Theory and equations

Penman–Monteith combination equation

With meteorological data, the potential evapotranspiration is calculated with the Penman–Monteith combination equation Monteith (1981); Monteith and Unsworth (1990) in the form of FAO Irrigation and Drainage Paper 56 Allen et al. (1998):

ET_p = (1/λ) [ Δ (R_n − G) + ρ_a c_p (e_s − e_a) / r_a ] / [ Δ + γ (1 + r_c/r_a) ]

Symbol	Meaning
ET_p	potential evapotranspiration [mm/d], converted to the project's units
λ	latent heat of vaporization, λ = 2.501 − 0.002361 T [MJ/kg]
Δ	slope of the saturation vapor pressure curve [kPa/°C]
R_n	net radiation at the surface [MJ/m²/d]
G	soil heat flux [MJ/m²/d], neglected for daily values
ρ_a c_p	volumetric heat capacity of the air
e_s − e_a	vapor pressure deficit [kPa]
r_a, r_c	aerodynamic and canopy resistance [s/m]
γ	psychrometric constant [kPa/°C], γ = 0.0016286 P/λ, with the atmospheric pressure P = 101.3 [(293 − 0.0065 z)/293]^5.253 at the altitude z [m]

Vapor pressure

The saturation vapor pressure is the mean of e°(T_max) and e°(T_min), with e°(T) = 0.6108 exp[17.27 T/(T + 237.3)], and Δ the mean of the slopes 4098 e°(T)/(T + 237.3)² at T_max and T_min. The actual vapor pressure e_a follows from the relative humidity or is given directly Allen et al. (1998).

Aerodynamic and canopy resistance

r_a = ln[(z_m − d)/z_om] ln[(z_h − d)/z_oh] / (k² u_z)

r_c = r_l / LAI_active = 200 / LAI

with the measurement heights of wind z_m and humidity z_h, the zero-plane displacement d = 0.667 h_c, the roughness lengths z_om = 0.123 h_c and z_oh = 0.1 z_om of a crop of height h_c, the von Kármán constant k = 0.41 and the wind speed u_z; r_l = 100 s/m and LAI_active = 0.5 LAI Allen et al. (1998). The LAI comes from the crop data or from the crop height (clipped grass: 0.24 h_c with h_c in cm; alfalfa: 5.5 + 1.5 ln h_c).

Radiation

The net radiation is the difference of the net short-wave and the net long-wave radiation:

R_n = (1 − a) R_s − R_nl

R_s = (a_s + b_s n/N) R_a

R_nl = σ [(T_max,K⁴ + T_min,K⁴)/2] (a_l + b_l √e_a) f

a is the albedo, R_s the solar radiation (measured, or from the relative sunshine duration n/N after Ångström (1924); a_s = 0.25, b_s = 0.5), R_a the extraterrestrial radiation from the latitude and the day of the year, N the day length, σ the Stefan–Boltzmann constant, a_l + b_l √e_a the net emissivity (0.34, −0.14 Allen et al. (1998)) and f the cloudiness factor: f = a₁ n/N + b₁ from the sunshine duration (0.9, 0.1 Doorenbos and Pruitt (1977)), or f = a_c R_s/R_so + b_c from the ratio of the measured to the clear-sky radiation (1.35, −0.35 Allen et al. (1998)).

Partitioning

ET_p is divided into potential transpiration and evaporation with the soil cover fraction SCF = 1 − exp(−k LAI) (partitioning) Ritchie (1972); the canopy interception is subtracted from the transpiration (interception).

Literature cited in this topic

Allen, R. G., L. S. Pereira, D. Raes, and M. Smith (1998). Crop Evapotranspiration – Guidelines for Computing Crop Water Requirements. FAO Irrigation and Drainage Paper 56, Food and Agriculture Organization of the United Nations, Rome, 300 pp. https://www.fao.org/4/x0490e/x0490e00.htm
Doorenbos, J., and W. O. Pruitt (1977). Guidelines for Predicting Crop Water Requirements. FAO Irrigation and Drainage Paper 24, Food and Agriculture Organization of the United Nations, Rome, 144 pp.
Monteith, J. L. (1981). Evaporation and surface temperature. Quarterly Journal of the Royal Meteorological Society 107(451): 1–27. doi:10.1002/qj.49710745102
Monteith, J. L., and M. H. Unsworth (1990). Principles of Environmental Physics. 2nd ed., Edward Arnold, London, 291 pp.
Ritchie, J. T. (1972). Model for predicting evaporation from a row crop with incomplete cover. Water Resources Research 8(5): 1204–1213. doi:10.1029/WR008i005p01204
Ångström, A. (1924). Solar and terrestrial radiation. Quarterly Journal of the Royal Meteorological Society 50(210): 121–126. doi:10.1002/qj.49705021008

All sources: References.