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SAFIR
Safe and High Quality Food Production using Low Quality Waters and Improved Irrigation
Systems and Management (SAFIR)
Contract-No. FOOD-CT-2005-023168
A Specific Targeted Research Project
under the Thematic Priority ' Food Quality and Safety '
WP 3 Single Crop Modelling
D3_2 Final version and documented models available
Due date: 30-04-09
Actual submission date: 30-04-09
Start date of project: 01-10-05 Duration: 48 months
Deliverable Lead contractor: Natural Environment Research Council
Participant(s) (Partner short names) DIAS, CER, LIFE-KU, NERC, UB, CAU,
CAAS
Author(s) in alphabetic order: Ragab Ragab
Contact for queries: Natural Environment Research Council
Centre for Ecology and Hydrology
Maclean Building
Crowmarsh Gifford
Wallingford, OXON
OX10 8BB, UK
United Kingdom
[email protected]
Dissemination Level:
(PUblic, Restricted to other Programmes Participants,
REstricted to a group specified by the consortium,
COnfidential only for members of the consortium)
PU
Deliverable Status: Revision 1.0
Project co-funded by the European Commission within the Sixth Framework Programme (2002-2006)
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Contents
The Daisy model description (Summary) ...................................................................................................................... 3
2D Soil Physics .............................................................................................................................................................. 3
Changes to the crop model .......................................................................................................................................... 3
New SVAT model .......................................................................................................................................................... 3
The SALTMED model description (Summary)............................................................................................................ 5
Default Data in the Databases ...................................................................................................................................... 6
Data Requirements ....................................................................................................................................................... 6
Annex 3.1 Water movement in Daisy model ................................................................................................................. 9
Annex 3.2 Daisy 2D code development ........................................................................................................................ 52
Annex 3.3 The stomata-photosynthesis model and the sunlit-shadow radiation model in DAISY ........................ 53
Annex 3.4 Estimating root density in Daisy ................................................................................................................ 75
Annex 3.5 ABA in Daisy................................................................................................................................................ 85
Annex 3.6 Soil Vegetation Atmosphere Transfer (SVAT) model.............................................................................. 89
Annex 3.7 Detailed description of the processes in the SALTMED........................................................................ 100
Annex 3.8 SALTMED model frames (user Interface) and Examples of outputs................................................... 111
Annex 3.9 Test of the new Daisy model ..................................................................................................................... 130
Annex 3.10 Some selected simulation results using the SALTMED model ............................................................ 141
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The Daisy model description (Summary) Daisy is a field scale model for simulating nitrogen, carbon and water dynamics. It has been used
for assessing both production and environmental impact of farm management. In the SAFIR
project the model has been extended in order to be able to simulate the effect of PRD irrigation in
row crops. The complete model (including documentation) can be downloaded from the Daisy
homepage:
http://code.google.com/p/daisy-model/
The changes can roughly be divided into three areas: soil physics, the crop model, and the soil
vegetation atmosphere transfer system.
2D Soil Physics
The original Daisy model is based on a vertical 1D description of the field, with finite difference
solutions to Richard’s equation for water flow, convection-dispersion for solute (nutrients and
pesticides) transport, and the heat transfer equation. To support row crops and PRD irrigation, a
2D description of the field has been introduced based on finite volume solutions to the same
equations. These numerical schemes have been documented in Annex 3.1. The principles for
integrating them with the existing Daisy code can be found in Annex 3.2
Changes to the crop model
The original Daisy photosynthesis model is based on radiation intensity (Goudriaan and Laar,
1978), with separately calculated water and nitrogen stress factors. The canopy is divided into 30
layers, with a single light extinction coefficient. To better support the effect of ABA on
production, a new photosynthesis model based on Farquhar et al (1980) and Ball et al (1987), with
Stomata conductance model coupled as described by Collatz et al., 1991, has been added. This is
supplemented by a new light distribution model that includes sunlit and shaded leaves, as well as
the effect of the sun angle on diffuse radiation, as per de Pury and Farquhar (1997). This work is
described in Annex 3.3.
A new 2D root distribution model has been included for row crops. It is based on a simple 2D
extension of the empirical relationship found in Gerwitz and Page, 1974, as detailed in Annex 3.4.
Binding the two together is the plant hormone ABA. The ABA generation is based on work by Liu
et al. (2008). Its implementation in Daisy, as well as the effect on stomata conductivity, is
described in Annex 3.5.
New SVAT model
To support the more complex photosynthesis model, a new SVAT (Soil Vegetation Atmosphere
Transfer) model has been introduced. The model divides the canopy into sunlit and shaded leaves,
and a set of equations describing the energy flow between laves, canopy air, soil surface, and the
above canopy atmosphere is introduced. By solving this equation system, we can find the
temperature of sunlit and shaded laves, canopy air humidity, as well as transpiration. Since the
temperature and humidity affects photosynthesis and stomata conductance, and stomata
conductance is an important element of the equation system, an iterative process is used.
The equation system is described in Annex 3.6. The components of the equation system are too
numerous to mention here, but most can be found in Houborg (2006), on which this work is based.
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References:
Ball, J.T., Woodrow, I.E. and Berry, J.A. (1987) A model predicting stomatal conductance and its
contribution to the control of photosynthesis under different environmental conditions. In:
I.Biggins (Editor), Progress in Photosynthesis Research. Martinus Nijhoff Publishers,
Netherlands, pp. 221-224.
Collatz, G.J., Ball, J.T., Grivet, C. and Berry, J.A. (1991) Physiological and Environmental-
Regulation of Stomatal Conductance, Photosynthesis and Transpiration - A Model That
Includes A Laminar Boundary-Layer. Agricultural and Forest Meteorology, 54(2-4): 107-136.
de Pury, D.G.G. and Farquhar, G.D. (1997) Simple scaling of photosynthesis from leaves to
canopies without the errors of big-leaf models. Plant Cell and Environment, 20(5): 537-557.
Farquhar, G.D., Caemmerer, S.V. and Berry, J.A. (1980) A Biochemical-Model of Photosynthetic
Co2 Assimilation in Leaves of C-3 Species. Planta, 149(1): 78-90
Gerwitz, A., and Page, E. R (1974) An empirical mathematical model to describe plant root
systems.The Journal of Applied Ecology 11, 773-781
Goudriaan, J., Laar, H.H. van (1978) Calculation of daily totals of the gross CO2 assimilation of
leaf canopies. Journal of Agricultural Science (Netherlands) 26(4) p. 373-382
Houborg, R (2006) Inferences of key environmental an vegetation biophysical controls for use in
regional-scale SVAT modeling using Terra and Agua MODIS and weather prediction data.
PhD dissertation, Institute of Geography, University of Copenhagen,
Liu, F., Song, R., Zhang., X., Shahnazari, A., Andersen, M. N., Plauborg, F., Jacobsen, S-E. and
Jensen, C. R. (2008) Measurement and modelling of ABA signalling in potato (Solanum
tuberosum L.) during partial root-zone drying. Environmental and Experimental Botany 63,
385-391
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The SALTMED model description (Summary)In this part a summary of SALTMED model processes will be given, however more detailed
description of the processes and the main equations are given in Annex 3.7. SALTMED model user
interface and examples of input and output are given in Annex 3.8.
The SALTMED model was designed to include a number of physical processes acting
simultaneously under field conditions. It was also designed to be, generic, physically based, and
friendly to use. The model contained the following key processes:
1. Evapotranspiration: several options to calculate the evapotranspiraion that include:
Penman-Monteith equation according to the modified version of FAO - 56 (1998) with average
seasonal value of surface conductance.
Penman Monteith equation with options to specify the surface conductance. The latter is
calculated by different methods:
A. Applying Penman – Monteith equation using stomata Conductance calculated
from environmental parameters: According to Jarvis 1976 and modified by Korner
et al. (1995). It is based on multiplication of maximum stomata conductance by the
relative effects of environmental stress factors such as Vapour Pressure Deficit,
VPD, temperature, soil water availability and radiation.
B. Applying Penman – Monteith equation using stomata Conductance calculated
from the Absecic Acid, ABA and leaf water potential
The equation suggested by Tardieu et al. 1993 was implemented. The equation is
based on minimum stomata conductance, leaf water potential, Absecic Acid
concentration, and other fitting coefficients.
C. Applying Penman – Monteith equation using average value of stomata
Conductance
D. Applying Penman – Monteith equation using measured daily value of stomata
Conductance
2. Modelling Crop Growth, Biomass / Dry Matter production and Yield
The crop growth, biomass / dry matter production and yield have been calculated based on:
radiation, photosynthetic efficiency, water uptake, air temperature, leaf nitrogen content,
leaf area index, respiration losses and the harvest index.
3. Modelling Soil Nitrogen Dynamics
Soil nitrogen dynamics based largely on the approach of Johnsson et al. 1987 in SOIL – N
model have been adapted and coded. The model takes into account the different external N-
sources as:
1-Dry deposition from the atmosphere
2-Wet deposition with rainfall
3- Commercial Fertilizers add as dry chemicals (Urea, Ammonium based fertilizer,
Nitrate based Fertilizer and mixed Ammonium Nitrate Fertilizer etc.)
4- Commercial fertilizers added with the irrigation water
(Fertigation) these are as above (Urea, Ammonium based fertilizer, Nitrate based
Fertilizer and mixed Ammonium Nitrate Fertilizer etc.)
5- Manure
6- Incorporated crop residues of previous crop on ploughing day before sowing
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The model implemented the following processes:
• Mineralization, Immobilization, Nitrification, Denitrification
• N-Leaching
• Plant N Uptake
.
4. Modelling Soil Temperature
Soil temperature has been calculated from air temperature according to the approach of
Kang et al. 2000 and Zheng et al. 1993. It is based on average air temperature, damping
ratio, thermal diffusivity as a function of soil water, air and mineral content, Leaf Area
Index, LAI and litter fraction.
5. Modelling water uptake: Plant water uptake was calculated according to Cardon and Letey
(1992) taking into account the water stress and the salinity stress in case of using saline
water.
6. Modelling soil water and solute movement: The water flow in soil was described
mathematically using Richard's equation. The movement of solute in the soil system was
described by the convection–dispersion equation. Under irrigation from a trickle line
source, the water and solute transport can be viewed as two-dimensional flow. In the model,
sprinkler, flood and basin irrigation are described by one-dimensional flow equations.
Furrow and trickle line source are described by 2-dimensional equations. Trickle point
source is described by cylindrical flow equations.
Default Data in the Databases
The model has 3 built-in databases:
Crop database (based largely on the FAO 1992 & 1998), contains different crops, trees and shrubs
(>200) from different regions, duration of each growth stage, sowing and harvest dates, Kc & Kcb
values for each growth stage, maximum height and maximum rooting depth. The model uses Kcb as
it runs on a daily time step.
Soils database: Contains the hydraulic characteristics and solute transport parameters of more
than 40 different soil types.
Irrigation system database: Contains information on the wetting fraction and the frequency of
application of the irrigation systems
Data Requirements
Plant characteristics: these include for each growth stage; the crop coefficient, Kc , Kcb, root depth
and lateral expansion, crop height and maximum / potential final yield observed in the region
under optimum conditions.
Soil characteristics: include depth of each soil horizon, saturated hydraulic conductivity, saturated
soil water content, salt diffusion coefficient, longitudinal and transversal dispersion coefficient,
initial condition of : soil moisture, NO3-N, NH
4-N and salinity in each soil layer, tabulated data of
soil moisture versus soil water potential and soil moisture versus hydraulic conductivity.
Meteorological data: include daily values of temperature (maximum), temperature (minimum),
relative humidity, total or net radiation, wind speed, and daily rainfall.
Water management data: include the date and amount of irrigation water and fertilizers applied
(fertigation) and the salinity of applied irrigation.
Nitrogent fertilization data: This includes amount and date of dry fertilizers added, dry and wet N
deposition, initial soil humus and litter contents.
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Model parameters: include the number of compartments in both vertical and horizontal direction,
tortuosity parameters, diffusion parameters, uptake parameters, position of plant relative to
irrigation source and the maximum time step for calculation.
Model run: The model runs at maximum time step of 200 seconds and output values on daily basis.
The model calculates the water and solute movement on grid square basis. The default grid size is 4
by 4 cm. The model considers different plant positions from the irrigation source and
accommodates rainfed as well as subsurface irrigation including deficit irrigation and Partial Root
Drying, PRD.
Model structure and equations used to describe each processes are given in Ragab, 2002.
The model is friendly and easy to use benefiting from the WindowsTM
environment. The model is
freely available at: http://www.safir4eu.org
Published SALTMED References:
Ragab, R. 2002. A holistic generic integrated approach for irrigation, crop and field
management: the SALTMED model. Environmental Modelling and Software 17: 345-361.
R. Ragab, (Editor), 2005. Advances in integrated management of fresh and saline water for
sustainable crop production: Modelling and practical solutions. International Journal of
Agricultural Water Management (Special Issue), volume 78- Issues 1-2, pages 1-164. Elsevier,
Amsterdam. The Netherlands.
Supporting development References
Cardon, E.G., Letey, J., 1992. Plant water uptake terms evaluated for soil water and
solute movement models. Soil Sci. Soc. Am. J. 56, 1876-1880.
FAO, 1998. Crop evapotranspiration, Irrigation and Drainage Paper No 56. Rome, Italy.
Jarvis, P. G. 1976. The iterepretation of the variations in leaf water potential and stomatal
conductance found in canopies in the field. Philosophical. Transactions of the Royal
Society. B273:593-610.
Pleijel, H., Danielsson, H., Vandermeiren, K., Blum, C., Colls, J, and Ojanpera, K. 2002. Stomatal
conductance and ozone exposure in relation to potato tuber yield-results from the European
CHIP programme. European J.of Agronomy, 17:303-317.
Tardieu, F, Zhang, J. and Gowing, D. J. G. 1993. Stomatal control by both [ABA] in the xylem sap
and leaf water status: a test of a model for droughted or ABA-fed field-grown maize. Plant,
Cell and environment .16:413-420.
Eckersten, H and Jansson, P,.- E. 1991. Modelling water flow, nitrogen uptake
and production for wheat. Fertilizer Research 27: 313-329.
Johnsson, H., Bergstrom, L and Jansson, P.-E.. 1987. Simulated nitrogen dynamics and losses in a
layered agricultural soil. Agriculture, Ecosystems and Environment, 18:333-356.
Wu, L., McGechan, M., B., Lewis, D. R., Hooda, P. S., and Vinten, A., J., A. 1998. Parameter
selection and testing the soil nitrogen dynamics model SOILN. Soil Use and Management,
14: 170-181
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Kang, S., Kim, S., Oh, S. and Lee, D. 2000. Predicting spatial and temporal patterns of soil
temperature based on topography, surface cover and air temperature. Forest Ecology and
Management 136:173-184.
Marshall, T.J., Holmes, J. W., and Rose, C.W. (editors). 1996. Soil Physics ( 3rd
edition) , 358-376.
Cambridge University Press. Cambridge, UK.
Zheng, D., Hunt, Jr., Running, S.W. 1993. A daily soil temperature model based on air temperature
and precipitation for continental applications. Climate Research 2: 183-191.
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Annex 3.1 Water movement in Daisy model
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4
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19
Page 20
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2
[
1
K(ψi)+
1
K(ψj)
]
;</BM?
N-351#!-)$ (#2& '().$ !(#! 5'-$4 !(& (#+3)$-" #2&+#4& "#$ (#2& !(& &@&"! !(#!
.#!&+ ,+#"!-"#117 $)! "#$ %& !+#$',)+!&* -$ ')3& "#'&' .-!( '(#+, 4+#*-&$!' -$
!(& ,+&''5+& ,)!&$!-#1' .(-"( "#$ )""5+' -$ '-!5#!-)$' .-!( &2#,)+#!-)$ #$*
1#7&+&* ')-1/ O0 !(#! +&#')$ -' !(& #+-!(3&!-" 3&#$ "()'&$ #' *&0#51!A
Kij =1
2[K(ψi) + K(ψj)] ;</B>?
<=
21
Page 22
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JP
22
Page 23
!"# $!" %&'( '# $!" )*+'# ,&'#$ '% -#%+$-*+$") .ψ < 0/ $!" %&(-$'&# 0&**"%,&#)%$& $!" %&(-$'&# 1&* +# -#)*+'#") %&'(2 31 $!" %&'( '% %+$-*+$") .ψ > 0/ $!" )*+'#%*"4&5"% 6+$"* 1*&4 $!" %&'( 4+$*'7 !"#0" ψ = 023# $!" #-4"*'0+( 4&)"(8 $!" )*+'# ,'," '% )"%0*'9") +% + ,&'#$2 :!" )*+'#
,&'#$% %!+(( 9" ,(+0") '# $!" '#$"*'&* &1 + 0"(( +#) 0+##&$ 9" ,(+0") +$ 0"(( ");"%2
<&* &9$+'#'#; + #-4"*'0+( %$+9(" %&(-$'&# '$ '% '# $!" 9";'##'#; &1 + #"6
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$!" 4"+# 5+(-" '% ,&%'$'5" $!" ,*"%%-*" '# $!" )*+'# 0"(( '% 1&*0") $& ?"*&2 @1$"*
"+0! $'4" %$", + 4+%% 9+(+#0" 1&* "+0! &1 $!" )*+'# 0"((% '% 4+)" $& 0+(0-(+$"
$!" +4&-#$ &1 )*+'#") 6+$"*2
:"%$ %'4-(+$'&#% %!&6 $!+$ $!" 0&)" 9&$! '% +9(" $& $-*# &# $!" )*+'# 6!"#
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<';-*" A2B %!&6% $!" *"%-($% 1*&4 + %'4-(+$'&# 6'$! +# +C-'$+*) 9&-#)+*= 0&#D
)'$'&# +#) + )*+'#2 :!" -,,"* 9&-#)+*= !+% + #& E-7 0&#)'$' $!-% $!" &#(=
%-,,(= &1 6+$"* '% $!*&-;! $!" +C-'$+*)2 @% '$ 0+# 9" &9%"*5")8 $!" 4+$*'7
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AH
23
Page 24
!"!# $%&' &%%&()*&+,
!"!- .*/%)*&+, 012/3/
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./8/'.0 &' $4/ 6#$2%7 82/00"2/ 8&$/'$%#9 %' $4/ '/%:43&2%': 1/990* <= #00/639%':
/!"#$%&' ()*+,- >&2 i = 1, 2 · · ·N ? $4/ 82&39/6 1#' 3/ 52%$$/' #0 # &2.%'#2=
.%@/2/'$%#9 /!"#$%&' (AB - &' $4/ >&26C
Qdθ
dt= E(ψ)ψ + F(ψ) ()*DE-
54/2/ Q %0 # .%#:&'#9 6#$2%7 5%$4 Q(i, i) = |Qi| #'. θ
= θ
(ψ)* E(ψ)ψ %0
$4/ #00/639= &> Dij #'. DDij′ #'. Gij ? GD
ij′ ? BNij′ #'. Si #2/ #00/639/. %' F(ψ)*
F4/ /!"#$%&' %0 0&9;/. %' $4/ $%6/ .&6#%' "0%': $4/ 3#1G5#2. "9/2 6/$4&.C
Qθn+1,m+1 − θn
∆t= E(ψn+1,m)ψn+1,m + F(ψn+1,m) ()*DH-
I' &2./2 $& :/$ 2%. &> θ
#$ %$/2#$%&' 0$/8 m + 1? $4/ 6%7/. >&26"9#$%&' 3=J/9%# ! "#$ ()HHK- %0 #889%/.* I' $4/ 6%7/. >&26"9#$%&'? $4/ 5#$/2 1&'$/'$ #$
$%6/ 0$/8 n+1 #'. %$/2#$%&' 0$/8 m+1 %0 #882&7%6#$/. 3= # F#=9&2 /78#'0%&'C
θn+1,m+1
= θn+1,m
+dθ
dψ|n+1,m (ψn+1,m+1 − ψn+1,m)
= θn+1,m
+ Cn+1,m(ψn+1,m+1 − ψn+1,m)
()*LK-
54/2/ C = ∂θ
/∂ψ %0 $4/ 08/1%M1 5#$/2 1#8#1%$= >"'1$%&'* F4/ $%6/ ./2%;#$%;/
&> θ
1#' $4/' 3/ #882&7%6#$/. #0C
∂θ
∂t≈
θn+1,m+1
− θn
∆t=
θn+1,m+1
− θn+1,m
∆t+
θn+1,m
− θn
∆t
≈ Cn+1,m ψn+1,m+1 − ψn+1,m
∆t+
θn+1,m
− θn
∆t
()*L)-
F4"0? $4/ %$/2#$%;/ 014/6/ %0
(
1
∆tQC(ψn+1,m) − E(ψn+1,m)
)
ψn+1,m+1 =
F(ψn+1,m) +1
∆tQC(ψn+1,m)ψn+1,m +
1
∆tQ
(
θn − θn+1,m)
()*L+-
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&2 .='#6%1#99= 0%R/ &> $4/ $%6/ 0$/80? ∆t* S&2 $4/ 9#0$ 14&%1/? $4/ 0%R/ &> ∆t./8/'.0 &' 4&5 .%T1"9$ %$ %0 $& &3$#%' # 0&9"$%&'* O 82&1/."2/ 3#0/. &' 0#6/
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)L
24
Page 25
!"! # $%&'() *+,-&(+. &/01.(2-/
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=
θr + θs−θr
[1+|αψ|n]m *'& ψ < 0
θs *'& ψ ≥ 03678L:
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K = KsS1/2e [1 − (1 − S1/m
e )m]2 3678O:
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27 =#$ A'$UA.$!"+ 0&$ "#$! *5!A".'!+
'* +'., G0"$& A'!"$!") θ
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.!Q,"&0".'!) 0,+' A0! @$ G&.""$! 0+ 0 %'G$& +$&.$+ .! t1
27 =#$ 0++51%".'!+ *'& "#$
"#$'&() .+ 0 '!$SC.1$!+.'!0, -$&".A0, V'G .!"' 0 #'1'/$!'5+ +'., +$1.S.!Q!."$
+'., A',51!) .!.".0,,( G."# 5!.*'&1 G0"$& A'!"$!"7 =#$ A515,0".-$ .!Q,"&0".'! .+
$2%&$++$C 0+
I =
+∞∑
n=1
Antn2
3678N:
6N
25
Page 26
!"#" A1 = S $% &!" '(&") #"("#""* %'#+&$,$&- .% *"/)"* $) 0!$1$+ 2345467 8!"
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)2
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%$1& 1'.C 2,.) K");9!&")H 34MN6 !"#" Ks = 4.96 9CO*.-H θs = 0.396 9C3/9C3H
θr = 0.131 9C3/9C3H α = 0.00423 9C−1
.)* n = 2.067
P 9')%&.)& %$J" '( ∆tQ 3O5N *.- !.% <"") .++1$"* $) &!" IRS &"%& %$C;1.>
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U)$&$.11-H $& .% %!' ) &!.& &!" +' "# %"#$"% %'1;&$') 9.) <" .++1$"* ('# )')>
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26
Page 27
0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4−500
−450
−400
−350
−300
−250
−200
−150
−100
−50
0
θ [cm3/cm
3]
Philip
FVM
!"#$% &'() *+,-./!0,- ,+1 23 45-#/!5+ 65$ 7%$/!0,- 6,--!+" 8%,1 !+9-/$,/!5+'
:8% 45-#/!5+ !4 485;+ 65$ t = 1/5, 2/5, 3/5, 4/5 ,+1 1 · t
'
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' *4 !/ 0,+ ?% 5?4%$7%1= /8% 45-#/!5+4 ,$% ,-@54/ !1%+/!0,- %A0%>/ 65$
t = t
BC'DECC 1,.4F ;8%$% /8% %G%0/4 56 /8% 4-!"8/-. %,$-!%$ %@>/.!+" >5+1%1
;,/%$ !+ /8% 23 4!@#-,/!5+ !+4/,+/-. %G%0/4 /8% ;,/%$ 05+/%+/ >$59-%4'
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√
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!"#$%& '
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%$&!)#2"4 #!/2')@
θ
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+ θ"
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− θ#$
) 9A;G=
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%$&!)#2"4 ?2"+;
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= K(θ!
