Potential Evapotranspiration

Potential evapotranspiration or PE is a measure of the ability of the atmosphere to remove water from the surface through the processes of evaporationand transpiration assuming no control on water supply.

Actual evapotranspiration or AE is the quantity of water that is actually removed from a surface due to the processes of evaporation and transpiration.

Scientists consider these two types of evapotranspiration for the practical purpose of water resource management. Around the world humans are involved in the production of a variety of plant crops. Many of these crops grow in environments that are naturally short of water. As a result, irrigation is used to supplement the crop's water needs. Managers of these crops can determine how much supplemental water is needed to achieve maximum productivity by estimating potential and actual evapotranspiration. Estimates of these values are then used in the following equation:

crop water need = potential evapotranspiration - actual evapotranspiration


The following factors are extremely important in estimating potential evapotranspiration:

  • Potential evapotranspiration requires energy for the evaporation process. The major source of this energy is from the Sun. The amount of energy received from the Sun accounts for 80% of the variation in potential evapotranspiration.
  • Wind is the second most important factor influencing potential evapotranspiration. Wind enables water molecules to be removed from the ground surface by a process known as eddy diffusion.
  • The rate of evapotranspiration is associated to the gradient of vapor pressure between the ground surface and the layer of atmosphere receiving the evaporated water.

Thornthwaite equation (1948)

PET = 16 \left(\frac{L}{12}\right)\left(\frac{N}{30}\right)\left(\frac{10\, T_a}{I}\right)^\alpha
Where
PET is the estimated potential evapotranspiration (mm/month)
T_a is the average daily temperature (degrees Celsius; if this is negative, use 0) of the month being calculated
N is the number of days in the month being calculated
L is the average day length (hours) of the month being calculated
\alpha = (6.75 \times 10^{-7}) I^3 - (7.71 \times 10^{-5}) I^2 + (1.792 \times 10^{-2}) I + 0.49239
I = \sum_{i=1}^{12} \left(\frac{T_{ai}}{5}\right)^{1.514} is a heat index which depends on the 12 monthly mean temperatures T_{ai}.[1]
Somewhat modified forms of this equation appear in later publications (1955 and 1957) by Thornthwaite and Mather. [2]

Numerous variations of the Penman equation are used to estimate evaporation from water, and land. Specifically the Penman-Monteith equation refines weather based potential evapotranspiration (PET) estimates of vegetated land areas.[1] It is widely regarded as one of the most accurate models, in terms of estimates.[citation needed]
The original equation was developed by Howard Penman at the Rothamsted Experimental Station, Harpenden, UK.
The equation for evaporation given by Penman is:
E_{mass}=\frac{m R_n + \rho_a c_p \left(\delta e \right) g_a }{\lambda_v \left(m + \gamma \right) }
where:
m = Slope of the saturation vapor pressure curve (Pa K-1)
Rn = Net irradiance (W m-2)
ρa = density of air (kg m-3)
cp = heat capacity of air (J kg-1 K-1)
ga = momentum surface aerodynamic conductance (m s-1)
δe = vapor pressure deficit (Pa)
λv = latent heat of vaporization (J kg-1)
γ = psychrometric constant (Pa K-1)
which (if the SI units in parentheses are used) will give the evaporation Emass in units of kg/(m²·s), kilograms of water evaporated every second for each square meter of area.
Remove λ to obviate that this is fundamentally an energy balance. Replace λv with L to get familiar precipitation units ETvol, where Lv=λvρwater. This has units of m/s, or more commonly mm/day, because it is flux m3/s per m2=m/s.
This equation assumes a daily time step so that net heat exchange with the ground is insignificant, and a unit area surrounded by similar open water or vegetation so that net heat & vapor exchange with the surrounding area cancels out. Some times people replace Rn with and A for total net available energy when a situation warrants account of additional heat fluxes.
temperaturewind speedrelative humidity impact the values of mgcp, ρ, and δe.

Shuttleworth (1993)

In 1993, W.Jim Shuttleworth modified and adapted the Penman equation to use SI, which made calculating evaporation simpler.[2] The resultant equation is:
E_{mass}=\frac{m R_n + \gamma * 6.43\left(1+0.536 * U_2 \right)\delta e}{\lambda_v \left(m + \gamma \right) }

where:
Emass = Evaporation rate (mm day-1)
m = Slope of the saturation vapor pressure curve (kPa K-1)
Rn = Net irradiance (MJ m-2 day-1)
γ = psychrometric constant = \frac{0.0016286 * P_{kPa}} {\lambda_v} (kPa K-1)
U2 = wind speed (m s-1)
δe = vapor pressure deficit (kPa)
λv = latent heat of vaporization (MJ kg-1)
Note: this formula implicitly includes the division of the numerator by the density of water (1000 kg m-3) to obtain evaporation in units of mm d-1

Some useful relationships

δe = (es - ea) = (1-relative humidity)es
es = saturated vapor pressure of air, as is found inside plant stoma.
ea = vapor pressure of free flowing air.
es, mmHg = exp(21.07-5336/Ta), approximation by Merva, 1975[3]
Therefore m= \Delta =\frac{d e_s}{d T_a} = \frac{5336}{T_a^2} e^{\left(21.07 - \frac{5336}{T_a}\right)}, mmHg/K


Ta = air temperature in kelvins

 

Like the Penman equation, the Penman–Monteith equation (after Howard Penman and John Monteith) predicts net evapotranspiration (ET), requiring as input daily mean temperature, wind speed, relative humidity and solar radiation. 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 and Agriculture Organization (FAO) standard methods for modeling evapotranspiration use a Penman–Monteith equation.[1] The standard methods of the American Society of Civil Engineers 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, 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.[3] Gaps in knowledge of such are filled by educated assumptions, until more specific data accumulates.
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 wiltair pollution effects, and salinity). Models of native vegetation cannot assume crop management to avoid recurring stress.

Equation

 \overset{\text{Energy flux rate}}{\lambda_v E=\frac{\Delta (R_n-G) + \rho_a c_p \left( \delta e \right) g_a } {\Delta + \gamma \left ( 1 + g_a / g_s \right)}} ~ \iff ~  \overset{\text{Volume flux rate}}{ET_o=\frac{\Delta (R_n-G) + \rho_a c_p \left( \delta e \right) g_a } { \left( \Delta + \gamma \left ( 1 + g_a / g_s \right) \right) L_v }}
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)
ETo = 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, or specific humidity (Pa)
ga = Conductivity of air, atmospheric conductance (m s−1)
gs = 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 rc 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 rigorous treatment of g_s perhaps varying within each day. Penman's equation was derived to estimate daily ET from daily averages.


This also explains relations used to obtain  \delta q  &  \Delta  in addition to assumptions key to reaching this simplified equation.

Priestley–Taylor

The Priestley–Taylor 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, \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 \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 \alpha values.[5]

 

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