Woltjer's theorem

In plasma physics, Woltjer's theorem states that force-free magnetic fields for a closed system with constant $\alpha$ , 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 $\mathbf {B}$ equation is

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

The helicity ${\mathcal {H}}$ invariant is given by

{\frac {d{\mathcal {H}}}{dt}}=0.

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

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

.

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.

[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.

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See also

References