A Minimum Energy Fit Method to Reconstruct Photospheric Velocity and Magnetic Diffusivity in Active Regions from Observed Magnetograms and Dopplergrams

We introduce MEF-R, a generalization of the minimum energy fit (MEF; Longcope, Astrophys. J.612, 1181, 2004) to a non-ideal (resistive) gas. The new technique requires both vector magnetograms and Doppler velocities as input fields. However, in the case of active regions observed only with the Michelson–Doppler Imager (MDI) onboard the Solar and Heliospheric Observatory (SOHO) such as AR 9077, we have only access to line-of-sight magnetograms. We reconstruct two-dimensional maps of the magnetic diffusivity η(x,y) together with velocity components v_x(x,y), v_y(x,y), and v_z(x,y) under the linear force-free magnetic field approximation. Computed maps for v_z(x,y) very well match the Doppler velocities v_r(x,y). We find the average value 〈η〉≈10⁸ m² s⁻¹ with a standard deviation of ≈ 10¹⁰ m² s⁻¹. Such high values of η(x,y) are to be expected at some places since our magnetic diffusivity is actually eddy-diffusivity. Inside AR 9077, the maps of η(x,y) do not resemble closely the maps from classical models of the magnetic diffusivity, but they are closer to η as a function of temperature than to η as a function of electric current density.