Modelling surface resistance from climatic variables?
Isabel Alves
*, Luis Santos Pereira
Department of Agricultural Engineering, Technical University of Lisbon, Tapada da Ajuda, 1399 Lisbon, Portugal
Accepted 15 February 1999
Abstract
For the Penman±Monteith equation to be used to predict crop evapotranspiration in a one-step approach, methodologies for determining surface resistance (rs) must be available. One usual
approach to the modelling ofrsis to compute it by inverting the Penman±Monteith equation and
then relate it to the most important environmental variables (radiation, temperature, vapour pressure deficit) using the multiplicative model of Jarvis. In this paper, some results obtained for lettuce are presented to illustrate the pitfalls of this approach. It is shown that the same environmental variables and the same functional forms that are used in the Jarvis model are already considered when calculatingrsas the residual term. One cannot thus expect to get a better insight on the behaviour of rswith the multiplicative model. Also, asrsincludes information on the transport conditions inside
the canopy and thus, is dependent on wind speed (or, indirectly, on the aerodynamic resistance), procedures that only contemplate stomatal functioning may be not adequate. The interactions betweenrsand latent heat flux are also discussed and indicate that future studies should be focused
on the determinism and quantification of the energy partitioning.#2000 Elsevier Science B.V. All rights reserved.
Keywords: Evapotranspiration; Penman±Monteith equation; Aerodynamic resistance; Surface resistance
1. Introduction
Knowledge of crop evapotranspiration is necessary in many different situations. In particular, planning and management both at the project and farm level rely on accurate estimates of crop water needs. Penman (1948) was the first to combine the energy balance with the expressions that describe heat fluxes to derive a method to estimate the vapour
* Corresponding author. Tel.: +351-1-363-8161; fax: +351-1-362-1575
E-mail address:isabelmalves@isa.utl.pt (I. Alves)
flux from a free water surface and, later, a vegetated surface. Monteith (1965), in an attempt to better characterise water loss by plants, introduced some modifications, resulting in the well known Penman±Monteith equation. Despite the better physical formulation of the Penman±Monteith approach, the FAO publication by Doorenbos and Pruitt (1977) based on the original Penman equation was then widely adopted together with crop coefficients, while the Monteith equation was regarded mostly as a theoretical rather than practical approach.
Only in recent years, the Penman±Monteith equation has gained a renewed interest to predict crop evapotranspiration in a one-step approach, which could better represent crop water loss than the traditional approach, based on reference evapotranspiration and a crop coefficient (Allen et al., 1999). But to be used predictively, methodologies for determining the aerodynamic resistance (ra) and the bulk surface resistance (rs) must be available. These variables have already been parameterized for an hypothetical reference crop closely resembling grass of uniform height, actively growing and not short of water and reference evapotranspiration may now be calculated using the Penman± Monteith equation (Allen et al., 1994a, 1999). However, methodologies still need to be developed regarding other crops.
2. Theoretical background
Aerodynamic resistance is usually evaluated with an expression that is derived from turbulent transfer and assuming a logarithmic wind profile (see Thom, 1975 or Monteith and Unsworth, 1990), with the form (for neutral conditions):
ra
ln zÿd=zOHln zÿd=zO
k2u z
(1)
wheredis zero plane displacement height (m),zois roughness length for momentum (m),
zoHis roughness length for heat (m),kis von Karman constant anduzis wind speed (m/s)
at heightz(m), also where the measurements of temperature and humidity are made. For its practical application, necessary parameters (d and zo), if not measured, can be
estimated (Brutsaert, 1982; Perrier, 1982; Shaw and Pereira, 1982). However, as discussed in Alves et al. (1998), the assumption that heat and vapour escape from the canopy from the level dzoH, as it is implied in Eq. (1), can be questioned. In
alternative,racan be calculated from the top of the canopy (hc) to the reference height,
using (Perrier, 1975; Stockle and Kjelgaard, 1996):
ra
ln zÿd= hcÿdln zÿd=zO
k2u z
(2)
Surface resistance is more complex and several procedures have been proposed for its derivation. Plant physiologists consider it to be a purely physiological parameter that accounts for the stomatal control of transpiration. Stomata have been carefully studied and the factors that determine their functioning are well known. Some of them, like
radiation (R) (either solar radiation or PAR), temperature (T) and vapour pressure deficit (D) are those that govern the physical process of evaporation. Others, like soil (or plant) water potential (y) represent the true physiological control by stomata which takes place mainly in water stress conditions. Other factors, like the age of the leaf, the previous history of water stress of the plant and the position of the leaf in the plant, are also important but less quantifiable.
