an evaluation of the priestley and taylor equation and the...

Estimation ofAreaŒvapoiranspirationCPvoceeàings of a workshop held at Vancouver, B.C., Canada, August 1987). IAHS Publ. no. 177,1989. 8 9

An evaluation of the Priestley and Taylor equation and the complementary relationship using results from a mixed-layer model of the convective boundary layer

K.G. McNaughton and T.W. Spriggs Plant Physiology Division, D.S.I.R. Private Bag, Palmerston North New Zealand

Abstract A simple model of the daily development of a convective boundary layer is used to calculate daytime regional evaporation rates for conditions observed on 9 days at Cabauw in The Netherlands and a range of assumed values of surface resistance. The results are compared with evaporation rates calculated using the Priestley and Taylor (1972) (P&T) equation, and they are used to test the complementary relationship (CR) proposed by Bouchet (1963). Some support is found for the P&T equation, though the calculations show that the coefficient in the P&T equation varies somewhat among days, and the original value of 1.26 is probably too low. The calculations give no support to the CR. Rather it is shown that there is a faulty basic premise in CR theory: it is not true that large-scale advection (interpreted here as entrainment of air from the capping inversion) is independent of the surface energy balance.

Une évaluation de l'équation de Priestley et Taylor et de la relation complémentaire à partir des résultats obtenus sur un modèle de couche mixte de la couche limite convective

Résumé On s'est servi d'un simple modèle du développement journalier de la couche limite convective pour.calculer les taux d'évaporation régionale dans les conditions observées pendant 9 jours à Cabauw (Pays-Bas) ainsi que la gamme des valeurs assumées de la résistance de la surface. On a comparé les résultats aux taux d'évaporation obtenus à l'aide de l'équation de Priestley et Taylor (1972) (P&T). On s'est alors servi des mêmes résultats pour vérifier la validité de la relation complémentaire (R.c.) avancée par Bouchet (1963). Les résultats corroborent un peu l'équation de P&T bien que nos calculs démontrent que le coefficient de cette équation varie quelque peu de jour en jour et que sa valeur originelle de 1,26 est probablement trop basse. Par contre, nos calculs ne soutiennent aucunement la R.c. Bien au contraire, on a démontré que la théorie de la R.c. est fondée sur une hypothèse erronée: il est faux d'affirmer que l'advection à grande échelle (que l'on interprète ici comme l'entraînement de l'air par l'inversion au sommet de la couche limite planétaire) est indépendante du bilan énergétique de la surface.

90 K.G. McNaughton and T.W. Spriggs

INTRODUCTION

Many methods have been used to estimate evaporation from regions using standard climate data, but most of these were not designed for the purpose. Their foundations are either wholly empirical or rest on the theory of transport processes and energy exchanges in only the bottom few meters of the atmosphere. Two exceptions to this rule are the formula of Priestley & Taylor (1972) (P&T) and the various procedures based on the 'complementary relationship' (CR) proposed by Bouchet (1963) and developed by Morton (1965, 1983).

The P&T and CR methods were designed as large-scale relationships. They are based on physical arguments about processes in the whole of the turbulent planetary boundary layer, up to a thousand meters or so. Priestley & Taylor (1972) dealt with regions measuring tens of thousands of square kilometers and their arguments concerned the relative sizes of advective and radiant energy inputs to land areas of that size. Bouchet (1963) directed his arguments at the somewhat smaller scale of "at least some tens of hectares", but Morton (1965) realized that the CR was more appropriate to the regional scale. The complementary relationship has now been placed explicitly in the context of processes in the whole planetary boundary layer (Morton, 1983), so it should apply to areas of at least a thousand square kilometers.

Though both the P&T and CR methods are based on plausible physical rationales, and both have had some empirical success; neither can be derived completely from physical principles. Doubts remain about their reliability and generality. The purpose of this paper is to re-examine the P&T and CR methods by comparing results calculated from them against results from a model which simulates the surface energy balance in a growing convective planetary boundary layer (CBL) (McNaughton &. Spriggs, 1986). Before doing this we give a brief account of both the arguments leading to the P&T and CR equations and of the model of the CBL developed by McNaughton & Spriggs (1986).

