Comparison of bowen ratio and aerodynamic estimates of evapotranspiration

Agricultural and Forest Meteorology, 49 (1990) 243-256

243

Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands

C O M P A R I S O N OF B O W E N R A T I O A N D A E R O D Y N A M I C E S T I M A T E S OF E V A P O T R A N S P I R A T I O N *

P. PIERI I and M. FUCHS 2

IINRA, Laboratoire de Bioclimatologie, Pont de-la-Maye (France) 2ARO, Institute of Soils and Water, Bet Dagan (Israel) (Received January 4, 1989; revision accepted July 18, 1989)

ABSTRACT Pieri, P. and Fuchs, M., 1990. Comparison of Bowen ratio and aerodynamic estimates of evapotranspiration. Agric. For. Meteorol., 49: 243-256. The energy balance of an irrigated cotton field was determined using the Bowen ratio to partition between sensible and latent heat flux density. These data verified estimates of the sensible heat flux density derived from either profiles of wind and temperature at six levels, or from a simplified aerodynamic method using wind and temperature differences between two levels only. The full profile provided good estimates of the sensible heat flux density, but the diurnal course lagged by ~ 1 h behind the values obtained from the Bowen ratio. The aerodynamic properties of the crop did not appear to be affected by the row structure of the canopy. Results of the simplified method varied with the selection of levels between which the differences were measured. Flux densities determined across some of the layers consistently matched those obtained by the Bowen ratio, but objective criteria for making the proper choice could not be defined.

INTRODUCTION

Automated irrigation systems can be operated to match the crop water requirements. Their use becomes optimal when application is closely adjusted to the soil water deficit. Correct irrigation scheduling is therefore essential to efficient management of the water resources. Automated systems are costly; their economic justification is the increased production resulting from the high frequency water application which they make possible. This operating mode requires reliable daily estimates of evaporation based on relatively simple measurements, to be carried out routinely under field conditions. Meteorological estimates of the water flux in the atmosphere are not affected by the small scale field variability and therefore provide a representative value *Contribution from the Agricultural Research Organization, The Volcani Center, Bet Dagan, Israel, No. 2518E, 1988 series.

0168-1923/90/$03.50

© 1990 Elsevier Science Publishers B.V.

244

P. P I E R I A N D M. F U C H S

of evapotranspiration. However, sustained operation of the instrumentation for long periods is technically difficult. The purpose of the present study was to compare the theoretical validity, instrumentation complexity and performance of three meteorological determinations of evapotranspiration. The first method selected is the surface energy balance using the Bowen ratio to partition between latent and sensible heat flux density. It is based on the assumption that the eddy diffusivities for sensible and latent heat flux are equal. The Bowen ratio can then be calculated from the measurement of the temperature and humidity gradient B = ~AT/Ae

(1 )

where y is the psychrometric constant, and A T and Ae are the mean temperature difference and water vapor difference, respectively, between two levels of free streaming air. T and e must be measured at the same height and preferably on the same sample of air. The similarity assumption implies that the same eddies transport heat and vapor, and consequently that sources of heat and vapor on the surface are uniformly distributed. During the early stages of the cotton crop development, most of the sensible heat is generated in the interrow bare soil whereas vapor formation occurs in the foliage. This horizontal heterogeneity may invalidate the similarity assumption (Garratt, 1978). In order to overcome the heterogeneity caused by the horizontal distribution of the foliage, the measurements must be made sufficiently high above the surface. Nevertheless, they must remain close enough to the ground to be within the air surface layer. The thickness of this layer grows by 1 m per 100 m of upwind uniform fetch (e.g., Brutsaert, 1982 ). As the temperature and humidity of the turbulent air stream fluctuate, their gradient is obtained as a time average over an interval sufficiently long to include all eddies contributing significantly to the vertical transport, but avoiding non-turbulent fluctuations like the diurnal cycle or the effect of intermittent clouds. An interval of 30 min was found to be appropriate. The latent heat equivalent of evapotranspiration is E=(Rn-G)/(I+B)

