The two-plate initial boundary-value problem of Parker is reviewed, treating the relaxation of a 3D magnetic field prescribed with an arbitrary topology to a terminal force-free field in a cold, viscous, electrically perfect fluid conductor. Anchored by their foot-points at the perfectly conducting rigid plates, the relaxing field preserves its topology. The Parker magnetostatic theorem states that for most prescribed field topologies, the terminal field must embed current sheets. The elements of this theorem are reviewed and analyzed to relate this initial boundary-value problem to (i) the variational problem for a force-free field of a given topology and (ii) the direct construction of a force-free field in terms of its pair of Euler flux functions. New insights and understanding are presented on the theorem as the compelling basis of the Parker theory of solar coronal heating.

In the 60 years after E. N. Parker's prediction of the existence of the solar wind and the magnetic origin of coronal heating, space-borne and ground-based observations have built a conceptually complete phenomenology of the corona as a fully ionized hydromagnetic atmosphere responding in step to the global magnetic reversals of the Sun in 11-year cycles. This phenomenology is reviewed with the theoretical ideas it motivated, describing the photospheric emergence of new-cycle magnetic fluxes of a reversed polarity into the corona, ubiquitous coronal heating, hydromagnetic self-organization, explosive energy release, and the breaking of self-confinement into flows of expansion winds and episodic ejections of magnetic structures. High electrical and thermal conductivities at coronal million-degree temperatures have central roles. The corona obeys a hemispherical rule independent of magnetic cycle, that self-organized structures have a statistical preference for left- and right-handed magnetic twists, respectively, in the northern and southern hemispheres relative to the rotational axis. It is pointed out, perhaps for the first time, that this hemispherical rule is a hydromagnetic implication of the Parker (1955a,b) dynamo, straightforward to deduce graphically from his book Cosmical Magnetic Fields (1979). Solar physics has reached a broad-brush physical understanding of the corona solar-wind system as the prototype of a universal astrophysical phenomenon.

Magnetic field topology frozen in ideal magnetohydrodynamics (MHD) and its breakage in near-ideal MHD are reviewed in two parts, clarifying and expanding basic concepts. The first part gives a physically complete description of the frozen field topology derived from magnetic flux conservation as the fundamental property, treating four conceptually related topics: Eulerian and Lagrangian descriptions of three dimensional (3D) MHD, Chandrasekhar-Kendall and Euler-potential field representations, magnetic helicity, and inviscid vortex dynamics as a fluid system in physical contrast to ideal MHD. A corollary of these developments clarifies the challenge of achieving a high degree of the frozen-in condition in numerical MHD. The second part treats field-topology breakage centered around the Parker Magnetostatic Theorem on a general incompatibility of a continuous magnetic field with the dual demand of force-free equilibrium and an arbitrarily prescribed, 3D field topology. Preserving field topology as a global constraint readily results in formation of tangential magnetic discontinuities, or, equivalently, electric current-sheets of zero thickness. A similar incompatibility is present in the steady force-thermal balance of a heated radiating fluid subject to an anisotropic thermal flux conducted strictly along its frozen-in magnetic field in the low-β limit. In a weakly resistive fluid the thinning of current sheets by these general incompatibilities inevitably results in sheet dissipation, resistive heating and topological changes in the field notwithstanding the small resistivity. Strong Faraday induction drives but also macroscopically limits this mode of energy dissipation, trapping or storing free energy in self-organized ideal-MHD structures. This property of MHD turbulence captured by the Taylor hypothesis is reviewed in relation to the Sun’s corona, calling for a basic quantitative description of the breakdown of flux conservation in the low-resistivity limit. A cylindrical initial-boundary value problem provides specificity in the general MHD ideas presented.

