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Evaporation from water, snow and ice

Williams, G. P.

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CANADA D I V I S I O N O F B U I L D I N G

CouNcu-R E S E A CouNcu-R C H

EVAPORATION

FROM

WATER.

SNOW AND ICE

BY

G. P. wf LLIAMS

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I P R O C E E D T N G S N o . ? . M A R c H 1 9 6 r . R E P R I N T E D F R O M O F H Y D R O L O G Y S Y M P O S I U M P . 3r - 52, DtscussroN P. 53 .54.

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E V A P O R A T I O N F R O M W A T E R , S N O W A N D I C E G . P . W i l t i a r n s l

SYNOPSIS

The basic theoretical differencee between evaporation frorn snow,

ice and water surfaces are outlined. Some estimated energy budget

analyses are made to ehow the order-of-rnagnitude of evaporation rates

frorn theee eurfacee wlder Canadian conditions. Some rneaeurernents of

evaporation from snow and water surfaces made in Canada are preaented

to confirrn the theoretical coneiderations.

I N T R O D U C T I O N

The main purpose of this sympoaium is rtto provide an opportunity

for Canadian researchers in evaporation to exchange inforrnation and

v i e w s , a n d t o e n c o u r a g e t h e u s e o f t h e b e s t r n e t h o d s a v a i l a b l e b y e n g i n e e r s , f o r e s t e r s , a g r i c u l t u r i s t s , r n e t e o r o l o g i s t s a n d o t h e r s , f o r d e a l i n g w i t h e v a p o r a t i o n " ( $ ) . W i t h t h i s a i m i n r n i n d , t h i s p a p e r h a s b e e n w r i t t e n t o p r o v i d e b a c k g r o u n d m a t e r i a l f o r t h e d i s c u s s i o n s , p a r t i c u l a r l y f o r t h o s e dealing with the baeic ideas concerning evaporation from saturatcd s u r f a c e s .

In evaporation investigations there are two points o{ view. The

f i r s t m i g h t b e t e r r n e d t h e r t r e g e a r c h " p o i n t o f v i e w w h e r e , w i t h s p e c i a l

instrurnentation and adequate etaff, investigations are u.ndertaken to

i n c r e a s e t h e k n o w l e d g e o f t h e e v a p o r a t i o n p r o c e s s a n d t o d e v e l o p o r c h e c k

theories of evaporation. The second rnight be terrned the iloperational,,

p o i n t o f v i e w w h e r e , w i t h e x i s t i n g r e c o r d s a n d e x i s t i n g t e c h n i q u e s , a t t e m p t c are rnade to estimate evaporation rates for epecified surfaces under given

couditions. This second point of view ie ernphasized in this paper by (a)

outlining the energy balance apprech to the evaporation problem,

(b) presenting energy budget calculatiotrs for eome typical conditions in C a n a d a , ( c ) o u t l i n i n g b r i e f t y t h e m a e e t r a n s f e r a p p r o a c h , a n d ( d ) p r e s e n t -ing sorne lirnited observationa of evaporation from snow and water

eurfacee and comparing them with one of the available empirical formulae.

E N E R G Y B A L A N C E A P P R O A C H

One form of the energy balance equation at an evaporating surface

l. Snos and lce Section, Divieion of Building Reaearch, National

Regearch Councit, Ottawa, Ontario.

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i e a a f o l l o w s :

O e + O 8 w * O l * + Q s + Q 6 = 0

where Qu r evaPorative heat,

Oe* = net short-wave radiatioD,

Qlt = tret long-w3've radiatioD,

Os = heat conducted to the surface from below (change ln

the total heat stored in the ice or water, includiag

the heat aaeociated wit} the freezing of water or

melting of stro\r or ice), and

Oc 3 convective heat (eeneible heat).

If the varioue terms in thie equation can be measured or calculated,

Q6 can be estimated. Aa Penrnan (16) points out, I'the fundameatal bagis

of the energy balance approach ie unchallenged, the challenge ig to our

abitity to measqre or estimate all the quantitiea needed to exploit the

priacipte of the conservation of energy." Therefore, the problem of

meaauring or estimating the different pararneters of the energy balance

equation for snow, ice, and water eurfacee will be emphaaized. The

energy that is available becauee of precipitation, ground water, or river

flour has been ignored in this diacussion.

Qsw, the Energy Available from Short-wave Radiation

The amount of short-wave radiation abeorbed at an evaPorating

s u r f a c e , Q s s 7 , e e u a l s R s w ( l - r )

where Rsw = short-wave radiation reaching the aurface, and

r : albedo, sometirnee given as the percentage of

6hort-wave radiation reflected.

