Examination of scaling relationships involving penetration distance at thebottom of a stellar convective envelope

A number of studies in the recent past have proposed a variety of scaling r elationships among the penetration depth (Delta(d)) at the bottom of a conv ective region, the vertical velocity of the fluid (V-z), and the input flux (F-b). While a relationship of the form Delta(d) proportional to V-z(3/2) has been proposed by Schmitt and coworkers on the basis of the equations of motion for buoyant plumes, Zahn proposed a similar relationship based on s caling arguments. The relationships involving Delta(d), V-z, and the input flux are based on recent two-dimensional numerical simulations by Hurlburt and coworkers. All these scalings were recently looked into by Singh, Roxbu rgh, & Chan, who performed full. three-dimensional simulations of turbulent compressible convection for a stable-unstable-stable sandwich configuratio n. In the present study, we numerically solve the full set of Navier-Stokes equations in three dimensions in order to study the behavior of convective motions penetrating into the bottom stable layer. We take up a series of m odels differing in resolution or mesh size and aspect ratio with a view to examine, in greater detail, the scaling relationships between the penetrati on distance and other flow parameters.