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The transpiration rate of a single tree — numerical simulations
EtOLOGIICllL momunG ELSEVIER
Ecological Modelling 75/76 (1994) 321-330
T h e transpiration rate o f a single tree - numerical simulations Ulrike Janssen
*
Institute of Meteorology and Climatology, University of Hannover, Herrenhiiuserstr. 2, 30419 Hannover, Germany
Abstract
A numerical simulation model for the calculation of the transpiration of single trees has been developed. The required input data, like meteorological conditions as well as the flow configuration in and around the tree are provided by other, already existing models. Stem position and tree shape relate a certain leaf area to each grid volume of the three-dimensional model. This spatial heterogeneity allows the study of the radiation regime as well as the heat and water vapour exchange within the tree crown. Stomatal resistances are estimated in dependence of environmental factors. The individual leaf temperatures are calculated by solving an energy balance equation. Some model results, demonstrating the sensitivity of different input parameters on the transpiration rate, are presented. Key words: Numerical simulation; Transpiration; Tree
I. Introduction
L a n d s u r f a c e - a t m o s p h e r e interactions influence local, regional and global climate to a high degree. O n e of the most important processes affecting t e m p e r a t u r e and moisture distribution in the lower a t m o s p h e r e is the transpiration. During the two field experiments H I B E '88 and '89 large data sets were collected by satellite, aeroplane and in-situ measurements, which should contribute to a better understanding of the various exchange processes. T h e area, called " H i l d e s h e i m e r B6rde", where the experiments were p e r f o r m e d is located southeast of the city of Hannover. T h e site is characterized by an almost h o m o g e n e o u s terrain with
* Corresponding author. Present address: Ludwigstr. 24, 63067 Offenbach, Germany. 0304-3800/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-3800(93)E0139-T
322
U. Janssen/ Ecological Modelling 75 / 76 (1994) 321-330
respect to relief and land use. A large part of the area is cultivated with wheat and sugar beet. There are no closed forest canopies, but single trees which stand in small groups or along the streets as alley trees. During the experiments the trees were not taken into account, so that their contribution to the transpiration has to be estimated by means of a numerical model. Until now the numerical models used in meteorology to estimate transpiration rates of vegetation are mainly one-dimensional (e.g. Deardorff, 1978). One may distinguish between "big-leaf" and "multiple-layer" models. Two- and three-dimensional simulations were performed by Yamada (1982), Cionco (1985) or Gross (1988). The array model of Wang and Jarvis (1990) and the model of Rose (1984) are first attempts for modelling the transpiration of individual plants. Measurements on single leaves or single trees were carried out by Thorpe (1978), Lafleur and Rouse (1990) and Leuning and Foster (1990). Their estimations are based on the application of different versions of the P e n m a n - M o n t e i t h equation. There are several factors which influence the transpiration rate of a plant: 1. the meteorological conditions, like air temperature, air moisture and wind velocity; 2. the water supply characterized by precipitation amount, soil properties and root characteristics; 3. the incoming solar radiation, which depends on the position of the sun, cloud cover and air turbidity; 4. the optical properties of the leaves: reflectivity, emissivity, extinction coefficient and leaf inclination angle; 5. the stomatal response. In the transpiration model all these factors have to be considered in an appropriate way.
2. Description of the model As input data for the transpiration model we need the meteorological conditions of the undisturbed environment. They are described by the one-dimensional version of the boundary-layer model FITNAH. Vertical profiles of wind velocity, temperature and moisture are obtained with a temporal resolution of 1 h (for detailed description see e.g. Gross, 1991). The flow configuration in and around the tree is calculated with a diagnostic, mass-consistent wind model, which was originally designed for complex building structures (R6ckle, 1990). It was modified in order to consider porous obstacles, where porosity results from leaf density, and to treat every possible tree shape. Corresponding to the leaf density, the wind will be attenuated and the cavity zone on the lee side will develop in a more or less accentuated way. A soil-water model is not yet implemented, so that effects on transpiration due to scarcity of water are not taken into consideration. According to the requirements the transpiration model has to be three-dimensional. The real tree with its individual distribution of leaves and branches is represented by a geometrical body like a cone, a cylinder or a sphere. This body is
U. Janssen/ Ecological Modelling 75 / 76 (1994) 321-330
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subdivided into grid volumes. One leaf with a certain leaf area and leaf inclination is related to each of these volumes. The transpiration rate of the complete tree is the sum of the individual water vapour fluxes of each leaf. These fluxes are assumed to be constant for a period of 1 h. The specific moisture of the leaf s t is a function of the leaf temperature T~; it is supposed to be equal to the saturation moisture at the leaf temperature:
sz=s?(Tt). (1) The individual l e a f temperatures are obtained by solving an energy balance equation. It consists of three terms: net radiation and turbulent fluxes for sensible and for latent heat: RN + QH + Qv = 0. (2) Other terms like heat storage or energy exchange due to photosynthesis are neglected. The numerical solution is done by a Newton-Raphson iteration:
O T, = T/
OQ/OTt,
(3)
where Tt is the actual leaf temperature and T[ is the leaf temperature at the previous iteration step. Q is the sum of all energy fluxes:
Q = R N + QH + Qv. (4) Net radiation R N consists of the short-wave and the long-wave radiation regime. As the process of photosynthesis is not considered here, the radiation spectrum is only subdivided into these two parts. The short-wave radiation regime will first of all be determined by the incoming solar radiation, which is a function of sun position, cloud cover and air turbidity. On its way through the tree, the sun beam will be attenuated by reflection, extinction and scattering. All of these processes are taken into account in the balance equation for the short-wave radiation: S = So(1 - a f ) [ e x p ( - k s . L r ) + f ( w ) ] . (5) The incoming radiation S o is reflected with a corresponding albedo of the foliage af. For the calculation of the extinction, the foliage is treated as a homogeneous turbid layer with the extinction coefficient for short-wave radiation ks, which is a function of the leaf inclination angle. L r denotes the leaf area index in the direction of the sun beam. Campbell (1988) developed an expression for k s where different leaf angle distributions may be considered. The scattering function f ( w ) and the scattering coefficient w are taken from the work of Ross and Nilson (in Anderson, 1969). The balance for the long-wave radiation consists of four different components: the long-wave irradiation of: (1) the atmosphere, (2) the soil, (3) the surrounding leaves, and (4) the considered leaf itself: R i = Rat m exp( - k t "Lup ) + Rsoil exp( - k t • Ldown )
i-1 + E Rlij +
j=l
N E
j=i+l
Rlij - 2ef~rTt 4.
