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Agricultural and Forest Meteorology 152 (2012) 3143

Contents lists available at SciVerse ScienceDirect

Agricultural and Forest Meteorology

journal homepage: www.elsevier .com/ locate /agr formet

The aerodynamics ofpan evaporation

Wee Ho Lim a , Michael L. Roderick a,b, , Michael T. Hobbins a,c , Suan Chin Wong a , Peter J. Groenevelda ,Fubao Sun a, Graham D. Farquhar a

a Research School of Biology, The Australian NationalUniversity, Canberra, ACT 0200,Australiab Research School of Earth Sciences, The AustralianNational University, Canberra, ACT 0200,Australiac Colorado Basin River Forecast Center, NationalWeather Service, National Oceanic andAtmospheric Administration,Salt Lake City, UT 84116, USA

a r t i c l e i n f o

Article history:Received 12 April 2011

Received in revised form 10 August 2011

Accepted 19 August 2011

Keywords:

Pan evaporation

Vapour transfer

Boundary layer theory

Aerodynamics

a b s t r a c t

In response to worldwide observations reporting a decline in pan evaporation over the last 3050 years,we developed an instrumented US Class A pan that replicated an operational pan at Canberra Airport in

Australia. The aim of the experimental facility was to investigate the physics ofpan evaporation under

non-steady state conditions. By monitoring the water level at 5-min intervals we were able to calculate

the evaporation rate and thereby determine the short-term mass balance of the pan. Over the same

time intervals, we also monitored (short- and long-wave) radiation, temperature (air, water surface, bulk

water, inner and outer pan wall), atmospheric pressure as well as the air vapour pressure and the wind

speed at a standard reference height (2 m above ground level). The experimental pan was operated for

three years (20072010).

In this paper, we develop a framework for quantifying vapour transfer by coupling Ficks First Law of

Diffusion with boundary layer theory. This approach adequately represented pan evaporation measure-

ments over short time intervals (half-hourly) under non-steady state conditions provided that surface

temperature measurements, that account for the substantial cooling associated with evaporation, are

available. It involved estimating the boundary layer thickness and other properties ofair above the evap-

orating surface for a pan. Our results are consistent with the envelope oftheoretical curves concept for

the wind function introduced by Thom et al. (1981).

2011 Elsevier B.V. All rights reserved.

1. Introduction

Pan evaporation is the evaporation from a standard water-filled

dish and is the most widely used physical measure of the evapora-

tive demand of the atmosphere. Pan evaporation measurements

have been widely used in agricultural meteorology due to their

simplicity, low cost and proven ease of application for irrigation

scheduling (Stanhill, 2002). Due to the widespread applications,

evaporation pans in various forms have been deployed in many

regions for at least the last several decades (Brutsaert, 1982). Anal-

ysis of worldwide pan evaporation data has found changes, mostly

declines (Roderick et al., 2009a,b) despite the trend of rising globalaverage airtemperature. This has becomeknown as the panevapo-

ration paradox (Peterson et al., 1995; Brutsaertand Parlange, 1998;

Roderick and Farquhar, 2002), because it has occured concurrently

with the global warming. Reduction in irradiance (Roderick and

Farquhar, 2002) and wind speed appear to have been major causes

of this phenomenon (Roderick et al., 2007).

Corresponding author at: Research School of Biology, The Australian NationalUniversity, Canberra, ACT 0200, Australia.

E-mail address:[email protected](M.L. Roderick).

For pans located above the ground, the surface area for heat

transfer is larger than for mass transfer (Kohler et al., 1955; Riley,

1966). Consequently, moststudies haveemployeda pan coefficient,

typically 0.7 for a US Class A pan (Stanhill, 1976), by which panevaporation is multiplied to givea valuemore representativeof nat-

uralevaporation, to account for this largely radiativeeffect (Linacre,

1994). In terms of theaerodynamics of panevaporation, Thomet al.

(1981) examined the roles of free and forced convection. Rotstayn

et al. (2006) subsequently developed the PenPan model of pan

evaporation by combining the radiative model ofLinacre (1994)

with the aerodynamic model ofThom et al. (1981). The PenPan

model has been used to attribute the cause of changes in pan evap-oration in both observations (Roderick et al., 2007; Shuttleworth

et al., 2009) andin climate models(Johnson and Sharma, 2010). The

derivation of the PenPan model assumed steady state conditions

that require a typical integration period of around 1 week in sum-

mer and upto 1 month inwinter (Roderick et al., 2009a). However,

the radiative forumulation (Linacre, 1994) that underlies the Pen-

Pan model, whilst physically based, has never been experimentally

tested. Further, the aerodynamic formulation (Thom et al., 1981)

has not, to our knowledge, been subject to an independent exper-

imental test. These experimental tests have a high priority given

the prominent role attributed to declines in windspeed (stilling)

0168-1923/$ see front matter 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.agrformet.2011.08.006
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32 W.H. Limet al. / Agricultural andForestMeteorology 152 (2012) 3143

and/or solar radiation in explanations of the world-wide decline in

pan evaporation (Roderick et al., 2007, 2009b).

