MODELLING MAXIMAL EVAPOTRANSPIRATION 

AND CROP COEFFICIENT

 

Yi Luo, Z. OuYang, Q. Zhao, Qiang Yu

Yucheng Comprehensive Experimental Station

Institute of Geographical Sciences and Natural Resources, the Chinese Academy of Sciences, Beijing, China

 

 

Abstract

 

A non-water-stress evapotranspiration (NWSE) model was established and validated with experimental data of maize site at hourly interval and was extended to estimate the daily maximal evapotranspiration (ETm) of winter wheat without further validation. It was found that evapotranspiration estimated by NWSE model was systematically higher than that estimated by Penman-Monteith formula, but the discrepancy diminished gradually as LAI increased. Crop coefficient was mathematically derived using the estimated ETm by NWSE model and Penman-Monteith formula. Surface soil wetness was set at different percentages of field capacity when applying NWSE model. Closest agreement was found between the experimentally obtained crop coefficient in Shandong province of China and the mean of the estimated values by NWSE model when surface soil wetness was set as 60% of field capacity. Meanwhile, it was found that obvious variation of crop coefficient exists according to the statistical analysis of the simulation results of 14 years. The average of variation and variation coefficient during the simulation period are 0.12 and 0.13 respectively. This demonstrates the necessity and feasibility of applying the proposed method to derive crop coefficient mathematically.

 

 

Introduction

 

Evapotranspiration data are of great importance in irrigation planning and scheduling as well as in water resource allocation. Evapotranspiration from a cropped soil is dependent not only on the meteorological conditions, but also on factors related to crop and to soil. In estimating crop evapotranspiration, the two-step combination method is often used by irrigation controllers, consultants and schedulers.

As the first step, reference evapotranspiration and crop coefficient are combined to estimate crop evapotranspiration under non-water-stress condition, defined as the maximal evapotranspiration hereafter, ETm 

                                                                                         (1)

where ETr is the reference evapotranspiration, Kc is the crop coefficient.

As the second step, water stress coefficient is introduced to account for the influence of water status in root zone upon evapotranspiration, i.e.

                                                                                           (2)

where ET is the actual crop evapotranspiration, Ks is the water stress coefficient.

Rewrite equation (1), crop coefficient is then defined as

                                                                                               (3)

Equation (3) forms foundation of experimental work of crop coefficient.

The two-step combination approach is widely adopted around the world[Y. Luo1] . With regard to difficulties existing in experimental work determining crop coefficient and selection of crops in applying the two-steps approach, the following work is focused primarily on the first step. Problems involved in the second step are beyond discussion of this paper. 

In China, Kc is determined experimentally according to definition given by equation (3). In the early 1980s, experimental works in defining Kc of winter and spring wheat, maize, paddy rice and other main crops produced in different climatic zones of China were undertaken extensively (Chen and Guo, 1986), in which about 200 experimental sites representing different climatic patterns were chosen and there were more than one thousand research personnel involved. The experimental work had lasted three years. As a result, contour map of Kc of the main crops was made and published. This has formed the basis of choice of values of crop coefficient in crop water management work to this day.  Experimental definition of Kc is costly, time-consuming and site specific.

Crop evapotranspiration consists of two parts: soil evaporation and crop transpiration. The ratio between these two component changes with canopy development, surface soil wetness, and weather conditions. Generally in China, soil evaporation and crop transpiration are not distinguished and are taken as a whole instead in defining Kc. So Kc is easily affected by surface soil wetness in transferring it from one situation to another. The Food and Agriculture Organization (FAO) proposed a crop coefficient scheme which divides Kc into two parts: basal crop coefficient (Kc with minimal soil evaporation) and soil evaporation coefficient. The soil evaporation part can be further modified according to surface soil wetness (Allen et al., 1996). This increases the flexibility of Kc to different rainfall and irrigation situations. Yet, this crop coefficient scheme has not been adopted in China.

