Magnetic helicity
This article's tone or style may not reflect the encyclopedic tone used on Wikipedia. (December 2022) |
In
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.
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
Magnetic helicity has units of magnetic flux squared: Wb2 (webers squared) in SI units and Mx2 (maxwells squared) in Gaussian Units.[14]
Current helicity
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
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
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:
In
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:
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.
See also
References
- ISBN 978-1-118-66447-6. Retrieved 2021-01-18.
- ^ S2CID 121478573.
- ISBN 978-94-010-6603-7. Retrieved 2020-10-08.
- ISSN 0004-637X.
- ^ S2CID 17015601.
- ^ Brandenburg, A. (2009). "Hydromagnetic Dynamo Theory". . rev #73469.
- S2CID 16261037.
- ISSN 0004-637X.
- PMID 10970583.
- ^ S2CID 45460069.
- ^ a b c d
Berger, M.A. (1999). "Introduction to magnetic helicity". S2CID 250734282.
- ISBN 9781118664476.
- S2CID 119518712.
- ^ Huba, J.D. (2013). NRL Plasma Formulary (PDF). Washington, D.C.: Beam Physics Branch Plasma Physics Division Naval Research Laboratory. Archived from the original (PDF) on 2019-06-30.
- ^ a b
Subramanian, K.; Brandenburg, A. (2006). "Magnetic helicity density and its flux in weakly inhomogeneous turbulence". S2CID 323935.
- S2CID 53126623.
- PMID 16590226.
- ISSN 0034-6861.
- S2CID 3746018.
- ISSN 0031-9007.
- ISSN 1070-664X.
- S2CID 8309837.
- ISSN 0004-637X.
- ISSN 0004-637X.
External links
- A. A. Pevtsov's Helicity Page
- Mitch Berger's Publications Page