Directory UMM :Data Elmu:jurnal:A:Agricultural Water Management:Vol42.Issue3.Jan2000:

Agricultural Water Management 42 (2000) 371±385

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 of rs is 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
calculating rs as the residual term. One cannot thus expect to get a better insight on the behaviour of
rs with the multiplicative model. Also, as rs includes 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
between rs and 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)
0378-3774/00/$ ± see front matter # 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 8 - 3 7 7 4 ( 9 9 ) 0 0 0 4 1 - 4

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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 ˆ

ln‰…z ÿ d†=zOH Šln‰…z ÿ d†=zO Š
k 2 uz

(1)

where d is zero plane displacement height (m), zo is roughness length for momentum (m),
zoH is roughness length for heat (m), k is von Karman constant and uz is wind speed (m/s)
at height z (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 d ‡ zoH, as it is implied in Eq. (1), can be questioned. In
alternative, ra can be calculated from the top of the canopy (hc) to the reference height,
using (Perrier, 1975; Stockle and Kjelgaard, 1996):
ra ˆ

ln‰…z ÿ d†=…hc ÿ d†Šln‰…z ÿ d†=zO Š
k 2 uz

(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

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373

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

gst ˆ g…R† ˆ gmax 1 R=…2 ‡ 3 R† Jarvis …1976†;

(3a)

gst ˆ g…T† ˆ gmax f1 ÿ 4 …T ÿ Tmax †2 g Jones …1983†;

(3b)

gst ˆ g…D† ˆ gmax =…1 ‡ 5 D† Kaufmann …1982†; and

(3c)

gst ˆ g… † ˆ gmax f1 ÿ exp‰ÿa6 … ÿ

max †Šg

Jarvis …1976†:

(3d)

Stomatal conductance (gst ˆ 1=rst ) is commonly preferred in these studies as the
functions g only 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,
where gst is regressed against several independent variables to give an equation of the
form:
gst ˆ a1 ‡ a2 R ‡ a3 D ‡ a4 T ‡




(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
gst ˆ gmax g…R†g…T†g…D†g… †

(5)

that considers that the influence of the different variables is independent and their effects
multiplicative. Functions g 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
(Paw U and Meyers, 1989; Baldocchi et al., 1991; Rochette et al., 1991) using the

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`top down' approach:

rs ˆ ra




a cp D
s
ÿ1 ‡

E

(6)

where s is 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 kgÿ1 8Cÿ1), 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
case D will 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:
rs
r
(7)
ˆa ‡b
ra
ra
where a and b are parameters to be determined experimentally, and r*, the climatic

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375

resistance, is computed as (Pereira et al., 1999)
r ˆ

s ‡
a cp D
s
…Rn ÿ G†

(8)

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 determining rs are 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 of rs calculated 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 388570 N, longitude 88320 W,
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 sativa var. capitata cv Saladin). Planting was made on 28 May with a density of
8 plants/m2 on a 0.5 ha field (50 m  100 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.

Table 1
Soil textural analysis, lettuce trial
Depth (cm)

Textural analysis (%)
Coarse sand

Fine sand

Loam

Clay

0±30
30±50
50±80

65.4
67.9
74.7

25.3
22.8
16.1

5.2
5.6
6.4

4.1
3.7
2.8

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Table 2
Soil water characteristics, lettuce trial
Depth (cm)

Moisture content (% V/V)a
Field capacity

Wilting point

0±10
10±20
20±30
30±40
40±50

16.5
18.7
17.2
13.9
11.9

4.2
3.7
3.8
4.4
5.6

a

Measured in situ.

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 of 0.028C. 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 were d/hc ˆ 0.67 and
zo/hc ˆ 0 .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:
Ri ˆ

g…t1 ÿ t2 † z1 ÿ z2
T
u1 ÿ u22

(9)

where g is the acceleration of gravity (m/s2), T is average absolute temperature (K)
between levels z1 and z2 (m) in the atmosphere, where wind speeds u1 and u2 (m/s) and air
temperatures t1 and t2 (8C) are measured. Surface resistance was calculated by inversion
of the Penman±Monteith model using Eq. (6). Only values obtained for |Ri| < 1 were

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Fig. 1. Representation of the lettuce crop.

retained when calculating rs allowing to use no corrections for stability. Errors in ra due to
not considering stability corrections in these conditions were less than 10% which finally
led to errors in rs less than 5%, as shown by the analysis of a set that included all the
necessary data.
Latent heat flux lE was 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 of rs. 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).
As rs contains 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
Table 3
Day-time climatic conditions during the trial (25 June±30 July, 1992).
Variable

Units

Range of values

Wind speed (u)
Rn - G
Air temperature (T)
Vapour pressure deficit (D)

m/s
W/m2
8C
Pa

0.8±4.8
0±595
11.3±36.0
100±2810

Mean value
1.8
350
27.2
1500

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Fig. 2. Daily evolution of surface resistance (rs).