Reference evapotranspiration incorporates the majority of the effects of changes in weather into the ETr estimate and hence Kc varies predominately with the specific crop characteristics and only a little with climate, this enables the transfer of standard values of Kc between locations and climates (Allen et al., 1996).  In transferring crop coefficient between and climates, it has been assumed that basal Kc and /or Kc for full canopy will be universally valid if the variation in weather elements is accounted for by reference evapotranspiration. Annandale and Stockle (1994) made a sensitivity analysis of fluctuations of crop evapotranspiration coefficients with weather elements to test the validity of this assumption. The analysis has shown that environmental factor do not affect latent heat exchange in the reference crop in the same way as they affect that of the target crop. Kc values of crops, that are taller and/or present a lower canopy resistance than the reference crop, are sensitive to changes in air temperature, air vapor density, wind speed, and also solar radiation, although to a lesser degree. So, cautions should be taken in transferring Kc values from one location or climate pattern to another.

The crop coefficient approach can be used to provide useful estimates of crop water requirements. However, as water becomes a more limiting resource and irrigation water application becomes more precise, crop water requirements will need to be estimated more accurately (Annandale and Stockle, 1994). Concerned about cost and time-consumption of crop coefficient experiment and variation of Kc with environmental elements, direct estimation of the maximal crop evapotranspiration without resorting to the two step approach of estimating actual crop evapotranspiration should be encouraged. Meanwhile, deriving crop coefficient mathematically as an alternative way of determining it experimentally is also an encouraging research topic.

Efforts have been made to derive crop coefficient mathematically. DUrso and Santin (1996) derived Kc by using Penman-Monteith formula to estimate both the maximal crop evapotranspiration and reference evapotranspiration. When calculating the maximal crop evapotranspiration, canopy resistance was set at its minimal representing the non-water stress condition. Annandale and Stockle (1994) simulated the crop coefficient under full canopy condition to investigate the fluctuation of crop coefficient with weather elements. They developed an energy-balance model to estimate latent and sensible heat flux over canopy. Again, canopy resistance was set at its minimal.

The crop coefficient is influenced by canopy development, which determines the partitioning of radiation into the fraction intercepted by the canopy and that reaching the soil surface. The energy reaching the soil surface is available for soil evaporation. During early growth, when canopy does not cover the soil completely and the amount of intercepted radiation is low, Kc is particularly sensitive to management factors such as irrigation frequency and method, and soil factors as hydraulic conductivity and water content near the soil surface. Penman-Monteith formula assumes that all radiation is available for canopy transpiration, and it cannot account for soil evaporation process. Thus, under partial canopy cover conditions, Penman-Monteith formula will very possibly overestimate crop transpiration when surface soil is very dry and underestimate the crop transpiration when surface soil is wet.

The objectives of this paper are twofold: (1) to establish a model for direct estimation of maximal evapotranspiration which can be used for any growth season of a crop; (2) to have a trial in deriving crop coefficient mathematically on the basis of modeled maximal evapotranspiration. As a comparison, Penman-Monteith formula will also be employed to estimate the maximal evapotranspiration and crop coefficient. In calculating crop coefficient, reference evapotranspiration will be estimated by Penman-Monteith formula as proposed by FAO.

 

 

NWSE Model Construction and Validation

 

NWSE model is constructed using framework of Shuttleworth (1985) depicted in Fig. 1. Radiation is partitioned into a fraction intercepted by crop canopy and one reaching soil surface in the following way:

                                                                                           (4)

                                                                        (5)

                                                                               (6)

where Rn is net radiation over canopy in Wm-2, Rnc is the fraction intercepted by canopy in Wm-2, Rns is the fraction reaching soil surface, in Wm-2, a is the extinction coefficient.

Figure1.  Graphical representation of the framework of NWSE model (adapted after Shuttleworth,1985)

Net radiation over the canopy is partitioned into total latent and sensible heat fluxes.

                                                                                             (7)

where LE is the total latent heat flux, L is the vaporization constant, E is the total evapotranspiration above canopy, and H is the total sensible heat flux.

