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2:2.2.1 Penman-Monteith Method

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, $r_c$ .

Soil Heat Flux

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.

Aerodynamic Resistance

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:

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

$z_{om}= 0.058*(h_c)^{1.19}$ $h_c>200cm$ 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.

Canopy Resistance

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):

$r_c=r_\Box/(0.5*LAI)$ 2:2.2.8

where $r_c$ is the canopy resistance (s m $^{-1}$ ), $r_{\Box}$ 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:

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

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

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

For hypostomatous leaves the effective stomatal resistance is:

$r_{\Box}=r_{\Box-ad}=r_{\Box-ab}$ 2:2.2.11

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

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

where $g_{\Box}$ 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:

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

where $r_c$ is the canopy resistance (s m $^{-1}$ ), $g_{\Box}$ 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:

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

where $g_{\Box,CO_2}$ 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:

Combined Term

For wind speed in m s, Jensen et al. (1990) provided the following relationship to calculate :

2:2.2.19

where 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 for the reference crop. Using this canopy height, the equation for aerodynamic resistance (2:2.2.3) simplifies to:

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 as a function of canopy height. For nonclipped grass and alfalfa greater than 3 cm in height:

2:2.2.21

where is the leaf area index and 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:

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.

2:2.2.1 Penman-Monteith Method

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

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

Canopy Resistance

$r_c=r_\Box/(0.5*LAI)$ 2:2.2.8

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

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

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

For hypostomatous leaves the effective stomatal resistance is:

$r_{\Box}=r_{\Box-ad}=r_{\Box-ab}$ 2:2.2.11

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

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

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

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

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

where $g_{\Box,CO_2}$ 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

$r_c=r_{\Box}*[(0.5*LAI)*(1.4-0.4*\frac{CO_2}{330})]^{-1}$ 2:2.2.15

SWAT+ will default the value of CO $_2$ concentration to 330 ppmv if no value is entered by the user. With this default, the term $(1.4-0.4*\frac{CO_2}{330})$ reduces to 1.0 and the canopy resistance equation becomes equation 2:2.2.8.

$g_{\Box}=g_{\Box,mx}*\lfloor1-\Delta g_{\Box,dcl}(vpd-vpd_{thr})\rfloor$ if $vpd>vpd_{thr}$ 2:2.2.16

$g_{\Box}=g_{\Box,mx}$ if $vpd \le vpd_{thr}$ 2:2.2.17

where $g_{\Box}$ is the conductance of a single leaf (m s $^{-1}$ ), $g_{\Box,mx}$ is the maximum conductance of a single leaf (m s $^{-1}$ ), $\Delta g_{\Box,dcl}$ is the rate of decline in leaf conductance per unit increase in vapor pressure deficit (m s $^{-1}$ kPa $^{-1}$ ), $vpd$ is the vapor pressure deficit (kPa), and $vpd_{thr}$ is the threshold vapor pressure deficit above which a plant will exhibit reduced leaf conductance (kPa). The rate of decline in leaf conductance per unit increase in vapor pressure deficit is calculated by solving equation 2:2.2.16 using measured values for stomatal conductance at two different vapor pressure deficits:

$\Delta g_{\Box,dcl}=\frac{(1-fr_{g,mx})}{(vpd_{fr}-vpd_{thr})}$ 2:2.2.18

where $\Delta g_{\Box,dcl}$ is the rate of decline in leaf conductance per unit increase in vapor pressure deficit (m s $^{-1}$ kPa $^{-1}$ ), $fr_{g,mx}$ is the fraction of the maximum stomatal conductance, $g_{\Box,mx}$ , achieved at the vapor pressure deficit $vpd_{fr}$ , and $vpd_{thr}$ is the threshold vapor pressure deficit above which a plant will exhibit reduced leaf conductance (kPa). The threshold vapor pressure deficit is assumed to be 1.0 kPa for all plant species.

Aerodynamic Resistance

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

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

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

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

$z_{ov} =0.1*z_{om}$ 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.