A magneto-hydrodynamic model of boundary layers at the core–mantle boundary (CMB) is derived and used to compute the viscous and electromagnetic torques generated by the Earth's nutation forcing. The predicted electromagnetic torque alone cannot account for the dissipation estimated from the observations of the free core nutation. The presence of a viscous boundary layer in the electromagnetic skin layer at the CMB, with its additional dissipative torques, may explain the geodetic data. An apparent Ekman number at the top of the core between 2 and 4 × 10−11 is inferred depending on the electrical conductivity of the mantle.