From studies in controlled chambers, where factors are varied independently, the forms of the individual functions are known. Examples of those functions are
gstg R gmax1R= 23RJarvis 1976; (3a)
gstg T gmaxf1ÿ4 TÿTmax2g Jones 1983; (3b)
gstg D gmax= 15DKaufmann 1982; and (3c)
gstg gmaxf1ÿexpÿa6 ÿ maxgJarvis 1976: (3d)
Stomatal conductance (gst1=rst) is commonly preferred in these studies as the
functionsgonly take values between 0 (most unfavourable condition, leading to complete closure of stomata) and 1 (optimal conditions). Maximum stomatal conductance (gmax)
depends on the morphology of the stomata and on their density on both sides of the leaves and thus is crop specific.
All these factors interact in a complex manner, especially because in the field they are not independent (the correlations between radiation and temperature or between temperature and vapour pressure deficit, for instance, are well known). The simplest approach to the modelling of stomatal functioning is probably by multiple regression, wheregst is regressed against several independent variables to give an equation of the
form:
gsta1a2Ra3Da4T (4)
Non-linear relations can be fitted by the use of higher-order polynomials. However, the best method to analyse stomatal functioning has been considered the model proposed by Jarvis (1976), of the form
gstgmaxg Rg Tg Dg (5)
that considers that the influence of the different variables is independent and their effects multiplicative. Functionsg are those obtained from controlled environment studies such as Eqs. (3a),(3b),(3c) and (3d).
Scaling resistances from leaf to canopy, which constitutes the `bottom up' approach to rs, is full of controversy. The standard procedure is to average stomatal resistance rst at different levels in the canopy, weighted by leaf area index (LAI) (Monteith,
1973). However, the values of rs determined this way even with measured stomatal
resistances seem to give good results only in very rough surfaces, like forests, and partial cover crops with a dry soil. On complete cover crops, especially when the soil is wet, average stomatal resistance can greatly depart, being normally lower, from
the values of rs obtained as a residual term of the Penman±Monteith equation
`top down' approach:
wheresis the slope of the vapour pressure curve (Pa/8C),is the psychometric constant (Pa/8C), a is the atmospheric density (kg/m3), cp is the specific heat of moist air
(J kgÿ18Cÿ1), lE is latent heat flux density (W/m2) and û is the Bowen ratio. This discrepancy has been regarded as to indicate that not all leaves actually contribute to the total water loss by the canopy. The concept of `effective' leaf area was, therefore, introduced and linked to radiation interception, the upper, well illuminated leaves being those that most contribute to transpiration.
However, as pointed out by Lhomme and Katerji (1985) and Alves et al. (1998), this kind of averaging, based on the electrical analogue of a parallel circuit, can only be made if the driving force (in the case of evaporation,D) at each node (leaf) is the same. This explains why good results can be obtained in forests and partial cover crops, as in this caseDwill show no or a small gradient inside the canopy. On the contrary, on complete cover crops or when there is free water (from dew or rain) or a wet soil, a humidity profile will exist, thus violating the main requirement for a parallel electric circuit: an equal driving force for all the elements. As a consequence, although individual stomatal resistances can be used in multi-layer models to determine total water flux, they cannot be used to compute a surface (bulk) resistance, that must always be back calculated from Eq. (6). Also a new concept of `effective' leaf area emerges: only the leaves in contact with unsaturated air will contribute to total water loss. Those leaves in contact with moist air (at the base of the canopy, especially if there is a wet soil) will only have a minor role. An apparent dependence on radiation arises because the top half of the canopy, where most radiation is absorbed, is also the layer that is in contact with the outer, unsaturated air.