Detailed physical investigations on pan evaporation have been

rare. Notably,Jacobs et al. (1998) have studied the sub-daily ther-

mal behaviour of a standard US Class A pan. They found that the

water in the pan was generally well-mixed. Their pan was located

on the ground and therefore in strong thermal contact with the

soil surface. We expect that the finding of a well-mixed water

body would also hold for a pan located on an elevated (

150mm)

wooden platform such as used in standard US Class A pan installa-

tions. The question of any depression in surface temperature due

to evaporation from a thin (1 mm or so) layer immediately belowthe surface (Ward and Stanga, 2001; Hisatake et al., 1993, 1995)

was not explicitly addressed because Jacobs et al. (1998) used a

Penman-style formulation for the energy balance where the sur-

face temperature was eliminated from the underlying equations.

More recently, the finding of well-mixed water within the pan has

also been reported based on experiments using an instrumentedUS

Class A pan (Martinez et al., 2006). In that research, the water sur-

face temperature was assumed to be uniform over a40mm deeplayer. As noted above, this is unlikely to be true for an evaporat-

ing surface (Ward and Stanga, 2001; Hisatake et al., 1993, 1995),

but again, the Penman-style formulation used in that research

(Martinez et al.,2006) effectivelyeliminated the watersurface tem-

perature from the equations. A second caveat applicable to that

study was that the external walls of the pan were insulated to min-

imise heat transfer and thereby simplify the analysis. Therefore,

that installation did not mimic standard evaporation pans.

To contribute to a better understanding of the world-wide

trends in pan evaporation, we have constructed an instrumented

experimental pan that replicates existing standard installations.

We installed specialised sensors onto a standard US Class A pan

as used by the Australian Bureau of Meteorology (BoM). The water

level, (short- andlong-wave)radiation, temperature (air,water sur-

face, bulk water, inner and outer pan wall), atmospheric pressure

as well as the air vapour pressure and the wind speedat a standard

reference height (2m above ground level) were all monitored at

5-min intervals for a three-year period (20072010).This paper describes the first step in the development of a new

parameterisation for a Penman-type combination equation for pan

evaporation. Here, we focus explicitly on the aerodynamic compo-

nent of pan evaporation. To do that, we develop a framework for

quantifying vapour transfer by coupling Ficks First Law of Diffu-

sionwith boundarylayer theory assuming thatsurfacetemperature

measurements are available.We investigatethe underlyingphysics

of mass transfer and test our theory using data collected at the

experimental pan. We relate our research to the ideas put forward

by Thom et al. (1981).

2. Theory

Section 2.1 summarises existing vapour transfer equations

employed in environmental applications. Section 2.2 derives a

vapour transfer equation based on Ficks First Law of Diffusion.

Section 2.3 describes the significance of boundary layer theory in

vapour transfer. Section 2.4 links the boundary layer theory with

Ficks First Law of Diffusion for vapour transfer and presents an

approach to quantifying vapour transfer from an evaporation pan.

2.1. Existing vapour transfer equations

Daltons Law assumes that vapour moves from high to low par-

tial pressure and is generally expressed as

E(es

(Ts

) ea

(Ta

))=f

v(es

(Ts

) ea

(Ta

)) (1)

where E [m s1] is the evaporation rate of liquid water in tradi-tional hydrologic units of depth per unit time, es(Ts) [Pa] is the

vapour pressure at the evaporating surface, ea(Ta) [Pa] is the air

vapour pressure at the same height that air temperature is mea-

sured at andfv [m s1 Pa1] is the aerodynamic function. (Note thates(Ts) is taken to be the saturated vapour pressure at the surface

temperature.)fv depends on both wind speed and the temperaturedifference between the evaporating surface and the air, especially

whenwindspeeds are low (Thom et al., 1981). For simplicity(Thomet al., 1981), fv has usually been taken as a function of wind speedand Eq. (1) becomes

E= f(u)(es(Ts) ea(Ta)) (2)

where f(u) is known as the wind function. In terms of the

widely used resistance terminology (Monteith,1965; Monteith and

Unsworth, 2008), Eq. (1) can be rewritten as

E= MwMa

awPa

(es(Ts) ea(Ta))ra

0.622 awPa

(es(Ts) ea(Ta))ra

(3)

where ra [s m1] is the aerodynamic resistance, Ma [kgmol1] is

the molecular mass of air, Mw [kgmol1] is the molecular mass ofwater, a [kgm3] is the density of air, w [kgm3] is the densityof liquid water and Pa [Pa] is the atmospheric pressure. In practice,

this is more or less equivalent to Eq. (2) since ra is mostly driven by

the wind (Chu et al., 2010). In subsequent sections we examine the

functional form of the relation between ra and wind speed.

