The penetrative convection induced by a localized heat source at the bottom surface oi a partially open enclosure with a stably, thermally stratified ambient medium is studied using a finite-difference method. The stratification that is considered here is a stable two-layered temperature variation. This flow is of interest in many thermal processes such as enclosure fires. The influence of the stratification on the flow is examined. The convective motion occurring in the unstable lower-layer is found to penetrate significantly into the stable upper-layer only for a weak stratification : But small penetration occurs at large stratification levels. In addition, the flow field reveals a multi-cellular pattern and the penetrative motion occurs in the form of nearly horizontal motion in the lowest part of the stable upper-layer. The thermal field is studied in detail and other important details on this penetrative convective transport are brought out.