LE and H can be expressed as

                                                                                      (8)

                                                                                       (9)

where ea and Ta are air vapor pressure and temperature at reference level respectively, eb and Tb  are air vapor pressure and temperature at top of canopy respectively, r is air density, Cp is iso-pressure specific heat capacity, ra is aerodynamic resistance from canopy top to the reference level. Substitute equation (8) and (9) into (7), we obtain

 

                                                                (10)

 

Radiation intercepted by canopy is available for latent and sensible heat flux from canopy.

                                                                                          (11)

where  LEc and Hc are latent and sensible heat flux from canopy respectively.

 LEc and Hc can be expressed in a way similar with equations (8) and (9) as

                                                                               (12)

                                                                                      (13)

where T1 is leaf surface temperature, r1 is leaf boundary layer resistance, rc is the bulk canopy resistance, es(T1) is saturated vapor pressure in leaf stomata at temperature T1 and can be given by

                                                                    (14)

Substitute equation (12)(13) into (11), we obtain

                                            (15)

Radiation reaching soil surface is available for latent, sensible and ground surface heat fluxes, i.e.

                                                                                   (16)

where LEs and Hs  are latent and sensible heat flux from soil surface, G is ground surface heat flux, negative when downward. LEs and Hs can be given by

                                                                           (17)

                                                                                      (18)

where T2 is surface temperature of soil, r2 is aerodynamic resistance from soil surface to top of canopy, rs is surface soil resistance, h2 is relative humidity of vapor in soil pore. h2 can given by

                                                                        (19)

where M and g and R are constantsM18.01510-3  kg/m3g=9.18  m/s2 R=8.314 J/molK, y  is soil matrix potential.

Substitute equation (17)(18) into(16), we obtain

                                              (20)

According to principle of mass conservation, following relationship can be derived

                                                                                        (21)

Substitute equation (6)(10)and (15) into (20), we can obtain

                                   (22)

Equations (8)(13)(19)and (22) form a closed equation set that consists of four undetermined variables eb, Tb, T2 and T1 . This is the so called NWSE modelResistance terms in NWSE model are detailed as follows.

Soil resistance rs

 Lin and Sun (1983) gave an empirical formula for soil resistance expressed as function of surface soil water content.

                                                                                       (23)

where qs is the saturated water content of surface soil, q is soil water content of the 5cm surface soil a, b1 and b2 are empirical constants with a=3.5, b1=33.5, b2=2.35.

 

Canopy resistance rc

The canopy resistance under condition of non-water stress was calculated after Allen et al. (1989):

                                                                                              (24)

where rsmin is the minimal leaf stomatal resistance, which is suggested by Monteith (1981) and Sharma (1985) as 100s/m.

 

Aerodynamic resistance ra

Assuming neutral stability conditions, ra can be computed as

                                                                  (4-25)

where Zm is the height of wind measurements, Zn is that of air temperature and humidity measurements, d is that of zero plane displacement of wind profile, Zom and Zon are the roughness length governing momentum transfer and heat and vapor transfers, k is von Karmans constant, and U is wind velocity measured at height Zm.

 

Boundary layer resistance of canopy leaf

 

Expression of conductance of canopy boundary layer can be given as

                                                                             (26)

where a=0.01ms-1/2u(Z) is the wind velocity at Z in canopy, WL is the mean leaf width. Wind profile in canopy can be given as

                                                                 (27)

where u(hc) is the wind velocity at top of canopy, hc is the crop canopy height, b is the  extinction coefficient of wind velocity in canopy.

Choudhury (1988) gave the average conductance in unit area of canopy as

                                                                               (28)

where L is the cumulated leaf area index  increasing from 0 at the top to LAI at the bottom of canopy[Y. Luo2] . Suppose uniform distribution of LAI from top to bottom of canopy, integrating equation (28) gives

                                             (29)

The boundary layer resistance of canopy can be obtained as

                                                                                               (30)

 

Aerodynamic resistance in canopy r2

Choudhury (1988) gave an expression of r2 as

                                                                                   (31)

where Zs is the roughness of surface soil, set as 0.01m, k(Z) is the momentum diffusion in canopy which is assumed as following formula in uniform canopy.

