Kronecker–Weber theorem: Difference between revisions

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In [[plasma physics]], '''magnetic helicity''' is the extent to which a [[magnetic field]] "wraps around itself". It is a generalization of the topological concept of [[linking number]] to the differential quantities required to describe the magnetic field. As with many quantities in electromagnetism, magnetic helicity (which describes magnetic field lines) is closely related to fluid mechanical helicity (which describes fluid flow lines).
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If magnetic field lines follow the strands of a twisted [[rope]], this configuration would have nonzero magnetic helicity; left-handed ropes would have negative values and right-handed ropes would have positive values.{{Clarifyme|date=February 2009}}
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Formally,
:<math> H = \int {\mathbf A}\cdot{\mathbf B}\,d^3{\mathbf r} </math>
 
where
:<math>{\mathbf B}</math> is the magnetic field strength
:<math>{\mathbf B}=\nabla\times{\mathbf A}</math>;
:<math>\mathbf A</math> is the [[vector potential]] of <math>{\mathbf B}</math>
 
Magnetic helicity has units of Wb<sup>2</sup> ([[Weber (unit)|weber]]s squared). It is a [[conservation law|conserved quantity]] in electromagnetic fields, even when [[magnetic reconnection]] dissipates [[energy]].   The concept is useful in [[Sun|solar dynamics]] and in [[hydromagnetic dynamo theory]].<ref>
{{cite journal
|last=Brandenburg |first=A.
|year=2009
|title=Hydromagnetic Dynamo Theory
|url=http://www.scholarpedia.org/wiki/index.php?title=Hydromagnetic_dynamo_theory&action=cite&rev=73469
|journal=[[Scholarpedia]]
|volume=2 |issue=3 |pages=2309
|doi=10.4249/scholarpedia.2309
|id=rev #73469
}}</ref> Magnetic helicity is a gauge-dependent quantity, because <math>\mathbf A</math> can be redefined by adding a gradient to it (gauge transformation). However, for perfectly conducting boundaries or periodic systems without a net magnetic flux, the magnetic helicity is gauge invariant. A gauge-invariant ''relative helicity'' has been defined for volumes with non-zero magnetic flux on their boundary surfaces<ref>
{{cite journal
|last1=Berger |first1=M.A.
|year=1999
|title=Introduction to magnetic helicity
|volume=41 |issue=12B |pages=167
|journal=[[Plasma Physics and Controlled Fusion]]
|bibcode=1999PPCF...41..167B
|doi=10.1088/0741-3335/41/12B/312
}}</ref> If the magnetic field is turbulent and weakly inhomogeneous a magnetic helicity ''density'' and its associated flux can be defined in terms of the density of field line linkages.<ref>
{{cite journal
|last=Subramanian |first=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= |pages=L71–L74
|arxiv=astro-ph/0509392
|bibcode=2006ApJ...648L..71S
|doi=10.1086/507828
}}</ref>
 
==References==
{{reflist}}
==External links==
* A. A. Pevtsov's [http://www4.nso.edu/staff/apevtsov/webhelicity.html Helicity on the Web] Page
* Mitch Berger's [http://emps.exeter.ac.uk/mathematics/staff/mab215/publications Publications] Page
 
[[Category:Physical quantities]]
[[Category:Plasma physics]]

Latest revision as of 00:54, 28 November 2014

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