This is a basic MHD study of the steady fall of an infinite, vertical slab of isothermal, resistive-viscous, compressible fluid across a dipped magnetic field in uniform gravity. This double-diffusion steady flow in unbounded space poses a nonlinear but numerically tractable, one-dimensional (1D) free-boundary problem, assuming constant coefficients of resistivity and viscosity. The steady flow is determined by a dimensionless number μ1 proportional to the triple product of the two diffusion coefficients and the square of the linear total mass. For a sufficiently large μ1, the Lorentz, viscous, fluid-pressure, and gravitational forces pack and collimate the fluid into a steady flow of a finite width defined by the two zero-pressure free-boundaries of the slab with vacuum. The viscous force is essential in this collimation effect. The study conjectures that in the regime μ1→0, the 1D steady state exists only for μ1∈Ω, a spectrum of an infinite number of discrete values, including μ1 = 0 that corresponds to two steady states, the classical zero-resistivity static slab of Kippenhahn and Schlüter [R. Kippenhahn and A. Schlüter, Z. Astrophys. 43, 36 (1957)] and its recent generalization [B. C. Low et al., Astrophys. J. 755, 34 (2012)] to admit an inviscid resistive flow. The pair of zero-pressure boundaries of each of the μ1→0 steady-state slabs are located at infinity. Computational evidence suggests that the Ω steady-states are densely distributed around μ1 = 0, as an accumulation point, but are sparsely separated by open intervals of μ1-values for which the slab must be either time-dependent or spatially multi-dimensional. The widths of these intervals are vanishingly small as μ1→0. This topological structure of physical states is similar to that described by Landau and Liftshitz [L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Addison-Wesley, Reading, MA, 1959)] to explain the onset of hydrodynamic turbulence. The implications of this MHD study are discussed, with an interest in the prominences in the solar atmosphere and the interstellar clouds in the Galaxy.

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.

An axisymmetric force-free magnetic field B(r, θ) in spherical coordinates is defined by a function r sin θBφ=Q(A) relating its azimuthal component to its poloidal flux-function A. The power law r sin θBφ=aA|A|1/n, n a positive constant, admits separable fields with A=An(θ)rn, posing a nonlinear boundary-value problem for the constant parameter a as an eigenvalue and An(θ) as its eigenfunction [B. C. Low and Y. Q Lou, Astrophys. J. 352, 343 (1990)]. A complete analysis is presented of the eigenvalue spectrum for a given n, providing a unified understanding of the eigenfunctions and the physical relationship between the field's degree of multi-polarity and rate of radial decay via the parameter n. These force-free fields, self-similar on spheres of constant r, have basic astrophysical applications. As explicit solutions they have, over the years, served as standard benchmarks for testing 3D numerical codes developed to compute general force-free fields in the solar corona. The study presented includes a set of illustrative multipolar field solutions to address the magnetohydrodynamics (MHD) issues underlying the observation that the solar corona has a statistical preference for negative and positive magnetic helicities in its northern and southern hemispheres, respectively; a hemispherical effect, unchanging as the Sun's global field reverses polarity in successive eleven-year cycles. Generalizing these force-free fields to the separable form B=H(θ,φ)rn+2 promises field solutions of even richer topological varieties but allowing for φ-dependence greatly complicates the governing equations that have remained intractable. The axisymmetric results obtained are discussed in relation to this generalization and the Parker Magnetostatic Theorem. The axisymmetric solutions are mathematically related to a family of 3D time-dependent ideal MHD solutions for a polytropic fluid of index γ = 4/3 as discussed in the Appendix.

This is a follow-up investigation of a magnetic flux cancellation event at a polarity inversion line (PIL) on the Sun observed with the spectropolarimeter on board Hinode. Anomalous circular polarization (Stokes V) profiles are observed in the photosphere along the PIL at the cancellation sites. Kubo et al. previously reported that the theoretically expected horizontal fields between the canceling opposite-polarity magnetic elements in this event are not detected at granular scales. We show that the observed anomalous Stokes V profiles are reproduced successfully by adding the nearly symmetric Stokes V profiles observed at pixels immediately adjacent to the PIL. This result suggests that these observed anomalous Stokes V profiles are not indications of a flux removal process, but are the result of either a mixture of unresolved, opposite-polarity magnetic elements or the unresolved width of the PIL, at an estimated resolution element of about 0 ''.3. The hitherto undetected flux removal process accounting for the larger-scale disappearance of magnetic flux during the observing period is likely to also fall below resolution.

