The penetrative convection induced by a localized heat source at the bottom of a partially open enclosure, while the background ambient medium is stably and thermally stratified, is considered. A rectangular cavity with a partially open side is assumed to be adjacent to a long corridor which is stably stratified. The stable thermal stratification that is of interest is a two-layered one or a linear temperature stratification. A control-volume finite-difference method, in stream function and vorticity formulation, is employed for the solution. The influence of the stratification parameter is examined in the laminar flow regime by considering a range of stratification levels. Heat transfer results, mass outflow rate, penetration distance, velocity and temperature distributions are presented to quantify the penetrative convection. It is found that, for a step ambient temperature profils, the convective motion occurring in the lower unstable layer penetrates significantly into the stable upper layer only for a week stratification. Small penetration occurs at large stratification levels. The flow field reveals a multi-cellular pattern. Furthermore, it is observed that 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. Other important details on this penetrative transport are brought out.