Physics:Force-free magnetic field

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Short description: Approximation in plasma physics
The magnetic field in the Sun's corona is often approximated as a force-free field.

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., [math]\displaystyle{ \beta \ll 1 }[/math]. 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 [math]\displaystyle{ \mathbf{B} }[/math] can be expressed as

[math]\displaystyle{ \mathbf{j} \times \mathbf{B} = \mathbf{0}, }[/math]
[math]\displaystyle{ \nabla \cdot \mathbf{B} = 0, }[/math]

where

[math]\displaystyle{ \mathbf{j} = \frac{1}{\mu_0}\nabla \times \mathbf{B} }[/math]

is the current density and [math]\displaystyle{ \mu_0 }[/math] is the vacuum permeability. Alternatively, this can be written as

[math]\displaystyle{ (\nabla \times \mathbf{B}) \times \mathbf{B} = \mathbf{0}, }[/math]
[math]\displaystyle{ \nabla \cdot \mathbf{B} = 0. }[/math]

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 [math]\displaystyle{ \phi }[/math]:

[math]\displaystyle{ \mathbf{B} = -\nabla\phi. }[/math]

The substitution of this into [math]\displaystyle{ \nabla \cdot \mathbf{B} = 0 }[/math] results in Laplace's equation, [math]\displaystyle{ \nabla^2\phi = 0, }[/math] 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., [math]\displaystyle{ \mu_0 \mathbf{j} = \alpha \mathbf{B} }[/math] where [math]\displaystyle{ \alpha }[/math] is a scalar function known as the force-free parameter or force-free function. This implies that

[math]\displaystyle{ \nabla \times \mathbf{B} = \alpha\mathbf{B}, }[/math]
[math]\displaystyle{ \mathbf{B} \cdot \nabla\alpha = 0. }[/math]

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 [math]\displaystyle{ \alpha }[/math] is constant everywhere, the field is called a linear force-free field (LFFF). A constant [math]\displaystyle{ \alpha }[/math] allows for the derivation of a vector Helmholtz equation

[math]\displaystyle{ \nabla^2\mathbf{B} = -\alpha^2 \mathbf{B} }[/math]

by taking the curl of the nonzero current density equations above.

Nonlinear force-free field

When the force-free parameter [math]\displaystyle{ \alpha }[/math] 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 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]

See also

References

  1. ↑ 1.0 1.1 1.2 Wiegelmann, Thomas; Sakurai, Takashi (December 2021). "Solar force-free magnetic fields". Living Reviews in Solar Physics 18 (1): 1. doi:10.1007/s41116-020-00027-4. https://link.springer.com/content/pdf/10.1007/s41116-020-00027-4.pdf. Retrieved 18 May 2022. 
  2. ↑ Bellan, Paul Murray (2006). Fundamentals of plasma physics. Cambridge: Cambridge University Press. ISBN 0521528003. 
  3. ↑ Parker, E. N. (2019). Cosmical Magnetic Fields: Their Origin and Their Activity. Oxford: Clarendon Press. ISBN 978-0-19-882996-6. 




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