NRM – Module 8
Lecture notes

Temporal dynamics in
weather and its influence on CO_{2} assimilation in crops

October 2004, V. Venus 
CONTENTS
Chapters
Figure 1.
Observed and modeled average daily progression of air temperature.
Figure 2, Modeled
daily progress of incident global radiation (daily total of 16 MJ m2 d1).
Figure 3. Generic
AMAXtotemperature response curves (Versteeg and van Keulen, 1986).
Daily totals or daily averages are convenient to work with and provide a first but rough impression of the weather conditions under which a given crop develops. For some plant processes this daily time interval not a bad match. For example, the development rate of a crop can be estimated quite well by making use of the average air temperature. After all, it is not possible to measure accurately the stages of development from day to day, which means that the effect of temperature must be integrated over at least a week. It is then usual to make use of the concept of 'temperature sum' (TSUM) or 'degree days'. This value is calculated as the integral of the value with which the (average) temperature exceeds a certain lower threshold value (e.g. 5 °C), below which no development occurs at all. This procedure implicitly assumes a linear response above the lower threshold.
However, for a number of reasons these summation methods may present a distorted image of the actual situation. Some of the most important reasons are:
Because of these kinds of effects, it is often necessary to take account of the daily progress of the weather conditions when modeling crop growth.
At a standard height (1.5 m), the air temperature exhibits a characteristic daily cycle with a minimum at sunrise and a maximum at 13.30 h (solar time, Figure 3.3). In dry climates this maximum tends to occur a little later. Of course, deviations from this average pattern can occur as a result of, for example, the passage of a cold front. However, on average the daily cycle can be described reasonably well when the minimum and maximum temperatures are known. The accuracy with which this should be done depends on the purpose.
If the mutual position of the radiation and temperature wave is not important (little interaction between temperature and radiation), a simple linear progression between a maximum and a minimum, e.g. at 13.30 h in the afternoon and 6.00 h in the morning, would be sufficient. A further improvement is to round off the points by using a simple sine curve.
If we require a reasonable representation of both the phase and the form, we will have to use a sinusoidal progression during the daytime and a decreasing exponential curve at night (Parton and Logan, 1981; Geurts and Engelen, 1983). The transition from one curve to the next takes place at sunrise (minimum temperature) and sunset. The section in the period of light (daytime) follows practically one sine wave in time according to:
T_{a} = T_{min} + (T_{max}  T_{min})
• sin( π (t_{h}_{ } 12 + d/2) / (d + 2p) ) [^{°}C] (3.9)
Where:
T_{a} is the air temperature (12  d(2 < th < 12 + d12) (°C),
T_{min}_{ }the daily minimum temperature (^{°}C),
T_{max} the daily maximum temperature (^{°}C),
d the daylength (Eqn 3.6) in hours,
t_{h}_{ } the time in hours (solar time), and
p time duration between solar noon and maximum temperature (h).
At sunrise (th = 12  d/2), the air temperature equals Tmin. The maximum temperature is reached when th equals 12 +p h solar time. The default value for p is 1.5 h. The sinusoidal curve is followed until sunset. Then a transition takes place to an exponential decrease, proceeding to the minimum temperature of the next day. To plot this curve correctly, we first need the starting point, the temperature at sunset (Tsset). This can be found by substituting the correct time, 12 + d/2, in Eqn 3.9. The equation for the nocturnal temperature progression is now:
T_{a} = T_{min}^{
}T_{sset} • exp(n /TC) + (T_{sset}  T_{min}) •
exp( (t_{h}^{} T_{sset})
/TC) (3.10)
1  exp(n /TC)
In this equation, n stands for the duration of the night, i.e. 24  d. The value of the nocturnal time coefficient (TC) is approximately 4 hours.
Figure 1. Observed and modeled average daily progression of air temperature.
Legend: observed air temperature for the months December, March, June and September in De Bilt, The Netherlands is indicated by the solid lines. The broken line is the result of Eqns. 3.9 and 3.10 (Kropff and Laar, 1993).
Listing 3.3, Appendix I shows the procedure for reconstructing the diurnal course of ambient air temperature based on daily totals.
Light energy directly/indirectly influences many secondary plant processes. A list of these processes includes: plant growth, seedling regeneration, vertical structure and crown shape of forest stands, leaf morphology, and uptake and emission of trace gases that participate in biogeochemical cycling and atmospheric chemistry.
