Penetrative convection spanning multiple scale heights is studied within a simple stellar envelope consisting of three layers: a convectively unstable middle layer bounded above and below by stably stratified polytropes. Two-dimensional numerical simulations are used to investigate the fully compressible nonlinear motions that ensue. The cellular flows display prominent downward-directd plumes surrounded by broader regions of upflow. Such asymmetry arises because pressure fluctuations accentuate buoyancy driving in the concentrated plumes and can even lead to weak buoyancy braking in the surrounding ascending flows. As the plumes plunge downward into a region of stable stratification, they serve to excite a broad spectrum of internal gravity waves there. The induced waves are not passive, for they feed back upon the plumes by deflecting them sideways, thereby modulating the amplitude of the convection in time even in the unstable layer. The penetrative motions that billow upward into the upper stable zone are distinctly weaker, and they cascade back downward toward the unstable zone over a broad horizontal scale. The strong excitation of gravity waves by the convection has implications for gradual mixing deep within a star.
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Hurlburt et al. (1986) studied this question.