# Magnetic helicity

The helicity of a smooth vector field defined on a domain in 3D space is the standard measure of the extent to which the field lines wrap and coil around one another.[1] As to magnetic helicity, this "vector field" is magnetic field. 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).

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,

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

where

${\displaystyle H}$ is the helicity of the entire magnetic field
${\displaystyle {\mathbf {B} }}$ is the magnetic field strength
${\displaystyle \mathbf {A} }$ is the vector potential of ${\displaystyle {\mathbf {B} }}$ and ${\displaystyle {\mathbf {B} }=\nabla \times {\mathbf {A} }}$
${\displaystyle d^{3}{\mathbf {r} }}$ is the differential volume element for the volume integral

Magnetic helicity has units of Wb2 (webers squared) in SI units and Mx2 (maxwells squared) in Gaussian Units.[2]

It is a conserved quantity in ideal magnetohydrodynamics (zero resistivity), and still remains conserved in a good approximation even with a small but finite resistivity, in which case magnetic reconnection dissipates energy. The concept is useful in solar dynamics and in dynamo theory.[3] [4]

Magnetic helicity is a gauge-dependent quantity, because ${\displaystyle \mathbf {A} }$ can be redefined by adding a gradient to it (gauge transformation). However, for perfectly conducting boundaries or periodic systems without a net magnetic flux, the magnetic helicity is gauge invariant. A gauge-invariant relative helicity has been defined for volumes with non-zero magnetic flux on their boundary surfaces.[5] 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.[6]