The amount of solar energy at the outer extremity of the atmosphere varies with the latitude of the site and the time of year. The daily progress of the incident global radiation has a sinusoidal shape which varies with the seasons (see Figure_4). The global radiation (S) from the sun (roughly between 350 and 2000 nm wavelength) incident on the earth's surface is the product of the incident radiation outside the atmosphere (So) and the atmospheric transmissivity (Ta) which is obviously very dependent upon the degree of cloudiness (see Figure_3).
Figure 2, Observed daily progress of incident diffuse and global radiation (daily total of 9.76 MJ m2 d1).
In application of radiation modeling to satellite data, incident values need to be interpreted to arrive at daily integrated values and visa versa in order to make rigid model input requirements commensurate with scant satellite observations. Listing 3.4, Appendix II shows the procedure for reconstructing incident global radiation based on a daily total. Based on this model the daily progress of incident global radiation has been reconstructed for Julian Day (JD) 110, 1988 as depicted in Figure 1.
Figure 3, Modeled daily progress of
incident global radiation (daily total of 16 MJ m2 d1).
Table 1, Distribution of Solar energy by Waveband (Monteith, Unsworth et al.).
Waveband (Nm) 
Energy (%) 
0300 
1.2 
300400,
ultraviolet 
7.8 
400700,
visible/PAR 
39.8 
7001500, near
infrared 
38.8 
1500 to
infinity 
12.4 
Approximately half the total global
radiation (R_{g}) is part of the photosynthetic
photon flux density (PPFD) ranging from 0.38 to 0.71 µm, better known as photosynthetically active radiation (PAR). It provides the
energy for photosynthesis. Light in this visible waveband contains energy
across the primary colors of purple, blue, green, yellow, orange, red. In
contrast to the indicative values in the abovelisted
table, the amount of solar radiation in this band is not constant and
varies between 21 and 46 per cent.
The spectral quality of light incident to the ground is affected by selected absorption by water vapor gases and aerosols. Water vapor is a strong absorber of light with wavelengths of 100, 1400, 1600 and 1900 nm. Ozone absorbs ultraviolet light (λ < 300 nm), CO_{2} absorbs strongly in the 2750 and 4250 nm bands and oxygen absorbs 690 and 760 nm sunlight (Bonhomme, 1993). The spectral quality of sunlight is mainly determined by the direct/diffuse fraction of sunlight and the solar elevation angle. As a quick rule of thumb, the PAR:Rg, for total radiation, is rather conservative, being on the order of 0.46 to 0.50 (these computations assume that PAR conversion factor between quanta and energy is 4.6 umol quanta per J) In Figure 4, the zonal distribution of a five year mean PAR values, as well as the corresponding conversion factors (CF), are illustrated.
Figure 4, Zonal distribution of five year averaged PAR (19831988), and corresponding conversion factors (Pinker and Laszlo, 1992).
Ross (1975) provides an algorithm for the conversion factor between PAR and Rg as a function of the direct and diffuse components (this algorithm is for daily integrated fluxes of solar energy):
On a clear day with 10% diffuse radiation, f_{par:rg} is 0.438. On a cloudy day with 90% diffuse radiation, f_{par:rg} is 0.582.
On an hourly basis, the PAR:Rg ratio, for direct radiation, ranges from 0.2 to 0.43 as the solar elevation (Y) goes from 0 to 40 degrees. It then remains relatively constant with higher elevation angles. For diffuse radiation, this ratio ranges from about 0.6 to 0.75. Analytically, these functions take the following 3^{rd} order polynomial form:
ƒ PAR_{direct}:R_{direct}_{ }= 0.1714 Y^{3}  0.5917 Y^{2} + 0.6535 Y + 0.2002_{}
ƒ PAR_{diffuse}:R_{diffuse}_{: }= 0.1118 Y^{3}  0.3859 Y^{2} + 0.4262 Y
+ 0.6001
where Y
is the solar elevation angle in radians.
The photosynthetic activity of a leaf is a function of the light intensity and the maximum rate of assimilation (AMAX).
At canopy level, the daily gross CO_{2} assimilation (DTGA, kg CO_{2} ha^{1} d^{1}) can be approximated by integrating the abovementioned instantaneous leaf processes at different heights in the canopy at different times of the day. DTGA is
Light
intensity
Light use efficiency at low light intensity (EFF) decreases by only 1% for every degree of temperature increase in C3 plants, and even less in C4 plants. For practical purposes EFF is assumed constant with a value of some 0.5 kg ha1 h1/J m2 s1 (de Wit, Goudriaan et al., 1987).
Maximum
rate of assimilation
The maximum rate of assimilation, AMAX (in kg ha1 h1), is codetermined by leaf temperature and the CO_{2 }assimilation capacity at light saturation (AMX, kg CO_{2} ha^{1} h^{1}). AMX varies with crop species (and even cultivars) and is much greater for C4 crops (AMX = 88) than for C3 crops (AMX = 27  50) because of enzymatic differences. Whether stomata are active or inactive (in assimilating CO_{2}) depends also on the nitrogen content of a leaf, which is indirectly related to the age of a leaf.
