Introduction
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 the gravitation, the only long range
interaction known and thus capable of creating large scale field
structures. These fields are induced by the motion of ionized
matter, the plasma, which is present in various forms nearly
everywhere in the universe. The properties 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 the large
astrophysical scales, the plasmas and their magnetic fields are
adequately described by a fluid theory called magnetohydrodynamics.
In this theory, which is closely related to hydrodynamics, the
plasma is a highly conducting fluid, the flow of which can induce
magnetic fields which in turn are acting 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 of the fields huge amounts of magnetic energy can be
stored. It is, however, a typical property of astrophysical
plasmas, that the dynamics of magnetic fields is alternating
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. Such events are
frequently observed on the surface of the sun and a wealth of new
and impressive observations has been recently made by spacecraft
like Yohkoh and SOHO.
Processes such as reconnection, however, are also important in the
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 lights ( Aurora ) in the polar
regions and so called geomagnetic storms. The storage and release
of magnetic energy in complex field structures is also important
for the 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. After a first very dynamic phase in the
research and modeling of magnetohydrodynamical plasmas, which was
very successful with comparatively simple models, now more complex
problems are encountered. Especially, observations show an immense
complexity in the structure of magnetic fields, which cannot be
described by simple models anymore. There is therefore an urgent
need for an systematic framework, which determines the crucial
quantities with respect to which a certain situation should be
analyzed.
 An
X-ray image of the Sun reveals
the complex structure of magnetic
fields because the plasma is closely
tied to magnetic field lines.
A larger version of the image showing
more details.
(Image made by the Soft-X-Ray telescope of the
YOHKOH satellite 1992)
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 part 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 greatest
interest to characterize the entangled structure of magnetic fields
and for instance calculate the energy stored in this configuration.
Such a conversion of measures from single lines to vector fields
was indeed successful for simple cases, and there are many hints
that methods of differential geometry and topology may help for the
conversion of higher invariants as well. In a completely new
approach this methods 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 in
close analogy to splitting of knots, which makes us confident that
the global dynamics of magnetic and electromagnetic fields can be
characterized with the help of such topological quantities as
well.
Links to popular descriptions of
this and neighboring fields of research.
Projects
- Development of measures of complexity for magnetic and moreover
electromagnetic fields
- Investigation of the relation of these measures to invariants
of knots
- Investigation of topological properties of magnetic and
electromagnetic fields and their dynamics, especially with respect
to critical phenomena like magnetic reconnection.
- Representation and visualization of electromagnetic fields
- Application of the theory to astrophysical plasmas, especially
to the problem of heating the solar corona.
Results
The following computer presentations show some of our results. For a more detailed documentation please see our list of Publications .
- Topological methods in Fluid dynamics by G. Hornig (PDF)
- Processes at magnetic null points by V.S. Titov (PDF)
- Magnetic Connectivity by V.S. Titov (PDF)
- Topological invariants of higher order by C. Mayer (PDF)
- Linking in Four Dimensions by H.v. Bodecker (PDF)
- New solutions for reconnective magnetic annihilation with curvilinear geometry by E. Tassi (PDF)
- Non-ideal MHD Properties of Magnetic Flux Tubes in the Solar Photosphere by J. Kleimann (PDF)
- Magnetic Helicity under Reconnection by G. Hornig (PDF)
- Maximum information from magnetic helicity by G. Hornig (PDF)
Diploma and PhD Thesis
- Magnetische Kopplung zwischen Chromosphaere und Konvektionszone der Sonne Diploma Thesis by J. Kleimann (PDF 0.9MB, German)
- Zur Interpretation der Novikov-Invarianten in der Fluiddynamik Diploma Thesis by H. v. Bodecker (PDF 1MB, German)
- Topological link invariants of magnetic fields PhD-Thesis by Christoph Mayer (PDF 7MB)
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