Atmospheric column mass ($m=p_s/g$)

The mass per unit horizontal area above a level in an atmosphere. With constant gravitational acceleration, hydrostatic balance gives $m=(p_s-p_{\mathrm{top}})/g$. This relation connects an observed surface pressure to the total reservoir mass without assuming an isothermal temperature profile.

For a dry background inventory of column weight $P_d$, a mixed layer of fixed pressure thickness $\Delta p$, and saturated surface vapor partial pressure $E$, put $x=E/P$ and let $ϵ$ be the vapor-to-background molar mass ratio. The vapor mass fraction is $ϵx/[1+(ϵ-1)x]$, so hydrostatic balance and mass conservation give the displayed quadratic equation. Physical solutions require $x\le 1$ and a nonnegative upper-layer pressure. At $x=1$, $P=E=P_d+\Delta p$. A real positive algebraic root beyond that value need not be physically admissible.