(6)
U, Janssen / Ecological Modelling 75 / 76 (1994) 321-330
324
kt is the extinction coefficient for long-wave radiation and is assumed to be 0.2" k s (Baldocchi et al., 1984). Lup and Ldow, are the leaf area index of the foliage above and below the considered leaf respectively. The influence of the jth on the ith leaf is:
Ruj=eftrTtj
4 biAxAyAz
~
e x p [ - k l ' L ( x i ~ ) ] cos Oij
(7)
The jth leaf irradiates according to the Stefan-Boltzmann law EfcrT~4. From this irradiation the ith leaf will only receive a small part, which corresponds to the ratio of its own leaf area b i A x A y A z to the surface of the hemisphere with radius xi~. bi is the spatial heterogeneous leaf area density and xi~ denotes the distance between the two leaves considered at the moment. Furthermore the angle Oi~ and the amount of foliage L(x~j) between the two leaves will reduce the radiation reaching the ith leaf. The turbulent fluxes of sensible heat are given by:
Te-T
QH = Ocp -
,
(8)
rH
with p being the density and Cp the specific heat of the air. Te and Tt are the environmental and leaf temperature respectively and r/4 is the aerodynamic resistance for heat, r/_/ is composed by the resistances for forced and flee convection. For their parameterization Braden (1982) uses results fom similarity theory. The constants were obtained by Pearman et al. (1972) during their measurements in plant canopies. rHfor c =
rmree
94~/d/u
454[ d 11J4 Cf
[ Tvt
Tve [
(9) (10)
with cf = 0.8, d being the leaf size, u the wind velocity and Tvt and TEe the virtual temperature of leaf and environment (T V = T(1 + 0.608. s)). The resulting resistance is obtained by a parallel connexion of the two individual resistances: r/_/=
- - + - rHforc
(11)
Ell free
Analogous to Eq. 8 the latent heat flux is written as:
Qv = Q L v
S e -- SI
r v + rst
,
(12)
with L V the latent heat for evaporation, s e and s t the specific moisture of environmental air and of the leaf. r E, the aerodynamic resistance for water vapour, is parameterized analogous to Eq. 9 by:
r v = 87~/d/u.
(13)
u. Janssen/ EcologicalModelling 75/ 76 (1994)321-330
325
O n e of the most important factors affecting transpiration is the behaviour of the stomata, which is a function of environmental conditions. T h e stomatal resistance rst or the inverse value, the stomatal c o n d u c t a n c e gst, is given as a direct function of m i n i m u m and maximum c o n d u c t a n c e and as a function of external conditions:
rst = (gs,) -1 = (gmin +gmax'fS "fT'fD) -1 (14) The modifiers f for the individual variables (S: short-wave radiation, T: temperature, and D: water v a p o u r saturation deficit) range between 0 and 1. They are given by the following relations: S - -
fs-
/
1000
al + S
1000+a 1
,
fD = 1 -- a 2 • ae T-Tmin
fT
{
TTpi----~min
with a I = 160.4 W / m e,
(15)
with a 2 = 0.1825 kPa - l
(16)
rmax-T Tmax _ Topt
(17)
T h e y are a combination of the models of L o h a m m e r et al. (1980) and D o l m a n and V a n D e n Burg (1988). As was m e n t i o n e d at the beginning, a s o i l - w a t e r model is not yet included and, until now, no attempts were m a d e to consider different species in the stomata submodel.
3. Sensitivity analysis and first results If a new model is developed, the first thing to do is to investigate how sensitive the model is to changes of certain input data. In Fig. 1 various input parameters were increased or diminished by 10%. Air humidity and the stomatal resistance
.: (/g0
+4. +2.
s :T,
F,[
,-,i.
i
q 111'/. ',+10% ]+lO¢Z~ +10% I+]0'}~. ~,+lO*/~ ',+ID% ',+lOVo ]+IO'X. | 10'~, --ltr/,, -i ii'; -10%, -10%- -10%, -i0%, 10%, -10
1
"
E
1
-i
U IU
n n U
U
Fig. 1. Percentual change of diurnal transpiration rate as a consequence of a 10% increase (left bar) and a 10% decrease (right bar) of the respective input parameters, u: wind speed, S: short-wave radiation, TI: leaf temperature, FF leaf area, if: mean leaf inclination angle, e: emissivity, k,: extinction coefficient, r H, rv: aerodynamic resistance for heat and water vapour, f: relative humidity, r,t: stomatal resistance.
