Abstract
Magnetic fields may relax dissipatively to the minimum energy force-free condition whenever they are not constantly created or distorted. We review the axially symmetric solutions for force-free magnetic fields, especially for the non-linear field. A new formulation for the scale invariant state is given. Illustrative examples are shown. Applications to both stellar coronas and galactic halos are possible. Subsequently we study whether such force-free fields may be sustained by classical magnetic dynamo action. Although the answer is `not indefinitely’, there may be an evolutionary cycle wherein the magnetic field repeatedly relaxes to the minimum energy condition after a period of substantial growth and distortion. Different force-free dynamos may coexist at different locations. Helicity transfer between scales is studied briefly. A dynamo solution is given for the temporal evolution away from an initial linear force-free magnetic field due to both α 2 and ω terms. This can be used at the sub scale level to create a `delayed’ α effect.
Highlights
There has been some progress both in the measurement of galactic halo magnetic fields ([1,2]), and in the analytic theory of corresponding mean field galactic dynamos
The case a = 2 has no scale invariant solution. This implies that a global constant of Dimensions equal to specific angular momentum can not be associated with a scale invariant, force-free magnetic field
The velocity helicity is fundamental to the turbulent dynamo. It is worth considering how this operates in a force-free magnetic field, but we begin with a general discussion of magnetic helicity, provided as always there is no electrostatic field
Summary
There has been some progress both in the measurement of galactic halo magnetic fields ([1,2]), and in the analytic theory of corresponding mean field galactic dynamos. We show that the Low and Lou solutions are a sub-set of our scale invariant solutions, and make a detailed correspondence between the two approaches We use both the general and the scale invariant formulations to present some intriguing examples that may apply either to solar active regions, or to the MHD ‘bubbles’ or ‘domes’ detected over the nuclei of spiral galaxies (e.g., NGC 3079 [25]). In the latter case a detailed comparison with ([22]) is given. A brief appendix considers the linear force-free steady dynamo in an approach compatible with the non linear examples and shows an example of the field
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