) 9A;M=
AA
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!"! #$ %& '(" ()*+,#- . ./%*#.' 0 ') 1#%.' /% ,$ #$"* 1/+ '(" 2,'"+ 3/0"3"%'
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%$'",* /1 θ! 7(#$ K!
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=
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= 0
K(θ"
) − K(θ#$"
) 1/+ θ!
≥ 09:!;<
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6" .,-.#-,'"* ,$
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q!
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4+ 3,+) 2,'"+5 C
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! 7("
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Γ% →%!
/+ , $/#+." % '(" $"./%*,+) */3, %5 −Γ%!→%
Γ% →%!
= −Γ%!→%
=
α →!
(C
− C!
) 1/+ C
≥ C!
α!→
(C
− C!
) 1/+ C
< C!
9:!IJ<
7(" +,'"$ 1/+ 3/0 %& $/-#'"$ 1+/3 C
'/ C!
5 α →!
$ %/' %"."$$,+ -) ">#,- '/
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5 α!→
!
7(" 3,$$ 6,-,%." 1/+ '(" $/-#'" .,% 6" "B4+"$$"* ,$8
∂(ρbCa)
∂t+
∂(θ
C
)
∂t+
∂(θ!
C!
)
∂t+
∂(θ"&
C"&
)
∂t= −∇ · j
−∇ · j!
−∇ · j"&
− Γ%
9:!II<
2("+" ρb $ '(" $/ - 6#-F *"%$ ') ,%* Ca $ '(" ./%."%'+,' /% % '(" ,*$/+6"*
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• !"#$%&'(
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3&%/ %/# .'&* .'*+%&'(9 :/# 8+; '< .'*+%# $ ( 6# !#.$,&6#! .=
j
= q
C
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1,'$#..9 :/# )'"#)#(% 60 !&-+.&'( *&7# 1,'$#..#. $ ( 6# #;1,#..#! .=
j
= −θ
D∇C
, D =
[
Dxx Dxz
Dzx Dzz
]
2>9?F4
3/#,# D &. %/# !&.1#,.&'( %#(.', 2', ) %,&;49 :/# $'(.#C+#($# &. %/ % %/#
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$'($#(%, %&'(9 :/# #*#)#(%. &( D ,# '<%#( $ *$+* %#! .=
Dxx = αL
v2 ,x
|v
|+ αT
v2 ,z
|v
|+ D∗
Dzz = αL
v2 ,z
|v
|+ αT
v2 ,x
|v
|+ D∗
Dxz = Dzx = (αL − αT )v ,xv
,z
|v
|
2>9?H4
3/#,# D∗&. %/# )'*#$+* , !&-+.&'(9 :/# ,#.% '< %/# %#,). ,# ,&.&(G <,') %/#
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%, (."#,. * !&.1#,.&'(9 :/# $ *$+* %&'( '< %/# !&.1#,.&'( %#(.', &. 6 .#! '( v
(! ('% v = q/θ9
:/# )'*#$+* , !&-+.&'( $ ( 6# $ *$+* %#! .=
D∗ = τD0 2>9?I4
3/#,# D0 &. %/# !&-+.&'( $'#J$&#(% <', %/# .'*+%# &( <,## 3 %#, (! τ &. %/#
%',%+'.&%0 < $%',9 5. ( #; )1*# @&**&(G%'( (! K+&,7 2?LM?4 .+GG#.%#!=
τ =θ7/3
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(! ('% %/# %/# %'% * ) %,&; 3 %#, $'(%#(%
θ!
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Page 34
!"#$ %$$!&'%(#) *'(+ $!,-(# .!/#.#0( 12 q
%0) θ
3 (+# #,#.#0($ !4 θ
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1# #5 "#$$#) %$6
θ
Dxx = αL
q2 ,x
|q
|+ αT
q2 ,z
|q
|+ D0
θ10/3
θs
θ
Dzz = αL
q2 ,z
|q
|+ αT
q2 ,x
|q
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θ10/3
θs
θ
Dxz = θ
Dzx = (αL − αT )q ,xq
,z
|q
|
789:;<
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= θ
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v
− θ
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789:?<
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C
)
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n · (C
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) = n · j
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7898:<
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'$ (+#
&!0�("%('!0 !4 (+# H!*9 G$ ,!*#" 1!-0)%"2 &!0)'('!0 '$ j
= n · q
C
= q
C
!4(#0 -$#)9 L0 1!(+ &%$#$ '( '$ %$$-.#) (+%( (+# )'M-$'!0 &"!$$'0E (+# 1!")#" '$
K#"!9
N-..%"'K#)3 (+# "!1,#. (! 1# $!,/#) 4!" )#(#".'0%('!0 !4 (+# &!0�("%('!0
!4 $!,-(# '0 Ω '$6
θ
∂C
∂t + C
∂θ
∂t = −∇ · (C
q
− θ
D∇C
) − Γ!
'0 Ω
n · (C
q
− θ
D∇C
) = j
!0 ∂Ω1
C
= C "#
!0 ∂Ω2
78988<
8O
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≤ 1, I"##)*!"& ./1 JK#)1 6LMMN89
O& *./ ".%$(+ @" *)/*(31"1 &!.& D""-$/7 &!" =)3#./& /30@"# () "# &!./ )/"
$% P3%& . Q3"%&$)/ )B %34*$"/&(+ %0.(( &$0"%&"-%9 J3& $% $& -)%%$@(" &) 0.D" .
0"%! !$*! 3/1"# .(( *$#*30%&./*"% -#"'"/&% &!.& &!" I"*("& /30@"# #.$%"% )'"#
>R9 :!" I"*("& /30@"# B)# &!" C) $/ &!" S51$#"*&$)/ $%<
(I"
)x =q!,x∆x
θ!
Dxx=
q!,x∆x
αLq ,xq
,x
|q| + αTq ,zq
,z
|q| + D0θ10/3
θs
<q!,x∆x
αTq ,xq
,x+q ,zq
,z
|q
|
=q!,x∆x
αT |q!
|≤
∆x
αT
6>9>T8
!"#" $& $% .%%30"1 &!.& αL ≥ αT 9 :!" %.0" -#)*"13#" *./ )B *)3#%" @" 3%"1
&) "'.(3.&" (I"
)z9 O& *./ &!"/ @" *)/*(31"1 &!.& &!" 0.S$030 I"*("& /30@"# $%
() "# &!./ ∆x/αT 9 OB &!" ()/7$&31$/.( 1$%-"#%$'$&+ $% T *0 ./1 &!" &#./%'"#%.( $%
LULV )B &!" ()/7$&31$/.( ./1 &!" 0.S$030 .(() "1 I
"
$% > *./ $& @" *)/*(31"1
&!.& &!" 0.S$030 %$1" ("/7&! )B &!" "("0"/&% %!.(( @" .--#)S$0.&"(+ L *09
:!$% $(( #"%3(& $/ . '"#+ ;/" 0"%!9
J"%$1"% &!" 3-%&#".0 "$7!&$/7 0"&!)1 $& $% -)%%$@(" &) *!))%" @"& ""/ > %&.5
@$($W$/7 0"&!)1%<
L9 O/&#)13*$/7 "S&#. 1$23%$)/ $/ &!" %&#".0($/" 1$#"*&$)/ %) I
"
=
≤ γ9 X!"#"γ $% &!" -"#B)#0./*" $/1"S9
>9 Y.#+$/7 &!" %$W" )B ∆t %) I"
=
≤ γ9
O& $% )B *)3#%" .(%) -)%%$@(" &) 1"%"("*& ./+ %&.@$($W$/7 0"&!)1%9 :!" (.%&
%&.@$($W$/7 0"&!)1 $% %&#.$7!& B)# .#1, @3& &!" ;#%& 1"%"#'"% $&% ) / %3@%"*&$)/<
!"#$%&'(# )'*+,'-(
G)# -#.*&$*.( %$&3.&$)/ .#" &!"#" )B&"/ %&.@$($&+ %) ()/7 I
"
=
≤ γ !"#" 2 ≤ γ ≤10 6I"##)*!"& ./1 JK#)1, LMMN8 !$*! 3/1"# .(( *$#*30%&./*"% $% ("%% #"%&#$*&$'"&!./ D""-$/7 @)&! I
"
≤ 2 ./1 =
≤ 19 γ $% *.(("1 &!" *)#+!#'$%,) -%.)/9
O/ &!" %&#".0($/" 1$23%$)/ $% .**)#1$/7 &) I"##)*!"& ./1 JK#)1 6LMMN8 .11"1
>Z
36
Page 37
!"# $%%&'&!($) )!(*&'+%&($) %& ,#- &!( '! ,-#.#(' '/$' 0
!
!"#$%$ &'%! ()% *)&+
$%, -%!.&!/",*% #,0%12 3)% "00#(#&,"4 4&,5#(60#,"4 0#$-%!$#&,7 αL #$ *"4*64"(%0
"$8
αL =
|v
|∆tγ − αL − D∗
|v
| , .&! αL + D∗
|v
| < |v
|∆tγ
0, .&! αL + D∗
|v
| ≥|v
|∆tγ
9:2:;<
!"#$%$!& !'(!(
3& #,'%$(#5"(% ()% $("=#4#(> &. ()% ,6/%!#*"4 /&0%4 #$ " $#/-4% $>$(%/ /&0%4%02
3)% $#(6"(#&, )%!% #$ " &,%+0#/%,$#&,"4 *&46/,7 )&!#?&,("4 *&46/, @#() $(%"0>+
$("(% @"(%! A&@ @#() -&!% '%4&*#(> v2 3)%!% #$ ,& $%*&,0"!> @"(%!7 θ"
()6$ v#
=v2 3)% 0#B6$#&,7 D 9=&() /&4%*64"! 0#B6$#&, ",0 )>0!&0>,"/#* 0#$-%!$#&,<
#$ 5#'%,2 C&! ()% -!%$%,( (%$( #$ ()% $&46(% ,&,+"0$&!-#,52 C&! " $*%,"!#& @#()
4#,%"! "0$&!-#,5 9*&,$(",( !%("!0"(#&, ."*(&!< ()% "0$&!-(#&, )"'% " $("=#4#?#,5
%B%*(7 ()6$ .&! 4#,%"! "0$&!-#,5 $&46(%$ ()% /&0%4 #$ %1-%*(%0 (& =% /&!% $("=4%2
3)% "0'%*(#&,+0#$-%!$#&, %D6"(#&, #, &,% 0#/%,$#&, *", =% @!#((%, "$8
∂C#
∂t= D
∂2C#
∂x2− v
∂C#
∂x9:2:E<
@)%!% v = q/θ2 3)% #,#(#"4 *&,0#(#&, #$ " ?%!& *&,*%,(!"(#&, #, ()% @)&4% *&46/,8
C#
(x, 0) = 0 9:2:F<
G( ()% 4%.( =&6,0"!> #$ ()% $&46(% A61 *&,$(",(2
(
−D∂C
#
∂x+ vC
#
)∣
∣
∣
∣
x=0
= vC#$%
9:2:H<
3)% $&46(#&, *", ()%, "**&!0#,5 (& '", I%,6*)(%, ",0 G4'%$ 9JHF:< =%
@!#((%, "$8
CJ(x, t) =1
2C
#$%
erfc
[
x − vt
2(Dt)1/2
]
+
(
v2t
πD
)
exp
[
−(x − vt)2
4Dt
]
−1
2C
#$%
(1 +vx
D+
v2t
D) exp(vx/D) erfc
[
x + vt
2(Dt)1/2
] 9:2KL<
C&! ()% $#/64"(#&,$ #$ /"0% " @"(%! A&@ $#(6"(#&, @#() $(%"0> $("(% A&@
@#() ()% *)&$%, -&!% @"(%! '%4&*#(> v = 1 */M)&6!2 C#$%
#$ .&! ()% $#/-4#*#(>
*)&$%, (& J ",0 D = 0.05 */2M)&6!2 C&! ()% N"#$> $#/64"(#&,$ #$ ()% '#!(6"4
$ *&46/, JL */ @#0% ",0 J */ )#5)2 O, ()% 0&/"#, #$ 5%,%!"(%0 " !%564"!
/%$) @#() JLL %D6"44> 4"!5% %4%/%,($7 %"*) @#() ∆x = 0.1 */2 P#() ∆t *)&$%,(& J )&6! "!%
!
= 10 ",0 Q
= 27 #2%2 Q
!
= 202 3)% ,6/%!#*"4 -"!"/%(%!7 ω#$ $%( (& JM:7 #2%2 " !",R+S#*)&4$&, $*)%/%2
T, U56!% :2J #$ ()% ","4>(#*"4 $&46(#&, *&/-"!%0 @#() ,6/%!#*"4 $&46(#&,$ @#()
",0 @#()&6( 6-$(!%"/ @%#5)(#,5 *&!!%$-&,0#,5 (& α = 1 ",0 α = 0.57 !%$-%*+(#'%4>2 V(!%"/4#,% 0#B6$#&, ",0 (#/%$(%- !%06*(#&, )"$ ,&( =%%, "--4#%02 G$ #(
:F
37
Page 38
!" #$ %#&$'($)* !'$ +,$ -.//0$& &./".1 !"+02 &3!00$' -,$" !4402."/ 54&+'$!3
-$./,+."/6 7,$ %"02 )'!-#! 8 &$$3& +% #$ &0./,+02 3%'$ "53$'. !0 ).95&.%"
%34!'$) -.+, +,$ "53$'. !0 &%05+.%" :%' α = 0.56 ;" +,$ :%00%-."/ !&$& 54<
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
x
C
Analytical
Regular
Upstream
=./5'$ >6?@ A"!02+. !0 &%05+.%" %34!'$) -.+, "53$'. !0 &%05+.%" -.+, '$/50!'
-$./,+."/ Bα = 0.5C !") 54&+'$!3 -$./,+."/ Bα = 1.0C6 7,$ &%05+.%"& !'$ &,%-"
!& %" $"+'!+.%" !& :5" +.%" %: D !:+$' &.350!+.%" %: E ,%5'&6 =%' +,$ ! +5!0 !&$
!'$ v = 1 3F,%5'* D = 0.05 3
2F,%5'6 =%' +,$ "53$'. !0 &.350!+.%"& !'$
∆t = ? ,%5' !") ∆x = 0.1 3* .6$6 G
= 2 !") H
!
= 106
&+'$!3 -$./,+."/ ,!& "%+ #$$" !440.$) Bα = 0.5C6
;" 1/5'$ >6> .& +,$ !"!02+. !0 &%05+.%" &,%-"6 I$&.)$& .& +,$ "53$'. !0 &%05<
+.%" &,%-" :%' +,$ !&$&@ "% &+!#.0.J!+.%"* +.3$&+$4 '$)5 +.%" !") &+'$!30."$
).95&.%"6 =%' #%+, +,$ +.3$&+$4 '$)5 +.%" !") &+'$!30."$ ).95&.%" 3$+,%) .&
+,$ 4$':%'3!" $ .")$D* γ = 10 ,%&$"6 =%' +,$ &.350!+.%" -.+,%5+ !"2 &+!#.<
0.J!+.%" !'$ +,$ -.//0$& &./".1 !"+6 7,$ -.//0$& !'$ &3!00$' :%' +,$ &.350!+.%"
-.+, &+'$!30."$ ).95&.%"6 I2 %34!'."/ -.+, +,$ !"!02+. !0 &%05+.%" !" +,$
!)).+.%"!0 ).95&.%" #$ %#&$'($)6 7,$ !)).+.%"!0 ).95&.%"& $9$ +.($02 '$)5 $)
+,$ G
<"53#$' :'%3 >K +% ?K6 7,$ '$3!."."/ /'!4, &,%-& +,$ &.350!+.%" -.+,
+,$ +.3$&+$4 '$)5 +.%" &+!#.0.J."/ 3$+,%) -,$'$ +,$ &.J$ %: ∆t .& ,!"/$) &%
G
H
!
≤ 106 L$'$ +,$ &.J$ %: +,$ +.3$&+$4 .& '$)5 $) :'%3 ∆t = 1 ,%5' B"%
&+!#.0.J!+.%"C +% K6M ,%5'6 N9$ +.($02 +,$ H
!
<"53#$' .& '$)5 $) :'%3 ?K +% M*
>O
38
Page 39
!"! #$% &'" ($)*+&,& $- . +."/ 0 & )". .$ ),-1 & )".&"*. 2$% ,**%$3 ),&"41
0 & )". .$ 4$-5 6789& )":! 6$)*,%"/ ; &' &'" -+)"% (,4 .$4+& $- ; &'$+&
.&,< 4 =,& $- ,%" ; 554". %"/+("/> <+& ',?" ,4.$ .),44"% ;,?"4"-5&'!
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
x
C
Analytical
No stabilization
Timestep reduction, γ=10
Streamline diffusion, γ=10
@ 5+%" 0!0A B-,41& (,4 .$4+& $- ($)*,%"/ ; &' / C"%"-& -+)"% (,4 .$4+& $-.!
D'" .$4+& $-. ,%" .'$;- ,. ($-("-&%,& $- ,. #+-(& $- $# 3 ,#&"% . )+4,& $- $# E
'$+%.! @$% &'" -+)"% (,4 .$4+& $- ; &'$+& .&,< 4 =,& $- ,%" 7
= 2 ,-/ 6
!
= 10!
F- G5+%" 0!H . &'" ,-,41& (,4 ,-/ -+)"% (,4 .$4+& $-. .'$;-! D'" -+)"% (,4
.$4+& $-. ,%" ($)*+&"/ +. -5 / C"%"-& *"%#$%),-(". -/"3".! D'" *"%#$%),-("
-/"3> γ . (',-5"/ ,**41 -5 & )".&"* %"/+(& $-! F& (,- <" $<."%?"/ &',& &'"
; 554". ,%" . 5- G(,-& #$% γ = 10> <+& #$% γ = 5 2,-/ 4$;"%: &'" . =" $# &'"
; 554". .""). &$ <" ,(("*&,<4" #$% )$.& *+%*$.".!
F- G5+%" 0!E . &'" ,-,41& (,4 .$4+& $- .'$;-! B4.$ &'" -+)"% (,4 .$4+& $-. #$%
?,%1 -5 *"%#$%),-(". -/"3". ,%" .'$;-! D'" *"%#$%),-(" -/"3> γ . (',-5"/
,**41 -5 .&%",)4 -" / C+. $-! D'" ; 554". ,%" %"/+("/ ;'"- +. -5 , 4$; ?,4+"
$# γ> <+& ($)*,%"/ ; &' &'" ,-,41& (,4 .$4+& $-> &'" .&""*-".. $# &'" #%$-& .
%"/+("/ /%,),& (,441!
HI
39
Page 40
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
x
C
Analytical
Timestep reduction, γ=10
Timestep reduction, γ=5
Timestep reduction, γ=2
!"#$% &'() *+,-./!0,- 12-#/!2+ 0234,$%5 6!/7 5!8%$%+/ +#3%$!0,- 12-#/!2+1 29:
/,!+%5 #1!+" /!3%1/%4 $%5#0/!2+ 6!/7 ;,$.!+" 4%$<2$3,+0% !+5%=' 2$ /7% 1!3#-,:
/!2+ ,$% v = 1 03>72#$ ,+5 D = 0.05 032>72#$' 2$ /7% +#3%$!0,- 1!3#-,/!2+1
!1 ∆x = ?'@ 03 ,+5 ∆t !1 $,+"!+" <$23 @>@? 72#$ Aγ = 2B /2 @>& 72#$ Aγ = 10B'
(@
40
Page 41
0 2 4 6 8 100
0.2
0.4
0.6
0.8
1
1.2
1.4
x
C
Analytical
Streamline diffusion, γ=10
Streamline diffusion, γ=5
Streamline diffusion, γ=2
!"#$% &'() *+,-./!0,- 12-#/!2+ 0234,$%5 6!/7 +#3%$!0,- 12-#/!2+1 28/,!+%5
#1!+" 1/$%,3-!+% 5!9#1!2+ 6!/7 :,$.!+" 4%$;2$3,+0% !+5%<' 2$ /7% ,+,-./!=
0,- 12-#/!2+ ,$% v = 1 03>72#$ ,+5 D = 0.05 03
2>72#$' 2$ /7% +#3%$!0,-
1!3#-,/!2+1 ,$% ∆x = ?'@ 03 ,+5 ∆t = @ 72#$'
A&
41
Page 42
!"! #$$%& '()*+,&- .(*+/0/(*
!" #$$"% &'#()*%+ ,'()-.-'( )"/,%-&"/ .!" 0'1"0"(. '2 /'3#." *$$3-") *. .!"
/#%2*," .!*. 0'1"/ -(.' .!" /'-34 5. *3/' )"/,%-&"/ .!" 0'1"0"(. '2 /'3#."/ '#.
2%'0 .!" )'0*-( -2 .!" 6*."% -/ 7'6-(8 '#. *. .!" #$$"% &'#()*%+4
52 .!" /'-3 /#%2*," ('. -/ $'()")9 .!" 7#: -(.' .!" /'-3 -/ 0'1") &+ *)1",.-'(
-(.' .!" )'0*-(9 6-.! .!" 6*."%4 52 .!" /#%2*," -/ $'()") 6-.! 6*."% ,'(.*-(-(8
* /'3#."9 .!" /'3#." -/ *3/' 0'1") /'3"3+ &+ *)1",.-'(4 ; 3-..3" 0'%" *,,#%*." )"<
/,%-$.-'( ,'#3) !*1" &""( '&.*-(") -2 .!" &'#()*%+ ,'()-.-'( 6*/ -0$3"0"(.")
*/ * =-%-,!3". ,'()-.-'( 6!-,! *3/' *33'6") .!" /'3#." .' &" 0'1") &+ )-/$"%/-'(
*() )->#/-'( -(.' .!" /'-34 5. -/ *//"//") .!*. .!" "%%'% -/ -(/-8(-?,*(.4
@!"( .!" 6*."% 0'1-(8 '#. '2 .!" )'0*-(4 A' /'3#." -/ 2'33'6-(8 .!" 6*."%4
!-/ -/ 2'% 0'/. /'3#."/ * 8'') &'#()*%+ ,'()-.-'( < *. 3"*/. 2'% "1*$'%*.-'(
$%',"//"/4 B'% * /-.#*.-'( 6-.! 3-C#-) 6*."% -/ 3"*1-(8 .!" /'-3 .!%'#8! .!" #$<
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-(."()") 2'% .!*. D-() '2 /,"(*%-'/4 E#00*%-F") .!" #$$"% &'#()*%+ ,'()-.-'(
,*( &" ":$%"//") */G
(
−θ
Dzz∂C
∂z+ qzC
)∣
∣
∣
∣
z=z !"#
=
q!"#
C$"%&
, q!"#
< 0
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> 0HI4JKL
6!"%" z$"%&
-/ .!" z<,''%)-(*." '2 .!" /'-3 /#%2*,"9 qz -/ .!" 7'6 H$'/-.-1" #$<
6*%)/L *() q!"#
-/ .!" 7#: '#. '2 .!" )'0*-( H("8*.-1" 2'% 7#: -(.' .!" )'0*-(L4
C$"%&
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A#0"%-,*33+ *33 .!" .+$"/ '2 #$$"% &'#()*%+ ,'()-.-'(/ -/ -0$3"0"(.") */ ":<
$3-,-. A"#0*(( ,'()-.-'(/9 -4"4 .!" /'3#." 0'1"0"(. '1"% .!" &'#()*%+ -/ -()"<
$"()"(. '2 .!" /'3#." ,'(,"(.%*.-'(/ -( .!" )'0*-(4
!"!" 1(2%& '()*+,&- .(*+/0/(*
!" 3'6"% &'#()*%+ ,'()-.-'( )"/,%-&"/ .!" 0'1"0"(. '2 /'3#." .!%'#8! .!"
