The Circulation and Closure of Electric Currents in the Solar Atmosphere

How much information about vertical and horizontal currents flowing inthe solar atmosphere can we learn from a single vector magnetogramtaken at the photosphere? It is well-known that one can determine theradial current density by taking the curl of the magnetic fieldcomponents parallel to an idealised photospheric surface. Here, weinvestigate the additional information that can be learned abouthorizontal currents flowing in the atmosphere above the photosphere.In particular, we study the radially integrated horizontal currentdistribution, expressed as a surface current density. In ourformalism, this surface-current density is projected onto thephotospheric surface, even though it represents currents flowing atall heights in the atmosphere. Horizontal currents reflect twosources: (1) currents flowing around radial magnetic distributions;and (2) horizontal currents that originate as radial currents at thephotosphere, become horizontal above it, and return to the photosphereagain as radial currents. Our aim is to derive two-dimensional``maps'' of how currents flow through the solar atmosphere that can bederived from vector magnetic field measurements at a single height.