The radial average-power distribution and normalized average power of orbital-angular-momentum (OAM) modes in a vortex Gaussian beam after passing through weak-to-strong atmospheric turbulence are theoretically formulated. Based on numerical calculations, the role of the intrinsic mode index, initial beam radius and turbulence strength in OAM-mode variations of a propagated vortex Gaussian beam is explored, and the validity of the pure-phase-perturbation approximation employed in existing theoretical studies is examined. Comparison between turbulence-induced OAM-mode scrambling of vortex Gaussian beams and that of either Laguerre-Gaussian (LG) beams or pure vortex beams has been made. Analysis shows that the normalized average power of OAM modes changes with increasing receiver-aperture size until it approaches a nearly stable value. For a receiver-aperture size of practical interest, OAM-mode scrambling is severer with a larger mode index or smaller initial beam radius besides stronger turbulence. Under moderate-to-strong turbulence condition, for two symmetrically-neighboring extrinsic OAM modes, the normalized average power of the one with an index closer to zero may be greater than that of the other one. The validity of the pure-phase-perturbation approximation is determined by the intrinsic mode index, initial beam radius and turbulence strength. It makes sense to jointly control the amplitude and phase of a fundamental Gaussian beam for producing an OAM-carrying beam.