3'6"% &'#()*%+4 52 .!" 6*."% !*/ * 2%"" )%*-(*8" ,'()-.-'(9 .!"%" -/ * 7#: ,'()-<
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.!"%" -/ /$",-?") * 8%'#()6*."% .*&3" '% *C#-.*%) &'#()*%+ ,'()-.-'(9 -4"4 $%"/<
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A#0"%-,*33+ .!" A"#0*(( &'#()*%+ ,'()-.-'(/ -/ -0$3"0"(.")9 "-.!"% -0$3-,-.
'% ":$3-,-. < -0$3-,-. 6!"( .!" 6*."% 7#: -/ '#.6*%)") 2%'0 .!" 3'6"% &'#()*%+
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42
Page 43
!!"#$ %&' #"(#&(%) %$"( $! %*& #"(#&(%) %$"( "+%!$'& %*& '", $(-
.*& /$)$#*0&% 1"+(' )2 #"('$%$"( $! $,30&,&(%&' ! ( &430$#$% 5&+, (( 1"+('6
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0""3 $(!$'& & #* %$,&!%&3 :$%* ="00":$(; $(#)& !&' #",3+% %$"( 0 %$,&!- ?( %*&
: %&) ,"8&,&(% !$,+0 %$"(!> %*& $%&) %$"( 0""3 : ! +('&) 00 #$)#+,!% (#&! (
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F
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%*& 1"+(' )2 $(%" %*& #&00 $( %$,&!%&3-
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'$<&)&(#& 1&%:&&( #&00 (' 1")'&) # ( 1& %) (!3")%&' 12 '$<+!$"( "8&) %*&
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." 3)&8&(% %" 0 );& #",3+% %$"( 0 %$,&!> %*& %$,&!%&3 # ( ("% 1& 0":&) %* (
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#"(#&(%) %$"(! % %*& #&00! % %*& 1"+(' )2> %*& #",3+% %$"(! $( %*& %$,&!%&3
)& )&3& %&' (' %*& !"0+%& $! "(02 ,"8&' 12 '8&#%$"( "8&) %*& 1"+(' )2- .*$!
# ( * 33&( $= %*& ,$($,+, 8 0+& "= %*& !$D& "= %*& %$,&!%&3! $! #*"!&( %" *$;*-
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!" #$ %$&%'(($" !)*
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44
Page 45
!" #$%&'($) (#
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Page 46
0 10 20 30 40 50 60 70 80 90 1000
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Page 52
Annex 3.2 Daisy 2D code developmentA major requirement for every programming project involving Daisy for the last 10 years has been that the
added functionality of the project should me merged into the mainstream code. When individual project
requirements conflicted, such as when they specified different models for the SAME process, both models
would be implemented as a user selectable choice. When the program framework didn't allow this, the
program framework would have to be modified to allow it. This approach is significantly more expensive to
implement than to branch the mainstream code for each project. However, the advantage to the individual
projects is even more significant, each project will benefit from improvements made in all the other projects.
The major requirement to Daisy in SAFIR is the 2D transport of water and solutes in the soil. This
requirement goes against the existing framework in one major way, and also requires a generalisation in
other areas. The main conflict was that Daisy was organised so the transport code was close to the turnover
code. This has been changed to all transport functionality now are part of the SAME selectable "Movement"
component. This component also contains the discretization, which has been generalized so the
discretization framework now in theory can handle 1, 2 or 3 dimensions. Only 1D and a simple 2D
discretization have been implemented so far. The old, 1D functionality of Daisy is available if the user
chooses the "vertical" Movement model. A new 2D model named "rectangle" supporting a simple,
rectangular grid of vertical and horizontal lines has been implemented as a proof-of-concept of the new
framework.
A stand-alone prototype for coupled 2D transport has also been developed, and needs to be integrated in the
new Daisy framework. Heat and solute transport has not yet been developed for anything but the old 1D
solution, but they have been ported to the new framework so the "vertical" model now has functionality
nearly identical to the old code.
52
Page 53
Annex 3.3 The stomata-photosynthesis model and the sunlit-shadow radiation
model in DAISY
Introduction
Photosynthesis is the conversion of CO2 to organic compounds in the presence of light. The chloroplasts of a
plant cell are the seat of photosynthesis and they are present only in the cells of the green parts of the plant.
Photosynthesis can be conveniently treated as three separate components: 1) light reactions, in which radiant
energy is absorbed and used to generate the high energy compounds ATP and NADPH; 2) dark reactions,
which include the biochemical reduction of CO2 to sugars using high energy compounds generated in the
light reactions; and 3) supply of CO2 from the ambient air to the site of reduction in the chloroplast.
Plants can be classified into at least three major groups on the basis of the biochemical pathway by which
they fix CO2, the C3, C4, and CAM. The latter will not be described in this context. The C4 photosynthesis
differs from C3 in several biochemical and physiological properties and C4 plants lack several features of C3
plants that are associated to photorespiration. Both C3 and C4 plants use the enzyme ribulose biphosphate
carboxylase (RuBP or Rubisco) for the primary fixation of CO2; however, the Rubisco reaction is
compartmented differently and.
The sun/shade radiation model in Daisy is inspired by the sun/shade model of de Pury and Farquhar (1997).
The sun/shade model of de Pury and Farquhar (1997) is a single-layer model which describes the sun and
shaded leaves separately. In the sun/shade model of de Pury and Farquhar (1997) the angel of incidence on
leaves is not considered. Instead the partitioning between the sunlit and shaded fractions of the canopies is
changed every time step. In Daisy, the canopy is divided in several layers with equal leaf area index. The
cumulative absorbed irradiance (from the top of the canopy) in the sun/shade model is calculated for each
canopy layer.
A number of mechanistic models of photosynthesis and stomatal conductance at the leaf level has been
developed and widely used that derive from the C3 photosynthesis model of Farquhar et al. (1980) and the
empirical stomatal conductance model of Ball et al. (1987). Boegh et al. (1987) and Collatz et al. (1991)
have implemented these two models combined with the leaf energy balances for both C3 and C4 plants.
These interacting models are solved by a numerical method, the Newton Raphsons method. There are two
models, based on the Farquhar-Ball-Collatz models, for C3 and for C4 plants implemented in the DAISY
code, the FC-C3 and the FC-C4 model, respectively.
53
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Absorbed irradiance of the canopies
Each plant community has a unique spatial pattern for displaying photosynthetic surfaces and to capture
photosynthetic active radiation (PAR). The photosynthetic quantum flux, I, is often the major factor
determining the rate of carbon dioxide (CO2) assimilation of individual leaves. Only about 50% of global
radiation is PAR.
eq. 1
,0 totalTotalI c R ,
c: Fraction of radiation which is PAR (0.5).
Conversion factor to convert daylight from units W m-2
to mol m-2
s-1
( = 46 10-6
mol s-1
W-1
(McCree, 1981)).
Rtotal: Global radiation [W m-2
].
I(Total,0): Total PAR per unit ground area above the canopy (mol m-2
s-1
).
In the sun-shade model, irradiance absorbed by sunlit leaves is calculated as absorbed beam plus absorbed
diffuse and scattered beam (total PAR). The irradiance absorbed by shaded leaves is calculated as absorbed
diffuse and absorbed scattered beam. Diffuse and scattered radiations are assumed isotropic and beam
radiation is unidirectional. If the global diffuse radiation is given in the climate input file, the diffuse PAR
above the canopy, I(d,0), is given by:
eq. 2
,0d dI c R ,
Rd: Global diffuse radiation [W m-2
].
I(d,0): Diffuse PAR per unit ground area above the canopy (mol m-2
s-1
).
and the photosynthetic quantum flux I for beam PAR above the canopy, I(b,0), is given by:
eq. 3
,0b total dI R R c ,
Diffuse radiation model (DifRad) If the global diffuse radiation is not given as an input driving variable, the diffuse radiation model (DifRad)
in DAISY calculates the fraction of the total PAR that is diffuse PAR by the principles described de Pury
and Farquhar (1997). It is furthermore assumed, that the fraction of the total PAR that is diffuse, equals the
fraction of total global radiation that is diffuse. Direct beam PAR calculated from extra-terrestrial PAR is
given by:
eq. 4
_ sinm
b optimal exI R ,
Ib_optimal: Beam PAR under a cloudless sky [W m-2
].
: Atmospheric transmission coefficient of PAR (0,72)
m: Optical air mass [unit less].
Rex: Extraterrestrial radiation [W m-2
].
: solar elevation angle [radians].
where the optical air mass is given by:
eq. 5
sin0P
Pm ,
P: Atmospheric pressure [Pa].
P0: Atmospheric pressure at sea level (1.013 105
Pa).
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Page 55
The diffuse PAR under a cloudless sky is given by:
eq. 6
sin1_ e
m
aoptimald IfI ,
Id_optimal: Diffuse PAR under a cloudless sky [W m-2
].
fa: Forward scattering coefficient of PAR in atmosphere (0,426).
An expression for the fraction of diffuse radiation (fd) of the total attenuated radiation for cloudless skies is
then given by:
eq. 7
optiamlboptiamld
optiamld
dII
If
__
_,
fd: fraction of diffuse radiation under a cloudless sky [unit less].
Id_optimal: Diffuse PAR under a cloudless sky [W m-2
].
At nighttime the sinus function of the solar elevation angle, sin , becomes negative and then it is assumed
that all radiation is diffuse by setting fd = 1.0. The global diffuse radiation is then given by eq. 8, which is
used to calculated the diffuse and beam PAR according to:
eq. 8
totalRfR dd ,
Rd: Global diffuse radiation [W m-2
].
Rtotal: Global radiation [W m-2
].
This model was originally developed for the decrease of short wave radiation. Short wave radiation has
different scattering and absorption than PAR but, as assumed by de Pury and Farquhar (1997), it is assumed
in the DifRad model that the process is similar for PAR.
Distribution of irradiance in the canopyAbsorption of I depends i.e. on leaf orientation, leaf arrangement in the canopy, sun elevation in the sky,
changes in spectral distribution of I through the canopy, and multiple reflections of I within the canopy. To
describe the penetration of diffuse, beam and scattered PAR in the canopy it is assumed that the decrease of
I down into a canopy is analogous to absorption of light by chlorophyll or other pigments in a solution,
which is described by Beer’s law (Nobel, 1991).
The sunlit leaf area fraction (or the sun fleck penetration), fsun, i, of canopy layer, i, is given by:
eq. 9
( , ) expsun i b if k L ,
Li: Cumulative leaf-area index from top of canopy to layer i (unit less, L = 0 for i = 0 (top), L = Lc for i = n (bottom)).
kb: Extinction coefficient of beam radiation (unit less).
55
Page 56
The extinction coefficient of beam radiation is given by:
eq. 10
06250 sin ,0.8
06250 sin ,sin
50
,
,.
kb ,
: The solar elevation (radiance).
However, when the sinus function to the solar elevation is close to zero, kb may reach unrealistic values.
Therefore kb has a maximum value when the sinus function returns zero, negative values, or larger the
0.0625 which corresponds to about 4°.
The penetration of sun fleck in the canopy at two different development stages is shown in Figure 1.
Increasing leaf area index through the canopy layers decrease the penetration of sun fleck.
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.1 0.2 0.3 0.4 0.5
Leaf area index in layer i , L (total,i)
su
n fle
ck, f
(sun,i)
))
0.0
0.2
0.4
0.6
0.8
1.0
1.2
0 0.5 1 1.5 2
Leaf area index in layer i , L (total,i)
Figure 1. The sun fleck penetration in the canopy layers given by eq. 9 and eq. 10 (using sin = 0.87) at two different
leaf area index of the total canopy (Lc). Left: Lc = 0.5. Right: Lc = 2.3.
The sunlit leaf area index, L(sun,i), of canopy layer, i, is then given by:
eq. 11
1,exp exp /b i b i bsun i
L k L k L k ,
Li: Cumulative leaf-area index from top of canopy to layer i (unit less, L = 0 for i = 0 (top), L = Lc for i = n (bottom)).
While the shaded leaf area index, L(sh,i), in the canopy layer is:
eq. 12
, ,ish i sun iL L L ,
L(sun,i): Cumulative leaf area index of sunlit fraction in canopy layer i (unit less).
L(sh,i): Cumulative leaf area index of shaded fraction in canopy layer i (unit less).
The cumulative sunlit and total leaf area indexes in the canopy layers, at two leaf area index of the canopy,
are shown in Figure 2. In the early stage of crop development, where the total leaf area index is low (top,
Figure 2) all the leaves in the canopy are sunlit. However, at later development stage, where the total leaf
area index is increased the shaded fractions of the leaves increases (bottom, Figure 2).
56
Page 57
Figure 2. The cumulative sunlit leaf area index, L(sun,i) and shaded leaf area index, L(sh,i) as function of total leaf area
index, L(total,i), through the canopy layers given by eq. 11 and eq. 12 at two different leaf area index of the total canopy
(Lc). Left: Lc = 0.5. Richt: Lc = 2.3.
Sunlit leavesBoth sunlit leaves and shaded leaves absorb diffuse and scattered diffuse irradiance. The total absorbed
irradiance is the sum of absorbed direct beam irradiance, the diffuse irradiance and the scattered beam
irradiance for the sunlit fraction. The cumulative (from top of the canopy to the actual layer) direct absorbed
beam (without scattering) irradiance, I(b,i), is given by:
eq. 13
, ,01 1 exp b ib i b
I I k L ,
I(b,0): Beam quantum flux per unit ground area above the canopy (mol m-2
s-1
).
L: Cumulative leaf-area index from top of canopy (unit less, L = 0 when i = 0 (top), L = Lc when i = 30).
The cumulative (from top of the canopy to the actual layer) quantum flux of scattered beam irradiance, I(bs,i),
is given by:
eq. 14
´´
, ,0 ´
1 exp 21 1 exp 1
2
b ib
cb b b ibs i b
b b
k LkI I k k L
k k
k´b: Modified extinction coefficient of beam radiation due to scattering (unit less).
cb: Canopy reflection coefficient for diffuse PAR ( cb = 0,029 (de Pury & Farquhar, 1997)).
The modified extinction coefficient of beam radiation due to scattering is affected by the leaf scattering.
eq. 15
1´
bb kk ,
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Page 58
The cumulative (from top of the canopy to the actual layer) quantum flux of diffuse irradiance, I(d,i), is given
by:
eq. 16
´´
, ,0 ´1 1 exp d
cd d b id i d
d b
kI I k k L
k k,
I(d,0): Diffuse quantum flux per unit ground area above the canopy (mol m-2
s-1
).
k´d: Extinction coefficient of diffuse and scattered PAR radiation (unit less, k´d: = 0,719 (de Pury & Farquhar, 1997)).
cd: Canopy reflection coefficient for diffuse PAR ( cd = 0,036 (de Pury & Farquhar, 1997)).
Li: Cumulative leaf-area index from top of canopy to layer i (unit less, L = 0 for i = 0 (top), L = Lc for i = n (bottom)).
The cumulative (from top of the canopy to the actual layer) quantum flux of irradiance absorbed by sunlit
leaves, I(sun,i), and that absorbed by shaded leaves, I(sh,i), are calculated separately by assuming that diffuse,
scattered diffuse, and scattered beam irradiance reach all leaves.
For the sunlit leaves the total quantum flux in each canopy layer is calculated as the sum of eq. 13, eq. 14,
and eq. 16:
eq. 17
, , , ,sun i b i bs i d iI I I I ,
I(bs,i): Scattered beam quantum flux per unit ground area in the canopy (mol m-2
s-1
).
I(b,i): Direct beam quantum flux per unit ground area in the canopy (mol m-2
s-1
).
I(d,i): Diffuse quantum flux per unit ground area in the canopy (mol m-2
s-1
).
The absorbed irradiances of the sunlit fractions in the canopy layers, at two different leaf area index of the
canopy, Lc, are shown in Figure 3. It is seen that for the sunlit fraction of the leaves the most dominating
type of irradiance which is absorbed is the direct beam fraction even at high leaf area index. However, the
sunlit leaf area fraction decrease through the canopy layers which is not shown in Figure 3. The diffuse and
scattered radiation remains relatively small through the canopy in this example. If the shaded fraction
increases, as during cloudy conditions, then the diffuse and scattered radiation also increases.
0
20
40
60
80
100
0 0.1 0.2 0.3 0.4 0.5
Leaf area index in layer i , L(total,i)
Quantu
m f
lux, I
(m
ole
m-2
s-1
)
0
20
40
60
80
100
0 0.5 1 1.5 2 2.5
Leaf area index in layer i , L(total,i)
I_Sunlit I_beam I_scattered beam I_diffuse
Figure 3. The absorbed quantum flux of the sunlit fractions, I(sun,i), as a function of the cumulative leaf area index, Li ) in
the canopy layers given by eq. 13 - eq. 17 at two different leaf area index of the total canopy (Lc). Left: Lc = 0.5. Right:
Lc = 2.3.
58
Page 59
Shaded leaves
The irradiance absorbed by the shaded leaf area of the canopy is calculated as the difference between the
total irradiance absorbed by the canopy, I(total,i), (eq. 18) and the irradiance absorbed by the sunlit leaf area,
I(sun,i) (eq. 17). The total quantum flux absorbed by the canopy, I(total,i), is given by:
eq. 18
´ ´
, ,0 ,01 1 exp 1 1 expcb b i cd d itotal i b d
I I k L I k L ,
The quantum flux absorbed by the shaded leaf area, I(sh,i) is then given by:
eq. 19
, , ,sh i total i sun iI I I ,
The absorbed quantum flux of the total, sunlit and shaded parts are shown in Figure 4.
Figure 4. The actual quantum flux of absorbed irradiances (total, sunlit and shaded) as a function of the total leaf area
index in each canopy layer, Li, at two different leaf area index of the total canopy (Lc). Top: Lc = 0.5. Bottom: Lc = 2.3.
Photosynthesis and stomata conductance model
A number of mechanistic models of photosynthesis and stomata conductance at the leaf level has been
developed and widely used (e.g. Boegh et al., 2002; Leuning, 1995; Collatz et al., 1991; Sellers et al., 1996)
that derive from the C3 photosynthesis model of Farquhar et al. (1980) and the empirical stomatal
conductance model of Ball et al. (1987). These interacting models are solved by a numerical method, the
Newton Raphsons method, described by Collatz et al. (1991).
The majority of plant uses the C3 pathway including all the temperate cereals (wheat, barley, etc) root crops
(e.g. potato and sugar beet), and leguminous species (beans, etc.). Another pathway, the C4 pathway, is
important in certain agricultural (and natural) systems. The C4 pathway is important for agricultural crops
like corn, sugar cane, and certain grasses for pasture as Sudan grass.
There are two Farquhar-Ball-Collatz models of photosynthesis for C3 and for C4 plants implemented in the
DAISY code, the FC-C3 and the FC-C4 model, respectively.
Stomatal sub-model
Complex physiological mechanisms adjust the opening of stomata in response to changes in environmental
conditions which affect the stomatal conductance of leaves and the canopy. The stomatal response to
environmental and physiological factors is modeled according to the empirical model developed by Ball et
59
Page 60
al. (1987). The model describes stomatal conductance, gs, as linearly related to CO2 assimilation rate, A, and
relative humidity, hs, and CO2 partial pressure, s, at the leaf surface. To improve the description of stomatal
behavior at low CO2 concentrations, s is replaced with s minus the CO2 compensation point, *, according
to Leuning (1995). The stomatal conductance, gs, for the sunlit or shaded fraction, f, and canopy layer, i,
(noted as (f,i)) is given by:
eq. 20
, ,
, ,*
, ,
,
, ,
0
0
totf i s f i
f i f i
s f i f i
s f i
f i f i
A P hwsf m b A
g
b A
,
wsf: Water stress factor (unit less, wsf = 1).
m: Empirical vegetation constant (m = 9 for wheat, m = 11 for soybean unitless).
b(f,i): Stomatal intercept factor, b(f,i) = b L(f,i) (b = 0.01 mol m-2
s-1
).
L(f,i): Cumulative leaf area index (unit less).
hs: The relative humidity at the leaf surface calculated by eq. 21 (unit less). *(f,i): CO2 compensation point of photosynthesis (Pa).
A(f,i): The net photosynthesis rate (mol m-2
s-1
).
s(f,i): The leaf surface partial CO2 pressure (Pa).
Ptot: The atmospheric pressure (100000 Pa).
The stomata conductance for the influx of CO2 and the simultaneous efflux of water are directly linked to
two vegetation-dependent coefficients (m, b). The two vegetation-dependent coefficients, m and b, have
been parameterized by Wang and Leuning (1998) and Ball and Berry (1982) for wheat and soybean,
respectively. The water stress factor, wsf, may be given as a function of ABA.
The stomatal model is merged with diffusion equations for water vapor flux through leaf boundary layer and
stomata. The humidity at the leaf surface, hs, is given, according to Collatz et al. (1991), by solving the
following quadratic (eq. 21) by the second root:
eq. 21
, 2
, , , , , ,* *
_, , ,
0sf tot sf totf i f,i a
s bw s bwf i f i f i f i f i f i
s s l satf i f i f i
w mP A w mP A eh b g h g b
e,
gbw(f,i): Leaf boundary-layer conductance of water (mol m-2
s-1
).
ea: Actual vapor pressure in the air (Pa).
el_sat: Saturated vapor pressure at the leaf surface given by eq. 24 (Pa).
where the actual air vapor pressure, ea, is given by:
eq. 22
sataaa ehe _ ,
ha: The relative humidity of the air (mol mol-1
).
60
Page 61
The saturated vapor pressure, esat, at the leaf surface or in the air, is according to the code of Collatz et al.
(1991):
eq. 23
6790.4985exp 54.8781919 5.02808ln 273.15
273.15sat a
a
e TT
,
Ta: Air temperature (°C).
The CO2 partial pressure in the leaf interior, i, for C3 and C4 plants, are given by Collatz et al. (1991) and
Collatz et al. (1992), respectively:
eq. 24
, ,
, ,
, ,
1.6 1.4b f i s f i
a totf i f i
s f i b f i
g gi P A
g g, for C3
, ,
,
1.6i a totf i f i
s f i
P Ag
, for C4
i(f,i): CO2 partial pressure in leaf interior (Pa).
A(f,i): Net rate of photosynthesis (mol m-2
s-1
).
gs(f,i): Stomatal conductance of leaves given by eq. 20 (mol m-2
s-1
).
gb(f,i): Leaf boundary-layer conductance (mol m-2
s-1
).
a: The air partial CO2 pressure (35 Pa).
Photosynthesis of C3 leaves
Leaf assimilation (or gross photosynthetic) rate of C3 leaves is described as the minimum of two limiting
rates, wc and we, which are functions that describe the assimilation rates as limited by the efficiency of the
photosynthetic Rubisco capacity, wc, the amount of PAR absorbed, we. Thus, the net leaf photosynthetic
rate, A, is given by:
eq. 25
, , , ,min ; c ef i f i f i f i
A w w R ,
A(f,i): Net photosynthesis (mol m-2
s-1
).
R(f,i): Leaf respiration (mol m-2
s-1
) given by eq. 34.
wc(f,i): Rubisco limited rate of assimilation (mol m-2
s-1
).
we(f,i): Light limited rate of assimilation (mol m-2
s-1
).