Given the difficulties of the `bottom up' approach the alternative could be using the `top down' procedure, where surface resistance (or its inverse gs) is in the first step
determined from back calculation with Eq. (6) or equivalent and relationships are found with the main environmental variables. Stewart (1988) was probably the first to apply the Jarvis model (Eq. (5)) to the whole canopy, with parameters of Eqs. (3a),(3b),(3c) and (3d) or similar been adjusted using multivariate analysis. This approach has been since widely adopted and considered `the' standard procedure (Itier and Brunet, 1996). However, although this model can give good results, being able to explain as much as 70% of the total variance, its use is year and site specific, significant errors arising when different data sets are used. Stewart (1988) concluded that `surface conductance depends on additional variable (. . .) or that the dependence of surface conductance on the included variables changed from year to year'.
A different approach was adopted by Katerji and Perrier (1983) who proposed the linear model:
resistance, is computed as (Pereira et al., 1999)
This model, whose inputs are simple climatic variables, has already been applied to grass and alfalfa (Katerji and Perrier, 1983), wheat (Perrier et al., 1980), tomato (Katerji et al., 1988) and rice (Peterschmitt and Perrier, 1991) but has not been extensively used. In summary, the Penman±Monteith equation can only be used to directly predict crop evapotranspiration, which is needed for an adequate water management, if accurate methodologies for determiningrsare available. In a previous work (Alves et al., 1998) problems in deriving surface resistance from stomatal resistance values were addressed. The objective of this paper is to show the pitfalls of the alternative modelling of surface resistance from climatic parameters based on values ofrscalculated with the `top down' approach, using micrometeorological data gathered in a field trial over a summer crop (lettuce).
3. Materials and methods
3.1. Site and crop characteristics
Field trials were conducted at an Experimental Station belonging to National Institute for Agricultural Research (INIA) located at Coruche (latitude 388570N, longitude 88320W, altitude 30 m), some 80 km north-east from Lisbon, Portugal. The Station has a total area of 42.5 ha and is located inside an irrigated area of several hundred hectares, in the Sorraia Valley Irrigation Project.
During the summer of 1992, a trial was conducted with an iceberg lettuce crop
(Lactuca sativavar.capitatacv Saladin). Planting was made on 28 May with a density of
8 plants/m2 on a 0.5 ha field (50 m100 m) of a sandy soil (see characteristics in Tables 1 and 2). Neighbouring plots, totalling an approximate area of 8 ha, were cultivated with tomato and bell pepper. The crop was drip irrigated almost every day, mostly during the night or early morning, so maintaining root zone permanently near field capacity. Each drip line irrigated two plant rows (Fig. 1). Fertilization was made in order to optimise plant growth. Measurements were made between 25 June and 30 July when the crop completely covered the soil and plant height varied from 0.15 to 0.20 m.
3.2. Instrumentation and data acquisition
The energy balance was measured by a net radiometer (Schenk) at 1.5 m height and south oriented. Air temperature and humidity were measured at the heights of 0.85 and 1.46 m with psychrometers, made from ventilated, double-shielded copper-constantan
thermocouples, with an accuracy of0.028C. An anemometer (Young, model 12102D)
was placed at 1.63 m over the soil. A wind direction sensor (Vector Instruments, model W200P), at the same height, was also used.
These instruments were installed in a measurement tower that was placed near the edge of the plot, and connected to a Campbell Scientific 21X datalogger that scanned the sensors every second and stored the average values at 10 min intervals.
3.3. Data handling
Only the values recorded during the periods when the wind direction was adequate in order to provide a sufficient (80 m) fetch were kept. This fetch was assessed by the theory of Elliot (1958), validated by Munro and Oke (1975), and shown adequate for the measurements to be made inside the constant flux layer, thus being representative of the surface below, as discussed in detail in Alves et al. (1998).