(2)

where R n is the net radiation and G is the soil heat flux density. The singularity for B -- - 1 occurs when R n - G = 0 and energy flux densities are small. Excluding the singular periods from daily summation of eq. 2 does not introduce serious errors. The Bowen ratio method has been thoroughly tested in the past and its validity as a reference evapotranspiration measurement has been well established (Fritschen, 1966; Fuchs and Tanner, 1970; Sinclair et al., 1975). The major difficulty associated with the approach is the instrumentation, which must detect temperature and vapor pressure differences of the same magnitude as the bias of the sensors. The common procedure for minimizing this error is

E S T I M A T E S OF E V A P O T R A N S P I R A T I O N

245

to regularly interchange the position of the sensors during the sampling time interval. The continuous operation of such a system for an entire growing season requires special technical skills. The major difficulty is the maintenance of a properly functioning humidity sensor. The second method is based on the aerodynamic determination of the sensible heat flux density

H= -p%u, T,

(3)

where p is the density of air and % is the specific heat of air. The friction velocity, u , , is derived from the wind profile measurement

u , = ku/{ln[ (z-D)/Zo] - qXm}

(4)

where k= 0.41 is the von Karman constant, u is the wind velocity at height z above the ground, D is the zero speed displacement height caused by the canopy, zo is the roughness length of the surface and ~gmis the diabatic effect on m o m e n t u m transport. The temperature, T , , is determined similarly from the temperature profile measurement

T . --k( T - To) / {ln[ ( z - D ) /zo] - gxh}

(5)

Here, T is the air temperature at height z, To is the air temperature extrapolated to z = D + Zo and ~h is the diabatic effect for heat. The contribution of thermally induced mixing in the transport properties of the surface air layer is accounted for by the diabatic term which is related to the Monin-Obukhov length, L, defined as

L = T u , 2/kgT,

(6)

where g is the gravitational acceleration. The diabatic influence function has been formulated in many ways (Brutsaert, 1982). The formulation used here is that proposed by Dyer and Hicks (1970) 0m ---- ( 1 - 1 6 ~ ) -0.25

(7)

-- (1-- 16()-°'5°

(8)

where ~= ( z - D ) / L , 0m and ~ and are for m o m e n t u m and heat, respectively. It offers the possibility of determining by simple, closed form integration (Paulson, 1970) ~m =2In[ (1 + x ) / 2 ] + I n [ ( 1 + x 2 ) / 2 ] -- 2arctan (x) +7r/2 ~h ----ln[ (1 + x2)/2]

(9) (10)

where x = 1/0m. Equations 4 and 5 require data for u and T at three levels, but accuracy is improved when a least squares procedure smooths profiles from several levels of measurements. As the diabatic effects are functions of u , and T , , the solutions are iterative.

246

P. PIERI AND M. FUCHS

~v Regression of A u versus In(zr/zn , !AT versus In(zr/z~ 1

r----

t

I kJ/

-U*o

index ]k ~t h i-~

n : 2,.

,6

Ho

k g Hi 1 ]

Regression of Z~uversus In(~nA n 1)-*(~o)+~(~n AT versus n(~n/t n ,)-*(~o)+~'(~

,) ,!

Fig. 1. Flow chart of the iterative solution of the profile equations.

The numerical procedure is explained in Fig. 1. Starting with an initial value D-- 0 and H = 0 ( L = ov ), wind speed and temperature differences between successive heights are linearly regressed against the natural logarithm of the corresponding height ratios. The slope of the resulting line produces the first estimates of u . and H, which are used to compute the initial L. The iteration starts with these values and continues until H remains within 0.01 of its previous value. Convergence is achieved within 40 iterations at most. The parameter D results from an optimization procedure seeking the value for which In ( z - D ) produces a linear fit to u with the least squares error. This procedure is an inner loop within the iterative solution of the sensible heat flux density equation. Evapotranspiration is determined as the residual term of the energy balance E=Rn-G-H

(11)

This approach eliminates the difficult measurement of the water vapor gradient in the air. However, the similarity assumption for momentum and heat transport for non-homogeneous surfaces is questionable and involves empiri-

ESTIMATES OF EVAPOTRANSPIRATION

247

cal corrections (Garratt and Hicks, 1973). The diabatic effects introduce additional errors in the method. From an instrumentation viewpoint, it is difficult to maintain continuous operation of wind and temperature profile measurements in a sprinkler-irrigated field. The mast holding the instruments must be dismantled to allow passage of the sprinkler line. A simplified version of the method has been proposed by Itier (1980) and Riou (1982) based on the measurements of Au and AT, i.e., requiring wind speed and temperature at two levels, z-- zl and z = Ze, only. In this version, the aerodynamic method is easier to apply in routine field conditions

H=KIATAu [ 1 - K 2 A T / ( A u ) 2] o75

(12)

with

K~ =pcp[k/ln(a) ]2 K2 = 16z.

