# Penman-Monteith and Priestley-Taylor Evaporation

This document synthesizes the information required to determine evapotranspiration (ET) using both the FAO’s version of the Penman-Monteith (PM) equation (Allen et al. 1998), as well as the Priestley-Taylor (PT) equation. An online version of the FAO report is available here.

Where appropriate, additional references are made when the FAO documentation lacks clarity, does not provide a means of calculating a particular value required to estimate ET, or I think a particular value estimate could be improved. I tried my best to document or determine where a particular formula comes from, but it’s not always clear, so where there is a lack of certainty I provided either websites with the information or question marks when I’m unsure of a reference’s attribution.

## Penman-Monteith

The full Penman-Monteith ET equation is (Allen et al. 1998, Allen 2005):

${\lambda_v ET}_{pm,o} = \frac{\Delta \cdot (R_n - G) + \rho_a \cdot c_p \cdot \bigg(\frac{e_s - e_a}{r_a}\bigg)}{\Delta + \gamma \cdot \bigg(1 + \frac{r_s}{r_a}\bigg)}$

Where:
$$ET_{pm,o}$$: Penman-Monteith evapotranspiration in m day-1
$$\lambda_v$$: Volumetric latent heat of vaporization, 2453 MJ m-3
$$\Delta$$: Slope of the saturation vapor pressure-temperature curve (kPa C-1)
$$R_n$$: Net radiation (MJ m-2 day-1)
$$G$$: Soil heat flux density (MJ m-2 day-1)
$$\rho_a$$: Air density for a given air pressure (kg m-3)
$$c_p$$: Specific heat of air (MJ kg-1 C-1)
$$e_s$$: Saturation vapor pressure (kPa)
$$e_a$$: Actual vapor pressure (kPa)
$$r_a$$: Air resistance (day m-1)
$$\gamma$$: The psychrometric constant (kPa C-1)
$$r_s$$: Surface resistance (day m-1)

Under saturated conditions, like those associated with lakes and wetlands (when the water table is at or near the soil surface) surface resistance can be approximate as being at or near 0, simplifying the PM equation to (after rearranging for ET only and converting to mm day-1):

${ET}_{pm} = \frac{\Delta \cdot (R_n - G) + \rho_a \cdot c_p \cdot \bigg(\frac{e_s - e_a}{r_a}\bigg)}{\lambda_v \cdot (\Delta + \gamma)} \cdot {1,000}$

In this formulation:
$$ET_{pm}$$: Penman-Monteith evapotranspiration in units of mm day-1

The value of 1,000 converts m day-1 to mm day-1.

Typically, Rn and G are reported in W m-2 by some in-situ measurement device, but equations (1) and (2) require data be in units of MJ m-2 day-1. The conversion between these units is: 1 W m-2 = 0.0864 MJ m-2 day-1.

For more on converting between different units common in evapotranspiration studies: http://www.fao.org/docrep/X0490E/x0490e0i.htm#annex%201.%20units%20and%20symbols

Net radiation is often measured in the field, or by using some approximation based on elevation, longitude, latitude, etc.

### Soil heat flux

Soil heat flux is a function of both the energy leaving the surface and stored in the surface layer:

$G = G_d + S$

Where:
$$G_d$$: The heat flux of the soil at a given depth (W m-2)
$$S$$: Energy stored in the soil between the surface and a given depth (W m-2)

Generally, Gd is measured with a heat flux plate, while S is determined using physical relationships, such as:

$S = \frac{\Delta T_s \cdot C_s \cdot d}{t}$

Where:
$$\Delta T_s$$: Change is soil temperature over a given time period (C)
$$C_s$$: The heat capacity of moist soil (J -3 C-1)
$$d$$: Depth of the heat flux plate (m); generally within 10 cm of the soil surface
$$t$$: The time period of interest (seconds)

In turn, the heat capacity of moist soil can be represented via:

$C_s = \rho_b \cdot C_d + \theta_v \cdot \rho_w \cdot C_w$

Where:
$$\rho_b$$: Bulk density of the soil kg m-3
$$C_d$$: Specific heat of dry soil (J kg-1 C-1)
$$\theta_v$$: Volumetric water content [-]
$$\rho_w$$: Density of water kg m-3
$$C_w$$: Specific heat of water (J kg-1 C-1)

Some additional information related to equations (3)-(5), as well as their practical concerns, are provided in Campbell Scientific’s manual for the HFP01: https://s.campbellsci.com/documents/au/manuals/hfp01.pdf

### Aerodynamic resistance

Aerodynamic resistance, $$r_a$$, can be determined via (Allen et al. 1998):

$r_a = \frac{\ln\bigg[\frac{z_m-d}{z_{om}}\bigg] \cdot \ln\bigg[\frac{z_h-d}{z_{oh}}\bigg]}{86,400 \cdot k^2 \cdot U_z}$

Where:
$$z_m$$: Height of the wind measurement (m)
$$z_h$$: Height of the humidity measurement (m)
$$d$$: Zero plane displacement height (m); Effectively the height at which vegetation is “stiff” (doesn’t move); See Dingman (2002) for more details.
$$z_{om}$$: Roughness length governing momentum transfer (m)
$$z_{oh}$$: Roughness length governing transfer of heat and vapor (m)
$$k$$: von Karman’s constant, 0.41 [-]
$$U_z$$: Wind speed at a height of zm (m s-1)

Common values for some of the above variables:
$$d = h \cdot 2/3$$
$$z_{om} = h \cdot 0.123$$
$$z_{oh} = z_{om} \cdot 0.1$$

Where $$h$$ is the height of the vegetation (m).

