Force-Free Magnetic Fields
E N C Y C LO P E D IA O F A S T R O N O M Y AN D A S T R O P H Y S I C S
Force-Free Magnetic Fields
In MAGNETOHYDRODYNAMICS (MHD), which describes the
fluid behavior of PLASMAS, the magnetic field B exerts a
force, the Lorentz force, given by F = (1/c)J × B , in
terms of the field and its electric current density J =
(c/4π )∇ × B , where c is the speed of light. In many
circumstances, the electrical conductivity of the plasma
is so large as to be taken infinite, in which case electric
currents change only by the inductive effect of the flowing
plasma, with negligible Ohmic decay. In the absence of
plasma motion, electric currents, once set up, persist with
no decay. A particular form of such an equilibrium state is
one in which the current density is parallel to the magnetic
field, as described by the equation
∇ × B = αB.
(1)
The Lorentz force vanishes in the plasma and the magnetic
field is said to be force free. In equation (1), the
proportionality factor α is allowed in general to vary in
space, subject to the condition
B · ∇α = 0
(2)
requiring α to be constant on a line of force, in order that
Maxwell’s equation
∇·B =0
(3)
is satisfied.
Force-free fields are expected in the outer solar
atmosphere where the plasma is electrically highly
conducting but is so tenuous that its pressure and other
body forces may be neglected compared with the typical
strength of the Lorentz force that the magnetic field and
its electric current are capable of producing. The Lorentz
force F may be written as F = −∇(B 2 /8π) + (1/4π)
(B · ∇)B , in terms of a pressure force and a tension force,
which mutually cancel when the magnetic field is force
free. The magnetic pressure B 2 /8π is a measure of the
magnitude of the Lorentz force the field is capable of
exerting. Force-free magnetic fields are therefore expected
when the ratio of the gas pressure to the magnetic pressure,
denoted by β in the plasma physics literature, is much
smaller than unity.
Equations (1) and (2) pose a nonlinear problem for the
magnetic field except when the function α is a constant.
In this exceptional case, equation (2) is trivially satisfied,
and solving equation (1), for some constant value of α =
α0 , may be transformed to solving the vector Helmholtz
equation
∇2 B + α02 B = 0
(4)
subject to suitable boundary conditions formulated for
a physical problem.
The astrophysical and plasma
physics literature contains many works on the solutions
to this linear problem. The general nonlinear problem
is not readily treated, with few known solutions. In the
following, we describe briefly some of the general physical
properties of force-free magnetic fields.
The nature of the Lorentz force is such that the
magnetic pressure force may be balanced by the magnetic
tension force only in some bounded region of space.
The magnetic pressure has an outward exertion which
cannot be globally confined by the magnetic tension force
alone. This is a fundamental property expressed by
Chandrasekhar’s virial relation:
1
1 2
B2
dV =
B r · dS − (B · r )(B · dS ) (5)
4π ∂V 2
V 8π
valid for any force-free field B in some volume V with
boundary surface ∂V . If the force-free field is selfconfining, we could take the volume V to be all space,
in which case the surface integral in equation (5) would
vanish since the field intensity would fall with distance
as fast as a dipole magnetic field. A contradiction then
arises from equation (5) unless B = 0. This classical result
implies the impossibility of creating a magnetically selfconfining plasma without rigid walls in the laboratory. In
the astrophysical context, this property shows that forcefree magnetic fields need to be anchored to regions where
the magnetic field exerts nonzero Lorentz force on the
embedding plasma.
In the solar atmosphere, only the magnetic field in
the thin layer called the photosphere can be measured
with any useful spatial resolution. Equation (5) offers a
means of estimating the energy of the magnetic field in the
atmosphere above, taken in the force-free approximation,
in terms of the measured vector field at the photosphere.
In this popular model, the volume V is often taken to
be the space above the photosphere treated as an infinite
plane. A basic motivation of this modeling work is to
demonstrate the amount of free energy that can be stored
in the parallel electric currents of a force-free field. The
stored energy is believed to be the origin of the enormous
amount (∼1031–32 erg) liberated in a large SOLAR FLARE. The
application of force-free fields to the solar atmosphere has
served to illustrate observed magnetic structures and to
illuminate basic physical points. Application with the
use of measured photospheric vector fields as inputs is
an involved and difficult numerical simulation effort, as
a result of nonlinearity, difficulties associated with the
proper posing of the boundary conditions and the ill-posed
nature of the mathematical problem.
In a perfect electrical plasma conductor, the magnetic
flux is frozen into the plasma so that the field topology
cannot change in time.