(g/T)ln(a)

(13) (14)

z . = [(Zl - D ) (z2 - D ) ]o.~

(15)

a = (z2 -D)/(zl

(16)

-D)

The value of D was taken from the wind profile m e a s u r e m e n t in near neutral stability conditions, but results show that D could be predicted from crop height data. Several numerical approximations are introduced in this approach and its validity has not yet been established for the heterogeneous conditions of a structured row crop like cotton. METHODS The experiments were carried out in a flat cotton field, 400 × 1300 m, in Israel's coastal plain (latitude 32 ° 20' N, longitude 34 ° 45'E ). A self-propelled sprayer line could irrigate its entire area within 8 days. Cotton (Gossypium barbadense L. var. 'Pima') was planted in rows 0.96 m apart at a density of 89 000 per ha. The field was surrounded by other irrigated cotton fields. The instrumentation was installed on a spot with a daytime fetch of at least 300 m and a night-time fetch of 100 m (Fig. 2 ). The line irrigated a 200-m band over the length of the field. After reaching the end, it pivoted 180 ° to the other half of the field without delivering water. Irrigation was restarted with the line progressing in the opposite direction. This movement led to a time interval of 4-8 days between the irrigations of the two halves of the field. The consequent change in soil moisture status could temporarily reduce the fetch over which humidity is uniform to 100 m. The output of all meteorological sensors was recorded digitally in a data logger (Campbell Scientific, Logan, UT, U.S.A., Model CR7). Sampling frequency was 0.1 Hz with an average taken every 15 min. The device used for

248

P. PIERI AND M. FUCHS

E DAY-TIM

~

~ I\

irrigotionline /-- experirnento[ NIGHT-TIME -WIND DIRECTIONS

N t~ ROW ~

~-" ~ = 34°

DIRECTION . I00 rn

S Fig. 2. Layout of experimentalsite and direction of prevalent winds. measuring the Bowen ratio has two ventilated wet and dry bulb single-junction copper constantan thermocouples. Their position is interchanged every 15 min by rotation around a central axis. A detailed description of the apparatus has already been published (Fuchs et al., 1987). The rotation is triggered by the data logger which recorded the data after allowing a 2-min pause for sensor equilibration. Wind speed and air temperature profiles were taken from six levels. Wind was measured with sensitive cup anemometers (Science Associates, Princeton, NJ, U.S.A. ) equipped with an optical rotation counting system; their starting velocity is 0.20 m s- 1. The thermometers consisted of a four-junction copper constantan thermopile with its reference inserted in a polystyrene-insulated massive brass cylinder weighing 2 kg. The temperature of the brass cylinder was measured with a single-junction copper constantan thermocouple using the electronic cold junction of the data logger. Measurements were made of the temperature differences between each level and the third level from the bottom. The temperature difference between the third level and the brass cylinder was also recorded. This procedure optimizes the performance of the data logger and achieves temperature difference mea-

