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Topological Magnetohydrodynamics and Astrophysics

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Encyclopedia of Complexity and Systems Science

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Definition of the Subject

Many important processes in astrophysics involve magnetic fields, from solar flares and eruptions (Fig. 1) to the formation of galactic jets. Magnetic fields are often highly structured by their field lines; a particularly striking example of this can be seen in the fibrous appearance of clouds in the solar atmosphere (Fig. 10). These linear structures can be twisted, kinked, and interlinked. Several geometrical quantities exist which measure the amounts of such structural features. When some of these geometrical quantities are left unchanged by some set of deformations of the field, they are called topological invariants. Astrophysicists have paid particular attention to the magnetic helicity integral, because it is approximately conserved in highly conducting plasmas. In our own solar system, magnetic helicity is transported from the interior of the sun, through the solar surface and atmosphere and into...

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Abbreviations

Corona:

The atmosphere of the sun or a star. The solar corona is generally much hotter (at temperatures up to 2 million K) than the photosphere (solar surface), due to magnetic heating.

Crossing number:

The projection of a three‐dimensional curve onto a plane will exhibit a certain number of crossings (where the projected curve passes over itself). This is the crossing number of the curve; in general the crossing number depends on the orientation of the plane as well as the original three‐dimensional curve. The average crossing number averages over all possible planar projections. This removes the dependence on orientation, but there is still a dependence on the geometry of the curve. Given arbitrary distortions of a curve, without letting the curve pass through itself, there will be a minimum number of crossings, the minimum crossing number. For closed curves (knots), and for collections of curves (links) the minimum crossing number provides a measure of topological complexity. Crossing numbers can also be defined for vector fields in terms of the crossings seen between all pairs of field lines.

Force-free field:

A force-free magnetic field does not impart any magnetic forces. In the absence of other forces, such as pressure or gravity, a magnetized fluid in equilibrium will possess a force-free field.

Helicity integral:

The helicity integral \( { H(\mathbf{V}, \mathbf{W}) } \) of two vector fields V and W measures the net linking of the field lines of V with those of W. Common examples in a fluid with magnetic field B and vorticity \( { \boldsymbol{\omega} } \) include the kinetic helicity \( { H(\boldsymbol{\omega}, \boldsymbol{\omega}) } \), which measures the self linking of the vorticity ω; the magnetic helicity \( { H(\mathbf{B}, \mathbf{B}) } \), which measures the self-linking of magnetic field B with itself; and the cross‐helicity \( { H(\mathbf{B}, \boldsymbol{\omega}) } \), the linking between the magnetic field with the vortex field.

Linking number:

The linking number L of two oriented closed curves measures how much they intertwine about each other. If the two curves are projected onto a plane, L equals one half the number of (signed) crossings. The linking number is invariant to arbitrary deformations of the curves, as long as the two curves do not pass through each other.

Magnetohydrodynamics (MHD):

Magnetohydrodynamics studies the evolution of a fluid carrying electrical currents and subjected to magnetic forces. Typical examples are liquid metals and ionized plasmas. In the latter case, MHD neglects small-scale effects, such as those arising from fluctuations about charge neutrality or from details of particle trajectories. Ideal MHD assumes a perfectly conducting plasma. The magnetic lines of force in ideal MHD are convected by the fluid motions without slipping through the fluid, breaking, or passing through each other.

Photosphere:

The surface of the sun. Near the photosphere there is a steep change in pressure, density, and optical depth. Also the temperature reaches a minimum (on average 5 800 K).

Reconnection:

Reconnection occurs when field lines change their connectivity. In a simple reconnection event two bundles of field lines meet in a small region. The field lines are cut in this region, allowing the two pieces of a field line from the first bundle to connect to two corresponding pieces from the second bundle. Reconnection events in nature may involve several simple events.

Twist number:

The twist number \( { \mathcal{T}_\mathrm{w} } \) applies to ribbons and tubes. For ribbons, it measures the extent to which one side of the ribbon rotates about the other. For magnetic flux tubes, it measures how much field lines within the tube rotate about the axis of the tube.

Winding number:

Consider two curves extending between parallel planes. The winding number \( { \mathrm{w} } \) measures the angle (divided by \( { 2 \pi } \)) through which the two curves rotate about each other.

Writhe:

The writhe number \( { \mathcal{W}_\mathrm{r} } \) is a property of a single curve. It measures three‐dimensional geometrical structure, and can change if the curve is distorted. For a ribbon, the writhe of the axis of the ribbon plus the twist of an edge of the ribbon about the axis equals the linking number between the edge and the axis (Călugăreanu theorem).

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Author information

Authors and Affiliations

  1. Mathematics, University of Exeter, Devon, UK

    Mitchell A. Berger

  2. Mullard Space Science Laboratory, UCL, London, UK

    Mitchell A. Berger

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  1. Mitchell A. Berger
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Berger, M.A. (2009). Topological Magnetohydrodynamics and Astrophysics. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_557

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