Water transport from soil through plant to atmosphere: a lumped-parameter model

Agricultural Meteorology, 16(1976) 171--184 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands

WATER TRANSPORT FROM SOIL THROUGH A LUMPED-PARAMETER MODEL

PLANT TO ATMOSPHERE:

C. W. ROSE 1, G. F. BYRNE and G. K. HANSEN 2 Division o f Land Use Research, CSIRO, Canberra A.C.T. (Australia)

(Received March 25, 1975; accepted December 29, 1975)

ABSTRACT Rose, C. W., Byrne, G. F. and Hansen, G. K., 1976. Water transport from soil through plant to atmosphere: a lumped-parameter model. Agric. Meteorol., 16: 171--184. A lumped-parameter model has been developed which allows the simulation of water transport from soil to atmosphere through a single plant or a crop. This model accepts the fact that there is current uncertainty as to whether the dominant liquid-phase resistance to water lies in the soil or the plant, and no assumptions are made on this question. The model is based on the Ohm's law analogue and on two experimentally derived functions. The model structure is thought to be generally applicable although these two empirical relationships are derived from an experiment with cotton plants in pots. The model may be suitable for the simulation of transpiration on a daily basis in a variety of soil and atmospheric environments, and also within-daily changes in transpiration and water potentials. One such simulation is compared with those of other models, and with other experiments. INTRODUCTION T h e a v a i l a b i l i t y o f w a t e r t o p l a n t s , l i n k e d as i t is w i t h n u t r i e n t a v a i l a b i l i t y , is a p o w e r f u l e n v i r o n m e n t a l f a c t o r in a n y e c o s y s t e m . C o n t r o l o f t r a n s p i r a t i o n is u l t i m a t e l y e x e r c i s e d b y t h e p l a n t t h r o u g h a n i n c r e a s e in t h e d i f f u s i v e r e s i s t a n c e o f s t o m a t a t o w a t e r in t h e gas p h a s e ( V a n d e n H o n e r t , 1 9 4 8 ; R a w l i n s , 1963). This resistance has a general (though not necessarily unique) relations h i p t o w a t e r p o t e n t i a l in t h e l e a f (~I'l). An analogue of Ohm's law (Van den Honert, 1948) has been commonly u s e d t o r e l a t e w a t e r p o t e n t i a l s in l e a f ( ~ l ) a n d s o i l (kOs) t o t r a n s p i r a t i o n r a t e (T): ~I'l = ~Ps - T ( R s + R p )

(1)

w h e r e R s a n d R p a r e r e s i s t a n c e s t o w a t e r f l o w in s o i l a n d p l a n t , r e s p e c t i v e l y . 1 Present address: School of Australian Environmental Studies, Griffith University, Queensland, Australia. 2 Present address: The Hydrotechnical Laboratory, Copenhagen, Denmark.

172 R s and Rp are conceived as dominantly liquid-phase resistances, and Rp therefore does not include stomatal resistance. It is implicitly assumed that the change to the vapour phase occurs at the site of ~ l , though this is clearly an oversimplification. Because of the dependence of stomatal aperture on q:! it is to be expected that ~IJ1 and T would necessarily be correlated. Conclusions as to the behaviour of Rs or Rp therefore cannot be drawn solely from the observed behaviour of ~I'l and T since the value of • at the interface between root and soil, as distinct from ~ s in the bulk soil, is unknown. In the use of eq.1, most authors have followed a method developed by Gardner and Ehlig (1962) for partitioning (Rs + Rp) into its soil and plant components. They assume Rs to be negligible when ~s is high (i.e. less negative) (say ~ - 1 0 J kg_l), thus allowing the calculation of Rp using eq.1. Resistance Rp is then assumed constant, so the considerable increase in resistance sum as ~s decreases is automatically ascribed to an increase in R s. This procedure ensures the conclusion that except at high ~s, Rs >> Rp. Soil physics theory is used to give support to this result (e.g. Gardner and Ehlig, 1963), an assumption being that only some small fraction of the r o o t length takes up water. Newman (1969) made an alternative assumption, namely that the entire root system is an effective sink for water, and used the same soil physics theory to reach the opposing conclusion that Rp >~ Rs, unless effective root densities and/or soil water potentials are very low indeed. The work of Hansen (1974) and Byrne et al. (1976) also supports Newman's conclusions. Further understanding at the process level is therefore desirable to clarify the relative importance of R s and Rp, and the principles governing the uptake of water by roots. Without awaiting this clarification, progress can be made towards a predictively useful model for simulating the pattern of transpiration in response to environmental variables by making the following broad assumptions. (1) We accept eq.1 as the basis of the model. (2) The sum of the resistances R s and Rp is regarded as one total resistance since the components cannot be reliably separated in general. (3) Water flow is characterized in terms of potentials at points at which potentials can be measured, i.e., at the plant leaf and in the soil. Profiles are therefore treated as points rather than as distributed parameters. Experimental data are used to derive functional relationships required in the model. The consequences and implications of the model so derived are then explored, and compared with other data and models.

