Magnetic helicity

In plasma physics, magnetic helicity is a measure of the linkage, twist, and writhe of a magnetic field.<ref>Cantarella, Jason; Deturck, Dennis; Gluck, Herman; Teytel, Mikhail (2013-03-19). "Influence of Geometry and Topology on Helicity". Magnetic Helicity in Space and Laboratory Plasmas. Washington, D. C.: American Geophysical Union. pp. 17–24. doi:10.1029/gm111p0017. ISBN 978-1-118-66447-6. Retrieved 2021-01-18.</ref><ref name=":6">Moffatt, H. K. (1969-01-16). "The degree of knottedness of tangled vortex lines". Journal of Fluid Mechanics. 35 (1): 117–129. Bibcode:1969JFM....35..117M. doi:10.1017/s0022112069000991. ISSN 0022-1120. S2CID 121478573.</ref> In ideal magnetohydrodynamics, magnetic helicity is conserved.<ref name=":5" /><ref name=":4" /> When a magnetic field contains magnetic helicity, it tends to form large-scale structures from small-scale ones.<ref name=":1">Frisch, U.; Pouquet, A.; LÉOrat, J.; Mazure, A. (1975-04-29). "Possibility of an inverse cascade of magnetic helicity in magnetohydrodynamic turbulence". Journal of Fluid Mechanics. 68 (4): 769–778. Bibcode:1975JFM....68..769F. doi:10.1017/s002211207500122x. ISSN 0022-1120. S2CID 45460069.</ref> 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, as three-dimensional turbulent flows in ordinary fluid mechanics tend to "destroy" structure, in the sense that large-scale vortices break 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,<ref name=":0"> Berger, M.A. (1999). "Introduction to magnetic helicity". Plasma Physics and Controlled Fusion. 41 (12B): B167–B175. Bibcode:1999PPCF...41B.167B. doi:10.1088/0741-3335/41/12B/312. S2CID 250734282.</ref> a large magnetic structure in the Solar System.

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 very good approximation. For example, magnetic helicity dynamics are important in analyzing solar flares and coronal mass ejections.<ref>Low, B. C. (1996). "Magnetohydrodynamic Processes in the Solar Corona: Flares, Coronal Mass Ejections and Magnetic Helicity". Solar and Astrophysical Magnetohydrodynamic Flows. Dordrecht: Springer Netherlands. pp. 133–149. doi:10.1007/978-94-009-0265-7_7. ISBN 978-94-010-6603-7. Retrieved 2020-10-08.</ref> Magnetic helicity is present in the solar wind.<ref>Bieber, J. W.; Evenson, P. A.; Matthaeus, W. H. (April 1987). "Magnetic helicity of the Parker field". The Astrophysical Journal. 315: 700. Bibcode:1987ApJ...315..700B. doi:10.1086/165171. ISSN 0004-637X.</ref> Its conservation is significant in dynamo processes.<ref name=":2">Blackman, E.G. (2015). "Magnetic Helicity and Large Scale Magnetic Fields: A Primer". Space Science Reviews. 188 (1–4): 59–91. arXiv:1402.0933. Bibcode:2015SSRv..188...59B. doi:10.1007/s11214-014-0038-6. S2CID 17015601.</ref><ref> Brandenburg, A. (2009). "Hydromagnetic Dynamo Theory". Scholarpedia. 2 (3): 2309. Bibcode:2007SchpJ...2.2309B. doi:10.4249/scholarpedia.2309. rev #73469.</ref><ref>Brandenburg, A.; Lazarian, A. (2013-08-31). "Astrophysical Hydromagnetic Turbulence". Space Science Reviews. 178 (2–4): 163–200. arXiv:1307.5496. Bibcode:2013SSRv..178..163B. doi:10.1007/s11214-013-0009-3. ISSN 0038-6308. S2CID 16261037.</ref><ref>Vishniac, Ethan T.; Cho, Jungyeon (April 2001). "Magnetic Helicity Conservation and Astrophysical Dynamos". The Astrophysical Journal. 550 (2): 752–760. arXiv:astro-ph/0010373. Bibcode:2001ApJ...550..752V. doi:10.1086/319817. ISSN 0004-637X.</ref> It also plays a role in fusion research, for example in reversed field pinch experiments.<ref>Escande, D. F.; Martin, P.; Ortolani, S.; Buffa, A.; Franz, P.; Marrelli, L.; Martines, E.; Spizzo, G.; Cappello, S.; Murari, A.; Pasqualotto, R. (2000-08-21). "Quasi-Single-Helicity Reversed-Field-Pinch Plasmas". Physical Review Letters. 85 (8): 1662–1665. Bibcode:2000PhRvL..85.1662E. doi:10.1103/physrevlett.85.1662. ISSN 0031-9007. PMID 10970583.</ref>

