B. C. Low

Magnetohydrodynamic processes in the solar corona: Flares, coronal mass ejections, and magnetic helicity*

The magnetized, million‐degree solar corona evolves in cycles of about 11 years, in dynamical response to newly generated magnetic fluxes emerging from below to eventually reverse the global magnetic polarity. Over the larger scales, the corona does not erupt violently all the time. Violent events like the flares and episodic ejections of material into interplanetary space occur frequently, several times a day, but they often originate in long‐lived magnetic structures that form continually throughout the solar cycle. In this paper, the creation, stability, and eventual eruption of these structures are discussed from basic principles, drawing on recent advances in observation and theory. A global view is offered in which different pieces of observation relate physically, with distinct roles for the conservation of magnetic helicity and the release of magnetic energy in dissipated and ordered forms.

Complex magnetohydrodynamic bow shock topology in field-aligned low-β flow around a perfectly conducting cylinder

Two-dimensional ideal magnetohydrodynamic (MHD) simulations are presented that demonstrate several novel phenomena in MHD shock formation. The stationary symmetrical flow of a uniform, planar, field-aligned, low-β and superfast magnetized plasma around a perfectly conducting cylinder is calculated. The velocity of the incoming flow is chosen such that the formation of fast switch-on shocks is possible. Using a time marching procedure, a stationary bow shock is obtained, composed of two consecutive interacting shock fronts. The leading shock front has a dimpled shape and is composed of fast, intermediate and hydrodynamic shock parts. A second shock front follows the leading front. Additional intermediate shocks and tangential discontinuities are present in the downstream part of the flow. The intermediate shocks are of the 1–3, 1–4, 2–4 and 1=2–3=4 types. This is a confirmation in two dimensions of recent results on the admissibility of these types of shocks. Recently it has also been shown that the 1=2–3=...

Topological complexity and tangential discontinuity in magnetic fields

This is a study of the topological magnetostatic problem. A magnetic field embedded in a perfectly conducting fluid and rigidly anchored at its boundary has a specific topology invariant for all time. Subject to that topology, the force-free state of such a field generally requires the presence of tangential discontinuities (TDs). This property proposed and demonstrated by Parker [Spontaneous Current Sheets in Magnetic Fields (Oxford University Press, New York, 1994)] is explained in terms of (i) the overdetermined nature of the magnetostatic partial differential equations nonlinearly coupled to the integral equations imposing the field topology and (ii) the hyperbolic nature of the partial differential equation for the twist function α of the force-free field. The mathematical analysis elucidates a basic incompatibility between preserving a complex field topology and attaining equilibrium, if analyticity is assumed. Physics avoids this incompatibility via TD formation as a natural consequence of perfect conductivity. The study relates TD formation to topological complexity in two-dimensional and three-dimensional fields, as well as the topological connectivity and geometric shape of the field domain. Mathematical points made are given physical interpretations, but important topological concepts for understanding spontaneous TDs have remained incomplete. As an application, examples are presented to define twisted and untwisted potential fields found in simply and multiply connected domains, clarifying a confusion in several recent publications. Appendix A treats the expression of the frozen-in condition by a continuum of conserved, total generalized helicities. Appendix B reports briefly on concurrent developments showing that a published objection to the theory of spontaneous TDs is based upon a misunderstanding of the theory.

On spontaneous formation of current sheets: Untwisted magnetic fields

This is a study of the spontaneous formation of electric current sheets in an incompressible viscous fluid with perfect electrical conductivity, governed by the magnetohydrodynamic Navier–Stokes equations. Numerical solutions to two initial value problems are presented for a three-dimensional, periodic, untwisted magnetic field evolving, with no change in magnetic topology under the frozen-in condition and at characteristic fluid Reynolds numbers of the order of 500, from a nonequilibrium initial state with the fluid at rest. The evolution converts magnetic free energy into kinetic energy to be all dissipated away by viscosity so that the field settles into a minimum-energy, static equilibrium. The solutions demonstrate that, as a consequence of the frozen-in condition, current sheets must form during the evolution despite the geometric simplicity of the prescribed initial fields. In addition to the current sheets associated with magnetic neutral points and field reversal layers, other sheets not associat...

