- Source: 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, 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.,
β
≪
1
{\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
B
{\displaystyle \mathbf {B} }
can be expressed as
j
×
B
=
0
,
{\displaystyle \mathbf {j} \times \mathbf {B} =\mathbf {0} ,}
∇
⋅
B
=
0
,
{\displaystyle \nabla \cdot \mathbf {B} =0,}
where
j
=
1
μ
0
∇
×
B
{\displaystyle \mathbf {j} ={\frac {1}{\mu _{0}}}\nabla \times \mathbf {B} }
is the current density and
μ
0
{\displaystyle \mu _{0}}
is the vacuum permeability. Alternatively, this can be written as
(
∇
×
B
)
×
B
=
0
,
{\displaystyle (\nabla \times \mathbf {B} )\times \mathbf {B} =\mathbf {0} ,}
∇
⋅
B
=
0.
{\displaystyle \nabla \cdot \mathbf {B} =0.}
These conditions are fulfilled when the current vanishes or is parallel to the magnetic field.
= Zero current density
=If the current density is identically zero, then the magnetic field is the gradient of a magnetic scalar potential
ϕ
{\displaystyle \phi }
:
B
=
−
∇
ϕ
.
{\displaystyle \mathbf {B} =-\nabla \phi .}
The substitution of this into
∇
⋅
B
=
0
{\displaystyle \nabla \cdot \mathbf {B} =0}
results in Laplace's equation,
∇
2
ϕ
=
0
,
{\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.,
μ
0
j
=
α
B
{\displaystyle \mu _{0}\mathbf {j} =\alpha \mathbf {B} }
where
α
{\displaystyle \alpha }
is a scalar function known as the force-free parameter or force-free function. This implies that
∇
×
B
=
α
B
,
{\displaystyle \nabla \times \mathbf {B} =\alpha \mathbf {B} ,}
B
⋅
∇
α
=
0.
{\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
α
{\displaystyle \alpha }
is constant everywhere, the field is called a linear force-free field (LFFF). A constant
α
{\displaystyle \alpha }
allows for the derivation of a vector Helmholtz equation
∇
2
B
=
−
α
2
B
{\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
α
{\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.: 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.
See also
Woltjer's theorem
Chandrasekhar–Kendall function
Magnetic helicity
References
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.
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