A spatial direct numerical simulation of an axisymmetric buoyant thermal plume is presented. The governing flow field equations at the centerline are put into a special form to circumvent the axis singularity associated with the cylindrical coordinates and the high order accuracy of the numerical scheme is preserved at the centerline. Boundary conditions associated with the spatial DNS of open-boundary buoyant flows and compatible with the modern nondissipative high-order finite difference schemes have been developed. The fluid exhibits a periodic oscillatory motion known as the puffing phenomenon, which is the formation and convection of vortex at the near field of the plume. Budgets of the vorticity transport are determined to examine the mechanisms leading to the puffing phenomenon. The analysis on vorticity transport shows that vorticity is created mainly by the gravitational term which is due to the interaction between the radial density gradients and gravity at the initial stage of the establishment of the puffing structure, while the baroclinic torque dominates the vorticity transport when the flow is established. Density stratification in the radial direction close to the plume base is found to be essential to the development of the buoyant flow instability. Simulations with different initial temperature ratios reveal that entrainment close to the plume base is enhanced at a higher temperature ratio despite the fact that the puffing structures and the plume pulsation frequency only vary very weakly with the initial temperature ratio. The predicted puffing frequencies are in agreement with the values from experimental correlations for fire and isothermal helium/air plumes.