61
Page 62
The transition from one limitation to another appears to be somewhat gradual in reality, it is more correct to
estimate A by solving the following quadratic (eq. 26) by the first root (Collatz et al., 1991):
eq. 26
2
( , ) ( , ) ( , ) ( , ) ( , ) ( , ) 0f i c f i e f i f i c f i e f iA w w A w w ,
: Empirical curvature constant (0.95).
is an empirical constants that describing the transition between limitations, and are typical close to one
(Collatz et al., 1991). Solving eq. 26, the net photosynthetic rate is given by:
eq. 27
2
( , ) ( , ) ( , ) ( , ) ( , ) ( , )
( , ) ( , )
4
2
c f i e f i c f i e f i e f i c f i
f i f i
w w w w w wA R ,
0102030405060708090
100
0 10 20 30 40 50
Overall photosynthesis [mmol m-2
s-1
]
Lim
ite
d r
ate
s [m
mo
l m
-2 s
-1]
we wc
0102030405060708090
100
0 10 20 30
Overall photosynthesis [mmol m-2
s-1
]
Figure 5. Relation between the light limited rate, we, and the Rubisco limited rate, wc, as a function of the overall
photosynthesis (A+R) with the parameter settings: Ta = 25 °C, i = 25 Pa, and Vmax25-2
s-1
. Left: Itotal,i =-2
s-1
, Right: Itotal,i-2
s-1
.
Figure 5 shows the relation between the light limited rate, we, and the Rubisco limited rate, wc, as a function
of the overall photosynthesis. When the quantum light flux is high, wc limits the photosynthesis (Fig. 5 left).
On the contrary, when the quantum light flux is low, we limits the photosynthesis (Fig. 5 right). The
assimilation rate limited by the efficiency of the photosynthetic Rubisco capacity, wc, is given by:
eq. 28
*
,
, ,
,
i f i
c f i m f i
cli f i
w VK
,
Vm(f,i): Photosynthetic Rubisco capacity (mol m-2
s-1
).
i(f,i): CO2 partial pressure in leaf interior (Pa).*: CO2 compensation point of photosynthesis (3.69 Pa at 25 °C (de Pury and Farquhar, 1997)).
62
Page 63
where the parameter Kcl is given by:
eq. 29
o
cclK
OKK 21 ,
Kc: Michaelis-Menten constant of Rubisco for CO2 (40.4 Pa at 25 °C (de Pury and Farquhar, 1997)).
Ko: Michaelis-Menten constant of Rubisco for O2 (24800 Pa at 25 °C (de Pury and Farquhar, 1997)).
O2: O2 partial pressure in leaf interior (20.5 103
Pa (de Pury and Farquhar, 1997)).
and Vm is the maximum catalytic capacity of Rubisco per unit leaf area (Farquhar et al., 1980).
The light limited rate of photosynthesis, we, is given by:
eq. 30
*
, ,
, , *
, ,2
i f i f i
e f i f i
i f i f i
w J ,
J(f,i): Rate of electron transport (mol m-2
s-1
).
The rate of electron transport, J, depends on the absorbed irradiance, Ile, and an empirical constant, . The
constant describes the non-linear curvature of leaf electron transport responds to irradiance (Farquhar et
al., 1980). The rate of electron transport J is estimated by solving the following quadratic (eq. 31) by the
small root:
eq. 31
2
, , , , , ,0le m le mf i f i f i f i f i f i
J I J J I J ,
Ile(f,i): PAR effectively absorbed (mol m-2
s-1
).
J(f,i): Rate of electron transport (mol m-2
s-1
).
Jm(f,i): Potential rate of electron transport (mol m-2
s-1
).
: Empirical constant, curvature of leaf responds to irradiance (0.7 (de Pury and Farquhar, 1997)).
where the potential rate of electron transport, Jm, is given by:
eq. 32
, ,2.1 m mf i f i
J V ,
The photosynthetic active radiation (PAR) effectively absorbed by the leaf, Ile, is given by Collatz et al.
(1991):
eq. 33
, ,le f i f iI I ,
: Fraction of PAR effectively absorbed (unit less, 0.08 (Collatz et al., 1991)).
I(f,i): Absorbed irradiance given by eq. 17 and eq. 19 (mol m-2
s-1
).
63
Page 64
0
10
20
30
40
50
0 10 20 30 40 50 60
Intercellular CO2 partial pressure [Pa]
Overa
ll p
ho
tosynth
esis
ssss
[m
ol m
-2 s
-1]
0
5
10
15
20
0 500 1000 1500 2000
Absorbed irradiance [ mol m-2
s-1
]
Overa
ll p
ho
tosynth
esis
ss
[m
ol m
-2 s
-1]
Figure 6. The effect of a) the intercellular partial pressure of CO2 and b) the absorbed quantum flux of leaf C3
photosynthesis. The parameter settings: The temperature T = 25 °C, and Vmax-2
s-1
in eq. 42. In a): The
absorbed irradiance (quantum flux density) by the leaf Itotal,0-2
s-1
, in b): The parameters of the
intercellular partial pressure of CO2 i = 25 Pa.
The plot of photosynthesis in Figure 6 is plotted against the intercellular partial pressure of CO2 in the
absence of stomata limitations, and the absorbed quantum flux density.
A central process in cellular metabolism is respiration, the oxidation of sugar to CO2 and water. With
respiration, cells obtain the useful chemical energy, adenosine triphosphate (ATP), from sugar in order to
maintenance life and growth. The leaf respiration rate, R is proportional to the photosynthetic Rubisco
capacity (de Pury & Farquhar, 1997):
eq. 34
, ,0.0089mf i f i
R V ,
R(f,i): Leaf respiration per unit leaf area (mol m-2
s-1
).
Figure 7 compare the Farquhar leaf respiration calculated by eq. 34 with the overall respiration calculated in
Daisy for different crop components. In Daisy the respiration is divided in maintenance and growth
respiration. The overall respiration calculated by eq. 34 is comparable to the total respiration of all the
contribution for growth and maintenance respiration in Daisy. Therefore the net photosynthesis equilibrated
with stomata conductance is calculated by eq. 25. The net photosynthesis is thereafter re-calculated by
adding the respiration given by eq. 34 and subtracting the overall respiration which is the sum of growth and
maintenance respiration for root, storage organ, leaf and stem compartment.
a)
b)
64
Page 65
Maintenance and growth respiration in Daisy
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
(g C
O2 m
-2 h
-1)
RootMaintRespirationSOrgMaintRespirationLeafMaintRespirationStemMaintRespirationRootGrowthRespirationSOrgGrowthRespirationStemGrowthRespirationLeafGrowthRespiration
Farquhar respiration
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
01-0
3-9
8
15-0
3-9
8
29-0
3-9
8
12-0
4-9
8
26-0
4-9
8
10-0
5-9
8
24-0
5-9
8
07-0
6-9
8
21-0
6-9
8
05-0
7-9
8
19-0
7-9
8
02-0
8-9
8
16-0
8-9
8
30-0
8-9
8
(g C
O2 m
-2 h
-1)
Figure 7. Top: The maintenance and growth respiration calculated by different crop components in Daisy. Bottom:
Farquhar “leaf respiration” calculated by eq. 34.
Photosynthesis of C4 plants
The photosynthetic model of C4 plants is based on the model developed by Collatz et al. (1992) and predicts
photosynthesis as a function of temperature, photosynthetic active quantum flux density, CO2 pressure and
relative humidity at the leaf surface. The important adjustable parameters in the C4 model are the capacities
of Rubisco and PEP carboxylase to fix CO2 which can be estimated from leaf photosynthetic responses to
light and CO2. The C4 photosynthesis model link the C3 photosynthesis in the bundle sheath chloroplast with
a carbon pump driven by the activity of PEP carboxylase in the mesophyll leaf cells. Carbon derived from
intercellular CO2 is fixed into C4 acids in the mesophyll, transported to the bundle sheath cells and released
as CO2. Leakage of inorganic carbon from the bundle sheath cells to the intercellular spaces occurs because
there is a large gradient in CO2 concentration created by the pump. The steady state balance of these
transport processes, the net leaf photosynthetic rate, A, is given by:
eq. 35
, , , , ,min c e sf i f i f i f i f i
A w ; w ; w R ,
A(f,i): Net photosynthesis (mol m-2
s-1
).
R(f,i): Leak of CO2 from the bundle sheath (mol m-2
s-1
).
65
Page 66
wc(f,i): CO2 limited rate of assimilation (mol m-2
s-1
).
we(f,i): Light limited rate of assimilation (mol m-2
s-1
).
ws(f,i): Rubisco limited rate of assimilation (mol m-2
s-1
).
At rate limiting light intensities, the efficiency of CO2 fixation with respect to absorbed light (quantum
yield) determines the rate of photosynthesis. The light dependent rate is given by:
eq. 36
, ,e absf i f iw a I ,
aabs: Leaf absorbtivity to PAR (aabs = 0.86 (Collatz et al., 1992)).-1
(Collatz et al., 1992)).
I(f,i): Absorbed irradiance given by eq. 17 and eq. 19 (mol m-2
s-1
).
At low CO2 concentrations, empirical studies show that net photosynthesis, A, increases linearly from the
compensation point (near zero Pa) to rate saturation which occur at an intercellular CO2 partial pressure of
about 10 Pa. Thus, the CO2 limited flux given by Collatz et al. (1992):
eq. 37
,
,
i f i
c Tf i
tot
w kP
,
kT: Initial slope of photosynthetic CO2 response (0.6 mol m-2
s-1
(Collatz et al., 1992)).
(f,i): CO2 partial pressure in leaf interior (Pa).
Ptot: The atmospheric pressure (100000 Pa).
Empirical observations show when wc and we are not limiting, then the rate of assimilation approaches a rate,
ws, that is largely independent of CO2 and light. The rate under these conditions is controlled by the capacity
for CO2 fixation by Rubisco:
eq. 38
, ,s mf i f iw V ,
Vm(f,i): Photosynthetic capacity per unit leaf area given by eq. 52 (mol m-2
s-1
).
The transition from one limitation to another appears to be somewhat gradual and therefore the
photosynthesis is estimated by solving the following quadratics (eq. 39 and eq. 40) by the first root:
eq. 39
2
, , , , , ,0s e s ef i f i f i f i f i f i
M w w M w w ,
M(f,i): The flux determined by Rubisco and light (mol m-2
s-1
).
: Curvature parameter (0.83 (Collatz et al., 1992)).
where the curvature parameter, , gives a gradual transition between the light limited and Rubisco limited
flux. The limitation on the overall rate M and the CO2 limited flux, wc, the likewise expressed as a quadratic:
eq. 40
2
, , , , ,0c cf i f i f i f i f i
A A M w M w ,
A(f,i): The flux determined by M and CO2 (mol m-2
s-1
).
: Curvature parameter (0.93 (Collatz et al., 1992)).
66
Page 67
where the curvature parameter, , gives a gradual transition between M and the CO2 limited flux.
Photosynthetic capacity
Rubisco is the most abundant protein in leaves of C3 plants, constitution up to half the total leaf protein. For
this reason it plays a crucial role in the nitrogen economy of plants. To estimate photosynthetic active
nitrogen (N) the non-functional and critical limits of N is estimate. The non-functional N is considered
structural N, and not used in photosynthesis. The N content above critical is considered luxury N uptake,
and also not used in photosynthesis. Hence, the photosynthetic active Rubisco nitrogen, Np, is given by:
eq. 41
, 0
,
0, 0
a n a n c n
p c n a n c n
a n
N N N N N N
N N N N N N N
N N
,
Np: Photosynthetic active Rubisco associated nitrogen (mol m-2
).
Na: Actual leaf nitrogen (mol m-2
).
Nc: Critical (luxury) leaf nitrogen (mol m-2
).
Nn,: Non-functional (structural) leaf nitrogen (mol m-2
).
where the actual leaf nitrogen content in the canopy, Na, is given by the crop production component in
DAISY. The critical, Nc, and non-functional, Nn, limits are given by the CropN component in DAISY.
The Rubisco N distribution with depth in the canopy layer Np(f,i) can be defined by different functions. An
example could be the exponential distribution given by Boegh et al. (2002) as described in section 0.
The maximum leaf Rubisco capacity in each layer, Vmaxi, at 25 °C is given by:
eq. 42
max 2 5( , ) ( , )f i n p f iV N ,
Vmax25,i: The maximum leaf Rubisco capacity at 25 °C (mol m-2
s-1
).
n: Ratio of measured Rubisco capacity to leaf nitrogen (0.116 mol mol-1
s-1
for wheat (Boegh et al., 2002)).
The canopy photosynthetic capacity in the canopy is described as a function of the Rubisco N distribution:
eq. 43
( , ) max ( , ) ( , )m f i f i p f iV V N ,
Vmax(f,i) The maximum leaf Rubisco capacity at leaf temperature given by eq. 52 (mol m-2
s-1
).
The partitioning of leaves into sunlit and shaded fractions is continually changing throughout the day. The
calculation of the photosynthetic capacity is affected by these separate fractions.
Sunlit leaves
The photosynthetic capacity of the sunlit leaf fraction, VmSun,i, of each canopy layers is given by:
eq. 44
67
Page 68
,
max( , ), ,
,
1 exp n b sun i
sun im sun i sun i
n b sun i
k k LV L V
k k L,
kb: Extinction coefficient of beam radiation (unit less, given by eq. 10).
Vm(sun,i): The canopy photosynthetic capacity in sunlit leaves layer i (mol m-2
s-1
).
Vmax(sun,i): The maximum leaf Rubisco capacity in sunlit leaves at temperature Ta given by eq. 52 (mol m-2
s-1
).
Shaded leaves
Photosynthetic capacity of the shaded leaf fraction, VmSh, of each canopy layers is given by:
eq. 45
, , ,m sh i m total i m sun iV V V ,
Vm(sun,i): The Rubisco photosynthetic capacity to leaf nitrogen in the sunlit leaf fraction (mol m-2
s-1
).
Vm(sh,i): The Rubisco photosynthetic capacity to leaf nitrogen in the shaded leaf fraction (mol m-2
s-1
).
Figure 8 show the different fractions of the photosynthetic capacity in the canopy together with the nitrogen
distribution.
0
50
100
150
200
250
300
0 0.1 0.2 0.3 0.4 0.5
Cummulative leaf area index in layer i , L i
Ca
no
py p
ho
tosynth
etic_
cap
acity (
mm
ol m
-2 s
-1)
0
50
100
150
200
250
nitro
ge
n d
istr
ibu
tio
n
(mm
ol m
-2)
0
50
100
150
200
250
300
0 0.5 1 1.5 2
Cummulative leaf area index in layer i , L i
Ca
no
py p
ho
tosynth
etic_
cap
acity (
mm
ol m
-2 s
-1)
0
50
100
150
200
250
nitro
ge
n d
istr
ibu
tio
n
(mm
ol m
-2)
Vmsun Vmsh Np
Figure 8. The canopy photosynthetic capacity distributions (left axis) and the nitrogen distribution of Rubisco N (right
axis) in the canopy layers. Lc = 2.3 and NC = 64.3 mg g-1
.
68
Page 69
Canopy nitrogen distribution sub-model (N-dist)The maximum leaf Rubisco capacity in each layer is defined as function of photosynthetic active nitrogen.
The distribution of photosynthetic active nitrogen in the canopy can be described by different functions. An
example could be the Ndist exp model which is an exponential distribution according to Boegh et al.,
(2002):
eq. 46
, , , ,0 ,exp np f i p f f i
N N k L ,
kn: Coefficient of leaf nitrogen allocation in a canopy (0,713 (Boegh et al., 2002)).
L(f,i): Cumulative leaf-area index.
Np(f,i): Photosynthetic active Rubisco associated nitrogen distribution (mol m-2
).
N p,(f,0): Photosynthetic active Rubisco associated nitrogen in the top of the canopy (mol m-2
).
where the photosynthetic active nitrogen (Rubisco N) in the top of the canopy, Np,(f,0), is given by:
eq. 47
, ,0
,1 exp
n p
p f
n f i
k NN
k L,
Np: Photosynthetic active Rubisco associated nitrogen (mol m-2
) given by eq. 41.
Figure 9 show the distribution of photosynthetic active nitrogen (Rubisco N) in the canopy where the Np,C =
248 mmol m-2
and Np,0 = 221 mmol m-2
.
0
50
100
150
200
250
0 0.1 0.2 0.3 0.4 0.5
Cummulative leaf area index in layer i , L i
nitro
ge
n d
istr
ibu
tio
n
(mm
ol m
-2)
0
50
100
150
200
250
0 0.5 1 1.5 2
Cummulative leaf area index in layer i , L i
Np Ns
Figure 9. The nitrogen distribution of Rubisco N (left axis) as a function of the cumulative leaf area index in the canopy
layers (Li) with Np,c = 248 mmol m-2
. Left: Lc = 0.5. Right: Lc = 2.3.
Temperature dependencies
Several of the photosynthetic- and stomatal-parameters depend on the temperature of the leaf. For C3 plants
this includes the parameters Kc, Ko, m, and Jm. For C4 plants it concerns the parameters kT and Vm.
kT
The pseudo-first order rate constant with respect to CO2, kT, is given by (Collatz et al., 1992):
eq. 4825
10
10
Ta
kTk kQ , for C4
Ta: Air temperature (°C).
69
Page 70
kT: The pseudo-first order rate constant with respect to CO2 (mol m-2
s-1
).
k: Rate constant (0.6 mol m-2
s-1
).
Q10k: The Q10 parameter of k (1.8).
Kc, Ko, and *
For the C3 photosynthesis model, the parameters Kc, Ko, and are adjusted for the effect of temperature by
the Arrhenius function (de Pyry and Farquhar, 1997). For the C4 photosynthesis model, only the is used
in the model and adjusted by the Arrhenius function:
eq. 49
,
, 25,
( 25)exp(
298 273
a x a
T x x
a
E Tk k
R T, for C3 and C4
kT,x: Parameter x at T °C.
k25,x: Parameter x at 25 °C.
Ea,x: Activation energy for parameter x (x = Kc, Ko,*).
Ta: Air temperature (°C).
Activation energies of the model parameters adjusted for temperature dependencies by eq. 49 and values at
25 °C are listed in Table 1 and defined as default values in the Daisy model.
Table 1 Activation energies and values at 25 °C of the model parameters adjusted for temperature dependencies by eq.
49 listed in de Pury & Farquhar (1997).
Parameter Ea k25
(J mol-1
) (Pa)*
29000 3.69
Ko 36000 24800
Kc 59400 40.4
Jm
The parameter Jm, for calculation of the electron-transport limited rate of photosynthesis, is also adjusted by
temperature according to de Pury & Farquhar (1997). However, for temperatures below 10 C° the
temperature function is reduced with a linear function. Below 4 C° Jm = 0:
eq. 50
25
,
2981 exp
273,15 298 298exp
273,15 298 273,151 exp
273,15
a a Jm
m m
a a
a
S H
T E Rj J
R T S T H
R T
70
Page 71
eq. 51
, 10
4, 4 10
6
0, 4
T
m a
am m a
a
j T C
TJ j T C
T C
, for C3
Ta: Air temperature (°C).
Jm25: Jm at 25 °C is given by 2.1 Vmax25(f,i) (de Pury and Farquhar, 1997).
R: Universal gas constant (8.314 J mol-1
K-1
).
H: Curvature parameter of Jm (220000 J mol-1
(de Pury and Farquhar, 1997)).
S: Electron transport temperatures response parameter (710 J mol-1
K-1
(de Pury and Farquhar, 1997)).
Ea,Jm: Activation energy for Jm (37000 J mol-1
(de Pury and Farquhar, 1997)).
Vm
In the C3 photosynthesis model, the maximum photosynthetic Rubisco capacity is adjusted for the
temperature dependency by a function defined by Harley et al. (1992) and partly by Bernacchi et al. (2001):
eq. 52
, max
max
, max 25 ,
, max
exp273,15
273,151 exp
273,15
a V
V
a
m f i f i
a d a V
a
EC
R TV V
Sv T E
R T
, for C3
Vmax25(f,i): The maximum leaf Rubisco capacity given by eq. 42 (mol m-2
s-1
).
Ea,Vmax: Activation energy for Vmax (65330 J mol-1
(Bernacchi et al., 2001)).
Eda,Vmax: Deactivation energy for Vmax (202900 J mol-1
(Harley et al., 1992)).
C,Vmax: Temperature scaling constant for Vmax (26,350, (Bernacchi et al., 2001)).
Sv: Entropy term for Vmax (650 J mol-1
K-1
(Harley et al., 1992)).
In the C4 photosynthesis model, the effect of temperature on the photosynthetic Rubisco capacity is given by
Collatz et al. (1992):
eq. 5325
10
10max 25
,1 exp 0 3 40 1 exp 0 2425 15
Ta
Vm
m f i
a a
V QV
+ . T - + . -T, for C4
Ta: Air temperature (°C).
Vmax25(f,i): The maximum leaf Rubisco capacity given by eq. 42 (mol m-2
s-1
).
Q10Vm: The Q10 parameter of Vm (2.4).
The effect of temperature on the leaf C3 parameters are show in Figure 10 together with the effect of
temperature on the overall photosynthesis.
71
Page 72
Figure 10. The effect of temperature on leaf C3 photosynthesis parameters: a) Jm and Vm, b) Kc, Ko, Kcl given by eq. 29,
and *, c) the overall photosynthesis and respiration calculated by Farquhar. The intercellular partial pressure of CO2 is -2
s-1
and Vmax-2
s-1
in eq.
42.
Calculation procedure of the photosynthesis-conductance model
There is a strong interaction between the photosynthesis and the stomatal sub-models in this system. The
Daisy code is constructed to obtain a numerical solution, using an initial guess for the stomatal conductance
(gs = (Li-1 -Li)/5 mol s-1
m-2
), and the CO2 partial pressure in the leaf interior ( i = 0.5 a = 17.5 Pa), the code
calculates the leaf temperature, photosynthesis, gs and i by iterations using the Newton-Raphson method
until i is stable. Newton-Raphson Method states that if x = r is an approximation to f(x) = 0, then a better
solution is given by:
eq. 54
rf
rfrr new
,
72
Page 73
When estimating i, the function f(x) is given by:
eq. 55
, ,1i if i f i
f x t t ,
i(f,i): CO2 partial pressure in leaf interior in the sunlit or shaded fraction, f, in canopy layer i (Pa).
t: Time step (hour).
where i for C3 plants is calculated using eq. 24. The derivates of f follows:
eq. 56
, ,
1 1 1.6 1.4 1tot
s f i b f i
f x P dxg t g t
,
gs(f,i): Stomatal conductance of leaves (mol m-2
s-1
).
gb(f,i): Leaf boundary-layer conductance (mol m-2
s-1
).
where dx for C3 plants is given by:
eq. 57
, 2
,
*clm f i
i clf i
Kdx V
K
, if wc < we
, 2
,
*3
2 *m f i
i f i
dx J , if wc we
Vm(f,i): Photosynthetic capacity per unit leaf area in the sunlit or shaded fraction, f, in canopy layer i (mol m-2
s-1
).
i(f,i): CO2 partial pressure in leaf interior in the sunlit or shaded fraction, f, in canopy layer i (Pa).
Kcl: The effective Michaelis-Menten coefficient CO2 (Pa).*: CO2 compensation point of photosynthesis (Pa).
Then the Newthon-Rapson solution to i(f,i)(t) is then given by:
eq. 58
, ,
, ,
, ,
11
1 1 1.6 1.4 1
i if i f i
i if i f i
tot
s f i b f i
t tt t
P dx tg t g t
, for C3
Which is used to calculate the partial pressure of CO2 in stomata. For C4 plants, the same consideration can
be done giving:
eq. 59
, ,
, ,
,
11
1 1.6 1
i if i f i
i if i f i
tot
s f i
t tt t
P dx tg t
, for C4
73
Page 74
where the dx for C4 plants is given by:
eq. 60
ififif
ififT
wcMA
MAktdx
,,,
,,
2, for C4
kT: Initial slope of photosynthetic CO2 response (k = 0.6 mol m-2
s-1
(Collatz et al., 1992)).
References
Ball,J.T. and Berry,J.A., 1982. The Ci/Cs ratio: A basis for predicitong stomatal control of photosynthesis.
In: pp. 88-92.
Ball,J.T., Woodrow,I.E. and Berry,J.A., 1987. A model predicitng stomatal conductance and its contribution
to the control of photosynthesis under different environmental conditions. In: I.Biggins (Editor),
Progress in Photosynthesisi Research. Martinus Nijhoff Publishers, Netherlands, pp. 221-224.