THE PRIESTLEY AND TAYLOR EQUATION

Priestley & Taylor (1972) asked what controls evaporation from wet land areas at the scale of the grids used in computer solutions of numerical weather-forecasting models. These grids are typically several hundred kilometers on a side. At this scale, Priestley & Taylor argued, radiant energy receipt must dominate over advective effects since "radiation received will increase as the square of gridpoint separation, whereas advective effects will increase more or less linearly because the differences in horizontal fluxes of heat and vapour at the upwind and downwind edges of an area will not continue to increase indefinitely as the edges are moved farther apart."

Having decided that advection could be ignored, Priestley & Taylor were left with the problem of how to apportion the radiant energy into sensible and latent heat fluxes. Earlier, Priestley (1959) had argued that the atmosphere in contact with a wet surface would remain saturated if the evaporation rate (E) is given by

E = T-JT(Rn-G) (1)

where Rn is the net radiation, G is the flux of heat conducted into the ground and e = sX/Cp, where s is the slope of the saturation specific humidity curve, dq*/dT, X is the latent

An evaluation of the Priestley and Taylor equation 91

heat of vaporization of water and cp is the specific heat of air at constant pressure. Priestley & Taylor (1972) now noted that (1) is not the only possibility; the flux (EH - XE) can take any value provided that it remains constant with height (H is sensible heat). They concluded that "the solution will not be definitive until some further restrictive condition can be identified."

Priestley & Taylor (1972) were forced to proceed empirically. They asked whether (1) was still a principal component of evaporation from a wet region, and they looked for a coefficient (a) to use in (1). A value of a = 1.26 was found to fit data from several sources. They wrote their equation for evaporation from wet regions as

XE-1.26 j -J -^Rn-G) (2)

This equation has achieved wide support and is often used as a measure of 'potential evaporation' in the absence of local advection. It often gives good estimates of evaporation even for areas much smaller than 'regions' and from vegetation that, though well watered, is certainly not wet (e.g. Davies & Allen, 1973).

Priestley & Taylor had examined the energy exchanges over the sides of a box, with its walls along the grid lines, bottom at the ground, and top at the upper limit of the air directly interacting with the surface. With hindsight, we now see that the missing condition was a statement of the energy exchanges at the top of the box - the interaction between the planetary boundary layer and the atmosphere at large.

THE COMPLEMENTARY RELATIONSHIP

The arguments presented by Bouchet (1963), and adopted and reworked in a series of papers by Morton (1965, 1969, 1975, 1983), were rather different. Bouchet noted that heat and moisture released at the surface (H and XE respectively) modify the temperature and humidity (T and q respectively) of the air mass above. He argued that the 'potential evaporation' measured over a region is as much the effect of evaporation as its cause. Bouchet's new idea was that the rise in potential evaporation observed above an area as it dries out might actually be used to measure the real evaporation rate from that area.

Bouchet argued: if, for a reason independent of energy supply E is reduced below the potential rate appropriate to a wet region (Epo) then an amount of energy (Q) would be released so that

XEpo - XE = Q (3)

Since this change within the air mass over the area leaves net radiation almost unaltered, the only important effects will be on temperature, humidity and turbulence, leading to a change in potential evaporation (Ep). If the changes do not modify the transfers of energy between the modified air mass and that beyond, this released energy, Q, should just equal the increase in XEP, So without modification of the initial climate from an energy point of view, and especially without modification of the original oasis effects, one has

XEp = XEpo + Q (4)

whence, with (3)

fc + fcp — C. fcpo (5)

K.G. McNaughion and T.W. Spriggs

Equation (5) is the 'complementary relationship'. It is the central proposition in the CR method. Developments of the idea by Bouchet (1963), Morton (1965, 1983) and Brutsaert & Strieker (1979) have centred on finding suitable definitions for Epo and Ep.

The arguments for the CR do not amount to a rigorous physical derivation, and they do not lead to a natural specification of how Ep should be evaluated. Whatever the ambiguities in the meaning of terms, the assumption that adveciive processes are not important is clearly fundamental. If the total energy supply to the region were to change with the evaporation rate then G would be unequal in (3) and (4). If advective effects were to be reduced as evaporation decreases, perhaps because air within the region becomes drier and more like the air blowing in from a neighbouring region, then E + Ep would be less than 2 Epo.

