Penman-Monteith

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Like the Penman equation, the Penman-Monteith equation requires daily mean temperature, wind speed, relative humidity, and solar radiation to predict net evapotranspiration. Other than radiation, these parameter are implicit in the derivation of $Δ$ , $c p$ , and $δ q$ , if not conductances below.

The United Nations Food & Agriculture Organization (FAO) standard methods for modeling Evapotranspiration (ET) use a Penman-Monteith equation^[1]. The ASCE standard methods modify that Penman-Monteith equation for use with an hourly time step. The SWAT model is one of many GIS integrated hydrologic models^[2] estimating ET using Penman-Monteith equations.

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, eg. rain & stream flow. In addition to weather uncertainties, the Penman-Monteith equation is sensitive to vegetation specific parameters, eg. stomatal resistance or conductance^[3]. Gaps in knowledge of such are filled by educated assumptions, until more specific data accumulates.

Various forms of crop coefficients (K_c) account for differences between specific vegetation modeled and a Reference Evapotranspiration (RET or ET₀) standard. Stress coefficients (K_s) account for reductions in ET due to environmental stress (eg. 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.

[edit] Equation

$\overset{\text{Energy flux rate}}{\lambda_v E=\frac{\Delta R_n + \rho_a c_p \left( \delta q \right) g_a } {\Delta + \gamma \left ( 1 + g_a / g_s \right)}} ~ \iff ~ \overset{\text{Volume flux rate}}{ET_o=\frac{\Delta R_n + \rho_a c_p \left( \delta q \right) g_a } { \left( \Delta + \gamma \left ( 1 + g_a / g_s \right) \right) \lambda_v }}$

λ_v = Latent heat of vaporization. Energy required per unit mass of water vaporized. (J/g)

L_v = Volumetric latent heat of vaporization. Energy required per water volume vaporized. (L_v = 2453 MJ m^-3)

E = Mass water evapotranspiration rate (g s^-1 m^-2)

ET_o = Water volume evapotranspired (m³ s^-1 m^-2)

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

R_n = Net irradiance (W m^-2), the external source of energy flux

c_p = Specific heat capacity of air (J kg^-1 K^-1)

ρ_a = dry air density (kg m^-3)

δe = vapor pressure deficit, or specific humidity (Pa)

g_a = Conductivity of air, atmospheric conductance (m s^-1)

g_s = Conductivity of stoma, surface conductance (m s^-1)

γ = Psychrometric constant (γ ≈ 66 Pa K^-1)

(Monteith, 1965)^[4]:

Note: Often resistances are used rather than conductivities.

$g_a = \tfrac{1}{ r_a} ~ ~ \And ~ ~ g_s = \tfrac{1}{ r_s} = \tfrac{1}{ r_c}$

where r_c refers to the resistance to flux from a vegetation canopy to the extent of some defined boundary layer.

Also note that $g s$ varies over each day, and in response to conditions as plants adjust such traits as stoma openings. Being sensitive to this parameter value, the Penman-Monteith equation obviates the need for more more rigorous treatment of $g s$ perhaps varying within each day. Penman's equation was derived to estimate daily ET from daily averages.

A derivation of this equation may be found at http://biomet.ucdavis.edu/evapotranspiration/PMDerivation/PMD.htm
This also explains relations used to obtain $δ q$ & $Δ$ in addition to assumptions key to reaching this simplified equation.

[edit] See also

[edit] References

^ Allen, R.G.; Pereira, L.S.; Raes, D.; Smith, M. (1998). Crop Evapotranspiration—Guidelines for Computing Crop Water Requirements, FAO Irrigation and drainage paper 56. Rome, Italy: Food and Agriculture Organization of the United Nations. ISBN 92-5-104219-5. Retrieved on 2007-11-24.
^ Examples of GIS integrated continuous simulation hydrologic models [ http://cobweb.ecn.purdue.edu/~aggrass/models/hydrology.html]
^ Beven, K. (1979) A sensitivity analysis of the Penman-Monteith actual evapotranspiration estimates. J. Hydrol. 44, 169-190.
^ Monteith, J.L. (1965) Evaporation and environment. Symp. Soc. Exp. Biol. 19, 205-224
Obtained from Forest Hydrology and Watershed Management – Hydrologie Forestiere et Amenagement des Bassins Hydrologiques (Proceedings of the Vancouver Symposium, August 1987, Actes du Co11oque de Vancouver, Aout 1987):IAHS-AISH Publ.no.167,1987. pp. 319-327

Jarvis, P.G. (1976) The interpretation of the variations in leaf water potential and stomatal conductance found in canopies in the field. Phil. Trans. R. Soc. Lond. B. 273, 593-610.