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Magnetic helicity

This page is about helicity in magnetic fields. For fluid mechanical helicity, see helicity

In plasma physics, magnetic helicity is the extent to which a magnetic field "wraps around itself". It is a generalization of the topological concept of linking number to the differential quantities required to describe the magnetic field. As with many quantities in electromagnetism, magnetic helicity (which describes magnetic field lines) is closely related to fluid mechanical helicity (which describes fluid flow lines).

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If magnetic field lines follow the strands of a twisted rope, this configuration would have nonzero magnetic helicity; left handed ropes would have negative values and right handed ropes would have positive values.

Formally,

$H=\int {\mathbf A}\cdot{\mathbf B}\,d^3{\mathbf r}$

where

${\mathbf B}$ is the magnetic field strength

${\mathbf B}=\nabla\times{\mathbf A}$ ;

$\mathbf A$ is the vector potential of ${\mathbf B}$

Magnetic helicity is a conserved quantity. It is conserved in electromagnetic fields, even when magnetic reconnection dissipates energy. The concept is useful in solar dynamics and in hydromagnetic dynamo theory.

Magnetic helicity is a gauge-dependent quantity, because A can be redefined by adding a gradient to it (gauge transformation). However, for periodic or perfectly conducting boundaries the magnetic helicity is gauge invariant. If the magnetic field is turbulent and weakly inhomogeneous a magnetic helicity density and its associated flux can be defined in terms of the density of field line linkages.