The term 'regional evaporation' is used in the sense that evaporation is from an area large enough that the effects of local advection can be ignored, so any effects of horizontal air mass transfers from neighbouring areas is excluded by definition. However there is another way that the air within a 'region' can be modified by air originating outside that region, and that is by the vertical movement down to the surface from the large-scale air masses brought in by the planetary weather systems. This could loosely be thought of as large-scale advection.

Bouchet (1963) acknowledged that advection on "cyclonic" and "planetary" scales was likely, but neglected this in his further development. Morton (1983) considered the possibility of these vertical interactions and, though he made no calculations, he concluded that the direct effects of large-scale weather systems on potential evaporation are negligible except through their effects on radiant energy and rainfall.

Thus the arguments for both the P&T and the CR equations are similar in that they both exclude local advection by definition, but both rely on unproven assumptions about the vertical interaction between the air near the ground and the atmosphere at large. The CR theory relies on the restrictive requirement that the effects of these vertical interactions must be, if not actually negligible, at least independent of the surface energy budget. The P&T equation requires that the effects of vertical interactions on the evaporation rate, though not necessarily small or independent of the surface energy balance, should be independent of the normal variations in large-scale atmospheric conditions.

A MIXED-LAYER MODEL FOR REGIONAL EVAPORATION

During the day over land the sensible heat flux at the ground is usually directed upwards and the atmosphere is unstable. Convective motions develop a little way above the ground, with air rising in heated walls, then plumes until the rise of these is halted by opposing buoyancy forces in a stable capping layer. The layer thus contained between the ground and the capping inversion is called the convective boundary layer (CBL). This is the layer that interacts directly with the ground and which receives the heat and vapour released at the ground.

A major characteristic of CBLs is daily growth. A CBL grows by eroding the base of its capping inversion and incorporating this air into its own bulk. We can ask what factors control the rate of entrainment, how the temperature and humidity of the entrained air alters those within the CBL, and how this alters the surface energy balance. In doing so we describe the vertical interaction between the CBL and the air comprising the


inversion. By this route we can address the important questions about the physical basis of the P&T and CR equations.

During the first hour or so after sunrise the CBL is often shallow, perhaps only 100 m deep, with the remnants of a ground-based inversion formed during the previous night as the capping inversion. However the CBL soon grows through this layer and, for the greater part of the day, the capping inversion is one created by weather processes at "cyclonic" and "planetary" scales. The properties of this inversion are independent of the energy balance of the local region.

We use the model of the CBL developed by McNaughton & Spriggs (1986) for our analysis of regional evaporation. This very simple mode! was developed to describe the energy balance within a growing CBL. Though simple, it has been shown to perform quite well; it is quite adequate for the comparisons made here. Only a brief outline is given below.

The CBL is represented as a well-mixed layer of air of thickness (h), potential temperature (8m) and specific humidity (qm). Above the CBL is an inversion with known temperature and humidity profiles 9s(z) and qs(z), respectively. Air from the inversion is entrained into the CBL as it grows. Between the bulk of the CBL and the ground is a thin' layer of air where mixing is not perfect and significant gradients of temperature and humidity may occur. The ground is assumed to be completely covered by vegetation so that transpiration is the only important component of evaporation.

The model is specified by a set of equations. These are: a budget equation for heat storage in the CBL

p c p h ^ = H + pcp(es - 9 m ) * | (6)

and a budget equation for vapour

p h ^ a = E + p(qs - q m ) ^ ; (7)

the energy balance and the big-leaf equations to specify the lower boundary, viz,

H + XE = Rn - G (8)

and

E = f [q*(6c) - qc] (9)

where p is the density of air, qc is the specific humidity at the canopy surface and q*(8c) is the saturation specific humidity at the canopy temperature 8C, and rc is the canopy resistance of the vegetation covering the uniform regional surface; and equations for transport of sensible heat and vapour across the surface layer given by


H = ^ ( 9 0 - e m ) (10) •as

and

E=f-(qc-qm) (11) 'as

where ras is the surface-layer resistance, for which the relationship

cr'"(^) <12)

was used, where L is the Obukhov length and u, is the friction velocity.

