#### Simple approach with response functions

Actual transpiration is calculated in two
steps to account for possible compensatory uptake of water by roots in layers with no
water stress if there are roots in other layers that are exposed to water
stress. The actual transpiration is given as:

(3.25)

where *f*_{umov} is the degree of compensation,
*E*_{ta}^{*} is the uptake without any account for
compensatory uptake and *E*_{tp}^{*} is the potential transpiration with eventual reduction due to
interception evaporation. The compensatory uptake is
distributed to the layers where no water stress occurs and in accordance with
the relative fraction of the roots in these layers. In a first step the
*E*_{ta}^{*} is calculated as the result of possible
stresses at each depth and finally integrated as:

(3.26)

where *n*_{r}
is the layer with the deepest roots, *r(z)* is the relative root density
distribution, *z*_{r} is root depth and
*f**(**ψ**(z))*, *f(**π* *(z))* and *f(**T(z))
*are response
functions for soil water potential, soil osmotic potential and soil temperature.
Root density may be expressed by root length per unit soil volume, or by any
other pertinent measure of roots.

Reduction because of dry soil is supposed to act through
the stomatal mechanism and xylary tissue resistance, which both have shown to be
very sensitive to the demand rate. The water potential response function,
*f**(**ψ**(z))*, has been given a simple
analytical form in the dry range:

(3.27)

where *p*_{1},
*p*_{2}_{ }and_{
}*ψ*_{c}_{ }are parameters (Jansson, 1981). See
viewing function Soil moisture
response, simple response function. If the soil water potential is reaching the
wilting point, *ψ*_{wi}_{lt}, the uptake is assigned to be zero from that
horizon. An additional response function, *f*_{θ}, correspond to the normal need
of oxygen supply to fine roots and it has been given as:

(3.28)

where *p*_{ox}_{ }is an empirical parameter and
*S*_{ox} is a critical saturation range defined as:

(3.29)

when the soil moisture, *θ,* is above
the critical soil moisture threshold, *θ*_{ox}_{. }The value of *θ*_{ox }is calculated as the
difference between the water content at saturation, *θ*_{s}, and the minimum air content, given as a parameter,
*θ*_{Amin}. In case *θ* is less than
the *S*_{ox}, *S*_{ox}_{ }is given a value of
zero, which means that the response function is equal to unity, i.e. the maximum
value.

Reduction because of low soil temperatures acts primarily
through a lowered conductivity between root surface and xylem and is, thus,
responding to temperature at each depth. There are different ways of estimating
the soil temperature response, *f(**T(z))**, *which is determined by the
switch Temperature
response. By
choosing “none”, there will be no reduction water uptake due to soil
temperature:

(3.30)

The second option “Double-exponential”, is an analytical
form of the soil temperature response, *f(**T(z))**, *which was proposed by Axelsson
& Ågren (1976):

(3.31)

where *t*_{WA} and *t*_{WB} are parameters. *T*_{trig} is the trigging temperature (see below). See
viewing functions Soil
temperature response, plant resistance and Soil
temperature response, double-exponential.

A single-exponential function for the temperature response,
*f(**T(z))*, can also be used:

(3.32)

where *t*_{WD} is a parameter. *T*_{trig} is the trigging temperature (see below). See
viewing functions Soil
temperature response, plant resistance and Soil
temperature response, single-exponential.

The forth alternative is to use a polynomial function for
the temperature response, *f(**T(z))*:

(3.33)

where *t*_{WD} and *t*_{WE} are parameters. *T*_{trig} is the trigging temperature (see below). See
viewing functions Soil
temperature response, plant resistance and Soil
temperature response, polynomial.

The trigging temperature, *T*_{trig}, can either be a static parameter,
*t*_{WC}, or a function of
air temperature (see switch Trigging Temperature). In the
latter case the accumulated daily average air temperature above a threshold
temperature determines the trigging temperature:

(3.34)

where *t*_{WC} and *t*_{WF} are parameters. *T*_{sumplant} is the accumulated sum of air temperatures above a
critical temperature, *t*_{crit} (see Description of
Plant).

The switch Salt Influence governs
reduction of water uptake due to soil salinity. If the salt influence is set to
be added to pressure head, the osmotic pressure, *π(z)*, is added to the soil water potential,
*ψ(z)*, in eq (3.27). If this option is chosen the salinity
response function, *f(**π* *(z))*, in eq (3.26) will be put to unity. Alternatively
the salt influence can be included as an independent response function by
choosing “Add multiplicative response” or “Add minimum response”. This response
function was proposed by van Genuchten et al(van Genuchten, 1983; van Genuchten
& Hoffman, 1984; van Genuchten & Gupta, 1993) as:

(3.35)

where *r*_{i}(*∆z)* is the relative root
distribution, and *π*_{c} and *p*_{π}_{ }are empirical parameter values.
See viewing function Soil salinity response. The “Add
Multiplicative response” option will multiply the response function for
salinity, *f(**π* *(z))*, with the other response functions for
water and temperature as written in eq (3.26). On the other hand if the “Add minimum
response” option is chosen, the smallest of the two response functions for soil
moisture and salt, will instead be used in determining the water uptake,
modifying eq (3.26)slightly into:

(3.36)