The task of finding a flow at the top of the core that fits geomagnetic observations at the Earth's surface is an underdetermined problem. Weighted regularized least squares together with the tangential geostrophy hypothesis strongly constrain the flow, inhibiting both medium- to small-scale flow features from emerging and low-latitude currents from crossing the geographic equator. In this study, we solve the inverse problem in the frozen flux approximation, using a weak regularization and alleviating the tangential geostrophy constraint. Two classes of solutions are found (I and II), both fitting the data within the errors but nonetheless showing very distinct features. Using, as a further criterion, the ability of the flow solutions to reproduce the decade variations of the length of day does not allow us, in general, to give a clear preference to one class over another. For typical flows from the whole set of solutions described above, we identify the regions on the core-mantle boundary (CMB) where tangential geostrophy is violated, and we quantify those deviations. Assuming that they are due to Lorentz forces acting on the fluid, we compute preliminary charts of electrical density currents (J) on the CMB, consistent with the frozen flux and insulating mantle hypotheses. The most pronounced features in these charts are concentrated over the hemisphere roughly centered beneath the Atlantic Ocean. In light of thermodynamical arguments, class II flows seem closer to real surface core flows. We compute an upper limit JRMS ≈ 0.4 A/m2 for geomagnetically visible density currents related to these flows.