This leaves only the rate of growth of the CBL (dh/dt) to be specified and here a number of choices are possible. McNaughton & Spriggs (1986) tried two alternative formulations for dh/dt. One was based on a parameterization of the budget of turbulent kinetic energy in the CBL (Driedonks, 1982) and the other given by

JËL . _ t k _ (13 ) dt pCphTv '

which is simpler but more approximate. It was found that both choices gave similar values of regional evaporation. The simpler formulation (13) is used in the calculations below. In (13), Hv is the virtual heat flux (Hv = H + 0.07ÀE) and v is the strength of the inversion at the top of the CBL, given by the gradient of potential virtual temperature (8V = 8(1 + 0.61 q)) immediately above the inversion base but with a small correction for moisture content.

Using this model, evaporation rates can be calculated when all of the external conditions are specified. The early-morning temperature and humidity in the mixed layer, (9mo and qm0, respectively) are needed; the profiles of temperature and humidity in the inversion (9s(z) and qs(z), respectively) must be given for heights up to the full height of CBL growth; and the forcing functions (Rn - G), u* and rc must be specified throughout the day.

The model has been tested (McNaughton & Spriggs, 1986) using 9 days of data from the site of the K.N.M.I. tower at Cabauw in the Netherlands (Driedonks, 1981). Simulated rates of evaporation agreed very well with measured values. Inversion strengths recorded at Cabauw varied from very weak at 0.8°C krrr1, to very strong at 21.9°C km~1. These simulations were initialized using conditions recorded shortly after dawn, and they were continued through the hours when the CBL was growing rapidly, typically until 3 pm.

Figs. 1 and 2 show an example of the inputs to the model and the results of a simulation for day 77217 of the Cabauw data set.


S »ooo

r £ 400

S * 200

1S 20 2» Potential Temperature (C)

R„-G

B 12 Time (houro)

Ai q«.

Specific Humidity (g kg"1)

Time (houra)

F/G. 1. Data inputs used in simulations of evaporation and CBL height growth for Cabauw day 77217. Initial profiles of temperature (0s) and specific humidity (qs) are from observations at 0545 hrs, and the time course of the forcing functions (Rn - G), rc and ra are from measurements made at the Cabauw tower site. Data are from Driedonks (1981).

500

• 5 3 0 0

c Q X

U-

s _

c UJ

100

Cabauw Day 77217

Observed Rn-G , ' W

Simulated XE

8 12

Time (hours) 16

FIG. 2. Results of a simulation of evaporation for Cabauw day 77217 using the mixed layer model of McNaughton & Spriggs (1986). Data inputs are as shown in Fig. 1. Simulations ( solid line ) generally agree very well with observations ( open circles).


CALCULATIONS OF E, Ep AMD Epo

Values of Epo were calculated lor each of the 9 days at Cabauw by repeating the simulations as described above, but with r0 set to zero. Epo is the evaporation rate that would have occurred if all initial conditions and forcing functions had been the same as on those days at Cabauw, but the surface was wet. Epo is not quite the same as the evaporation rate that would have occurred had the ground at Cabauw truly been wet for, in that case, with cooler ground the net radiation would have been a little larger; the initial saturation deficit would have been smaller also. However, the value of Epo calculated here is quite in keeping with the model proposed by Bouchet, who assumed that the energy supply does not depend on the evaporation rate.

Values of E can also be calculated for any other values of rc, and so repeated simulations can be performed with a range of r0 values to find the relationship between E and rc for each day. With larger rc values the CBL sometimes grew to a height greater than that for which profiles of 9S and qs were defined in the Cabauw data set. In these cases the top segments of the profiles were simply extrapolated linearly upwards as far as necessary.

Finally, values of Ep can be calculated by any chosen meteorological formula, such as Penman's equation (Penman, 1948), given by

XE = r f T ( R n - G ) + r ^ | 7 r (14)

where

r *= (1 , 2 0 0 54u) SmA <15>

and the saturation deficit (D) and wind speed (u) in m s-1, are both calculated using values for 2 meters above ground. This is possible because the solutions of the mixed-layer model give the temperature and humidity throughout the CBL for each day. Thus 9, q, and u at 2 meters are known through each day and Ep can be calculated and averaged over the simulation period of each day. Ep can be found as a function of rc by repeating the simulations for a range of values of rc.