                                                                 (32)

where z is the extinction coefficient. By integration, r2 can be get finally as

                                      (33)

Equation (10), (15), (20) and (22), along with the resistance terms, form the NWSE model. The driving force input of NWSE model includes meteorological data at reference level, height, LAI and mean leaf width of the crop, and surface soil moisture content. State variables include surface soil temperature, average leaf temperature of canopy, temperature and vapor pressure of air in canopy. Latent and sensible heat fluxes from soil surface, canopy, and above canopy can be obtained from the solution of the model.

The established NWSE model was validated with meteorological data collected from maize field at hourly intervals. Diurnal courses of change in wet and dry bulb temperature of air, wind speed, and net radiation were recorded at reference level 2m above the ground every two hours. Leaf stomatal resistance and temperature were measured with Li-6200 every two hours during daytime. Three stems, three leaves at top, middle and bottom of each stem were measured each time. Canopy resistance was calculated by scaling the measured stomatal resistance of leaf up to canopy. Therefore, whether well watered or not, water stress was accounted for by the canopy resistance. Canopy temperature was taken as the average of leaf measurements. Surface soil water content was measured by gravitational sampling at ten oclock in the morning and four oclock in the afternoon. Van Genuchtens (1980) model was employed to derive soil matrix potential from the measured soil moisture content. Surface soil temperature and ground heat fluxes were measured every two hours day and night. LAI in experimental period was 4.5, and mean width of maize leaf was measured as 7cm. Height of maize was 185cm. The field experiment was carried out at Yucheng Comprehensive Experimental Station of Chinese Academy of Science, located in northwest plain of Shandong province, China.

Fig.2 shows the simulated and the observed canopy temperatures. They agree very well with each other during daytime. Because of the lack of observed canopy temperature during nighttime, no comparison was made.

 

Figure 2.  Simulated diurnal courses of changes in canopy temperature and measured

 

Fig.3 shows the comparison of the simulated and observed surface soil temperatures. Close agreement was got during nighttime. But large discrepancies were found with the maximal one of 4C during time around noon.

Figure3.  Simulated diurnal courses of changes in surface soil temperatures and measured

 

From the comparison shown in Fig. 2 and Fig. 3, it can be tentatively concluded that NWSE model and parameters taken in simulation are rational.

 

 

ETm Estimation with NWSE Model and Penman-Monteith Formula

 

NWSE model is extended to estimate daily ETm. As a comparison, Penman-Monteith formula is employed to estimate ETm as well. ETm estimated by NWSE model is called NWSE ETm hereafter, and ETm estimated by Penman-Monteith formula called P-M ETm.

A series of daily climatic record of 14 years collected from YCES was used to drive the computations with both NWSE model and Penman-Monteith formula. ETm estimation was performed for winter wheat growth season during greening and post-harvesting. LAI and height of winter wheat used in the NWSE model and Penman-Monteith formula were determined as shown in Fig.4 based on years of observation. The mean leaf width of winter wheat was taken as linear function of time from 0.2cm to 1.2cm when LAI was at its maximum, then remains unchanged to end of the simulation period. Surface soil moisture content used to drive the NWSE model was assumed at three levels, i.e., 50%, 60%, and 90% of field capacity. NWSE model was run at each level. In estimating ETm, daily average of ground heat flux is simply taken as zero. Canopy resistance is determined by equation (24).