The arnount of ehort-wave radiation that ia reflected, r, variea with

t.Lc type of wat€r, aaow or ice, and aleo with the elevatioD of the sun,

eapecially in the case of water. There ia enough agreernent arnong varioua

inveetigators to give the folloring approximate valuee:

r(atbedo) N e w s n o w . . . 0 . 8 0 - 0 . 9 0 O l d e n o w . . . . 0 . 6 0 - 0 . 8 0 M e l t i n g 6 n o w . 0 . 4 0 - 0 . 6 0 I c e . . . . ; . . . 0 . 4 0 - 0 . 5 0 W a t e r . 0 . 0 5 - 0 . 1 5

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E V A P O R A T I O N F R O M WATER SNOW AND ICE

These values indicate that coneiderably leee ghort-wave radiation is reflected frorn an ice surface than from a anow surface and that

considerably less ehort-wave radiation ie reflected from a water surface

t h a n f r o m a n l c e s u r f a c e .

Cornparative values showing the seasonal variation of Qso, have been estimated for two latitudes in Canada and are given in Figure l.

The average insolation for two stations, Ottawa (latitude approxi-m a t e l y 4 5 ' N ) , a n d A k l a v i k ( l a t i t u d e a p p r o x i r n a t e l y 6 8 " N ) were obtained f r o m e s t i m a t e s o f a v e r a g e i n s o l a t i o n g i v e n b y Mateer (13). The average reflection values for snow, ice and water used for these two latitudes are t h o s e g i v e n b y B u d y k o ( 6 ) .

Figure I illustratee the differencea in the energy available frorn

s h o r t - w a v e r a d i a t i o n a t t h e t w o s i t e s . D u r i n g the winter period, october to May, the average energy available frorn short-wave radiation at Aklavik is a.ppr_oximately 32 cat/craz/z+ hr for snow surfaces, compared

with 7? cal/crn?/24 hr for the winter period Novernber to April at ottawa.

During the summer period, June to September, the average energy a v a i l a b l e f r o m s h o r t - w a v e r a d i a t i o n for water surfaces at Aklavik ie

approxirnately 333 cal/crnZ/24 hr, compared with 415 cal/crnT/24 hr f,ot

t h e p e r i o d M a y t o O c t o b e r a t O t t a w a . Q l * , L o n g - w a v e R a d i a t i o n

In detailed etudies where adequate records are availabte, the long-wave radiation cornponents are broken down into incoming long-long-wave radiation from the atmosphere, reflected long,-wave radiation, and long-wave radiation emitte'd by the surface.

- w h e r e a d e q u a t e r e c o r d s a r e n o t availabre, empirical formurae

have been devetoped to estirnate net long-wave radiation frorn etandard

rrreteorological obeervatione. one type of equation propoaed by Brunt (5)

i a a s f o l l o w s :

Q l r " . q T a 4 ( a + b e ) f ( n / N )

where c : Stefan-Boltzrnann constant

T a = m e a n a i r t e m p e r a t u r e e = r n e a n v a p o u r p r e s s u r e o f air a , b , = c o n s t a n t a

f(n/N) = cloudineee factor

I n r e g a r d t o t h i s t y p e o f f o r m u l a B r u n t states: rrhese equations are beet r e g a r d e d a s e m p i r i c a l f o r m u l a e , w h i c h , within the limits of vapour

p r e s s u r e w h i c h o c c u r i n the atmoephere, livg a reagonable repreeentation

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of the average alnount of radiation in given conditione. No auch formulae

can give accurate representation of individual observationa which

frequently show wide variatione of R (incoming long-wave radiation) for

t h e s a m e v a l u e s o f T a n d e . r l

To show the approxirnate value that can be expected for the

loag-wave radiation term Q1*, mean rnonthly values were calculated for the

Ottawa latitude, using the empirical formula proposed by Brunt, with

appropriate constants, and mean monthly values for air temperature and

a i r v a p o u r p r e s s u r e . T h e c a l c u t a t e d v a l u e s f o r Q q s , v a r i e d f r o r n 2 2 0 t o

250 ca|/crnZ/Z4 hr for clear sky conditione. There was no rnarked

seasonal trend, as the increaae of vapour Pressure in the 6urluner waa

offset by the decrease of air ternperature in the winter. A preliminary

analysis of sky conditions in the Ottawa area indicated that the rnean value

of long-wave radiation, e1* should be reduced to about l2A cal/crnZ/24

hr for average sky conditiona.

Observations made at nightby the author at Ottawa for the period

October I to Novernber 30, 1960, give an average value of 5.75 caL/crr^21\t

(138 cal/cmL/24 hrl for the net long-rpave radiation from a water eurface.

Some average values taken frorn various sources that can be used for cornparison are as followE:

Open snow fielda

Snow ia pine staada, 4O

S i e r r a N e v a d a Average value, Lake

Ktlmrningen, Swedea

Average catculated value

Lake Ontario

Average values Lake l/tead and Lake Hefner

Ql*, cal,/cmz/24 hr r 5 5 t23 9 6 t 5 0 - 1 6 z Sorirce D . H . M i l l a r ( 1 4 ) D . H . M i l l a r ( 1 4 ) J o h n s s o n ( l 0 ) B r u c e a n d R o d g c r e ( 4 ) L a k e M e a d S t u d i e g ( 1 9 l

Radiometers have been developed to mcaaure the combined net

short-wave and long-wave radiation. Eecauae theae instrurnents are still

at the state of development where they require practically constant

auPer-vigion in order to yield reliable resulte, however, they aie not suitable

for routine observation programs. It ia only in special studies that such

instrurnents have found ueeful application to date. For more general

application of the energy balance equation to evaPoration, long-wave radiation will at preaent have to be calculated ueing formulae such as that d e v e l o p e d b y B r u n t .

