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 vectorin solar active regions. For the first time, both the field-aligned and thecross-field velocity components are reconstructed. The cross-field velocitysolution accounts for the changes of the vertical magnetic field seen between apair of successive active-region vector magnetograms via the ideal inductionequation. The field-aligned velocity is obtained using the Doppler velocity andthe calculated cross-field velocity. Solving the ideal induction equation invector magnetograms measured at a given altitude in the solar atmosphere is anunder-determined problem. In response, our general formalism allows the use ofany additional constraint for the inductive cross-field velocity to enforce aunique solution in the induction equation. As a result, our methodology cangive rise to new velocity solutions besides the one presented here. Toconstrain the induction equation, we use a special case of the minimumstructure approximation that was introduced in previous studies and is alreadyemployed here to resolve the $180^o$-ambiguity in the input vectormagnetograms. We reconstruct the inductive velocity for three active regions,including NOAA AR 8210 for which previous results exist. Our solutionbelievably reproduces the horizontal flow patterns in the studied activeregions but breaks down in cases of localized rapid magnetic flux emergence orsubmergence. Alternative approximations and constraints are possible and can beaccommodated into our general formalism.