173 DEVELOPMENT OF A LUMPED-PARAMETER MODEL OF WATER TRANSPORT

Experimental Development of the model required data covering a wide range of soil and plant water conditions. Data were obtained from experiments of Lang and Gardner (1970) and further unpublished data by Lang from the same series of controlled-environment experiments on 10--12 weeks old c o t t o n plants grown in four different soils which were allowed to dry out. Since previously published and unpublished data are from the same experiments these are not distinguished, except where desirable (as in Figs.1 and 3). Plants were raised in 20-1itre drums in a glasshouse under constant day and night temperatures. Natural changes in radiation and day length resulted in variations in plant height of some 50%. Plants were then transferred to controlled-environment cabinets and in different experiments temperatures ranged from 15 ° to 35°C, and the saturation deficit of the air in the cabinets ranged from 12 to 47 mbar (1200--4700 Pa). Two or three days after watering, the saturation deficit of the environment was increased by increasing the air temperature and the flux of water out of the plant was measured over a period of about two hours. Water potentials in soil and plant were measured frequently using psychrometric techniques. Plant water potentials of enclosed attached leaves were measured to give the potential in the stem xylem (~x), rather than in the leaves (~1). The rate of transpiration (kg s-1) was measured by weighing individual drums, and the maximum rate measured under any given set of environmental conditions was denoted Tin, defined in more general terms below. In an experiment with any particular combination of temperature, humidity, and plant size, Tm was obtained with the highest soil water potentials. It follows from eq.1 that resistances per unit plant are in units of J s kg -2, where water potentials are in J kg -1. T H E M O D E L AND ITS R E L A T I O N S H I P S

It is a generalization of much experimentation that transpiration from a crop is largely determined by the environment until water stress increases b e y o n d a limit when both plant and environmental factors interact to modify transpiration. This interaction between plant and environmental factors has been expressed (e.g., by Gardner and Ehlig, 1963) as a relationship between (T/Tm) and ~x (or ~1). The maximum transpiration rate Tm of a plant or crop in a particular environment may be defined as the transpiration rate at maximum stomatal conductivity. Fig.1 shows (T/Tm) as a function of ~x, with the relationship hand fitted, somewhat as an envelope, to the subset of data published by Lang and

174

0.8

,o

.x.

.....

0.6 p-E

0"4

i

x

i

0"2 x

x"

o-o 2000 4000 -L~ x (J kg -1 )

Fig.1. Relation between the ratio of actual to maximum transpiration rates (T/Tm) and water potential in the shoot xylem (~x) of cotton. The relationship used in the model was the line hand fitted to data of Lang and Gardner (1970) (x). Other data (e) are from the same experiment by Lang (unpublished).

Gardner (1970). The exact form of this relation is uncertain, especially at very low values of ~x. This relationship between transpiration rate T, an evaporative descriptor of the aerial environment (Tin), and ~ x may be represented mathematically as a function with two linear elements:

T/Tm f ( ~ x ) =

(2)