U. Janssen /Ecological Modelling 75/76 (1994) 321-330
326
8)
b)
160
180"
120
t20~
80
80-
40
40" 21.4
21.5
20
21.8
d;tt,,
40 80 80 rel. humidity (%)
100
d) 180
18o I
120
1201
80
801
40
401 1
2 3 4 5 8 wind velocity (m/s)
7
8
18 2 0
38 4 8 50 80 78 leaf inclination (*)
80
90
=) 180
160-
120
120-
80
80
40
40 8
8
I0 12 leaf size (era)
14
0.2
0.4
0.6
0.8
1.8
cloud cover
Fig. 2. Percentual change of daily transpiration related to a reference simulation when varying(a) date, (b) relative humidity, (c) wind speed, (d) leaf inclination angle, (e) leaf size and (f) cloud cover. influence transpiration to a high degree, as a change of 10% results in a variation of transpiration in the same order of magnitude. The incident solar radiation still causes a difference of approximately 4%, but all the other changes in transpiration do not exceed 2%. Fig. 2 shows the dependence of the daily transpiration rate on a wider range of input parameters. The 100% value is always assigned to a reference simulation run. The effect of the sun position and the day length, expressed by a variation of the date, is to be seen in Fig. 2a. From spring to summer transpiration may double if we assume the leaf area and the leaf properties to be constant over the whole period. Air humidity has a still more accentuated effect (Fig. 2b). Wind velocity causes a rapid decrease in transpiration when less than 1 m / s and only a small increase at higher wind speeds (Fig. 2c). Transpiration increases if leaves have a nearly horizontal or vertical inclination, while it decreases for oblique positions (Fig. 2d). Leaf size almost does not affect the transpiration rate (Fig. 2e) and transpiration decreases proportionally to the increase in cloud cover (Fig. 2f). Fig. 3 shows the diurnal course of transpiration for different air temperatures. Note the decrease in transpiration due to stomatal closure around noon for curve b.
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U. Janssen / Ecological Modelling 75/76 (1994) 321-330
~2
b)
8
I0
12
14
16
18
t (h)
Fig. 3. Diurnal course of transpiration with noon air temperatures of 292 (a) and 397 K (b).
At last two case studies were performed for two different alleys in the experimental area. The first one is oriented n o r t h - s o u t h and is bordered by linden trees that have a small, sphere-shaped and dense crown with a diameter of about 4 m. The other, east-west-oriented street is bordered by young birch trees. The crowns have an ellipsoidal shape and are only very sparsely leafed. Leaves are much smaller than commonly observed, which may indicate root damage or long-lasting dry spells. The leaf area densities were assumed to be 1 and 0.2 m 2 / m 3 for the linden and the birch respectively. The leaf inclinations were prescribed to be planophile for the linden and erectophile for the birch. The simulations were run for the 13th June 1988. The observed meteorological conditions are shown in Fig. 4. In Fig. 5a the diurnal course of the transpiration rate in g / s for a single linden and a single birch tree is shown. These rates correspond to 49.5 l / d for the linden tree or 1.2 1 d -1 m -2 leaf area. For the birch the respective values are 20.2 1/d or 3.6 1 d -1 m -2 leaf area. The factor 3 in the daily transpiration rate is a consequence of the sparse crown of the birch trees, where only weak extinction of radiation occurs and wind speed is reduced only to a small degree. The decrease in transpiration around noon is due to the decrease of the large wind speed at this time of the day (see Fig. 4). The different pronunciation of morning and afternoon peaks is due to the temperature course (see also Fig. 4). In order to explain the
3
6
9
12
15
18
21
24
t (h)
Fig. 4. Diurnal course of temperature ( June 1988 at the experimental site.
), relative humidity (. --) and wind speed ( - - - )
on 13th
328
U. Janssen / Ecological Modelling 75/76 (1994) 321-330
a) 3"0 t
"-~ LO
9'
12
b)
3"0 t ~2.5
,,''". ,'
.,
15
t (h)
"..
1B'
C)
~ o.71 VO. 5
,-"
"i.~0.3 0.4 9
12
15
t (h)
",,
,, ,'" 9
iB
"',, ", 12
t (h)
15
18
d) .~0.6 ~0.5
~o4
- 0.3-
-
" -- ~ ~ ~ ~'~ /
_~ 0.2 ~0.! 9
12
15
18
t (h) Fig. 5. Diurnal course of transpiration in g / s for (a) a single linden tree ( ) and a single birch tree ( - - - ) , (b) for linden at a mean leaf inclination angle of 10 ( ), 0 ( - - - ) und 80 ° ( - - - - - ) , (c) for birch with a mean leaf inclination angle of 80 ( ) and 0 ° ( - - - ) , (d) for single birch tree ( ) as well as for the alley birch on the northern side of the street ( - - - ) and for the alley birch on the northern side of the street if shading is considered ( - - - - - - ) .
increased transpiration of birch in the evening, different leaf inclinations for both trees were investigated (Fig. 5b and 5c). Transpiration shows to increase for both trees in the evening if leaf inclination is assumed to be erectophile. Finally the effect of neighbouring trees is studied. The wind field of the linden tree is only influenced by the other alley trees during noon. At this time the wind blows from easterly directions. The trees on both sides of the street reduce transpiration by 4%. For the birch alley wind field modification occurs only from 8:00 to 10:00 and from 14:00 to 16:00 h. Although the birch trees only have a sparse crown, the effect on transpiration is more accentuated, the one on the windward side being reduced by 8% and the one on the lee side even by 14% (Fig. 5d). It seems that, as radiation is not attenuated as strong as in the linden case, the wind field plays a dominant role.