ESTIMATES OF EVAPOTRANSPIRATION

249

surements with a precision close to 0.001 ° C. However, the absolute temperature accuracy was not better than 0.1 °C, as specified by the electronic reference junction of the data logger. Anemometer and thermometer performance was checked prior to the experiment, on a horizontal stand. The net radiation was measured by a net pyrradiometer (Swissteco, Melbourne, Victoria, Australia) recalibrated for short-wave sensitivity against a Precision Spectral pyranometer (Eppley, Newport, RI, U.S.A. ) by the shading technique. Its polyethylene domes were inflated with nitrogen. The pressure was regulated by a 0.1-m column of liquid paraffin. Four soil heat flux plates (Fuchs and Hadas, 1973 ), buried 0.05 m below the soil surface and connected in parallel, measured the soil heat flux density. The experiment was carried out in 1986, from DOY (day of year) 145 to D OY 213, with brief interruptions during the passage of the irrigation line over the plot. The plant height increased from 0.37 to 1.12 m during this period. The instrumentation level was periodically adjusted to plant growth. The levels of the two psychrometers measuring the Bowen ratio were kept 0.60 m apart, the lowest remaining 0.30 m above plant height. The heights for wind and temperature profiles were logarithmically distributed between 0.30 m above plant height and the height z=4.0 m or z = 4.5 m above the ground, at the beginning and end of the experiment, respectively. Additional weather data were available from an automated meteorological station located in a fenced clearing in a neighboring cotton field. The distance between the station and the measurement plot was < 1000 m. The geometric characteristics of the rows and L A I (leaf area index) were recorded periodically (Figs. 3 and 4). L A I was obtained from direct measurements of the area of all the leaves from 12 randomly selected plants using an optical area meter (Licor, Lincoln, NE, U.S.A. ). The sampling procedure caused some inconsistencies among the results. Nevertheless, Fig. 3 shows that the experiment characterizes the period of rapid L A I development until DOY 180, when it reaches a nearly constant value between 3 and 3.5.

3.o 4.0

.J

2.0

LO

o

I 150

170

I

I

190 210 DAY OF YEAR

Fig. 3. Seasonalchangeof the cotton crop leafarea index.

i 230

250

P. P I E R I

1.2

I

I

I

I

I

I

I

1.0

7

AND M. FUCHS

12 -l0

0.8 ~

0.8

-

0.6

- 0.6

0.4 i

~

DISPLACEMENT(m) 0 RELATIVE VOLUME

V

0,2 0 140

150

I 160

L 170

I L 180 190 DAY OF YEAR

I 200

I 210

°>

0.4 "~ J LLI Q:~ 02 0 220

Fig. 4. Seasonal evolution of the geometric characteristics of the cotton crop row structure.

RESULTS AND DISCUSSION The roughness length, Zo, and the displacement height, D, associated with the aerodynamic method were derived from the analysis of the wind and temperature profile measurements. The roughness length, Zo, is obtained by extrapolation of the wind profile to the height above D where the speed is 0. Values of Zo and D had a very distinct diurnal variation with a very sharp transition between day and night (Fig. 5 ). The cause of this separation should be associated with atmospheric stability. The temperature profiles and the energy balance clearly indicate that the surface air layer is unstable during most of the daylight hours; at night, the layer is stable. Airflow in the region is dominated by a sea breeze regime with moderate to strong daytime westerly winds. Night-time is characterized by low wind speed from the east. The selected diabatic model is known to provide a poor account of the transport process within the air surface layer when wind speed is low and thermal stratification is stable (Brutsaert, 1982); therefore the values of Zo and D deriving from profiles measured during the night are considered erroneous. However, as night-time evapotranspiration is low, this particular problem of the aerodynamic method has no practical importance. The correlation between wind speed and direction, and atmospheric stability did not allow us to investigate whether the angle between air stream and row has an effect on Zo and D. Daytime values of Zo are ~ 10% of the average crop height. The seasonal profile of D (Fig. 4) follows very closely that of the row size development. The determined value is close to the 2/3 crop height as derived from theory (Cowan, 1968; Kondo, 1971 ) and from other measurements (Stanhill and Fuchs, 1968). The scatter diagram of Zoversus D, normalized to crop height, for the profiles measured during periods of near neutral stability (L < - 1 0 0 m) in Fig. 6, shows a cluster of points similar to that found above

ESTIMATES OF EVAPOTRANSPIRATION

251

1.0

0.8

!