Mathematical definition

Generally, the helicity <math>H^{\mathbf f}</math> of a smooth vector field <math>\mathbf f</math> confined to a volume <math>V</math> is the standard measure of the extent to which the field lines wrap and coil around one another.<ref>Cantarella, Jason; Deturck, Dennis; Gluck, Herman; Teytel, Mikhail (1999). "Influence of geometry and topology on helicity[J]". Magnetic Helicity in Space and Laboratory Plasmas. Geophysical Monograph Series. pp. 17–24. doi:10.1029/GM111p0017. ISBN 9781118664476.</ref><ref name=":6" /> It is defined as the volume integral over <math>V</math> of the scalar product of <math>\mathbf f</math> and its curl, <math>\nabla\times{\mathbf f}</math>:

<math> H^{\mathbf f} = \int_V {\mathbf f} \cdot \left(\nabla\times{\mathbf f}\right)\ dV . </math>

Magnetic helicity

Magnetic helicity <math>H^{\mathbf M}</math> is the helicity of a magnetic vector potential <math>{\mathbf A}</math> where <math>\nabla \times {\mathbf A}={\mathbf B}</math> is the associated magnetic field confined to a volume <math>V</math>. Magnetic helicity can then be expressed as<ref name=":2" />

<math> H^{\mathbf M} = \int_V {\mathbf A}\cdot{\mathbf B}\ dV . </math>

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.<ref>Brandenburg, Axel; Subramanian, Kandaswamy (2005). "Astrophysical magnetic fields and nonlinear dynamo theory". Physics Reports. 417 (1–4): 1–209. arXiv:astro-ph/0405052. Bibcode:2005PhR...417....1B. doi:10.1016/j.physrep.2005.06.005. ISSN 0370-1573. S2CID 119518712.</ref>

Magnetic helicity has units of magnetic flux squared: Wb² (webers squared) in SI units and Mx² (maxwells squared) in Gaussian Units.<ref>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.</ref>

Current helicity

The current helicity, or helicity <math>M^{\mathbf{J}}</math> of the magnetic field <math>\mathbf{B}</math> confined to a volume <math>V</math>, can be expressed as

<math> H^{\mathbf J} = \int_V {\mathbf B}\cdot{\mathbf J}\ dV </math>

where <math> {\mathbf J} = \nabla \times {\mathbf B} </math> is the current density.<ref name=":3"> Subramanian, K.; Brandenburg, A. (2006). "Magnetic helicity density and its flux in weakly inhomogeneous turbulence". The Astrophysical Journal Letters. 648 (1): L71–L74. arXiv:astro-ph/0509392. Bibcode:2006ApJ...648L..71S. doi:10.1086/507828. S2CID 323935.</ref> Unlike magnetic helicity, current helicity is not an ideal invariant (it is not conserved even when the electrical resistivity is zero).