Bernacchi,C.J., Singsaas,E.L., Pimentel,C., Portis,A.R. and Long,S.P., 2001. Improved temperature
response functions for models of Rubisco-limited photosynthesis. Plant Cell and Environment,
24(2): 253-259.
Boegh,E. and Soegaard,H., 2004. Remote sensing based estimation of evapotranspiration rates. International
Journal of Remote Sensing, 25(13): 2535-2551.
Boegh,E., Soegaard,H., Broge,N., Hasager,C.B., Jensen,N.O., Schelde,K. and Thomsen,A., 2002. Airborne
multispectral data for quantifying leaf area index, nitrogen concentration, and photosynthetic
efficiency in agriculture. Remote Sensing of Environment, 81(2-3): 179-193.
Collatz,G.J., Ball,J.T., Grivet,C. and Berry,J.A., 1991. Physiological and Environmental-Regulation of
Stomatal Conductance, Photosynthesis and Transpiration - A Model That Includes A Laminar
Boundary-Layer. Agricultural and Forest Meteorology, 54(2-4): 107-136.
Collatz,G.J., Ribas-Carbo,M. and Berry,J.A., 1992. Coupled Photosynthesis-Stomatal Conductance Model
for Leaves of C4 Plants. Australian Journal of Plant Physiology, 19(5): 519-538.
de Pury,D.G.G. and Farquhar,G.D., 1997. Simple scaling of photosynthesis from leaves to canopies without
the errors of big-leaf models. Plant Cell and Environment, 20(5): 537-557.
Farquhar,G.D., Caemmerer,S.V. and Berry,J.A., 1980. A Biochemical-Model of Photosynthetic Co2
Assimilation in Leaves of C-3 Species. Planta, 149(1): 78-90.
Harley,P.C., Thomas,R.B., Reynolds,J.F. and Strain,B.R., 1992. Modeling Photosynthesis of Cotton Grown
in Elevated Co2. Plant Cell and Environment, 15(3): 271-282.
Leuning,R., 1995. A Critical-Appraisal of A Combined Stomatal-Photosynthesis Model for C-3 Plants. Plant
Cell and Environment, 18(4): 339-355.
McCree,K.J., 1981. Photosynthetically active radiation. In: O.L.Lange, P.S.Nobel, C.B.Osmond, and
H.Zeigler (Editors), Physiological Plant Ecology. Vol 12A, Encycclopedia of plant physiology.
Springer-Verlag, Berlin, pp. 41-55.
Nobel,P.S., 1991. Physicochemical and environmental plant physiology. Academic Press, San Diego.
Sellers,P.J., Randall,D.A., Collatz,G.J., Berry,J.A., Field,C.B., Dazlich,D.A., Zhang,C., Collelo,G.D. and
Bounoua,L., 1996. A revised land surface parameterization (SiB2) for atmospheric GCMs .1. Model
formulation. Journal of Climate, 9(4): 676-705.
Wang,Y.P. and Leuning,R., 1998. A two-leaf model for canopy conductance, photosynthesis and
partitioning of available energy I: Model description and comparison with a multi-layered model.
Agricultural and Forest Meteorology, 91(1-2): 89-111.
74
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Annex 3.4 Estimating root density in Daisy
75
Page 76
!"#$%"#&' "() *++" ,)&!#"- .*+$ *++" ,*- $%"")*%&, !#/) !" #$"%&%'(!)∗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z;$% " %$< "! 9' &'7 %)9'& 98Lz = L0 e−az 015(+'%' L0 )7 *+' %$$* &'!7)*8 "* *+' 7$)= 7:%;" '6 a )7 " &)7*%)9:*)$! <"%">'*'%6"!& z )7 *+' &'<*+ 9'=$( 7$)= 7:%;" '?@' +'%' "77:>' *+"* *+' &'!7)*8 )7 :!);$%>=8 &)7*%)9:*'& $! *+' +$%)-$!*"=<="!'6 "! "77:><*)$! *+"* ;")=7 ()*+ '?/? %$( %$<7?A+' <"%">'*'%7 a "!& L0 ()== 9$*+ B"%8 ()*+ *)>'? C$% " <%$&: *)$! $%)'!*'&7)>:="*)$! >$&'= =)D' E")78 0F"!7'! '* "=?6 1221G H9%"+">7'! "!& F"!7'!6 IJJJ56)* "! 9' >$%' $!B'!)'!* *$ 7<' );8 *+' &'!7)*8 )! *'%>7 $; " :>:="*'& %$$*&%8 >"**'% Mr "!& *$*"= %$$* &'<*+ dc6 "7 &'7 %)9'& )! F"!7'! '* "=? 0122J5 $%*+' ;$==$()!/?@' &'K!' *+' %$$* &'<*+ "* *+' =$('7* &'<*+ (+'%' *+' %$$* &'!7)*8 )7 "*"9$B' 7<' )K'& *+%'7+$=& Lm? L8 )!7'%*)!/ *+)7 )! 0156 (' /'*
Lm = Ldc= L0 e−adc 0I5@' $!B'%* *+' %$$* >"77 *$ %$$* ='!/*+ lr 98 "77:>)!/ *+' 7<' )K %$$*='!/*+ Sr )7 " D!$(! $!7*"!* 0%"*+'% *+"! B"%8)!/ ()*+ &'<*+5
lr = Sr Mr 0M5∗ !""#$%!&'(&) *+,-!". /*01 23454555563 78*(91 '*($:'(&*.<=9.'<1
76
Page 77
!" #$#%& '$$# &"()#! *+ %&+$ #!" *(#")'%& $, #!" '$$# -"(+*#. $/"' #!" 0'$1&"lr =
∫
∞
0Lz dz =
∫
∞
0L0 e−az dz =
L0
a2345. *(+"'#*() #!" "60'"++*$( 7" )"# ,$' L0 ,'$8 234 *( 294 7" )"#
Lm = lr a e−adc 2:4;, 7" +<=+#*#<#" W = −adc %(- *+$&%#" #!" >($7( /%&<"+ $( #!" '*)!# +*-"#!*+ )*/"+ <+WeW = −Lm
dc
lr2?4 !" +$&<#*$( #$ #!*+ "@<%#*$( 7*#! '")%'- #$ W !%00"(+ #$ =" #!" -"1(*#*$( $,#!" A%8="'#BC ,<( #*$( 2E<&"'F GHIJK A%8="'#F GH:I4L !" ,<( #*$( $( #!" &",#!%(- +*-" $, #!" "@<%#*$( *+ -"0* #"- $( 1)<'" GL
BML:MML:GGL:9
9L:J
B: B3 BJ B9 BG M GN*)<'" GO WeWP*( " 7" ($7 >($7 #!" /%&<" ,$' W8 7" %( 1(- #!" -"+*'"- -"(+*#. 0%B'%8"#"'+ L0 %(- a =. +<=+#*#<#*() =% >a = −W/dc 2H4
L0 =Lm
e−adc= Lmeadc 2I4 ! "#$%&' )*+#,'*- ,* WC" +#%'# =. -*/*-*() #!" ,<(#*$(+ *(#$ 8$($#$(* *(#"'/%&+ =. 1(-*() #!" -"'*/%B#*/"
dW eW
dW= eW + WeW 2Q49
77
Page 78
!" "#$%&'()eW + WeW = 0 *+,-!%. ()" .(/$&'()0 W = −11 !" "234"..'() W eW '. 5" 4"%.')7 8"/(9 −1 %)5') 4"%.')7 %8(:" −11 !$.0 W = 0 '. % 7/(8%/ ;')';$;1<') " limQ→−∞ W eW = 0 9" 7"& % .')7/" .(/$&'() 9!") −Lm
dc
lr'. "2% &/=%& &!" 8(&&(; 3(')& *−1e−1-0 &9( 9!") '& '. %8(:" *'& '. )":"4 3(.'&':"-0 %)5)()" 9!") '& '. 8"/(91 !" /%&"4 .'&$%&'() (44".3()5. &( &!" %." 9!"4" &!"4"%4" ').$> ")& 4((& lr &( .%&'.?= &!" ;')';%/ 4((& 5").'&= Lm 9'&!') &!" 7':")4((& @()" dc1A(&! .(/$&'(). %4" :%/'50 8$& 4"34".")& 5'B"4")& 5'.&4'8$&'().1
• !" .(/$&'() ?(4 W < −1 4"34".")&. % /%47" a 3%4%;"&"41 C4(; *D- 9"."" &!'. %/.( ;"%). L0 '. /%47"1 !$.0 &!" .(/$&'() (44".3()5. &( % 4((&@()" 9'&! % !'7! 5").'&= )"%4 &!" &(3 &!%& 5" 4"%.". 4%3'5/= &( Lm %&&!" 8(&&(; (? &!" 4((& @()"0 %)5 ()&')$". &( 5" 4"%." .( ()/= % .;%// ()&4'8$&'() &( &!" &(&%/ 4((& /")7&! ?4(; 8"/(9 !" 4((& @()"1• !" .(/$&'() ?(4 W > −1 *%)5 &!$. .;%// :%/$". (? a %)5 L0- (44".3()5.&( % /(9 4((& 5").'&= )"%4 &!" &(3 &!%& 5" 4"%.". ./(9/=0 %)5 &!$. 7':". %/%47"4 ()&4'8$&'() &( &!" &(&%/ 4((& /")7&! ?4(; 8"/(9 &!" 4((& @()"1E. &!" &(&%/ 4((& /")7&! ') 4"%.".0 34"..')7 W &(9%45. 0 (4 −∞0 &!" 5'B"4") "8"&9"") &!" .(/$&'(). 74(91 F!") &!"4" '. G$.& ")($7! 4((&. &( .%&'.?= &!" ()&4%')&. %& W = −10 &!" &9( .(/$&'(). ():"47". &( ()"1 E. 9" /'H" ($4 4((&.&( .&%= ;(.&/= 9'&!') &!" 4((& @()"0 9" !((." &!" .(/$&'() ?(4 W < −11 F" %) &!$. I)5 W )$;"4' %//= $.')7 J"9&()K. ;"&!(5 %)5 %) ')'&'%/ 7$".. (? −21 !" #$%$&'( )*+,&- !" 5'.&4'8$&'() ') *+- ';3/'". % 74%5$%/ 5" 4"%." (? 4((&. 7(')7 &(9%45.0 8$&)":"4 4"% !')7 @"4(1 !"4" %4" &9( 34(8/";. 9'&! &!'.1 !" I4.& ()" '. ";L34' %/0 ?(4 .(;" .('/. '& 5(".)K& ;%& ! 9!%& 9" (8."4:"0 4%&!"4 &!%) % 74%5$%/5" 4"%."0 &!"4" '. .!%43 5" 4"%." %& % .3" 'I 5"3&!0 %. &!" 4((&. %4" $)%8/"&( 3")"&4%&" ?$4&!"4 5(9) 1 !" (&!"4 ()" '. 34% &' %/0 &(( /%47" % 4((& @()";%H". (;3$&%&'() ';34% &' %/1 !" 9%= 9" ;(5"/ &!" I4.& '..$" '. &( 5':'5" &!" 4((& 5"3&! ')&( % 4(3.3" 'I %)5 .('/ ')5"3")5")& 3(&")&'%/ 4((& 5"3&! dc0 %)5 .('/ .3" 'I %)5 4(3')5"3")5")& ;%2';$; 4((& 5"3&! ds1 !" % &$%/ 4((& 5"3&! da '. &!") &!".!%//(9".& (? &!"." &9(1
da = min(dc, ds) *++-F" )(9 4"%&" % ;(5'I"5 4((& 5").'&= ?$) &'() L∗
z 8= 5"I)')7 '& &( @"4(8"/(9 da0 %)5 % Lz . %/"5 &( 34"."4:" ;%.. 8%/%) " %8(:"1L∗
z =
k∗Lz '? z ≤ da
0 '? z > da
*+M-9!"4"k∗ =
lr∫ da
0 Lz dz*+N-N
78
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!"# #$%&'() *$ ! +,$*%-.#/ !"# %"&'0- 2( 3-# ,'*- 2 ,$4 ,$+ 4' ! 2 4$ 3'.-(#'$(2% .$3-% *5 2##".'() !2 !- +%2( # 2,- 3-(#'%5 +2 6-3 '( !- ,$4/ 7", #- $(3 3'.-(#'$( x '# !$,'8$( 2%9$, $)$(2% $ !- ,$4/ :!- ,$$ 3-(#' 5 2 2 #+- '; +$'( 2( *- 3-($ -3 Lz,x92(3 4- !$$ - $,')$ #$ L0,0 '# !- !- ,$$ 3-(#' 5 '( !- $+ $< !- ,$4/ =--;)",- >/:7?7@ A(#-, ;)",- !-,-/ B')",- >@ C,$+ ,$4/0- !-( 3-;(- !- <$%%$4'() ,$$ 3# ,'*" '$(Lz,x = L0,0 e−azze−axx DEFG4!-,- az 2(3 az $( ,$% !- 3-(#' 5 3- ,-2#- '( !- 4$ 3'.-(#'$(#/ !" #$%&$%' ()* +,-,.*(*-/:$ ;(3 !- +2,2.- -,# az9 ax 2(3 L0,09 4- 2##".- 2# *-<$,- !2 !- ,$$ 3-+ !2(3 ,$$ .2## '# 6($4(9 2(3 ($4 233' '$(2%%5 !2 !- ,$$ ,23'"# wc '# 6($4(/0- 3-;(- !- ,$$ 8$(- 3-+ ! dc $ *- !- 3-+ ! ,')! *-%$4 !- ,$4 Dx = 0G4!-,- !- ,$$ 3-(#' 5 '# Lm/ H# x = 0 '# !- +%2 - 4!-,- DEFG +,-3' # !-!')!-# 3-(#' 59 !- 2&-,2)- ,$$ 3-(#' 5 2 !2 3-+ ! 4'%% *- 4-%% *-%$4 Lm/='.'%2,'%59 4- 3-;(- !- ,23'"# wc 2# !- !$,'8$( 2% 3'# 2( - <,$. !- ,$4 4!-,- !- ,$$ 3-(#' 5 2 !- #",<2 - Dz = 0G
Lm = Ld,0 = L0,r DEIG:!- $ 2% ,$$ %-() ! $( $(- #'3- $< !- ,$4 DlRG9 4!' ! 4- 2##".- '# 6($4(<,$. $", ,$+ .$3-%9 '# !- '( -),2% $< !- ,$$ 3-(#' 5 $&-, !- !2%< +%2(-lR =
∫
∞
0
∫
∞
0 Lz,x dz dx=
∫
∞
0
∫
∞
0 L0,0 e−azze−axx dz dx
=L0,0
azax
DEJG:!"# DEFG 2( *- ,-4,' -(Lz,x = lR az ax e−azze−axx DEKGL5 "#'() DEKG '( DEIG 4- )-
Lm = lR az ax e−azd DEMGLm = lR az ax e−axr DENG:!"# e−azd = e−axr $,
ax =dc
wc
az D>OGF79
Page 80
! "#$%&'"#( )*+, "# )-., /% (%'Lm = lR az
d
raz e−azd )*-,01 /% $23$'"'2'%
Q = −azd )**,4#5 "$674'% '8% 9#6/# :472%$ 6# '8% &"(8' $"5%; '8"$ (":%$ 2$<Q2 eQ = Lm
d r
lR)*=,>8% 7%1' 84#5 $"5% %?@&%$$"6# "$ "772$'&4'%5 "# A(2&% =B C#7"9% )D,; #6365!36'8%&%5 '6 (":% '8% $672'"6# '6 )*=, 4 #4E%B
++BF--BF**BF=
G-+ GH GD GI G* +J"(2&% =< Q2eQ ! "#$%&' )*+#,'*- ,* QK% $'4&' 3! 5":"5"#( '8% 12#'"6#$ "#'6 E6#6'6#" "#'%&:47$ 3! A#5"#( '8% 5%&":4G'":%d(Q2 eQ)
dQ= 2QeQ + Q2eQ )*I,>8% %M24'"6#
2QeQ + Q2eQ = 0 )*F,84$ '/6 $672'"6#$; Q = 0 4#5 Q = −2; 4#5 '8% %?@&%$$"6# Q2 eQ "$ "# &%4$"#(3%76/ −2; 5% &%4$"#( 3%'/%%# −2 4#5 0; 4#5 "# &%4$"#( 436:% 0B >82$; Q = 0 "$4 76 47 )4#5 "# '8"$ 4$% 47$6 (76347, E"#"E2E; 4#5 Q = −2 "$ 4 76 47 E4?"E2EBK% 4&% #6' "#'%&%$'%5 "# @6$"'":% :472%$ 16& Q; '8%! 6&&%$@6#5 '6 #%(4'":%:472%$ 16& az; 4#5 '8% $"E@7"A 4'"6# "# )-D, 4&% 6#7! :47"5 "1 az > 0BF80
Page 81
!" $ limQ→−∞ Q2 eQ = 0 %$ &$' ( )!"&*$ "$&('!+$ ),*-'!," %.$" Lmd rlR
!)$/( '*0 (' '.$ ',1 1,!"' 222e−234 '%, %.$" !' !) )5(**$6 2!' !) "$+$6 "$&('!+$34("7 ","$ %.$" !' !) *(6&$68 9.$ *('$6 )!'-('!," ,66$)1,"7) ', '.$ ()$ %.$6$'.$6$ (6$ !")-: $"' 6,,' lR ', )('!);0 '.$ 5!"!5(* 6,,' 7$")!'0 Lm %!'.!" '.$&!+$" 6,,' <,"$ d r8=,'. "$&('!+$ ),*-'!,") (6$ +(*!74 >-' 6$16$)$"' 7!?$6$"' 7!)'6!>-'!,")8• 9.$ ),*-'!," ;,6 Q < −2 6$16$)$"') ( *(6&$ az 2("7 '.-) (*), ax3 1(6(5@$'$68 A6,5 2BC3 %$ )$$ '.!) (*), 5$(") L0,0 !) *(6&$8 9.-)4 '.$ ),*-'!," ,66$)1,"7) ', ( 6,,' <,"$ %!'. ( .!&. 7$")!'0 "$(6 '.$ $"'$6 '.(' 7$@ 6$()$) 6(1!7*0 ', Lm "$(6 '.$ $7&$ ,; '.$ 6,,' <,"$4 ("7 ,"'!"-$) ',7$ 6$()$ ), ,"*0 ( )5(** ,"'6!>-'!," ', '.$ ','(* 6,,' *$"&'. ;6, ,-')!7$'.$ 6,,' <,"$8• 9.$ ),*-'!," ;,6 Q > −2 2("7 '.-) )5(** +(*-$) ,; ax4 ax ("7 L0,03 ,66$)1,"7) ', ( *,% 6,,' 7$")!'0 "$(6 '.$ $"'$6 '.(' 7$ 6$()$) )*,%*04("7 '.-) &!+$) ( *(6&$6 ,"'6!>-'!," ', '.$ ','(* 6,,' *$"&'. ;6,5 ,-')!7$'.$ 6,,' <,"$8D) '.$ ','(* 6,,' *$"&'. !" 6$()$)4 16$))!"& Q ',%(67) 0 ,6 −∞4 '.$ 7!?$6$" $>$'%$$" '.$ ),*-'!,") &6,%8 E.$" '.$6$ !) F-)' $",-&. 6,,') ', )('!);0 '.$ ,"'6(!"') (' Q = −24 '.$ '%, ),*-'!,") ,"+$6&$) ', ,"$8 D) %$ *!G$ ,-6 6,,')', )'(0 5,)'*0 %!'.!" '.$ 6,,' <,"$4 %$ .,,)$ '.$ ),*-'!," ;,6 Q < −28E$ (" H"7 Q "-5$6! (**0 -)!"& I$%',"J) 5$'.,7 ("7 (" !"!'!(* &-$)) ,;
−38 A6,5 '.(' %$ (" H"7 az ;6,5 2KK34 ax ;6,5 2KL34 ("7 L0,0 ;6,5 2BC38 !" #$%&'(%) *+,-M; '.$ 6,%) (6$ *,)$ $",-&.4 '.$ 6,,' )0)'$5) %!** ,+$6*(1 () ).,%" ," H&-6$ N8M; R !) '.$ 7!)'(" $ >$'%$$" 6,%)4 ("7 %$ ())-5$ (" !"H"!'$ "-5>$6 ,;!7$"'! (* 6,%)4 '.!) (" >$ $/16$))$7 >0 '.$ $O-('!,"L∗
z,x =
∑
∞
i=0(Lz,x+iR + Lz,R+iR−x) !; x < R/2
0 !; x ≥ R/22KC3P)!"& 2BN3 ("7 '.$ 6-*$) ;,6 &$,5$'6! )$6!$)$) %$ (" 6$%6!'$ '.$ H6)' ()$', &$' 6!7 ,; '.$ )-5
∑
∞
i=0(Lz,x+iR + Lz,R+iR−x)
= L0,0 e−azz∑
∞
i=0(e−ax(x+iR) + e−ax(R+iR−x))
= L0,0 e−azz(e−axx∑
∞
i=0 e−axiR + e−ax(R−x)∑
∞
i=0 e−axiR)
= L0,0 e−azz(e−axx + e−ax(R−x))∑
∞
i=0 e−axiR)
= L0,0 e−azz(e−axx + e−ax(R−x))∑
∞
i=0((1e)axR)i
=L0,0 e−azz(e−axx+e−ax(R−x))
1− 1e
axR
2KQ3C
81
Page 82
! " # $ % & ' ( ) !
*! *" *# *$ *%
+,- .//0123450 .//016780 .//01
934:.7 $; <57 =>?=31 [email protected] 057 A310?- 7 8./C ? ./, 0/ 057 C3A@/3-0 D70,77-30 ?-A 057 ./, 0/ 301 .3450* <57 E>?=31 31 057 .//0 A7-130E 8/. .//01 /.343-?03-4 3-? 1@7 3F ./,* <57 0/@ G3-7 [email protected] 057 .//01 8./C 057 ./, 3017G8* <57 -7=0G3-7 057 .//01 8./C 057 ./, 0/ 057 .3450* H-A 057 G?10 G3-7 057 .//01 8./C 057 ./,0/ 057 G780* I- 057/.EJ ?GG 057 ./,1 /- 057 F7GA ,3GG /-03D:07 1/C7 .//01 0/ 0573-07.K?G* <57 .//0 A7-130E 3- 057 3-07.K?G ,3GG D7 057 1:C /8 ?GG 057 3-A3K3A:?G /-0.3D:03/-1*
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!" #$%%&'( )*+,**' +-* ./0*12 ! "#$%& %'(! )# *!)+', #$* #*'-',+% &'.)*'/$)'#, "0!, '-,#*',- )0! 1 &'2!,.'#,3 ! #$&,5) &# )0+) "0!, %##(',- #,%6 +) )0! *##) .6.)!2 7#* + .',-%! *#"8 +. ')'. ',9,')!%6 "'&! +,& )0$. 0+. +, +:!*+-! &!,.')6 #7 ;!*#3 <#"!:!*8 '7 "! %##(+) )0! *##). #7 .',-%! *#"8 "! -!)Lz =
2∫
∞
0 Lz,x dx
R=>?@ ! 2$%)'A%6 /6 )"# +. "! +..$2! )0! )"# .'&!. #7 )0! *#". +*! '&!,)' +%3 B6',)!-*+)',- )# ∞ *+)0!* )0+, C$.) R/2 "! &# ', %$&! *##). 7*#2 #$).'&! )0!*#"3 <#"!:!*8 /! +$.! )0! .6.)!2 0+. +, ',9,')! ,$2/!* #7 '&!,)' +% *#".8 )0!+2#$,) #7 *##). 7*#2 )0! *#A #$).'&! '). #", *#" '. !1+ )%6 )0! .+2! +. )0!+2#$,) #7 *##). 7*#2 #)0!* *#". ',.'&! )0! *#" "! +*! !1+2',',-3D,.!*)',- =EF@ +,& =E@ ', =>?@ "! -!)
