Thornthwaite equation (1948)

${\displaystyle PET=16\left({\frac {L}{12}}\right)\left({\frac {N}{30}}\right)\left({\frac {10T_{d}}{I}}\right)^{\alpha }}$ Where

${\displaystyle PET}$ is the estimated potential evapotranspiration (mm/month)

${\displaystyle T_{d}}$ is the average daily temperature (degrees Celsius; if this is negative, use ${\displaystyle 0}$) of the month being calculated

${\displaystyle N}$ is the number of days in the month being calculated

${\displaystyle L}$ is the average day length (hours) of the month being calculated

${\displaystyle \alpha =(6.75\times 10^{-7})I^{3}-(7.71\times 10^{-5})I^{2}+(1.792\times 10^{-2})I+0.49239}$

${\displaystyle I=\sum _{i=1}^{12}\left({\frac {T_{m_{i}}}{5}}\right)^{1.514}}$ is a heat index which depends on the 12 monthly mean temperatures ${\displaystyle T_{m_{i}}}$.^[1]

Somewhat modified forms of this equation appear in later publications (1955 and 1957) by Thornthwaite and Mather. ^[2]

Penman equation (1948)

The Penman equation describes evaporation (E) from an open water surface, and was developed by Howard Penman in 1948. Penman's equation requires daily mean temperature, wind speed, air pressure, and solar radiation to predict E. Simpler Hydrometeorological equations continue to be used where obtaining such data is impractical, to give comparable results within specific contexts, e.g. humid vs arid climates.

FAO 56 Penman–Monteith equation (1998)

The Penman–Monteith equation refines weather based evapotranspiration (ET) estimates of vegetated land areas. This equation was then derived by FAO for retrieving the potential evapotranspiration ₀. ^[3] It is widely regarded as one of the most accurate models, in terms of estimates.

${\displaystyle ET_{o}={\frac {0.408\Delta (R_{n}-G)+{\frac {900}{T}}\gamma u_{2}\delta e}{\Delta +\gamma (1+0.34u_{2})}}}$

ET₀ = Potential evapotranspiration, Water volume evapotranspired (mm day⁻¹)

Δ = Rate of change of saturation specific humidity with air temperature. (Pa K⁻¹)

R_n = Net irradiance (MJ m⁻² day⁻¹), the external source of energy flux

G = Ground heat flux (MJ m⁻² day⁻¹), usually equivalent to zero on a day

T = Air temperature at 2m (K)

u_2 = Wind speed at 2m height (m⁻¹)

δe = vapor pressure deficit (kPa)

γ = Psychrometric constant (γ ≈ 66 Pa K⁻¹)

N.B.: The coefficient 0.408 and 900 are not unitless but account for the conversion from energy values to equivalent water depths: radiation [mm day⁻¹] = 0.408 radiation [MJ m⁻² day⁻¹].

Priestley–Taylor

The Priestley–Taylor equation was developed as a substitute to the Penman–Monteith equation to remove dependence on observations. For Priestley–Taylor, only radiation (irradiance) observations are required. This is done by removing the aerodynamic terms from the Penman–Monteith equation and adding an empirically derived constant factor, ${\displaystyle \alpha }$.

The underlying concept behind the Priestley–Taylor model is that an air mass moving above a vegetated area with abundant water would become saturated with water. In these conditions, the actual evapotranspiration would match the Penman rate of potential evapotranspiration. However, observations revealed that actual evaporation was 1.26 times greater than potential evaporation, and therefore the equation for actual evaporation was found by taking potential evapotranspiration and multiplying it by ${\displaystyle \alpha }$. The assumption here is for vegetation with an abundant water supply (i.e. the plants have low moisture stress). Areas like arid regions with high moisture stress are estimated to have higher ${\displaystyle \alpha }$ values.^[4]

The assumption that an air mass moving over a vegetated surface with abundant water saturates has been questioned later. The lowest and turbulent part of the atmosphere, the atmospheric boundary layer, is not a closed box, but constantly brings in dry air from higher up in the atmosphere towards the surface. As water evaporates more easily into a dry atmosphere, evapotranspiration is enhanced. This explains the larger than unity value of the Priestley-Taylor parameter ${\displaystyle \alpha }$. The proper equilibrium of the system has been derived and involves the characteristics of the interface of the atmospheric boundary layer and the overlying free atmosphere.^[5][6]