Force-free magnetic field

A force-free magnetic field is a magnetic field that arises when the plasma pressure is so small, relative to the magnetic pressure, that the plasma pressure may be ignored, and so only the magnetic pressure is considered. For a force free field, the electric current density is either zero or parallel to the magnetic field. The name "force-free" comes from being able to neglect the force from the plasma.
Basic equations.
Neglecting the effects of gravity, the Navier–Stokes equation for a plasma, in steady state, reads
formula_1
where formula_2 is the thermal pressure, formula_3 is the magnetic field and formula_4 is the electric current. Assuming that the gas pressure formula_5 is small compared to the magnetic pressure, i.e.,
formula_6
then the pressure term can be neglected. Here formula_7 is the magnetic permeability of the plasma. Therefore,
formula_8.
This equation implies that:
formula_9. e.g. the current density is either
zero or parallel to the magnetic field, and where formula_10 is a spatial-varying function
to be determined. Combining this equation with Maxwell's equations:
formula_11
formula_12
and the vector identity:
formula_13
leads to a pair of equations for formula_10 and formula_15:
formula_16
formula_17
Physical examples.
In the corona of the Sun, the ratio of the gas pressure to the magnetic pressure can locally be of order 0.01 or lower, and in these regions the magnetic field can be described as force-free.
Mathematical limits.
and