Potential evapotranspiration (PET) was a concept originally introduced by Thornthwaite (1948) as part of a climate classification scheme. He defined PET is the rate at which evapotranspiration would occur from a large area uniformly covered with growing vegetation that has access to an unlimited supply of soil water and that was not exposed to advection or heat storage effects. Because the evapotranspiration rate is strongly influenced by a number of vegetative surface characteristics, Penman (1956) redefined PET as “the amount of water transpired ... by a short green crop, completely shading the ground, of uniform height and never short of water”. Penman used grass as his reference crop, but later researchers (Jensen et al., 1990) have suggested that alfalfa at a height of 30 to 50 cm may be a more appropriate choice.

Numerous methods have been developed to estimate PET. Three of these methods have been incorporated into SWAT+: the Penman-Monteith method (Monteith, 1965; Allen, 1986; Allen et al., 1989), the Priestley-Taylor method (Priestley and Taylor, 1972) and the Hargreaves method (Hargreaves et al., 1985). The model will also read in daily PET values if the user prefers to apply a different potential evapotranspiration method.

The three PET methods included in SWAT+ vary in the amount of required inputs. The Penman-Monteith method requires solar radiation, air temperature, relative humidity and wind speed. The Priestley-Taylor method requires solar radiation, air temperature and relative humidity. The Hargreaves method requires air temperature only.

Studies in canopy resistance have shown that the canopy resistance for a well-watered reference crop can be estimated by dividing the minimum surface resistance for a single leaf by one-half of the canopy leaf area index (Jensen et al., 1990):

$r_c=r_l/(0.5*LAI)$ 2:2.2.8

where $r_c$ is the canopy resistance (s m$^{-1}$), $r_{l}$ is the minimum effective stomatal resistance of a single leaf (s m$^{-1}$), and $LAI$ is the leaf area index of the canopy.

The distribution of stomates on a plant leaf will vary between species. Typically, stomates are distributed unequally on the top and bottom of plant leaves. Plants with stomates located on only one side are classified as hypostomatous while plants with an equal number of stomates on both sides of the leaf are termed amphistomatous. The effective leaf stomatal resistance is determined by considering the stomatal resistance of the top (adaxial) and bottom (abaxial) sides to be connected in parallel (Rosenburg et al., 1983). When there are unequal numbers of stomates on the top and bottom, the effective stomatal resistance is calculated:

$r_{l}=\frac{r_{l -ad}*r_{l-ab}}{r_{l-ab}+r_{l-ad}}$ 2:2.2.9

where $r_{l}$ is the minimum effective stomatal resistance of a single leaf (s m$^{-1}$), $r_{l-ad}$ is the minimum adaxial stomatal leaf resistance (s m$^{-1}$), $r_{l-ab}$ and is the minimum abaxial stomatal leaf resistance (s m$^{-1}$). For amphistomatous leaves, the effective stomatal resistance is:

$r_{l}=\frac{r_{l-ad}}{2}=\frac{r_{l-ab}}{2}$ 2:2.2.10

For hypostomatous leaves the effective stomatal resistance is:

$r_{l}=r_{l-ad}=r_{l-ab}$ 2:2.2.11

Leaf conductance is defined as the inverse of the leaf resistance:

$g_{l}=\frac{1}{r_{l}}$ 2:2.2.12

where $g_{l}$ is the maximum effective leaf conductance (m s$^{-1}$). When the canopy resistance is expressed as a function of leaf conductance instead of leaf resistance, equation 2:2.2.8 becomes:

$r_c=(0.5*g_l*LAI)^{-1}$ 2:2.2.13

where $r_c$ is the canopy resistance (s m$^{-1}$), $g_{l}$ is the maximum conductance of a single leaf (m s$^{-1}$), and $LAI$ is the leaf area index of the canopy.

For climate change simulations, the canopy resistance term can be modified to reflect the impact of change in CO$_2$ concentration on leaf conductance. The influence of increasing CO$_2$ concentrations on leaf conductance was reviewed by Morison (1987). Morison found that at CO$_2$ concentrations between 330 and 660 ppmv, a doubling in CO$_2$ concentration resulted in a 40% reduction in leaf conductance. Within the specified range, the reduction in conductance is linear (Morison and Gifford, 1983). Easterling et al. (1992) proposed the following modification to the leaf conductance term for simulating carbon dioxide concentration effects on evapotranspiration:

$g_{l,CO_2}=g_{l}*[1.4-0.4*(CO_2/330)]$ 2:2.2.14

where $g_{l,CO_2}$ is the leaf conductance modified to reflect CO$_2$ effects (m s$^{-1}$) and CO$_2$ is the concentration of carbon dioxide in the atmosphere (ppmv).