L0 e−az = 2R
∫
∞
0 L0,0 e−azze−axxdx
=2L0,0 e−azz
R
∫
∞
0 e−axxdx= L0,0 e−azz 0−1
−ax
=2L0,0
Raxe−azz
=>G@H# "! -!)
az = a =IJ@L0,0 =
1
2axRL0 =IE@
L0 =2L0,0
axR=I>@
+. )0! !K$+)'#, )# $.! "0!, ."') 0',- /!)"!!, )0! #,! +,& )"# &'2!,.'#,+%&!. *'A)'#,.3 ?83
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84
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Annex 3.5 ABA in Daisy
85
Page 86
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Annex 3.6 Soil Vegetation Atmosphere Transfer (SVAT) model
A SVAT model simulates the exchange of gases and energy between the canopy-soil-system and the
atmosphere. This SVAT model is based on the resistance/conduction concept and considers three sources
viz. soil, sunlit and shaded leaves. The SVAT model considers the following surface fluxes:
atm S c snl c shl cH H H H (0.1)
atm S c snl c shl cE E E E (0.2)
atm S c snl c shl cF F F F (0.3)
where H, E, and F denote ecosystem fluxes of sensible heat [W m-2
], water vapour [kg m-2
s-1
], and CO2-2
s-1
]
respectively. The subscripts describes the pathway of the fluxes; atm, S , snl , and shl , are the flux into the
atmosphere, the flux from the soil to the canopy air space, the flux from the sunlit leaves to the canopy air space, and
the flux from the shaded leaves to the canopy air space. The conceptualization for the exchange of latent heat or water
vapour is shown in Fig. 1.
Latent heat flux from
soil surface
Atmosphere: wind, temperature and vapor pressure
Aerodynamic resistance
, ,z a au T e
ar
Shade
,b snlr
Sunlit
sE
cTce
,b shlr,s snlr,s shlrsnlT shlT
Canopy:– Air space temperature
– Air space vapor pressure
– Boundary resistance for sunlit and shaded leaves
Leaves:– Leaf temperature for
sunlit and shaded leaves
– Stomata resistance for sunlit and shaded leaves
Figur 1 Conceptual model of the exchange of latent heat (water vapour)
Latent heat fluxes [W m-2
] are obtained by multiplying the corresponding water vapour flux by the latent
heat of vaporization, [J kg-1
]:
3149000 2370 aT (0.4)
where aT is the air temperature [K].
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The model is based on the principles of conservation of energy and matter, i.e.:
abs soil o soil soil S c S cR L G H E (0.5)
abs snl o snl snl c snl cR L H E (0.6)
abs shl o shl shl c shl cR L H E (0.7)
where eq. (1.5), eq. (1.6), and eq. (1.7) are energy balances of the soil, sunlit leaves and shaded leaves,
respectively. The term on the left side of the equations represents the absorbed radiation of the considered
part of the system and Gsoil is the ground heat flux at the surface. The model considers the absorption of
photosynthetically active radiation (PAR), near infrared radiation (NIR), and long-wave thermal radiation
separately. Adding the three radiation components in (1.5), eq. (1.6), and eq. (1.7) and subtracting the
emission of long-wave radiation from each fraction (o soilL ,
o snlL , and o shlL ) yields the net radiation of
canopy-soil system. Gsoil [W m-2
] is estimated by:
1
1
hsoil S z
kG T T
z (0.8)
where sT and 1zT is the soil surface temperature [K] and the soil temperature at the depth 1z [m],
respectively, and hk is the thermal conductivity of the soil [W m-1
]. 1zT , 1z , and hk are obtained from the
soil temperature model. For simplicity the values from the beginning of the considered time-step are used.
S cE in eq. (1.5) is estimated by the soil water dynamics model and surface model. S cE represents the
combined evaporation through the soil surface and from water stored at the soil surface. Again, the value
from the beginning of the considered time-step is used.
The term S cF in eq. (1.3) comprises soil respiration (soil microbial biomass and plant roots) and is obtained
from the soil organic matter model and the plant growth model. snl c shl cF F comprises photosynthesis of
sunlit and shaded leaves in combination with the respiration of the above ground plant biomass. Values are
obtained from the photosynthesis model and the plant growth model. The photosynthesis model considers
sunlit as well as shaded leaves and in linked to the SVAT through a common dependence of stomata
conductance.
Absorption of radiative energy Shortwave radiation comprises photosynthetically active radiation (PAR) and near infrared radiation (NIR).
In the model they each contribute 50% to the solar radiation and they are treated in the same way. They only
differ in respect to optical properties, Table 1.
The canopy cover fraction as function of the leaf area index of aiL is estimated as:
1 exp 0.5can aif L (0.9)
Shortwave radiation, PAR or NIR, (tI , W m
-2) is divided into a direct beam (
bI , W m-2
) and a diffuse (dI ,
W m-2
) component, i.e.:
1b dif tI f I (0.10)
d dif tI f I (0.11)
where diff is the diffuse fraction of the global radiation.
Shortwave radiation absorbed by a canopy with a leaf area index of aiL and a spherical leaf distribution is
estimated as:* *
, ,1 1 exp 1 1 expabs can d d c s d ai b b c s b aiI I k L I k L (0.12)
where parameters is obtained from Table 1 and Table 2.
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Page 91
Shortwave radiation absorbed by the sunlit fraction of the canopy is: * * * *
, ,1 1 2
1 exp
abs snl d d c s d d b b b c s b b b b b b b
ai
I I k F k k I k F k k I a k F k F k
F x xL x
(0.13)
and the sunlit fraction of the leaf area index, aiL , is estimated as:
1 exp b ai
snl
b ai
k Lf
k L (0.14)
Shortwave radiation absorbed by the shaded fraction of the canopy is:
abs shl abs can abs snlI I I (0.15)
Shortwave radiation absorbed by the soil is:
, ,1 1abs soil d d c s b b c s abs canI I I I (0.16)
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Table 1. Optical properties of leaves and soil surface.
PAR NIR Long-wave
Leaf absorptance 0.80 0.17
Soil surface reflectance s 0.10 0.18
Leaf emissivity l 0.98
Soil emissivity l 0.95
Table 2. Canopy and canopy-soil parameters assuming spherical leaf distribution. aiL is the leaf area index
of the canopy
Parameter Equation
Direct-beam extinction coefficient for black leaves. =
sun elevation angle
0.5
sinbk
Direct-beam extinction coefficient *
b bk k
Direct-beam canopy reflectance ,
11 exp 2
1 1
bb c
b
k
k
Direct-beam canopy-soil reflectance
, *
,
,
,, *
,
,
exp 21
1 exp 21
b c s
b c b ai
b c s
b c sb c s
b c b ai
b c s
k L
k L
Diffuse radiation extinction coefficient for black leaves. 2
0
ln
2 exp sin cos
d
d
ai
d b ai
kL
k L d
Diffuse radiation extinction coefficient *
d dk k
Diffuse radiation canopy-soil reflectance
, *
,
,
,, *
,
,
exp 21
1 exp 21
b c s
b c d ai
b c s
b c sb c s
b c d ai
b c s
k L
k L
Absorbed long-wave radiation:
1abs soil can iL f L (0.17)
abs snl can snl iL f f L (0.18)
1abs shl can snl iL f f L (0.19)
where iL is the incoming long-wave radiation from the atmosphere. The total (PAR, NIR and long-wave)
absorbed radiation by soil, sunlit and shaded leaves can now be estimated as: PAR NIR
abs soil abs soil abs soil abs soilR I I L (0.20)
PAR NIR
abs snl abs snl abs snl abs snlR I I L (0.21)
PAR NIR
abs shl abs shl abs shl abs shlR I I L (0.22)
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The absorbed net radiation for the soil is estimated as: 44
4 3
1 1
1 1 4
n soil abs soil o soil abs soil can s s abs soil can s a s a
Eq R
abs soil can s a can s a s a abs soil soil s a
R R L R f T R f T T T
R f T f T T T R G T T
(0.23)
where is Stefan-Boltzmann constant (5.67 10-8
W m-2
K-1
), 41Eq
abs soil abs soil can s aR R f T is
absorbed equilibrium net radiation for the soil [W m-2
] and 31 4R
soil can s aG f T is soil radiative
conductance [W m-2
K-1
]. Correspondingly, the absorbed net radiation for sunlit and shaded leaves can be
written as: Eq R
n snl abs snl snl snl aR R G T T (0.24)
Eq R
n shl abs shl shl shl aR R G T T (0.25)
where 4Eq
abs snl abs snl can snl s aR R f f T and 41Eq
abs shl abs shl can shl s aR R f f T is absorbed
equilibrium net radiation for the sunlit and shaded leaves, respectively, and 34R
snl can snl s aG f f T and
31 4R
shl can snl s aG f f T is soil radiative conductance for sunlit and shaded leaves, respectively.
Sensible heat fluxes
The sensible heat fluxes (atmH ,
soil cH ,snl cH , and
shl cH ) are estimation of as:
H
atm p a a c a atm c aH c g T T G T T (0.26)
H H
soil c p a soil c s c soil c s cH c g T T G T T (0.27)
H H
snl c p a snl c snl c snl c snl cH c g T T G T T (0.28)
H H
shl c p a shl c shl c shl c shl cH c g T T G T T (0.29)
where pc is the specific heat of air (1005 J kg-1
K-1
), a
is the density of air [kg m-3
], ag ,
H
soil cg ,H
snl cg ,
andH
shl cg are conductances corresponding to the fluxes [m s-1
] , cT , snlT , and shlT is canopy air temperature,
and temperature of sunlit and shaded leaves, respectively. The estimation of the conductances are given in
sections below.
Latent heat fluxes
The latent heat fluxes (atmE ,
snl cE , and shl cE ) are estimation of as:
p a a w
atm c a atm c a
c gE e e G e e (0.30)
* *
w
p a snl c w
snl c snl c snl c snl c
c gE e T e G e T e (0.31)
* *
w
p a shl c w
shl c shl c shl c shl c
c gE e T e G e T e (0.32)
where is psychrometer constant,w
ag ,w
snl cg , andw
shl cg are conductances corresponding to the fluxes [m s-
1] ,
cT ,snlT , and
shlT is canopy air temperature, and temperature of sunlit and shaded leaves, respectively.
*e T is saturation vapour pressure at the temperature T .The estimation of the conductances are given in
sections below.
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Page 94
Introducing the Penman approximation eq. (1.31) and eq. (1.32) can be rewritten:*w
snl c snl c a snl a cE G e T s T T e (0.33)
*w
shl c shl c a shl a cE G e T s T T e (0.34)
where s is slope of the saturation vapour pressure curve vs. temperature. The estimation of the conductances
are given in sections below.
Conservation of energy Considering the tree components, soil, sunlit and shaded leaves, separately yield tree conservation of energy
equations, viz. eq. (1.5), eq. (1.6), and eq. (1.7). These equations may now be rewritten. Introducing eq.
(1.23), eq. (1.8), and eq. (1.27), in eq. (1.5) yields:
1
1
Eq R Hhabs soil soil s a S z soil c s c soil c
kR G T T T T G T T E
z (0.35)
Similarly, introducing eq. (1.24), eq. (1.28), and eq. (1.33), in eq. (1.6) yields;*Eq R H w
abs snl snl snl a snl c snl c snl c a snl a cR G T T G T T G e T s T T e (0.36)
And introducing eq. (1.25), eq. (1.29), and eq. (1.34), in eq. (1.7) yields:*Eq R H w
abs shl shl shl a shl c shl c shl c a shl a cR G T T G T T G e T s T T e (0.37)
Introducing eq. (1.26), eq. (1.27), eq. (1.28), and eq. (1.29), in eq. (1.1) yields:H H H H
atm c a soil c s c snl c snl c shl c shl cG T T G T T G T T G T T (0.38)
And introducing eq. (1.30), eq. (1.33), and eq. (1.34), in eq. (1.2) yields:* *w w w
atm c a snl c a snl a c shl c a shl a c soil cG e e G e T s T T e G e T s T T e E
(0.39)
Determination of state variables and fluxes
Assuming that the state variables sT , cT , snlT , shlT , and ce are the only unknown variables then they can be
found by solving eq. (1.35) through (1.39). When sT , cT , snlT , shlT , and ce are known then the system is
determined and the appropriate sensible heat, latent heat, and raditive fluxes can be calculated. However,
several of the conductances depend on the unknown state variables. Hence, the equations must be solved in
an iterative manner.
Aerodynamic ResistanceDisplacement height (d [m]) and roughness length for momentum (z0 [m]) is according to Shuttleworth and
Gurney (1990) estimated as:
41.1 ln 1veg d aid h c L (0.40)
0
0
0.3 0.0 0.2
0.3 0.2 1.5
veg p ai p ai
veg p ai
z h c L c Lz
h d c L (0.41)
where vegh is vegetation height [m], aiL is leaf area index, pc is an effective drag coefficient 0.07pc ,
and 0z is roughness length for momentum for bare soil.
The roughness length for heat is
0 0 7hz z (0.42)
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Page 95
The aerodynamic stability indicator ( ) is according to Choudhury (1986) estimated as:
0
2
5 r a
a z
z d g T T
T u (0.43)
where rz screen-height [m], g is acceleration due to gravity (9.82 m s
-2),
aT and zu are air temperature (K)
and wind speed (m s-1
) at screen-height, and 0T is surface temperature (K). 0 corresponds to a stable
atmosphere while 0 corresponds to an unstable atmosphere.
Aerodynamic resistance, ra [s m-1
] between canopy source height (canopy point) and reference (screen)
height above the canopy is estimated by:
* *2
0 0
0 0
3 42
1ln ln for 0
ln ln
for 01
r r
z h
ar r
h
z
z d z d
u z z
r z d z d
z z
u
(0.44)
where:
2
2
0
*
0 0
4 1 ln
2 1
ln 2 ln
ra a
r ra
h
z d
z
z d z d
z z (0.45)
When * 5 , then * is set to -5 (Choudhury, 1986) .
Aerodynamic conductance ag is estimate as:
1a
a
gr
(0.46)
The aerodynamic conductance depends on the following state variables: vegetation heightvegh , leaf area
index aiL , wind speed zu , air temperature aT , and surface temperature 0T .
Boundary layer conductance of sunlit and shaded leaves
Diffusivity of heat, hD [m s
-1], water vapor,
wD [m s-1
], and CO2, CD [m s-1
]:
1.81
5 2 -1101300 Pa1.869 10 m s
273.16 K
ah
surf
TD
P (0.47)
1.81
5 2 -1101300 Pa2.178 10 m s
273.16 K
aw
surf
TD
P (0.48)
1.81
5 2 -1101300 Pa1.38110 m s
273.16 K
aC
surf
TD
P (0.49)
where surfP is air pressure at the surface (101300 Pa unless otherwise specified) and aT is air temperature
(at screen-height) [K]).
95
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Leaf boundary layer conductance for heat, eq. (1.50), water vapor, eq. (1.51), and CO2, eq. (1.52), due to
free convection (Houborg, 2006):
3 2 14l l a aH
lbf h
l
gw T T Tg D
w (0.50)
0.5 hypostomotous leaves
amphistomatous leaves
H
lbf w hw
lbf H
lbf w h
g D Dg
g D D (0.51)
2CO H
lbf lbf C hg g D D (0.52)
where g is acceleration due to gravity (9.82 m s-2
), lw is leaf width (0.05 m unless otherwise specified),
is molecular viscosity (1.815 2 -11.327 10 101300 Pa 273.16 K m ssurf aP T ), and
lT is leaf
temperature [K] (sunlit or shaded leaves).
Leaf boundary layer conductance for heat, eq.(1.53), water vapor, eq.(1.54), and CO2, eq.(1.55), due to
forced convection (Houborg, 2006):
exp0.006
z u aiH
lbu
l
u k Lg
w (0.53)
0.5 hypostomotous leaves
amphistomatous leaves
H
lbu w hw
lbu H
lbu w h
g D Dg
g D D (0.54)
2CO H
lbu lbu C hg g D D (0.55)
where zu is wind speed (m s-1
) at screen-height, uk parameter describing the vertical variation of wind speed
within the canopy ( 0.5uk , Houborg (2006)), and aiL is leaf area index.
Applying the “big leaf” approach, leaf boundary layer conductance is up-scaled to canopy level according to
Wang and Leuning (1998). The total boundary conductance at canopy level for heat for sunlit leaves, eq.
(1.56) , and shaded leaves, eq. (1.57), due to the combined effect of free and forced convection is:
1 exp 0.5
0.5
u b aiH snl H H
b snl ai lbf snl lbu
u b
k k Lg L g g
k k (0.56)
1 exp 0.51 exp 0.5
0.5 0.5
u b aiu aiH shl H H
b shl ai lbf shl lbu
u u b
k k Lk Lg L g g
k k k (0.57)
where snl
aiL and shl
aiL are the leaf area index of sunlit and shaded leaves, respectively; H
lbf snlg and H
lbf shlg are
leaf boundary layer conductance for heat of sunlit and shaded leaves, respectively (eq. (1.50), lT leaf
temperature of corresponding sunlit and shaded leaves); and bk is extinction coefficient for black leaves in
direct-beam irradiance.
w
b snlg and w
b shlg are estimated by eq. (1.56) and eq. (1.57), respectively by substituting the appropriate leaf
boundary layer conductances. Similarly, 2CO
b snlg and 2CO
b shlg are estimated.
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The boundary layer conductances depend on the following state variables: total leaf area index, aiL , sunlit
leaf area index,snl
aiL , shaded leaf area index,shl
aiL , wind speed zu , air temperature aT , and leaf
temperature of sunlit snlT and shaded leaves shlT .
The conductance for sensible heat from sunlit and leaf surface into the canopy air is given by eq. (1.58) and
eq. (1.59), repectively:H H
snl c b snlg g (0.58)
H H
shl c b shlg g (0.59)
And the conductance for latent heat from the stomata of sunlit and leaf surface into the canopy air is given
by eq. (1.60) and eq. (1.61), repectively:w w
w b snl s snlsnl c w w
b snl s snl
g gg
g g (0.60)
w ww b shl s shlshl c w w
b shl s shl
g gg
g g (0.61)
where w
s snlg and w
s shlg is the bulk stomata conductance for sunlit and shaded leaves, respectively [m s-1
].
Soil aerodynamic conductance
The conductance for transport of heat between the soil surface and a height of the canopy point (H
soil cg , [m
s-1
]) can according to Norman et al. (1995) be estimated as:
0.004 0.012H
snl c sg u (0.62)
where su is a wind speed characterizing the conditions in the canopy air space just above the soil surface [m
s-1
]. su is estimatet as:
2 3 1 3 1 3
0.05exp 1
0.28
s c
veg
ai veg m
u u ah
a L h l
(0.63)
where cu is a wind speed at the top of the canopy [m s
-1],
aiL is leaf area index, vegh is vegetation height [m],
and ml is mean leaf size [m] given by four times the leaf area divided by the perimeter. If we assume
elliptical leaves and that the major axis equals the leaf width lw [m] and the minor axis equals ½ lw then we
get 1.26m ll w .
The stability is characterized by the Monin-Obukhov length (moL [m]):
3
*a amo
s a
r u TL
g T T (0.64)
where ar is aerodynamic resistance [s m
-1],
*u is friction velocity [m s-1
], aT is air temperature (at screen-
height) [K], is von Karman’s constant ( 0.41), g is acceleration due to gravity (9.82 m s-2
), and sT is
surface temperature (K). moL is positive in stable conditions and negative in unstable conditions.
97
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The friction velocity is:
*
0ln
z
r
uu
z d z (0.65)
where zu is wind speed (m s
-1) at screen-height,
rz screen-height [m], d is displacement height [m], and z0 is
roughness length for momentum [m].
The wind speed at the top of the canopy cu [m s-1
] is estimated as:
0
0
0
0
2 2
0.25
lnfor 0
ln 4.7
lnfor 0
ln
1 1ln 2arctan
2 2 2
1 16
veg
z mo
r r mo
c
veg
z mo
r
r
mo
h d zu L
z d z z d L
uh d z
u Lz d z
y yy
z dy
L
(0.66)
where vegh is vegetation height [m].
The soil aerodynamic conductance depends on the following state variables: vegetation heightvegh , leaf
area index aiL , wind speed zu , air temperature aT , and soil surface temperature sT .
Canopy physiological conductance for CO2 transferThe Ball-Berry-Leuning model (molar conductance [mol m
-2 s
-1]) is described in the Annex on “The
stomata-photosynthesis model and the sunlit-shadow radiation model in DAISY”. The model has been
modified in order to take into account the effects of ABA. The modified Ball-Berry-Leuning model (molar
conductance [mol m-2
s-1
]) for leaf layer i is:
2'
, 0 1expCO n
s i ABA
s s
Ag g m c
c h (0.67)
where 0g is a stomatal intercept factor, m is an empirical vegetation constant,
nA is the net leaf
photosynthesis rate [mol m-2
s-1
], sc is the leaf surface CO2 concentration [mol m
-3], the CO2
compensation point [mol m-3
],sh is the relative humidity at the leaf surface,
1is an empirical constant and
ABAc is the ABA concentration in the xylem sap.
The physiological conductance can be converted to [m/s] by:
2 2'
, ,
CO CO
s i s i
RTg g
P (0.68)
98
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Up-scaling from leaf to canopy yields the stomata conductance for water vapor:
2
2
,
,
2 hypostomotous leaves
amphistomatous leaves
LAI
LAI
nCO
w C ai s i
iw
s nCO
w C ai s i
i
D D L g
g
D D L g
(0.69)
where LAIn is the number of leaf layers and aiL is the size of the leaf layer.
ReferencesChoudhury, BJ, Reginato, RJ, and Idso, SB (1986). An analysis of infrared temperature observations over
wheat and calculation of latent heat flux. Agricultural and Forest Meteorology, 37, 75-88.
Houborg, R (2006) Inferences of key environmental an vegetation biophysical controls for use in regional-
scale SVAT modeling using Terra and Agua MODIS and weather prediction data. PhD
dissertation, Institute of Geography, University of Copenhagen,
Norman, JM, Kustas, WP, and Humes, KS (1995) Source approach for estimating soil and vegetation energy
fluxes in observations of directional radiometric surface temperature. Agricultural and Forest
Meteorology, 77, 263-293.
Shuttleworth, WJ and Gurney, RJ (1989). The theoretical relationship between foliage temperature and
canopy resistance in sparse crops. Quarterly Journal of the Royal Meteorological Society, 116,
497-519.
99
Page 100
Annex 3.7 Detailed description of the processes in the SALTMED The first version of the SALTMED model has been described in detail in Ragab (2002) with some
examples of applications. The SALTMED model includes the following key processes:
evapotranspiration, plant water uptake, water and solute transport under different irrigation
systems, nitrogen dynamics and dry matter & biomass production. A brief description of the above
mentioned processes will be given in the following sections.
1. Evapotranspiration, water uptake and Water & solute flow equations
Evapotranspiration
Evapotranspiration has been calculated using the Penman-Monteith equation according to the
modified version of FAO (1998) in the following form:
)34.01(
)(273
900)(408.0
2
2
U +
eeUT
+GR
= ET
an
o
S
(1a)
where ETo is the reference evapotranspiration, (mm day-1
), Rn is the net radiation,
(MJ m-2
day-1
), G is the soil heat flux density, (MJ m-2
day-1
), T is the mean daily air temperature at
2 m height, (oC ), is the slope of the saturated vapour pressure curve, (kPa
oC
-1), is the
psychrometric constant, 66 Pa oC
-1, es is the saturated vapour pressure at air temperature (kPa), ea is
the prevailing vapour pressure (kPa), and U2 is the wind speed at 2 m height (m s-1
) . The calculated
ETo here is for short well-watered green grass. In this formula, a hypothetical reference crop with an
assumed height of 0.12 m, a fixed surface resistance of 70 s m-1
and an albedo of 0.23 were
considered.