2.2. Ficks First Law of Diffusion

Ignoring complications of inhomogeneity, a one-dimensional

form of Ficks First Law of Diffusion (see Fick (1995) for a trans-

lation) can be written as

J= DdCdz

(4)

whereJ[molm2 s1] is the flux density,D [m2 s1] is the diffusioncoefficient, C[molm3] is the molar concentration andz[m] is thedistance. (Note: the negative sign implies that J is positive when

diffusion is toward the lower concentration.) For vapour transfer,

Eq. (4) can be expressed in a finite form assuming an ideal gas

J= DvR

(ea(Ta)/Ta) (es(Ts)/Ts)z

= DvR

(es(Ts)/Ts) (ea(Ta)/Ta)z

(5)

where Dv [m2 s1] is the diffusion coefficient for water vapour inair, R [J mol1 K1] is the ideal gas constant, Ts [K] is the water sur-face temperature, Ta [K] is the air temperature and z [m] is theboundary layer thickness (see details in Sections 2.3 and 2.4). For

typical environmental conditions,

es(Ts)

Ts ea(Ta)

Ta es(Ts) ea(Ta)

Ta,

and Eq. (5) can be simplified as

J 1R

DvTa

(es(Ts) ea(Ta))z

(6)

This is Ficks First Law of Diffusion in Daltons form for vapour

transfer (see Appendix D for relation to formulations commonly

used in plant physiology). It should be noted that Dv is not a con-stant, but increases with Ta (Gilliland, 1934) and varies inversely

withPa (e.g., Monteith and Unsworth, 2008; Rohsenow et al., 1985).

An equation to calculate Dv as a function ofTa and Pa is given in

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W.H. Limet al. / Agricultural andForestMeteorology 152 (2012) 3143 33

Fig.1. A schematic diagram of thethreshold model adopted herefor the variationin vapour pressure (e) with height (z) above the evaporating surface.

After Leighly (1937) and Machin (1964, 1970).

Appendix A. We can express Eq. (6) in units of depth per unit time

(i.e., E) by incorporating Mw and w ,

E= MwRw

DvTa

(es(Ts) ea(Ta))z

(7)

This is thegeneral form of FicksFirstLaw of Diffusionfor vapour

transfer based on an ideal gas. By comparing Eqs. (1) and (7) we see

that the aerodynamic function (fv in Eq. (1)) is

fv =

Mw

Rw

Dv

Ta

1

z(8)

and the conductancegm,v [mol m2 s1] is given by

gm,v=1

R

DvPaTa

1

z(9)

2.3. Boundary layer theory

Thickness of the boundary layer in air has been measured

directly at various times and by various methods. Its order of

magnitude is from a few millimeters down to some small frac-

tion of a millimeter. Within it, gradients of vapor concentration,

temperature, and velocity are linear. Leighly (1937)

Current conceptions of vapour transfer are for a thin (bound-ary) layer of air above the evaporating surface with transport of

vapour across that layer by molecular diffusion (e.g., Giblett, 1921,

pp. 473474; Penman, 1948) and this is confirmed by observa-

tion (Doe, 1967). Further, the boundary layer thickness is known

to decrease as the wind speed increases (Machin, 1964, 1970). The

treatment consistent with those experimental results holds that

the vapour pressure at the top of the boundary layer is the same as

the vapour pressure at a reference height (e.g., 2m above ground

level) (Leighly, 1937). The simplified threshold model is depicted

in Fig. 1.

To ensure that the framework adopted here can be applied at

other evaporation pans we make the initial assumption that mea-

surements of vapour pressure, air temperature and wind speed

will be available at the reference height. Hence, the challenge is

to estimate the boundary layer thickness zusing those availablemeasurements.

2.4. Boundary layer thickness

Vapour transfer can be conceived as due to free or forced con-

vection, or a mixture of both, often called mixed convection

(Monteith and Unsworth, 2008). It is difficult to specify precise

boundaries between these various convection regimes. Here, wepropose a convenient structure for estimating zwithout a pri-

ori selection of the thresholds. Following boundary layer theory

(Hisatake et al., 1993,1995) we formulatethe boundary layerthick-

ness as

z=f(Re,L) ReqL (10)

where Re is the Reynolds number (dimensionless) and L [m] is the

characteristic length of the evaporating surface. For a cylindrical

evaporation pan, we assume that L is the diameter (1.21m for a

US Class A pan). Here q is a dimensionless constant (range: 0 to

1.0). Conventionally, q is 0.5 for laminar flow over a flat surface(Schlichting, 1960).