Influence
of temperature on the CO2 assimilation capacity at light saturation
The CO_{2 }assimilation capacity at light saturation AMAX (in kg ha1 h1) is illustrated by the figure below and strongly temperature dependent:
Figure 5. Generic AMAXtotemperature response curves
(Versteeg and van Keulen, 1986).
Legend: I = C3 crops in cool and temperate climates; II = C3 crops in warm climates; III = C4 crops in warm climates; IV = C4 crops in cool climates.
μ
These highly simplified AMAXtotemperature (AMAXTEMP) response curves can be analytically solved and for instantaneous processes the following (polynomial) approximations hold:
ƒ AMAXTEMP_{I }: = 0.0046 T_{c}^{2}
+ 0.1646 T_{c}^{ }  0.4506
ƒ AMAXTEMP_{II
}: = 0.0024 T_{c}^{2} + 0.1364 T_{c}^{
}  0.9376
ƒ AMAXTEMP_{III}:
= 6E05 T_{c}^{3} 
0.008 T_{c}^{2} + 0.3229
T_{c}_{ }  2.9443
ƒ AMAXTEMP_{IV
}: = 2E06 T_{c}^{4}
+ 0.0003 T_{c}^{3}  0.0152 T_{c}^{2} + 0.3525 T_{c}  1.996
Where T_{c}_{ }is the canopy temperature in Celsius.
Influence
of plant and leaf age on the maximum rate of assimilation
The photosynthetic capacity of individual
leafs and plants are also affected by their age. For maize (Zea Mays L.), Dwyer and Stewart (1986) showed that as leaf age increased past full leaf expansion, photosynthesis decreased and the curves flattened to approach an asymptote at lower
irradiance levels. Linear regressions of photosynthesis at irradiance (I) = 2000 [mu] E m2 s1 against plant and
leaf age had high correlations (r = 0.91 and 0.92, respectively). Nitrogen (N_{2})
also plays an important role in enzymatic processes by which plants are able to
extract CO_{2} from the atmospheric boundary layer. This ageing process
can be explained by the efficiency by which nitrogen is redistributed; over the
life duration of a leaf this process involves costs that are higher than the
benefits in increased photosynthesis (Field,
1983). The effect of ageing of
the canopy is introduced by a multiplication factor (AMAXRDS) defined as a function of the development
stage. After emergence this factor increases from 0 to 1 as leafs are fully expanded. Hereafter,
AMAXRDS
decreases gradually
until a minimum of 0,5 is attained. In other words, the
moment when AMAX attains
a less then optimal value is determined by the rate at which a given crop
develops as determined by the weather conditions until the moment of simulation.
This can take the following form:
AMAX = AMX * AMAXTEMP * AMAXRDS
Exercise:
sensitivity analysis
In the upscaling of crop growth simulation using RS we would like to prioritize our activities by identifying the most important driving and/or state variable within a model of crop growth. Your task is to investigate the sensitivity of two versions of the same CO2 assimilation model, both using a 3point Gaussian integration of the canopy but one for instantaneous and the other for describing daily integrated CO2 assimilation processes. This sensitivity analysis will reveal which of the driving variables (temperature, PAR, etc.) or state variables (LAI) are most ‘important’ and at what timestep remote sensing should be able to provide estimate of this driving variable.
Input
data: primary and derived parameters
The Excel worksheet “input data.xls” lists both the instantaneous (green columns), as well as the derived daily total (red colums) radiation measurements. The column “Qs_mast” contains the Global short wave radiation, which is the incident short wave radiation (radiation between 0,2 and 3 µm) and comprises the direct and diffuse components. It is measured using a Pyranometer. “Qdiff_mast” contains the diffuse short wave radiation, which is only the diffuse component of the incident short wave radiation. It also is measured using Pyranometer, but now combined with a shadow ring. The Direct short wave radiation can be inferred by subtracting the diffuse component from the Global short wave radiation.
Model:
instantaneous and daily integrated CO2 assimilation
For daily integrated CO2 assimilation use the “DAYASS.exe”.
For instantaneous CO2 assimilation use the “INSTASS.exe”.
REFERENCES
Bonhomme, R. (1993). "The solar
radiation: characterization and distribution in the canopy." Crop Structure and
Light Microclimate. INRA.: 1728.
de Wit, C. T., J. Goudriaan, H. H. van Laar, F. W. T. Penning de Vries, R.