02 0 1.0

"~ 0.6 o

0.4

~1

~

II

I

0.6

~i' i~'~ I

~ I~,,,,,

-

0.4 3:

o

o

0 N

I

o

o

0.2

--

o

co

o

o

o

o

0 12 0 12 j,o 4 , , , 0.6 0.8 1.0 HOUR (I S.TI D/H Fig. 5. Time variations of displacement height, D, a n d roughness length, %, for DOY 177-180. 0

12

Fig. 6. Scatter diagram of roughness length, %, versus displacement height, D, determined in near neutral stability conditions (L < - 100 m, where L is the M o n i n - O b u k h o v length).

uniform canopies. Consequently, the effect of row structure on the air flow characteristics was not significant. The partitioning of the energy balance by the Bowen ratio is the meteorological method involving the fewest assumptions regarding the transport coefficients in the atmospheric surface layer. For this reason, it was taken as the reference for the comparison of the other methods. A typical diurnal course of the energy balance terms is shown in Fig. 7. The latent heat flux density is the dominant dissipation term during the daytime. Condensation occurs during the night to wet the foliage, but evaporation starts as soon as net radiation is positive, using all the available radiant energy until 7 a.m. As the plant surface dries, evapotranspiration deviates below net radiation and reaches a plateau around 9 a.m., when free water on the foliage has been evaporated completely. The figure shows that the contribution of sensible heat flux density is more important during the afternoon than in the morning. Another example of the dominant evaporation during the early morning followed by a midday plateau is shown in Fig. 8. As this result was obtained independently by the aerodynamic method, it corroborates the findings presented in Fig. 7. The diurnal course of sensible heat flux density by the aerodynamic methods on the same day (Fig. 9) confirms the asymmetry between morning and afternoon. The sensible heat flux density derived from the complete profile lags behind that obtained by the Bowen ratio. This feature was found on most of the days. The profile extends

252

P. PIERI AND M. FUCHS

oG~

~-" 600 E

~

soo

•

~- 4oo z 300

--

Rn

v

H

a

E

50O

ea

40O

E

• Rn

aE

3o0

z 200

200 J

~ w z

I00

_J u._

-

O-

o

w qO0

I

qO0

0

4

8

12 16 HOUR (I.S.T.)

20

24

I

8 12 HOUR (I.S.T.)

0

J

16

20

Fig. 7. D i u r n a l v a r i a t i o n s of t h e e n e r g y b a l a n c e t e r m s b a s e d on B o w e n r a t i o m e a s u r e m e n t s for D O Y 186. R n = n e t r a d i a t i o n , G= soil h e a t flux d e n s i t y , H = s e n s i b l e h e a t flux d e n s i t y , E = l a t e n t h e a t flux d e n s i t y . Fig. 8. D i u r n a l v a r i a t i o n s of Rn ( n e t r a d i a t i o n )

a n d E { l a t e n t h e a t flux d e n s i t y ) for D O Y 156.

160 z

x D_

Method '~ Bowen rotio o Complete oerodynomic Simplified oerodynomic

,zo

80

~4o W0 ~ z w m

-4O -80

1

I

I

I

4

8

12

16

vl 20

I 24

HOUR (I.S.T)

Fig. 9. Sensible heat flux densities for DOY 186 calculated by the Bowen ratio, the complete aerodynamic and the simplified aerodynamic methods. to a height of 4.5 m, whereas the Bowen ratio is measured on a 0.60-m air layer 1 m above the crop surface. The time lag is at least 3600 s and the average heat flux density difference is > 20 W m -2. The presumed heat flux density divergence in the 3-m air layer separating the Bowen ratio and profile sampling heights is 20 W m -2 divided by 3 m, or 24 000 J m -3 h -1. The corresponding air temperature change rate is ~ 20 ° C h - 1. This rate is one order of magnitude higher than the morning heating or afternoon cooling rate of the 3-m air layer, therefore the discrepancy between heat flux density estimates cannot be attributed to heat flux density divergence. Experimental accuracy was insufficient to detect a possible violation of the assumption implied by eqs. 4 and 5 that fluxes across the surface layer are constant with height. The comparison of sensible heat flux density estimates was repeated for an earlier date, when more sensible heat was generated because the canopy was