Incorporating this modification into equation 2:2.2.8 gives

When calculating actual evapotranspiration, the canopy resistance term is modified to reflect the impact of high vapor pressure deficit on leaf conductance (Stockle et al, 1992). For a plant species, a threshold vapor pressure deficit is defined at which the plant’s leaf conductance begins to drop in response to the vapor pressure deficit. The adjusted leaf conductance is calculated:

The Penman-Monteith equation combines components that account for energy needed to sustain evaporation, the strength of the mechanism required to remove the water vapor and aerodynamic and surface resistance terms. The Penman-Monteith equation is:

$\lambda E=\frac{\Delta*(H_{net}-G)+\rho_{air}*c_p*[e^o_z-e_z]/r_a}{\Delta+\gamma*(1+r_c/r_a)}$ 2:2.2.1

where $\lambda E$ is the latent heat flux density (MJ m$^{-2}$ d$^{-1}$), $E$ is the depth rate evaporation (mm d$^{-1}$), $\Delta$ is the slope of the saturation vapor pressure-temperature curve, $de/dT$ (kPa ˚C$^{-1}$), $H_{net}$ is the net radiation (MJ m$^{-2}$ d$^{-1}$), $G$ is the heat flux density to the ground (MJ m$^{-2}$ d$^{-1}$), $\rho_{air}$ is the air density (kg m$^{-3}$), $c_p$ is the specific heat at constant pressure (MJ kg$^{-1}$ ˚C$^{-1}$), is the saturation vapor pressure of air at height $z$ (kPa), $e_z$ is the water vapor pressure of air at height $z$ (kPa), $\gamma$ is the psychrometric constant (kPa ˚C$^{-1}$), $r_c$ is the plant canopy resistance (s m$^{-1}$), and $r_a$ is the diffusion resistance of the air layer (aerodynamic resistance) (s m$^{-1}$).

For well-watered plants under neutral atmospheric stability and assuming logarithmic wind profiles, the Penman-Monteith equation may be written (Jensen et al., 1990):

$\lambda E_t=\frac{\Delta*(H_{net}-G)+\gamma*K_1*(0.622*\gamma*\rho_{air}/P)*(e^o_z-e_z)/r_a}{\Delta+\gamma*(1+r_c/r_a)}$ 2:2.2.2

where $\lambda$ is the latent heat of vaporization (MJ kg$^{-1}$), $E_t$ is the maximum transpiration rate (mm d$^{-1}$), $K_1$ is a dimension coefficient needed to ensure the two terms in the numerator have the same units (for $u_z$ in m s$^{-1}$, $K_1$ = 8.64 x 104), and $P$ is the atmospheric pressure (kPa).

The calculation of net radiation, $H_{net}$, is reviewed in Chapter 1:1. The calculations for the latent heat of vaporization, $\lambda$, the slope of the saturation vapor pressure-temperature curve, $\Delta$, the psychrometric constant, $\gamma$, and the saturation and actual vapor pressure, $e^o_z$and $e_z$, are reviewed in Chapter 1:2. The remaining undefined terms are the soil heat flux, $G$, the combined term $K_1 0.622 \lambda\rho/P$, the aerodynamic resistance, $r_a$, and the canopy resistance, $r_c$.

For wind speed in m s$^{-1}$, Jensen et al. (1990) provided the following relationship to calculate $K_1*0.622\lambda\rho/P$:

$K_1 *0.622*\lambda*\rho/P=1710-6.85*\overline T_{av}$ 2:2.2.19

where $\overline T_{av}$ is the mean air temperature for the day (˚C).

To calculate potential evapotranspiration, the Penman-Monteith equation must be solved for a reference crop. SWAT+ uses alfalfa at a height of 40 cm with a minimum leaf resistance of 100 s m$^{-1}$ for the reference crop. Using this canopy height, the equation for aerodynamic resistance (2:2.2.3) simplifies to:

$r_a=\frac{114.}{u_z}$ 2:2.2.20

The equation for canopy resistance requires the leaf area index. The leaf area index for the reference crop is estimated using an equation developed by Allen et al. (1989) to calculate $LAI$ as a function of canopy height. For nonclipped grass and alfalfa greater than 3 cm in height:

$LAI=1.5*ln(h_c)-1.4$ 2:2.2.21

where $LAI$ is the leaf area index and $h_c$ is the canopy height (cm). For alfalfa with a 40 cm canopy height, the leaf area index is 4.1. Using this value, the equation for canopy resistance simplifies to:

$r_c=49/(1.4-0.4*\frac{CO_2}{330})$ 2:2.2.22

The most accurate estimates of evapotranspiration with the Penman-Monteith equation are made when evapotranspiration is calculated on an hourly basis and summed to obtain the daily values. Mean daily parameter values have been shown to provide reliable estimates of daily evapotranspiration values and this is the approach used in SWAT+. However, the user should be aware that calculating evapotranspiration with the Penman-Monteith equation using mean daily values can potentially lead to significant errors. These errors result from diurnal distributions of wind speed, humidity, and net radiation that in combination create conditions which the daily averages do not replicate.

