Determining the electric field (E-field) distribution on the Suns photosphere is essential for quantitative studies of how energy flows from the Suns photosphere, through the corona, and into the heliosphere. This E-field also provides valuable input
for data-driven models of the solar atmosphere and the Sun-Earth system. We show how Faradays Law can be used with observed vector magnetogram time series to estimate the photospheric E-field, an ill-posed inversion problem. Our method uses a poloidal-toroidal decomposition (PTD) of the time derivative of the vector magnetic field. The PTD solutions are not unique; the gradient of a scalar potential can be added to the PTD E-field without affecting consistency with Faradays Law. We present an iterative technique to determine a potential function consistent with ideal MHD evolution; but this E-field is also not a unique solution to Faradays Law. Finally, we explore a variational approach that minimizes an energy functional to determine a unique E-field, similar to Longcopes Minimum Energy Fit. The PTD technique, the iterative technique, and the variational technique are used to estimate E-fields from a pair of synthetic vector magnetograms taken from an MHD simulation; and these E-fields are compared with the simulations known electric fields. These three techniques are then applied to a pair of vector magnetograms of solar active region NOAA AR8210, to demonstrate the methods with real data.
We describe the computational techniques employed in the recently updated Fourier local correlation tracking (FLCT) method. The FLCT code is then evaluated using a series of simple, 2D, known flow patterns that test its accuracy and characterize its errors.
Estimates of velocities from time series of photospheric and/or chromospheric vector magnetograms can be used to determine fluxes of magnetic energy (the Poynting flux) and helicity across the magnetogram layer, and to provide time-dependent boundary
conditions for data-driven simulations of the solar atmosphere above this layer. Velocity components perpendicular to the magnetic field are necessary both to compute these transport rates and to derive model boundary conditions. Here, we discuss some possible approaches to estimating perpendicular flows from magnetograms. Since Doppler shifts contain contributions from flows parallel to the magnetic field, perpendicular velocities are not generally recoverable from Doppler shifts alone. The induction equations vertical component relates evolution in $B_z$ to the perpendicular flow field, but has a finite null space, meaning some ``null flows, e.g., motions along contours of normal field, do not affect $B_z$. Consequently, additional information is required to accurately specify the perpendicular flow field. Tracking methods, which analyze $partial_t B_z$ in a neighborhood, have a long heritage, but other approaches have recently been developed. In a recent paper, several such techniques were tested using synthetic magnetograms from MHD simulations. Here, we use the same test data to characterize: 1) the ability of the induction equations normal component, by itself, to estimate flows; and 2) a tracking methods ability to recover flow components that are perpendicular to $mathbf{B}$ and parallel to contours of $B_z$. This work has been supported by NASA Heliophysics Theory grant NNG05G144G.