Traditionally, Re is calculated using the free stream velocity

of the air and adjusted using a numerical factor (e.g., Eq. 2.2

in Schlichting (1960)). Conceptually, the numerical factor is an

attempt to estimate the wind velocity immediately adjacent to

the evaporating surface. Importantly, the numerical factor ofRe

will depend on the height at which wind velocity is measured. An

alternative formulation is to calculate Re using the wind velocity

immediately adjacent to the evaporating surface. To develop such

an expression forRe we note that

Re inertial forcesviscous forces

= ausLa

(11)

where us [m s1] is a three-dimensional wind velocity immedi-

ately adjacent to the evaporating surface anda [kgm1 s1] is thedynamic viscosity of air. (Note that a is a function of air tem-perature, see Appendix B.) Here, us

f(uV, uH) where uV [m s

1]and uH [m s

1] are the vertical (analogous to free convection) andhorizontal (analogous to forced convection) components respec-

tively. When uVuH, us is dominated by the vertical component,i.e., free convection dominates. Alternatively, when uHuV, us isdominated by the horizontal component, i.e., forced convection

dominates. When uV and uH are of similar magnitude, mixed con-

vection occurs.

In our experiment, the wind velocity components above the

evaporating surface (i.e.,uVanduH) arenot measured directly.Here

we assume that uH is some fraction of the horizontal wind speed at

the reference height, urefas follows

uH= nuref (12)

wheren is a dimensionless constant(range: 0 to1.0)anduref[m s1]

is the horizontal wind speed measured at the reference height, e.g.,

2 m above groundlevel. Using thestandard theorybased on thever-

tical gradient in air density (see Appendix C for details), we derive

that uV can be calculated as

uV = kuV,C (13)

where k is another dimensionless constant (range: 0) and uV,C[m s1] is the characteristic speed of air movement in the verticaldirection.

One way of combining uH and uV to estimate us is

us =uV

+ uH1/

(14)

where is a dimensionless constant (range: 1 to ). For exam-

ple, is equivalent to the assumption that us is the maximum

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of the free or forced convection component (McAdams, 1954; Ball

et al., 1988). Alternatively, setting = 1 implies that us is the sum

of free and forced convection components (Adams et al., 1990).

Intermediate values between these extremes change the relative

contributions ofuVand uH to us.

Combining Eqs. (10)(14) the boundary layer thickness is given

by

z= a[(kuV,C) + (nuref)]1/

L

aq

L (15)

Substituting Eq. (15) into Eq. (7), the final form of the evapora-

tion equation is

E= MwRw

DvTa

(es(Ts) ea(Ta))

[(a[(kuV,C) + (nuref)]

1/L)/a]

qL

(16)

3. Materials and methods

3.1. Field installation

The experimental US Class A Pan (with bird guard) was located

at the BoM field station at Canberra Airport (Australia, Latitude:

35.3S, Longitude: 149.2E, elevation 578m) and was directlyadjacent (5 m) to a BoM operational US Class A pan. Once oper-ational, our experimental pan was replenished with water daily

(at 9am local time) by the BoM duty officer following standard

operating procedure. We commenced installation in September

2006 and the pan was operational from early 2007 until 20 January

2010.

The experimental pan facility is depicted in Fig. 2. The pan

was equipped with a water-level sensor (see details below) that

enabled us to calculatethe mass balance at 5-min intervals. In addi-

tion, Pt100 temperature sensors (Type GW2105, dimension 2 mm

2.3 mm, Degussa, Hanau, Germany) were located on the interiorand exterior pan wall at the four cardinal compass points, and at

three different levels (25, 100 and 175mm from the bottom), to

characterise the thermal dynamics of the pan. Temperature sen-sors were also placed at the same three levels at the centre of

the pan to record the bulk water temperature. The temperature

of the water surface was measured using an infrared thermome-

ter (Model: M50-1C-06-L, Mikron Instrument Co. Inc., Oakland, NJ,

USA) (Fig. 2b).

The evaporation rate was calculated from water height mea-

surements made using a magnetostrictive linear displacement

transducer (MLDT) (MagneRule Plus, MRU-4001-015, Schawitz

Sensors, Hampton, VA, USA) with a spherical float. The float was

installed in a stilling well connected to one side of the pan. After

initial experimentation, and contrary to the manufacturers spec-

ifications, we found that the output of the MLDT was sensitive to

variationsin ambient temperature. To overcome thatlimitation,we

attached a proportional-integral-derivative (PID) controlled heaterto the casing of the sensor head to maintain a constant tempera-

ture of 40 C. The resolution of the MLDT for our installation was10m.