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EVAPORATION FROM WATER, SNOW AND ICE 35

Q e , H e a t S t o r a g e T e r m

The amount of heat transferred from the underlying material to or away frorn the evaporating surface in a given time is called in this PaPer the rate of change of heat storage (Qs). To understand the conditions under which this heat storage energy canbe irnportant to the evaPoration process, some thermal properties of enow, ice and water are briefly c o n s i d e r e d .

As the therrnal conductivity of ice is approxirnatety ten tirnes that of average snow, the heat available for evaporation by conduction through ice is about ten times that available by conduction through enow eubject to a sirnilar temperature gradient. The thermal conductivity of ice is about five times that of still water. In the case of water. however, heat can be brought to the surface by convective or turbulent transfer as well as by conduction. The convective heat transfer can be large incornpariaon with the heat brought to the surface by pure conduction.

Priestley ( l7) has considered that the total change in heat content of varioug rnedia subject to a temperature change at their surface is proportional to pc ,,lk , where k is the thermal diffusivity of the rnediurn, p ie the density and c is the specific heat. He considers that the "rate of heat flux consequent on an imposed ternperature reSirne at the boundary willbe directly proportional to pc^r/1 " and for this reason defines this quantity as the 'rconductive capacity'r of the medium. He hae given the following valuea of pc,,,/k for various media:

New enow. Old anow I c e . . Still water Stirred water g r e a t s t a b i l i t y . . . moderately stable pc ./t 0 . 0 0 2 0 . 0 1 2 0 . 0 5 0 . 0 3 9 ( a ) ( b ) 0 . 3 2 ? 0 ( c ) h o m o g e n e o u s , s t r o n g c u r r e n t a . . 1 7 . 0

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To illustrate further the influence different media, some values obtained by of heat storage for different rnedia in the

of the conductive capacity of the writer for the rate of change O t t a w a a r e a a r e a 6 f o l l o w s :

A v e r a q e T e m p e r a t u r e G r a d i e n t - *--

"C'/cr,t

12 in. of snow

12 in. of ice (no enow)

o p e n w a t e r ( a v e r a g e depth 20 ft) w h e r e 46 1 6 5 o . 2 o . l 0 . 0 1 o r l e s s

The value ofthe heat etorage terrn depends not only on the conductive capacity but alao on the temperature gradient in the media. At certain timeg whenthe snow layer ie thin or when ice ie first forming, the ternperature gradient rnay be quite large and the resulting heat flow to the surface correspondingly significant.

Q..-Energy Lost or Gained by Convection Frorn the Air

The direct meaaurenrent of O., the convective component, is moet difficult. Such measurernenta have been undertaken for special studies only and are not available in Canada for the more general application of t h e e n e r g y b a l a n c e e q u a t i o n . F o r p r a c t i c a l a p p l i c a t i o n s , a r e l a t i o n between evaporation and convection is assumed which enables the convec-tive component Q. to be related directly to the evaporaconvec-tive cornponent Qu. B o w e n ( 3 ) d e r i v e d s u c h a r e l a t i o n s h i p w h i c h h a s b e c o r n e k n o w n a s B o w e n ' s ratio.

By assurning that the convective heat tranefer and vapour transfer p r o c e s s e s a r e i d e n t i c a l , t h e e x p r e s s i o n d e r i v e d i s :

o ,9t- = 5.9-"--ral

Q e ( e " - e " ) R - B o w e n r B r a t i o , K ! con8tant, To = surface ternperature, T a ; a i r t e m p e r a t u r e .

€s : vapour pressure of air at the surface, and

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E V A P O R A T I O N F R O M W A T E R . S N O W A N D I C E 1 7

T h e g e n e r a l v a l i d i t y o f t h i s e x p r e B s i o D h a s b e e n t h e s u b j e c t o f

much discussion. The Lake Hefner investigators have stated (18): "The

Bowen ratio appeare to be eufficiently accurate for cornputing energy

utilized by evaporation for rnost conditions. In exceptional conditions, for

exarnple, when evaporation ratea are srnall and the difference in vapour pressure of the atrnoephere and that of air aaturated at the surface

-water ternperature approaches instrumental accuracy, the Bowen ratio

is inadequater'.

Usuatly the vapour preaaure differences over snow and ice are small, so that caution shouldbe exercised in applying the Bowen ratio to relate convective and evaporative heat losses under such conditions. One of the difficult factorB to mea6ure in Bowen's ratio is surfaie

t e m p e r a t u r e , p a r t i c u l a r l y o f s n o w a n d i c e e u r f a c e s w h € r e t h e t e m p e r a t u r e -rneasuring device norrnally rnust be exposed to the incorning ehort-wave

radiation. Available evidence doee indicate that when a reliable value for

B o w e n ' g r a t i o c a n b e c a l c u l a t e d , i t c a n b e u s e d t o e v a l u a t e t h e c o r n p a r a -tive rnagnitude of evaporation and convection heat losaes, if valuee are averaged over a period of a month or several daya.