Values of d (zero plane displacement height) and zo (roughness length) were
determined in a parallel study, using wind profiles obtained in neutral conditions and linear regression techniques. Values used in subsequent calculations wered/hc0.67 and
zo/hc0 .126.
Aerodynamic resistance was calculated according to Eq. (2). Stability conditions of the atmosphere were evaluated using the Richardson number (Ri) in its finite difference form computed as:
where g is the acceleration of gravity (m/s2), T is average absolute temperature (K) between levelsz1andz2(m) in the atmosphere, where wind speedsu1andu2(m/s) and air
temperaturest1andt2(8C) are measured. Surface resistance was calculated by inversion
of the Penman±Monteith model using Eq. (6). Only values obtained for |Ri| < 1 were
Table 2
Soil water characteristics, lettuce trial
Depth (cm) Moisture content (% V/V)a
Field capacity Wilting point
retained when calculatingrsallowing to use no corrections for stability. Errors inradue to
not considering stability corrections in these conditions were less than 10% which finally led to errors inrsless than 5%, as shown by the analysis of a set that included all the necessary data.
Latent heat fluxlEwas computed using the û (Bowen ratio) method. As the objective was to analyse the situations where crop evapotranspiration was maximal, only the values of |û| < 0.3 were used. Soil heat flux (G) was estimated to be 10% of the measured net radiation, following the studies of Clothier et al. (1986) on closed canopies. All necessary parameters were calculated with the algorithms proposed by Allen et al. (1994b).
Average conditions during the trial are presented in Table 3.
4. Results and discussion
Fig. 2 presents the daily evolution ofrs. Since the data are relative to a short period of time and weather conditions remained stable from day to day, values are fairly consistent and show, besides the abrupt fall/rise observed in early morning/late afternoon, a steady increase throughout the day. This pattern has also been reported by other authors and deduced by Monteith (1995b).
Asrscontains physiological information one expects it to depend on the same variables that control the opening of the stomata (radiation, temperature, vapour pressure deficit and, for non well watered conditions, water potential). The most important factor is radiation, as shown in Fig. 3. Net radiation was used since a close linear relationship with
Fig. 1. Representation of the lettuce crop.
Table 3
Day-time climatic conditions during the trial (25 June±30 July, 1992).
Variable Units Range of values Mean value
Wind speed (u) m/s 0.8±4.8 1.8
Rn- G W/m2 0±595 350
Air temperature (T) 8C 11.3±36.0 27.2
global solar radiation (Rg) was found (Rn0.65Rg-38.0,r20.995). According to Jones
(1983), maximal stomatal opening is obtained when solar radiation exceeds 200 W/m2 (which corresponds, in these experimental conditions, to approximatelyRn> 100 W/m2).
The abrupt fall/rise of rs that is observed in early morning/late afternoon is then in accordance to the light induced opening/closure of the stomata. The hysteresis that typically is found in the relationrsversusRn(the values in the morning being lower than
the ones in the afternoon for the same level of radiation) can be clearly seen in Fig. 3. This can be explained by the fact that other weather variables influencingrs, like temper-ature (T) and vapour pressure deficit (D), also have hysteretic relationships withRn(Fig. 4).
In addition, as temperature andDremain more or less constant during the afternoon (see Fig. 4), it is possible to find then a close, hyperbolic, dependence ofrsonRnalone (Fig. 5),
which follows the behaviour of the individual stomata (Eq. (3a)). A similar trend of the dependence ofrson solar radiation is presented by Tolk et al. (1996) for corn.
One would then be encouraged to evaluate the response ofrsto the other variables,D andT, that influence stomatal opening in non-water stressed conditions. For that, and
Fig. 2. Daily evolution of surface resistance (rs).
following a similar procedure as is used in controlled chamber studies, were chosen subsets of the data in which all variables, besides the one being studied, remain nearly constant. The results are presented in Figs. 6 and Fig. 7. There is a linear relationship betweenrsandD(Fig. 6), which is compatible with the behaviour of the single stomate (cf. Eq. (3c)). The effect of temperature (Fig. 7) is less clear as it is difficult to separate the effect of other variables, in particularD, which exhibits a strong correlation withT. However, results in Fig. 7 agree with the trend represented by Eq. (3c) forT <Tmax.
