Force-free magnetic 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., ${\displaystyle \beta \ll 1}$. 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 ${\displaystyle \mathbf {B} }$ can be expressed as

${\displaystyle \mathbf {j} \times \mathbf {B} =\mathbf {0} ,}$

${\displaystyle \nabla \cdot \mathbf {B} =0,}$

where

${\displaystyle \mathbf {j} ={\frac {1}{\mu _{0}}}\nabla \times \mathbf {B} }$

is the current density and ${\displaystyle \mu _{0}}$ is the vacuum permeability. Alternatively, this can be written as

${\displaystyle (\nabla \times \mathbf {B} )\times \mathbf {B} =\mathbf {0} ,}$

${\displaystyle \nabla \cdot \mathbf {B} =0.}$

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 ${\displaystyle \phi }$:

${\displaystyle \mathbf {B} =-\nabla \phi .}$

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

${\displaystyle \nabla \times \mathbf {B} =\alpha \mathbf {B} ,}$

${\displaystyle \mathbf {B} \cdot \nabla \alpha =0.}$

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

${\displaystyle \nabla ^{2}\mathbf {B} =-\alpha ^{2}\mathbf {B} }$

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

Nonlinear force-free field

When the force-free parameter ${\displaystyle \alpha }$ 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.^{[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]

See also

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

References