In addition, standard meteorological measurements included

radiation, wind speed, air temperature, air vapour pressure and

atmospheric pressure. All components of the radiation balance

(incoming andoutgoingshort- and long-waveradiation) weremea-

sured using a Kipp & Zonen CNR 1 Net Radiometer attached to

a swinging-arm (Fig. 2). Most of the time, the swinging arm was

parked to the southeast of the pan with the downward sensors fac-

ing the ground. At 5-min intervals, a motor swung the arm over the

centre of the pan where the downward sensors sampled radiation

from the water surface for a 20-s period. Forty radiometer readings

aretakenin each directional swing.(A paper focusing on theenergy

balance ofpan evaporationis in preparation.) Wind speedwas mea-

sured using a cup anemometer at 2 m above ground level (u2). In

addition,a mast wasinstalled 5 m away from the experimentalpan

to enable installation of temperature, vapour pressure and atmo-

spheric pressure sensors and a 2D ultrasonic anemometer (Wind

Observer II, Gill Instrument Ltd.). The 2D ultrasonic anemome-

ter was located at the same height as the cup anemometer to

enable us to check the performance of the cup anemometer. Atmo-

spheric pressure was measured with a Vaisala Pressure Transmitter

(Model: PTB101B). Air temperature and air vapour pressure were

measured with a Vaisala Humitter (Type 50Y, Vaisala, Helsinki,

Finland).Air temperature,vapour pressure andthe water level sen-

sor were all calibrated in the laboratory and periodically checked

on-site after field installation. All analog sensors signals were con-

vertedvia a 16-bit analog-to-digital converter,averagedover 5-min

intervals and stored in a single-board computer.

3.2. Data sampling

Short-term oscillations in the water level were apparent in the

5-min data. To avoid the high-frequency noise, all water level data

were aggregated, and the resulting evaporation rate calculated, at

half-hourlyintervals. Thechange in water level dueto dailyrefilling

(at9amlocaltime)was takeninto account. All othermeteorologicaldata were resampled to half-hourly intervals.

From the database, we identified 160 days of elite data that

had no missing half-hourly totals and included 40 (rainless) days

each in spring, summer, autumn, andwinter, respectively. The final

elite database contains (= 160 days 48 samples per day) a totalof 7680 half-hourly measurements.

3.3. Parameter estimation and validation

We split the datafromthe 160 daysintotwo databases. The first

subset (60 days, including 15 days in each of the four seasons) was

used to estimate the model parameters. The second subset (100

days, including 25 days in each of the four seasons) was used to

validate the model.While theoptimumvalue of (Eq. (16)) was >2, theeffect on the

fit to the data was not strong, and we chose=2 based ona vecto-

rial combination assumption (Adams et al., 1990). The parameters

(k, n, q) were estimated using 2880 half-hourly measurements (=

60 days 48 samples per day) using a least squares optimisationapproach. To do that we initially assumed values of the parame-

ters and then computed pan evaporation Epan for that parameter

combination. Many possible combinations were tested using an

automated computer algorithm and the parameter combination

with the lowest root mean square error (RMSE) was selected.

4. Results

4.1. Meteorological data, calibration andvalidation

The wind speed measured by the cup anemometer (u2, also

known as the (horizontal) wind speed hereafter) was compared

to measurements by the 2D ultrasonic anemometer. We used half-

hourly data from all 160 days (7680 samples) (Fig. 3). At low wind

speeds (

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Fig. 2. Experimentalpan installationat Canberra Airport BoMstation. (a)The US Class A pan is 1.21m in diameter (4ft) and 0.254m in height (10 in.), equipped with instru-

ments measuring (short- and long-wave) radiation, wind, temperature(water surface, bulk water, inner and outer panwall) and water level. (b) The infrared thermometer

installation (inside thewhite PVC pipe facingthe watersurface in thepan) measuring long-waveradiation emitted by thewatersurface.

Fig. 3. Comparison of wind speed measurements from the cup anemometer and the 2D ultrasonic anemometer over half-hourly intervals at the experimental pan (7680

data points,y=1.07x0.44, R2 =0.97, RMSE= 0.42ms1).

The least squares estimates of the parameters are k=0.20,

n= 0 .10, and q=0.64 (half-hourly samples: 2880, regressionof estimated versus observed Epan: slope = 0.86, inter-

cept=6.7106

mm s1

, R2

=0.82, RMSE=3.1105

mm s1

).

The estimate for q is close to the previously noted value of0.5for laminar flow. The estimate ofn is also sensible in that it makes

the horizontal component of wind velocity immediately above the

surface only10% of the wind speed at2 m above ground level. With

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Fig. 4. Pan evaporation Epan (observed versus estimated), conductancegm,v (observed versus estimated per Eq. (17)), vapour pressures (es(Ts), ea(Ta)), temperatures (Ts , Ta),

u2 (wind speedat 2m aboveground level) and atmospheric pressure Pa for theexperimental pan fordiurnal cyclesin: spring (a andb), summer (c and d),autumn (e andf),

and winter (g and h).

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Fig. 4. (Continued ).

