Simple Electric Currents with Tangled Magnetic Fields

Abstract:

We consider analytically and numerically the geometry and topology of magnetic field lines, tubes and surfaces for the simple electric current systems exemplified by two ring currents. Local and global properties demonstrate ample and not expected diversity with regular and irregular field lines when the symmetry is not high. Most of field lines usually is not closed, but open and irregular with both ends lost in the finite volume. The manifold of regular closed field lines is continuum only in the case of a sufficiently high symmetry. Otherwise, this manifold is countable as in the classical case of I. Tamm (linear wire with a planar concentric ring current) representing axially symmetric imbedded tori with rational and irrational lines. Deformed magnetic surfaces and tubes do not exist globally as a rule in such cases and can be only local characteristics contrary to existing textbook cartoons. It is due to absence of independent integrals. Algebraic simplicity of formulae for the electric currents does not guarantee per se the simplicity of the field line geometry. The overall situation is usually similar to dense Kantor sets of rational and irrational numbers. We demonstrate this thesis by our numerical calculations. Hidden symmetry arguments and separation of variables discussed for the possible finding of integrals. New classification of zero points for potential magnetic fields developed using harmonic expansions up to the second and higher orders in the distance. The results are useful in the magnetic reconnection theory for different approximations and their validations in space and laboratory plasmas such as tokomaks etc. The conclusion that magnetic field chaos can naturally arise with simple electric current configurations.