Force-free magnetic field explained

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, the electric current density is either zero or parallel to the magnetic field.

Definition

When a magnetic field is approximated as force-free, all non-magnetic forces are neglected and the Lorentz force vanishes. For non-magnetic forces to be neglected, it is assumed that the ratio of the plasma pressure to the magnetic pressure—the plasma β—is much less than one, i.e.,

\beta \ll 1 . With this assumption, magnetic pressure dominates over plasma pressure such that the latter can be ignored. It is also assumed that the magnetic pressure dominates over other non-magnetic forces, such as gravity, so that these forces can similarly be ignored.

In SI units, the Lorentz force condition for a static magnetic field

B can be expressed asj x B = 0 , \nabla ⋅ B = 0 , wherej = (1)/(\mu _{0}) \nabla x B is the current density and \mu _{0} is the vacuum permeability. Alternatively, this can be written as(\nabla x B ) x B = 0 , \nabla ⋅ B = 0 . These conditions are fulfilled when the current vanishes or is parallel to the magnetic field.^[1]

Zero current density

\phi :B = - \nabla\phi. The substitution of this into \nabla ⋅ B = 0 results in Laplace's equation, \nabla ^{2\phi} = 0 , which can often be readily solved, depending on the precise boundary conditions. In this case, the field is referred to as a potential field or vacuum magnetic field.

Nonzero current density

If the current density is not zero, then it must be parallel to the magnetic field, i.e.,

\mu _{0} j = \alpha B where \alpha is a scalar function known as the force-free parameter or force-free function. This implies that\nabla x B = \alpha B , B ⋅ \nabla\alpha = 0 . The force-free parameter can be a function of position but must be constant along field lines.

Linear force-free field

When the force-free parameter

\alpha is constant everywhere, the field is called a linear force-free field (LFFF). A constant \alpha allows for the derivation of a vector Helmholtz equation\nabla ^{2B} = - \alpha ^{2} B by taking the curl of the nonzero current density equations above.

Nonlinear force-free field

When the force-free parameter

\alpha depends on position, the field is called a nonlinear force-free field (NLFFF). In this case, the equations do not possess a general solution, and usually must be solved numerically.^{[2] [3]}

Physical examples

In the Sun's upper chromosphere and lower corona, the plasma β can locally be of order 0.01 or lower allowing for the magnetic field to be approximated as force-free.^{[4] [5] [6]}

See also

• Woltjer's theorem
• Chandrasekhar–Kendall function
• Magnetic helicity

Further reading

• Book: Marsh . Gerald E. . Force-free magnetic fields: solutions, topology and applications . 1996 . World Scientific . 981-02-2497-4 . 10.1142/2965.

Notes and References

1. Wiegelmann . Thomas . Sakurai . Takashi . Solar force-free magnetic fields . Living Reviews in Solar Physics . December 2021 . 18 . 1 . 1 . 10.1007/s41116-020-00027-4 . 232107294 . 18 May 2022.
2. Book: Bellan . Paul Murray . Fundamentals of plasma physics . 2006 . Cambridge University Press . Cambridge . 0521528003.
3. Book: Parker . E. N. . Cosmical Magnetic Fields: Their Origin and Their Activity . 2019 . Clarendon Press . Oxford . 978-0-19-882996-6.
4. Amari . T. . Aly . J. J. . Luciani . J. F. . Boulmezaoud . T. Z. . Mikic . Z. . Reconstructing the Solar Coronal Magnetic Field as a Force-Free Magnetic Field . Solar Physics . 1997 . 174 . 129–149 . 10.1023/A:1004966830232 . 1997SoPh..174..129A.
5. Low . B. C. . Lou . Y. Q. . Modeling Solar Force-Free Magnetic Fields . The Astrophysical Journal . March 1990 . 352 . 343 . 10.1086/168541 . 1990ApJ...352..343L.
6. Peter . H. . Warnecke . J. . Chitta . L. P. . Cameron . R. H. . Limitations of Force-Free Magnetic Field Extrapolations: Revisiting Basic Assumptions . Astronomy & Astrophysics . November 2015 . 584 . 10.1051/0004-6361/201527057 . 2015A&A...584A..68P . 1510.04642.