Birkeland Currents: A Force-Free Field-Aligned Model

Birkeland Currents: A Force-Free Field-Aligned Model

http://www.ptep-online.com/2015/PP-41-13.PDF

Donald E. Scott
Department of Electrical Engineering (Retired), University of Massachusetts, Amherst, MA, USA.
e-mail: dascott3@cox.net The fundamental vector calculus definition of a force-free, field-aligned current in space is expanded in cylindrical coordinates, to directly obtain the Bessel partial differential equation, that specifies the magnetic field created by such a current. This result is often
called the Lundquist solution. A simple but detailed derivation is included here. The
physical properties, of the resulting intricate magnetic field structure, are described. The
cause of its characteristic counter-rotation, and counter-flows are identified. The describing
equations, are put into state-variable form, and a step-wise approximation is applied.
This solution reveals the primary effect of the force-free parameter, α, as being a scale
factor of radial distance. We show that: 1) both the axial and azimuthal magnetic and
current density components, cyclically reverse their directions with radial distance, (it's the dualistic nature of reality, everything has it's opposite form), from the central axis of the current; 2) the magnetic field extends farther from the central axis within a force-free field, than it would if produced by a current, in a long straight conductor. The total magnetic field magnitude and current density, are shown to vary inversely as the square root of r. For large r, outside the plasma, the azimuthal magnetic field is shown to vary as 1/r. These results are shown to be consistent with laboratory and astronomical observations