Topological interpretation

The name "helicity" is due to the fact that the trajectory of a fluid particle in a fluid with velocity <math>\boldsymbol v</math> and vorticity <math>\boldsymbol{\omega}=\nabla \times \boldsymbol{v}</math> forms a helix in regions where the kinetic helicity <math>\textstyle H^K=\int \mathbf v \cdot \boldsymbol{\omega} \neq 0</math>. When <math>\textstyle H^K > 0</math>, the resulting helix is right-handed and when <math>\textstyle H^K < 0</math> it is left-handed. This behavior is very similar to that found with respect to 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.<ref name=":0" /> Where linking numbers describe how many times curves are interlinked, magnetic helicity describes how many magnetic field lines are interlinked.<ref name=":2" />

Examples of curves with varying values of writhe and twist. Magnetic helicity measures the sum of these two quantities for magnetic field lines. The sum is conserved under all transformations where curves are not cut or joined.

Magnetic helicity is proportional to the sum of the topological quantities twist and writhe for magnetic field lines. Twist is the rotation of the flux tube around its own 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.<ref name=":1" /><ref>Linkmann, Moritz; Sahoo, Ganapati; McKay, Mairi; Berera, Arjun; Biferale, Luca (2017-02-06). "Effects of Magnetic and Kinetic Helicities on the Growth of Magnetic Fields in Laminar and Turbulent Flows by Helical Fourier Decomposition". The Astrophysical Journal. 836 (1): 26. arXiv:1609.01781. Bibcode:2017ApJ...836...26L. doi:10.3847/1538-4357/836/1/26. ISSN 1538-4357. S2CID 53126623.</ref>

Ideal quadratic invariance

In the late 1950s, Lodewijk Woltjer and Walter M. Elsässer discovered independently the ideal invariance of magnetic helicity,<ref name=":4">Woltjer, L. (1958-06-01). "A Theorem on Force-Free Magnetic Fields". Proceedings of the National Academy of Sciences. 44 (6): 489–491. Bibcode:1958PNAS...44..489W. doi:10.1073/pnas.44.6.489. ISSN 0027-8424. PMC 528606. PMID 16590226.</ref><ref name=":5">Elsasser, Walter M. (1956-04-01). "Hydromagnetic Dynamo Theory". Reviews of Modern Physics. 28 (2): 135–163. Bibcode:1956RvMP...28..135E. doi:10.1103/revmodphys.28.135. ISSN 0034-6861.</ref> 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

<math> \frac{\partial {\mathbf B}}{\partial t} = \nabla \times ({\mathbf v} \times {\mathbf B}),\quad \frac{\partial {\mathbf A}}{\partial t} = {\mathbf v} \times {\mathbf B} + \nabla\Phi, </math>

respectively, where <math> \nabla\Phi </math> is a scalar potential given by the gauge condition (see § Gauge considerations). Choosing the gauge so that the scalar potential vanishes, <math>\nabla \Phi = \mathbf{0}</math>, the time evolution of magnetic helicity in a volume <math>V</math> is given by:

<math>\begin{align}

\frac{\partial H^{\mathbf M}}{\partial t} &= \int_V \left( \frac{\partial {\mathbf A}}{\partial t} \cdot {\mathbf B} + {\mathbf A} \cdot \frac{\partial {\mathbf B}}{\partial t} \right) dV \\ &= \int_V ({\mathbf v} \times {\mathbf B}) \cdot{\mathbf B}\ dV + \int_V {\mathbf A} \cdot \left(\nabla \times \frac{\partial {\mathbf A}}{\partial t}\right) dV . \end{align}</math> The dot product in the integrand of the first term is zero since <math>{\mathbf B}</math> is orthogonal to the cross product <math>{\mathbf v} \times {\mathbf B}</math>, and the second term can be integrated by parts to give

