Cloud base

Two numbers off a thermometer, or off any METAR, give you the height the cumulus will form at. It is the first thing anybody wants to know about a soaring day, and it takes one subtraction and one division — but the divisor is the interesting part, and almost nobody is told where it comes from.

h = (T − T_d) ÷ 2.43 × 1000

h

cumulus cloud base above the surface, in feet

T

surface temperature, °C

T_d

surface dew point, °C

2.43

the rate the spread closes with height, °C per 1,000 ft

In Fahrenheit the divisor becomes 4.37, or 228 ft per °F. The “spread over 2.5” version everybody quotes is this with the divisor rounded up, which is why it reads a little low.

Where the divisor comes from

Divide the spread by two and a half, multiply by a thousand. Every soaring textbook prints it and almost none of them say what 2.5 is. It is not a fudge factor and it is not empirical — it is the difference between two lapse rates, and once you have seen that, the whole thing stops being a rule and becomes arithmetic you could have done yourself.

A parcel of air lifted off a warm field cools as it rises, because it expands against the falling pressure around it and does work doing so. For unsaturated air that rate is fixed by physics rather than by the weather: the dry adiabatic lapse rate, 9.8 K per kilometre, which is almost exactly 3 °C per 1,000 ft.

The parcel's dew point falls too, which surprises people — but it falls far more slowly, about 0.55 °C per 1,000 ft. The parcel carries the same water it started with, so its mixing ratio is conserved; what changes is that the pressure drops, which lowers the temperature at which that water would condense.

3.0 − 0.55 = 2.43 °C per 1,000 ft

3.0

dry adiabatic lapse rate, how fast the parcel cools

0.55

dew-point lapse rate, how fast its dew point falls

2.43

how fast the gap between them closes

Two numbers chasing each other. The temperature is falling five and a half times faster than the dew point, so the gap shuts, and the height at which it shuts is the height the cloud forms at. Nothing about the weather enters into it.

So the calculation is: how far does the parcel have to go before a gap closing at 2.43 °C per thousand feet has used up the spread it started with? That is one division. The height it lands on is the lifting condensation level, and for a convective day it is where the cumulus sit.

Espy worked this out in 1841 and expressed it as 125 metres per degree of spread. In feet that is 410, against the 412 this page computes from the lapse rates above — a two-foot disagreement after nearly two centuries, which is about as settled as meteorology gets.

Why cloud base climbs through the day

Dew point is a property of the airmass, not of the afternoon. Barring a front or an advected change, it sits roughly where it was at dawn. The temperature does not — it climbs all morning.

Which means the spread opens at exactly the rate the temperature rises, and cloud base goes up with it: roughly 1,000 feet for every 2.4 °C of heating. A morning that starts 18 °C over a 10 °C dew point has its base near 3,300 ft; the same air at 27 °C puts it near 7,000. Nothing about the moisture changed. That is why the first cumulus of the day are often disappointingly low and why waiting an hour is so frequently the right call.

It also explains the opposite case. If the base is not climbing as the temperature does, the dew point is rising too — moisture is being advected in, or the ground is wet and evaporating. That is a day that will over-develop rather than one that will improve.

The thing this cannot tell you

This gives you the height a parcel would condense at if it got there. It says nothing whatever about whether it gets there.

An inversion a couple of thousand feet up will cap the convection well below the condensation level, and the answer will be a cloud base for cloud that never forms. A stable layer does the same thing more gently. That is what a sounding is for, and it is the reason a forecast built on a sounding beats two surface numbers every time. What the two surface numbers give you is a ceiling on the day — the cumulus cannot be higher than this, and they are often lower.

The other honest limitation is that the dew-point lapse rate is not a constant. It varies with temperature and pressure, and 0.55 °C per 1,000 ft is a good working value rather than a law. At the margins — a very cold day, a very high field — the answer drifts. It drifts by a few per cent, not by a factor, so it remains the right first number and the wrong last one. A real forecast model takes the whole sounding into account, and the comparison of those is a separate page.