On the possibility of electric-current sheets in dense formation

This is a mathematical study of the formation of tangential discontinuities, or current sheets, in a magnetic field evolving in an electrically perfectly conducting fluid in response to deformation of its domain, an effect first treated by Hahm and Kulsrud [Phys. Fluids 28, 2412 (1985)]. Explicit examples are presented of three-dimensional, untwisted fields, anchored to the boundary, that cannot assume a force-free state in the absence of current sheets. The underlying physics of this process is as described by the Parker theory of spontaneous current sheets, namely, that for most continuous magnetic fields in complex three-dimensional geometry, there is an incompatibility between the preservation of field topology and point-by-point force balance to achieve equilibrium. This incompatibility is removed through discontinuous plasma displacements that produce magnetic tangential discontinuities. In contrast to the twisted magnetic fields central to the Parker theory, fixing the connectivity between the anch...

Characteristic analysis of a complex two-dimensional magnetohydrodynamic bow shock flow with steady compound shocks

A simple, compact, and systematic derivation is given of the characteristic properties of the magnetohydrodynamic (MHD) equations with two independent variables (time-dependent MHD in the xt plane and steady MHD in the xy plane), based on the symmetrizable Galilean invariant form of the equations and using a matrix approach. A numerically obtained stationary planar field-aligned MHD bow shock flow with interacting shocks is then analyzed in terms of hyperbolic and elliptic regions, steady xy characteristics, limiting lines, and allowed shock transitions. With the help of this analysis, a wave structure present in the bow shock flow can be interpreted as a double steady compound shock. This interpretation is based on the complete analogy demonstrated in our analysis, between the xy characteristic structure of this novel steady compound shock and the xt characteristic structure of the well-known time-dependent MHD compound shock.

Absolute magnetic helicity and the cylindrical magnetic field

The different magnetic helicities conserved under conditions of perfect electrical conductivity are expressions of the fundamental property that every evolving fluid surface conserves its net magnetic flux. This basic hydromagnetic point unifies the well known Eulerian helicities with the Lagrangian helicity defined by the conserved fluxes frozen into a prescribed set of disjoint toroidal tubes of fluid flowing as a permanent partition of the entire fluid [B. C. Low, Astrophys. J. 649, 1064 (2006)]. This unifying theory is constructed from first principles, beginning with an analysis of the Eulerian and Lagrangian descriptions of fluids, separating the ideas of fluid and magnetic-flux tubes and removing the complication of the magnetic vector potential’s free gauge from the concept of helicity. The analysis prepares for the construction of a conserved Eulerian helicity, without that gauge complication, to describe a 3D anchored flux in an upright cylindrical domain, this helicity called absolute to distin...

Magnetostatic atmospheres possessing identical invariants of ideal magnetohydrodynamics

A physical analysis is presented of two distinct families of two-dimensional (2D) analytical solutions for isothermal periodic magnetostatic atmospheres in uniform gravity, one arrived at by Dungey and the other arrived at by Low and Manchester. It is demonstrated that particular members of the two families of 2D equilibria may be generated from the same planar state by plasma displacements which move the system through continuous sequences of equilibria while ensuring flux freezing. The two families of solutions both possess undulating magnetic field lines but geometrically different flux surfaces. The Dungey solutions can be created from a planar state by an undulating deformation whose spatial variation is along the field lines. By contrast, the 2D plane of variation of the Low-Manchester solutions lies at an angle to the field lines of the planar state. As a result, a mixed mode of undulating, interchange and shearing displacements must be made to the planar state to produce the more complex 2D state....

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 integ...