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., . 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 can be expressed as
These conditions are fulfilled when the current vanishes or is parallel to the magnetic field.[1]
Zero current density
If the current density is identically zero, then the magnetic field is the gradient of a magnetic scalar potential:
The substitution of this into results in Laplace's equation, 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., where is a scalar function known as the force-free parameter or force-free function. This implies that
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 is constant everywhere, the field is called a linear force-free field (LFFF). A constant allows for the derivation of a vector Helmholtz equation
by taking the curl of the nonzero current density equations above.
Nonlinear force-free field
When the force-free parameter 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.[1][2][3]: 50–54
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.[1][4][5][6]
^ a b cWiegelmann, Thomas; Sakurai, Takashi (December 2021). "Solar force-free magnetic fields" (PDF). Living Reviews in Solar Physics. 18 (1): 1. doi:10.1007/s41116-020-00027-4. S2CID 232107294. Retrieved 18 May 2022.
^Bellan, Paul Murray (2006). Fundamentals of plasma physics. Cambridge: Cambridge University Press. ISBN 0521528003.
^Parker, E. N. (2019). Cosmical Magnetic Fields: Their Origin and Their Activity. Oxford: Clarendon Press. ISBN 978-0-19-882996-6.
^Amari, T.; Aly, J. J.; Luciani, J. F.; Boulmezaoud, T. Z.; Mikic, Z. (1997). "Reconstructing the Solar Coronal Magnetic Field as a Force-Free Magnetic Field". Solar Physics. 174: 129–149. Bibcode:1997SoPh..174..129A. doi:10.1023/A:1004966830232.
^Low, B. C.; Lou, Y. Q. (March 1990). "Modeling Solar Force-Free Magnetic Fields". The Astrophysical Journal. 352: 343. Bibcode:1990ApJ...352..343L. doi:10.1086/168541.
^Peter, H.; Warnecke, J.; Chitta, L. P.; Cameron, R. H. (November 2015). "Limitations of Force-Free Magnetic Field Extrapolations: Revisiting Basic Assumptions". Astronomy & Astrophysics. 584. arXiv:1510.04642. Bibcode:2015A&A...584A..68P. doi:10.1051/0004-6361/201527057.
Further reading
Marsh, Gerald E. (1996). Force-free magnetic fields: solutions, topology and applications. World Scientific. doi:10.1142/2965. ISBN 981-02-2497-4.