Direct numerical simulation (DNS) and a linear analysis of the global instability of a buoyancy layer have been performed. The spatially developing basic flow under consideration is induced by a vertical heated flat plate immersed in a thermally stratified medium. It is revealed numerically that, depending on the modified Grashof number, the disturbed flat-plate boundary-layer flows may not relax to the basic state but instead oscillate with an intrinsic frequency. The front of globally unstable waves in numerical simulations agrees very well with the position of marginal absolute instability, and the dominant frequencies in the oscillating region are identical and tuned to the marginal absolute frequency derived from the local linear dispersion relation based on the unperturbed basic state. The front of the nonlinear global modes is thus of a pulled type in this buoyancy-driven flow system.