Contex. Solar tornados are dynamical, conspicuously helical magnetic structures that are mainly observed as a prominence activity.Aims. We investigate and propose a triggering mechanism for the solar tornado observed in a prominence cavity by SDO/AIA on September 25, 2011.Methods. High-cadence EUV images from the SDO/AIA and the Ahead spacecraft of STEREO/EUVI are used to correlate three flares in the neighbouring active-region (NOAA 11303) and their EUV waves with the dynamical developments of the tornado. The timings of the flares and EUV waves observed on-disk in 195 angstrom are analysed in relation to the tornado activities observed at the limb in 171 angstrom.Results. Each of the three flares and its related EUV wave occurred within ten hours of the onset of the tornado. They have an observed causal relationship with the commencement of activity in the prominence where the tornado develops. Tornado-like rotations along the side of the prominence start after the second flare. The prominence cavity expands with the accelerating tornado motion after the third flare.Conclusions. Flares in the neighbouring active region may have affected the cavity prominence system and triggered the solar tornado. A plausible mechanism is that the active-region coronal field contracted by the "Hudson effect" through the loss of magnetic energy as flares. Subsequently, the cavity expanded by its magnetic pressure to fill the surrounding low corona. We suggest that the tornado is the dynamical response of the helical prominence field to the cavity expansion.

This chapter contains sections titled: Introduction Taylor Relaxation Ergodic Lines of Force Anchored Magnetic Fields Open Domains Magnetic Self-Confinement Solar Coronal Mass Ejections Conclusion

This paper treats the relaxation of a magnetic field into a minimum-energy force-free state in a cold (pressure-less) viscous fluid, under the frozen-in condition of perfect electrical conductivity and letting the viscosity-dissipated energy be completely lost. A non-Newtonian fluid in popular use is studied in relation to the Newtonian viscous fluid, as two alternative numerical means to (1) construct force-free fields representing solar coronal structures in realistic geometry and (2) investigate the Parker theory of spontaneous formation of electric current sheets as a basic MHD process. Faraday's induction equation imposes an independent condition on the fluid velocity at rigid, perfectly conducting boundaries. This boundary condition is quite compatible with Newtonian mechanics but not with the non-Newtonian fluid model where velocity is equated to the Lorentz force with a free, positive multiplicative-factor. This defining property gives rise to unphysical or artificial singularities not previously known that are completely distinct from the physically admissible singularities representing the current sheets of the Parker theory. In particular, the non-Newtonian fluid takes a magnetic field with neutral points from any one of a continuum of initial states into an unphysical state instead of the proper force-free end-state accessible by Newtonian relaxation. The validity of previously published MHD results based on this non-Newtonian fluid, including some counterclaims against the Parker theory, is dubious. Investigating the Parker theory requires numerical relaxation models capable of anticipating and accurately describing inevitable current-sheet singularities. By including a weak resistivity to dissipate the inevitable current sheets as they form, the field can change topology intermittently to seek a terminal force-free state free of singularities. The minimum-energy state of this more complete model corresponds to the long-lived relaxed structures in the corona.

AbstractAn enormous solar tornado was observed by SDO/AIA on 25 September 2011. It was mainly associated with a quiescent prominence with an overlying coronal cavity. We investigate the triggering mechanism of the solar tornado by using the data from two instruments: SDO/AIA and STEREO-A/EUVI, covering the Sun from two directions. The tornado appeared near to the active region NOAA 11303 that produced three flares. The flares directly influenced the prominence-cavity system. The release of free magnetic energy from the active region by flares resulted in the contraction of the active region field. The cavity, owing to its superior magnetic pressure, expanded to fill this vacated space in the corona. We propose that the tornado developed on the top of the prominence due to the expansion of the prominence-cavity system.

