Magnetic helicity is a significant concept in the analysis of astrophysical systems, where the resistivity may be very low, so that magnetic helicity is conserved to a good approximation. In practice, magnetic helicity dynamics are important in analyzing
solar flares and
coronal mass ejections.[3] Magnetic helicity is present in 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.[5][6][7][8][9]
When a magnetic field contains magnetic helicity, it tends to form large-scale structures from small-scale ones.[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
vorticesbreak up into smaller ones, until dissipating through
viscous effects into heat. Through a parallel but inverted process, the opposite happens for magnetic vortices, where small helical structures with non-zero magnetic helicity combine and form large-scale magnetic fields. This is visible in the dynamics of the
heliospheric current sheet,[11] a large magnetic structure in the
Solar System.
Mathematical definition
Generally, the helicity of a smooth
vector field confined to a volume is the standard measure of the extent to which the field lines wrap and coil around one another.[12][2] It is defined as the
volume integral over of the scalar product of and its
curl, :
Magnetic helicity
Magnetic helicity is the helicity of a
magnetic vector potential where is the associated
magnetic field confined to a volume . Magnetic helicity can then be expressed as[5]
Since the magnetic vector potential is not
gauge invariant, the magnetic helicity is also not gauge invariant in general. As a consequence, the magnetic helicity of a physical system cannot be measured directly. In certain conditions and under certain assumptions, one can however measure the current helicity of a system and from it, when further conditions are fulfilled and under further assumptions, deduce the magnetic helicity.[13]
The current helicity, or helicity of the magnetic field confined to a volume , can be expressed as
where is the
current density.[15] Unlike magnetic helicity, current helicity is not an ideal invariant (it is not conserved even when the
electrical resistivity is zero).
Gauge considerations
Magnetic helicity is a gauge-dependent quantity, because can be redefined by adding a gradient to it (
gauge choosing). 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.[11]
Topological interpretation
The name "helicity" is because the trajectory of a fluid particle in a fluid with velocity and
vorticity forms a
helix in regions where the
kinetic helicity. When , the resulting helix is right-handed and when it is left-handed. This behavior is very similar to that found concerning magnetic field lines.
Regions where magnetic helicity is not zero can also contain other sorts of magnetic structures, such as helical 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.[5]
Magnetic helicity is proportional to the sum of the
topological quantities
twist and
writhe for magnetic field lines. The twist is the rotation of the flux tube around its axis, and
writhe is the rotation of the flux tube axis itself. Topological transformations can change twist and writhe numbers, but conserve their sum. As
magnetic flux tubes (collections of closed magnetic field line loops) tend to resist crossing each other in magnetohydrodynamic fluids, magnetic helicity is very well-conserved.
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]
Properties
Ideal quadratic invariance
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:
respectively, where is a
scalar potential given by the
gauge condition (see
§ Gauge considerations). Choosing the gauge so that the scalar potential vanishes, , the time evolution of magnetic helicity in a volume is given by:
The
dot product in the integrand of the first term is zero since is orthogonal to the cross product , and the second term can be integrated by parts to give
where the second term is a surface integral over the boundary surface of the closed system. The dot product in the integrand of the first term is zero because is orthogonal to The second term also vanishes because motions inside the closed system cannot affect the vector potential outside, so that at the boundary surface since the magnetic vector potential is a continuous function. Therefore,
and magnetic helicity is ideally conserved. In all situations where magnetic helicity is gauge invariant, magnetic helicity is ideally conserved without the need for the specific gauge choice
Magnetic helicity remains conserved in a good approximation even with a small but finite resistivity, in which case
magnetic reconnection dissipates
energy.[11][5]
Inverse transfer
Small-scale helical structures tend to form larger and larger magnetic structures. 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.[19][20][21][22][23][24] As a consequence, the presence of magnetic helicity is a possibility to explain the existence and sustainment of large-scale magnetic structures in the Universe.
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 (where is the Fourier coefficient at the
wavevector of the magnetic field , and similarly for , the star denoting the
complex conjugate). The "realizability condition" corresponds to an application of
Cauchy-Schwarz inequality, which yields:
with the magnetic energy spectrum. To obtain this inequality, the fact that (with the
solenoidal part of the Fourier transformed magnetic vector potential, orthogonal to the wavevector in Fourier space) has been used, since . The factor 2 is not present in the paper[10] since the magnetic helicity is defined there alternatively as .
One can then imagine an initial situation with no velocity field and a magnetic field only present at two wavevectors and . We assume a fully helical magnetic field, which means that it saturates the realizability condition: and . 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 second equality for energy comes from the fact that we consider an initial state with no kinetic energy. Then we have the necessarily . Indeed, if we would have , then:
which would break the realizability condition. This means that . In particular, for , the magnetic helicity is transferred to a smaller wavevector, which means to larger scales.