We present the results of a series of high-resolution, three-dimensional numerical experiments that investigate the nature of turbulent compressible convective motions extending from a convection zone into a stable layer below. In such convection, converging flows in the near-surface cellular convecting network create strong downflowing plumes that can traverse the multiple scale heights of the convection zone. Such structures can continue their downward motions beyond the convecting region, piercing the stable layer, where they are decelerated by buoyancy braking. If these motions mix entropy to an adiabatie state below the convection zone, the process is known as penetration; otherwise it is termed overshooting. We find that in three-dimensional turbulent compressible convection at the parameters studied, motions generally overshoot a significant fraction of the local pressure scale height but do not establish an adiabatie penetrative region, even at the highest Peclet numbers considered. This is mainly due to the low filling factor of the turbulent plumes. The scaling of the overshooting depth with the relative stability S of the two layers is affected by this lack of true penetration. Only an S~(-1) dependence is exhibited, reflecting the existence of a thermal adjustment region without a nearly adiabatie penetration zone. Rotation about a vertical axis decreases the depth of overshooting, owing to horizontal mixing induced by the rotation. For rotation about an inclined axis, turbulent rotational alignment of the strong downflow structures decreases the overshooting further at mid-latitudes, but the laminar effects of cellular roll solutions take over at low latitudes. Turbulent penetrative convection is quite distinct from its laminar counterpart and from the equivalent motions in a domain confined by impenetrable horizontal boundaries. Although overshooting would not be so deep in the solar case, the lack of true penetration extending the adiabatie region may explain why helioseismic inferences show little evidence of the expected abrupt change between the convection zone and the radiative interior. These results may also provide insight into how overshooting motions can provide a coupling between the solar convection zone and the tachocline.
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