<math> \frac{\partial H^{\mathbf M}}{\partial t} = \int_V \left(\nabla \times {\mathbf A}\right) \cdot \frac{\partial {\mathbf A}}{\partial t}\ dV + \int_{\partial V} \left({\mathbf A} \times \frac{\partial {\mathbf A}}{\partial t}\right) \cdot d\mathbf{S} </math>

where the second term is a surface integral over the boundary surface <math>\partial V</math> of the closed system. The dot product in the integrand of the first term is zero because <math> \nabla \times {\mathbf A} = {\mathbf B} </math> is orthogonal to <math> \partial {\mathbf A}/\partial t .</math> The second term also vanishes because motions inside the closed system cannot affect the vector potential outside, so that at the boundary surface <math> \partial {\mathbf A}/\partial t = \mathbf{0} </math> since the magnetic vector potential is a continuous function. Therefore,

<math> \frac{\partial H^{\mathbf M}}{\partial t} = 0 , </math>

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 <math> \nabla \Phi = \mathbf{0} . </math>

Magnetic helicity remains conserved in a good approximation even with a small but finite resistivity, in which case magnetic reconnection dissipates energy.<ref name=":0" /><ref name=":2" />

Inverse transfer property

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 has first been proposed by Uriel Frisch and collaborators<ref name=":1" /> and has been verified through many numerical experiments.<ref>Pouquet, A.; Frisch, U.; Léorat, J. (1976-09-24). "Strong MHD helical turbulence and the nonlinear dynamo effect". Journal of Fluid Mechanics. 77 (2): 321–354. Bibcode:1976JFM....77..321P. doi:10.1017/s0022112076002140. ISSN 0022-1120. S2CID 3746018.</ref><ref>Meneguzzi, M.; Frisch, U.; Pouquet, A. (1981-10-12). "Helical and Nonhelical Turbulent Dynamos". Physical Review Letters. 47 (15): 1060–1064. Bibcode:1981PhRvL..47.1060M. doi:10.1103/physrevlett.47.1060. ISSN 0031-9007.</ref><ref>Balsara, D.; Pouquet, A. (January 1999). "The formation of large-scale structures in supersonic magnetohydrodynamic flows". Physics of Plasmas. 6 (1): 89–99. Bibcode:1999PhPl....6...89B. doi:10.1063/1.873263. ISSN 1070-664X.</ref><ref>Christensson, Mattias; Hindmarsh, Mark; Brandenburg, Axel (2001-10-22). "Inverse cascade in decaying three-dimensional magnetohydrodynamic turbulence". Physical Review E. 64 (5): 056405. arXiv:astro-ph/0011321. Bibcode:2001PhRvE..64e6405C. doi:10.1103/physreve.64.056405. ISSN 1063-651X. PMID 11736099. S2CID 8309837.</ref><ref>Brandenburg, Axel (April 2001). "The Inverse Cascade and Nonlinear Alpha‐Effect in Simulations of Isotropic Helical Hydromagnetic Turbulence". The Astrophysical Journal. 550 (2): 824–840. arXiv:astro-ph/0006186. Bibcode:2001ApJ...550..824B. doi:10.1086/319783. ISSN 0004-637X.</ref><ref>Alexakis, Alexandros; Mininni, Pablo D.; Pouquet, Annick (2006-03-20). "On the Inverse Cascade of Magnetic Helicity". The Astrophysical Journal. 640 (1): 335–343. arXiv:physics/0509069. Bibcode:2006ApJ...640..335A. doi:10.1086/500082. ISSN 0004-637X.</ref> 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<ref name=":1" /> is repeated here, which is based on the so-called "realizability condition" on the magnetic helicity Fourier spectrum <math> \hat{H}^M_{\mathbf k} = \hat{\mathbf A}^*_{\mathbf k} \cdot \hat{\mathbf B}_{\mathbf k} </math> (where <math> \hat{\mathbf B}_{\mathbf k} </math> is the Fourier coefficient at the wavevector <math> {\mathbf k} </math> of the magnetic field <math> {\mathbf B} </math>, and similarly for <math> \hat{\mathbf A} </math>, the star denoting the complex conjugate). The "realizability condition" corresponds to an application of Cauchy-Schwarz inequality, which yields: <math display="block"> \left|\hat{H}^M_{\mathbf k}\right| \leq \frac{2E^M_{\mathbf k}}{|{\mathbf k}|} ,</math> with <math display="inline"> E^M_{\mathbf k} = \frac{1}{2} \hat{\mathbf B}^*_{\mathbf k}\cdot\hat{\mathbf B}_{\mathbf k} </math> the magnetic energy spectrum. To obtain this inequality, the fact that <math> |\hat{\mathbf B}_{\mathbf k}|=|{\mathbf k}||\hat{\mathbf A}^\perp_{\mathbf k}| </math> (with <math> \hat{\mathbf A}^\perp_{\mathbf k} </math> the solenoidal part of the Fourier transformed magnetic vector potential, orthogonal to the wavevector in Fourier space) has been used, since <math> \hat{\mathbf{B}}_{\mathbf k} = i {\mathbf k} \times \hat{\mathbf{A}}_{\mathbf k} </math>. The factor 2 is not present in the paper<ref name=":1" /> since the magnetic helicity is defined there alternatively as <math> \frac{1}{2} \int_V {\mathbf A} \cdot {\mathbf B}\ dV </math>.