The aerodynamic resistance to sensible heat and vapor transfer, $r_a$, is calculated:

$r_a=\frac{ln[(z_w-d)/z_{om}]ln\lfloor(z_p-d)/z_{ov}\rfloor}{k^2u_z}$ 2:2.2.3

where $z_w$ is the height of the wind speed measurement (cm), $z_p$ is the height of the humidity (psychrometer) and temperature measurements (cm), $d$ is the zero plane displacement of the wind profile (cm), $z_{om}$ is the roughness length for momentum transfer (cm), $z_{ov}$ is the roughness length for vapor transfer (cm), $k$ is the von Kármán constant, and $u_z$ is the wind speed at height $z_w$ (m s$^{-1}$).

The von Kármán constant is considered to be a universal constant in turbulent flow. Its value has been calculated to be near 0.4 with a range of 0.36 to 0.43 (Jensen et al., 1990). A value of 0.41 is used by SWAT+ for the von Kármán constant.

Brutsaert (1975) determined that the surface roughness parameter, $z_o$, is related to the mean height ($h_c$) of the plant canopy by the relationship $h_c/z_o$ = $3e$ or 8.15 where e is the natural log base. Based on this relationship, the roughness length for momentum transfer is estimated as:

$z_{om} = h_c/8.15 =0.123*h_c$ when $h_c \le 200cm$ 2:2.2.4

$z_{om}= 0.058*(h_c)^{1.19}$ when $h_c>200cm$ 2:2.2.5

where mean height of the plant canopy ($h_c$) is reported in centimeters.

The roughness length for momentum transfer includes the effects of bluff-body forces. These forces have no impact on heat and vapor transfer, and the roughness length for vapor transfer is only a fraction of that for momentum transfer. To estimate the roughness length for vapor transfer, Stricker and Brutsaert (1978) recommended using:

$z_{ov} =0.1*z_{om}$ 2:2.2.6

The displacement height for a plant can be estimated using the following relationship (Monteith, 1981; Plate, 1971):

$d=2/3*h_c$ 2:2.2.7

The height of the wind speed measurement, $z_w$, and the height of the humidity (psychrometer) and temperature measurements, $z_p$, are always assumed to be 170 cm.

Soil heat storage or release can be significant over a few hours, but is usually small from day to day because heat stored as the soil warms early in the day is lost when the soil cools late in the day or at night. Since the magnitude of daily soil heat flux over a 10- to 30-day period is small when the soil is under a crop cover, it can normally be ignored for most energy balance estimates. SWAT+ assumes the daily soil heat flux, $G$, is always equal to zero.

Priestley and Taylor (1972) developed a simplified version of the combination equation for use when surface areas are wet. The aerodynamic component was removed and the energy component was multiplied by a coefficient, $\alpha_{pet}$ = 1.28, when the general surroundings are wet or under humid conditions

$\lambda E_o=\alpha_{pet}*\frac{\Delta}{\Delta+\gamma}*(H_{net}-G)$ 2:2.2.23

where $\lambda$ is the latent heat of vaporization (MJ kg$^{-1}$), $E_o$ is the potential evapotranspiration (mm d$^{-1}$), $\alpha_{pet}$ is a coefficient, $\Delta$ is the slope of the saturation vapor pressure-temperature curve, $de/dT$ (kPa ˚C$^{-1}$), $\gamma$ is the psychrometric constant (kPa ˚C$^{-1}$), $H_{net}$ is the net radiation (MJ m$^{-2}$ d$^{-1}$), and $G$ is the heat flux density to the ground (MJ m$^{-2}$ d$^{-1}$).

The Priestley-Taylor equation provides potential evapotranspiration estimates for low advective conditions. In semiarid or arid areas where the advection component of the energy balance is significant, the Priestley-Taylor equation will underestimate potential evapotranspiration.

The Hargreaves method was originally derived from eight years of cool-season Alta fescue grass lysimeter data from Davis, California (Hargreaves, 1975). Several improvements were made to the original equation (Hargreaves and Samani, 1982 and 1985) and the form used in SWAT+ was published in 1985 (Hargreaves et al., 1985):

2:2.2.24

where is the latent heat of vaporization (MJ kg), is the potential evapotranspiration (mm d), is the extraterrestrial radiation (MJ m d), is the maximum air temperature for a given day (°C), is the minimum air temperature for a given day (°C), and is the mean air temperature for a given day (°C).

Table 2:2-2: SWAT+ input variables used in potential evapotranspiration calculations summarized in this section.

Users are now able to read in daily measured or estimated potential evapotranspiration (PET) values from an external data source for each climate station. However, the user still has the option to calculate PET using the Penman Monteith, Priestley-Taylor, or Hargreaves method. The equations used to generate PET data in SWAT+ are reviewed in Chapter 2:2.2.

Using measured or estimated PET allows the model to use real observed data, which can improve accuracy and reliability. By integrating measured data, the model can account for local variations in climatic conditions that might not be captured by theoretical methods. By incorporating this functionality, the model enhances its capability to simulate real-world conditions, making it a valuable tool for hydrological studies and water resource management. SWAT+ input variables that pertain to measured or estimated PET are summarized in Table 2:2-3.

Table 2:2-3: SWAT+ input variables that pertain to measured or estimated PET.