Fig. 1

Fluid motions near the core–mantle boundary under the tangentially magnetostrophic constraint. Upper and lower figures show fluid motions at \(r={r}_{1}\) and at \(r={r}_{2}\), respectively, for a \(\sigma =1\times {10}^{5}~\mathrm{S}~{\mathrm{m}}^{-1}\), b \(\sigma =1\times {10}^{6}~\mathrm{S}~{\mathrm{m}}^{-1}\), and c \(\sigma =1\times {10}^{7}~\mathrm{S}~{\mathrm{m}}^{-1}\) at the epoch of 2010. Arrows show the horizontal flows, and color contours denote upwellings and downwellings given by \({\nabla }_{H}\cdot {{\varvec{V}}}_{H}\). For \(\rho =1.1\times {10}^{4}~\mathrm{kg}~{\mathrm{m}}^{-3}\), \(\Omega =7.29\times {10}^{-5}~\mathrm{rad}~{\mathrm{s}}^{-1}\), and a root-mean-square value of the radial magnetic field at the CMB, \({B}_{r}\approx 0.2~\mathrm{mT}\), \(\Lambda =\sigma {B}_{r}^{2}/\rho\Omega \approx 0.005\) for \(\sigma ={10}^{5}~\mathrm{S}~{\mathrm{m}}^{-1}\), \(\Lambda \approx 0.05\) for \(\sigma ={10}^{6}~\mathrm{S}~{\mathrm{m}}^{-1}\), and \(\Lambda \approx 0.5\) for \(\sigma ={10}^{7}~\mathrm{S}~{\mathrm{m}}^{-1}\)