Appendix F — Application Penman Monteith method

Description of Penman-Monteith method after Allen et al. (1998).

The original form of the Penman-Monteith equation can be written as (Monteith 1965, 1981):

\[ ET_p = \frac{\frac{\Delta_v} {\lambda_w} \left(R_n - G\right)+\frac{p_1 \rho_{air} c_{air}} {\lambda_w } \frac{e_{sat} - e_a} {r_{air}}} {\Delta_v \gamma_{air} \left(1+ \frac{r_{crop}} {r_{air}} \right)} \tag{F.1}\]

where \(ET_p\) is the potential transpiration rate of the canopy (mm d^-1), \(\Delta_v\) is the slope of the vapour pressure curve (kPa \(^\circ\)C^-1), \(\lambda_w\) is the latent heat of vaporization (J kg^-1), \(R_n\) is the net radiation flux at the canopy surface (J m-2 d^-1), \(G\) is the soil heat flux (J m^-2 d^-1), \(p_1\) accounts for unit conversion (=86400 s d^-1), \(\rho_{air}\) is the air density (kg m^-3), \(C_{air}\) is the heat capacity of moist air (J kg-1 \(^\circ\)C^-1), \(e_{sat}\) is the saturation vapour pressure (kPa), \(e_a\) is the actual vapour pressure (kPa), \(\gamma_{air}\) is the psychrometric constant (kPa \(^\circ\)C^-1), \(r_{crop}\) is the crop resistance (s m^-1) and \(r_{air}\) is the aerodynamic resistance (s m^-1).

To facilitate analysis of the combination equation, an aerodynamic and radiation term are defined:

\[ ET_p = ET_{rad} + ET_{aero} \tag{F.2}\]

where \(ET_p\) is potential transpiration rate of crop canopy (cm d^-1), \(ET_{rad}\) is the radiation term (cm d^-1) and \(ET_{aero}\) is the aerodynamic term (cm d^-1).

The radiation term equals:

\[ ET_{rad} = \frac{\Delta_v \left(R_n - G\right)} {\lambda_w \left(\Delta_v+\gamma_{air}^*\right)} \tag{F.3}\]

where the modified psychrometric constant (kPa \(^\circ\)C^-1) is:

\[ \gamma_{air}^* = \gamma_{air} \left(1+\frac{r_{crop}} {r_{air}}\right) \tag{F.4}\]

The aerodynamic term equals:

\[ ET_{aero} = \frac{p_1 \rho_{air} c_{air} \left(e_{sat} - e_a\right)} {\lambda_w r_{air} \left(\Delta_v \gamma_{air}^* \right) r_{air}} \tag{F.5}\]

Many meteorological stations provide mean daily values of air temperature \(T_{air}\) (\(^\circ\)C), global solar radiation \(R_s\) (J m^-2 d^-1), wind speed \(u_0\) (m s-1) and air humidity \(e_{act}\) (kPa). These basic meteorological data are used to apply the Penman Monteith equation.

Radiation term

The net radiation \(R_n\) (J m^-2 d^-1) is the difference between incoming and outgoing radiation of both short and long wavelengths. It is the balance between the energy adsorbed, reflected and emitted by the earth’s surface:

\[ R_n = \left(1-\alpha_r \right) R_s-R_{nl} \tag{F.6}\]

where \(\alpha_r\) is the reflection coefficient or albedo (-) and \(R_{nl}\) is the net longwave radiation (J m^-2 d^-1). The albedo is highly variable for different surfaces and for the angle of incidence or slope of the ground surface. It may be as large as 0.95 for freshly fallen snow and as small as 0.05 for a wet bare soil. A green vegetation cover has an albedo of about 0.20-0.25 (Bruin 1998). SWAP will assume in case of a crop \(\alpha_r\) = 0.23, in case of bare soil \(\alpha_r\) = 0.15.

The earth emits longwave radiation, which increases with temperature and which is adsorbed by the atmosphere or lost into space. The longwave radiation received by the atmosphere increases its temperature and, as a consequence, the atmosphere radiates energy of its own. Part of this radiation finds its way back to the earth’s surface. As the outgoing longwave radiation is almost always greater than the incoming longwave radiation, the net longwave radiation \(R_{nl}\) represents an energy loss. Allen et al. (1998) recommend the following formula for the net longwave radiation:

\[ R_{nl} = \sigma_{sb} \left[\frac{T_{max}^4+T_{min}^4} {2}\right] \left(0.34 - 0.14 \sqrt{e_{act}} \right) \left(0.1+0.9 N_{rel}\right) \tag{F.7}\]

