Cloud Base Height from Temperature and Dew Point

A cumulus base is a subtraction. On a day when the sun heats the ground, a parcel of surface air rises and cools, and the cooling brings its temperature down towards its dew point. Where the two meet, water vapour begins to condense and cloud begins to form. The formula $h \approx 125\,(T - T_d)$ turns that meeting point into one subtraction: the spread between the surface temperature $T$ and the surface dew point $T_d$, times about 125 metres for every kelvin of it. A small spread means moist air and a low base; a wide spread means dry air and a base high above the ground.

Where the 125 comes from. The rising dry air cools about 9.8 K per kilometre, while its dew point falls only about 1.8 K per kilometre, so the temperature closes on the dew point at about 8 K per kilometre, and each kelvin of spread takes about 125 metres of climb to remove. The page's sources are Espy's 1836 essay on meteorology, Romps' 2017 exact expression for the lifting condensation level, and Bolton's 1980 computation of equivalent potential temperature. Bolton's formula for the temperature at saturation is the reference the rule is checked against.

Reading the answer. The result is a height above the ground at the station, good to a few percent rather than to the metre. Take 30 °C with a dew point of 18 °C, a 12 K spread: the formula gives 1,500 m, while the FAA rule (1000 ft for every 4.4 °F of spread) gives about 4,900 ft, or 1,493.5 m, so the formula is 0.4 % high. The Navy aerographer's rule, in which temperature and dew point close at 4.5 °F per 1,000 ft, turns 85 °F and 67 °F into 4,000 ft, or 1,219.2 m, against 1,250 m from the formula, which is 2.5 % high. Against Bolton's formula, 25 °C and 15 °C give 1,258 m, and the page's 1,250 m is 0.6 % low.

Limits. The formula describes cumulus formed by surface heating, not layered cloud or fog, and the dew point must not be above the temperature. It also runs backward: the temperature is about $T_d + h/125$ and the dew point about $T - h/125$, useful when the cloud base is observed. When only the relative humidity is known, the dew-point page supplies $T_d$ first.