To make further progress with the lumped parameter model of eq.1, a relationship between (R s + Rp) and ~s might be used, and such a relationship has been c o m m o n l y assumed (e.g., Gardner and Ehlig, 1962; Cowan, 1965). However, this relationship is not well defined for the data from Lang's experiments as is shown in Fig.2. Figs.3 and 4 show the same resistance sum plotted against ~ x and T, respectively. Though obviously non-linear, the relationship of (Rs + Rp) with q~x or T is tighter than with ~s (Fig.2). This conclusion is also supported by the work of Hansen (1974, 1975). The published subset of data in Fig.3 is shown fitted by an exponential t y p e of relationship. On the statistical basis of tightness of correlation either ~ x or T could be chosen as a suitable independent variable (Figs.3, 4). Some conceptual basis can be given for selecting either parameter as the assumed independent variable in a function for total resistance, and models based on either have similar characteristics. Hence, a decision can be made on the basis of convenience. Since ~ x is the independent variable in eq.2, the relationship of total resistance to ~ x (Fig.3) will be used in the model. Thus: R s + Rp = F(~x) = exp(13.9 • 10 -4 [kOxl + 16.3)

(3)

175

o i

== 3 # +

C

E

E E D e_ BCcEE D C C CE r..: r DDI~. C CB E CC C D )BD ~ ECCE C C

0 ~

CE...C

0

A A C

o o

cD

1000

2000 -~s

(J

3000 k( :1-I )

Fig.2. R e l a t i o n b e t w e e n t h e s u m o f soil a n d p l a n t resistance (R s + R p ) a n d soil w a t e r p o t e n t i a l (kl's). S a m e d a t a s o u r c e as Fig.1. A = T m ~ 8, B = 8 < T m ~ 14, _< T~n ( 20, D = 2 0 < T m ( 26, E = T m ~ 26, w h e r e T m is here e x p r e s s e d in u n i t s o f kg s- • 10 -6.

×

~"

x

x

E

...~.: x. •" o

~

x..

"

x~ ,. .~. x x • •

Q

"

L

1000

•

2000 -~s

(J

Q

~

,

3000

4000

kg -1 )

Fig.3. Relation between resistance sum (R s + Rp) and xylem potentia] (dJx). Fitted by regression o f In (R s + Rp) on (-~Irx) for data subset of Lang and Gardner (1970) (x). Other data s h o w n (e).

176

2 -~

10

-

05

-

t

0

* 0

,±__

ts.

t

1

_

t~

2 T

(kg

sec

•

• 3

1 ×

-

J 4

10-5)

Fig.4. Relation between resistance sum (R s + Rp) and transpiration rate (T). Same data source as Fig.l, hand fitted.

The numerical expression for F ( ~ x ) in eq.3 is obtained by fitting the subset of data published by Lang and Gardner (1970), consistent with f(~x) in Fig.1. The lumped-parameter model is defined by eqs.2 and 3, together with eq.1 expressed as: ~x = ~s - T(Rs +

Rp)

(4)

Substituting from eqs.2 and 3 into 4, the model can be expressed in one equation: ~x = ~Ps- Tm f(~Px) F(kOx)

(5)

It is assumed that Tm and ~s are known or separately obtained• To obtain ~s this lumped model needs to be linked with another expressing conservation of mass of water in the soil (e.g., Fitzpatrick and Nix, 1969; Rose et al., 1972), which then allows a lumped value of ~s for the soil profile explored by roots to be obtained via soil moisture characteristics. If it is also assumed that the experimentally determined functions of f(q2x) and F ( ~ x ) are generally applicable, then eq.5 can be solved (iteratively or otherwise) for the only known q~x- This allows T to be calculated from eq.2. The model is illustrated in flow-chart form in Fig.5 using the symbol conventions of Forrester (1968). (Briefly, material flows are solid lines, information flows dashed, boxes levels, valve symbols rate controllers, circles auxiliary calculations, and the cloud symbol externals to the system).

177

~

-

/,,"

@

Transpiration

T=T m f

rate

(~)

" "

I~x= L~s-T m f(~x)F(l~x)

i

IT

(;?) """"'"~..... t Amount of water il~ the root

[

profile

l

Rate of soil water replenish

Fig.5. The lumped-parameter model in flow chart form using the conventions of Forrester (1968).

Relationships between xPx and ~s

Eq.5 implies that the relationship between ~ x and ~s depends on Tin, but the form of their relationship is not obvious because: (1) eq.5 is implicit in ~Px; (2) the function F(~Px) is non-linear. These relationships between ~ x and ~s are shown in Fig.6, where in the limiting case of Tm = 0, ~Px = ~Ps. Note that ~ x decreases very rapidly for quite small decreases in ~Ps at higher values of Tm. It is experimentally very difficult to investigate the form of the relationship in conditions of extreme stress as illustrated in the region A of Fig.6 though the exact form in this region of the relationship is of little practical concern. At some low value of ~Ps, ~ x may continue to decline without further drop in ~s (unlike the curves shown).

