FAO-56 Penman-Monteith Reference Evapotranspiration (Daily)

Reference evapotranspiration is the water, in millimetres a day, that a well-watered stretch of short grass would lose to the air by evaporation and transpiration. The grass is a hypothetical reference crop with an assumed height of 0.12 m, a fixed surface resistance of 70 s/m and an albedo of 0.23, so the number depends on the weather and not on a particular crop. It is the standard against which crop water needs are scaled by a crop coefficient. The equation is equation 6 of chapter 4 of FAO Irrigation and Drainage Paper 56 (Allen, Pereira, Raes and Smith, 1998).

The numerator has two terms. The radiation term, $0.408\,\Delta\,(R_n - G)$, uses the net radiation at the surface minus the heat going into the soil, which is the energy available to evaporate water; the constant 0.408 brings it to millimetres of water a day. The aerodynamic term, $\gamma\,\dfrac{900}{T + 273}\,u_2\,(e_s - e_a)$, is the drying power of the air: the wind speed $u_2$ at 2 m times the vapour pressure deficit, saturation minus actual vapour pressure, so drier air and stronger wind both raise it. Here $T$ is the mean daily air temperature at 2 m in degrees Celsius, $\Delta$ is the slope of the saturation vapour pressure curve and $\gamma$ is the psychrometric constant, both in kilopascals per degree Celsius. The denominator, $\Delta + \gamma(1 + 0.34\,u_2)$, scales the sum.

The paper works an example for 6 July at Uccle near Brussels: $R_n = 13.28$ MJ/m² a day, $G = 0$, a mean temperature of 16.9 °C, a wind of 2.078 m/s at 2 m, a deficit of 0.589 kPa, $\Delta = 0.122$ and $\gamma = 0.0666$ kPa per degree. The radiation term contributes 2.81 mm a day and the aerodynamic term 1.07, a total of 3.88 that the paper rounds to 3.9 mm/day. The page gives 3.8789, since the paper's intermediate numbers are rounded to two digits.

The page takes $\Delta$ and $\gamma$ as numbers; the paper's equations 13 and 8 give them from the temperature and the pressure, and the library's saturation-vapour-pressure page is where $e_s$ and the slope come from. The wind must be the value at 2 m, so a wind measured at another height has to be brought to 2 m first; the paper uses its equation 47 to move a 10 m wind. The net radiation and the soil heat flux are daily totals, and the equation is for daily data, while the paper gives an hourly form separately.