where \(\sigma_{sb}\) is the Stefan-Boltzmann constant (4.903 10^-3 J K^-4 m^-2 d^-1), \(T_{min}\) and \(T_{max}\) are the minimum and maximum absolute temperatures during the day (K), respectively, \(e_{act}\) is the actual vapour pressure (kPa), and \(N_{rel}\) is the relative sunshine duration. The latter can be derived from the measured global solar radiation \(R_s\) and the extraterrestrial radiation \(R_a\) (J m^-2 d^-1), which is received at the top of the Earth’s atmosphere on a horizontal surface:

\[ N_{rel} = \left(\frac{R_s} {R_a} - a\right) \frac{1} {b} \tag{F.8}\]

where \(a\) and \(b\) are so-called Angstrom coefficients which depend on the local climate. For international use Allen et al. (1998) recommend \(a\) = 0.25 and \(b\) = 0.50.

The extraterrestrial radiation \(R_a\) depends on the latitude and the day of the year. \(R_a\) is calculated with:

\[ R_a = \frac{G_{sc}} {\pi} d_r \left[ \omega_s \sin\left(\varphi\right)\sin\left(\delta\right)+\cos\left(\varphi\right)\cos\left(\delta\right)\sin\left(\omega_s\right)\right] \tag{F.9}\]

where \(G_{sc}\) is the amount of solar radiation striking a surface perpendicular to the sun’s rays at the top of the Earth’s atmosphere, called the solar constant (J m^-2 d^-1), \(d_r\) is the inverse relative distance Earth-Sun (-), \(\omega_s\) is the sunset hour angle (rad), \(\varphi\) is the latitude (rad) and \(\delta\) is the solar declination (rad). The inverse relative distance Earth-Sun and the solar declination are given by:

\[ d_r = 1+ 0.033 \cos \left(\frac{2\pi} {365} J\right) \tag{F.10}\]

\[ \delta = 0.409 \sin \left(\frac{2\pi} {365} J - 1.39\right) \tag{F.11}\]

where \(J\) is the number of the day in the year (1-365 or 366, starting January 1). The sunset hour angle expresses the day length and is given by:

\[ \omega_s = \arccos \left[-\tan\left(\varphi\right) \tan\left(\delta\right)\right] \tag{F.12}\]

Aerodynamic term

Latent heat of vaporization, \(\lambda_w\) (J g^-1), depends on the air temperature \(T_{air}\) (\(^\circ\)C) (Harrison 1963):

\[ \lambda_w = 2501 - 2.361 T_{air} \tag{F.13}\]

Saturation vapour pressure, \(e_{sat}\) (kPa), also can be calculated from air temperature (Tetens 1930):

\[ e_{sat} = 0.611 \exp \left(\frac{17.27 T_{air}} {T_{air} + 237.3}\right) \tag{F.14}\]

The slope of the vapour pressure curve, \(\Delta_v\) (kPa \(^\circ\)C^-1), is calculated as (Murray 1967):

\[ \Delta_v = \frac{4098 e_{sat}} {\left(T_{air}+237.3\right)^2} \tag{F.15}\]

The psychrometric constant, \(\gamma_{air}\) (kPa C-1), follows from (Brunt 1952):

\[ \gamma_{air} = 1.63 \frac{p_{air}} {\lambda_w} \tag{F.16}\]

with \(p_{air}\) the atmospheric pressure (kPa) at elevation \(z_0\) (m), which is calcula¬ted from (Burman et al. 1987):

\[ p_{air} = 101.3 \left(\frac{T_{air,K} - 0.0065 z_0} {T_{air,K}}^5.256\right) \tag{F.17}\]

Employing the ideal gas law, the atmospheric density, \(\rho_a\) (g cm^-3), can be shown to depend on p and the virtual temperature \(T_{vir}\) (K):

\[ \rho_{air} = 3.48610^{-3} \frac{p_{air}} {T_{vir}} \tag{F.18}\]

where the virtual temperature is derived from:

\[ T_{vir} = \frac{T_{air,K}} {1-0.378 \frac{e_{act}} {p_{air}}} \tag{F.19}\]

The heat capacity of moist air, \(C_{air}\) (J g^-1 \(^\circ\)C^-1), follows from:

\[ C_{air} = 0.622 \frac{\gamma_{air} \lambda_w} {p_{air}} \tag{F.20}\]

Aerodynamic resistance

The aerodynamic resistance \(r_{air}\) depends on the wind speed profile and the roughness of the canopy and is calculated as (Allen et al. 1998):

\[ r_{air} = \frac{\ln \left(\frac{z_m - d} {z_{0m}}\right) \ln\left(\frac{z_h - d} {z_{0h}}\right)} {\kappa_{vk}^2 u} \tag{F.21}\]

where \(z_m\) is height of wind speed measurements (m), \(z_h\) is height of temperature and humidity measurements (m), \(d\) is zero plane displacement of wind profile (m), \(z_{0m}\) is roughness parameter for momentum (m) and \(z_{0h}\) is roughness parameter for heat and vapour (m), \(\kappa_{vk}\) is von Karman constant = 0.41 (-), \(u\) is wind speed measurement at height \(z_m\) (m s^-1).

