov'ev, respectively. The second one is constructed by using Palumbo's theory of isodynamic equilibria, for which the magnetic pressure is constant on each flux surface. Both types of ST2s have their magnetic field confined within a bounded region, respectively, of spherical and toroidal shapes. A much more general ST2 can be obtained by juxtaposing n + q elementary ST2s, with n of the first type and q of the second type, in such a way that the magnetic regions do not pairwise overlap. The specific equilibria we obtain by superposition thus have no external field extending to infinity, and may be three dimensional (3D), which invalidates a recent nonexistence conjecture. Moreover, they may be arranged to contain force-free regions. Our superposition method can be considered as a 3D generalization of the axisymmetric splitting method previously developed by Kutvitskii & Solov'ev.

We consider magnetostatic equilibria in which a bounded region D containing a magnetized plasma is either fully confined by a field-free external medium - magnetic bubble equilibria (MBEqs) - or is confined by both such a medium and line-tying in a dense plasma region magnetic tower equilibria (MTEqs). We first establish some of their general properties. In particular, we derive a series of useful integral equalities relating the magnetic field and the thermal pressures inside and outside D, respectively. We use them to prove the non-existence of an axisymmetric MBEq with a purely poloidal field, and to discuss some recent results of Braithwaite on MBEq formation by relaxation from an initial non-equilibrium state. We next present two families of exact analytical axisymmetric MBEqs with, respectively, spherical and toroidal shapes. The first family is extracted from Prendergast's model of a self-gravitating magnetized body, while the second one is constructed by using Palumbo's theory of isodynamic equilibria, for which both magnetic and thermal pressures take constant values on any flux surface. MTEqs with a large variety of structures are thus obtained in a simple way: we start from an arbitrary MBEq and just consider the part of it above a given plane cutting the bubble D. For MBEqs and MTEqs in either family, we compute in closed form most of the interesting physical quantities (such as energy, magnetic helicity and twist). Our results are expected to be useful for building up simple models of several astrophysical objects (such as X-ray cavities in the intracluster medium, jets emitted by disc accreting compact objects, eruptive events in stellar coronae and their ejecta).

In the context of coronal mass ejections triggering, we reconsider the class of models in which the evolution of an active region (AR) is driven by imposed boundary motions converging toward the polarity inversion line (PIL). We introduce a new model problem in which there is a large-scale flow with a diverging structure on the photosphere. This flow is reminiscent of that of the well-known moat flow around each of the two spots of a bipolar AR and transports only part of the magnetic flux toward the PIL. It is thus more compatible with observations than the one used in our previous study, which forced the whole positive and negative polarity parts of the AR approaching each other. We also include a diffusion term associated with small-scale turbulent photospheric motions, but keep the associated diffusivity at a low value in the particular study described here. We show that the evolution of an initial sheared force-free field first leads to the formation of a twisted flux rope which stays in equilibrium for some time. Eventually, however, the configuration suffers a global disruption whose underlying mechanism is found by energetic considerations to be nonequilibrium. It begins indeed when the magnetic energy becomes of the order of the energy of an accessible partially open field. For triggering an eruption by converging flows, it is thus not necessary to advect the whole AR toward the PIL, but only its central part.

The low solar corona is dominated by the magnetic field which is created inside the sun by a dynamo process and then emerges into the atmosphere. This magnetic field plays an important role in most structures and phenomena observed at various wavelengths such as prominences, small and large scale eruptive events, and continuous heating of the plasma, and therefore it is important to understand its three-dimensional properties in order to elaborate efficient theoretical models. Unfortunately, the magnetic field is difficult to measure locally in the hot and tenuous corona. But this can be done at the level of the cooler and denser photosphere, and several instruments with high resolution vector magnetographs are currently available (THEMIS, Imaging Vector Magnetograph (IVM), the Advanced Stokes Polarimeter (ASP), SOLIS, HINODE, Solar Dynamics Observatory (SDO), or will be shortly availableby future telescopes such as EST and solar missions as SOLAR-ORBITER. This has lead solar physicists to develop an approach which consists in "Reconstructing" the coronal magnetic field from boundary data given on the photosphere. We will present some of the issues we encountered in solving this problem numerically as well our recent progress and results.

We establish some general properties of a 3D isodynamic magnetostatic equilibrium admitting a family of nested toroidal flux surfaces. In particular, we use the virial theorem to prove a simple relation between the total pressure (magnetic + thermal) and the magnetic pressure on each flux surface, and we derive some useful consequences of the latter. We also show the constancy on each rational surface of two integrals along magnetic lines. As a simple application of our results, we show the nonexistence of an equilibrium with vanishing toroidal current, and of an equilibrium with closed lines.

Janse & Low have most recently addressed the following question. Consider a cylindrical domain containing a simple topology potential magnetic field threading its lower and upper horizontal faces, and a perfectly conducting plasma. Suppose that this domain is made to slowly contract or expand in the vertical direction, so driving the field into a quasi-static evolution through a series of force-free configurations. Then are these configurations smooth, or do they contain current sheets (CSs)? We reexamine here their three-step argument leading to the conclusion that CSs form most generally. We prove analytically that the field has to evolve through "topologically untwisted" and "nonpotential" configurations, thus confirming the first two steps. However, we find the third step-leading to the conclusion that a smooth untwisted force-free field is necessarily potential-to be very disputable.