The system under study is an Ekman boundary layer in a circular vortex bounded by a plane horizontal surface which rotates around the axis in an absolute sense. It is found that the vertical motion forced by a turbulent Ekman layer near the axis will tend to zero with the radius. In contrast, it is known from the Greenspan-Howard theory that the vertical motion is constant near the axis in the laminar case. Implications for the spin-up mechanism and for the structure of typhoons are pointed out.