View source for Magnetic helicity

{{Short description|Measure of magnetic field topology}}
{{Tone|date=December 2022}}
In [[plasma physics]], '''magnetic helicity''' is a measure of the linkage, twist, and writhe of a [[magnetic field]].<ref>{{Cite book|last1=Cantarella|first1=Jason|chapter=Influence of Geometry and Topology on Helicity|date=2013-03-19|url=http://dx.doi.org/10.1029/gm111p0017|title=Magnetic Helicity in Space and Laboratory Plasmas|pages=17–24|place=Washington, D. C.|publisher=American Geophysical Union|isbn=978-1-118-66447-6|access-date=2021-01-18|last2=Deturck|first2=Dennis|last3=Gluck|first3=Herman|last4=Teytel|first4=Mikhail|doi=10.1029/gm111p0017}}</ref><ref name=":6">{{Cite journal|last=Moffatt|first=H. K.|date=1969-01-16|title=The degree of knottedness of tangled vortex lines|url=http://dx.doi.org/10.1017/s0022112069000991|journal=Journal of Fluid Mechanics|volume=35|issue=1|pages=117–129|doi=10.1017/s0022112069000991|bibcode=1969JFM....35..117M|s2cid=121478573 |issn=0022-1120}}</ref> In [[ideal magnetohydrodynamics]], magnetic helicity is [[Conservation law|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">{{Cite journal|last1=Frisch|first1=U.|last2=Pouquet|first2=A.|last3=LÉOrat|first3=J.|last4=Mazure|first4=A.|date=1975-04-29|title=Possibility of an inverse cascade of magnetic helicity in magnetohydrodynamic turbulence|url=http://dx.doi.org/10.1017/s002211207500122x|journal=Journal of Fluid Mechanics|volume=68|issue=4|pages=769–778|doi=10.1017/s002211207500122x|bibcode=1975JFM....68..769F|s2cid=45460069 |issn=0022-1120}}</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 [[Vortex|vortices]] [[Energy cascade|break up]] into smaller ones, until dissipating through [[Viscosity|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">
{{cite journal|last1=Berger|first1=M.A.|date=1999|title=Introduction to magnetic helicity|journal=[[Plasma Physics and Controlled Fusion]]|volume=41|issue=12B|pages=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 flare]]s and [[coronal mass ejection]]s.<ref>{{Cite book|last=Low|first=B. C.|chapter=Magnetohydrodynamic Processes in the Solar Corona: Flares, Coronal Mass Ejections and Magnetic Helicity|date=1996|url=http://dx.doi.org/10.1007/978-94-009-0265-7_7|title=Solar and Astrophysical Magnetohydrodynamic Flows|pages=133–149|place=Dordrecht|publisher=Springer Netherlands|doi=10.1007/978-94-009-0265-7_7|isbn=978-94-010-6603-7|access-date=2020-10-08}}</ref> Magnetic helicity is present in the [[solar wind]].<ref>{{Cite journal|last1=Bieber|first1=J. W.|last2=Evenson|first2=P. A.|last3=Matthaeus|first3=W. H.|date=April 1987|title=Magnetic helicity of the Parker field|journal=The Astrophysical Journal|volume=315|pages=700|doi=10.1086/165171|bibcode=1987ApJ...315..700B|issn=0004-637X|doi-access=free}}</ref> Its conservation is significant in [[Dynamo theory|dynamo]] processes.<ref name=":2">{{cite journal|last=Blackman|first=E.G.|date=2015|title=Magnetic Helicity and Large Scale Magnetic Fields: A Primer|journal=[[Space Science Reviews]]|volume=188|issue=1–4|pages=59–91|arxiv=1402.0933|bibcode=2015SSRv..188...59B|doi=10.1007/s11214-014-0038-6|s2cid=17015601}}</ref><ref>
{{cite journal|last=Brandenburg|first=A.|date=2009|title=Hydromagnetic Dynamo Theory|journal=[[Scholarpedia]]|volume=2|issue=3|pages=2309|bibcode=2007SchpJ...2.2309B|doi=10.4249/scholarpedia.2309|id=rev #73469|doi-access=free}}</ref><ref>{{Cite journal|last1=Brandenburg|first1=A.|author2-link=Alexandre Lazarian|last2=Lazarian|first2=A.|date=2013-08-31|title=Astrophysical Hydromagnetic Turbulence|url=http://dx.doi.org/10.1007/s11214-013-0009-3|journal=Space Science Reviews|volume=178|issue=2–4|pages=163–200|arxiv=1307.5496|doi=10.1007/s11214-013-0009-3|bibcode=2013SSRv..178..163B|s2cid=16261037|issn=0038-6308}}</ref><ref>{{Cite journal|last1=Vishniac|first1=Ethan T.|last2=Cho|first2=Jungyeon|date=April 2001|title=Magnetic Helicity Conservation and Astrophysical Dynamos|journal=The Astrophysical Journal|volume=550|issue=2|pages=752–760|doi=10.1086/319817|arxiv=astro-ph/0010373|bibcode=2001ApJ...550..752V|issn=0004-637X|doi-access=free}}</ref> It also plays a role in [[Fusion power|fusion research]], for example in [[reversed field pinch]] experiments.<ref>{{Cite journal|last1=Escande|first1=D. F.|last2=Martin|first2=P.|last3=Ortolani|first3=S.|last4=Buffa|first4=A.|last5=Franz|first5=P.|last6=Marrelli|first6=L.|last7=Martines|first7=E.|last8=Spizzo|first8=G.|last9=Cappello|first9=S.|last10=Murari|first10=A.|last11=Pasqualotto|first11=R.|date=2000-08-21|title=Quasi-Single-Helicity Reversed-Field-Pinch Plasmas|url=http://dx.doi.org/10.1103/physrevlett.85.1662|journal=Physical Review Letters|volume=85|issue=8|pages=1662–1665|doi=10.1103/physrevlett.85.1662|pmid=10970583|bibcode=2000PhRvL..85.1662E|issn=0031-9007}}</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>{{cite book| last1=Cantarella | first1=Jason | last2=Deturck | first2=Dennis | last3=Gluck | first3=Herman | last4=Teytel | first4=Mikhail|chapter=Influence of geometry and topology on helicity[J]|url=http://onlinelibrary.wiley.com/book/10.1029/GM111|title=Magnetic Helicity in Space and Laboratory Plasmas| series=Geophysical Monograph Series |year=1999|pages=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 (mathematics)|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>{{Cite journal|last1=Brandenburg|first1=Axel|last2=Subramanian|first2=Kandaswamy|title=Astrophysical magnetic fields and nonlinear dynamo theory|url=http://dx.doi.org/10.1016/j.physrep.2005.06.005|journal=Physics Reports|year=2005|volume=417|issue=1–4|pages=1–209|doi=10.1016/j.physrep.2005.06.005|arxiv=astro-ph/0405052|bibcode=2005PhR...417....1B|s2cid=119518712|issn=0370-1573|via=}}</ref>