It is not clear that Penman's equation is the best measure of Ep for use in the CR method. Here we take the view that it is at least a widely used and often successful measure of potential evaporation in other applications, so it is a useful starting point. Further discussion on this issue is given below, when the complementary relationship is tested using the computed results.

COMPARISONS WITH THE PRIESTLEY AND TAYLOR EQUATION

Fig. 3 shows the relationship between the calculated P&T coefficient a and canopy resistance for each of the 9 days. Average evaporation rates over the simulated period of each day were calculated using constant daily values of rc ranging from 0 to 10,000 s rrH, with all other inputs as observed at Cabauw. Values of e were calculated using the


temperatures calculated at 2 meters. Values of a were then calculated as E/Eeq, where Eeq is the expression on the right hand side of (1).

1 10 100 1000 Canopy Resistance (s m - ' )

FIG. 3. Priestley- Taylor coefficient a calculated for a range of rc for each of the 9 days of the Cabauw data set. For each day all inputs were unchanged from the values observed at Cabauw except rc, which was varied over the range shown.

For individual days the calculated evaporation rate declines fairly slowly with rc as the input value of rc increases from 0 to about 60 s rrr1. In six of the nine cases the results for low canopy resistances agree with the P&T value of a = 1.26 within ± 10%. In three cases a is significantly greater than 1.26. These values were for the days with the coolest temperature profiles (0s(z)) above the CBL. The humidity profiles (qs(z)) were amongst the driest also and net radiation totals were low on these days. The strengths of the capping inversions on these three high-a days ranged from 1.5°C krrH to 6.5°C krrH and were similar to values on other days.

These results show that the P&T equation is not an exact relationship. Indeed this was already known from a sensitivity analysis of the mixed-layer model (McNaughton & Spriggs, 1986). Though our tests span a limited range of conditions, they are enough to provide examples of plausible conditions where evaporation from wet regions is signficantly different to that given by the P&T equation (2).

Evaporation declined at higher values of rc in a similar way on all days. The spread of a values for any value of rc is not very large, perhaps giving rise to hopes that useful estimates of a might be made from canopy resistance data alone. However, potential temperatures within the inversions at Cabauw covered but a narrow range - from about 12°C to 25°C. A wider range of conditions may well have given a wider spread of a values.

These simulations are for the hours of the day when the CBL grew rapidly at Cabauw. Calculations over full days would include night-time conditions with a ground-


based inversion and with XE > (Rn - G). Values of a for full 24-hour periods would probably be larger than those shown in Fig. 3.

What the calculations do show is that a values tend to be similar over a range of weather conditions, and to change only a small amount over the range of surface resistance values from 0 to about 60 s rrH. Such values of rc are characteristic of well-watered, rapdily-growing, full-cover agricultural crops and improved pastures. The results do give some guarded support to the empirical value of a = 1.26 for calculations of evaporation in the daylight hours for this range of values of rc . We could also support a slightly higher value of a when the surface is wet (cf. Brutsaert & Strieker, 1979). Values of a are smaller for vegetation with larger canopy resistances, such as forests, even when such higher values do not indicate water stress.

COMPARISONS WITH THE COMPLEMENTARY RELATIONSHIP

Fig. 4 shows the calculated values of E/Epo and Ep/Epo plotted against canopy resistance using data from the first day of the Cabauw data set, 77217. Also shown is the value of 0.5(E + Ep)/Epo which CR theory proposes should be a horizontal line on Fig. 4, equal to one for all rc. The values of E used to construct Fig. 4 are the same as those used in the calculations for Fig. 3, but this time they are normalized using Epo rather than Egq. Thus E/Epo is identically equal to one when rc = 0.

The simulations for each value of rc give different solutions for the sensible heat flux and so different results for the rate of growth of the CBL. Fig. 5 shows the height of the CBL through the day calculated using canopy resistance values of 10,100 and 1,000 s m-1. Sensible heat fluxes are larger for larger rc, so the CBL grows more rapidly and entrains more dry air from the capping inversion. This leads to greater saturation deficits within the CBL when rc is larger, as shown for day 77217 in Fig. 6.

