Astronomy:Woltjer's theorem
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In plasma physics, Woltjer's theorem states that force-free magnetic fields in a closed system with constant force-free parameter [math]\displaystyle{ \alpha }[/math] 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] The force-free field strength [math]\displaystyle{ \mathbf{B} }[/math] equation is
- [math]\displaystyle{ \nabla \times \mathbf{B} = \alpha \mathbf{B}. }[/math]
The helicity [math]\displaystyle{ \mathcal{H} }[/math] invariant is given by
- [math]\displaystyle{ \frac{d\mathcal{H}}{d t} =0. }[/math]
where [math]\displaystyle{ \mathcal{H} }[/math] is related to [math]\displaystyle{ \mathbf{B}=\nabla\times \mathbf{A} }[/math] through the vector potential [math]\displaystyle{ \mathbf{A} }[/math] as below
- [math]\displaystyle{ \mathcal{H} = \int_V \mathbf{A}\cdot\mathbf{B}\ dV = \int_V \mathbf{A} \cdot (\nabla \times \mathbf{A}) \ dV. }[/math]
See also
References
- ↑ Woltjer, L. (1958). A theorem on force-free magnetic fields. Proceedings of the National Academy of Sciences, 44(6), 489-491.
- ↑ Chiuderi, C., & Velli, M. (2016). Basics of Plasma Astrophysics. Springer.
- ↑ Moffatt, H. K. (1978). Field generation in electrically conducting fluids. Cambridge University Press, Cambridge, London, New York, Melbourne.
- ↑ Sturrock, P. A. (1994). Plasma Physics: an introduction to the theory of astrophysical, geophysical and laboratory plasmas. Cambridge University Press.
- ↑ 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.
- ↑ Kholodenko, A. L. (2013). Applications of contact geometry and topology in physics. World Scientific.
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