Magnetic helicity has units of [[magnetic flux]] squared: Wb<sup>2</sup> ([[Weber (unit)|webers]] squared) in [[International System of Units|SI units]] and Mx<sup>2</sup> ([[Maxwell (unit)|maxwells]] squared) in [[Gaussian units|Gaussian Units]].<ref>{{Cite book|last=Huba|first=J.D.|title=NRL Plasma Formulary|year=2013|url=http://www.nrl.navy.mil/ppd/sites/www.nrl.navy.mil.ppd/files/pdfs/NRL_FORMULARY_13.pdf|archive-url=https://web.archive.org/web/20190630224315/http://www.nrl.navy.mil/ppd/sites/www.nrl.navy.mil.ppd/files/pdfs/NRL_FORMULARY_13.pdf|archive-date=2019-06-30|access-date=|location=Washington, D.C.|publisher=Beam Physics Branch Plasma Physics Division Naval Research Laboratory}}</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">
{{cite journal|last1=Subramanian|first1=K.|last2=Brandenburg|first2=A.|year=2006|title=Magnetic helicity density and its flux in weakly inhomogeneous turbulence|journal=[[The Astrophysical Journal Letters]]|volume=648|issue=1|pages=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" />
[[File:Circular DNA Supercoiling.png|thumb|Examples of curves with varying values of [[writhe]] and [[Twist (mathematics)|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 (mathematics)|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 tube]]s (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 [[Hydrodynamical helicity|fluid mechanical helicity]], the corresponding quantity for fluid flow lines, and their dynamics are interlinked.<ref name=":1" /><ref>{{Cite journal|last1=Linkmann|first1=Moritz|last2=Sahoo|first2=Ganapati|last3=McKay|first3=Mairi|last4=Berera|first4=Arjun|last5=Biferale|first5=Luca|date=2017-02-06|title=Effects of Magnetic and Kinetic Helicities on the Growth of Magnetic Fields in Laminar and Turbulent Flows by Helical Fourier Decomposition|journal=The Astrophysical Journal|volume=836|issue=1|pages=26|arxiv=1609.01781|doi=10.3847/1538-4357/836/1/26|bibcode=2017ApJ...836...26L|s2cid=53126623|issn=1538-4357 |doi-access=free }}</ref>

