Based on the generalized Huygens-Fresnel principle and the expression of the cross spectral density function, the integral form of intensity distribution of partially coherent Gaussian-Schell vortex beams propagating in the atmospheric turbulence is derived. A complete analytic expression is obtained by some special integral processings. The impacts of transmission distance, turbulence intensity and beam parameters on the beam intensity distribution are discussed. The results show that the intensity distribution of the partially coherent vortex beams in the atmospheric turbulence is gradually transformed into the Gaussian distribution by the hollow distribution with increase of the transmission distance. When the topological charge number is zero, the intensity distribution keeps Gaussian distribution with increase of the transmission distance. In terms of invariant turbulence intensity, the smaller the topological charge number is, the shorter the coherent length is, and the longer the waist width is, then the shorter the transmission distance is needed when the intensity distribution is transformed from hollow distribution to Gaussian distribution.