I t s h o u l d b e a t r e e s e d t h a t u n l e s s Q s o r Q . c a n b e r n e a s u r e d i n d e p e n d e n t l y , t h e e r r o r s i n m e a s u r i n g o r e s t i m a t i n g a l l t h e c o m p o n e n t s i n t h e e n e r g y b a l a n c e e q u a t i o n a r e a c c u r r r u l a t e d i n t h e r e m a i n d e r Q " * Q s . T h u s , i f Q c a n d Q " a r e e v a l u a t e d b y a s e u r n i n g t h a t B o w e n ' a r a t i o i e v a l i d , i t b e c o m e s v e r y d i f f i c u l t t o a E B e s s t h e a c c u r a c y o f t h e e a t i m a t e f o r Q . which is obtained. E s t i r n a t e d E n e r g y B u d g e t f o r a S m a l l L a k e a t O t t a w a

To illustrate further the relation between evaporation and the

other components of the energy exchange, the energy budget of a small l a k e i n t h e O t t a w a a r e a i 8 p r e s e n t e d . I t w a s a s s u r n e d i n t h i s a n a l y s i s

that no energy was brought into the lake by precipitation or groundwater

flow, and that the lake had no 6now on the ice during the winter.

Figure 2 shows the estimated monthly values of the components of

the energy balance. In calculating the various components rnany

assurnp-tione had to be made. Therefore the values obtained rnuet be considered

as approximate, but they do show on an annualbaeis the relative

importance o{ the different componente of the energy exchange.

The approxirnate short-wave terrn was obtained frorn the average v a l u e e u e e d i n F i g u r e l . F o r t h e p u r p o s e s o f i l l u s t r a t i o n t h e l o n g - w a v e r a d i a t i o n w a s c o n e i d e r e d c o n e t a n t a t 1 2 0 c a t / c m Z / Z 4 l n r .

The rate of change of the heat etorage waa estirnated during the i c e - f r e e p e r i o d f r o m a v e r a g e v a l u e s g i v e n b y J o h n s s o n f o r L a k e

Klimmingen (f0), and (rorn aome Ereasurernents rnade by the author on

the rate of heat loae from a small lake in the Ottawa area. The heat used

to melt the ice and the heat loea during the period of ice formation were estirnated frorn typical ratea of ice growth and melting meaeured in

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the Ottawa area.

The surn of Q" * Q. waa obtained by adding or subtracting the rate of change of heat Btorage from the calculated net radiation. An average value wae obtained for Bowenr e ratio by direct calculation during thoee

periods when recorda were available and by estimation during the perioda when records were incornplete. Tlre average value of Bowenrg ratio obtained wag then ueed to separate the Q. and Clu cofirPonents. The valueg obtained for the term Qc compare reasonably well with thoae given by Johnssonfor Lake Ktamrningen during the ice-free period.

The reaults of t}rie approximate analyeis are shown in Figure 2. The eetirnate of tlre radiation componenta, though satisfactory to ghow seaeonal trende, cogld be greatly in error, particularly for a aingle month. If the calcutated net radiationwaa in error by l0 per cent. the error would be of the sarne magnitude as the convective term. Therefore unless the radiation terrno can be meaeured accurately, not much

confidence can be placed iu the convective terrn Q.. But even if the convective term is out by ae much ae l0O per cent, the trend iu the evaPo-ration term will stitt be repreaeDtative of true evaporation, particularly during the surnmer months when the evaporation component Q. ie large cornpared to Qq.

The heat Btorage terrn is eignificant for a small lake. If ice thickness and water ternperature recorde are available, the rate of change of the heat otorage can be estirnated with reasonable accuracy.

Atthough, on the basis of this preliminary analyeie, the convective cornponent ie likely to be much ematler than the evaporative cornponent, during the apring rnelt period when warm air is paaeing over the melting ice or cold water, the convective component rnay becorne more Bignificant.

Figure 2 indicatee that during the apring, about the game arnount of energy is used in evaporating ice ag in rnelting it. Other studies (7, l) have ehown that the arnount of energy uaed in melting is ueually much greater than the energy used in evaporation. It ahould be pointed out that ae it requiree approximately 80 cal/gtarn to mett ice cornpared with 6?5 cal to sublimate it, the mass of ice melted is large cornpared to the mase evaporated. It ateo ahould be pointed out that an ice cover rarely remains clear of gnow throughout a winter aeason. If an ice cover becornes snow covered the amount of radiation absorbed ia reduced, the heat flow through the cover is also reduced, arid the evaporation component witl be leaeened. Cornparieon of Energy Balance of Pond,.Srnall Lake and Large Lake

To ahow the effect of depth of water on the energy balance, the energy budget for a emall lake ia cornpared with the energy budget of a ehallow tank and a large deep lake. Figure 3 ahows this comparison of the approximate energy budgete for theae three sizes of water bodiea.