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Fig. 5. Estimated versus observed Epan by integrating half-hourly results to a daily basis for 100 days: (a) Ts available (y=1.04x+0.39, R2 =0.98, RMSE=0.51mmd1); (b)

assume Ts =Ta (y= 1.56x+1.43, R2 =0.91, RMSE=3.06mmd1).

Fig. 6. Temperature depression of water surface at half-hourly intervals for the experimental pan (7680 data points). (a) Frequency distribution ofTs Ta (binsize= 0.1K).(b) Frequency distribution ofTs Tw (binsize= 0.1K). (c) Observed Epan versus Ts Ta .

those results, the pan evaporation Epan (semi-empirical equation

from Eq. (16)) is

Epan =MwRw

DvTa

(es(Ts) ea(Ta))a

(0.20uV,C)2 + (0.10u2)2L

/a

0.64L

(17)

Note that the numerical value ofL is 1.21m, i.e., the diameter of

a US Class A pan.

We subsequently used Eq. (17) to estimate Epan for the remain-

ing (4800) half-hourly totals. (The resulting half-hourly totals are

compared with measurements over eight typical days in Fig. 4.)

The half-hourly totals were then summed into 100 daily totals and

compared with measurements (Fig. 5a). The model explained 98%

of the observed variance in the daily Epan with an overall RMSE of

0.51mmd1 giving us confidence in the parameterisation.

4.2. Water surface temperature and evaporative cooling

The model parameters and results to date were derived using

our (infrared) measurement of the water surface temperature Ts.

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Fig.7. Half-hourlyboundary layerthickness z(perEq. (15);= 2,k= 0.20,n= 0.10,

q=0.64) versus u2 (wind speed at 2m above ground level) for the experimentalpan (7680 data points).

Fig. 8. Half-hourly aerodynamic functionfv (per Eq. (17)) versus u2 (wind speed at

2 m aboveground level)for theexperimental pan (7680 data points).

In practical applications, and especially for evaluating the histori-

cal records, Ts is unknown. Further, inspection ofFig. 4 shows that

temperature of the water surfaceTs can be quite different from thatof the air Ta, as would be expected for a freely evaporating surface.

The following question arises: canEpan be estimatedaccuratelywith-

out measurementsof Ts in theabsence of radiationmeasurements? To

answer this, we assumed that the water surface was at the air tem-

perature (i.e., Ts = Ta) and accordingly recalculated the saturated

vapour pressure at the surface. The results, using totals from the

100 elite-days, show that this is a bad assumption for a purely

aerodynamic formulation of evaporation (Fig. 5b). In general, the

estimateof dailyEpan basedontheassumptionthat Ts =Ta was much

larger than the observations at high rates ofEpan. That result means

that the water surface is cooler than the air when Epan is high and

implies evaporative cooling. The same phenomenon is visible in

Fig. 4, where Ts is substantially lower than Ta in the mid-afternoon

when Epan tends to be highest.

To investigate further, we examined the relationships between

Ts, Ta and the bulk water temperature Tw (taken as the average of

measurements at 25, 100 and 175mm below the water surface).

The bulk water was found to be well mixed (results not shown) but

the evaporating surface was often up to 3 K warmer or 5K cooler

than the bulk water, with an overall average of around 2K cooler

than the bulk water (Fig. 6b). We also found that the evaporating

surface could be up to 5K warmer or 11K cooler than the air, and

was on average, 1K cooler than the air over the 160 days (Fig. 6a).

In general, the temperature of the water surface was much lower

than the air when Epan was high (Fig. 6c), confirming our earlier

deductions about evaporative cooling. In summary, the magnitude

of the surface cooling due to evaporation was substantial.

4.3. Estimates of boundary layer thickness and aerodynamic

function

Estimates of boundary layer thickness zusing the estimatedparameter values are plotted as a function of wind speed at 2m

above ground level (u2) in Fig. 7. For u2> 1 m s1, the results are

dominated by forced convection with z in the range 14 m m.For u2< 1 m s

1, zis highlyvariablewithinan envelope constraintimposed by the free convection regime.

The resulting aerodynamic function has been computed usingall available half-hourly data (Fig. 8). The overall features of the

aerodynamic function are consistent with the ideas put forward by

Thom et al. (1981, Figs. 2 and 5). In particular, at low wind speeds

(u2 < 1 m s1) we see the variation in the aerodynamic function due

to a mixture between free and forced convection. At higher wind

speeds when forced convection dominates, the relation collapses

tobe u20.64, which is nearly a straight line.

4.4. Estimates of pan evaporation without a bird guard

Our experimental pan used the same bird guard as used by the

Australian BoM (Fig. 2). The effect is to reduce the mass transfer

by

7% in comparison to a similar pan without a bird guard (van

Dijk, 1985). For comparative purposes, it is useful to estimate theimpact of the bird guard. On the boundary layer formulation used

here (Fig. 1), the bird guard would affect the evaporation rate by

reducing the wind speed near the evaporating surface and thereby

increasingthe boundarylayer thicknessz. Hence,we canincorpo-rate that effectby adjusting thenumerical value of thenparameter.