U. Janssen/Ecological Modelling 75/76 (1994) 321-330
329
4. Conclusions A n e w n u m e r i c a l m o d e l for th e e s t i m a t i o n o f the t r a n s p i r a t i o n of single plants has b e e n i n t r o d u c e d . T h e new aspect of this t h r e e - d i m e n s i o n a l m o d e l is that single plants of every c o n f i g u r a t i o n can be t r e a t e d . W e are in t h e position to c o n s i d e r a sunny and a shady part of t h e plant, which implies a h o r i z o n t a l l y i n h o m o g e n e o u s distribution o f r a d i a t i o n r e g i m e a n d leaf t e m p e r a t u r e and, c o n s e q u e n t l y , a t h r e e d i m e n s i o n a l distribution of t r a n s p i r a t i o n within the individual t r e e crown. D u r i n g the field e x p e r i m e n t s no data c o n c e r n i n g the t r ees w e r e collected, so that a v a l i d a t i o n o f t h e m o d e l was not yet possible. But the results p r o v e d to be consistent with o t h e r values f o u n d in l i t e r a t u r e (see e.g. Butler, 1976; T h o r p e , 1978). D u e to contacts at I S E M '92 a v a l i d a t i o n with h azel h e d g e r o w s is to be p e r f o r m e d and will be p u b l i s h e d soon.
References Anderson, M.C., 1969. A comparison of two theories of scattering of radiation in crops. Agric. Meteorol., 6: 399-405. Baldocchi, D.D., Matt, D.R., Hutchison, B.A. and McMillen, R.T., 1984. Solar radiation within an oak-hickory forest: an evaluation of the extinction coefficients for several radiation components during fully-leafed and leafless periods. Agric. For. Meteorol., 32: 307-322. Braden, H., 1982. Simulationsmodell fi)r den Wasser-, Energie- und Stoffhaushalt in Pflanzenbest~inden. Bet. Inst. Meteorol. Klimatol. Univ. Hannover, No. 23. Butler, D.R., 1976. Estimation of the transpiration rate in an apple orchard from net radiation and vapour pressure deficit measurements. Agric. Meteorol., 16: 277-289. Campbell, G.S., 1988. Extinction coefficients for radiation in plant canopies calculated using an ellipsoidal inclination angle distribution. Agric. For. Meteorol., 36: 317-321. Cionco, M.R., 1985. Modeling windfields and surface layer wind profiles over complex terrain and within vegetative canopies. In: B.A. Hutchison and B.B. Hicks (Editors), The Forest-Atmosphere Interaction. Reidel, Dordrecht, The Netherlands, pp. 97-114. Deardorff, J.W., 1978. Efficient prediction of ground surface temperature and moisture, with an inclusion of a layer of vegetation. J. Geophys. Res., 83: 1889-1903. Dolman, A.J. and Van Den Burg, G.J., 1988. Stomatal behaviour in an oak canopy. Agric. For. Meteorol., 43: 99-108. Gross, G., 1988. A numerical estimation of the effects on local climate in the area of the Frankfurt International Airport. Beitr. Phys. Atmos., 61: 219-231. Gross, G., 1991. Anwendungsm6glichkeiten mesoskaliger Simulationsmodelle dargestellt am Beispiel Darmstadt. 1: Wind-und Temperaturfelder. Meteorol. Rundsch., 43:97-112. Lafleur, P.M. and Rouse, W.R., 1990. Application of an energy combination model for evaporation from sparse canopies. Agric. For. Meteorol., 49: 135-153. Leuning, R. and Foster, I.J., 1990. Estimation of the transpiration by single trees: comparison of a ventilated chamber, leaf energy budgets and a combination equation. Agric. For. Meteorol., 51: 63-86. Lohammer, T., Larsson, S., Linder, S. and Falk, O., 1980. Simulation models of gaseous exchange in Scots pine. Ecol. Bull., 32: 505-523. Pearman, G.I., Weaver, H.L. and Tanner, C.B., 1972. Boundary layer heat transfer coefficients under field conditions. Agric. Meteorol., 10: 83-92. R6ckle, R., 1990. Bestimmung der Str6mungsverh~iltnisse im Bereich komplexer Bebauungsstrukturen. Dissertation, TH Darmstadt.
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U. Janssen / Ecological Modelling 75 / 76 (1994) 321-330
Rose, C.W., 1984. Modelling evapotranspiration: an approach to heterogeneous communities. Agric. Water Manage., 8: 203-221. Thorpe, M.R., 1978. Net radiation and transpiration of apple trees in rows. Agric. Meteorol., 19: 41-57. Wang, Y.P. and Jarvis, P.G., 1990. Description and validation of an array model - MAESTRO. Agric. For. Meteorol., 51: 257-280. Yamada, T., 1982. A numerical model study of turbulent airflow in and above a forest canopy. J. Meteorol. Soc. Jpn., 60: 439-454.