A relationship betweenrs and racan also be found (Fig. 8). Though wind speed is normally ignored when modelling stomatal behaviour, plant physiologists do recognise that wind may induce closure of stomata (Salisbury and Ross, 1992). This could explain the behaviour shown in Fig. 8. Also,rsis not a purely physiological parameter but also includes physical processes, namely those related to vapour transfer inside the canopy (Perrier, 1975; Alves et al., 1998) which will surely be affected by the transport conditions in the atmosphere and hencera.
Fig. 4. Relationship between air temperature and vapour pressure deficit (D) and net radiation (Rn). 14 July, 1992.
Having shown the similar behaviour relative to the main environmental variables of the single stomate and surface resistance, the modelling ofrsusing the multiplicative model of Jarvis (1976) (Eq. (5)), as first done by Stewart (1988), could be considered the logical next step. However, the above relationships derive, obviously, from the use of Eq. (6) to calculatersin the first place and they could easily be anticipated. Considering that
ERnÿG
1 (10)
then Eq. (6) can be modified into
rsra
s
ÿ1
1 acpD
RnÿG
(11)
which becomes the essential relationship to deriversfrom micrometeorological data.
Fig. 6. Relationship between surface resistance (rs) and vapour pressure deficit (D) forRn> 500 W/m2and ra<40 s/m.
This equation shows that when temperature, vapour pressure deficit, the Bowen ratio û and wind are constant (as it occurs in the field during the afternoon, and is made in controlled chamber studies) it results a simple relationship between rs and net radiationRn:
rsf Rn a
b Rn
(12)
witha andb being constants. This is the equation of a hyperbola, as the one drawn in Fig. 5 and is equivalent to Eq. (3a).
If radiation, temperature and wind speed are made to remain constant and the crop is well watered (ûconstant), one gets
rsf D abD (13)
as in Fig. 6, which parallels the behaviour of a single stomate (cf. Eq. (3c)).
It is well established that as water availability decreases the energy partitioning between lE andH is altered, with an increase in H, and hence û. In fact, û has been sometimes used as an indicator of water stress (Peterschmitt and Perrier, 1991; Frangi et al., 1996), thus replacing the soil (or plant) water potentialy. In this way, also the effect of water stress onrs is considered in Eq. (11) through û.
A relationship betweenrsandracan also be deduced from Eq. (11), by keeping vapour pressure deficit and radiation constant or varying by the same relative amount for a well watered crop (ûconstant). A linear relationship is then obtained
rsf ra arab (14)
as represented in Fig. 8. Many researchers fail to admit that the transfer of vapour inside
the canopy is one of the components of surface resistance, which is dependent on wind speed or, indirectly, onra. As a consequence, approaches to the modelling of rs often disregard this physical component, which may account for some of the variability problems experienced by Stewart (1988) and is in accordance with one of his conclusions, that there was probably an additional variable that was `missing' in the modelling.
It is then concluded that Eq. (11) considers all the same variables (R, T, D and, indirectly,y, as û can be seen as a water stress indicator) and the same functional forms that describe stomatal behaviour and that are used in the model of Jarvis (1976) (Eq. (5)). In Table 4, a summary of the individual functions ofrson the environmental variables is presented, which shows that they are to vary according to the weather conditions. They will thus change from day to day, season to season and from year to year, as also concluded by Stewart (1988). Furthermore, being non-transferable, they are of negligible predictive use. Eq. (11) can even question the validity of the multiplicative model that in fact has no theoretical background to support it. The assumption that weather variables act independently is, in particular, most doubtful.
