Recent Magnetic Helicity Papers and Reviews

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 as the 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.

BASIC PROPERTIES OF MUTUAL MAGNETIC HELICITY

Abstract

We derive the magnetic helicity for configurations formed by flux tubes contained fully or only partially in the spatial domain considered (called closed and open configurations, respectively). In both cases, magnetic helicity is computed as the sum of mutual helicity over all possible pairs of magnetic flux tubes weighted by their magnetic fluxes. We emphasize that these mutual helicities have properties which are not those of mutual inductances in classical circuit theory. For closed configurations, the mutual helicity of two closed flux tubes is their relative winding around each other (known as the Gauss linkage number). For open configurations, the magnetic helicity is derived directly from the geometry of the interlaced flux tubes so it can be computed without reference to a ground state (such as a potential field). We derive the explicit expression in the case of a planar and spherical boundary. The magnetic helicity has two parts. The first one is given only by the relative positions of the flux tubes on the boundary. It is the only part if all flux tubes are arch-shaped. The second part counts the integer number of turns each pair of flux tubes wind about each other. This provides a general method to compute the magnetic helicity with discrete or continuous distributions of magnetic field. The method sets closed and open configurations on an equal level within the same theoretical framework.

INTRODUCTION TO MAGNETIC HELICITY

Abstract

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 MHD 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 Poynting's 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.

Figure 4.

In the interior of the sun, the equator rotates faster than the poles. Differential rotation provides a strong source of helicity injection.

RATE OF HELICITY PRODUCTION BY SOLAR ROTATION

Abstract

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 of 10⁴⁵ Mx² in coronal mass ejections each cycle. Both the a effect (arising from helical turbulence) and the W effect (arising from differential rotation) should contribute to the hemispheric chirality. We show that the W 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 x 10⁴⁷ Mx² . For comparison, we estimate the a 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.

Figure 3. Helicity Transfer into the sun from magnetogram and solar rotation data.

Helicity transfer into the southern interior (predominantly positive curve)and northern interior (predominantly negative curve). The units are 10⁴⁰ Mx² /day. The differences in magnitude between the two curves go up to 5 x 10⁴² Mx² /day.

MAGNETIC HELICITY IN SPACE PHYSICS

Abstract

The origins of magnetic helicity go back to Gauss in the early 19th century. This chapter traces the early history of magnetic helicity in the 1950s to the 1980s. We discuss the relation to field topology and to minimum energy Taylor states. The approximate conservation of helicity during reconnection is outlined. Also, we discuss how helicity is defined in open volumes and how helicity can be transferred across boundaries.

Figure 1. Two tubes with linking number -3.

MAGNETIC HELICITY AND FILAMENTS

Abstract

Some of the most dramatic images of prominences show helical structure. Helical structure, as well as other structural features such as twist, shear, and linking, can be quantified using helicity integrals. This paper reviews how the calculation of helicity may be applied to prominence models. Recent observations indicate that the sign of helicity in an active region depends on which hemisphere the region is in. The source of this asymmetry is an important problem in solar physics. The total helicity of each hemisphere obeys a Poynting-like theorem which describes how helicity is transferred across the photosphere and the equator. Estimating this helicity transfer may help us in understanding the helicity balance of the sun.

Figure 4. Mutual helicity.

For the tubes on the left, the mutual helicity is H_AB = (a - b ) F _A F _B / p . On the right, H_AB = (g + d ) F _A F _B / p .

MAGNETIC HELICITY IN A PERIODIC DOMAIN

Abstract

Sometimes, ideally conserved quantities in turbulent flows can cascade to larger length scales as well as smaller length scales. Any quantity which approaches the largest length scale available will be strongly affected by the boundary conditions. Often turbulence is modeled in a domain with periodic boundary conditions. In three dimensions, such a domain is a compact manifold without boundary called a 3-torus. A 3-torus is multiply connected, unlike many other domains of interest (e.g., the interior of a sphere). This difference in topology affects the structure of the fields contained inside. For example, there may be streamlines or magnetic field lines which do not close upon themselves but which stretch across the entire domain. Such periodic lines do not exist in nonperiodic geometries. This paper asks whether the presence of periodic lines can change the dynamics of the fluid. Recently, Stribling et al. [1994] examined MHD turbulence with a mean magnetic field and periodic boundary conditions. They noted that the magnetic helicity of the fluctuating field decreases in time. In a closed domain such behavior is readily explained. In particular, the helicity of the fluctuating field only measures the self-linking of the fluctuating field. The conserved total helicity, however, also measures mutual linking between the fluctuating field and the mean field. The decrease of the fluctuating helicity indicates a transference of helicity to this linking. Unfortunately in a periodic domain this explanation does not work. We show that in general, the total helicity is not even definable, much less conserved. The helicity exists only if all toroidal field lines close upon themselves without stretching across the domain. Even in this case helicity has unusual properties. For example, a simple sequence of reconnections can convert a flux tube with right handed twist into a tube with left handed twist. This flips the sign of the helicity. Taylor relaxation is not valid when such processes occur. We present a new helicity-like quantity which can be defined even when the usual helicity does not exist. This quantity measures how much the toroidal field links both interior and exterior mean flux. It is an ideal invariant when the fluid velocity has no z component.

Figure 2. How to Turn a Flux Rope Inside Out

[The diagram shows a single transverse field line in a twisted flux tube. The domain is periodic. From Figure 2a to 2b part of the field moves through the left nonboundary x = 0 and then is pinched. From Figure 2b to 2c reconnection creates periodic field lines. From Figure 2c to 2d a further distortion through the top nonboundary occurs. Next to Figure 2e: reconnection restores closed field lines. From Figure 2e to 2f, to display the final state clearly, the flux tube is translated diagonally to the center of the figure].