The value of 86,400 in the denominator converts seconds to days.

### Saturation vapor pressure

Saturation vapor pressure of water, $$e_s$$, can be estimated via (Buck 1981):

$e_s = [1.0007 + (3.46 \times 10^{-5} \cdot P)] \cdot 6.1121 \cdot \exp \frac{17.502 \cdot T_a}{240.97 + T_a}$

Where:
$$P$$: Air pressure (kPa)
$$T_a$$: Air temperature (C)

### Vapor pressure

Actual vapor pressure, $$e_a$$, is determined via (Dingman 2002):

$e_a = RH \cdot e_s$

Where:
$$RH$$: Relative humidity [-]

### Slope of the saturation vapor pressure curve

The slope of the saturation vapor pressure curve, $$\Delta$$, is determined by the following relationship (Dingman 2002):

$\Delta = \frac{4098 \cdot e_s}{(T_a + 237.3)^2}$

### Air density

The following equation was extracted from: https://www.engineeringtoolbox.com/density-air-d_680.html

Air density, $$\rho_a$$, uses standard relationships that can be determined from the Ideal Gas Law. The following accounts for both moist and dry components of the atmosphere (Green and Perry 2008?):

$\rho_a = \frac{\frac{P \cdot 1,000}{R_a \cdot (T_a + 273.15)} \cdot (1 + X)}{1 + X \cdot \frac{R_v}{R_a}}$

where:
$$R_a$$: The gas constant for dry air, 286.9 (J kg-1 K-1)
$$R_v$$: The gas constant for water vapor, 461.5 (J kg-1 K-1)
$$X$$: Humidity ratio (kg water vapor / kg dry air)

The value $$P \cdot 1,000$$ converts kPa to Pa, while $$T_a + 273.15$$ converts Celsius to Kelvin.

The ratio of Ra to Rv is ≈ 0.622.

The humidity ratio, $$X$$, (aka, moisture content, mixing ratio) can be determined via (Green and Perry 2008?):

$X = \frac{R_a / R_v \cdot e_a}{P - e_a}$

### Specific heat of air

Specific heat equation is from: https://en.wiktionary.org/wiki/humid_heat.

Specific heat of air, $$c_p$$, can be estimated as (Green and Perry 2008?):

$c_p = \frac{1.005 + 1.82 \cdot X}{1000}$

Note that the above is a slight modification from the source equation, to ensure the final units are in MJ kg-1 C-1 and not kJ kg-1 C-1.

### Psychrometric constant

The psychrometric constant, $$\gamma$$, is determined from the following (Dingman 2002): $\gamma = \frac{c_p \cdot P}{\frac{R_a}{R_v} \cdot \lambda_m}$

Where:
$$\lambda_m$$: latent heat of vaporization for a mass, 2.453 MJ kg-1

## Priestley-Taylor

The Priestley-Taylor (PT) equation is similar to the Penman-Monteith method, but simplified. That is, vapor deficit and convection terms are reduced to a single empirical constant, α. The PT model can be represented as follows:

$ET_{pt} = \alpha \cdot \frac{\Delta \cdot (R_n - G)}{\lambda_v \cdot (\Delta + \gamma)} \cdot 1,000$

Where:
$$ET_{pt}$$: Priestley-Taylor evapotranspiration in units of mm day-1
$$\alpha$$: An empirical constant accounting for the vapor pressure deficit and resistance values [-]

Typically, α is 1.26 for open bodies of water, but has a wide range of values from less than 1 (humid conditions) to almost 2 (arid conditions).

Similar to the PM model, the same terms and relationships (e.g., equations in the Penman-Monteith section of this document) can be used to generate ETpt.

### Bibliography

Allen, RG, LS Pereira, D Raes, & M Smith. 1998. Crop evapotranspiration-Guidelines for computing crop water requirements-FAO Irrigation and drainage. paper 56. FAO, Rome, 300, 6541.

Allen, RG. 2005. The ASCE standardized reference evapotranspiration equation. Amer Society of Civil Engineers.

Buck AL. 1981. New equations for computing vapor pressure and enhancement factor. Journal of Applied Meteorology 20 (12): 1527-1532 DOI: 10.1175/1520-0450(1981)020<1527:NEFCVP>2.0.CO;2.

Dingman, SL, 2002. Physical hydrology. Waveland Press.

Green, DW & RH Perry. 2008. Perry’s chemical engineers’ handbook. McGraw-Hill.

#### Technical Bibliography

Script used to generate numbers to the right of display equations:

https://stackoverflow.com/questions/35026405/auto-number-equations-in-r-markdown-documents-in-rstudio

To prevent a number from showing up use \nonumber in the equation. For ex, $$E=mc^2\nonumber$$ results in:

$E=mc^2\nonumber$

Updated: