Reconstruction of an Inductive Velocity Field Vector from Doppler Motions and a Pair of Solar Vector Magnetograms

We outline a general methodology to infer the inductive velocity field vector in solar active regions. For the first time, both the field-aligned and the cross-field velocity components are reconstructed. The cross-field velocity solution accounts for the changes of the vertical magnetic field seen between a pair of successive active-region vector magnetograms via the ideal induction equation. The field-aligned velocity is obtained using the Doppler velocity and the calculated cross-field velocity. Solving the ideal induction equation in vector magnetograms measured at a given altitude in the solar atmosphere is an under-determined problem. In response, our general formalism allows the use of any additional constraint for the inductive cross-field velocity to enforce a unique solution in the induction equation. As a result, our methodology can give rise to new velocity solutions besides the one presented here. To constrain the induction equation, we use a special case of the minimum structure approximation that was introduced in previous studies and is already employed here to resolve the $180^o$-ambiguity in the input vector magnetograms. We reconstruct the inductive velocity for three active regions, including NOAA AR 8210 for which previous results exist. Our solution believably reproduces the horizontal flow patterns in the studied active regions but breaks down in cases of localized rapid magnetic flux emergence or submergence. Alternative approximations and constraints are possible and can be accommodated into our general formalism.