Assuming all else is held constant, we find a decrease in n of close

to 10% will reduce Epan by around 7%. In summary, in the absence

of a bird guard, the parameter estimates are k= 0.20, n= 0.11,

q=0.64.To compare our adjusted formulation for a pan without a bird

guard with previous research, we plotted the aerodynamic func-

tion fv as a function of wind speed (2 m above ground level) u2for various differences between the water surface temperature Ts

and the air temperatureTa under typical conditions (Pa =101.3kPa,ea(Ta)=1kPa, Ta = 293.15K (20

C)). The resulting envelope of the-oretical curves (for differentTs Ta) (Fig. 9a) is consistent with theconcepts proposed by Thom et al. (1981, Figs. 2 and 5). This enve-

lope coversthe majority(butnot all) of aerodynamic functionsused

in previous studies when free or mixed convection dominates (i.e.,

u2 < 1 m s1); and has a linear or near-linear form of the classical

wind functions under high wind speeds when forced convection

dominates (Fig. 9b).

5. Discussion and summary

The theoretical formulation derived here is based on the idea

of vapour transport, predominantly by diffusion, from the liquid to

vapour phases across a well-defined boundary layer (Fig. 1). This

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Fig. 9. Aerodynamic functionfv versus u2 (wind speed at 2m above ground level) for pan evaporation under typical conditions (Pa = 101.3 kPa, ea(Ta)=1kPa, Ta =293.15K

(20 C)). (a) Our model without a bird guard (per Eq. (16); = 2, k=0.20, n=0.11, q=0.64) for various differences between water surface and air temperature; (b) fv fromprevious studies.

is well established in laboratory studies (Doe, 1967; Machin, 1964,

1970) and the resulting formulation (Eq. (7) has a direct physical

interpretation. Hence, the utility of this approach rests on whether

the conceptual framework (Fig. 1) is useful.

The assumption that= 2 is basedon theideaof vector average

of the fluxes associated with the two physical processes (Adams

et al., 1990). In contrast, the idea of taking a maximum value of

free or forced convection (McAdams, 1954; Ball et al., 1988) means

having , which might be useful over very short time inter-vals (e.g., minutes); yet could be less appropriate over longer time

intervals (e.g., half hours) since a mixture of both free and forced

convection is most likely to be the case under outdoor conditions.

Typical wind function approaches of the formf(u) =a+bu implic-

itly assume the free convection component (a) to be a constant(Adams et al., 1990). This is similar to setting = 1 and q=1 inEq. (16).

Once was set = 2, the remaining parameters were estimated

using the measurement database consisting of half-hourly data for

60 days. The resulting parameters(k=0.20, n=0.10, q=0.64) werecloseto broad expectations. The estimatedkuV,Crange(00.6 m s

1)is within our expected range ( 1 m s1), forced convection dominates, and

the air above the pan is quickly replaced by the surrounding air.

Under those conditions, z is predominantly a function of wind

speed. However, under low wind speeds (u2 < 1 m s1), free con-

vection dominates, and other factors also play important roles in

determiningz. In particular, the spatial gradientof temperature inthe vertical direction (a surrogate for density difference) becomes

important at low wind speeds.

Our theory (Eq. (17)) and subsequent results (Fig. 8) show that

a unique wind function does not exist. It also suggests a depen-

dence on atmospheric pressure.Thus it would be interesting to test

whether the formulation presented here (Fig. 9a) could be applica-

ble under a wider set of conditions, such as at high altitude sites

(Blaney, 1960; Giambelluca and Nullet, 1992) where the atmo-spheric pressure is substantially reduced. Our formulation assumes

that the water surface is always close to the top of the pan and

thereby avoids the shelter effect (Chu et al., 2010). This is most

easily achieved in operational settings by refilling the pan each

day.

Measuring the water surface temperature Ts (in absence of

radiation measurements) proved to be important for accurately

estimating Epan using the aerodynamic approach when the evapo-

ration rate was high and the associated evaporative cooling of the

surface was at a maximum (Figs. 5 and 6). On average, the evap-

orating water surface was cooler than both the air and the bulk

water (Fig.6a and b), as foundpreviously in laboratory experiments

(Hisatake et al., 1995, Fig. 5). This implies that the very thin layer of

surface water, is, on average, a net absorber of sensible heat fromboth the air above it and from the bulk water below it. We were

surprised by the magnitude of this cooling effect.

Acknowledgements

We thank the BoM staff; Tony McCarthy, Ross Hearfield, Kirsty

Rhind, Nigel Smedley, David Pottage, Neil McArthur and Kenn Batt

for their help in maintaining our experimental pan at the Canberra

Airport and Liang Li for his contribution in setting up the exper-

imental pan database. We acknowledge the Australian Research

Council (ARC) for the financial support of this study through the

grant DP0879763. We are grateful to two anonymous reviewers

for helpful comments.

