  We present spatial-resolved measurements of the columnar vortex structures in
rotating Rayleigh-B\'enard convection. The scaled radial profiles of the
azimuthal velocity $u_{\phi}(r)$ and vertical vorticity ${\omega}(r)$ of the
vortices are analyzed and compared with the predictions of the asymptotic
theory. The results reveal that the asymptotic theory predicts accurately
$u_{\phi}(r)$ and ${\omega}(r)$ in the geostrophic convection regime, but
extension of the theory in the weak rotation regime is needed to interpret the
rotation-dependence of the experimental data. Our measurements of the mean
velocity, vorticity of the vortices, and the strength of the vortex shield
structure all indicate a flow transition from weekly rotating convection to
geostrophic convection. Results of the parameter values for the transition are
in agreement with the scaling relationship obtained from previous heat-transfer
measurements.