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EVAPORATION FROM lryATER, SNOW AND ICE

The energybudget for the small lake ia the eame as in the

preceding analysie. The calculated energy budget for Lake Ontario

preaented by Brucc and Rodgcrr (4) waa uaed aa rr cxanrple of a large deep lake. The rnonthly energy budget of the ernalt tank was eatimated

by rneans of the 6arne general approximations used to obtain the energy

budget of a small lake. The net radiation energy terma were assurned to be the aame a6 for the small lake. whenever possible, meaaured valuee

of the heat atorage term and values of Bowenra ratio were obtained frorn

obgervations rnadc at Ottawa and uaed in the calculatione.

Figure 3 ehows that the rate of change ofheat atorage and

convection terma are relatively emau for the tank and that net radiation

ie approxirnately equal to, and in phase with, evaporation. It ia for this

reaaon that varioug auttors (15) working with saturated goil are able to

eatirnate evapotranspiration from measured aet radiation.

In the case of the emalt lake the heat Btorage term becomes rnore

significant. During the period septernber to Novernber the energy frorn

heat storage is available for evaporation resulting in higher evaporation than frorn the grnall tank. In the caae of the very large ind deep lake, the heat storage term becomee very important and cauaes the evaporation to be out of phaee with the net radiation.

If it is deeired to relate evaporation frorn a ernall pond or class A pan with the eiraporation from a larger, deeper body by meana of a

co-efficient, it can only be done with qualificatibne. For exarnple, from

Figure 3, the relationbetween the small tank and small lake follows a

regular pattern from April to Auguet which might be predictable. During

the fall rnonths, however, evaporation frorn the emall lake ie tikely to exceed evaporation from the srnall pond and the coefficient wotild have to

be adjusted. If evaporation frorn a small tank ie used to eetirnate

evapora-tion frorn a very large lake, the coefficient wouldvary from month to

month and it ia unlikely that the relationehip would be predictable except p e r h a p e o n a y e a r l y b a a i s .

with the records avaitable it wae not possible to preaent rnore complete observationa on the effect of the heat storage terrn on evapora-tion, but it is congidered that what has been preaented is a fair deecription

of what occura. Mansfietd ( l2), in Australia, haa made a rrore detailed

analyais of the influence of depth, and hence of heat atorage, on the an'ual

cycle of evaporation. In hie analyeia the calculated monthly evaporation

for a water body 104 cm deep was approximately Z months out of phase

with the maxirnurn evaporation for a water body 103-crn deep, and

ipproxi-mately Zj monthe out of phaee with a *"t"r-U-"jy-a%;-;"jp.

MASS TRANSFER APPROACH

Nurnerous maas traaafer equatione have beeo derived for

eatirnating evaporation. A review (z) of theae equationo waa undertaken

(13)

i[ that inweatigatioa. Ooly a few aapectg of rnaea tra.Dsltort formulae, related directly to tbeir uee, will be briefly cooeidered in tJlie PaPcr.

Several rnaaa traaafer formulae require meaaurerneat of the vapour preeaure and wind velocity profitea;observatioDawhich are ueually aot available. Often, tbe ditlference in wind speed and vapour pressure between two heighta io very amall aod thia fact inposeo etriugent require-ments on the instrunentation and t|re erperience of, tbe obeerver.

Sirnpler forure of magg traagfer formulae have been developed where evaporation is asaunred proportional to a functioa of average wind speed and tJre dilfereace betweea ttre vapour pressure of tte air at tbe evaporating eurf,ace aod tte vapour presaure of tte air at some level above t.be eurface. One sirnplified era,raple of such a formula ie givea below as a baeis for discussion:

F _ r f ( u ) ( e r - e r l

E - I \ T

E . evaporatiom rate, f(u) = firactioa of rrind velocity, f{-zal = roughneaa pararneter,

e6 = vapour Preaaurc at eutfacc'

e" = vapourpreEaureofair at eane levelabone the aurfaceraud K = conetant (which ilcludea air dencity a'-l air preeeurc terma!

The differeace between e, and e" ie uaually fairly large eo that thc requirernent m the accuracy of the meaauring inatrurnenta i8 nd 3o oevere ae it is when ttaing to deterrriae the wapour preature profile ia the free air etreann. But the deterrnhatio of e, doea presed problem. due in part to the large gradient in vapour pressute which cap exist at the

surface aad the dilficulty aqaetirrrea encountered in defining the poaition of the surface. Oftea the vapour presaure at the surface ia determined by asgurni.g it equal to the gaturated vapour pressure correeponding to the surface ternperature. Thie, of course, Oren creates ttre problern of accurately rneaeuriag the aurface ternpe rature.