== Ideal quadratic invariance ==
In the late 1950s, [[Lodewijk Woltjer]] and [[Walter M. Elsasser|Walter M. Elsässer]] discovered independently the [[Conservation law|ideal invariance]] of magnetic helicity,<ref name=":4">{{Cite journal|last=Woltjer|first=L.|title=A Theorem on Force-Free Magnetic Fields|date=1958-06-01|journal=Proceedings of the National Academy of Sciences|volume=44|issue=6|pages=489–491|doi=10.1073/pnas.44.6.489|pmid=16590226|pmc=528606|bibcode=1958PNAS...44..489W|issn=0027-8424|doi-access=free}}</ref><ref name=":5">{{Cite journal|last=Elsasser|first=Walter M.|date=1956-04-01|title=Hydromagnetic Dynamo Theory|url=http://dx.doi.org/10.1103/revmodphys.28.135|journal=Reviews of Modern Physics|volume=28|issue=2|pages=135–163|doi=10.1103/revmodphys.28.135|bibcode=1956RvMP...28..135E|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 fixing|gauge condition]] (see {{slink||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>{{Cite journal| last1=Pouquet|first1=A.| last2=Frisch|first2=U.| last3=Léorat|first3=J.| date=1976-09-24| title=Strong MHD helical turbulence and the nonlinear dynamo effect| url=http://dx.doi.org/10.1017/s0022112076002140| journal=Journal of Fluid Mechanics| volume=77| issue=2| pages=321–354| doi=10.1017/s0022112076002140| bibcode=1976JFM....77..321P| s2cid=3746018 |issn=0022-1120}}</ref><ref>{{Cite journal| last1=Meneguzzi|first1=M.| last2=Frisch|first2=U.| last3=Pouquet|first3=A.| date=1981-10-12| title=Helical and Nonhelical Turbulent Dynamos| url=http://dx.doi.org/10.1103/physrevlett.47.1060| journal=Physical Review Letters| volume=47| issue=15| pages=1060–1064| doi=10.1103/physrevlett.47.1060| bibcode=1981PhRvL..47.1060M| issn=0031-9007}}</ref><ref>{{Cite journal| last1=Balsara|first1=D.| last2=Pouquet|first2=A.| date=January 1999| title=The formation of large-scale structures in supersonic magnetohydrodynamic flows| url=http://dx.doi.org/10.1063/1.873263| journal=Physics of Plasmas| volume=6| issue=1| pages=89–99| doi=10.1063/1.873263| bibcode=1999PhPl....6...89B| issn=1070-664X}}</ref><ref>{{Cite journal| last1=Christensson|first1=Mattias| last2=Hindmarsh|first2=Mark| last3=Brandenburg|first3=Axel| date=2001-10-22| title=Inverse cascade in decaying three-dimensional magnetohydrodynamic turbulence | url=http://dx.doi.org/10.1103/physreve.64.056405 | journal=Physical Review E | volume=64 | issue=5 | page=056405| doi=10.1103/physreve.64.056405 |pmid=11736099 |arxiv=astro-ph/0011321 | bibcode=2001PhRvE..64e6405C| s2cid=8309837| issn=1063-651X}}</ref><ref>{{Cite journal| last=Brandenburg|first=Axel| date=April 2001| title=The Inverse Cascade and Nonlinear Alpha‐Effect in Simulations of Isotropic Helical Hydromagnetic Turbulence| journal=The Astrophysical Journal| volume=550| issue=2| pages=824–840| doi=10.1086/319783| arxiv=astro-ph/0006186| bibcode=2001ApJ...550..824B| issn=0004-637X| doi-access=free}}</ref><ref>{{Cite journal| last1=Alexakis|first1=Alexandros| last2=Mininni|first2=Pablo D.| last3=Pouquet|first3=Annick| date=2006-03-20| title=On the Inverse Cascade of Magnetic Helicity|journal=The Astrophysical Journal| volume=640| issue=1| pages=335–343| doi=10.1086/500082| arxiv=physics/0509069| bibcode=2006ApJ...640..335A| issn=0004-637X| doi-access=free}}</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 [[Wave vector|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|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 vector field|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 fixing|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]]

== References ==
{{reflist}}

== External links ==

* A. A. Pevtsov's [https://gonewithsolarwind.com/index.php/helical-gallery Helicity] Page
* Mitch Berger's [http://emps.exeter.ac.uk/mathematics/staff/mab215/publications Publications] Page

[[Category:Physical quantities]]
[[Category:Plasma parameters]]
[[Category:Astrophysics]]