Woltjer's theorem

In plasma physics, Woltjer's theorem states that force-free magnetic fields, for a closed system with constant ${\displaystyle \alpha }$, have the minimum magnetic energy and the magnetic helicity is invariant under this condition, named after Lodewijk Woltjer who derived in 1958[1][2][3][4][5][6]. The force-free field strength ${\displaystyle \mathbf {B} }$ equation is

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

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

References

1. Woltjer, L. (1958). A theorem on force-free magnetic fields. Proceedings of the National Academy of Sciences, 44(6), 489-491.
2. Chiuderi, C., & Velli, M. (2016). Basics of Plasma Astrophysics. Springer.
3. Moffatt, H. K. (1978). Field generation in electrically conducting fluids. Cambridge University Press, Cambridge, London, New York, Melbourne.
4. Sturrock, P. A. (1994). Plasma Physics: an introduction to the theory of astrophysical, geophysical and laboratory plasmas. Cambridge University Press.
5. Solov'ev, A. A. (1985). Woltjer's theorem and the force-free magnetic field stability problem. Byulletin Solnechnye Dannye Akademie Nauk SSSR, 1985, 55–62.
6. Kholodenko, A. L. (2013). Applications of contact geometry and topology in physics. World Scientific.