Rabbinge, H. van Keulen, W. Louwerse, L. Sibma and C. de Jonge (1987). Simulation of assimilation,
respiration and transpiration of
crops. Wageningen,
Pudoc.
Dwyer, L. M. and D. W. Stewart (1986). "Effect of leaf age
and position on net photosynthetic rates in maize (Zea
Mays L.)." Agricultural and Forest Meteorology 37(1): 2946.
Field,
C. (1983). "Allocating leaf nitrogen for the
maximization of carbon gain: leaf age as a control on the allocation
program." Oecologia 56(23):
341347.
Geurts, H. A. M. and A. F. V. v. Engelen (1983). Een rekenmodel dat het
verloop van de temperatuur over een etmaal berekent uit drie termijnmetingen
van de temperatuur. De Bilt, Knmi.
Kropff, M. and H. H. V. Laar (1993). CropWeed Interactions. [S.l.], C. A. B. Internat.
Monteith, J. L., M. H. Unsworth and A. Webb "Principles of environmental
physics."
Quarterly journal of the Royal Meteorological Society,
Vol. 120, No. 520 (1994), p. 1699.
Parton, W. J. and J.
A. Logan (1981). "A model for diurnal variation in soil and air
temperature." Agricultural Meteorology 23: 205216.
Pinker, R. T. and I. Laszlo,1992b. Global distribution of photosynthetically active radiation as observed from satellites. J.Climate, 5, 5665.
Ross, J.
(1975). "Radiative transfer in
plant communities." Monteith J.L., ed. Vegetation and the atmosphere. London:
Academic Press.: 1355.
Versteeg, M. N. and H.
van Keulen (1986). "Potential crop
production prediction by some simple calculation methods, as compared with
computer simulations." Agricultural Systems 19(4): 249272.
APPENDIX I Listing 3.3, Diurnal course of temperature
TITLE Exercise
3.3, Diurnal course of temperature (L33.FST)
INITIAL
CONSTANT PI =
3.1415926 PARAM LATT = 52.
FUNCTION TMAXTB
= 150.,20., 151.,22., 153., 21., 154., 21.7,
155.,25., 156.,25.
FUNCTION TMINTB
= 150.,12., 151.,13., 153., 13., 154., 12.,
155.,13., 156.,13.
* RAD converts from degrees into
RAD
= PI/180.
* Sine and cosine of latitude:
SINLAT = SIN(RAD*LATT)
COSLAT = COS(RAD*LATT)
* Maximal sine of declination:
SINDCM =
SIN(RAD*23.45)
TIMER STTIME = 151., FINTIM = 155., PRDEL = 0.04, DELT = 0.04
PRINT DAYL,
TEMP, HOUR, TMIN, TMINA, TMAXB,TMAX
DYNAMIC
• Sine and
cosine of declination (Eqns 3.4, 3.5): SINDEC =
SINDCM*COS(2.*PI*(TIME+10.)/365.)
COSDEC = SQRT(l.SINDEC*SINDEC)
The terms A and
B according to Eqn 3.3: A = SINLAT*SINDEC
B = COSLAT*COSDEC
Daylength according to Eqn
3.6:
DAYL = 12.*(1.+(2./PI)*ASIN(A/B)
)
HOUR = MOD(TIME,1.)*24.  hour/24.
DATE = MOD(TIME, 365.)
TMAXB = AFGEN(TMAXTB, DATEi.) TMAX = AFGEN(TMAXTB, DATE) TMIN = AFGEN(TMINTB, DATE) TMINA =
AFGEN(TMINTB, DATE+1.)
CALL TEMPS
(TMAXB,TMIN,TMAX,TMINA,DAYL,HOUR,TEMP)
TRANSLATIONGENERAL
DRIVER='RKDRIV' END
STOP
SUBROUTINE
TEMPS (TMAXB,TMIN,TMAX,TMINA,DAYL,HOUR,
$ TEMP)
IMPLICIT REAL
(AZ)
PARAMETER
(PI=3.14159, TC=4., P=1.5)
C Errors and warnings
IF (HOUR.LT.O.)