In presence of stomata / canopy surface resistance data, one could use the widely used equation
Penman-Monteith (1965) in the following form:
(1b)
where rs and ra are the bulk surface and aerodynamic resistances ( s m-1 ).
The rs can be measured or calculated from environmental and meteorological parameter or from the
Leaf water potential and Absicic Acid, ABA.
In the absence of meteorological data (temperature, radiation, wind speed etc.) and if Class A pan
evaporation data are available, the SALTMED model can use these data to calculate ETo according
to the FAO (1998) procedure. The model can also calculate the net radiation from solar radiation
according to the FAO (1998) procedure if net radiation data is not available.
r
)r + (1 +
r
e) - e(C + R
= E
a
s
a
spn
p
100
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The crop evapotranspiration ETc is calculated as:
)( ecboc KKETET (2)
where Kcb is the crop transpiration coefficient (known also as basal crop coefficient) and Ke is the
soil evaporation coefficient. The values of Kcb and Kc, (the crop coefficient) for each growth stage
and the duration of each growth stage for different crops are available in the model’s database.
These data can be used in the absence of measured values. Ke is calculated according to FAO
(1998). Kcb and Kc are adjusted according to FAO (1998) for wind speed and relative humidity
different from 2 m s-1
and 45% respectively. The SALTMED model runs with a daily time step and
uses Kcb and Ke. The latter are not universal and their values differ according to climatic conditions
and other factors.
Plant Water Uptake in the Presence of Saline Water
The Actual Water Uptake Rate
The formula adopted in the SALTMED model is that suggested by Cardon and Letey (1992),
which determines the water uptake S (d-1
) as:
S z tS t
a t h
t
z t( , )( )
( )
( )
( , )max
150
3 (3)
where
(4)
= 25/12L * (1 - z/L) for 0.2L < z (4a)
= 0.0 for z > L (4b)
where Smax(t) is the maximum potential root water uptake at the time t; z is the vertical depth taken
positive downwards, (z,t) is the depth-and time-dependent fraction of total root mass, L is the
maximum rooting depth, h is the matric pressure head, is the osmotic pressure head; 50 (t) is
the time-dependent value of the osmotic pressure at which Smax(t) is reduced by 50%, and a(t) is a
weighing coefficient that accounts for the differential response of a crop to matric and solute
pressure. The coefficient a(t) equals 50(t)/h50(t) where h50(t) is the matric pressure at which
Smax(t) is reduced by 50%.
The Maximum Water Uptake Smax (t). Smax (t)is calculated as:
Smax(t) = ETo (t)* Kcb (t) (5)
The values of h50 50 can be obtained from experiments or from literature such as FAO (1992).
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The Rooting Depth
The rooting depth was assumed to follow the same course as the crop coefficient Kc. Therefore, it
has been described by the following equation:
Root depth (t) = [Root depthmin + ( Root depthmax - Root depthmin )] * Kc (t) (6)
The maximum root depth is available either from direct measurements or from the literature.
The Rooting Width
Compared with rooting depth, there is a very little information in the literature on lateral extent of
the rooting systems of field crops over time. Therefore, a simple equation has been suggested as
follows:
Root width (t) = [Root width / Root depth] ratio * root depth (t) (7)
The [Root width/Root depth] ratio is dependant on the crop and soil type and other factors. It can
be obtained either from experimental data or from the literature. During the growth, new roots enter
new grid cells.
The model then calculates the water uptake only from those cells with roots. The model grid cells
are identified by 0, 1 or 2 . The value of 0 is associated with cells with no roots and 1 for cells fully
occupied with roots and 2 for cells with partial root presence. The model produces a data file
showing the two – dimensional root distribution for every day of the simulation.
Relative and Actual Crop Yield
The Relative Crop Yield, RY
Due to the unique and strong relationship between water uptake and biomass production, and
hence the final yield, the relative crop yield RY is estimated as the sum of the actual water uptake
over the season divided by the sum of the maximum water uptake (under no water and salinity
stress conditions) as:
),,(
),,(
max tzxS
tzxSRY (8)
where x, z are the horizontal and vertical coordinates of each grid cell that contain roots,
respectively.
The Actual Yield, AY
The actual yield, AY is simply obtainable by:
max*YRYAY (9)
where Ymax is the maximum yield obtainable in a given region under optimum and stress-free
condition. The other option to obtain the actual yield is by calculating the daily biomass
production and obtaining the actual yield from the harvest index times the total dry matter (see the
relevant section on crop growth and dry matter).
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Water and Solute Flow
The water flow in soils was described mathematically by the well-known Richard's equation. It is a
partial non-linear differential equation, partial in time and space. It is based on two soil physical
principles: Darcy's law and mass continuity. Darcy's law reads:
Where q is the water flux, K(h) is the hydraulic conductivity as a function of soil water pressure head,
Z is the vertical coordinate directed downwards with its origin at soil surface, and H is the hydraulic
Z+=H (11)
The vertical transient-state flow water in a stable and uniform segment of the root zone can be
described by a Richard's type equation as:
t zK
z
zSw( ) (12)
where is volume wetness; t is the time; z is the depth; K ( ) is the hydraulic conductivity (a
function of wetness); is the matrix suction head; and Sw is the sink term representing extraction
by plant roots. The movement of solute in the soil system, its rate and direction, depends greatly
on the path of water movement, but it is also determined by diffusion and hydrodynamic
dispersion. If the latter effects are negligible, solute flow by convection can be formulated (Hillel,
1977) as:
J qc v cc (13)
where Jc is the solute flux density; q is the water flux density of the water; c the concentration of
solute in the flowing water and v is the average velocity of the flow. The rate of a diffusion of a
solute (Jd) in bulk water at rest is related (by Fick’s law) to the concentration gradient as:
J D c xd o (14)
where D0 is the diffusion coefficient. In soil the diffusion coefficient, Ds , is decreased due to the
fact that the liquid phase occupies only a fraction of soil volume, and also due to the tortuous
nature of the path. It can therefore be expressed according to the following equation:
D Ds 0 (15)
7/3s
2 (16)
where is the tortuosity, an empirical factor smaller than unity, which can be expected to decrease
with decreasing convection flux
generally causes hydrodynamic dispersion too, an effect that depends on the microscopic non-
uniformity of flow velocity in the various pores. Thus a sharp boundary between two miscible
q = K(h)H
Z (10)
103
Page 104
solutions becomes increasingly diffuse about the mean position of the front. For such a case, the
diffusion coefficient has been found by Bresler (1975) to depend linearly on the average flow
velocity v , as follows:
D vh (17)
where is an empirical coefficient. By the combination of the diffusion, the dispersion and the
convection the overall flux of solute can be obtained as:
J D D c x v ch s (18)
If one takes the continuity equation into consideration, one-dimensional transient movement of a
non-interacting solute in soil can be expressed as:
c
t zD
c
z
qc
zSa s (19)
in which c is the concentration of the solute in the soil solution, q is the convective flux of the
solution, Da is a combined diffusion and dispersion coefficient, and Ss is a sink term for the solute
representing root adsorption/uptake.
Under irrigation from a trickle line source, the water and solute transport can be viewed as two-
dimensional flow and can be simulated by one of the following:
1) a “plane flow” model involving the Cartesian co-ordinates x and z. Plane flow takes place if one
considers a set of trickle sources at equal distance and close enough to each other so that their
wetting fronts overlap after a short time from the start of the irrigation.
2) a “cylindrical flow” model described by the cylindrical co-ordinates r and z.
Cylindrical flow takes place if one considers the case of a single trickle nozzle or a number of
nozzles spaced far enough apart so that overlap of the wetting fronts of the adjacent sources does
not take place. For a stable, isotropic and homogeneous porous medium, the two-dimensional flow
of water in the soil can be described according to Bresler (1975) as:
z
zK
zxK
xt
)( (20)
where x is the horizontal co-ordinate; z is the vertical-ordinate (considered to be positive
downward); K ( ) is the hydraulic conductivity of the soil. Considering isotropic and homogeneous
porous media with principal axes of dispersion oriented parallel and perpendicular to the mean
direction of flow, the hydrodynamic dispersion coefficient Dij can be defined as follows:
)(||/ sjiTLijTij DVVVVD (21)
where L is the longitudinal dispersivity of the medium; T is the transversal dispersivity of the
medium; ij is Kronecker delta (i.e., ij =1 if i = j and ij = 0 if i i and Vj are the ith and jth
components of the average interstitial flow velocity V respectively, V = (V2
x + V2
z )1/2
and Ds ( ) is
the soil diffusion coefficient as defined in Equation 15.
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Page 105
If one considers only two dimensions and substituting Dij , the salt flow equation becomes:
C
t xD
C
xD
C
zq C
zD
C
zD
C
xq Cxx xz x zz zx z (22)
In the model, sprinkler, flood and basin irrigation are described by one-dimensional flow equations
(e.g. Eqs 12 & 19). Furrow and trickle line source are described by 2-dimensional equations (e.g.
Eqs 20 & 22). Trickle point source is described by cylindrical flow equations obtained by replacing
x by the radius “ r” and rearranging Equations 20 and 22 as given by Bresler (1975) and Fletcher
Armstrong and Wilson (1983). The water and solute flow equations were solved numerically using
a finite difference explicit scheme (Ragab et al., 1984).
Soil Hydraulic Parameters
Solving the water and solute transport equations require two soil water relations, namely the soil
water content - water potential relation and the soil water potential - hydraulic conductivity
relation. They were taken according to van Genuchten (1980).
Drainage
The model has two options for drainage. Two options are available, either free drainage or an
impermeable layer at the bottom of the soil profile.
Example of the flow domain under drip irrigation
Bresler, E., 1975. Two-dimensional transport of solute during non-steady infiltration for a trickle
source. Soil Sci. Soc. Amer. Proc. 39, 604-613.
References
Cardon, E.G., Letey, J., 1992. Plant water uptake terms evaluated for soil water and
solute movement models. Soil Sci. Soc. Am. J. 56, 1876-1880.
105
Page 106
FAO, 1998. Crop evapotranspiration, Irrigation and Drainage Paper No 56. Rome,
Italy.
FAO, 1992. The use of saline waters for crop production. Irrigation and Drainage Paper No 48.
Rome, Italy.
Fletcher Armstrong, C., Wilson, T.V., 1983. Computer model for moisture distribution in stratified
soils under tickle source. Transactions of ASAE, 26: 1704-1709.
Hillel. D., 1977. Computer simulation of soil-water dynamics; a compendium of recent work. IDRC,
Otawa, Canada, 214pp.
Ragab, R., 2002. A holistic generic integrated approach for irrigation, crop and field management:
the SALTMED model. Environmental Modelling and Software, 17, 345-361.
Ragab, R., Feyen, J., Hillel, D., 1984. Simulating two-dimensional infiltration into sand from a
trickle line source using the matric flux potential concept. Soil Sci. 137, 120-127.
-dimensional transport model for variably saturated porous
media with major chemistry. Water Resources Research, 30, 1115-1133.
Van Genuchten, M. Th., 1980. A closed - form equation for predicting the hydraulic conductivity
of unsaturated soils. Soil Sci. Soc. of Am. J. 44, 892-898.
The approach used is very much based on the work of Eckersten and Jansson, 1991.
2. Crop Growth and Biomass production
1- 2/day = Net Assimilation “NA”
Net Assimilation “NA” = Assimilation ” A ” – Respiration losses ” R”
2- Assimilation rate,”A”per unit of area = E* I* f(Temp)* f(T)*f(Leaf-N)
g/m2/day
where
E = is the photosynthetic Efficiency, g dry matter / MJ (=~2.0)
I is the radiation input: = Rs (1- e–k*LAI
)
Rs is global Radiation, MJ/m2/day, k is extinction coefficient (~=0.6) and
LAI is the leaf area Index (m2/m
2).
Rs is given in climate data, LAI is interpolated in SALTMED
Assimilation rate,”A”per unit of area = E* I* (stress factors related to Temperature, Transpiration
and Leaf Nitrogen content)
Eckersten, H and Jansson, P,.- E. 1991. Modelling water flow, nitrogen uptake and production for
wheat. Fertilizer Research 27: 313-329.
References:
106
Page 107
The modelling approach is based on Tardieu et al. (1993).
3. Calculating the Stomata Conductance from the Absecic Acid, ABA
gs = gs minimum l ))
gs = Stomata conductance, mole/m2/sec
gs minimum = mimimum Stomata conductance (default 0.05 mole/m2/sec)
ABA = Absecic Acid concentration, daily values
(default 0.5 mmole/m3)
l = Leaf water potential in M pa, daily values,
(default -1.3 Mpa)
0.184 -2.69 -0.183
l are given as daily values.
Tardieu, F, Zhang, J. and Gowing, D. J. G. 1993. Stomatal control by both [ABA] in the xylem sap
and leaf water status: a test of a model for droughted or ABA-fed field-grown maize. Plant, Cell
and environment .16:413-420.
References
The modelling approach for stomatal conductance is based on the multiplicative model described
by Jarvis 1976 and modified by Korner et al. (1995).
4. Calculating the stomata Conductance from regression Equation:
Based on Jarvis (1967) and Korner (1994)
gs = gsmax * f(VPD) * f(T) * F(SW)*f(PAR)
gsmax = Maximum Stomata conductance
f (VPD) is the relative effect of the VPD on stomata conductance
f(T) is the relative effect of the Temperature on stomata conductance
f (SW) is the relative effect of the soil water content on stomata conductance
f(PAR) is the relative effect of the radiation on stomata conductance
The sum of
f(VPD) = 1- [(VPD min – VPD) / (VPD min- VPD max) ]
107
Page 108
VPD is calculated on daily basis from Temperature and RH data given as daily values in the input
file of the climate data .
VPD min and VPD max are either User input values or can be calculated first from the model code
as the first thing before a dynamic run.
F (T) = 1- [(T – Tminimum) /(Toptimum – T minimum)]2
T is daily average Temperature given as input in climate data file
Tminimum and Toptimum are user input, C
WP
- WP fc - WP WP FC
FC SAT
e obtained as average values of the fully rooted
squares.
WP, FC, SAT are Soil water content at wilting point, at Field Capacity and at saturation (or
Porosity) given as input in the soil data base.
F (PAR) = 1 - exp ( - *PFD)
t = 0.2
PFD is Photons flux density micromole /m2/sec (range : 0-900 , average is 450)
Emberson, L.D. Ashmore, M. R., and Cambridge, H.M. 1998. Development of Methodologies for
Mapping Level II Critical Levels of Ozone. DETR Report no. EPG 1/3/82. Imperial
College of London, London.
References
Jarvis, P. G. 1976. The iterepretation of the variations in leaf water potential and stomatal
conductance found in canopies in the field. Philosophical. Transactions of the Royal
Society. B273:593-610.
Körner, C. 1994. Leaf diffusive conductance in the major vegetative types of the globe. In Schulze,
E. D., and Calwell, M.M . (Editors), Ecophysiology of Photosynthesis. Ecological Studies,
vol.100. Springer Verlaag, Berlin, pp 463-490.
Pleijel, H., Danielsson, H., Vandermeiren, K., Blum, C., Colls, J, and Ojanpera, K. 2002. Stomatal
conductance and ozone exposure in relation to potato tuber yield-results from the
European CHIP programme. European Journal of Agronomy, 17:303-317.
108
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The top soil layer is the most biologically active layer where most of the organic matter
decomposition and mineralization takes place. The microbial activity is affected by soil
temperature of this layer. This temperature was found to be correlated to air temperature. The
approach used here is to infer the soil temperature of the top layer (ploughing layer) from the air
temperature based on the work of Kang et al. (2000) and Zheng et al. (1993).
5. Calculating Soil temperature from Air Temperature:
For air temperature “A” and soil temperature “ T ”, the relation can be described as:
For A j > Tj-1 (z):
Tj (z) = Tj-1 (z) + [Aj - Tj-1 (z)] * Exp [- s * p))0.5
] * Exp [ -k(LAIj + litterj)]
For A j j-1 (z):
Tj (z) = Tj-1 (z) + [Aj - Tj-1 (z)] * Exp [- ( ks * p))0.5
] * Exp [ -k(litterj)]
Aj is average Air Temperature at day “ j “ in °C .
This is calculated from Tmin and Tmax given as input in climate data file.
Tj-1 (z) is Soil temperature at day “ j-1 “ previous day at depth “z “ below soil surface, °C
Tj (z) is Soil temperature at day “ j “ and depth “z “ below soil surface, °C
Exp [- s * p))0.5
] is a damping ratio
ks is the thermal diffusivity as a function of soil water, air and mineral content. m2
s-1
ks = ( thermal conductivity/(bulk density* specific heat capacity)).
P: is period of either diurnal or annual temperature variation, z is in meters
LAI: is calculated already in the model on daily basis, Litter fraction is given as user input.
Kang, S., Kim, S., Oh, S. and Lee, D. 2000. Predicting spatial and temporal patterns of soil temperature
based on topography, surface cover and air temperature. Forest Ecology and Management
136:173-184.
References
Marshall, T.J., Holmes, J. W., and Rose, C.W. (editors). 1996. Soil Physics ( 3rd
edition) , 358-376.
Cambridge University Press. Cambridge, UK.
Zheng, D., Hunt, Jr., Running, S.W. 1993. A daily soil temperature model based on air temperature and
precipitation for continental applications. Climate Research 2: 183-191.
109
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This is very much based on SOIL N model of Johnsson et al. 1987. The following processes were
implemented in SALTMED:
6. Soil Nitrogen dynamics and Nitrogen uptake
•
•
Mineralization
•
Immobilization
•
Nitrification
•
Denitrification
•
Leaching
Plant N Uptake
Nitrogen input included dry and wet deposition, incorporation of crop residues, manure application,
chemical fertilizer application and with irrigation water as fertigtion.
Eckersten, H and Jansson, P,.- E. 1991. Modelling water flow, nitrogen uptake References
and production for wheat. Fertilizer Research 27: 313-329.
Johnsson, H., Bergstrom, L and Jansson, P.-E.. 1987. Simulated nitrogen dynamics and losses in a
layered agricultural soil. Agriculture, Ecosystems and Environment, 18:333-356.
Wu, L., McGechan, M., B., Lewis, D. R., Hooda, P. S., and Vinten, A., J., A. 1998. Parameter
selection and testing the soil nitrogen dynamics model SOILN. Soil Use and Management,
14: 170-181
110
Page 111
Annex 3.8 SALTMED model frames (user Interface) and Examples of outputs
Main input tabs
Model input can be saved and uploaded via a text file
111
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Climate data input
Evapotranspiration calculation options (1)
112
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Evapotranspiration calculation options (2)
Evapotranspiration calculation options (3)
113
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Evapotranspiration calculation options (4)
Evapotranspiration calculation options (5)
114
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Evapotranspiration calculation options (6)
Evapotranspiration calculation options (7)
115
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Evapotranspiration calculation options (8)
Irrigation input file (drip sub subsurface example)
116
Page 117
Irrigation input file (drip sub subsurface PRD example)
Crop growth input parameters (1)
117
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Crop growth input parameters (2)
Crop rotation option 1 (no rotation)
118
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Crop rotation option 2 (rotation)
Soil input parameters
119
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Soil temperature, N-Fertilizer input file, N-mineralization, N-transformation parameters initial
N condition and environmental input parameters
Soil initial condition- input parameters per soil layer
120
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Model input parameters
Output specifications for plotting output parameters
121
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Output selection options
Output selection folder
122
Page 123
Output example of Dry matter
Output example of plant –N uptake
123
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Output example of N-Leaching
Output example of soil moisture under drip irrigation
124
Page 125
Output example of soil salinity under drip irrigation
125
Page 126
Output example of soil moisture under subsurface drip irrigation
Output example of soil moisture under PRD drip irrigation
126
Page 127
Output example of soil moisture under PRD subsurface drip irrigation
Output example of soil moisture profile under PRD drip irrigation
127
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Output example of soil nitrogen profiles
Output example of Evapotranspiration
128
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Output example of irrigation+ rainfall
Output example of crop growth parameters
SALTMED MODEL Can be Downloaded at: http://www.safir4eu.org
The basis of SALTMED model can be found at:
Special Issu : J. Agric. Water Management,volume 98 (1-2), September, 2005, (Guest Editor, Ragab Ragab)
129
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Annex 3.9 Test of the new Daisy model
by Finn Plauborg1, Mikkel Mollerup
2, Per Abrahamsen
2, Fulai Liu
3, Bo Vangsø Iversen
1, Mathias N.
Andersen1, Christian R. Jensen
3, Søren Hansen
2
1. Department of Agroecology and Environment, Faculty of
Agricultural Sciences, University of Aarhus, Denmark
2. Dept. of Basic Sciences and Environment, Faculty of Life
Sciences, University of Copenhagen, Denmark
3. Dept. of Agriculture and Ecology, Faculty of Life Sciences,
University of Copenhagen, Denmark
New processes developed for the Daisy modelBased on several experiments in glasshouse and field conditions in Denmark new process sub-models have
been developed for implementation in Daisy. These new sub-models have been reported in scientific journal
papers, where the most important are mentioned below.
F. Liu, A. Shahnazari, M.N. Andersen, S.-E. Jacobsen, C.R. Jensen. 2006. Effects of deficit irrigation (DI)
and partial root drying (PRD) on gas exchange, biomass partitioning, and water use efficiency in potato.
Scientia Horticulturae 109: 113–117.
F. Liu, A. Shahnazari, M.N. Andersen, S.-E. Jacobsen, C.R. Jensen. 2006. Physiological responses of potato
(Solanum tuberosum L.) to partial root zone drying: ABA signalling, leaf gas exchange, and water use
efficiency. Journal of Experimental Botany 57: 3727-2735.
Shahnazari, F. Liu, M.N. Andersen, S.-E. Jacobsen, C.R. Jensen. 2007. Effects of partial root-zone drying
on yield, tuber size and water use efficiency in potato under field conditions. Field Crops Research 100:
117–124.
F. Liu, R. Song, X. Zhang, A. Shahnazari, M.N. Andersen, F. Plauborg, S.-E. Jacobsen, C.R. Jensen. 2008.
Measurement and modelling of ABA signalling in potato (Solanum tuberosum L.) during partial root-zone
drying. Environmental and Experimental Botany 63, 385-391.
Ahmadi, S.H., Andersen, M.N, Poulsen, R.T., Plauborg, F., Hansen, S. 2009. A Quantitative Approach for
Developing More Mechanistic Gas Exchange Models for Field Grown Potato: A New Insight into Chemical
and Hydraulic Signalling. Agricultural and Forest Meteorology 149, 1541-1551.
Liu, F., Andersen, M.N., Jensen, C.R. 2009. Capability of the 'Ball-Berry' model for predicting stomatal
conductance and water use efficiency of potato leaves under different irrigation regimes. Scientia
Horticulturae 122, 346-354.
These studies deals with ABA produced in the root system of potato when the crop is imposed to different
deficit regimes, e.g. Partial Root Drying (PRD). The results have been formulation of ABA production
functions which have been included into Daisy, Figs. 1 and 2.
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s (-kPa)
y = 26.625e-4.2816x
0
1000
2000
3000
4000
5000
6000
7000
-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0
s [MPa]
[AB
A]
[ng
/cm
3]
0gC
hmAg
s
sns
semm i
s
s
s
ns ABAc
hmAg expexp
ssn
s
s ABAhAc
mg expexpexpexp
Figure 1. ABA concentration (X-[ABA as a response to soil water potential (Liu et al., 2008).
Figure 2. ABA concentration (X-[ABA]) as a response to soil water potential Abrahamsen (pers. comm.
2009).
This response and its effect on gas exchange have lead to formulation of new single leaf models of stomatal
conductance (gs).
