Magnetic configurations related to the coronal heating and solar wind generation. II. Quantitative characterization of active-region magnetic structure based on the geometric property of force-free α

We investigate the geometric property of a scalar position function, the so-called force-free α, to see how it quantitatively characterizes the magnetic structure of solar active regions, which is reminiscent of the electrostatic potential whose geometric property provides information on the structure of electrified objects. Scatter plots of (α, ∇²α) show characteristic shapes representing twisted magnetic field configurations in two model active regions, one of which is formed by an emerging magnetic flux tube composed of strongly twisted field lines while the other is formed by a weakly twisted flux tube. By generalizing the concept of an axis field line in a uniformly twisted cylindrical flux tube, we introduce a topological object called the principal field line to represent magnetic structure reconstructed using a force-free field model. The principal field line and associated scatter plot of (α, ∇²α) provide a method reminiscent of active-region radiography, which could be used for quantitative classification of active regions with those magnetic configurations responsible for heating coronal plasmas, generating solar winds, and producing potentially harmful coronal explosive phenomena.