Mitchell A. Berger

The topological properties of magnetic helicity

The relation of magnetic helicity to the topological structure of field lines is discussed. If space is divided into a collection of flux tubes, magnetic helicity arises from internal structure within a flux tube, such as twist and kinking, and external relations between flux tubes, i.e. linking and knotting. The concepts of twist number and writhing number are introduced from the mathematical-biology literature to describe the contributions to helicity from twist about the axis of a flux tube, and from the structure of the axes themselves. There exists no absolute measure of the helicity within a subvolume of space if that subvolume is not bounded by a magnetic surface. However, a topologically meaningful and gauge-invariant relative measure of helicity for such volumes is presented here. The time derivative of this relative measure is calculated, which leads to an expression for the flow of topological structure across boundaries.

Rigorous new limits on magnetic helicity dissipation in the solar corona

Abstract The Cauchy-Schwarz inequality is employed to find geometry-independent limits on the magnetic helicity dissipation rate in a resistive plasma. These limits only depend upon the total energy of the plasma, the energy dissipation rate, and a mean diffusion coefficient. For plasmas isolated from external energy sources, limits can also be set on the minimum time necessary to dissipate a net amount of helicity ΔH. As evaluated in the context of a solar coronal loop, these limits strongly suggest that helicity decay occurs on a diffusion timescale which is far too great to be relevant to most coronal processes. Furthermore, rapid reconnection is likely to approximately conserve magnetic helicity. The dilliculties involved in determining the free energy residing in a magnetic structure (given the constraint of magnetic helicity conservation) are discussed.

MAGNETIC ENERGY AND HELICITY FLUXES AT THE PHOTOSPHERIC LEVEL

The source of coronal magnetic energy and helicity lies below the surface of the Sun, probably in the convective zone dynamo. Measurements of magnetic and velocity fields can capture the fluxes of both magnetic energy and helicity crossing the photosphere. We point out the ambiguities which can occur when observations are used to compute these fluxes. In particular, we show that these fluxes should be computed only from the horizontal motions deduced by tracking the photospheric cut of magnetic flux tubes. These horizontal motions include the effect of both the emergence and the shearing motions whatever the magnetic configuration complexity is. We finally analyze the observational difficulties involved in deriving such fluxes, in particular the limitations of the correlation tracking methods.

Rate of helicity production by solar rotation

In recent years, solar observers have discovered a striking pattern in the distribution of coronal magnetic structures: northern hemisphere structures tend to have negative magnetic helicity, while structures in the south tend to have positive magnetic helicity. This hemispheric dependence extends from photospheric observations to in situ measurements of magnetic clouds in the solar wind. Understanding the source of the hemispheric sign dependence, as well as its implications for solar and space physics has become known as the solar chirality problem. Rotation of open fields creates the Parker spiral which carries outward 1047 Mx2 of magnetic helicity (in each hemisphere) during a solar cycle. In addition, rough estimates suggest that each hemisphere sheds on the order 1045 Mx2 in coronal mass ejections each cycle. Both the α effect (arising from helical turbulence) and the Ω effect (arising from differential rotation) should contribute to the hemispheric chirality. We show that the Ω effect contribution can be captured in a surface integral, even though the helicity itself is stored deep in the convection zone. We then evaluate this surface integral using solar magnetogram data and differential rotation curves. Throughout the 22 year cycle studied (1976–1998) the helicity production in the interior by differential rotation had the correct sign compared to observations of coronal structures - negative in the north and positive in the south. The net helicity flow into each hemisphere over this cycle was approximately 4 × 1046 Mx2. For comparison, we estimate the α effect contribution; this may well be as high or higher than the differential rotation contribution. The subsurface helicity can be transported to the corona with buoyant rising flux tubes. Evidently only a small fraction of the subsurface helicity escapes to the surface to supply coronal mass ejections.

Introduction to magnetic helicity

This paper reviews several aspects of magnetic helicity, including its history from Gauss to the present, its relation to field structure, its role in Taylor relaxation, and how it is defined for sub-volumes of space. Also, its importance in solar physics will be discussed. Magnetic helicity quantifies various aspects of magnetic field structure. Examples of fields possessing helicity include twisted, kinked, knotted, or linked magnetic flux tubes, sheared layers of magnetic flux, and force-free fields. Helicity thus allows us to compare models of fields in different geometries, avoiding the use of parameters specific to one model. Magnetic helicity is conserved in ideal magnetohydrodynamics and approximately conserved during reconnection. Often, physical systems are described in terms of interacting parts: for example one might separate the solar magnetic field into an interior field and an atmospheric (coronal) field. We can obtain insight into how the parts of a magnetic system interact by describing how magnetic helicity is transferred from one part to another. This transfer is governed by an equation similar to Poyntings theorem for the transfer of energy through boundaries. In a confined volume, widespread reconnection may reduce the magnetic energy of a field while approximately conserving its magnetic helicity. As a result, the field relaxes to a minimum energy state, often called the Taylor state, where the current is parallel to the field. Such relaxation processes are important to both fusion and astrophysical plasmas. Recent observations show that structures in the northern hemisphere of the sun have predominantly negative helicity, and structures in the south have predominantly positive helicity. Helicity injection by differential rotation may explain this dependence.