- In cornparirrg evaporation frorn 6Dow or ice surfaces with water surfacee the vapour presaure differeace es - €a hae special significance. Becauae the rnaxirnurn ternperature that a snorp or ice surface caa attain is 32"F Ore rnaxirnurn vatrx)ur preosure at the Bnow or ice eurface caDnot be greater than the eaturated vapour preesure of air at 32"F which ig 6.1 rnbs. Hence evaporation frorn enow or ice caDnot take place ualess ttre dew point is below 32"F (ea ie less than 6.1 mbl.

(14)

EVAPORATION FROM WATER SNOW AND ICE Roughness Pararneter

The difficulty of evaluating the roughness pararneter f(Ze) and its irnportance in inass transfer formulae is stresaed in the Lake Hefner-studies (18) where it ia stated tJlat ttthe roughneea Pararneter of a natural water surface is a critical factor because the comPuted evaporation

depends eo much upon it. Values of Zofor oPeD water, generally the open ocean, have been variously quoted between 0.02 to 0.6 crn. Even if there were agreement a6 to value, it would not follow that the aarne value would apply to a small lake." Further in their report they 6tate: rrAlthough the values of Zo quoted above are based only on rneaaurementg near the centre of the lake, they have been conaidered applicable to the entire surface and used accordingly. There ia certainly a poseibility that Zo varies downwind, eince all the conventional surface characteristics (wave height, etc.) do eo."

Because roughness parameters depend on such features aa wave heights, height of snow drifts, and type of vegetation grorring above a anow cover, valueg of roughness pararnetera are likety to be quite variable. For exarrple, the values of the roughness parameler Zo found for water for one observational period during the Lake Hefaer study varied from 0 . 5 5 t o l . l 5 c m . T h e r o u g h n e a s p a r a t n e t e r s p r e s e n t e d b y P r i e a t l e y ( l ? ) for snow covera varied frorn 0.005 crn for srnooth 6now oD ahort graas to 0.10 cm for snow gurfacea on natural prairie. Because of this variability,evaporation forrnulae which require a value of roughnesa . parameter for eolution muet be treated with caution, particularly if the period of observation ie of short duratioa.

Mid-ocean and Mid-dese rt Conditions

In using maas transfer forrnulae a distinction should be made between the two extrernea under which evaporation takee place under field

conditions. Penman (15) has called thege two extrernes rrmid-oceanrr and rrmid-deserttt. He Etates ItThe first ia almost eeU-explanatory: it ie the state in which evaporatioa takes place from a limited area in the midst of an infinite sheet of the sarne kind of eurface. A equare kilometre ia mid-ocean satisfies thie condition, but it may be equally well satisfied by a field in the midst of an extended area of vegetation. At the other extrerne, ilrnid-desertrr conditiong are those in which a reaervoir or other

evaporating surface is surrounded by an infinite plane from which no evaporation tdkee place. rl

When evaporation is rneasured frorn anow covers uaing evaporation pans filled with snow and flush with the eurface, the rrrrrid-oceanrr condition ia closely approximated. If evaporation ie meagured from a Class rrAtt evaporation pan surrounded by arid land, however,the conditionapproachee the rrmid-de sertrr condition.

In the first caee, the air pasaing over the evaporation pan will not u-dergo appreciable rnodification even if the paa were of very large aize. In the aecoad caae, the size of the evaporatioa pao or body becomes a

(15)

consideration as the temperature and vapour preasure oJ the air are modified aa it paasea from the arid area over the wet evaporating gurface. Halstead and covey (8) have demonatrated that under extreme conditiona the rate of evaporation will vary appreciably with the eize of the body because of this effect.

Other authors (ll) suggest that the increase in vapour pressure of the air downwind acroea an open-water surface ia offset by an increaee in wind speed downwind from the leading edge of a lake and thus the rate of evaporation does not decrease appreciably downwiad across an open-w a t e r s u r f a c e .

General Considerations

Nurnerous empirical forrnulae have been developed which are similar in forrn to formulae based on lnaaa trangfer theory. uaually only the rnean wind epeed aad vapour pre6aure terma are conaidered. other variables such as surface roughness are incorporated into an evaporation constant. Theee empirical formulae neglect rnany factors but the need for formulae, using commonly avaitable records, makes their use wide-s p r e a d .

One condition not considered in using either empirical or rnass transfer formulae is the effect on evaporation when the wind ie strong enough to disturb the surface. Io exposed areas where the wind can move large quantities of anow or cauae Bpray to blow off wave topa, evaporation rate6 at leaet for short tirne intervale, might be considerably higher than a general evaporation equation would indicate.

One problem, corrurlon to all these forrnulae, is to obtain valuee of wind spegd, air vapour preaaure, and surface ternperature, which are representative of average values over the entire evaporating surface. If estimatee of evaporation on a short-term baeis are required, then the need of obtaining adequate values for these variablea ie critical. If long-term estirnate6 are required the problern ie not eo critical ae the

variability of these factors tend to average out.