Fig. 3. Relationship between surface resistance (rs) and net radiation (Rn).

global solar radiation (Rg) was found (Rn ˆ 0.65 Rg-38.0, r2 ˆ 0.995). According to Jones
(1983), maximal stomatal opening is obtained when solar radiation exceeds 200 W/m2
(which corresponds, in these experimental conditions, to approximately Rn > 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 relation rs versus Rn (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 influencing rs, like temperature (T) and vapour pressure deficit (D), also have hysteretic relationships with Rn (Fig. 4).
In addition, as temperature and D remain more or less constant during the afternoon (see
Fig. 4), it is possible to find then a close, hyperbolic, dependence of rs on Rn alone (Fig. 5),
which follows the behaviour of the individual stomata (Eq. (3a)). A similar trend of the
dependence of rs on solar radiation is presented by Tolk et al. (1996) for corn.
One would then be encouraged to evaluate the response of rs to the other variables, D
and T, that influence stomatal opening in non-water stressed conditions. For that, and

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379

Fig. 4. Relationship between air temperature and vapour pressure deficit (D) and net radiation (Rn). 14 July,
1992.

Fig. 5. Relationship between surface resistance (rs) and net radiation (Rn) in lettuce, during the afternoon
(14:00±20:00 hours).

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
between rs and D (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 particular D, which exhibits a strong correlation with T.
However, results in Fig. 7 agree with the trend represented by Eq. (3c) for T < Tmax.
A relationship between rs and ra can 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, rs is 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 hence ra.

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Fig. 6. Relationship between surface resistance (rs) and vapour pressure deficit (D) for Rn > 500 W/m2 and
ra 500 W/m2.

Having shown the similar behaviour relative to the main environmental variables of the
single stomate and surface resistance, the modelling of rs using 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
calculate rs in the first place and they could easily be anticipated. Considering that
E ˆ

Rn ÿ G
1‡

then Eq. (6) can be modified into


s
a cp D
rs ˆ ra  ÿ 1 ‡ …1 ‡ †

…Rn ÿ G†

(10)

(11)

which becomes the essential relationship to derive rs from micrometeorological data.

I. Alves, L.S. Pereira / Agricultural Water Management 42 (2000) 371±385

381

Fig. 8. Relationship between surface resistance (rs) and aerodynamic resistance (ra) for 1500 < D < 2000 a and
Rn > 500 W/m2.

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
radiation Rn:
rs ˆ f …Rn †  a ‡

b
Rn

(12)

with a and b 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 (û ˆ constant), one gets
rs ˆ f …D†  a ‡ bD

(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 and H 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 potential y. In this way, also the effect
of water stress on rs is considered in Eq. (11) through û.
A relationship between rs and ra can 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 (û ˆ constant). A linear relationship is then obtained
rs ˆ f …ra †  ara ‡ b

(14)

as represented in Fig. 8. Many researchers fail to admit that the transfer of vapour inside

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Table 4
Functional forms of the dependence of surface resistance (rs) on environmental variables
rs
rs
rs
rs

ˆ f …Rn † ˆ c1 ‡ c2 =Rn
ˆ f …D† ˆ c3 ‡ c4 D
ˆ f …ra † ˆ c5 ra ‡ c6
ˆ f …Rn ; D; ra † ˆ c7 ra ‡ c8 D=Rn

c1
c3
c5
c7

ˆ ra …s=
ÿ 1†
ˆ ra …s=
ÿ 1†
ˆ s=
ÿ 1
ˆ s=
ÿ 1

c2
c4
c6
c8

ˆ …1 ‡ †a cp D=…
†
ˆ …1 ‡ †a cp =…
Rn †
ˆ …1 ‡ †a cp D=…
Rn †
ˆ …1 ‡ †a cp =…
†

 ˆ 1-G/Rn.

the canopy is one of the components of surface resistance, which is dependent on wind
speed or, indirectly, on ra. 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 of rs on 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 obtaining rs by 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 of rs and 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 parameters a and b being
given by:
s
a ˆ …1 ‡ † s‡
s
b ˆ
 ÿ1

(15)

This model shows that the adjustment is actually made on b, 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 of a and b. 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

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383

Fig. 9. Results of the application of the model by Katerji and Perrier (1983), Eq. (7).