The parameters \(d\), \(z_{om}\) and \(z_{oh}\) are defined as:

\[ d = \frac{2}{3}h_{crop} \tag{F.22}\]

\[ z_{om} = 0.123 h_{crop} \tag{F.23}\]

\[ z_h = 0.1 z_{om} \tag{F.24}\]

with \(h_{crop}\) the crop height (cm).

A default height of 2.0 m is assumed for wind speed measurements (\(z_m\)) and a default height of 1.5 m for temperature and humidity measurements (\(z_h\)).

Meteorological stations generally provide 24 hour averages of wind speed measurements, according to international standards, at an altitude of 10 m.

To calculate \(r_{air}\), the average daytime wind (7.00 - 19.00 h) should be used. For ordinary conditions we assume (Smith 1990) for the average daytime wind speed (\(u_{0,day}\)):

\[ u_{0,day} = 1.33 u_0 \tag{F.25}\]

where is the measured average wind speed over 24 hours (m s^-1).

The wind speed is corrected with the following assumptions:

• a uniform wind pattern at an altitude of 100 meter;
• wind speed measurements are carried out above grassland;
• a logarithmic wind profile is assumed;
• below 2 meter wind speed is assumed to be unchanged with respect to a value at an altitude of 2.0 m; applying a logarithmic wind profile at low altitudes is not carried out due to the high variation below 2.0 m.

The logaritmic wind profile can be written as:

\[ u\left(2\right) = \frac{\ln \left(\frac{z_2-d} {z_{0m}}\right)} {\ln \left(\frac{z_1-d} {z_{0m}}\right)} u\left(1\right) \tag{F.26}\]

with \(u\) denoting wind speed (m s^-1), \(z_1\) and \(z_2\) are two heights (m), \(d\) is zero-plane displacement (m), and \(z_{0m}\) is the momentum roughness length (m).

We may assume the same wind velocity at 100 m height for the weather station and our actual field. The weather station is usually located at a grassland. Therefore we can write for the weather station:

\[ u\left(100\right) = \frac{\ln \left(\frac{z_{100}-d_{grass}} {z_{0m,grass}}\right)} {\ln \left(\frac{z_m-d_{grass}} {z_{0m,grass}}\right)} u_{0,day} \tag{F.27}\]

with \(u(100)\) the wind velocity at 100 m, \(u_{0,day}\) is the measured average daytime windspeed (m s^-1), \(z_{100}\) and \(z_m\) are the heights at 100 m and wind observation (m), \(d_{grass}\) is the grass zero-plane displacement (m), and \(z_{0m,grass}\) is the momentum roughness length for grass (m).

For the actual field for which we perform our simulation, we can write:

\[ u\left(100\right) = \frac{\ln \left(\frac{z_{100}-d_{act}} {z_{0m,act}}\right)} {\ln \left(\frac{z_{act}-d_{act}} {z_{0m,act}}\right)} u \tag{F.28}\]

with \(u\) the wind velocity at the top of the actual crop (m s^-1), \(z_{act}\) is the actual crop height (m), \(d_{act}\) is the zero-plane displacement of the actual crop (m), and \(z_{0m,act}\) is the momentum roughness length for the actual crop (m). The wind velocity \(u\) is input to the Penman-Monteith equation.

The above 2 equations yield the wind velocity at the top of the simulated crop:

\[ u = \frac{\ln \left(\frac{z_{act}-d_{act}} {z_{0m,act}}\right) \ln \left(\frac{z_{100}-d_{grass}} {z_{0m,grass}} \right)} {\ln \left(\frac{z_{100}-d_{act}} {z_{0m,act}}\right) \ln \left(\frac{z_m-d_{grass}} {z_{0m,grass}}\right)} u_{0,day} \tag{F.29}\]

Below 2.0 m height, the wind speed is not reduced. Therefore, \(z_{act}\) has a minimum value of 2.0 m.

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