178 4000

3000

.~

2000

Trn

(kg

s-lxl0

-6)

o: i

1000

i 1000

i 2000_~s( J

kg-1}

I 3000

Fig.6. Relationship between water potential in the sylem (~x) and the soil (~s), computed from the model for a range of values of maximum or potential transpiration rate (Tm).

Extension to a field crop The model is based on the Ohm's law analogue of eq.1 and the two experimentally obtained functions relating (T/Tm) and (R s + Rp) to ~x (eqs.2 and 3). These functions were obtained with a simulated c o m m u n i t y in a controlled environment cabinet. Since the water relations of field grown plants can be significantly different from those grown in controlled environments {e.g., Jordan and Ritchie, 1971), it is expected that the form of the functional relationships will be different for plants grown in either situation. However, in the remainder of the paper it will be assumed that the structure of the model is also applicable to a crop c o m m u n i t y in the field. In the field Tm could be regarded as the potential transpiration and calculated from standard meteorological observation (Penman, 1948). SIMULATION WITH THE MODEL Fig.7 shows the implications of this model plotted in the form used by Denmead and Shaw (1962) and others. There are general similarities between their relationships for a maize crop and those in Fig.7 for cotton. The behaviour of the model was investigated when subject to variation in the relationships f(~x) (Fig.l) and F(XPx) (Fig.3). This sensitivity analysis

179

08

06

k-

Tm ( g h - I )

04

Tm ( k g 5 - I x 10- 6 )

I O2

O0 I0

I 100

I 1000 -~s

(J

I0,000

kg-l}

Fig.7. Relation between the ratio of actual to maximum transpiration rate (T/Tm) and soil water potential (~s) computed from the model for a range of values of maximum transpiration rate (Tm). indicated th at appreciably different data in either function, singly or in combination, lead to unlikely or impossible changes in the predictions of the model. The sensitivity is n o t so great for the range of possible curves which could be fitted to the data in Figs.1 and 3 to give qualitatively different model predictions, but the analysis indicated a restricted degree of independence between the relations f ( ~ x) and F ( ~ x ) . SIMULATION OF WITHIN-DAILY CHANGES IN TRANSPIRATION AND WATER POTENTIALS The model was used t o investigate the variation in T in response t o the diurnal variation in Tm with declining soil moisture. Although the experimental relationships f ( ~ x ) and F ( ~ x ) were measured under steady d a y t i m e conditions, it is assumed that these relationships also hold when Tm varies through the day. For this simulation, Tm was taken to be zero at night, and to fluctuate during a 12-h p h o t o p e r i o d in a realistic manner according to the relation: Tm = 2Tm[1 + cos(4~t)] where Tm is the daily average value of Tm, and t is time in days, taking zero as midday. The midday m a x i m u m in Tm was taken as equal to a value (50 g h -~ or 14 • 10 -6 kg s -~) within the range of the experiments of Lang and Gardner ( 1970). Other values a d o p t e d for the illustrative simulation were an initial available volumetric water c o n t e n t of 0.09 cm 3 cm -3, and a soil volume of 20 litres. Soil moisture characteristics were the same as used by Lang and Gardner (1970).

180

Fig.8 shows some results of the simulation, in which actual transpiration T was equal to Tm throughout day 1, and all available water expired during day 8 (not shown). Minimum xPx decreases with time more rapidly than Vs. The rise in resistance at lower values of ~ x results in a depression in T during periods of higher Tm, with a midday depression evident by day 5 (Fig.8). Midday depression in transpiration in cotton has been observed under conditions of high stress (Slatyer, 1955), and also in the active layers of other crops (e.g., Begg et al., 1964). 20

~'-

× i

25

c~

==

F-

0

0

~ J ~ 500

~-

s

- ~ / ~ x

A 1000 c~ =c 1500 2000 2500

noon Day

1

Day

3

Day

5

Day

7

Fig.8. Actual transpiration rate (T), water potential in the soil (~s) and shoot x y l e m (~x), on four days in a drying cycle of seven from assumed initial conditions in response to a constant diurnal pattern of m a x i m u m or potential transpiration rate. C o m p u t e d using the model.

