Transformation of vortex Bessel beams during propagation in turbulent atmosphere is theoretically analyzed. Deforming influence of the random inhomogeneity of the turbulent medium on propagation of diffraction-free beams leads to disappearance of their invariant properties. In the given research, features of evolution of the spatial structure of distribution of mean intensity of vortex Bessel beams in turbulent atmosphere are analyzed. A quantitative criterion of possibility of carrying over of a dark central domain by vortex Bessel beams in a turbulent atmosphere is derived. The analysis of the behavior of several physical parameters of mean-level optical radiation shows that the shape stability of a vortex Bessel beam increases with the topological charge of this beam during its propagation in a turbulent atmosphere.