Gutschick and Simonneau (2002), Abrahamsen (pers. 2009)
gs = m An hs/cs exp (- [ABA]) + gs0
with m =17, = 0.095 [cm3/ng] and gs0= 0.15 [mol/m
2/s]
Liu et al. (2009)
Ahmadi et al. (2009)
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0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
17/06/07 00:00 27/06/07 00:00 07/07/07 00:00 17/07/07 00:00 27/07/07 00:00
gs (
mo
l/m
2/s
)
Simulated, FI
Simulated, PRD
Measured, FI
Measured, PRD
0
20
40
60
80
100
120
140
20/07/2007 00:00 20/07/2007 12:00 21/07/2007 00:00
AB
A [
ng
/cm
3]
Simulated, FI
Simulated, PRD
Measured, FI
Measured, PRD
Testing the new Daisy modelThe above sub-models have been implemented into the comprehensive Daisy model (see D3_2) and testing
have been carried out for potatoes in Denmark and fresh tomatoes in Crete, Greece
Results from potatoes in Denmark
The focus of semi-field experiments at Aarhus University was to create comprehensive datasets on gas
exchange and soil water dynamics in potatoes imposed to irrigation strategies, Full Irrigation, Deficit
Irrigation and Partial Root zone Drying. The findings are fully reported in a coming special issue in
Agricultural Water Management, and hence here only some few results are shown. Figure 3 shows the Daisy
modelled stomatal conductance (gs) under predict the field measurements in the late season. Still research is
needed to understand this difference, which for the moment is thought to be an effect of nitrogen on
photosynthesis maybe not yet well included in the model.
Figure 3. Daisy simulated stomatal conductance (gs) versus measured in field grown potatoes.
Another discussed still to be further researched is if ABA production functions needs to be soil texture
dependent, and morning and afternoon ABA not pictured by the model (Fig. 4) has any relevance. The latter
as hardly any production takes place early and late during the day.
Figure 4. Simulated and measured ABA concentration in the upper leaves of potatoes around 10, 14 and 18
hours 20 July 2007.
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0
1
2
3
4
5
6
7
8
08/04/07 28/04/07 18/05/07 07/06/07 27/06/07 17/07/07 06/08/07 26/08/07 15/09/07
DM
(t/
ha)
Tomato DM repl1
Leaf DM repl1
Leaf DM Daisy
Tomato DM Daisy
Leaf DM repl2
LeafDM repl3
Tomato DM repl2
Tomato DM repl3
Stem DM Daisy
Stem DM repl1
Stem DM repl2
Stem DM repl3
DS
0
0.5
1
1.5
2
2.5
3
08/04/07 28/04/07 18/05/07 07/06/07 27/06/07 17/07/07 06/08/07 26/08/07 15/09/07
m2/m
2 LAI Daisy
LAI repl1
LAI repl2
LAI repl3
0
10
20
30
40
50
60
08/04/07 28/04/07 18/05/07 07/06/07 27/06/07 17/07/07 06/08/07 26/08/07 15/09/07
kg
N/h
a Leaf N Daisy
Leaf N repl1
Leaf N repl2
Leaf N repl2
Results from fresh tomatoes at Crete, Greece
Data from 2007 was used for calibration and data from 2008 for verification of the Daisy model.
2007 data
Figure 5 shows (top) a good calibration of the Daisy model as measured and simulated dry matter (DM) in
fully irrigated fresh tomato fits well although total DM in the harvested tomatoes in the last harvest was 10%
over predicted. Measured and simulated leaf area index compares well (middle), as do measured and
simulated leaf nitrogen (bottom).
Figure 5. Fully irrigated fresh tomato, measured and simulated: (top) dry matter, (middle) leaf area index,
(bottom) leaf nitrogen.
Figure 6 shows the same comparison as in figure 5, but data from the PRD irrigated treatments. The
calibration from fully irrigated treatment performs very well and nice comparisons were obtained.
133
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0
1
2
3
4
5
6
7
8
08/04/07 28/04/07 18/05/07 07/06/07 27/06/07 17/07/07 06/08/07 26/08/07 15/09/07
DM
(t/
ha)
Tomato DM repl1
Leaf DM repl1
Leaf DM Daisy
Tomato DM Daisy
Leaf DM repl2
LeafDM repl3
Tomato DM repl2
Tomato DM repl3
Stem DM Daisy
Stem DM repl1
Stem DM repl2
Stem DM repl3
DS
0
0.5
1
1.5
2
2.5
3
08/04/07 28/04/07 18/05/07 07/06/07 27/06/07 17/07/07 06/08/07 26/08/07 15/09/07
m2/m
2
LAI Daisy
LAI repl1
LAI repl2
LAI repl3
0
10
20
30
40
50
60
08/04/07 28/04/07 18/05/07 07/06/07 27/06/07 17/07/07 06/08/07 26/08/07 15/09/07
kg
N/h
a
Leaf N Daisy
Leaf N repl1
Leaf N repl2
Leaf N repl2
Figure 6. PRD irrigated fresh tomato, measured and simulated: (top) dry matter, (middle) leaf area index,
(bottom) leaf nitrogen.
Figure 7 shows measured and simulated soil water content in the centre of the tomato row in depth 0-40 cm
(top) fully irrigated, (middle) right side of the tomato in the PRD treatment, and (bottom) the left side of the
tomato plant. The hydraulic parameters were not calibrated resulting in a slight over prediction, especially in
the fully irrigated treatment. From 7 July the over prediction may be caused by a too low simulated stomatal
conductance (cf. Fig. 3) indicating that this model developed in pot experiments needs to be assessed from
the field measurements. However a clear response in both measured and simulated soil water content were
observed in the PRD treatments.
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Full and DI irrigated subsurface - Mesurements only in Full irrigated
0
5
10
15
20
25
30
35
08/04/07 28/04/07 18/05/07 07/06/07 27/06/07 17/07/07 06/08/07 26/08/07 15/09/07
vo
l% H
2O
Daisy FI 0-40cm left simulated
Daisy FI 0-40cm right simulated
Left 0-40cm repl1
Left 0-40cm repl2
Right 0-40cm repl3
Right 0-40cm repl1
Right 0-40cm repl2
Left 0-40cm repl3
Daisy DI 0-40cm simulated
PRD subsurface right
0
5
10
15
20
25
30
35
08/04/07 28/04/07 18/05/07 07/06/07 27/06/07 17/07/07 06/08/07 26/08/07 15/09/07
vo
l% H
2O Right 0-40cm repl3
Right 0-40cm repl1
Right 0-40cm repl2
Daisy
PRD subsurface left
0
5
10
15
20
25
30
35
08/04/07 28/04/07 18/05/07 07/06/07 27/06/07 17/07/07 06/08/07 26/08/07 15/09/07
vo
l% H
2O Left 0-40cm repl1
Left 0-40cm repl2
Left 0-40cm repl3
Daisy
Figure 7. Measured and simulated soil water content. Fully irrigated (top), right site of the tomato plant
(middle), and left side (bottom).
Figure 8 shows simulated ABA concentration in the fully, deficit and the PRD treatments. The responses are
quite acceptable as the highest concentration of ABA was obtained in the deficit and next the PRD
treatments.
135
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0
25
50
75
100
125
150
175
200
08/04/07 28/04/07 18/05/07 07/06/07 27/06/07 17/07/07 06/08/07 26/08/07
ng
/cm
3
Crop ABa Daisy SubsSurf
DeficitCrop ABA Daisy SubsSurf
PRDCrop ABa Daisy SubSurf
Full
0
100
200
300
400
500
600
Irr SubSurf Full Irr Surf &
SubSurf DI
Irr SubSurf PRD Transp. SubSurf
Full
Transp. Surf
Deficit
Transp. SubSurf
Deficit
Transp. SubSurf
PRD
mm
H2O
Figure 8. Simulated ABA concentration in the top leaves of tomato in the fully, deficit and PRD treatments.
Figure 9 shows irrigation inputs (green bars) to the fully, deficit and PRD treatments, causing a simulated
transpiration (blue bars) slightly higher in fully irrigated compared with the PRD treatment.
Figure 9. Irrigation input in the 2007 treatments with tomato (green bars). Simulated transpiration in the
different treatments (blue bars).
Figure 10 shows tomato dry matter yield in simulated (purple bars) and measured (green bars) in the 2007
tomato experiments.
The above mentioned tendency of over prediction of the tomato yield can be found in both the fully and
PRD irrigated treatments, however the trend of lower yield in the PRD treatment was consistent in both
measured and simulated data.
136
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0
1
2
3
4
5
6
7
8
DM Yield SubSurf
Full Daisy
DM Yield SubSurf
Full measured
DM Yield Surf
Deficit Daisy
DM Yield Surf
Deficit measured
DM Yield SubSurf
Deficit Daisy
DM Yield SubSurf
Deficit measured
DM Yield SubSurf
PRD Daisy
DM Yield SubSurf
PRD measured
t/h
a
0
1
2
3
4
5
6
7
8
9
10
02/04/08 22/04/08 12/05/08 01/06/08 21/06/08 11/07/08 31/07/08 20/08/08 09/09/08
DM
(t/
ha)
Tomato DM repl1
Leaf DM repl1
Leaf DM Daisy
Tomato DM Daisy
Leaf DM repl2
LeafDM repl3
Tomato DM repl2
Tomato DM repl3
Stem DM Daisy
Stem DM repl1
Stem DM repl2
Stem DM repl3
DS
0
0.5
1
1.5
2
2.5
3
3.5
02/04/08 22/04/08 12/05/08 01/06/08 21/06/08 11/07/08 31/07/08 20/08/08 09/09/08
m2/m
2 LAI Daisy
LAI repl1
LAI repl2
LAI repl3
0
10
20
30
40
50
60
70
80
02/04/08 22/04/08 12/05/08 01/06/08 21/06/08 11/07/08 31/07/08 20/08/08 09/09/08
kg
N/h
a Leaf N Daisy
Leaf N repl1
Leaf N repl2
Leaf N repl3
0
1
2
3
4
5
6
7
8
9
10
02/04/08 22/04/08 12/05/08 01/06/08 21/06/08 11/07/08 31/07/08 20/08/08 09/09/08
DM
(t/
ha)
Tomato DM repl1
Leaf DM repl1
Leaf DM Daisy
Tomato DM Daisy
Leaf DM repl2
LeafDM repl3
Tomato DM repl2
Tomato DM repl3
Stem DM Daisy
Stem DM repl1
Stem DM repl2
Stem DM repl3
DS
0
0.5
1
1.5
2
2.5
3
3.5
02/04/08 22/04/08 12/05/08 01/06/08 21/06/08 11/07/08 31/07/08 20/08/08 09/09/08
m2/m
2
LAI Daisy
LAI repl1
LAI repl2
LAI repl3
0
20
40
60
80
100
02/04/08 22/04/08 12/05/08 01/06/08 21/06/08 11/07/08 31/07/08 20/08/08 09/09/08
kg
N/h
a
Leaf N Daisy
Leaf N repl1
Leaf N repl2
Leaf N repl3
Figure 10. Measured and simulated dry matter yield in the 2007 tomato treatments.
2008 data
Figure 11 shows measured and simulated dry matter (DM) in fully irrigated fresh tomato (top left) and PRD
irrigated fresh tomato (top right). Stem and leafs compare well whereas the fruit dry matter was highly over
predicted. It seems that the model needs some refinements as the tendency of over prediction was found also
in 2007, however the big difference in 2008 may be caused mainly by the observed miss flowering and
pollination not captured by the model. Measured and simulated leaf area index compares well (middle left
and right), as do measured and simulated leaf nitrogen (bottom left and right).
Figure 11. Fully irrigated (left) and PRD irrigated (right) fresh tomato, measured and simulated: (top) dry
matter, (middle) leaf area index, (bottom) leaf nitrogen.
Figure 12 shows measured and simulated soil water content in the centre of the tomato row in depth 0-40 cm
(top) fully irrigated, (middle) right side of the tomato in the PRD treatment, and (bottom) the left side of the
tomato plant. The hydraulic parameters were not calibrated resulting in a slight over prediction, especially in
the fully irrigated treatment. From 21 June the over prediction may be caused by a too low simulated
stomatal conductance indicating that this model developed in pot experiments needs to be assessed from the
137
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Full irrigated subsurface
0
5
10
15
20
25
30
35
02/04/08 22/04/08 12/05/08 01/06/08 21/06/08 11/07/08 31/07/08 20/08/08 09/09/08
vo
l% H
2O
Daisy FI 0-40cm left simulated
Daisy FI 0-40cm right simulated
Left 0-40cm repl1
Left 0-40cm repl2
Right 0-40cm repl3
Right 0-40cm repl1
Right 0-40cm repl2
Left 0-40cm repl3
PRD subsurface right
0
5
10
15
20
25
30
35
02/04/08 22/04/08 12/05/08 01/06/08 21/06/08 11/07/08 31/07/08 20/08/08 09/09/08
vo
l% H
2O Right 0-40cm repl3
Right 0-40cm repl1
Right 0-40cm repl2
Daisy
PRD subsurface left
0
5
10
15
20
25
30
35
02/04/08 22/04/08 12/05/08 01/06/08 21/06/08 11/07/08 31/07/08 20/08/08 09/09/08
vo
l% H
2O Left 0-40cm repl1
Left 0-40cm repl2
Left 0-40cm repl3
Daisy
field measurements. However a clear response in both measured and simulated soil water content were
observed in the PRD treatments.
Figure 12. Measured and simulated soil water content. Fully irrigated (top), right site of the tomato
plant (middle), and left side (bottom).
Figure 13 shows simulated ABA concentration in the fully and the PRD treatments. The responses are quite
acceptable as the highest concentration of ABA was obtained in the PRD treatments.
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0
25
50
75
100
125
150
175
200
02/04/08 22/04/08 12/05/08 01/06/08 21/06/08 11/07/08 31/07/08 20/08/08
ng
/cm
3
Crop ABA Daisy PRD
Crop ABa Daisy Full
0
100
200
300
400
500
600
Irr Full Irr DI Irr PRD Transp. Full Transp. Deficit Transp. PRD
mm
H2O
0
2
4
6
8
10
DM Yield Full
Daisy
DM Yield Full
measured
DM Yield
Deficit Daisy
DM Yield
Deficit
measured
DM Yield PRD
Daisy
DM Yield PRD
measured
t/h
a
Figure 13. Simulated ABA concentration in the top leaves of tomato in the fully and PRD treatments.
Figure 14 (top) shows 2008 irrigation inputs (green bars) to the fully and PRD treatments, causing a
simulated transpiration (blue bars) quite higher in fully irrigated compared with the PRD treatment, and
hence an important water saving was achieved. This was obtained without a dramatic decrease in fruit yield.
(Fig. 14, bottom).
Figure 14. (top) Irrigation input in the 2008 treatments with tomato (green bars). Simulated transpiration in
the different treatments (blue bars). (bottom) Tomato dry matter yield in simulated (purple bars) and
measured (green bars) in the 2008 tomato experiments.
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ConclusionsIn the Danish experiments the level of ABA measured in FI and PRD irrigated potatoes differed due to the
sampling method, field or glasshouse conditions?
The pot gs model seams not fully valid in field experiments, especially in the middle and late part of the
season, however as no measurements were carried out in the Crete experiment further studies is needed to
confirm this.
The new Daisy 2D-water soil water flow and leaf gas exchange equations for photosynthesis and stomata
conductance seems well implemented in the Daisy model. Calibration of the soil hydraulic parameters may
improve the comparisons to measured soil water content.
Increased intrinsic WUE at leaf level in pot potatoes with PRD was not consistently verified at canopy level
with the new Daisy for fresh tomatoes, however important trends of water saving without too high yield loss
was observed in the PRD treatments.
Further studies are needed to explore if ABA production function and gs models are different in tomatoes
compared with potatoes.
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Annex 3.10 Some selected simulation results using the SALTMED model
Yield simulation results
Tomatao yield , Italy 2006-2008
0
2
4
6
8
10
12
2006 2007 2008
Year
Yie
ld ,
DM
- t
ha
-1 PRD Observed
PRD Model
Drip Observed
Drip Model
Sprinkler Oberseved
Sprinkler Model
Italy observed and simulated tomato yield dry matter
y = 0.8068x + 1.2079
R2 = 0.9507
0
2
4
6
8
10
12
0 2 4 6 8 10 12
Observed yield, T/ha
Sim
ula
ted
yie
ld,
T/h
a
141
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Tomato Yield, Itlay 2006-2008
0
2
4
6
8
10
12
PRD20
06
Drip
2006
Spr
inkler
2006
PRD20
07
Drip
2007
Spr
inkler
2007
PRD20
08
Drip
2008
Spr
inkler
2008
Irrigation System
Yie
ld, t
ha
-1
0
100
200
300
400
500
600
700
Irri
ga
tio
n-
rain
fall, m
m
Yield oberved
Yield Model
irrigation, mm
Rain, mm
irri+rain, mm
Italy Potato Yield- DM, t ha-1
0
1
2
3
4
5
6
7
8
9
10
2007 2008
Year
Yie
ld -
DM
, t
ha-1 PRD observed
PRD Model
Drip observed
Drip Model
Sprinkler observed
Sprinkler Model
142
Page 143
Simulated and Observed Potato Yield- Italy, DM t ha-1
y = 0.8259x + 1.426
R2 = 0.9599
4
5
6
7
8
9
10
11
12
4 5 6 7 8 9 10 11 12
Observed
Sim
ula
ted
Potato-Italy 2007-2008
0
2
4
6
8
10
12
PRD07
Drip
07
Spr
inkler
07
PRD20
08
Drip
08
Spr
inkler
08
Irrigation system
Yie
ld-D
M, t
ha
-1
0
100
200
300
400
Irri
ga
tio
n-r
ain
falll, m
m
Yield oberved
Yield Model
Irr mm
Rain mm
irr+rain
143
Page 144
Crete Tomato
0
1
2
3
4
5
6
7
2006 2007 2008
Year
Yie
ld-D
M,
t h
a-1
Drip observed
Drip model
PRD observed
PRD model
Crete, Tomato yield, 2006-2008
0
1
2
3
4
5
6
7
Drip
2006
PRD
2006
Drip
2007
PRD
2007
Drip
2008
PRD
2008
Irrigation System
Yie
ld -
DM
, t
ha
-1
0
200
400
600
800
Irri
ga
tio
n -
Ra
infa
ll, m
m
Yield oberved
Yield Model
Irr mm
Rain mm
irr+rain
144
Page 145
Serbia potato yield - DM in t ha-1
0
2
4
6
8
10
12
2006 2007 2008
Year
Tu
ber
yie
ld-D
M,
t h
a-1 Drip FULL Model
Drip Full observed
Drip dificit Model
Drip dificit observed
PRD Model
PRD observed
Furrow Model
Furrow observed
Serbia Potato Yield 2006-2008
0
2
4
6
8
10
12
Drip
200
6
PRD 2
006
Furrow
200
6
Drip
200
7
Drip
Defic
it 200
7
PRD 2
007
Furrow
200
7
Drip
200
8
Drip
Defic
it 200
8
PRD 2
008
Furrow
200
8
Irrigation System
Yie
ld -
DM
, t
ha
-1
0
100
200
300
400
500
600
700
Yield Observed
Yield Model
rain
irrigation
Irri+rain
0
20
40
60
80
100
120
140
160
180
5/2/2008 5/22/2008 6/11/2008 7/1/2008 7/21/2008 8/10/2008 8/30/2008
kg N
/h
Total N in Tuber DM
simulate
Serbia. Total N in tuber DM measured and simulated data obtained from fully irrigated
potato crops with sand filter water treatment during 2008 season.
145
Page 146
0
2
4
6
8
10
12
5/2/2008 5/22/2008 6/11/2008 7/1/2008 7/21/2008 8/10/2008 8/30/2008
kg/h
a
Tuber DM
Saltmed simulate
Serbia Tuber DM measured and simulated data obtained from fully irrigated potato crops
with sand filter water treatment during 2008 season.
Soil Moisture and soil nitrogen modelling results
Potato, Italy PRD 2007 average
0
0.1
0.2
0.3
0.4
0.5
18/05/2007 28/05/2007 07/06/2007 17/06/2007 27/06/2007 07/07/2007Date
So
il m
ois
ture
Observed
Modelled
146
Page 147
Potato, Italy Drip 2008 average
00.05
0.10.15
0.20.25
0.30.35
0.4
02/05/2008 12/05/2008 22/05/2008 01/06/2008 11/06/2008 21/06/2008 01/07/2008Date
so
il m
ois
ture
Average obsModelled
Observed average Soil N versus Simulated Soil
0
10
20
30
40
50
60
12/0
6/200
7
14/0
6/200
7
16/0
6/200
7
18/0
6/200
7
20/0
6/200
7
22/0
6/200
7
24/0
6/200
7
26/0
6/200
7
28/0
6/200
7
30/0
6/200
7
02/0
7/200
7
04/0
7/200
7
06/0
7/200
7
08/0
7/200
7
10/0
7/200
7
12/0
7/200
7
14/0
7/200
7
16/0
7/200
7
18/0
7/200
7
20/0
7/200
7
22/0
7/200
7
24/0
7/200
7
26/0
7/200
7
Date
So
il N
- m
g l
-1
Av Observed
Model
Simulated and Observed Soil N (mg l -1) Drip irrigated Tomato, 2008 plot 4
0
5
10
15
20
25
30
35
40
45
50
03/0
7/200
8
05/0
7/200
8
07/0
7/200
8
09/0
7/200
8
11/0
7/200
8
13/0
7/200
8
15/0
7/200
8
17/0
7/200
8
19/0
7/200
8
21/0
7/200
8
23/0
7/200
8
25/0
7/200
8
27/0
7/200
8
29/0
7/200
8
31/0
7/200
8
02/0
8/200
8
04/0
8/200
8
06/0
8/200
8
08/0
8/200
8
10/0
8/200
8
12/0
8/200
8
14/0
8/200
8
16/0
8/200
8
18/0
8/200
8
20/0
8/200
8
Date
So
il N
, m
g l
-1
Plot4
model
147
Page 148
Crete 2006 Drip Sub PRD, 10cm
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
27-Jun 02-Jul 07-Jul 12-Jul 17-Jul 22-Jul 27-Jul 01-Aug 06-Aug 11-Aug 16-Aug 21-Aug
Date
So
il M
ois
ture
, m
3m
-3
Crete 2006 Drip Sub PRD, 30cm
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
27-Jun 02-Jul 07-Jul 12-Jul 17-Jul 22-Jul 27-Jul 01-Aug 06-Aug 11-Aug 16-Aug 21-Aug
Date
So
il M
ois
ture
, m
3m
-3
148
Page 149
Crete 2006 Drip Sub PRD, 40cm
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
27-Jun 02-Jul 07-Jul 12-Jul 17-Jul 22-Jul 27-Jul 01-Aug 06-Aug 11-Aug 16-Aug 21-Aug
Date
So
il M
ois
ture
, m
3m
-3
Crete 2006 Drip Sub Full, 10cm
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
27-Jun 07-Jul 17-Jul 27-Jul 06-Aug 16-Aug 26-Aug
Date
So
il M
ois
ture
, m
3m
-3
149
Page 150
Crete 2006 Drip Sub Full, 30cm
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
27-Jun 07-Jul 17-Jul 27-Jul 06-Aug 16-Aug 26-Aug
Date
So
il M
ois
ture
, m
3m
-3
Crete 2006 Drip Sub Full, 40cm
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
27-Jun 02-Jul 07-Jul 12-Jul 17-Jul 22-Jul 27-Jul 01-Aug 06-Aug 11-Aug 16-Aug 21-Aug
Date
So
il M
ois
ture
, m
3m
-3
150
Page 151
Crete 2007 Drip Sub Full, 40cm
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
28-Apr 18-May 07-Jun 27-Jun 17-Jul 06-Aug 26-Aug
Date
So
il M
ois
ture
, m
3m
-3
Crete 2007 Drip Sub PRD, 40cm
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
28-Apr 18-May 07-Jun 27-Jun 17-Jul 06-Aug 26-Aug
Date
So
il M
ois
ture
, m
3m
-3
151
Page 152
Crete 2008 Drip Sub Full, 30cm
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
28-Apr 18-May 07-Jun 27-Jun 17-Jul 06-Aug 26-Aug
Date
So
il M
ois
ture
, m
3m
-3
152