One can then imagine an initial situation with no velocity field and a magnetic field only present at two wavevectors <math> \mathbf p </math> and <math> \mathbf q </math>. We assume a fully helical magnetic field, which means that it saturates the realizability condition: <math> \left|\hat{H}^M_{\mathbf p}\right| = \frac{2E^M_{\mathbf p}}{|{\mathbf p}|} </math> and <math> \left|\hat{H}^M_{\mathbf q}\right| = \frac{2E^M_{\mathbf q}}{|{\mathbf q}|} </math>. Assuming that all the energy and magnetic helicity transfers are done to another wavevector <math> \mathbf k </math>, the conservation of magnetic helicity on the one hand and of the total energy <math> E^T = E^M + E^K </math> (the sum of magnetic and kinetic energy) on the other hand gives:

<math display="block"> H^M_{\mathbf k} = H^M_{\mathbf p} + H^M_{\mathbf q}, </math>

<math display="block"> E^T_{\mathbf k} = E^T_{\mathbf p}+E^T_{\mathbf q} = E^M_{\mathbf p}+E^M_{\mathbf q}. </math>

The second equality for energy comes from the fact that we consider an initial state with no kinetic energy. Then we have the necessarily <math> |\mathbf k| \leq \max(|\mathbf p|, |\mathbf q| ) </math>. Indeed, if we would have <math> |\mathbf k| > \max(|\mathbf p|,|\mathbf q| ) </math>, then:

<math display="block"> H^M_{\mathbf k} = H^M_{\mathbf p} + H^M_{\mathbf q} = \frac{2E^M_{\mathbf p}}{|\mathbf p|} + \frac{2E^M_{\mathbf q}}{|\mathbf q|} > \frac{2\left(E^M_{\mathbf p} + E^M_{\mathbf q}\right)}{|\mathbf k|} = \frac{2E^T_{\mathbf k}}{|\mathbf k|} \geq \frac{2E^M_{\mathbf k}}{|\mathbf k|}, </math>

which would break the realizability condition. This means that <math> |\mathbf k| \leq \max(|\mathbf p|,|\mathbf q| ) </math>. In particular, for <math> |{\mathbf p}| = |{\mathbf q}| </math>, the magnetic helicity is transferred to a smaller wavevector, which means to larger scales.

Gauge considerations

Magnetic helicity is a gauge-dependent quantity, because <math>\mathbf A</math> 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,<ref name=":3" /> 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.<ref name=":0" />

See also

• Woltjer's theorem

External links

• A. A. Pevtsov's Helicity Page
• Mitch Berger's Publications Page