## The Penman Monteith ET0 Equation

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The grass reference crop in the above definition can be fully described using the PM approach. The grass reference crop is probably, together with alfalfa, the best-studied crop regarding its aerodynamic and bulk stomatal characteristics [2, 21, 59, 60].

The aerodynamic resistance ra (Eq. 5.4) can be calculated easily because the zero plane displacement height can be estimated from the crop height h, d = 0.67h; the roughness height for momentum transfer can be given by zom = 0.123h; and the roughness height for heat and vapor transfer can be approximated by zoh = 0.1zom.

From the constant crop height h = 0.12 m, the following standardized grass reference parameters result: zero plane displacement height d = 0.08 m; roughness height for momentum transfer zom = 0.015 m; roughness height for heat and vapor transfer zoh = 0.0015 m.

Assuming a standardized height for wind-speed, temperature, and humidity measurements at 2.0 m (zm = zh = 2.0 m), and introducing the above parameters into Eq. (5.4), the following aerodynamic resistance results:

where ra is the aerodynamic resistance (s m-1), and U2 is the wind-speed measurement at 2-m height (ms-1).

When wind speed is measured at a height higher than 2.0 m, assuming the logarithmic profile of the wind (Eq. 5.3) the wind speed at 2.0 m (U2) can be obtained from the wind speed at height zm (Uz) by lnM

For the standard heights d = 0.08 m and zom = 0.015 m, the following simplified equation results:

4.87

When ra from Eq. (5.11) is combined with rs = 70 s m-1, the FAO-PM ET0 equation for 24-h periods becomes [59, 60, 75]:

0.408A(Rn - G) + y790273 ^fe - ej 1 ^ ET0 =-A + y (1 + 0.34U2)-' (5.14)

where ET0 is the reference ET (mmday-1); Rn is the net radiation at the crop surface (MJ m-2 day-1); G is the soil heat flux density (MJ m-2 d-1); T is the average temperature at 2-m height (°C); U2 is the wind speed measured at 2-m height (m s-1); (es - ea) is the vapor pressure deficit for measurements at 2-m height (kPa); A is the slope vapor pressure curve (kPa °C-1); y is the psychometric constant (kPa °C-1); 900 is the coefficient for the reference crop (kJ-1 kg K day-1) resulting from conversion of units and substitution of variables p, cp, and ra; 0.34 is the wind coefficient for the reference crop (kJ-1 kg K) resulting from the ratio rs/ra(70/208 = 0.34); and 0.408 is the value for 1/X with X = 2.45 MJ kg-1. Details on the derivation of this equation are given by Allen etal. .

To ensure the integrity of computations, the weather measurements for this equation must be taken above an extensive surface of green grass, with shading of the ground and no shortage of water.

The form of Eq. (5.14) is not very different from other Penman expressions except for the addition of T in the numerator and U2 in both the numerator and the denominator.

A form of the FAO-PM equation for calculating hourly ET0 is analyzed by Allen et al. .

The FAO-PM equation should not require local calibration or use of a localized wind function if wind speed is measured at a height of 2 m or is adjusted to this height [Eq. (5.13)].

No weather-based ET equation can be expected to predict ET perfectly under every climatic situation because of simplifications in formulation and errors in data measurement. Precision instruments under excellent environmental and biological management conditions probably will show that the FAO-PM equation (5.14) deviates at times from true measurements of grass ET0. However, it has been the opinion of the FAO Expert Consultation  that the hypothetical reference definition of the FAO-PM equation should be used as the definition for grass ET0 when deriving crop coefficients. This recommendation is based on the important need to standardize the ET0 concept and its use. 