Woltjer's Theorem
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In
plasma physics Plasma () is a state of matter characterized by the presence of a significant portion of charged particles in any combination of ions or electrons. It is the most abundant form of ordinary matter in the universe, mostly in stars (including th ...
, Woltjer's theorem states that
force-free magnetic field In plasma physics, a force-free magnetic field is a magnetic field in which the Lorentz force is equal to zero and the magnetic pressure greatly exceeds the plasma pressure such that non-magnetic forces can be neglected. For a force-free field, t ...
s in a closed system with constant force-free parameter \alpha represent the state with lowest
magnetic energy The potential magnetic energy of a magnet or magnetic moment \mathbf in a magnetic field \mathbf is defined as the mechanical work of the magnetic force on the re-alignment of the vector of the magnetic dipole moment and is equal to: E_\text = ...
in the system and that the
magnetic helicity In plasma physics, magnetic helicity is a measure of the linkage, twist, and writhe of a magnetic field. Magnetic helicity is a useful concept in the analysis of systems with extremely low resistivity, such as astrophysical systems. When resistiv ...
is invariant under this condition. It is named after Lodewijk Woltjer who derived it in 1958. A force-free magnetic field with flux density \mathbf satisfies :\nabla \times \mathbf = \alpha \mathbf where \alpha is a scalar function that is constant along field lines. The helicity \mathcal invariant is given by :\frac = 0 where \mathcal is related to \mathbf=\nabla\times \mathbf through the
vector potential In vector calculus, a vector potential is a vector field whose curl is a given vector field. This is analogous to a ''scalar potential'', which is a scalar field whose gradient is a given vector field. Formally, given a vector field \mathbf, a ' ...
\mathbf as below :\mathcal = \int_V \mathbf\cdot\mathbf\ dV = \int_V \mathbf \cdot (\nabla \times \mathbf) \ dV.


See also

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Chandrasekhar–Kendall function Chandrasekhar–Kendall functions are the eigenfunctions of the curl operator derived by Subrahmanyan Chandrasekhar and P. C. Kendall in 1957 while attempting to solve the force-free magnetic fields. The functions were independently derived by both ...
*
Hydrodynamical helicity In fluid dynamics, helicity is, under appropriate conditions, an Invariant (mathematics), invariant of the Euler equations (fluid dynamics), Euler equations of fluid flow, having a topological interpretation as a measure of Link (knot theory), li ...


References

Astrophysics Plasma theory and modeling {{plasma-stub