Figure 4.  Changes in LAI and height of winter wheat

 

Fig.5 shows the NWSE ETm, transpiration (Ec) and soil evaporation (Es) at surface soil water content 60% of field capacity in 1986. ETm increased with time generally because of the increase in evaporative demand in this area. Ec increased in a way as ETm. However, soil evaporation decreased with time in this period.  At early stage, soil evaporation accounted for the main part of the total ETm and as time went on, the ratio of soil evaporation to ETm decreased very quickly as shown in Fig.6. Fig.6 gives the ratio of soil evaporation to ETm at two surface soil wetness levels and the ratio of radiation reaching soil surface to net radiation. Es/ETm changed with time in a way very similar with Rns/Rn. At later part of the season, Es/ETm increased a little as LAI decreased. Meanwhile, Es/ETm was influenced by surface soil wetness. The higher the surface soil water content, the larger was the ratio. This indicates that at the early stage of canopy development, crop coefficient may be very sensitive to soil factors, especially as soil water content.

 

Figure 5.  Maximal evapotranspiration ETm, transpiration Ec and soil evaporation Es estimated by NWSE model

Figure 6.  Simulated ratio of soil evaporation to the maximal evoptranspiration with NWSE model at different soil wetness levels

 

Fig. 7 shows the comparison of NWSE ETm and P-M ETm of 1986. In the early part, there was discrepancy between them. NWSE ETm was larger than P-M ETm. However, in the later part, agreement between them was very close. Although none of these ETm estimation schemes has been validated by experimental data, NWSE model seems more reasonable than Penman-Monteith formula at the early stage of canopy development, because NWSE model takes account of soil evaporation but Penman-Monteith formula assumes simply all net radiation is used by the canopy.

Figure 7.  Comparison of maximal evapotranspiration estimated by Penman-Monteith formula  and NWSE model with soil wetness at 60% of field capacity.

 

Difference between NWSE ETm and P-M ETm at early stage of canopy development is systematic as shown in Fig. 8. According to correlation analysis of 14 year of data, the linear regression coefficient gets as high as 0.95 with P-M ETm 7% lower than NWSE ETm. Difference between them happened mainly during the early canopy development stage. 

Figure 8.  Regression analysis of maximal evapotranspiration estimated by Penman-Monteith formula and NWSE model with soil wetness at 60% of field capacity

 

 

Crop Coefficient and Analysis

 

Crop coefficient was derived mathematically based on sequence of climatic record from YCES for winter wheat during growth season from greening to harvest. ETr was estimated by Penman-Monteith formula as proposed by FAO. ETm was estimated by both NWSE model and Penman-Monteith formula. Crop coefficient estimated by NWSE ETm called NWSE Kc, while that estimated by Penman-Monteith ETm called P-M Kc.

Fig. 9 and Fig. 10 show the scatter points of daily NWSE Kc and P-M Kc of 14 years respectively. Kc grows up gradually from low LAI values to high values, and then goes down slightly with decreasing LAI values. Meanwhile, over the years, Kc fluctuates in a fairly wide range. Considering the same crop growth pattern adopted in estimating Kc of different years, the fluctuation can be attributed to variation of weather elements among years. This indicates that variation of weather elements cannot be fully accounted for by reference evapotranspiration. So, caution should be taken in applying Kc values that were obtained from only a few years of experimental data.

Figure 9.  Scattering points of P-M Kc of 14 years

 

Figure 10.  Scattering points of NWSE Kc of 14 years

 

The mean value of NWSE Kc at different surface soil wetness levels and P-M Kc was calculated from the 14 years Kc data sets. To get a clear comparison among the NWSE Kc, P-M Kc, and the experimental Kc, polynomial fitting was applied to the mean of NWSE Kc and P-M Kc as shown in Fig. 11. At different surface soil wetness levels, clear difference in NWSE Kc occurred at early stage of canopy development. The higher the surface soil moisture, the higher the Kc values were. P-M Kc was obviously smaller than NWSE Kc at an early stage of canopy development. But the difference diminished gradually as LAI increased. It can be seen that NWSE Kc at different surface soil wetness levels and P-M Kc agreed with each other very closely. The reason is that soil evaporation plays only minor role under full canopy, ETm values estimated by NWSE model and Penman-Monteith formula are very similar. Month average of Kc of Shandong province of winter wheat was shown in Fig. 11.  Comparing all Kc curves, it can be found that the NWSE Kc value obtained with surface soil wetness maintained at 60% of field capacity is closest to the experimentally derived one. In experiment of crop coefficient, soil moisture regime in root zone is delicately maintained between 60%~90% of field capacity. Because of fast depletion, 60% of field capacity or so of surface soil water content occurs much more frequently than 90%. Therefore, the results of 60% of field capacity seem more rational when taken to simulate the crop coefficient. Meanwhile, influence of surface soil moisture on Kc indicates that the basal crop coefficient concept is a good policy for distinguishing the effects of soil factor and crop factor in crop coefficient. The traditional approach of taking transpiration coefficient and evaporation coefficient as a whole should be revised in China. Caution should be taken in transferring the experimental Kc to situation different from the one at which the Kc was derived. For the obvious disadvantages of Penman-Monteith formula in cases where LAI is small, it is not suitable for ETm modeling for period with low LAI, and also the mathematical derivation of crop coefficient.

