We put the the discretised column of atmosphere in a 1 D coordinate system.
We focus our analysis on one arbitrary column. Let A be its cross sectional area and \(\Delta z\) the thickness.
(Click on any column to explore the dynamics !)
As can be seen, 3 forces are acting on the column.
Force due to atmospheric pressure from above, $$ F_1 = P_{i+1} \times A $$
Force due to weight of the column of air, $$ F_2 = \rho Agdz $$
Force due to atmospheric pressure from below, $$ F_3 = P_{i} \times A $$