Similar calculations can be made for each of the other 8 days of the Cabauw data set. The extremes of strength of the capping inversion occurred on days 78285, when the capping inversion was very strong at 22°C krrH, and 77262, when the inversion was very weak at 0.8°C krrH. Results from these two days are shown in Figs. 7 and 8. These two days are chosen for presentation because the effect of the surface energy balance on the temperature and humidity of the CBL is strongly influenced by the strength of the capping inversion, and the CR model showed extremes in performance on these two days. Other days displayed intermediate behaviour; day 77217 (Fig. 4) is an example.

On day 77217 Penman's equation gives a value of Ep that is about 14% less than Epo when rc = 0, while CR theory has the definitional requirement that Ep and Epo be equal over a wet regional surface. In search of a better definition of Ep, it would be possible to raise the Penman estimate by a factor of 1.16 on this day. Such an adjustment would also improve the fit on the other days. However such an adjustment would not be enough to satisfy the CR requirement that Ep/Epo = 2 at the dry end of the scale on day 77217.

Alternatively, it would be possible to select separate weightings for the equilibrium and aerodynamic terms of Penman's equation (14) and to force a fit between the data for day 77217 and the CR relationship at both the wet and dry ends of the range of rc. Indeed, with such 'local calibration' the CR could be applied on any day. However, our results show that different factors would be needed for different days because conditions in the CBL are strongly influenced by conditions above, in a way not allowed for in CR theory.


1.5

1.0

E/Ep,

0.5

0.0

—- 1 Cabauw Day 77217

.TTTr-?,

i

i -- r—

^ - - ' " V E , .

10 100 1000 Canopy Resistance (s m~1)

FIG. 4. CR diagram for Cabauw day 77217. Epo is the simulated evaporation calculated by setting rc to zero. E is the simulated evaporation rate for each r0 and Ep is calculated using Penman's equation with meteorological inputs from the simulations, appropriate to 2 meters above ground.

9 12 Time (hours)

15

FIG. 5. Calculated course of height growth of the CBL for Cabauw day 77217 from simulations using three different values of canopy resistance as input. At the higher values of rc a lower evaporation rate leads to a greater sensible heat flux and faster growth of the CBL


20

* 15

a

10

5 -

1 - 1 Cabauw Day 77217

-

~ yS

1 1

1

/ ^ 1000

__ ---— 100

r„=10 s m"'

-

~~

9 12 Time (hours)

15

FIG. 6.

FIG. 7.

Calculated daily course of saturation deficit for the same simulations as shown in Fig. 5. Greater rc input values lead to a larger saturation deficit in the mixed layer because air in the CBL receives more sensible heat flux and less latent heat from the surface at higher rc, and because more dry air is entrained into the CBL from the capping inversion with higher rc.

1.5

E/Epo 1.0

0.5

0.0

1 — -Cabauw Day 78285

-

i

i " ' r

^ " " \

10 100 1000 Canopy Resistance (s rrf1)

CR diagram for Cabauw day 78285 when the capping inversion was very strong at21.9°C km-1. CBL height growth was small in the simulations and changed conditions within the CBL strongly reflected the surface fluxes of heat and vapour. Calculated Ep is greatly increased at larger values of rc when the sensible heat flux was large and the latent heat flux small.


Cabauw Day 77262

1 10 100 1000 Canopy Resistance (s rrf1)

FIG. 8. CR diagram for Cabauw day 77262 when the capping inversion was very weak at 0.8°C km-1. The CBL grows very rapidly in these simulations and saturation deficit in the mixed layer of the CBL is closely controlled by the temperature and humidity of the air entrained from the inversion layer. Calculated values of Ep are fairly insensitive to changes in input values ofrc.