As shown in Fig. 8 and Fig. 9, there are obvious variations of Kc among years. Based on the simulated NWSE Kc, statistic analysis was performed. Fig. 12 shows standard deviation (solid line) and variation coefficient (open circle) of NWSE Kc during the simulation period. The average standard deviation during the simulation period is 0.12, and the variation coefficient 0.13. The statistical results are very close to that of experimental data analysis of Chen and Guo (1986).

 

 

 

Figure 11.  Comparison between NWSE Kc and P-M Kc, and the experimental one

 

Figure 12.  Standard deviation and variation coefficient of NWSE Kc when surface soil moisture is set as 60% of field capacity

 

 

Conclusion Remarks

 

A non-water stress evapotranspiration model was established with soil evaporation considered It was validated with field experimental data for maize at hourly scale. The values of canopy resistance in validation were obtained by scaling up the measured leaf stomatal resistance to canopy in validation. The simulated surface soil temperature and leaf canopy temperature agreed closely with the measured ones, indicating that the model is reasonable.

The established model was extended to estimate the daily values of the maximal evapotranspiration of winter wheat. Effects of surface soil wetness on soil evaporation and total evapotranspiration were analyzed. The results showed that the effects at early stage of canopy development were obvious. The ratio of soil evaporation to total evapotranspiration decreased with LAI negative-exponentially, and was larger with higher surface soil wetness. Penman-Monteith formula was also employed to estimate the maximal evapotranspiration. The values of canopy resistance for both the NWSE model and Penman-Monteith formula were set at the minimal leaf stomatal resistance. The ETm values estimated by NWSE model and Penman-Monteith formula were compared. It was found that systematic discrepancy existed between the ETm values estimated by the different two methods at stage of underdeveloped canopy.  The values of ETm estimated by the NWSE model were higher than that by Penman-Monteith formula. But the discrepancies diminished as LAI increased. It was tentatively concluded that the NWSE model was rational than Penman-Monteith formula in cases where the values of LAI were low, because the former took soil factors into account, but the later assumed that all net radiation was used for canopy.

Crop coefficients were mathematically derived with the ETm values estimated by the NWSE model and the Penman-Monteith formula. Reference evapotranspirations were calculated with Penman-Monteith formula as proposed by FAO. Crop growth pattern was taken to be the same when applying NWSE model and Penman-Monteith formula for all the climatic records of 14 years. Surface soil wetness was set at different levels of field capacity when NWSE model used. It was found that crop coefficient varied among years. This was attributed to variation of weather elements among years. Difference ocured between different surface soil wetness levels and different estimation method. But this difference diminished as values of LAI increased. Comparing the estimated Kc values  with the experimentally derived monthly average of Kc values, it was found that the closest agreement was obtained between the experimental one and the one estimated by NWSE model when surface soil wetness was set at 60% of field capacity. This was because assuming surface soil water content of 60% of field capacity was very similar with the case of experiment. Penman-Monteith formula estimated crop coefficient was far small than the experimental one at an early stage of canopy development, because it reduced the effects of soil factors on evapotranspiration.