If a magnetic field, not
necessarily in force equilibrium, is taken to be given and
is then deformed by displacing the embedding perfectly
conducting plasma to an extremum in its total magnetic
energy, the result would be the force-free field as described
by equations (1) and (2), where the scalar function α
takes a distribution in space that renders the solution B
topologically identical to the given initial field.
The variational problem stated above is difficult
to treat but has very interesting physical implications.
Copyright © Nature Publishing Group 2001
Brunel Road, Houndmills, Basingstoke, Hampshire, RG21 6XS, UK Registered No. 785998
and Institute of Physics Publishing 2001
Dirac House, Temple Back, Bristol, BS1 6BE, UK
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Force-Free Magnetic Fields
The solution to the variational problem may not
be unique, in which case the nonunique solutions
may be distinguished between the force-free fields
which are linearly stable and those which are linearly
unstable (see MAGNETOHYDRODYNAMIC INSTABILITIES). Linear
magnetohydrodynamic stability of force-free magnetic
fields has been treated extensively in the literature, with
the general result that the rigid anchoring of the force-free
field lines to the base of an atmosphere provides some
degree of stability so long the lines of force are not too
lengthy within the atmosphere. Most of the force-free
fields studied in linear stability analyses are one- or twodimensional fields. Extension to non-trivial variation in
three-dimensional space, except for the case of constant
α fields, is limited although some simple examples
have been found, including some with realistic geometry
suggestive of observed solar magnetic structures. These
examples were found to be linearly stable.
More interesting is the possibility that an extremum
in magnetic energy corresponds to a force-free field which
necessarily contains magnetic tangential discontinuities
in order to satisfy the requirement that the force-free
field has a prescribed field topology. Mathematically,
this is related to the fact that, in variational problems,
the set of smooth minimizing functions to approach an
extremum is not necessarily compact. If the extremum
magnetic field must have a tangential discontinuity and
is a minimum in energy, a magnetic field having the
same magnetic topology may physically relax into such
a state with the formation of the inevitable tangential
discontinuity. Physical plasmas have large but finite
electrical conductivities so that the formation of the
tangential discontinuity in such a plasma would proceed
until the thickness of the discontinuity is small enough
for the electrical resistivity, however weak, to dissipate
the sheet. This process is a consequence of high electrical
conductivity, which, remarkably, creates the physical
circumstance for inevitable dissipation of electric currents.
The idea was proposed by E N PARKER as the origin of plasma
heating under astrophysical conditions.
The turbulent MHD relaxation of plasmas at large
magnetic Reynolds’ numbers proceeds by the formation
of magnetic tangential discontinuities which dissipate by
MAGNETIC RECONNECTION (see also MAGNETOHYDRODYNAMICS:
MAGNETIC RECONNECTION AND TURBULENCE). With resistive
effects important only at these reconnection sites, the
change in magnetic field topology proceeds in such a way
that, as a measure of the complexity in the field topology,
the MAGNETIC HELICITY is conserved in the global volume of
the plasma, as opposed to the more stringent requirement
of no change in field topology. This global conservation
of magnetic helicity is a constraint that sets the extremum
in the total magnetic energy to be one of a force-free field
having a constant α = α0 , described by equation (4). The
value of α0 is then fixed such that the linear force-free
field has the conserved amount of total magnetic helicity.
This provides a physical basis for singling out the linear
force-free fields from the nonlinear ones. Application of
E N C Y C LO P E D IA O F A S T R O N O M Y AN D A S T R O P H Y S I C S
this theory due to B Taylor to the laboratory plasmas has
been very successful. Extending the application to solar
and astrophysical magnetic fields is not straightforward
because of the problem of gauge dependence of magnetic
helicity in the case of magnetic fields which thread across
the boundary of its physical domain.
Bibliography
Canfield R and Pevtsov A (ed) 1999 Magnetic Helicity
in Space and Laboratory Plasmas (Washington, DC:
American Geophysical Union)
Low B C 1988 Astrophys. J. 330 992
Parker E N 1994 Spontaneous Current Sheets in Magnetic
Fields (Oxford: Oxford University Press)
Priest E R 1982 Solar Magnetohydrodynamics (Dordrecht:
Reidel)
Copyright © Nature Publishing Group 2001
Brunel Road, Houndmills, Basingstoke, Hampshire, RG21 6XS, UK Registered No. 785998
and Institute of Physics Publishing 2001
Dirac House, Temple Back, Bristol, BS1 6BE, UK
Boon Chye Low
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