A THEORY OF ENTRAINMENT IN CONVECTIVE CURRENTS

Abstract

It is held that entrainment is a necessary dynamic consequence of the vertical stretching of an accelerated convective column. On this basis, equations are developed for the rate of entrainment, the vertical divergence and the lapse rate, for both unsaturated air and cloud air. It is assumed that a steady state exists, the cross section of the rising column is invariant with height, the entrained air is uniformly mixed with the rising air and the environment is at rest. The equations are integrated numerically over height in a number of selected cases. In unsaturated air, entrainment results in a lapse rate which is always greater than the dry adiabatic; if the environmental lapse-rate is superadiabatic, the lapse rate of the rising air is intermediate between the lapse rate of the environment and the dry adiabatic lapse-rate. In a cloud, entrainment results in a lapse rate intermediate between the environmental lapse-rate and the moist adiabatic lapse-rate. The lapse rate of the rising air increases as the relative humidity of the environment decreases. As a result of entrainment, the cloud liquid-water content increases with height at a significantly slower rate than would result from a simple lifting process. A decrease in the humidity of the environment reduces the rate of increase of liquid water with height, but it does not appear possible to ?dry out? the cloud even in a dry environment. The horizontal velocity-convergence is found to be of the order of 10?3 sec?1, and the computed vertical velocities in the cloud are in general agreement with those observed by the Thunderstorm Project. It is pointed out that the entrained air may be added through an ordered inflow, by turbulent exchange or by a combination of the two. It is assumed here that an ordered inflow occurs.