Fig.9 shows the integrated daily value of the ratio (T/Tm) as a function of the average daily value of available water content in the root profile. The simulation is well fitted by the dashed straight line; such an approximately linear decline has also been measured under a situation of approximately con-

lo l

~

09

5 6 /7

08

Day

no

//

~/

07

06

I

I

I

0.02 Daily

mean

I

I

0O4 available

water

I

~.__

0"06 content

0O 8

(crn 3 c m - 3 )

Fig.9. The integrated daily value of the ratio (T/Tm) for the seven day drying cycle of Fig.8, plotted against the average daily value of the available water content in the root zone. The dashed straight line is the linear approximation to it.

181

stant Trn when transpiration is limited by water availability (e.g., Slatyer, 1956; Bierhuizen, 1958; Denmead and 8haw, 1962; Gardner and Ehlig, 1963). DISCUSSION

The extensive data of Lang suggest that there is a closer correlation between (Rs + Rp) and water potential in the plant (Fig.3) than water potential in the soil (Fig.2). However, statistical support for this suggestion is weakened by the necessary correlation between q~s and qzx which is implied in eq.1. Acceptance of the relations in Figs.1 and 3 implies that Fig.2 is a family of relationships dependent on Tm, and despite considerable scatter this has support in the data. The data also suggest some possible dependence of the resistance sum on soil type. It follows that Lang's data do not strongly support the explicitly stated assumption of Gardner and Ehlig (1962, 1963) that Rs usually dominates the plant resistance Rp, since we would then expect (Rs + Rp) to be at least as closely related to qJs as it is to qJx. It was found that R s was still an order of magnitude lower than the value of fiRs + Rp), even in the driest soil (~s = - 1 , 6 2 0 J kg-1), in an analysis of Lang's data using the t h e o r y developed by Lang and Gardner (1970) and assuming that only 1/10th of the measured root length is effective in taking up water. Such calculations, provided the assumptions are valid, support Newman's (1969) conclusions that rather extreme conditions of dryness must exist for Rs to be the dominant resistance, except at low root densities (and associated high water fluxes per unit root length), or for certain sandy soils with exceptionally poor hydraulic conductivity characteristics on desaturation. However, limitations in our current understanding of the mechanisms involved make such conclusions speculative. Another possible difficulty in the interpretations of Gardner and Ehlig (1963) is the substantial potential drop measured across the plant. For example, they found this potential drop to be about 10 bar (1,000 J kg -1) in pepper plants for high values of qQ; Macklon and Weatherley {1965) measured about 5 bars (500 J kg -1) in Ricinus communis plants rooted in water; Begg and Turner (1970) measured up to 10 bars potential drop for field-grown tobacco plants. It would not be expected that this substantial potential drop across the plant, measured with ~os high or zero, would disappear as qJs is lowered, yet this is a necessary corollary to the assumption, as in Gardner and Ehlig (1962), that Rp ~ R s except at high values of ~s. Gardner and Ehlig {1962, 1963) have defined Rs as (b/k) where b is a geometric constant of the root system, and k the hydraulic conductivity of the soil. In experiments where a constant value of b was expected, Rawitz (1969) found that b varied by more than an order of magnitude. When taken together, these results and comments indicate the desirability of renewed attempts to investigate the relative magnitudes of Rs and Rp, though new techniques also appear to be required.