Ecological Modelling 75/76 (1994) 321-330
T h e transpiration rate o f a single tree - numerical simulations Ulrike Janssen
*
Institute of Meteorology and Climatology, University of Hannover, Herrenhiiuserstr. 2, 30419 Hannover, Germany
Abstract
A numerical simulation model for the calculation of the transpiration of single trees has been developed. The required input data, like meteorological conditions as well as the flow configuration in and around the tree are provided by other, already existing models. Stem position and tree shape relate a certain leaf area to each grid volume of the three-dimensional model. This spatial heterogeneity allows the study of the radiation regime as well as the heat and water vapour exchange within the tree crown. Stomatal resistances are estimated in dependence of environmental factors. The individual leaf temperatures are calculated by solving an energy balance equation. Some model results, demonstrating the sensitivity of different input parameters on the transpiration rate, are presented. Key words: Numerical simulation; Transpiration; Tree
I. Introduction
L a n d s u r f a c e - a t m o s p h e r e interactions influence local, regional and global climate to a high degree. O n e of the most important processes affecting t e m p e r a t u r e and moisture distribution in the lower a t m o s p h e r e is the transpiration. During the two field experiments H I B E '88 and '89 large data sets were collected by satellite, aeroplane and in-situ measurements, which should contribute to a better understanding of the various exchange processes. T h e area, called " H i l d e s h e i m e r B6rde", where the experiments were p e r f o r m e d is located southeast of the city of Hannover. T h e site is characterized by an almost h o m o g e n e o u s terrain with
* Corresponding author. Present address: Ludwigstr. 24, 63067 Offenbach, Germany. 0304-3800/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDI 0304-3800(93)E0139-T
322
U. Janssen/ Ecological Modelling 75 / 76 (1994) 321-330
respect to relief and land use. A large part of the area is cultivated with wheat and sugar beet. There are no closed forest canopies, but single trees which stand in small groups or along the streets as alley trees. During the experiments the trees were not taken into account, so that their contribution to the transpiration has to be estimated by means of a numerical model. Until now the numerical models used in meteorology to estimate transpiration rates of vegetation are mainly one-dimensional (e.g. Deardorff, 1978). One may distinguish between "big-leaf" and "multiple-layer" models. Two- and three-dimensional simulations were performed by Yamada (1982), Cionco (1985) or Gross (1988). The array model of Wang and Jarvis (1990) and the model of Rose (1984) are first attempts for modelling the transpiration of individual plants. Measurements on single leaves or single trees were carried out by Thorpe (1978), Lafleur and Rouse (1990) and Leuning and Foster (1990). Their estimations are based on the application of different versions of the P e n m a n - M o n t e i t h equation. There are several factors which influence the transpiration rate of a plant: 1. the meteorological conditions, like air temperature, air moisture and wind velocity; 2. the water supply characterized by precipitation amount, soil properties and root characteristics; 3. the incoming solar radiation, which depends on the position of the sun, cloud cover and air turbidity; 4. the optical properties of the leaves: reflectivity, emissivity, extinction coefficient and leaf inclination angle; 5. the stomatal response. In the transpiration model all these factors have to be considered in an appropriate way.
2. Description of the model As input data for the transpiration model we need the meteorological conditions of the undisturbed environment. They are described by the one-dimensional version of the boundary-layer model FITNAH. Vertical profiles of wind velocity, temperature and moisture are obtained with a temporal resolution of 1 h (for detailed description see e.g. Gross, 1991). The flow configuration in and around the tree is calculated with a diagnostic, mass-consistent wind model, which was originally designed for complex building structures (R6ckle, 1990). It was modified in order to consider porous obstacles, where porosity results from leaf density, and to treat every possible tree shape. Corresponding to the leaf density, the wind will be attenuated and the cavity zone on the lee side will develop in a more or less accentuated way. A soil-water model is not yet implemented, so that effects on transpiration due to scarcity of water are not taken into consideration. According to the requirements the transpiration model has to be three-dimensional. The real tree with its individual distribution of leaves and branches is represented by a geometrical body like a cone, a cylinder or a sphere. This body is
U. Janssen/ Ecological Modelling 75 / 76 (1994) 321-330
323
subdivided into grid volumes. One leaf with a certain leaf area and leaf inclination is related to each of these volumes. The transpiration rate of the complete tree is the sum of the individual water vapour fluxes of each leaf. These fluxes are assumed to be constant for a period of 1 h. The specific moisture of the leaf s t is a function of the leaf temperature T~; it is supposed to be equal to the saturation moisture at the leaf temperature:
sz=s?(Tt). (1) The individual l e a f temperatures are obtained by solving an energy balance equation. It consists of three terms: net radiation and turbulent fluxes for sensible and for latent heat: RN + QH + Qv = 0. (2) Other terms like heat storage or energy exchange due to photosynthesis are neglected. The numerical solution is done by a Newton-Raphson iteration:
O T, = T/
OQ/OTt,
(3)
where Tt is the actual leaf temperature and T[ is the leaf temperature at the previous iteration step. Q is the sum of all energy fluxes:
Q = R N + QH + Qv. (4) Net radiation R N consists of the short-wave and the long-wave radiation regime. As the process of photosynthesis is not considered here, the radiation spectrum is only subdivided into these two parts. The short-wave radiation regime will first of all be determined by the incoming solar radiation, which is a function of sun position, cloud cover and air turbidity. On its way through the tree, the sun beam will be attenuated by reflection, extinction and scattering. All of these processes are taken into account in the balance equation for the short-wave radiation: S = So(1 - a f ) [ e x p ( - k s . L r ) + f ( w ) ] . (5) The incoming radiation S o is reflected with a corresponding albedo of the foliage af. For the calculation of the extinction, the foliage is treated as a homogeneous turbid layer with the extinction coefficient for short-wave radiation ks, which is a function of the leaf inclination angle. L r denotes the leaf area index in the direction of the sun beam. Campbell (1988) developed an expression for k s where different leaf angle distributions may be considered. The scattering function f ( w ) and the scattering coefficient w are taken from the work of Ross and Nilson (in Anderson, 1969). The balance for the long-wave radiation consists of four different components: the long-wave irradiation of: (1) the atmosphere, (2) the soil, (3) the surrounding leaves, and (4) the considered leaf itself: R i = Rat m exp( - k t "Lup ) + Rsoil exp( - k t • Ldown )
i-1 + E Rlij +
j=l
N E
j=i+l
Rlij - 2ef~rTt 4.
