Constant-α Force-free Fields Research Articles

Cylindrical Taylor states conserving total absolute magnetic helicity

The Taylor state of a three-dimensional (3D) magnetic field in an upright cylindrical domain V is derived from first principles as an extremum of the total magnetic energy subject to a conserved, total absolute helicity Habs. This new helicity [Low, Phys. Plasmas 18, 052901 (2011)] is distinct from the well known classical total helicity and relative total helicity in common use to describe wholly-contained and anchored fields, respectively. A given field B, tangential along the cylindrical side of V, may be represented as a unique linear superposition of two flux systems, an axially extended system along V and a strictly transverse system carrying information on field-circulation. This specialized Chandrasekhar-Kendall representation defines Habs and permits a neat formulation of the boundary-value problem (BVP) for the Taylor state as a constant-α force-free field, treating 3D wholly-contained and anchored fields on the same conceptual basis. In this formulation, the governing equation is a scalar integro-partial differential equation (PDE). A family of series solutions for an anchored field is presented as an illustration of this class of BVPs. Past treatments of the constant-α field in 3D cylindrical geometry are based on a scalar Helmholtz PDE as the governing equation, with issues of inconsistency in the published field solutions discussed over time in the journal literature. The constant-α force-free equation reduces to a scalar Helmholtz PDE only as special cases of the 3D integro-PDE derived here. In contrast, the constant-α force-free equation and the scalar Helmholtz PDE are absolutely equivalent in the spherical domain as discussed in Appendix. This theoretical study is motivated by the investigation of the Sun's corona but the results are also relevant to laboratory plasmas.

GUIDING NONLINEAR FORCE-FREE MODELING USING CORONAL OBSERVATIONS: FIRST RESULTS USING A QUASI-GRAD-RUBIN SCHEME

Presently, many models of the coronal magnetic field rely on photospheric vector magnetograms but these data have been shown to be problematic as the sole boundary information for nonlinear force-free field (NLFFF) extrapolations. Magnetic fields in the corona manifest themselves in high-energy images (X-rays and EUV) in the shapes of coronal loops, providing an additional constraint that at present is not used due to the mathematical complications of incorporating such input into numerical models. Projection effects and the limited number of usable loops further complicate the use of coronal information. We develop and test an algorithm to use images showing coronal loops in the modeling of the solar coronal magnetic field. We first fit projected field lines with field lines of constant-\als force-free fields to approximate the three-dimensional distribution of currents in the corona along a sparse set of trajectories. We then apply a Grad-Rubin-like iterative technique to obtain a volume-filling nonlinear force-free model of the magnetic field, modifying method presented in \citet{Wheatland2007}. We thoroughly test the technique on known analytical and solar-like model magnetic fields previously used for comparing different extrapolation techniques \citep{Schrijver2006, Schrijver2008} and compare the results with those obtained by presently available methods that rely only on the photospheric data. We conclude that we have developed a functioning method of modeling the coronal magnetic field by combining the line-of-sight component of photospheric magnetic field with information from coronal images. Vector magnetograms over the full or partial photospheric boundary of the numerical domain could optionally be used.

MAGNETIC RELAXATION IN THE SOLAR CORONA

This is a mathematical study of the long-lived hydromagnetic structures produced in the tenuous solar corona by the turbulent, resistive relaxation of a magnetic field under the condition of extremely high electrical conductivity. The relaxation theory of Taylor, originally developed for a laboratory device, is extended to treat the open atmosphere where the relaxing field must interact with its surrounding fields. A boundary-value problem is posed for a two-dimensional model that idealizes the corona as the half Cartesian plane filled with a potential field (1) that is anchored to a rigid, perfectly conducting base and (2) that embeds a force-free magnetic field in the form of a flux-rope oriented horizontally and perpendicular to the Cartesian plane. The flux-rope has a free boundary, which is an unknown in the construction of a solution for this atmosphere. Pairs of magnetostatic solutions are constructed to represent the initial and final states of a flux-rope relaxation that conserve both the total magnetic helicity and total axial magnetic flux, using a numerical iterative method specially developed for this study. The collection of numerical solutions found provides an insight into the interplay among several hydromagnetic properties in the formation of long-lived coronal structures. In particular, the study shows (1) that the outward spread of reconnection between a relaxing flux-rope and its external field may be arrested at some outer magnetic flux surface within which a constant-α force-free field emerges as the minimum-energy state and (2) that this outward spread is complicated by an inward, partial collapse of the relaxing flux-rope produced by a loss of internal magnetic pressure.

