Magnetic helicity
When resistivity is low, magnetic helicity is conserved over longer timescales, to a good approximation.
Magnetic helicity dynamics are particularly important in analyzing solar flares and coronal mass ejections.
[3] Magnetic helicity is relevant in the dynamics of the solar wind.
[4] Its conservation is significant in dynamo processes, and it also plays a role in fusion research, such as reversed field pinch experiments.
[10] This process can be referred to as an inverse transfer in Fourier space.
This property of increasing the scale of structures makes magnetic helicity special in three dimensions, as other three-dimensional flows in ordinary fluid mechanics are the opposite, being turbulent and having the tendency to "destroy" structure, in the sense that large-scale vortices break up into smaller ones, until dissipating through viscous effects into heat.
This is visible in the dynamics of the heliospheric current sheet,[11] a large magnetic structure in the Solar System.
is the standard measure of the extent to which the field lines wrap and coil around one another.
As a consequence, the magnetic helicity of a physical system cannot be measured directly.
However, for perfectly conducting boundaries or periodic systems without a net magnetic flux, the magnetic helicity contained in the whole domain is gauge invariant,[15] that is, independent of the gauge choice.
A gauge-invariant relative helicity has been defined for volumes with non-zero magnetic flux on their boundary surfaces.
This behavior is very similar to that found concerning magnetic field lines.
Magnetic helicity is a continuous generalization of the topological concept of linking number to the differential quantities required to describe the magnetic field.
[11] Where linking numbers describe how many times curves are interlinked, magnetic helicity describes how many magnetic field lines are interlinked.
Topological transformations can change twist and writhe numbers, but conserve their sum.
As with many quantities in electromagnetism, magnetic helicity is closely related to fluid mechanical helicity, the corresponding quantity for fluid flow lines, and their dynamics are interlinked.
[10][16] In the late 1950s, Lodewijk Woltjer and Walter M. Elsässer discovered independently the ideal invariance of magnetic helicity,[17][18] that is, its conservation when resistivity is zero.
Woltjer's proof, valid for a closed system, is repeated in the following: In ideal magnetohydrodynamics, the time evolution of a magnetic field and magnetic vector potential can be expressed using the induction equation as respectively, where
The second term also vanishes because motions inside the closed system cannot affect the vector potential outside, so that at the boundary surface
Magnetic helicity remains conserved in a good approximation even with a small but finite resistivity, in which case magnetic reconnection dissipates energy.
This can be called an inverse transfer in Fourier space, as opposed to the (direct) energy cascade in three-dimensional turbulent hydrodynamical flows.
The possibility of such an inverse transfer was first proposed by Uriel Frisch and collaborators[10] and has been verified through many numerical experiments.
An argument for this inverse transfer taken from[10] is repeated here, which is based on the so-called "realizability condition" on the magnetic helicity Fourier spectrum
The "realizability condition" corresponds to an application of Cauchy-Schwarz inequality, which yields:
The factor 2 is not present in the paper[10] since the magnetic helicity is defined there alternatively as
We assume a fully helical magnetic field, which means that it saturates the realizability condition:
Assuming that all the energy and magnetic helicity transfers are done to another wavevector
, the conservation of magnetic helicity on the one hand and of the total energy
(the sum of magnetic and kinetic energy) on the other hand gives:
, the magnetic helicity is transferred to a smaller wavevector, which means to larger scales.
