Finite amplitude convection in a conditionally unstable stagnant air mass

The steady-state non-linear equations of motion and heat transport for a conditionally unstable stagnant air mass between two plane horizontal boundaries are solved as a power series of a parameter of instability for the region of vertically upward motion of a disturbance symmetric about a vertical axis. For such a disturbance, which is known from previous studies to have the dimension of a typical cumulus cloud, it is shown that the magnitudes of the different velocity components in the state of maximum development will depend on the assumed values of the kinematic coefficient of eddy viscosity and the Prandtl number. Examples are given with some assumed values of these parameters in which the first term of the power series is calculated. The calculated values of the maximum vertical velocity seem to be of the same order of magnitude as in a typical cumulus cloud of moderate intensity. It is also seen that the disturbance will develop a component of the tangential velocity independent of height, which will be of the second order.