Fig. 4

a The schematic view of the viscous boundary layer at the core–mantle boundary (after Matsushima 2015), b The RMS misfit for velocity at \(r_{b2} = 3428.5\) km (Eq. 42), where the gray horizontal dashed line denotes the assimilation window from 2004.50 to 2014.25, and c comparison between the real core surface flows at \(r_{b2}\) from the MCM3 model and the recovered velocity field for Case A1 (see Table 1 for details). Arrows denote the horizontal flow velocity field, while color contours show upwellings and downwellings given by horizontal divergence of the velocity field. The poloidal and toroidal scalar potentials of velocity are compared at \(r_{b2}\) in our data assimilation