The writhe of open and closed curves

Twist and writhe measure basic geometric properties of a ribbon or tube. While these measures have applications in molecular biology, materials science, fluid mechanics and astrophysics, they are under-utilized because they are often considered difficult to compute. In addition, many applications involve curves with endpoints (open curves); but for these curves the definition of writhe can be ambiguous. This paper provides simple expressions for the writhe of closed curves, and provides a new definition of writhe for open curves. The open curve definition is especially appropriate when the curve is anchored at endpoints on a plane or stretches between two parallel planes. This definition can be especially useful for magnetic flux tubes in the solar atmosphere, and for isotropic rods with ends fixed to a plane.

The Writhe of Helical Structures in the Solar Corona

Context. Helicity is a fundamental property of magnetic fields, conse rved in ideal MHD. In flux rope topology, it consists of twist and writhe helicity. Despite the common occurrence of helical structures in the solar atmosphere, little is known about how their shape relates to the writhe, which fraction of helicity is contain ed in writhe, and how much helicity is exchanged between twist and writhe when they erupt. Aims. Here we perform a quantitative investigation of these questions relevant for coronal flux ropes. Methods. The decomposition of the writhe of a curve into local and nonlocal components greatly facilitates its computation. We use it to study the relation between writhe and projected S shape of helical curves and to measure writhe and twist in numerical simulations of flux rope instabilities. The results are discussed with re gard to filament eruptions and coronal mass ejections (CMEs) . Results. (1) We demonstrate that the relation between writhe and projected S shape is not unique in principle, but that the ambiguity does not affect low-lying structures, thus supporting the established empirical rule which associates stable forward (reverse) S shaped structures low in the corona with positive (negative) helic ity. (2) Kink-unstable erupting flux ropes are found to trans form a far smaller fraction of their twist helicity into writhe helicity than o ften assumed. (3) Confined flux rope eruptions tend to show str onger writhe at low heights than ejective eruptions (CMEs). This argues against suggestions that the writhing facilitates the rise o f the rope through the overlying field. (4) Erupting filaments which are S shaped already before the eruption and keep the sign of their axis writhe (which is expected if field of one chirality dominates the source vol ume of the eruption), must reverse their S shape in the course of the rise. Implications for the occurrence of the helical kink instabi lity in such events are discussed. (5) The writhe of rising lo ops can easily be estimated from the angle of rotation about the direction of ascent, once the apex height exceeds the footpoint separation significantly. Conclusions. Writhe can straightforwardly be computed for numerical data and can often be estimated from observations. It is useful in interpreting S shaped coronal structures and in constrai ning models of eruptions.

A MORPHOLOGICAL STUDY OF HELICAL CORONAL MAGNETIC STRUCTURES

Magnetostatic solutions describing magnetic flux ropes in realistic geometry are used to study solar coronal structures observed to have sigmoidal forms in soft X-rays. These solutions are constructed by embedding a rope of helically symmetric force-free magnetic fields in an external field such that force balance is assured everywhere. The two observed sigmoidal shapes, the S shapes and the mirror-reflected S shapes referred to as Z shapes in this paper, are found in both hemispheres of the solar corona, but observations made over the last two solar cycles suggest that the Z and S shapes occur preferentially in the northern and southern solar hemispheres, respectively. Our study makes an identification of the sigmoidal high-temperature coronal plasmas with heating by the spontaneous formation of current sheets described by the theory of Parker. This process involves a tangential discontinuity developing across a ribbon-like, twisted flux surface through an interaction between a magnetic flux rope and the photosphere, under conditions of high electrical conductivity. In this identification, Z- and S-shaped sigmoids are associated with flux ropes with negative and positive magnetic helicities, respectively. This association is physically consistent with the conclusion, based independently on measurements of prominence magnetic fields, that magnetic flux ropes occur preferentially with negative and positive helicities in the northern and southern solar hemispheres, respectively.

Topological Ideas and Fluid Mechanics

The use of topological ideas in physics and fluid mechanics dates back to the very origin of topology as an independent science. In a brief note in 1833 Karl Gauss, while lamenting the lack of progress in the “geometry of position” (or Geometria Situs, as topology was then known I, gives a remarkable example of the relationship between topology and measurable physical quantities such as electric currents. He considers two inseparably linked circuits, each of them a copper wire with ends joined, and flowing electric current. Without comment he puts forward a formula that gives the relationship between the magnetic action induced by the currents and a pure number that depends only on the type of link, and not on the geometry. This number is a topological invariant now known as the linking number. The formula, as well as the very first studies in topology done by Johann Benedict Listing in 1847, became known to Kelvin (then William Thomson), James Clerk Maxwell and Peter Guthrie Tait in Britain.

SELF-ORGANIZED BRAIDING AND THE STRUCTURE OF CORONAL LOOPS

The Parker model for heating of the solar corona involves reconnection of braided magnetic flux elements. Much of this braiding is thought to occur at as yet unresolved scales, for example, braiding of threads within an extreme-ultraviolet or X-ray loop. However, some braiding may be still visible at scales accessible to TRACE or Hinode. We suggest that attempts to estimate the amount of braiding at these scales must take into account the degree of coherence of the braid structure. In this paper, we examine the effect of reconnection on the structure of a braided magnetic field. We demonstrate that simple models of braided magnetic fields which balance the input of topological structure with reconnection evolve to a self-organized critical state. An initially random braid can become highly ordered, with coherence lengths obeying power-law distributions. The energy released during reconnection also obeys a power law. Our model gives more frequent (but smaller) energy releases nearer to the ends of a coronal loop.