Magnetic field and plasma scaling laws:: Their implications for coronal heating models

In order to test different models of coronal heating, we have investigated how the magnetic field strength of coronal flux tubes depends on the end-to-end length of the tube. Using photospheric magnetograms from both observed and idealized active regions, we computed potential, linear force-free, and magnetostatic extrapolation models. For each model, we then determined the average coronal field strength, < B >, in approximately 1000 individual flux tubes with regularly spaced footpoints. Scatter plots of < B > versus length, L, are characterized by a flat section for small L and a steeply declining section for large L. They are well described by a function of the form log < B > = C-1 + C-2 log L + C-3/2 log (L-2 + S-2), where C-2 approximate to 0, -3 less than or equal to C-3 less than or equal to -1, and 40 less than or equal to S less than or equal to 240 Mm is related to the characteristic size of the active region. There is a tendency for the magnitude of C-3 to decrease as the magnetic complexity of the region increases. The average magnetic energy in a flux tube, < B-2>, exhibits a similar behavior, with only C-3 being significantly different. For flux tubes of intermediate length, 50 less than or equal to L less than or equal to 300 Mm, corresponding to the soft X-ray loops in a study by Klimchuk & Porter (1995), we find a universal scaling law of the form < B > proportional to L-delta, where delta = -0.88 +/- 0.3. By combining this with the Klimchuk & Porter result that the heating rate scales as L-2, We can test different models of coronal heating. We find that models involving the gradual stressing of the magnetic field, by slow footpoint motions, are in generally better agreement with the observational constraints than are wave heating models. We conclude, however, that the theoretical models must be more fully developed and the observational uncertainties must be reduced before any definitive statements about specific heating mechanisms can be made.