The Penman–Monteith equation approximates net evapotranspiration (ET) from meteorological data, as a replacement for direct measurement of evapotranspiration. The equation is widely used, and was derived by the United Nations Food and Agriculture Organization for modeling potential evapotranspiration ET0.
Significance
Evapotranspiration contributions are very significant in a watershed's water balance, yet are often not emphasized in results because the precision of this component is often weak relative to more directly measured phenomena, e.g. rain and stream flow. In addition to weather uncertainties, the Penman–Monteith equation is sensitive to vegetation specific parameters, e.g. stomatal resistance or conductance.
Various forms of crop coefficients (Kc) account for differences between specific vegetation modeled and a reference evapotranspiration (RET or ET0) standard. Stress coefficients (Ks) account for reductions in ET due to environmental stress (e.g. soil saturation reduces root-zone O2, low soil moisture induces wilt, air pollution effects, and salinity). Models of native vegetation cannot assume crop management to avoid recurring stress.
Equation
Per Monteith's Evaporation and Environment, the equation is:
λv = Latent heat of vaporization. Energy required per unit mass of water vaporized. (J g−1) Lv = Volumetric latent heat of vaporization. Energy required per water volume vaporized. (Lv = 2453 MJ m−3)
E = Mass water evapotranspiration rate (g s−1 m−2) ET = Water volume evapotranspired (mm s−1)
Δ = Rate of change of saturation specific humidity with air temperature. (Pa K−1) Rn = Net irradiance (W m−2), the external source of energy flux G = Ground heat flux (W m−2), usually difficult to measure cp = Specific heat capacity of air (J kg−1 K−1) ρa = dry air density (kg m−3) δe = vapor pressure deficit (Pa) ga = Conductivity of air, atmospheric conductance (m s−1) gs = Conductivity of stoma, surface or stomatal conductance (m s−1) γ = Psychrometric constant (γ ≈ 66 Pa K−1)
Note: Often resistances are used rather than conductivities.
where rc refers to the resistance to flux from a vegetation canopy to the extent of some defined boundary layer.
The atmospheric conductance ga accounts for aerodynamic effects like the zero plane displacement height and the roughness length of the surface. The stomatal conductance gs accounts for effect of leaf density (Leaf Area Index), water stress and concentration in the air, that is to say to plant reaction to external factors. Different models exist to link the stomatal conductance to these vegetation characteristics, like the ones from P.G. Jarvis (1976) or Jacobs et al. (1996).
Accuracy
While the Penman-Monteith method is widely considered accurate for practical purposes and is recommended by the Food and Agriculture Organization of the United Nations, errors when compared to direct measurement or other techniques can range from -9 to 40%.
Variations and alternatives
FAO 56 Penman-Monteith equation
To avoid the inherent complexity of determining stomatal and atmospheric conductance, the Food and Agriculture Organization proposed in 1998 a simplified equation for a reference evapotranspiration called potential evapotranspiration ET0. It is defined as the evapotranpiration for "[an] hypothetical reference crop with an assumed crop height of 0.12 m, a fixed surface resistance of 70 s m-1 and an albedo of 0.23." This reference surface is defined to represent "an extensive surface of green grass of uniform height, actively growing, completely shading the ground and with adequate water". The corresponding equation is:
ET0 = Potential evapotranspiration, Water volume evapotranspired (mm day−1) Δ = Rate of change of saturation specific humidity with air temperature. (Pa K−1) Rn = 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].
This potential evapotranspiration ET0 can then be used to evaluate the evapotranspiration rate ET from unstressed plant through crop coefficients Kc: ET = Kc * ET0.