Penman-Monteith

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 $Delta$, $c\_p$, and $delta\_q$, if not conductances below.

The United Nations Food & Agriculture Organization (FAO) standard methods for modeling Evapotranspiration (ET) use a Penman-Monteith equation [cite book |last=Allen |first=R.G. |coauthors=Pereira, L.S.; Raes, D.; Smith, M. |title=Crop Evapotranspiration—Guidelines for Computing Crop Water Requirements |url=http://www.fao.org/docrep/X0490E/x0490e00.HTM |accessdate=2007-11-24 |series=FAO Irrigation and drainage paper 56 |year=1998 |publisher=Food and Agriculture Organization of the United Nations |location=Rome, Italy |isbn=92-5-104219-5 ] . 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 [Examples of GIS integrated continuous simulation hydrologic models [ http://cobweb.ecn.purdue.edu/~aggrass/models/hydrology.html] ] 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 conductanceBeven, K. (1979) A sensitivity analysis of the Penman-Monteith actual evapotranspiration estimates. J. Hydrol. 44, 169-190.. 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.

Equation

:$overset\{\; ext\{Energy\; flux\; rate\{lambda\_v\; E=frac\{Delta\; R\_n\; +\; ho\_a\; c\_p\; left(\; delta\; q\; ight)\; g\_a\; \}\{Delta\; +\; gamma\; left\; (\; 1\; +\; g\_a\; /\; g\_s\; ight)~\; iff\; ~\; overset\{\; ext\{Volume\; flux\; rate\{ET\_o=frac\{Delta\; R\_n\; +\; ho\_a\; c\_p\; left(\; delta\; q\; ight)\; g\_a\; \}\{\; left(\; Delta\; +\; gamma\; left\; (\; 1\; +\; g\_a\; /\; g\_s\; ight)\; ight)\; lambda\_v$

:"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):δ"e" = 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) [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-AISHPubl.no.167,1987. pp. 319-327] :

Note: Often resistances are used rather than conductivities.
:$g\_a\; =\; frac\{1\}\{\; r\_a\}\; ~\; ~\; And\; ~\; ~\; g\_s\; =\; frac\{1\}\{\; r\_s\}\; =\; frac\{1\}\{\; r\_c\}$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 $delta\; q$ & $Delta$ in addition to assumptions key to reaching this simplified equation.

ee also

*John Monteith
*Evapotranspiration
*Penman equation
*Thornthwaite model

References

*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.