There were obvious variations of crop coefficient over years during the simulation period. The standard deviation and variation coefficient was found to be very close to the ones reported by Chen and Guo (1986) on the basis of experimental data.

It is concluded that the effects of variations of weather elements cannot be fully accounted for by reference evapotranspiration in crop coefficient estimation. Influences of soil factors and variations of weather elements upon crop coefficient give a caution about transferring experimentally determined crop coefficient to locations different from that they were originally derived at. 

 

 

Acknowledgments

 

This paper was supported by the National Sciences Foundation of China with grant no 49890330.

 

 

Appendix I. Reference

 

Allen, R. G., M. E. Jensen, J. L. Wright, R. D. Burman, 1989. Operational estimates of evapotranspiration. Agron. J., 81: 650-662

Allen, R. G., Smith, M., Pruitt, W. O., Pereria, L. S., 1996. Modifications to the FAO crop coefficient approach. Evapotranspiration And Irrigation Scheduling, Proceedings Of The International Conference, Nov. 3-6, 124-132

Annandale, J. G., Stockle, C. O., 1994. Fluctuation of crop evapotranspiration coefficients with weather: a sensitivity analysis. Irrigation Science, 15:1-7

Monteith, J. L., 1981. Evaporation and surface temperature. Quart. J. Roy. Meteorol. Soc., 107: 1-27

Sharma, M. L., 1985. Estimating evapotranspiration. Advances in irrigation, D. Hillel, Ed., Academic Press, Inc., 213-281

Chen, Y. M., Guo, S. L., 1986. Contour maps of water requirement of main crops in china (In Chinese). Chinese Agricultural Science Press.

Choudhury, B. J., Monteith, J. L., 1988. A four-layer model for the heat budget of homogeneous land surface. Q. J. R. Met. Soc., 114, 373-398

DUrso, G., Santini, A., 1996. A remote sensing and modeling integrated approach for the management of irrigation distribution systems. Evapotranspiration and Irrigation Scheduling, Proceedings of the International conference. Nov 3-6, 1996. p435-441

Lin, J. D., S. F. Sun, 1983. Moisture and heat flow in soil and theirs effects on bare soil evaporation. (In Chinese), Trans. Water Conservancy. 7:1-7.

Shuttleworth, W. J., Wallace, J. S., 1985. Evaporation from sparse crops - an energy combination theory. Quart. J. R. Met. Soc., 111, 839-855

Van Genuchten, M. Th., 1980. A closed-form equation for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J.. 44: 892-898

 

 

Appendix II. Notation

 

The following symbols are used in this paper:

r1                         =   boundary layer resistance of canopys/m;

r2                =   aerodynamic resistance in canopys/m;

ra                =   aerodynamic resistance from top of canopy to reference  levels/m;

Rn                       =   net radiationMJ/sm2

Rnc               net radiation intercepted by canopy, MJ/sm2;

Rns                     =   net radiation reaching to soil surface, MJ/sm2;

rs                 =   soil resistances/m;

T1                       =   leaf surface temperature;

T2                       =   surface soil temperature;

Ta                       =   air temperature at reference level;

Tk                       =   absolute temperatureK;

Wf                     =   field capacitym3/m3;

WL                    =   mean leaf widthm;

Zom             =   roughness length of momentum transfer, m;

Zon                     =   roughness length of heat transferm;

Zs               =   roughness length of soil surfacem;

D                =   slope of saturated vapor pressure curve, Kpa/C;

g                =   phycrometer constant, Kpa/C;

q                =   surface soil moisture contentm3/m3;

qs                      =   saturated soil moisture contentcm3/cm3;

r                =   air densityg/cm3;

y                =   soil matrix potential, m;

LAI              =  leaf area index;

hc =  crop height, m;

Ec =  crop transpiration, mm;

Es   =  soil evaporation, mm;

Hc  =  canopy heat flux, MJ/sm2;

Hs  =  soil heat flux, MJ/sm2;


 [Y. Luo1]Reference indicating the application of this approach in china and other countries

 [Y. Luo2]from top to bottom or bottom to top?