Under the strong inversion on day 78285 the CBL grew slowly in the simulations: with rc set to 1 s rrH it grew from 100 m at 0715 hrs to 279 m at 1415 hrs, while with rc set to 1000 s rrH it grew to a height of 468 m. For the lower canopy resistance, r0 = 1 s rrH, little dry air was entrained into the CBL and the surface latent heat flux was larger than the equilibrium rate so the potential saturation deficit in the mixed layer (Dm) remained low all day, never exceeding 1.8 g kpH. Potential saturation deficit is the saturation deficit that a parcel of air would have if brought adiabatically to the surface. With rc set to 1000 s rrH more dry air was entrained and the surface latent heat flux was much less than the equilibrium evaporation rate; Dm rose steadily to 8.9 g kg-1. The larger values of Dm are reflected in the larger values of daily Ep calculated for the higher input values of rc, as shown in Fig. 7. The results for day 78285 most nearly support the complementary relationship (5).

Under the weak inversion of day 77262 the CBL grew much more rapidly, from 50 m at 0615 hrs to 1115 m at 1545 hrs when rc was set to 1 s rrH and to 1984 m when rc was set to 1000 s rrH, but, in consequence, the potential saturation deficit took on a narrower range of values. The value of D within the inversion was about 8 g kg-1 and did not change much with height. Even with rc set to 1 s rrH a large volume of this air was entrained and Dm rose to 4.1 g kg-1. When rc was set to 1000 s rrH the rate of entrainment was very rapid and Dm reached 7.2 g kg-1 - still below the upper bound set by conditions within the inversion above. Thus the values of Dm were confined to a narrower range in these simulations and Ep also took on a narrower range of values, as shown in Fig. 8. The results fit the CR very poorly.


The saturation deficit in the mixed layer reflected the conditions within the inversion rather more and the surface fluxes rather less on the weak-inversion day, day-77262, than on the strong-inversion day 78285. Empirical weightings that could be applied to the terms of Penman's equation on the one day to force a fit to the requirements of the CR would not be appropriate to the other.

It may be that Ep should have been based on some definition other than Penman's equation. Much research on the CR has centred on the search for an acceptable definition of Ep; both conceptual and empirical arguments have been developed. Bouchet (1963), Morton (1965,1975,1983) and Brutsaert & Strieker (1979) all make suggestions, but none seems to be definitive. Now it seems that none is possible. Any feedback between the surface fluxes of sensible and latent heat and the regional air mass must manifest itself chiefly through a dependence of temperature and humidity on the surface fluxes. Our investigation using the mixed-layer model has shown that the dependence exists, but that it can be highly modified by the strength of the inversion capping the CBL. Our use of the Penman definition of Ep merely gives an example of the difficulty.

This failure of the CR is related to what we can now see as a fundamental error in its derivation. The assumption, in the theory, that changes in the surface energy balance do not modify transfers of energy between the air mass interacting with the surface and that beyond, is seen clearly to be invalid. Entrainment of air from a capping inversion into the CBL is strongly controlled by the surface energy balance. This leads to important modification of the saturation deficit in the CBL and, furthermore, the degree of this modification depends on the strength of the capping inversion and the potential saturation deficit of the air within that inversion. Morton (1983) is wrong in his claim that entrainment has little effect on evaporation.

GENERAL DISCUSSION

Developments in the meteorology of planetary boundary layers over the past 20 years, and the more recent integration of these results with evaporation theory, have given new insight into the processes that control evaporation at the regional scale. The models can be particularly simple for dry (i.e. no condensation), convective boundary layers (CBLs). Though they do not encompass a sufficient range of atmospheric conditions and require too much input data to give fully operational estimates of daily regional evaporation, they can be used to gain physical insight into simpler methods that are widely used.

An interesting feature of the results described in this paper is the reasonably small variation in the P&T a values calculated for the wet end of the range, both from day to day and from one value of surface resistance to another in the range from zero up to about 50 or 60 s rrH. This small variation occurs because over wet surfaces with constant available energy the CBL height growth is sensitively controlled within a negative feedback loop: lower E leads to higher H, greater entrainment of dry air, higher saturation deficit in the CBL and so, to complete the loop, to a higher evaporation rate. The direct effect of the surface fluxes on the saturation deficit of the CBL has an additional, but smaller effect on this feedback.

