Penetrative convection

Abstract

A penetrative convection problem which depends on the density maximum of water near 4 °C is solved numerically. When a layer of water has its lower boundary maintained at 0 °C and its upper boundary at some temperature above 4 °C, the layer will be divided into a lower convectively unstable region and an upper convectively stable region. Steady-state finite-amplitude solutions to this problem are obtained using the mean field approximation and free boundaries.Convective mixing alters the temperature structure of the layer so that the temperature of a large fraction of the layer is slightly below the temperature of maximum density, in agreement with laboratory measurements. The largest motions are found in a principal convective cell which extends from the lower boundary to a temperature of 7° or 8 °C. Above the principal cell one or more counter cells may form, depending on the temperature of the upper boundary. The velocities in the counter cells are substantially less than those in the principal cell and fall off rapidly going upward. When the temperature of the upper boundary is 7 °C or higher, convection first takes place at a finite amplitude and at a Rayleigh number less than that predicted by the linear theory. When the temperature of the upper boundary is 10 °C or higher, the upper boundary is no longer important dynamically to the system.As the Rayleigh number is increased above critical stability the velocities in the principal cell, heat transport and the distortion of the temperature field all increase. In addition, the principal cell becomes more slender and fills a greater fraction of the layer. Also, as the Rayleigh number increases the counter cells become more flattened, fill a smaller fraction of the layer, and the velocities in the counter cells decrease relative to those in the principal cell.The most important penetration of convective motions takes place in the form of nearly horizontal motions in the lowest part of the stable region, corresponding to the upper part of the principal cell. The velocities of these motions are a large fraction of the largest vertical velocities in the unstable region.