Deriving Potential Coronal Magnetic Fields from Vector Magnetograms
Abstract
The minimumenergy configuration for the magnetic field above the solar photosphere is curlfree (hence, by Ampère's law, also currentfree), so can be represented as the gradient of a scalar potential. Since magnetic fields are divergence free, this scalar potential obeys Laplace's equation, given an appropriate boundary condition (BC). With measurements of the full magnetic vector at the photosphere, it is possible to employ either Neumann or Dirichlet BCs there. Historically, the Neumann BC was used with available lineofsight magnetic field measurements, which approximate the radial field needed for the Neumann BC. Since each BC fully determines the 3D vector magnetic field, either choice will, in general, be inconsistent with some aspect of the observed field on the boundary, due to the presence of both currents and noise in the observed field. We present a method to combine solutions from both Dirichlet and Neumann BCs to determine a hybrid, "leastsquares" potential field, which minimizes the integrated square of the residual between the potential and actual fields. We also explore weighting the residuals in the fit by spatially uniform measurement uncertainties. This has advantages both in not overfitting the radial field used for the Neumann BC, and in maximizing consistency with the observations. We demonstrate our methods with SDO/HMI vector magnetic field observations of active region 11158, and find that residual discrepancies between the observed and potential fields are significant, and they are consistent with nonzero horizontal photospheric currents. We also analyze potential fields for two other active regions observed with two different vector magnetographs, and find that hybridpotential fields have significantly less energy than the Neumann fields in every case  by more than 10^{32} erg in some cases. This has major implications for estimates of free magnetic energy in coronal field models, e.g., nonlinear forcefree field extrapolations.
 Publication:

Solar Physics
 Pub Date:
 August 2016
 DOI:
 10.1007/s1120701609386
 arXiv:
 arXiv:1503.08754
 Bibcode:
 2016SoPh..291.1681W
 Keywords:

 Active regions;
 magnetic fields;
 Electric currents and current sheets;
 Magnetic fields;
 models;
 Sunspots;
 Astrophysics  Solar and Stellar Astrophysics
 EPrint:
 25 pages, 11 figures, under review