Thus the observation by Thomthwaite (Wilm, 1944), that "évapotranspiration is independent of the character of the plant cover, of soil type and of land utilization to the extent that it varies under normal (non water-stressed) conditions" is seen to be based on a real physical conservativeness in natural evaporation. This empirical generalization led Thomthwaite to propose the concept of 'Potential Evaporation'. It is now seen that the


concept does not express an exact truth, but was a working approximation which was quite understandable given the quality of the data with which Thornthwaite had to work. Our results explain why the concept of potential evaporation has been popular in agriculture, where unstressed crops usually have rc values less than 60 s rrH. They also explain why potential evaporation formulae have found much less favour in forestry, where rc values are usually greater than 60 s rrH.

There are therefore understandable limits to the accuracy within which any potential evaporation formula might be expected to work. The Priestley & Taylor equation is no exception. We have calculated a values with a spread of about 20% for evaporation from a wet region using the limited range of conditions observed at Cabauw. This spread of a values is acceptably small for many puposes. Further investigations using a wider range of conditions would appear to be justified. Perhaps these would confirm the suggestions from our work that the P&T equation should be applied only to the daytime (convective) hours, and that a should be increased slightly from the original 1.26. Of course if the equation were to be used for estimating evaporation from well-watered agricultural regions (i.e. rc in the range 20 < rc < 60 s rrr1) rather than wet regions then a = 1.26 might be close to optimal.

It should also be noted that the calculated values of Ep, though less than Epo by about 15%, are very good estimates of E in the 'agricultural' range of canopy resistance, 20 < rc < 60 s rrH. In this range Penman's equation performs a little better than the P&T equation, but, given the limited nature of our comparisons, the improvement is not significant.

The complementary relationship finds no support in our results. Rather, we find that we are able to generate case studies that are widely at variance with the requirements of the CR. This finding is supported by physical arguments which show that a basic assumption of CR theory, that the interaction between the CBL and the rest of the atmosphere does not depend on the surface energy balance, is not even approximately satisfied in the real atmosphere. Although Morton (1983) has achieved some impressive results in fitting many data sets by adjusting functions and parameters in his version of the CR, the prospects for finding any generally-valid formulation of the CR seems remote.

REFERENCES

Bouchet, R.J. (1963) Evapotranspiration réelle et potentielle, signification climatique. In: Int. Assoc. Sci. Hydro!., Proc. Berkeley, Calif. Symp. Publ. No. 62, 134-142.

Brutsaert. W. & Strieker, H. (1979) An advection-aridity approach to estimate actual regional evaporation. Wat. Resour. Res. 15,443-450.

Davies, J.Â. & Allen, CD. (1973) Equilibrium, potential and actual evaporation from cropped surfaces in southern Ontario. J. Appl. Meteorol. 12, 649-657.

Driedonks, A.G.M. (1981 ) Dynamics of the well-mixed atmospheric boundary layer. Scientific Report W.R. 81-2 K.N.M.I., De Bilt, The Netherlands.

Driedonks, A.G.M. (1982) Models and observations of the growth of the atmospheric boundary layer. Boundary-Layer Meteorol. 23, 283-306.

McNaughton, K.G. & Spriggs, T.W. (1986) A mixed-layer model for regional evaporation. Boundary-Layer Meteorol. 34, 243-262.

Morton, F.I. (1965) Potential evaporation and river basin evaporation. J. Hydraul. Div., Proc. Am. Soc. Civ. Eng. 91 (HY6), 67-97.


Morton, F.I. (1969) Potential evaporation as a manifestation of regional evaporation. Wat. Resour. Res. 5, 1244-1255.

Morton, F.I. (1975) Estimating evaporation and transpiration from climatological observations. J. Appl. Meteorol. 14,488-497.

Morton, F.I. (1983) Operational estimates of aerial évapotranspiration and their significance to the science and practice of hydrology. J. Hydrol. 66,1-76.

Penman, H.L. (1948) Natural evaporation from open water, bare soil, and grass. Proc. R. Soc. London, A 193, 120-145.

Priestley, C.H.B. (1959) Turbulent Transport in iha Lower Atmosphere. University of Chicago Press, Chicago.

Priestley, C.H.B. & Taylor, R.J. (1972) On the assessment of the surface heat flux and evaporation using large-scale parameters. Mon. Weather RevAQO, 81-92.

Wilm, H.G. (1944) Report of committee on evaporation and transpiration. Trans. Am. Geophys. Union 29, 258-262.

an evaluation of the priestley and taylor equation and the...

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