Magnetic Field and Plasma Scaling Laws: Their Implications for Coronal Heating Models

Abstract

In order to test di †erent models of coronal heating, we have investigated how the magnetic Ðeld strength of coronal Ñux tubes depends on the end-to-end length of the tube. Using photospheric magne-tograms 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 Ðeld strength, SBT, in approximately 1000 individual Ñux tubes with regularly spaced footpoints. Scatter plots of SBT versus length, L , are characterized by a Ñat section for small L and a steeply declining section for large L. They are well described by a function of the form log SBT \ C 1 ] C 2 log L ] C 3 /2 log (L2 ] S2), where and 40 ¹ S ¹ 240 Mm is related to the characteristic size of the C 2 B 0, [3 ¹ C 3 ¹ [1, active region. There is a tendency for the magnitude of to decrease as the magnetic complexity of the C 3 region increases. The average magnetic energy in a Ñux tube, SB2T, exhibits a similar behavior, with only being signiÐcantly di †erent. For Ñux tubes of intermediate length, 50 ¹ L ¹ 300 Mm, corresponding C 3 to the soft X-ray loops in a study by Klimchuk & Porter (1995), we Ðnd a universal scaling law of the form SBT P Ld, where d \ [0.88 ^ 0.3. By combining this with the Klimchuk & Porter result that the heating rate scales as L ~2, we can test di †erent models of coronal heating. We Ðnd that models involving the gradual stressing of the magnetic Ðeld, 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 deÐnitive statements about speciÐc heating mechanisms can be made.