182 The models of Gardner and Ehlig (1962, 1963) and of Cowan (1965) assume a constant plant resistance and make assumptions leading to the conclusion that Rs is much greater than Rp except at values of ~ s close to zero. Since no such assumptions were made in the model presented here, qualitative and quantitative differences might be expected between the two types of model in the predicted relationships between variables. One feature is the midday drop in transpiration predicted by our model (Fig.8). The above mentioned models do not show this feature, which has been observed for plants under severe stress (e.g., Slatyer, 1955; Begg et al., 1964). The fact that there are qualitative similarities (e.g., in daily performance, Fig.7) between experiments using the lumped parameter model, and that of Gardner and Ehlig (1962) based on different hypotheses, warns of the danger in drawing conclusions as to mechanisms simply because of such qualitative agreement. Fig.4 shows a decrease in resistance associated with increasing transpiration, also observed by Stoker and Weatherley (1971) and others for plants grown either in culture solution or in soil. The association between low T and high (R s + Rp) (Fig.4) can occur for two quite different reasons: either a low Tin, or high plant water stress (which, in Lang's data, occurs with moderate to low soil water potentials). Superficially it would appear to be a feature favouring plant survival for the resistance sum (R s + Rp) to increase with declining ~ x (Fig.3). If this feature were absent (and all other relations remained the same), water extraction from the soil would proceed further before transpiration was markedly reduced, reducing water reserves for survival. Barrs (197 0) measured resistances of cotton plants grown in culture solution, but otherwise using similar techniques to Lang. If taken as complementary to Lang's data for higher values of ~s, these data indicate a small rise of (Rs + Rp) in this region (Fig.3) as is indicated for other crop plants also (e.g., Hansen, 1971; Stoker and Weatherley, 1971). Thus, despite quantitative differences between species, available data suggest similarity in general form for the relation between (Rs + Rp) and ~ x , again possibly implying that Rp is an important resistance component. The lumped-parameter model described above used data from controlledenvironment experiments where ~s and density of rooting were uniform throughout the profile. Such uniformity is not c o m m o n in the field, where appreciable variations in the spatial and temporal patterns of water uptake by roots have been measured (e.g., Rijtema, 1965; Van Bavel et al., 1968; Rose and Stem, 1969). It is possible that the non-linear relationships used in the model are artifacts related to the lumped-parameter assumptions in the model. However, the realism of any distributed-parameter model of water uptake by roots based on the Ohm's law analogue is limited by current uncertainties concerning mechanisms. The marked response of the lumped-parameter model to changes in ~s

183

(Fig.6) shows that the effect of profile differences in ~s is likely to be most significant, a conclusion supported both by the field experiments mentioned above, and controlled environment studies (Reicosky et al., 1972). CONCLUSIONS

(1) Current models (including this lumped-parameter model) and presently available data do not permit allocation of the dominant liquid phase resistance to the soil or to the plant although evidence is accumulating that plant resistance is significant. (2) Despite this uncertainty it is possible to develop a lumped-parameter model of water transport from soil through plant to atmosphere. Though the structure of this model is thought to be generally applicable, it employs empirical relationships derived from a particular set of experiments. (3) Any extension of the lumped-parameter model to distributed-parameter form requires further clarification of the mechanisms involved. (4) It has been illustrated once again that qualitative similarities between model performance and experiment is a necessary but not sufficient condition for concluding that the model must represent the mechanisms concerned. ACKNOWLEDGMENTS

The authors are indebted to Dr A.R.G. Lang for help and advice in providing other previously unpublished data from the extensive series of controlledenvironment experiments described by Lang and Gardner (1970). The assistance of Mr J. Knight and Mrs A. Komarowski with calculations, and Mrs K. Haszler with computer programming and display is gratefully acknowledged. REFERENCES Barrs, H. D., 1970. Controlled environment studies of the effects of variable atmosphere water stress on photosynthesis, transpiration and water status of Zea mays L. and other species. UNESCO Symposium on Plant Response to Climatic Factors, Uppsala, Sweden. Begg, J. E. and Turner, N. C., 1970. Water potential gradients in field tobacco. Plant Physiol. (Lancaster), 45: 343--346. Begg, J. E., Bierhuizen, J. F., Lemon, E. R., Misra, D. K., Slatyer, R. O. and Stern, W. R., 1964. Diurnal energy and water exchanges in bulrush millet in an area of high solar radiation. Agric. Meteorol., 1: 294--312. Bierhuizen, J. F., 1958. Some observations on the relation between transpiration and soil moisture. Neth. J. Agric. Sci., 6: 94--98. Byrne, G. F., Begg, J. E. and Hansen, G. K., 1976. Cavitation and resistance to water flow in plants (submitted to Aust. J. Biol. Res.). Cowan, I. R., 1965. Transport of water in the soil-plant-atmosphere system. J. Appl. Ecol., 2" 221--239. Denmead, O. T. and Shaw, R. T., 1962. Availability of soil water to plants as affected by soil moisture content and meteorological conditions. Agron. J., 54" 385--390.

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