A phase singularity of the light field created by interference of two Gaussian singular beams which propagate in a weak and near ground turbulent atmosphere is analyzed by the Rytov approximation and the short-term averaging method of the dislocation-position. We demonstrate that an edge or circular dislocation may be formed by both parallel and coaxial or noncoaxial collimated beams with different or equal beam-width interfere. The edge or circular short-term wavefront dislocations of super position field depend on the atmospheric turbulence strength, beam propagation distance, amplitude ratio, dislocation of nesting vortices, and beam-width or beam-width ratio of the individual beams.