Lifting Condensation Level (Bolton's Saturation Temperature)

When air near the ground is lifted without exchanging heat, it cools until it reaches saturation, and from there it begins to make cloud. The height where that happens is the lifting condensation level, the base of the cumulus that forms when the ground heats the air. This page finds it from two readings at the surface: the air temperature $T$ and the dew point $T_d$, which should not be above the temperature.

The formula has two parts of different kinds. The first, $T_{LCL}$, is Bolton's empirical fit (1980, equation 15) for the temperature the parcel has when it reaches saturation, accurate to about a tenth of a kelvin. The second is physics: the dry adiabat cools a rising parcel by $g/c_p$, which is 9.76 K for each kilometre, so the cooling $T - T_{LCL}$ multiplied by $c_p/g$ is a height. Romps (2017) gives an exact expression for the same level, listed here as a second source.

In the worked example, air at 20 °C with a dew point of 10 °C gives 1,252 m. A check by a different route, lifting a parcel step by step from 1000 hPa with its mixing ratio held, finds saturation at 861.55 hPa and 280.944 K, at a height of 1252.1 m. The same route gives 1,883 m for 30 °C and 15 °C, and 624.5 m for 10 °C and 5 °C. The old rule of thumb, Espy's 125 metres for each kelvin of dew-point spread, gives 1,250 m for the first case, which differs by 0.2 %.

That rule agrees with this formula to within about 2 percent in ordinary weather (8 metres at 25 °C with a 10 K spread). The formula is the better answer when the air is very warm or very dry, where the rule drifts: at 40 °C with a 40 K spread the rule comes out about 100 metres too high. This formula describes cumulus formed by heating from the ground, not layered cloud or fog. The dew point can come from the relative humidity on the Magnus dew point page, and the cooling rate of the real air can be compared with the dry adiabat using the environmental lapse rate page.