STOP 'ERROR IN FUNCTION TEMP: HOUR < 0'
IF
(HOUR.GT.24.) STOP 'ERROR IN FUNCTION TEMP: HOUR > 24'
IF
(TMIN.GT.TMAX) STOP 'ERROR IN FUNCTION TEMP: TMIN > TMAX'
SUNRIS =
12.0.5*DAYL SUNSET = 12.+0.5*DAYL
IF
(HOUR.LT.SUNRIS) THEN
C Period A: Hour between midnight and sunrise
TSUNST = TMIN+(TMAXBTMIN)*SIN(PI*(DAYL/(DAYL+2.*P)))
NIGHTL =
24.DAYL
TEMPI =
(TMINTSUNST*EXP(NIGHTL/TC)+
$ (TSUNSTTMIN)*EXP((HOUR+24.SUNSET)/TC))/
$ (1.EXP(NIGHTL/TC))
ELSE IF
(HOUR.LT.12.+P) THEN
C Period B: Hour between sunrise and normal
time of TMAX
TEMPI = TMIN+(TMAXTMIN)*SIN(PI*(HOURSUNRIS)/(DAYL+2.*P))
ELSE IF
(HOUR.LT.SUNSET) THEN
C Period C: Hour between normal time of TMAX
and sunset
TEMPI = TMINA+(TMAXTMINA)*SIN(PI*(HOURSUNRIS)/(DAYL+2.*P))
ELSE
C Period D: Hour between sunset and midnight
TSUNST = TMINA+(TMAXTMINA)*SIN(PI*(DAYL/(DAYL+2.*P)))
NIGHTL =
24.DAYL
TEMPI =
(TMINATSUNST*EXP(NIGHTL/TC)+
S (TSUNSTTMINA)*EXP((HOURSUNSET)/TC))/
$ (1.EXP(NIGHTL/TC))
END IF
TEMP = TEMPI
RETURN
END
APPENDIX II Listing 3.4, Diurnal course of radiation
TITLE Exercise
3.8, Diurnal course of radiation (L34.FST)
INITIAL
CONSTANT
PI = 3.1415926
PARAM LATT
= 52.
* Daily global radiation in MJ/m2:
PARAM RDSUM
= 9.75E6
* RAD converts from degrees into
RAD
= PI/180.
* Sine and cosine of latitude:
SINLAT = SIN(RAD*LATT)
COSLAT = COS(RAD*LATT)
* Maximal sine of declination:
SINDCM =
SIN(RAD*23.45)
TIMER STTIME = 229.; FINTIM = 230.; PRDEL = 0.01; DELT = 1.
PRINT S, HOUR,
DAYL
DYNAMIC
* Sine and cosine of declination (Eqns 3.4, 3.5):
SINDEC = SINDCM*COS(2.*PI*(TIME+10.)/365.)
COSDEC = SQRT(1.SINDEC*SINDEC)
* The terms A and B according to Eqn 3.3:
A
= SINLAT*SINDEC
B
= COSLAT*COSDEC
* Daylength
according to Eqn 3.6:
DAYL
= 12.*(1.+(2./PI)*ASIN(A/B) )
HOUR
= AMOD(TIME,1.)*24.
* Sine of solar height:
SINB
= A*B*COS(2.*PI*(HOUR12.)/24.)
* Integral of sine of solar height (Eqn 3.7):
SININT = A*DAYL + (24.*B/PI)*COS((PI/2.)*((DAYL/12.)1.))
* Solar constant fluctuations (W/m2):
SC
= 1367.*(1.+0.033*COS(2.*PI*(TIME10.)/365.))
* Daily average atmospheric transmissivity:
ATMTR
= RDSUM/(SININT*3600.*SC)
* Incident global radiation in W/m2:
S
= MAX(0.,ATMTR * SC * SINB)*10.
TRANSLATION_GENERAL
DRIVER='RKDRIV'
END
This is termed "photosynthetic photon flux density" (PPFD) or "photosynthetically active radiation" (PAR) and is given in units of µmoles photons/m2 of surface area/s. You may also see reference to µEinsteins/m2/s, which is equivalent because an Einstein is a mole of photons.
Measurements of the temperature relations of plants require a measure of the energy content of the incident radiation, rather than the PPFD. The pyranometer is filtered to measure the total energy content (Watts/m2) of incident shortwave radiation (350  2000 nm). The relationship between the energy content and photon content of radiation depends upon the spectral distribution of the light source in question. Thus, appropriate conversions must take in account what light source is being measured (see table below). The same goes for conversions of lux (commonly seen in older literature) to PPFD.
Approximate conversion factors for various light sources for photosynthetically active (visible) radiation (400700 nm).
Light source
Daylight Metal Sodium Mercury White Incand.
Halide (HP) fluor.
To convert Multiply by
W m2 (PAR) to 4.6 4.6 5.0 4.7 4.6 5.0
µmol m2 s1 (PAR)
klux to 18 14 14 14 12 20
µmol m2 s1 (PAR)
klux to W m2 (PAR) 4.0 3.1 2.8 3.0 2.7 4.0
data courtesy of LiCor Co.,