The following diagram shows four cells sitting on a flat surface, just as four cells in our atmospheric simulation might sit upon the ground. We mark the center of mass of each with a cross-shaded circle. The cells each have mass M and base area A. The mass per unit area is m = M/A. Pressure drops from p in the lower cells to p−mg in the upper cells, where mg≪p.
The temperature of cell D is T. Cells B, C, and D have the same enthalpy. If D were to expand adiabatically from p to p−mg, it would cool to the same temperature as B and C. When gas expands adiabatically, TpR/Cp remains constant, where Cp is the heat capacity of the gas at constant pressure and R is the gas constant. When pressure scales by (1−mg/p), temperature scales by (1−mg/p)R/Cp. Because mg≪p, the temperature of the upper cells is very close to T(1−Rmg/pCp).
Cell A has been warmed by a small amount δT compared to cell D. Because it is warmer, it is taller than cell D by δh=mRδT/p. Cell B rises above cell C by a distance δh, and cell A itself rises by δh/2 compared to cell D
In our simulation program, we assume the temperature and pressure within each cell is uniform. Here we allow for variation in pressure from the top to bottom. The top is at p−mg/2 and the bottom is at p+mg/2, so that the total change from top to bottom is mg.
At the top of cell D, the pressure is p−mg/2, but the pressure within A at the same height is greater by mgδT/T. On the left side of our diagram is a graph of the excess pressure in cell A with altitude. Cell A pushes sideways into cell D with an average pressure mgδT/2T. Cell B pushes upon cell C with average pressure mgδT/T, which is twice as much. Cell B starts to push its way over the top of cell C, and in doing so places some of its own weight upon cell D. This additional weight causes the pressure at the base of D increases, and now cell D starts to push its way under cell A, forcing it upwards.
Thus cell A becomes buoyant, and the four cells will rotate clockwise to allow A to rise. Buoyancy manifests itself not only as an upward force upon the buoyant cell, but also as horizontal forces upon the cells above and beside the buoyant cell. The cells above slide out of the way, and the cells beside it push inwards. It is only after these horizontal movements have begun that the buoyant cell can rise.