Woltjer's theorem

In plasma physics, Woltjer's theorem states that force-free magnetic fields in a closed system with constant force-free parameter ${\displaystyle \alpha }$ represent the state with lowest magnetic energy in the system and that the magnetic helicity is invariant under this condition. It is named after Lodewijk Woltjer who derived it in 1958.^{[1][2][3][4][5][6]} A force-free magnetic field with flux density ${\displaystyle \mathbf {B} }$ satisfies

${\displaystyle \nabla \times \mathbf {B} =\alpha \mathbf {B} }$

where ${\displaystyle \alpha }$ is a scalar function that is constant along field lines. The helicity ${\displaystyle {\mathcal {H}}}$ invariant is given by

${\displaystyle {\frac {d{\mathcal {H}}}{dt}}=0}$

where ${\displaystyle {\mathcal {H}}}$ is related to ${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} }$ through the vector potential ${\displaystyle \mathbf {A} }$ as below

${\displaystyle {\mathcal {H}}=\int _{V}\mathbf {A} \cdot \mathbf {B} \ dV=\int _{V}\mathbf {A} \cdot (\nabla \times \mathbf {A} )\ dV.}$

See also

• Chandrasekhar–Kendall function
• Hydrodynamical helicity

References

1. Woltjer, L. (June 1958). "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. PMC 528606. PMID 16590226.
2. Chiuderi, C.; Velli, Marco; Chiuderi, Claudio (2015). Basics of Plasma Astrophysics. Milan: Springer. ISBN 978-88-470-5280-2.
3. Moffat, H. K. (1978). Magnetic Field Generation in Electrically Conducting Fluids. Cambridge: Cambridge University Press.
4. Sturrock, P. A. (1994). Plasma Physics: An Introduction to the Theory of Astrophysical, Geophysical and Laboratory Plasmas. Cambridge: Cambridge University Press. ISBN 9780521448109.
5. Solov'ev, A. A. (1985). "Woltjer's Theorem and the Force-Free Magnetic Field Stability Problem". Byulletin Solnechnye Dannye Akademie Nauk USSR. 1985: 55–62. Bibcode:1985BSolD1985...55S.
6. Kholodenko, Arkady L. (2013). Applications of Contact Geometry and Topology in Physics. New Jersey: World Scientific. Bibcode:2013acgt.book.....K. doi:10.1142/8514. ISBN 9789814412087.