Spatial Configuration of Highly Sheared Magnetic Structures in Active Region (NOAA 6659) in 1991 June

In this paper, we analyze first the configuration, especially the reversal features in the Hβ chromospheric longitudinal magnetograms in a very flare-active δ group (NOAA 6659). We discuss the possibility of large-scale reversal chromospheric magnetic structures overlapped on the photospheric umbrae after the comparison between the photospheric vector magnetograms and chromospheric longitudinal magnetograms and their forming height difference. We present the possible spatial configuration of the magnetic shear and the relationship with the electric current, and we and that some highly sheared magnetic features of opposite polarity in the active region deviated from the constant-α force-free field. The interaction of the newly emerging magnetic flux of opposite polarity with large-scale magnetic structures is the triggering mechanism of a series of powerful fiares formed in the lower solar atmosphere.

A model for magnetic energy storage and Taylor’s relaxation in the solar corona. I: Helicity-constrained minimum energy state in a half-cylinder

The problem of the existence of a minimum energy state is studied in the set ℋ of all the magnetic fields B: (i) occupying the half-cylinder D={r<R,z≳0}; (ii) having a normal component vanishing on the vertical part {r=R} of the boundary of D and taking given values Q(r) on its horizontal part {z=0}; (iii) having a relative helicity equal to a prescribed value H. It is first shown that the only field that may possibly be an energy minimizer in ℋ is the unique (and therefore axisymmetric) constant-α force-free field Bα contained in that set. Thus it is proved that Bα minimizes the energy indeed if and only if 0≤‖H‖≤Hc≤∞, where Hc is an estimated critical value. For Hc≤‖H‖≤∞, on the contrary, it is possible to construct in ℋ nonaxisymmetric fields with an energy smaller than that of Bα and no minimum energy state does exist. However, Bα still minimizes the energy for Hc ≤‖ H‖ ≤ Haxc (with possibly Haxc = ∞) if attention is restricted to the axisymmetric fields of ℋ. These results are used to put a limit on the validity of a popular model of the heating of the solar corona, in which the field of a coronal structure is supposed to release sporadically, by Taylor’s relaxation, a part of the energy it continuously extracts from the kinetic energy of the photospheric motions. It is argued that, as a consequence of the results above, one of the basic assumptions of the model breaks down when the field becomes highly sheared. It is speculated that, in such a situation, a completely new regime should set up, in which helicity and energy are continuously ejected at large distances by the system.

A two-dimensional model for a solar prominence: Effect of an external magnetic field

Using analytical approximations we study the effects of different external magnetic configurations on the half-width, mass, and internal magnetic structure of a quiescent solar prominence, modelled as a thin vertical sheet of cool plasma. Firstly, we build up a zeroth-order model and analyse the effects produced by a potential coronal field or a constant-α force-free field. This model allows us to obtain the half-width and mass of the prominence for different values of the external field, pressure and shear angle. Secondly, the effects of these external magnetic configurations on a two-dimensional model proposed by Ballester and Priest (1987) are studied. The main effects are a change of the half-width with height, an increase of the mass, a decrease of the magnetic field strength with height and a change in the shape of the magnetic field lines.

Computational modeling of magnetic fields in solar active regions

The magnetic field plays an important role in various solar activities. This paper reviews techniques for computational modeling of magnetic fields in solar active regions. The input data are photospheric magnetic fields supplied by magnetograph observations. The field above the photosphere is computed by assuming an equation for the magnetic field. Three classes of magnetic fields, namely current-free fields, constant-α force-free fields, and general force-free fields are considered. Their physical/mathematical significances and computational procedures are systematically presented.