Magnetic Taylor-Proudman constraint explains the flows into the Tangent Cylinder

Tangent Cylinders (TCs) have shaped our understanding of planetary dynamos and liquid cores. The Taylor-Proudman Constraint (TPC) due to planetary rotation creates these imaginary surfaces separating polar and equatorial regions but cannot explain the flows meandering through them. Here we establish and verify experimentally that magnetic fields aligned with rotation drive flows \emph{into} TCs, linked to the flows \emph{along} TCs by a \emph{magnetic} Taylor-Proudman constraint. This constraint explains and quantifies how magnetic fields reshape rotating flows in planetary interiors and magnetorotating flows in general.