Woltjer's theorem explained

In plasma physics, Woltjer's theorem states that force-free magnetic fields in a closed system with constant force-free parameter

\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 B satisfies\nabla x B = \alpha B where \alpha is a scalar function that is constant along field lines. The helicity l{H} invariant is given by(d l{H)/() } = 0where l{H} is related to B = \nabla x A through the vector potential A as belowl{H} = \int _{V} A ⋅ B dV = \int _{V} A ⋅ (\nabla x A ) dV.

See also

• Chandrasekhar–Kendall function
• Hydrodynamical helicity

Notes and References

1. Woltjer . L. . A Theorem on Force-Free Magnetic Fields . Proceedings of the National Academy of Sciences . June 1958 . 44 . 6 . 489–491 . 10.1073/pnas.44.6.489 . free . 16590226 . 1958PNAS...44..489W. 528606 .
2. Book: Chiuderi . C. . Velli . Marco . Chiuderi . Claudio . Basics of Plasma Astrophysics . 2015 . Springer . Milan . 978-88-470-5280-2.
3. Book: Moffat . H. K. . Magnetic Field Generation in Electrically Conducting Fluids . 1978 . Cambridge University Press . Cambridge .
4. Book: Sturrock . P. A. . Plasma Physics: An Introduction to the Theory of Astrophysical, Geophysical and Laboratory Plasmas . 1994 . Cambridge University Press . Cambridge . 9780521448109 .
5. Solov'ev . A. A. . Woltjer's Theorem and the Force-Free Magnetic Field Stability Problem . Byulletin Solnechnye Dannye Akademie Nauk USSR . 1985 . 1985 . 55–62 . 1985BSolD1985...55S.
6. Book: Kholodenko . Arkady L. . Applications of Contact Geometry and Topology in Physics . 2013 . World Scientific . New Jersey . 10.1142/8514 . 2013acgt.book.....K . 9789814412087 .