COMPARISON OF MEASURED EVAPORATION RATES FROM PANS AND SMALL TANKS

During the past few yeare the author has measured the evaporatioo frorn snow coverB and emall bodies of water in the ottawa area. Although this paper is not concerned with measurement, the records obtained are presented and cornpared with a pan evaporation forrnula to indicate the rnagnitude of evaporation ratee that can be expected from enow and water, t w o o f t h e s u r f a c e s t h a t h a v e b e e n u n d e r d i s c u s s i o n .

Evaporation Measurernent s

(16)

EVAPORATION FROM WATER S N O W A N D I C E

d e e p w i t h 3 5 5 s q c m a r e a ) f i l l e d w i t h s n o w , f l u e h w i t h t h e g n o w a u r f a c e a n d weighing the pans at different time intervals. The weight loae or gain during t}re time period waa alaurned to be due to evaporation or condeo-sation at t}re aurface during the period of obaervation. The aluminurn p.urs were painted white to reduce the abaorption of radiation frorn the aun.

Snow meaaurernenta were taken at Ottawa during the years 1956-5?, f 9 5 7 - 5 8 a n d 1 9 5 8 - 5 9 . I n a d d i t i o n r n e a s u r e r r e n t a w e r e t a k e n a t F t . F r a n c e ! , O n t a r i o , i n 1 9 5 7 - 5 8 , I 9 5 8 - 5 9 , a n d a t G l a c i e r , 8 . C . , i n L 9 5 7 - 5 8 , 1 9 5 8 - 5 9 . The work of Mr. O. R. Carlson at Ft. Francee (Departrnent of Northern Affairs and National Resources) and Mr. P. Schaerer at Glacier

(National Research Councit) ia gratefully acknowledged. The sitee at G l a c i e r a n d F t . F r a n c e s a r e c o n s i d e r e d t o b e e e m i - e h e l t e r e d b e c a u s e the rneasurernents were made in a cleared section of a general forest area. The eite at Ottawa'ie considered eerni-expoaed ae tJle obaervation area waa expoeed to drifting BDow.

There were m:rny daye when it was irnposaible to obtain reliable ttreaaurernents because of weather conditions and so the records are rather eporadic. The opportunity for rneaaurernent at Ft. Frances was greater than at Ottawa or Glacier and consequently there are rnore observations frorn thig site than frorn the other two, party'cularly during the colder rnonths when the rate of evaporation ie srnall. Figure 4 showe hiatograrns of the evaporatioD rates obgerved at the three otations.

At Ottawa evaporation loeses were observed frorn other surfaces a s w e l l . T h e a e i n c l u d e d t h e w a t e r s u r f a c e o f a c l a s e "Arrevaporation pan which waa aet up as part of the evaporation meaaurernent prograrn of the Meteorological Branch, Departrnent of Transport, water saturated peat moes placed in a 4-ft diarneter pan ( l2 in. deep) flush with the ground near the Claes ilArr pan eite, and a water aurface in a l0-ft diameter tank, lZ in. deep, placed flueh with the gror'-d at the eame location. The pane were located in a graesed field which was frequently wet and therefore the evaporation condition carr be considered as apfroaching ilmid-ocean'l rather than rrrnid-desertrr. The analyaia of the recorda obtained ia not yet cornplete but aome results are presented and cornpared with the obaerved lossea from the anow cover.

Analygis of Resulte

The average values of all rneagurements are ahown in Figure 5, where meaeured evaporatioD rates are plotted against the average vapour Pressure difference. Average values for the vapour pressure of air over the periods of obaervation were calculated by using the relative hurnidity and air ternPerature recorded at nearby etandard rneteorological atatioos. In calculatiDg air vapour preaaures for Ft. Frances. the dew-points recorded at Kenora were uged.

The vapour preogure of the air at the anow surface was calculated by aaeuming that the average sDow-surface tempcrature is equal to the

(17)

tnean air temperature. Studiea at Ottawa indicate that over a period of time average sno\p-surface ternperature equals average air ternperature. Whenever the snow wag melting the vapour pressure at the gurface wae assurned to be equal to 6.1 rnb. The vapour preesure at the other surfaces was obtained by asauming that it was equal to the eaturationvapour

Presaure of air at a temperature equal to the average, measured temper-ature of the surface.

Because of the difficulty of obtaining aatiefactory evaporation measurementa aDd of eatirnatiDg average vapour presaure differencea, it was realized that accurate correlations could not be expected. The snow evaporation measurementa in particular showed considerable scatter wheu plotted on a daily basig. The reeulte are presented, however, and

compared with a recent Penrnan formula for evaporation paas to show that if observations are averaged over a long enough period, reasonable estirnates of evaporation can be obtained when suitable records are available.

Wind speed recordg were not available for the enow obeervatioDa at Ft. Frances or Glacier. Records at the Ottawa eite indicated that the average wind speed at 2 metres was approxirnately l00 rniles per day for the observation periods. This value for the wind epeed wae used in the Penrnan forrnula given below:

Eo . o.3s

(eo - ed)

(0.5

+ ffil

which for a wind speed of 100 miles per day at two tnetreB airnplifies to: E o = . 0 3 9 A e

where Ae = vapour preaaure difference, (mb of Hg), and Eo r evaporatioa rate, ctn/24 }nr.