Some may question that the analysis presented is rather redundant and trying to show the obvious. However, it has been the common procedure (Adams et al., 1991; Cienciala et al., 1994; Granier and Loustau, 1994, among many others) to adjust a multiplicative model like Eq. (5), after obtainingrsby back calculation using Eq. (6) (or similar). It is hoped that it becomes clear that this kind of procedure, besides the computational complexity involved, gives no further insight on the determinism ofrsand thus is in fact a useless exercise.
On the other hand, Eq. (11) corresponds and gives a theoretical support to the already tested approach of Katerji and Perrier (1983) (Eq. (7)), with parametersa andb being given by:
This model shows that the adjustment is actually made onb, the only factor that is not readily available and needs to be estimated for practical uses. For well watered crops and for short periods of time, when the weather doesn't change too much from day to day, it is expected to have no great variation of the values ofaandb. This is in fact the case of Fig. 9, that shows that dispersion along the line of best fit is relatively small.
The behaviour of the stomata in response to the environment has been seen as a physiological response. In particular, the response of the stomata to radiation is considered to be intimately linked to the process of photosynthesis and this link is the basis of optimization models of stomatal conductance, as refered by Monteith (1995b). It
Table 4
Functional forms of the dependence of surface resistance (rs) on environmental variables
rsf Rn c1c2=Rn c1ra s=ÿ1 c2 1acpD= rsf D c3c4D c3ra s=ÿ1 c4 1acp= Rn rsf ra c5rac6 c5s=ÿ1 c6 1acpD= Rn rsf Rn;D;ra c7rac8D=Rn c7s=ÿ1 c8 1acp=
1-G/Rn.
may then be puzzling that one can retrieve from Eq. (11), that is derived from the physical laws that govern heat fluxes, the same (physiological?) behaviour of the single stomate.
The dependence ofrson û, and thus onlE, poses some problems. If it is necessary to have some knowledge on the energy partitioning to obtain a good estimate ofrsthen this may constitute a serious drawback to the direct use of the Penman±Monteith equation for crop evapotranspiration estimation. On the other hand, most of the research and focus on rsduring these last decades has been made on the premise thatrscontrolslErather than the opposite. A recent reanalysis by Monteith (1995a) of stomatal conductance data of single leaves, which were originally published demonstrating the dependence ofrstonD,
showed however that rst could be better described as a function of lE, of the form
gst1=rstaÿbE(which is equivalent to the relationship betweenrsandlEthat can be retrieved from Eq. (11) asrs/1/lE).
It appears that research made onrshas thus, gone a round circle and that our attention should be refocused on the determinism and quantification of the energy partitioning.
5. Conclusions
Eq. (11), which is mostly utilized to calculatersin a `top down' approach, can also be
used to show the dependence of rs on the environmental variables. Apart from a
physiological component, that responds to the main environmental variables (Rn,Dand
T) in the same manner individual stomata do, there is also an aerodynamic component to be considered. Purely physiological approaches to the estimation ofrsthat just consider stomatal resistance may therefore be not suitable. Eq. (11) or its equivalent form Eq. (7) thus contain all the information needed and can be regarded as a simpler alternative to the
Jarvis multiplicative approach for the modelling ofrsthat, despite being widely accepted, does not have any theoretical background. In addition, the Jarvis model has been conceived to approach the stomatal conductance determinism and not to explain or derive the (bulk) surface resistance of a full cover crop where the stomatal resistance is only one of the components. Eq. (11) also shows that a good prediction of rs requires a good
knowledge or estimation of the Bowen ratio b, which is in accordance with recent
findings that indicate that plant transpiration may be more the cause than the consequence of a given stomatal resistance. More research on the determinism of the energy partitioning is therefore required for further developments in the direct prediction of crop evapotranspiration.
Acknowledgements
The authors are grateful to INIA, who provided the field facilities and performed all the necessary cultural operations. This study was made in the framework of Project NATO-PO-Irrigation and Contract CE n.8001-CT91-0115 that provided the financial support. The authors also thank comments by Dr. Alain Perrier (INA, Paris-Grignon) and Dr. Bernard Itier (INRA, Thiverval-Grignon), France.
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