Precipitable Water of a Layer of Air

Precipitable water is the depth of liquid water that would stand in a vessel if all the water vapour in a layer of air were condensed and collected. The American Meteorological Society's Glossary of Meteorology defines it for a layer between pressures $p_1$ and $p_2$ as the integral of the mixing ratio over pressure, divided by the density of water and the acceleration of gravity. This page does the sum for one layer, using the mean mixing ratio $\bar r$ of the layer and its pressure thickness $\Delta p$; the whole column is then the sum of its layers.

Each factor has a plain meaning. The pressure thickness $\Delta p$ is the bottom pressure minus the top pressure, and it is also the weight of the air in the layer per unit area, so dividing by $g$ gives the mass of air per unit area. Multiplying by the mean mixing ratio, the mass of vapour per mass of dry air, gives the mass of vapour. Dividing by the density of water $\rho_w$ turns that mass per unit area into a depth, which is why the answer is a length. The mean has to be weighted by pressure: the same depth comes from a thick layer at a modest mixing ratio or a thin layer at a high one.

The worked example is Stull's. A layer of 10 g/kg between 100 and 50 kPa and a layer of 2 g/kg between 50 and 20 kPa together hold 57 millimetres of water, and a single layer of 80 kPa at the pressure-weighted mean of 7 g/kg gives the same. On the page, 7 g/kg through 800 hPa gives 57.1 mm. The lower layer on its own, 10 g/kg through 500 hPa, holds 50.99 mm. Run backwards, 57.1 mm in 800 hPa returns a mean mixing ratio of 7.0 g/kg.

The page uses $g = 9.80665$ m/s² and 1000 kg/m³ for water; Stull rounds $g$ to 9.8, so for the same case the page gives 57.104 mm where Stull's $g$ gives 57.14. Precipitable water measures how moist the air column is, not how much rain to expect: storms often rain more than the column holds because wind brings in more vapour. The sources are the Glossary of Meteorology and Stull's open textbook Practical Meteorology (2017), chapter 4.