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Progress in the fields of space physics, astronomy and astrophysics over the last decade increasingly reveals the significance of magnetic fields in these areas. The electromagnetic interaction is, together with gravitation, the only interaction known which is long range, and thus capable of creating large scale field structures. These fields are induced by the motion of ionized matter - plasma - which is present in various forms nearly everywhere in the universe. Its properties cover a wide range, from the very hot and dense plasmas of stars to the extremely diluted plasmas of the interstellar medium, which is only partially ionized.

On large astrophysical scales, plasmas and their magnetic fields are adequately described by a fluid theory called magnetohydrodynamics (MHD). In this theory (which is closely related to hydrodynamics) the plasma is a highly conducting fluid, the flow of which can induce magnetic fields. These magnetic fields act in turn on the fluid flow via Lorentz forces. This interaction of plasma and magnetic field can create an astonishing variety of structures, which often exhibit linked and knotted forms of magnetic flux. In these complex structures, huge amounts of energy can be stored in the magnetic field. It is, however, a typical property of astrophysical plasmas that the dynamics of magnetic fields alternates between an ideal motion, where all forms of knottedness and linkage of the field are conserved (topology conservation), and a kind of disruption of the magnetic structure, the so called magnetic reconnection. In the latter the magnetic structure breaks up and re-connects, a process often accompanied by explosive eruptions where enormous amounts of energy are set free.

still frame from movie An X17 flare as observed by the TRACE satellite on 28th October 2003. Click here for movie: Quicktime (6.7M).

Such dynamic explosive events are frequently observed in the atmosphere of the Sun, and a wealth of impressive observations has been recently made by spacecraft like Yohkoh, SOHO and TRACE. In an exciting time for the field of Solar Physics, two new observing spacecraft - STEREO and Hinode - have just been launched (in late 2006), which promise to provide observations in even greater detail of our enigmatic local star. Developing an understanding of the processes which go on on the Sun is key not only because all life on Earth depends ultimately on the Sun's energy, but also because it affords us a unique laboratory in which to investigate the workings of all stars (and other astrophysical objects such as accretion disks), as it is the only such object whose details we presently have any hope of resolving.

Strip showing movie frames A view of the outer solar atmosphere (the solar disk is blocked by an "occluding disk", and its size is marked by the white circle), showing a series of coronal mass ejections. Taken by the LASCO instrument on SOHO, May 1-31, 1998. Click here for movie: Mpeg (3.9M) or Quicktime (1.1M).

still frame from movie Movie made by the new Hinode satellite of a sunspot on the Sun's surface. Click here for movie: Mpeg (8.5M). (From the Hinode webpage.)

Processes such as magnetic reconnection are also of key importance in the immediate surroundings of the Earth. For instance, reconnection occurs at the magnetopause, where the solar wind encounters the magnetic field of the Earth, and also in the magnetotail, the wake of the earth's magnetic field in the solar wind. In both cases the electric fields induced by reconnection accelerate particles which in turn produce phenomena such as the northern and southern lights (Aurora) in the polar regions, as well as so called geomagnetic storms (see the movie below). The storage and release of magnetic energy in complex field structures is also important for dynamo theory, which investigates the origin and dynamics of magnetic fields in planets and stars. Furthermore, there is increasing strong evidence that complex magnetic fields play an important role in the dynamics and self-organization of matter in many distant astronomical objects such as pulsars, galaxies, and protogalactic clouds.

Strip showing movie frames Artist's impression of a coronal mass ejection (CME) from the Sun, impacting on the Earth's magnetosphere. Click here for movies: Mpeg (1.9M) or Quicktime (2.8M).(From the SOHO webpage, where larger versions may be found.)

An initial highly dynamic phase in the research and modelling of magnetohydrodynamical plasmas was very successful with comparatively simple models. Now, however, more complex problems are encountered. In particular, observations show an immense complexity in the structure of magnetic fields, which cannot be described by simple (often two-dimensional) models anymore. It is therefore of great importance to work towards an understanding of the complex three-dimensional structure of the plasma environment around us. In this endeavour, it is crucial to employ both analytical techniques in idealised configurations, and computational simulations, with each approach guiding the other.

As a result of the newly apparent complexity of astrophysical plasmas, there is an urgent need for a systematic framework, which determines the crucial quantities with respect to which a certain situation should be analyzed. Such a framework could be provided with the help of an interesting analogy between the structure of magnetic fields and the mathematical theory of knots. In this fast-growing area of topology, so called invariants are known which describe the linkage or knottedness of isolated lines, and thus represent a measure of complexity. Corresponding measures for (divergence-free) vector fields, or their field lines respectively, would be of the greatest interest to characterize the entangled structure of magnetic fields and for instance calculate the energy stored in such configurations. Such a conversion of measures from single lines to vector fields has indeed been successful for simple cases, and there are many hints that methods of differential geometry and topology may be of use for the conversion of higher invariants as well. In a completely new approach, this method could be generalized to the electromagnetic field tensor, i.e. the physically more precise description, which includes the electric field. This is suggested by a certain analogy in the underlying mathematical structure and represents an extraordinary, most interesting, and new approach to the understanding of electromagnetic fields. On the other hand it is very important not only to characterize these structures but also to understand their dynamics. Here, magnetic reconnection is closely analogous to the splitting of knots, which gives us confidence that the global dynamics of magnetic and electromagnetic fields can be characterized with the help of such topological quantities as well.