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 of rs on û, and thus on lE, poses some problems. If it is necessary to
have some knowledge on the energy partitioning to obtain a good estimate of rs then 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
rs during these last decades has been made on the premise that rs controls lE rather than
the opposite. A recent reanalysis by Monteith (1995a) of stomatal conductance data of
single leaves, which were originally published demonstrating the dependence of rst on D,
showed however that rst could be better described as a function of lE, of the form
gst ˆ 1=rst ˆ a ÿ bE (which is equivalent to the relationship between rs and lE that can
be retrieved from Eq. (11) as rs / 1/lE).
It appears that research made on rs has 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 calculate rs in 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, D and
T) in the same manner individual stomata do, there is also an aerodynamic component to
be considered. Purely physiological approaches to the estimation of rs that 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

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Jarvis multiplicative approach for the modelling of rs that, 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 NATOPO-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.

References
Adams, R.S., Black, T.A., Fleming, R.L., 1991. Evapotranspiration and surface conductance in a high elevation,
grass-covered forest clearcut. Agric. For. Meteorol. 56, 173±193.
Allen, R.G., Smith, M., Perrier, A., Pereira, L.S., 1994a. An update for the definition of the reference
evapotranspiration, ICID Bull. 43(2), 1±34.
Allen, R.G., Smith, M., Pereira, L.S., Perrier, A., 1994b. An update for the calculation of reference
evapotranspiration, ICID Bull. 43(2), 35±92.
Allen, R.G., Pereira, L.S., Raes, D., Smith, M., 1999. Crop evapotranspiration. Guidelines for computing crop
water requirements. FAO Irrigation and Drainage Paper 56, FAO, Rome.
Alves, I., Perrier, A., Pereira, L.S., 1998. Aerodynamic and surface resistances of complete cover crops: how
good is the `big leaf'?. Trans. ASAE 41(2), 345±351.
Baldocchi, D.D., Luxmoore, R.J., Hatfield, J.L., 1991. Discerning the forest from the trees: An essay on scaling
canopy stomatal conductance. Agric. For. Meteorol. 54, 197±226.
Brutsaert, W., 1982. Evaporation into the Atmosphere. R. Deidel, Dordrecht, Holland.
Cienciala, E., Eckersten, H., Lindroth, A., HaÈllgren, J.E., 1994. Simulated and measured water uptake by Picea
abies under non-limiting soil water conditions. Agric. For. Meteorol. 71, 147±164.
Clothier, B.E., Clawson, K.L., Pinter Jr, P.J., Moran, M.S., Reginato, R.J., Jackson, R.D., 1986. Estimation of
soil heat flux from net radiation during the growth of alfalfa. Agric. For. Meteorol. 37, 319±329.
Doorenbos, J., Pruitt, W.O., 1977. Crop water requirements. FAO Irrigation and Drainage Paper 24, FAO, Rome.
Elliot, W.P., 1958. The growth of the atmospheric internal boundary layer. Trans. Am. Geophys. Un. 39, 1048±
1054.
Frangi, J.-P., Garrigues, C., Haberstock, F., Forest, F., 1996. Evapotranspiration and stress indicator through
Bowen ratio method. In: Camp, C.R., Sadler, E.J., Yoder, R.E. (Eds) Evapotranspiration and Irrigation
Scheduling (Proc. Int. Conf., San Antonio, TX, 3±6 November), ASAE, St. Joseph, MI, pp. 800±805.
Granier, A., Loustau, D., 1994. Measuring and modelling the transpiration of a maritime pine canopy from sapflow data. Agric. For. Meteorol. 71, 61±81.
Itier, B., Brunet, Y., 1996. Recent developments and present trends in evapotranspiration research: A partial

