||Overshooting above a convection zone is a common occurrence in nature. On earth, this is a principal process to exchange constituents of the troposphere and the stratosphere. In stars, it can change the amount of nuclear fuel supply to a convective core and thus affect the evolutionary outcome. The extent of overshooting and the amount of mixing are important questions to address. However, as the situation is highly nonlinear and nonuniform, one has to resort to numerical computation. Recent calculations by Chan et al. (2010) have shown that gravity waves are generated in the overshoot region, and they are responsible for keeping the velocities not decaying for a large height into the stable region. However, the presence of velocities does not necessarily imply mixing; this is particularly true if the velocities are associated with waves. In this thesis, the dynamics and the mixing ability of a single plume penetrating above a convection zone is studied. It is found that the gravity waves generated by the plume do not last long. A rather longterm outcome of the penetration is the generation of a stack of very flat cells in the stable layer. The horizontal cell boundaries act as barriers to mixing. This is a novel description of the upward overshooting process.