On the Theory of Convective Currents

A theory of steady state, saturated, convective currents is presented which includes the transfer of heat and momentum by lateral diffusion as well as the systematic entrainment of environmental air. The system of equations governing the behavior of such convective currents is derived and numerically integrated for a variety of initial and environmental conditions. In agreement with basic physical principles, it is found that the cloud height, mass, vertical velocity, temperature excess over the environment, and liquid water content all increase with increasing (a) initial temperature, (b) environmental relative humidity, and (c) environmental lapse rate; but all decrease as the diffusion coefficient increases. Also the cloud height, vertical velocity, temperature excess, and liquid water content increase with increasing initial velocity: however, the total mass of air decreases with increasing initial velocity, reflecting a lower rate of entrainment.

Even with frictional drag in the form of momentum loss by diffusion, the updraft overshoots the level of zero buoyancy in the upper part of the cloud, giving rise to cloud temperatures at the cloud top which may be several degrees colder than the environment.

When applied to an upper-air sounding taken near the time and location of a thunderstorm, the model gives the right order of magnitude for the cloud height, vertical velocity, and temperature excess over environment for a diffusion coefficient of k₁ = .001 sec^–1. While the life cycle of a convective cloud is certainly not a steady-state phenomenon, the theory probably affords a fair approximation of the convective cloud during the updraft stage.