ESTIMATES OF EVAPOTRANSPIRATION

25~

small, thereby exposing dry bare soil (Fig. 10). The diurnal trend of the aerodynamic estimates and that derived from the Bowen ratio are similar. Cloudiness during the day caused variance of the solar radiation input and had a direct effect on each term of the energy balance. However, the aerodynamic determinations, which are independent of the radiation measurement, show the same fluctuations. The sensible flux density was estimated by the simplified aerodynamic method using the five successive air layers defined by the wind and temperature profile measurements. Each layer produced a different estimate of the sensible heat flux density (Fig. 11 ). The diurnal course based on a given couple of levels differed consistently from that obtained for the other couples. How-

,~"280 E ~240

Method v 8owen

--

>-

200

~ ~ ~

folio

o Complete oerodynomic Simplified oerodynomic

z

F116o ~2o so

40 ~'

o

z -40 ~o 0

I

I

4

8

I

1

-

I-

Vl

12 16 20 24 HOUR (I.S.T.) Fig. 10. Sensible heat flux densities for D O Y 153 calculated b y t b e B o w e n ratio, the complete aerodynamic

and the simplified aerodynamic

oJ 'E 40O

methods.

-

¢-

¢

~ 300 z w o

\

Zl IlO,S

Z2 I.I

+1 I.I Oil,5

2.,

"121'

o

1.5

~ 200 _J w I00 I ~z

0 I

8

I

I0

I

l

,

I

12 14 HOUR ( I. S.T.)

l

I

16

=

I

18

Fig. 11. Sensibleheat fluxdensitiesfor DOY 153 estimatedby the simplifiedaerodynamicmethod for fivesuccessiveair layersdefinedby the temperatureand windprofiles.

254

P. PIERI AND M. FUCHS

ever, the calculated fluxes did not decrease with height, as reported by Cellier and Brunet (1987). The convergence of the five curves to zero heat flux density towards morning (except for one spurious data point) and evening indicates that the instruments were exempt from systematic errors. As the differences become larger with increasing sensible heat flux, an incorrect mathematical assessment of the diabatic effect can be the cause of the discrepancies. Estimates derived for Layer 5 (top of the profile) and Layer 3 (third layer from the top) follow closely and consistently the measurements of the energy balance by the Bowen ratio. Data for Layer 3 are drawn in Figs. 9 and 10 to illustrate how well they match the determination based on the energy balance, hinting that the physical principle of the method is sound, but that its mathematical form is inappropriate. The three methods were used to determine evapotranspiration for several irrigation cycles. The results, summarized in Table 1, show the excellent agreement among them. Their absolute accuracy is quite suitable for estimating the amount of irrigation water. For the simplified aerodynamic method, Layer 3 was selected because of its fit to the energy balance measurements. As we were unable to determine objective independent criteria for operating this selection, the simplified method does not appear to be generalizable. The experimentation was conducted in a field where the irrigation regime did not allow the development of intense water stress. Daytime Bowen ratios were of the order of 0.0-0.3. In this range of values, the accuracy of the Bowen ratio method is optimum. As sensible heat flux is small, errors in the diabatic terms and in aerodynamic parameters, which may have large relative effects on H, are attenuated when E is obtained from eq. 11. The application of the TABLE1 Daily average evapotranspiration ( E T ) ( m m d a y - ~) determined by three micrometeorological methods for several irrigation cycles during the cotton growing season Period DOY

ET

E T 2b

E T 3c

152-159 160-167 173-175 177-186 190 195 209-212

5.16 5.58 6.76 6.71 6.87 6.07

4.88 5.30 6.95 6.70 6.16

4.55 5.34 7.00 5.93 6.89 5.81

Mean

6.19

6.00

5.92

Ia

aBowen ratio method, bAerodynamic m e t h o d using full profile. CAerodynamic m e t h o d simplified.

ESTIMATES OF EVAPOTRANSPIRATION

255

m e t h o d to m e a s u r e t h e w a t e r loss of d r o u g h t - s t r i c k e n v e g e t a t i o n still n e e d s to be i n v e s t i g a t e d .