I. Alves, L.S. Pereira / Agricultural Water Management 42 (2000) 371±385

385

survey. In: C.R. Camp et al. (Eds.), Evapotranspiration and Irrigation Scheduling (Proc. Int. Conf., San
Antonio, TX, 3±6 November), ASAE, St. Joseph, MI, pp. 1±20.
Jarvis, P.G., 1976. The interpretation of the variations in leaf water potential and stomatal conductance found in
canopies in the field. Phil. Trans. R. Soc. Lon. B. 273, 595±610.
Jones, H.G., 1983. Plants and microclimate: a quantitative approach to environmental plant physiology.
Cambridge University Press, Cambridge.
Katerji, N., Perrier, A., 1983. ModeÂlisation de l'eÂvapotranspiration reÂelle ETR d'une parcelle de luzerne: roÃle
d'un coefficient cultural. Agronomie 3(6), 513±521.
Katerji, N., Itier, B., Ferreira, M.I., 1988. Etudes de quelques criteÁres indicateurs de l'eÂtat hydrique d'une culture
de tomate em reÂgion seÂmi-aride. Agronomie 5, 425±433.
Kaufmann, M.R., 1982. Leaf conductance as a function of photosynthetic photon flux density and absolute
humidity difference from leaf to air. Plant Physiol. 69, 1018±1022.
Lhomme, J.P., Katerji, N., 1985. Une meÂthode de calcul de l'eÂvapotranspiration reÂelle aÁ partir de mesures
ponctuelles de reÂsistance stomatique et de tempeÂrature foliaire. Agronomie 5, 397±403.
Monteith, J.L., 1965. Evaporation and environment. Symposia of the Soc. Exp. Bio. 19, 205±234.
Monteith, J.L., 1973. Principles of Environmental Physics. Edward Arnold, London.
Monteith, J.L., 1995a. A reinterpretation of stomatal responses to humidity. Plant Cell Environ. 18, 357±364.
Monteith, J.L., 1995b. Accomodation between transpiring vegetation and the convective boundary layer. J.
Hydrol. 166, 251±263.
Monteith, J.L., Unsworth, M.H., 1990. Principles of Environmental Physics. Edward Arnold, London.
Munro, D.S., Oke, T.R., 1975. Aerodynamic boundary-layer adjustment over a crop in neutral stability.
Boundary-layer Meteorol. 9, 53±61.
Paw U, K.T., Meyers, T.P., 1989. Investigations with a higher-order canopy turbulence model into mean sourcesink levels and bulk canopy resistances. Agric. For. Meteorol. 47, 259±271.
Penman, H.L., 1948. Natural evaporation from open water, bare soil and grass. Proc. Royal Soc. 193, 120±145.
Pereira, L.S., Perrier, A., Allen, R.G., Alves, I., 1999. Evapotranspiration: review of concepts and future trends.
J. Irrigation and Drainage Eng. ASCE 125, 45±51.
Perrier, A., 1975. Etude physique de l'evapotranspiration dans les conditions naturelles. III - evapotranspiration
reÂelle et potentielle des couverts veÂgeÂtaux. Ann. Agronomiques 26, 229±243.
Perrier, A., 1982. Land surface processes: vegetation. In: Eagleson, P.S. (Ed.), Land Surface Processes in
Atmospheric General Circulation Models, pp. 395±448.
Perrier, A., Katerji, N., Gosse, G., Itier, B., 1980. Etude `in situ' de l'eÂvapotranspiration reÂelle d'une culture de
bleÂ. Agric. Meteorol. 21, 295±311.
Peterschmitt, J.M., Perrier, A., 1991. Evapotranspiration and canopy temperature of rice and groundnut in
southeast coastal India. Crop coefficient approach and relationship between evapotranspiration and canopy
temperature. Agric. For. Meteorol. 56, 273±298.
Rochette, P., Pattey, E., Desjardins, R.L., Dwyer, L.M., Stewart, D.W., Dube, P.A., 1991. Estimation of maize
(Zea mays L.) canopy conductance by scaling up leaf stomatal conductance. Agric. For. Meteorol. 54, 241±
261.
Salisbury, F.B., Ross, C.W., 1992. Plant Physiology, 4th ed., Wadsworth, Belmont, CA.
Shaw, R.H., Pereira, A.R., 1982. Aerodynamic roughness of a plant canopy: A numerical experiment. Agric.
Meteorol. 26, 51±65.
Stewart, J.B., 1988. Modelling surface conductance of pine forest. Agric. For. Meteorol. 43, 19±35.
Stockle, C.O., Kjelgaard, J., 1996. Parameterizing Penman±Monteith surface resistance for estimating daily crop
ET. In: Camp, C.R., Sadler, E.J., Yoder, R.E. (Eds.), Evapotranspiration and Irrigation Scheduling (Proc.
Int. Conf., San Antonio, TX, 3±6 November), ASAE, St. Joseph, MI, pp. 697±703.
Thom, A.S., 1975. Momentum, mass and heat exchange. In: Monteith, J.L. (Ed.) Vegetation and the
Atmosphere, vol. 1, Academic Press, pp. 57±109.
Tolk, J.A., Howell, T.A., Steiner, J.L., Krieg, D.R., 1996. Corn canopy resistance determined from whole plant
transpiration. In: Camp, C.R., Sadler, E.J., Yoder, R.E. (Eds.), Evapotranspiration and Irrigation Scheduling
(Proc. Int. Conf., San Antonio, TX, 3±6 November), ASAE, St. Joseph, MI, pp. 347±351.