Transpiration
Reference evapotranspiration \(ET0\) is calculated using the Penman-Monteith method, according to Allen et al. (1998). This method requires the diurnal minimum and maximum temperature, the water vapour pressure deficit, wind velocity, and total global radiation.
\(\Delta\) Slope of the vapour pressure curve \([kPa\,K^{-1}]\)
\(R_n\) Net radiation at the crop surface \([MJ\, m^{-2} \, d^{-1}]\)
\(G\) Soil heat flux density \([MJ\, m^{-2} \, d^{-1}]\)
\(T\) Mean daily air temperature at 2 m height \([^{\circ} C]\)
\(u_2\) Wind speed at 2 m height \([m \, s^{-1}]\)
\(e_s\) Saturation vapour pressure \([kPa]\)
\(e_a\) Actual vapour pressure \([kPa]\)
\(\gamma\) Psychrometric constant \([kPa \, K^{-1}]\)
\(r_a\) Atmospheric resistance \([s\, m^{-1}]\)
\(r_s\) Surface resistance \([s\, m^{-1}]\)
where
The surface resistance for the reference evapotranspiration assumes a 12 cm cut grass crop and is calculated using:
\(r_1\) Stomata resistance; 100 s m–1 \([s \, m^{-1}]\)
The leaf area index \(LAI\) and crop height \(h\) are considered for the surface resistance of the actual crop:
\(r_s\) Surface resistance \([s \, m^{-1}]\)
\(r_1\) Stomata resistance \([s \, m^{-1}]\)
\(LAI\) Leaf area index \([m^2 \, m^{-2}]\)
\(h_c\) Crop height \([m]\)
The crop height is calculated using:
\(h_{c\,max}\) Crop-specific maximum height \([m]\)
\(a, b\) Crop-specific parameter
\(DD_{relh}\) Relative crop development for height
where
\(DD_{acth}\) Actual temperature sum from emergence \([^{\circ}C \, d]\)
\(DD_{croph}\) Crop-specific total temperature sum up to maximum height \([^{\circ}C \, d]\)
Specific leaf weights to calculate \(LAI\) from the leaf biomass are given for each development stage and adjusted linearly to the relative phenological development.
The stomata resistance of the actual crop is calculated according to a suggestion of Yu et al. (2001):
\(r_1\) Stomata resistance \([s^{-1}]\)
\(C_s\) CO2 concentration outside the leaf (= \(C_a\)) \([\mu mol \, mol^{-1}]\)
\(A_g\) Gross CO2 assimilation rate \([kg \, CO_2 \, ha^{-1} \, d^{-1}]\)
\(e_a\) Actual air vapour pressure \([Pa]\)
\(e_s\) Saturation air vapour pressure \([Pa]\)
In a rough simplification, \(C_s\) is considered equal to the atmospheric CO2 concentration \(C_a\). Its seasonal dynamic from 1958 up until today is described using:
\(C_a\) Atmospheric CO2 concentration \([\mu mol \, mol^{-1}]\)
\(t_{dec}\) Decimal date
Additionally, this function carries forward the atmospheric CO2 concentration until 2100, similar to the assumption made in the IPCC A1B scenario (IPCC, 2007).
Crop-specific potential evapotranspiration is calculated using also crop-specific factors (\(K_c\)) during the crop’s growth period, and a factor for bare soil between harvest and emergence of the subsequent crop. The \(K_c\) factors are coupled to the crop’s developmental stages.
\(ET_p\) Potential evapotranspiration \([mm]\)
\(ET_0\) Reference evapotranspiration \([mm]\)
\(K_c\) ncrop-specific factor
\(I\) Evaporation from interception storage \([mm]\)
To what extent transpiration contributes to total evapotranspiration is determined by the ground coverage:
\(T_p\) Potential transpiration \([mm]\)
\(ET_p\) Potential evapotranspiration \([mm]\)
\(\beta\) Ground coverage
Transpiration is calculated layer-wise for water uptake from the respective layer, considering root distribution, efficiency, and possible oxygen deficit.
\(T_z\) Actual transpiration in layer \(z\) \([mm]\)
\(T_p\) Potential transpiration \([mm]\)
\(\omega_z\) Root water uptake efficiency in layer \(z\) (Fig. 1)
\(\Lambda_z\) Root length density in layer \(z\) \([m \, m^{-3}]\)
\(\zeta_o\) Stress factor oxygen deficit
Figure 1: Reduction function for transpiration (\(\tau_z\)) and for root water uptake efficiency (\(\omega_z\)) in dependency of water availability in the respective soil layer.
Given sufficient amounts of available water, it is removed from the soil layers according to the layer-wise calculated transpiration, from the soil surface downwards. A potential transpiration deficit is calculated for every layer:
\(\zeta_{Wpot}\) Potential transpiration deficit in layer \(z\) \([mm]\)
\(T_z\) Actual transpiration in layer \(z\) \([mm]\)
\(\Delta z\) Depth of soil layer \([m]\)
\(\theta\) Water content in soil layer \([m^3\, m^{-3}]\)
\(\theta_{PWP}\) Soil water content at permanent wilting point \([m^3\, m^{-3}]\)
At decreasing water contents water conductivity in the soil is reduced. The resulting reduction of transpiration in the soil layer is calculated as:
\(T_{red}\) Reduced transpiration in layer \(z\) \([mm]\)
\(T_z\) Actual transpiration in layer \(z\) \([mm]\)
\(\tau_z\) Reduction factor water availability (Fig.)
The actual transpiration deficit of the respective layer is calculated from the larger of both values. It will be used to update actual transpiration.
\(\zeta_{Wact}\) Actual transpiration deficit in layer \(z\) \([mm]\)
\(\zeta_{Wpot}\) Potential transpiration deficit in layer \(z\) \([mm]\)
\(T_{red}\) Reduced transpiration in layer \(z\) \([mm]\)
The total actual transpiration of the crop adds up from the actual transpiration of the layers:
\(T_a\) Actual transpiration \([mm]\)
\(T_z\) Actual transpiration in layer \(z\) \([mm]\)
\(z\) Layer number
\(z_{max}\) Lowest layer in soil profile
Crop growth is limited by water availability. Drought stress is indicated by the relation of actual to potential transpiration (Kersebaum, 1995).