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Fig. A.1. Diffusion coefficient Dv versus air temperature Ta at different values of

atmospheric pressurePa.

Appendix A. Diffusion coefficient of water vapour

The diffusion coefficient of water vapourDv is calculated basedon Pruppacher and Klett (1997):

Dv = 2.11

Ta273.15

1.94PoPa

105 [m2 s1] (A.1)

where Ta [K]is the air temperature,Pa [Pa] is the atmospheric pres-

sure, Po [Pa] is the atmospheric pressure at the mean sea level

(101.325 kPa). Fig. A.1 shows the change ofDv with Ta and Pa using

Eq. (A.1).

Appendix B. Dynamic viscosity of air

The dynamic viscosity of air a (assumed dry air for simplicity)is calculated based onJacobson (2005):

a = 1.8325

416.16

Ta + 120

Ta

296.16

1.5 105 [kgm1 s1] (B.1)

Although a is based on dry air instead of moist air, the differ-ence is small (Maxwell, 1866; Kestin and Whitelaw, 1964). Fig. B.1

illustrates the change ofa with Ta using Eq. (B.1).

Appendix C. Derivation of the speed of air in the vertical

direction above evaporating surface

The vertical circulation of air above the evaporating sur-

face is determined by the air density difference (Schlichting,1960; Incropera and DeWitt, 1990; Holman, 2002; Monteith

and Unsworth, 2008), which results from temperature gradients,

vapour concentrationgradients,or a combination of both(Monteith

and Unsworth, 2008).

In principle, the speed of air in the vertical direction can be

derived from Reynolds and Grashof numbers using dimensional

analysis. An equivalent Reynolds number in the vertical direction

(ReV) is the ratio of inertial forces (in the vertical direction) to vis-

cous forces, i.e.,

ReV =auVL

a(C.1)

where a [kgm3] is the density of air, uV [m s1] is the speed of

air in the vertical direction, L [m] is the characteristic length of the

Fig. B.1. Dynamic viscosity of aira versus air temperature Ta.

evaporating surface anda [kgm1 s1] is the dynamic viscosity ofair (calculation ofa is given in Appendix B).

The Grashof number (Gr) is equal to buoyancy forces times iner-

tia forces divided by the square of viscous forces. Since the origin

ofGr is in heat transfer studies (Karwe and Deo, 2003), it is com-

monly calculated based on a spatial temperature difference. Here,

we calculateGrby using thespatial density difference. The purpose

is to take into account both temperature and concentration (water

vapour and dry air) differences between the evaporating surface

and the reference height, i.e.,

Gr= 2ag(a/a)L

3

2

a

(C.2)

whereg[m s2] is the gravitational acceleration, a [kgm3] is theaverage air density between the evaporating surface and the refer-

ence heightanda [kgm3] is thespatialdifference in thedensityof air between the evaporating surface and the reference height.

Assuming constant atmospheric pressure between the evapo-

rating surface and the reference height, uV [m s1] can be derived

from Eqs. (C.1) and (C.2) as follows,

Re2V Gr, ReV Gr,

auVL

a

2ag(a/a)L3

2a,

uV gaa

L, uV = kgaa

L, uV = kuV,C (C.3)

where k is a dimensionless constant (range: 0) and uV,C [m s1]g(a/a)L

is the characteristic speed of air in the verti-

cal direction. We calculate a and a as

a =1

R

(Pa ea(Ta))Ma + ea(Ta)Mw

Ta

(Pa es(Ts))Ma + es(Ts)MwTs (C.4)

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Stanh ill, G. , 20 02. Is th e C lass A evapo ration pan still the most pr actical andaccurate meteorological method for determining irrigation water require-ments? Agric. For. Meteorol. 112, 233236, doi:10.1016/S0168-1923(02)00132-6.

Stokes, V.J., Morecroft, M.D., Morison, J.I.L., 2006. Boundary layer conductance forcontrasting leaf shapes in a deciduous broadleaved forest canopy. Agric. For.Meteorol. 139, 4054, doi:10.1016/j.agrformet.2006.05.011.

Thom, A.S.,Thony, J.L., Vauclin, M., 1981. On the proper employment of evaporationpans andatmometersin estimating potentialtranspiration.Quart.J. R.Meteorol.Soc. 107, 711736, doi:10.1002/qj.49710745316.

van Dijk, M.H., 1985. Reduction in evaporation due to the bird screen used in theAustralian class A pan evaporation network. Aust. Meteorol. Mag. 33, 181183.

Ward, C.A., Stanga, D., 2001. Interfacial conditions during evaporation or condensa-tion of water. Phys. Rev. E 64, 051509, doi:10.1103/PhysRevE.64.051509.

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