AbstractWe present SDO/AIA observations of a potentially novel type of prominence, called “funnel prominence”, that forms out of coronal condensation at magnetic dips.They can drain a large amount of mass (up to ~1015 g day−1) and may play an important role as return flows of the chromosphere-corona mass cycle.

We investigate the evolution of five granular-scale magnetic flux cancellations just outside the moat region of a sunspot by using accurate spectropolarimetric measurements and G-band images with the Solar Optical Telescope (SOT) aboard Hinode. The opposite-polarity magnetic elements approach a junction of the intergranular lanes and then collide with each other there. The intergranular junction has strong redshifts, darker intensities than the regular intergranular lanes, and surface converging flows. This clearly confirms that the converging and downward convective motions are essential for the approaching process of the opposite-polarity magnetic elements. However, the motion of the approaching magnetic elements does not always match with their surrounding surface flow patterns in our observations. This suggests that, in addition to the surface flows, subsurface downward convective motions and subsurface magnetic connectivities are important for understanding the approach and collision of the opposite-polarity elements observed in the photosphere. We find that the horizontal magnetic field appears between the canceling opposite-polarity elements in only one event. The horizontal fields are observed along the intergranular lanes with Doppler redshifts. This cancellation is most probably a result of the submergence (retraction) of low-lying photospheric magnetic flux. In the other four events, the horizontal field is not observed between the opposite-polarity elements at any time when they approach and cancel each other. These approaching magnetic elements are more concentrated rather than gradually diffused, and they have nearly vertical fields even while they are in contact each other. We thus infer that the actual flux cancellations are highly time-dependent events at scales less than a pixel of Hinode SOT (about 200 km) near the solar surface.

This second paper of the series investigates the transverse response of a magnetic field to the independent relaxation of its flux tubes of fluid seeking hydrostatic and energy balance, under the frozen-in condition and suppression of cross-field thermal conduction. The temperature, density, and pressure naturally develop discontinuities across the magnetic flux surfaces separating the tubes, requiring the finite pressure jumps to be compensated by magnetic-pressure jumps in cross-field force balance. The tangentially discontinuous fields are due to discrete currents in these surfaces, δ-function singularities in the current density that are fully admissible under the rigorous frozen-in condition but must dissipate resistively if the electrical conductivity is high but finite. The magnetic field and fluid must thus endlessly evolve by this spontaneous formation and resistive dissipation of discrete currents taking place intermittently in spacetime, even in a low-β environment. This is a multi-dimensional effect in which the field plays a central role suppressed in the one-dimensional (1D) slab model of the first paper. The study begins with an order-of-magnitude demonstration that of the weak resistive and cross-field thermal diffusivities in the corona, the latter is significantly weaker for small β. This case for spontaneous discrete currents, as an important example of the general theory of Parker, is illustrated with an analysis of singularity formation in three families of two-dimensional generalizations of the 1D slab model. The physical picture emerging completes the hypothesis formulated in Paper I that this intermittent process is the origin of the dynamic interiors of a class of quiescent prominences revealed by recent Hinode/SOT and SDO/AIA high-resolution observations.

We report the first analyses of SDO/AIA observations of the formation of a quiescent polar crown prominence in a coronal cavity. The He ii 304 Å (log Tmax  ∼ 4.8 K) data show both the gradual disappearance of the prominence due to vertical drainage and lateral transport of plasma followed by the formation of a new prominence 12 hr later. The formation is preceded by the appearance of a bright emission “cloud” in the central region of the coronal cavity. The peak brightness of the cloud progressively shifts in time from the Fe xiv 211 Å channel, through the Fe xii 193 Å channel, to the Fe ix 171 Å channel (log Tmax  ∼ 6.2, 6.1, 5.8 K, respectively) while simultaneously decreasing in altitude. Filter ratio analysis estimates the initial temperature of the cloud to be approximately log T ∼ 6.25 K with evidence of cooling over time. The subsequent growth of the prominence is accompanied by darkening of the cavity in the 211 Å channel. The observations imply prominence formation via in situ condensation of hot plasma from the coronal cavity, in support of our previously proposed process of magnetothermal convection in coronal magnetic flux ropes.