(6)
U, Janssen / Ecological Modelling 75 / 76 (1994) 321-330
324
kt is the extinction coefficient for long-wave radiation and is assumed to be 0.2" k s (Baldocchi et al., 1984). Lup and Ldow, are the leaf area index of the foliage above and below the considered leaf respectively. The influence of the jth on the ith leaf is:
Ruj=eftrTtj
4 biAxAyAz
~
e x p [ - k l ' L ( x i ~ ) ] cos Oij
(7)
The jth leaf irradiates according to the Stefan-Boltzmann law EfcrT~4. From this irradiation the ith leaf will only receive a small part, which corresponds to the ratio of its own leaf area b i A x A y A z to the surface of the hemisphere with radius xi~. bi is the spatial heterogeneous leaf area density and xi~ denotes the distance between the two leaves considered at the moment. Furthermore the angle Oi~ and the amount of foliage L(x~j) between the two leaves will reduce the radiation reaching the ith leaf. The turbulent fluxes of sensible heat are given by:
Te-T
QH = Ocp -
,
(8)
rH
with p being the density and Cp the specific heat of the air. Te and Tt are the environmental and leaf temperature respectively and r/4 is the aerodynamic resistance for heat, r/_/ is composed by the resistances for forced and flee convection. For their parameterization Braden (1982) uses results fom similarity theory. The constants were obtained by Pearman et al. (1972) during their measurements in plant canopies. rHfor c =
rmree
94~/d/u
454[ d 11J4 Cf
[ Tvt
Tve [
(9) (10)
with cf = 0.8, d being the leaf size, u the wind velocity and Tvt and TEe the virtual temperature of leaf and environment (T V = T(1 + 0.608. s)). The resulting resistance is obtained by a parallel connexion of the two individual resistances: r/_/=
- - + - rHforc
(11)
Ell free
Analogous to Eq. 8 the latent heat flux is written as:
Qv = Q L v
S e -- SI
r v + rst
,
(12)
with L V the latent heat for evaporation, s e and s t the specific moisture of environmental air and of the leaf. r E, the aerodynamic resistance for water vapour, is parameterized analogous to Eq. 9 by:
r v = 87~/d/u.
(13)
u. Janssen/ EcologicalModelling 75/ 76 (1994)321-330
325
O n e of the most important factors affecting transpiration is the behaviour of the stomata, which is a function of environmental conditions. T h e stomatal resistance rst or the inverse value, the stomatal c o n d u c t a n c e gst, is given as a direct function of m i n i m u m and maximum c o n d u c t a n c e and as a function of external conditions:
rst = (gs,) -1 = (gmin +gmax'fS "fT'fD) -1 (14) The modifiers f for the individual variables (S: short-wave radiation, T: temperature, and D: water v a p o u r saturation deficit) range between 0 and 1. They are given by the following relations: S - -
fs-
/
1000
al + S
1000+a 1
,
fD = 1 -- a 2 • ae T-Tmin
fT
{
TTpi----~min
with a I = 160.4 W / m e,
(15)
with a 2 = 0.1825 kPa - l
(16)
rmax-T Tmax _ Topt
(17)
T h e y are a combination of the models of L o h a m m e r et al. (1980) and D o l m a n and V a n D e n Burg (1988). As was m e n t i o n e d at the beginning, a s o i l - w a t e r model is not yet included and, until now, no attempts were m a d e to consider different species in the stomata submodel.
3. Sensitivity analysis and first results If a new model is developed, the first thing to do is to investigate how sensitive the model is to changes of certain input data. In Fig. 1 various input parameters were increased or diminished by 10%. Air humidity and the stomatal resistance
.: (/g0
+4. +2.
s :T,
F,[
,-,i.
i
q 111'/. ',+10% ]+lO¢Z~ +10% I+]0'}~. ~,+lO*/~ ',+ID% ',+lOVo ]+IO'X. | 10'~, --ltr/,, -i ii'; -10%, -10%- -10%, -i0%, 10%, -10
1
"
E
1
-i
U IU
n n U
U
Fig. 1. Percentual change of diurnal transpiration rate as a consequence of a 10% increase (left bar) and a 10% decrease (right bar) of the respective input parameters, u: wind speed, S: short-wave radiation, TI: leaf temperature, FF leaf area, if: mean leaf inclination angle, e: emissivity, k,: extinction coefficient, r H, rv: aerodynamic resistance for heat and water vapour, f: relative humidity, r,t: stomatal resistance.