ACKNOWLEDGMENTS

T h e i n s t r u m e n t a t i o n u s e d in t h i s r e s e a r c h was a d a p t e d to field c o n d i t i o n s b y Mr. V. F a l k e n f l u g . T h e o p e r a t i o n s in t h e field w e r e s u p e r v i s e d b y Mr. Y. C o h e n a n d Mr. E. B e c h o r . W e are g r a t e f u l for t h e a s s i s t a n c e p r o v i d e d b y Mr. E. K l e t t e r , E x t e n s i o n Officer of t h e H a d e r a region, a n d his staff, as well as for t h e c o o p e r a t i o n o f t h e c o t t o n c r e w f r o m K i b b u t z S e d o t Y a m . T h e r e s e a r c h was s u p p o r t e d in p a r t b y a g r a n t f r o m t h e C o m m i s s i o n of t h e E u r o p e a n C o m m u nity, Science a n d T e c h n o l o g y for D e v e l o p m e n t p r o g r a m .

LIST OF SYMBOLS

A ), p Oh ¢m ~Yh ~m a B % D E e g G H k K1 K2 L Rn

T T* To u* x z

difference operator psychrometric constant dimensionless height air density diabatic function for temperature profile diabatie function for wind profile diabatic correction for temperature profile diabatic correction for wind profile height ratio defined in eq. 16 Bowen ratio specific heat of air displacement height latent heat flux density water vapor pressure gravity soil heat flux density sensible heat flux density von Karman constant coefficient defined in eq. 12 coefficient defined in eq. 13 Monin-Obukhov length net radiation air temperature profile temperature surface temperature friction velocity 1/~m height above the ground

256 Z* Z0

P. PIER! AND M. FUCHS

mean height defined in eq. 15 roughness length

REFERENCES Brutsaert, W., 1982. Evaporation into the atmosphere. Reidel, Dordrecht, 299 pp. Cellier, P. and Brunet, Y., 1987. Flux gradient relationships above tall homogeneous vegetation canopies. Paper presented at the French-Israeli Symposium on Irrigation Scheduling, INRA, Bordeaux. Cowan, I.R., 1968. Mass, heat and momentum exchange between stands of plants and their atmospheric environment. Q. J. R. Meteorol. Soc., 94: 523-544. Dyer, A.J. and Hicks, B.B., 1970. Flux-gradient relationships in the constant flux layer. Q. J. R. Meteorol. Soc., 96: 715-721. Fritschen, L.J., 1966. Evapotranspiration rates of field crops determined by the Bowen ratio method. Agron. J., 58: 339-342. Fuchs, M. and Hadas, A., 1973. Analysis of the performance of an improved soil heat flux transducer. Soil Sci. Soc. Am. Proc., 37: 173-175. Fuchs, M. and Tanner, C.B., 1970. Error analysis of Bowen ratios measured by differential psychrometry. Agric. Meteorol., 7: 329-334. Fuchs, M., Cohen, Y. and Moreshet, S., 1987. Determining transpiration from meteorological data and crop characteristics for irrigation management. Irrig. Sci., 8: 91-99. Garratt, J.R., 1978. Flux profile relations above tall vegetation. Q. J. R. Meteorol. Soc., 104: 199211. Garratt, J.R. and Hicks, B.B., 1973. Momentum, heat and water vapor transfer to and from natural and artificial surfaces. Q. J. R. Meteorol. Soc., 99: 680-687. Itier, B., 1980. Une m~thode simplifi~e pour la mesure du flux de chaleur sensible. J. Rech. Atmos., 14: 17-34. Kondo, J., 1971. Relationship between the roughness coefficient and other aerodynamic parameters. J. Meteorol. Soc. Jpn., 49: 121-124. Paulson, C.A., 1970. The mathematical representation of wind speed and temperature profiles in the unstable atmospheric surface layer. J. Appl. Meteorol., 9: 857-861. Riou, C., 1982. Une expression analytique du flux de chaleur sensible en conditions superadiabatiques ~ partir de mesures du vent et de la temperature h deux niveaux. J. Rech. Atmos., 16: 15-22. Sinclair, T.R., Allen, L.H. and Lemon, E.R., 1975. An analysis of errors in the calculation of energy flux densities above vegetation by a Bowen ratio profile method. Boundary-Layer Meteorol., 8:129 139. Stanhill, G. and Fuchs, M., 1968. The climate of the cotton crop. Physical characteristics and microclimate relationships. Agric. Meteorol., 5: 183-202.