In this paper, we continue our theoretical studies addressing the possible consequences of magnetic helicity accumulation in the solar corona. Our previous studies suggest that coronal mass ejections (CMEs) are natural products of coronal evolution as a consequence of magnetic helicity accumulation and that the triggering of CMEs by surface processes such as flux emergence also have their origin in magnetic helicity accumulation. Here, we use the same mathematical approach to study the magnetic helicity of axisymmetric power-law force-free fields but focus on a family whose surface flux distributions are defined by self-similar force-free fields. The semi-analytical solutions of the axisymmetric self-similar force-free fields enable us to discuss the properties of force-free fields possessing a huge amount of accumulated magnetic helicity. Our study suggests that there may be an absolute upper bound on the total magnetic helicity of all bipolar axisymmetric force-free fields. With the increase of accumulated magnetic helicity, the force-free field approaches being fully opened up with Parker-spiral-like structures present around a current-sheet layer as evidence of magnetic helicity in the interplanetary space. It is also found that among the axisymmetric force-free fields having the same boundary flux distribution, the one that is self-similar is the one possessing the maximum amount of total magnetic helicity. This gives a possible physical reason why self-similar fields are often found in astrophysical bodies, where magnetic helicity accumulation is presumably also taking place.

This series of papers investigates the dynamic interiors of quiescent prominences revealed by recent Hinode and SDO/AIA high-resolution observations. This first paper is a study of the static equilibrium of the Kippenhahn-Schluter diffuse plasma slab, suspended vertically in a bowed magnetic field, under the frozen-in condition and subject to a theoretical thermal balance among an optically thin radiation, heating, and field-aligned thermal conduction. The everywhere-analytical solutions to this nonlinear problem are an extremely restricted subset of the physically admissible states of the system. For most values of the total mass frozen into a given bowed field, force balance and steady energy transport cannot both be met without a finite fraction of the total mass having collapsed into a cold sheet of zero thickness, within which the frozen-in condition must break down. An exact, resistive hydromagnetic extension of the Kippenhahn-Schluter slab is also presented, resolving the mass-sheet singularity into a finite-thickness layer of steadily falling dense fluid. Our hydromagnetic result suggests that the narrow, vertical prominence H a threads may be falling across magnetic fields, with optically thick cores much denser and ionized to much lower degrees than conventionally considered. This implication is discussed in relation to (1) the recent SDO/AIA observations of quiescent prominences that are massive and yet draining mass everywhere in their interiors, (2) the canonical range of 5-60 G determined from spectral polarimetric observations of prominence magnetic fields over the years, and (3) the need for a more realistic multi-fluid treatment.

Imaging solar coronal condensation forming prominences was difficult in the past, a situation recently changed by Hinode and the Solar Dynamics Observatory (SDO). We present the first example observed with the SDO/Atmospheric Imaging Assembly, in which material gradually cools through multiple EUV channels in a transequatorial loop system that confines an earlier eruption. Nine hours later, this leads to eventual condensation at the dips of these loops, forming a moderate-size prominence of similar to 10(14) g, to be compared to the characteristic 10(15) g mass of a coronal mass ejection (CME). The prominence mass is not static but maintained by condensation at a high estimated rate of 10(10) g s(-1) against a comparable, sustained drainage through numerous vertical downflow threads, such that 96% of the total condensation (similar to 10(15) g) is drained in approximately one day. The mass condensation and drainage rates temporally correlate with the total prominence mass. The downflow velocity has a narrow Gaussian distribution with a mean of 30 km s(-1), while the downward acceleration distribution has an exponential drop with a mean of similar to 1/6 g(circle dot), indicating a significant canceling of gravity, possibly by the Lorentz force. Our observations show that a macroscopically quiescent prominence is microscopically dynamic, involving the passage of a significant mass through it, maintained by a continual mass supply against a comparable mass drainage, which bears important implications for CME initiation mechanisms in which mass unloading is important.