U. Janssen /Ecological Modelling 75/76 (1994) 321-330
326
8)
b)
160
180"
120
t20~
80
80-
40
40" 21.4
21.5
20
21.8
d;tt,,
40 80 80 rel. humidity (%)
100
d) 180
18o I
120
1201
80
801
40
401 1
2 3 4 5 8 wind velocity (m/s)
7
8
18 2 0
38 4 8 50 80 78 leaf inclination (*)
80
90
=) 180
160-
120
120-
80
80
40
40 8
8
I0 12 leaf size (era)
14
0.2
0.4
0.6
0.8
1.8
cloud cover
Fig. 2. Percentual change of daily transpiration related to a reference simulation when varying(a) date, (b) relative humidity, (c) wind speed, (d) leaf inclination angle, (e) leaf size and (f) cloud cover. influence transpiration to a high degree, as a change of 10% results in a variation of transpiration in the same order of magnitude. The incident solar radiation still causes a difference of approximately 4%, but all the other changes in transpiration do not exceed 2%. Fig. 2 shows the dependence of the daily transpiration rate on a wider range of input parameters. The 100% value is always assigned to a reference simulation run. The effect of the sun position and the day length, expressed by a variation of the date, is to be seen in Fig. 2a. From spring to summer transpiration may double if we assume the leaf area and the leaf properties to be constant over the whole period. Air humidity has a still more accentuated effect (Fig. 2b). Wind velocity causes a rapid decrease in transpiration when less than 1 m / s and only a small increase at higher wind speeds (Fig. 2c). Transpiration increases if leaves have a nearly horizontal or vertical inclination, while it decreases for oblique positions (Fig. 2d). Leaf size almost does not affect the transpiration rate (Fig. 2e) and transpiration decreases proportionally to the increase in cloud cover (Fig. 2f). Fig. 3 shows the diurnal course of transpiration for different air temperatures. Note the decrease in transpiration due to stomatal closure around noon for curve b.
327
U. Janssen / Ecological Modelling 75/76 (1994) 321-330
~2
b)
8
I0
12
14
16
18
t (h)
Fig. 3. Diurnal course of transpiration with noon air temperatures of 292 (a) and 397 K (b).
At last two case studies were performed for two different alleys in the experimental area. The first one is oriented n o r t h - s o u t h and is bordered by linden trees that have a small, sphere-shaped and dense crown with a diameter of about 4 m. The other, east-west-oriented street is bordered by young birch trees. The crowns have an ellipsoidal shape and are only very sparsely leafed. Leaves are much smaller than commonly observed, which may indicate root damage or long-lasting dry spells. The leaf area densities were assumed to be 1 and 0.2 m 2 / m 3 for the linden and the birch respectively. The leaf inclinations were prescribed to be planophile for the linden and erectophile for the birch. The simulations were run for the 13th June 1988. The observed meteorological conditions are shown in Fig. 4. In Fig. 5a the diurnal course of the transpiration rate in g / s for a single linden and a single birch tree is shown. These rates correspond to 49.5 l / d for the linden tree or 1.2 1 d -1 m -2 leaf area. For the birch the respective values are 20.2 1/d or 3.6 1 d -1 m -2 leaf area. The factor 3 in the daily transpiration rate is a consequence of the sparse crown of the birch trees, where only weak extinction of radiation occurs and wind speed is reduced only to a small degree. The decrease in transpiration around noon is due to the decrease of the large wind speed at this time of the day (see Fig. 4). The different pronunciation of morning and afternoon peaks is due to the temperature course (see also Fig. 4). In order to explain the
3
6
9
12
15
18
21
24
t (h)
Fig. 4. Diurnal course of temperature ( June 1988 at the experimental site.
), relative humidity (. --) and wind speed ( - - - )
on 13th
328
U. Janssen / Ecological Modelling 75/76 (1994) 321-330
a) 3"0 t
"-~ LO
9'
12
b)
3"0 t ~2.5
,,''". ,'
.,
15
t (h)
"..
1B'
C)
~ o.71 VO. 5
,-"
"i.~0.3 0.4 9
12
15
t (h)
",,
,, ,'" 9
iB
"',, ", 12
t (h)
15
18
d) .~0.6 ~0.5
~o4
- 0.3-
-
" -- ~ ~ ~ ~'~ /
_~ 0.2 ~0.! 9
12
15
18
t (h) Fig. 5. Diurnal course of transpiration in g / s for (a) a single linden tree ( ) and a single birch tree ( - - - ) , (b) for linden at a mean leaf inclination angle of 10 ( ), 0 ( - - - ) und 80 ° ( - - - - - ) , (c) for birch with a mean leaf inclination angle of 80 ( ) and 0 ° ( - - - ) , (d) for single birch tree ( ) as well as for the alley birch on the northern side of the street ( - - - ) and for the alley birch on the northern side of the street if shading is considered ( - - - - - - ) .
increased transpiration of birch in the evening, different leaf inclinations for both trees were investigated (Fig. 5b and 5c). Transpiration shows to increase for both trees in the evening if leaf inclination is assumed to be erectophile. Finally the effect of neighbouring trees is studied. The wind field of the linden tree is only influenced by the other alley trees during noon. At this time the wind blows from easterly directions. The trees on both sides of the street reduce transpiration by 4%. For the birch alley wind field modification occurs only from 8:00 to 10:00 and from 14:00 to 16:00 h. Although the birch trees only have a sparse crown, the effect on transpiration is more accentuated, the one on the windward side being reduced by 8% and the one on the lee side even by 14% (Fig. 5d). It seems that, as radiation is not attenuated as strong as in the linden case, the wind field plays a dominant role.
