The Thornthwaite formula, step by step
Thornthwaite published his method in 1948 with one specific aim: to classify climates using nothing but mean monthly temperature, which was the data that existed everywhere in the world. It computes monthly potential evapotranspiration in three steps.
1. Annual heat index. The twelve months are summed as I = Σ (Ti / 5)1.514, taking 0 for months with a negative mean temperature. It measures how much thermal energy the year accumulates.
2. Exponent. a = 6.75·10⁻⁷·I³ − 7.71·10⁻⁵·I² + 1.792·10⁻²·I + 0.49239. The polynomial has no physical justification: it is the empirical fit Thornthwaite found on data from United States catchments.
3. Uncorrected PET and the day-length correction. PET = 16 · (10·T / I)a millimetres per month, and that figure is multiplied by a factor depending on latitude and month, because the twelve months have neither the same hours of daylight nor the same number of days. In Madrid that factor runs from 0.84 in December to 1.25 in July.
What the P − PET balance says
Subtracting the month’s potential evapotranspiration from its precipitation gives the figure that really describes a place’s climate for a plant or for a reservoir. A positive value is a surplus: there is water to spare, which infiltrates or runs to the channel. A negative value is a deficit: the atmosphere could evaporate more than falls as rain, and whatever grows there will have to draw on soil storage or irrigation. The typical Spanish pattern is surplus from November to March and deficit from May to September, and what distinguishes one region from another is not the annual total but how many months that deficit lasts and how deep it is.
Where it falls short
Using temperature alone is the method’s virtue and also its limit. Real evaporation depends as well on solar radiation, on air humidity and on wind, and where those three depart from the norm Thornthwaite fails in a known direction: it underestimates PET in arid, windy climates such as the Ebro valley or southeastern Spain, where dry air and the cierzo evaporate far more than temperature suggests.
That is why agronomy and river basin authorities do not use Thornthwaite to compute irrigation, but the Penman-Monteith equation in the version standardised by the FAO in its Paper 56, which incorporates all four variables and is the international reference for reference evapotranspiration. Thornthwaite remains useful for what it was invented for: comparing and classifying climates using a variable that exists in every long series in the world, including those from a century ago.