U. Janssen/Ecological Modelling 75/76 (1994) 321-330
329
4. Conclusions A n e w n u m e r i c a l m o d e l for th e e s t i m a t i o n o f the t r a n s p i r a t i o n of single plants has b e e n i n t r o d u c e d . T h e new aspect of this t h r e e - d i m e n s i o n a l m o d e l is that single plants of every c o n f i g u r a t i o n can be t r e a t e d . W e are in t h e position to c o n s i d e r a sunny and a shady part of t h e plant, which implies a h o r i z o n t a l l y i n h o m o g e n e o u s distribution o f r a d i a t i o n r e g i m e a n d leaf t e m p e r a t u r e and, c o n s e q u e n t l y , a t h r e e d i m e n s i o n a l distribution of t r a n s p i r a t i o n within the individual t r e e crown. D u r i n g the field e x p e r i m e n t s no data c o n c e r n i n g the t r ees w e r e collected, so that a v a l i d a t i o n o f t h e m o d e l was not yet possible. But the results p r o v e d to be consistent with o t h e r values f o u n d in l i t e r a t u r e (see e.g. Butler, 1976; T h o r p e , 1978). D u e to contacts at I S E M '92 a v a l i d a t i o n with h azel h e d g e r o w s is to be p e r f o r m e d and will be p u b l i s h e d soon.
References Anderson, M.C., 1969. A comparison of two theories of scattering of radiation in crops. Agric. Meteorol., 6: 399-405. Baldocchi, D.D., Matt, D.R., Hutchison, B.A. and McMillen, R.T., 1984. Solar radiation within an oak-hickory forest: an evaluation of the extinction coefficients for several radiation components during fully-leafed and leafless periods. Agric. For. Meteorol., 32: 307-322. Braden, H., 1982. Simulationsmodell fi)r den Wasser-, Energie- und Stoffhaushalt in Pflanzenbest~inden. Bet. Inst. Meteorol. Klimatol. Univ. Hannover, No. 23. Butler, D.R., 1976. Estimation of the transpiration rate in an apple orchard from net radiation and vapour pressure deficit measurements. Agric. Meteorol., 16: 277-289. Campbell, G.S., 1988. Extinction coefficients for radiation in plant canopies calculated using an ellipsoidal inclination angle distribution. Agric. For. Meteorol., 36: 317-321. Cionco, M.R., 1985. Modeling windfields and surface layer wind profiles over complex terrain and within vegetative canopies. In: B.A. Hutchison and B.B. Hicks (Editors), The Forest-Atmosphere Interaction. Reidel, Dordrecht, The Netherlands, pp. 97-114. Deardorff, J.W., 1978. Efficient prediction of ground surface temperature and moisture, with an inclusion of a layer of vegetation. J. Geophys. Res., 83: 1889-1903. Dolman, A.J. and Van Den Burg, G.J., 1988. Stomatal behaviour in an oak canopy. Agric. For. Meteorol., 43: 99-108. Gross, G., 1988. A numerical estimation of the effects on local climate in the area of the Frankfurt International Airport. Beitr. Phys. Atmos., 61: 219-231. Gross, G., 1991. Anwendungsm6glichkeiten mesoskaliger Simulationsmodelle dargestellt am Beispiel Darmstadt. 1: Wind-und Temperaturfelder. Meteorol. Rundsch., 43:97-112. Lafleur, P.M. and Rouse, W.R., 1990. Application of an energy combination model for evaporation from sparse canopies. Agric. For. Meteorol., 49: 135-153. Leuning, R. and Foster, I.J., 1990. Estimation of the transpiration by single trees: comparison of a ventilated chamber, leaf energy budgets and a combination equation. Agric. For. Meteorol., 51: 63-86. Lohammer, T., Larsson, S., Linder, S. and Falk, O., 1980. Simulation models of gaseous exchange in Scots pine. Ecol. Bull., 32: 505-523. Pearman, G.I., Weaver, H.L. and Tanner, C.B., 1972. Boundary layer heat transfer coefficients under field conditions. Agric. Meteorol., 10: 83-92. R6ckle, R., 1990. Bestimmung der Str6mungsverh~iltnisse im Bereich komplexer Bebauungsstrukturen. Dissertation, TH Darmstadt.
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Rose, C.W., 1984. Modelling evapotranspiration: an approach to heterogeneous communities. Agric. Water Manage., 8: 203-221. Thorpe, M.R., 1978. Net radiation and transpiration of apple trees in rows. Agric. Meteorol., 19: 41-57. Wang, Y.P. and Jarvis, P.G., 1990. Description and validation of an array model - MAESTRO. Agric. For. Meteorol., 51: 257-280. Yamada, T., 1982. A numerical model study of turbulent airflow in and above a forest canopy. J. Meteorol. Soc. Jpn., 60: 439-454.