Obeervationo obtained at Norrnan Welle, N.W.T. by Mr. R. Brown of the Divieion of Building Reaearch for the Meteorological Bra-ch

evaporatioa rneagurement program are aleo plotted on Figure 5. Although thie gite coutd be considered aa an exposed eite, the obgervationa are comparable to the observations made at Ottawa. If wind speed recorde had been available at all stations and ueed in the analysiu, bett€r correlations might have been achieved.

Otlrer autiore (ll) have confirmed that pan evaporation caabe reliably estimated from empirical functione involving wlnd epeed, dew-poiDt, and water ternperature. They have aleo pointed out that uaually when pan water temperature is obeerved, parr evaporatiotr is meaapred ao that the empirical firnctione are, in thie case, of limited usefulnese. If the

surface vapour preaaure canbe eatimated from mean air temperature as wae done in the caee of the enow obiervatione, then the empirical functioas derived caa be ueed to eetimate pan evaporation ueing only commonly available records of wind, dew point and air temperature. Peuman (16)

(18)

IVAPORATION FROM WATER SNOW AND ICE

has attempted to elirninate the need for water-ternperature obeervationa

through sirnultaneorrs solution of an ernpirical maea-tranafer equation and

an errergy bal.-cc equatioa. Of cou,rac, even if pan evaporation formulae

or recorda are available, there atill rernains the problem of obtaining a

euitable pan coefficient if they are to be used to eatimate evaporation

from lakee. Detailed digcueoiona of thia and other aspects of pan

evaporation are given in the Lake Mead Studies (19) and will, undoubtedly,

be the aubject of coasiderable diacuaoion in other papers at thig aympoeium.

CONCLUSIONS

It was pointed out that evaporation investigations fall into two

classee: thoge associated with the fundarnental evaporation process, and those aeaociated with the estimation of evaporation for practical

application. For the firet case, atringent requirernentg wil,l, be placed on

the quality and accuracy of the instruments and the experience of the

observer. To quote Penman (16) once again, ItThere ie the purely

acientific problern of the phyaica of the evaporation proceaa, the eolution

of which involvee complex ideae, complex experirnental. techniquea, and

may at times appear completely irrelevant to field problem6.r' Frequently

in evapotation etudies investigatora attack the evaporation problern from

a research point of view when the ingtrurnentation and facilities available

do not juatify auch an approach.

Comrnoa to almost every evaporation problem, be it an operational

or a research problern, is the difficulty of obtaining a direct measure of

evaporation to conrpare with any calculated value. Coneequently, it is

most difficult to put limite on the accuracy of evaporation estimates. If

estirnates are obtained by different methods there will be a choice of

estimate. The best egtimate will usually have to be decided oo the basis

of the records that were available for analyaie. For whatever forrnula or

rnet'hod is ueed, the accuracy of the estirnate will depend on the quality of

the basic recorda available. Thus, from the operational point of view at

the preaent tirne in Gaaada, the need is to measure and record, under fietd

conditiona, all the quantitiea needed to check and uae the metlrodg that are

now available for estirnating evaporation loseea.

ACKNOWLEDGMENTS

The aqthor ie indebted to Mr. D. Boyd and Mr. L. Gold for moet

helpful diacugsiona durlng the preparation of thie paper. Thia ia a

contribution from the Diwiaion of Building Reeearch, Natioaal Research

couucil, ottawa ard ir preacnted witlr the approval of the Director of the

Divisio'n.

(19)

REFERENCES

Ambach, If,. Investigation of the heat balance in the area of ablation

oa the Greenland Ice Cap. Records in Meteor., Geophye. and

Bioclirnatology, Inat. of Phys., Univ. of Innabruck, Ser. B,

V o l . 1 0 , N o . 3 . 1 9 6 0 .

Anderson, E.R., L.J. Anderson, and J.J. Marcia-o. A review of

evaporation theory aud developmeDt of inetrurrrentation, Lake

Mead water loee investigationa. Int. Rept., Navy Electronica

L a b . R e p t . N o . 1 5 9 . 1 9 5 0 .

Bowen, J.S. The ratio of heat logseg by coduction and by

evapora-tion from any water surface. Ptrya. Rev. 2? z 779-787. 1926.

Bruce, J.P., and G.K. Rodgere. The water balance of the Great Lakes

systern. Sympoaium on the Great Lakea, Am. Aseoc. for the

Advanc. of Sci., Chicago. Dec., 1959.

Brunt, D. Phyaical aad dynarnical rneteorology. Cambridge

Univeraity Pregs. 1952.

Budyko, M.I. The heat balance of the earthrg surface. (Tranalated

by N.A. Stepanova from Teplovoi balana zernnoi poverkhnoeti.

Gidrometeorologicheskoe izdate['atvo, Leningrad, lt56). U.S.

Dept. of Comrnerce, Waehington. 1958.

Diamorld, M. Evaporation or melt of 6Dow cover. U.S. Gorpe of

Engra., Snow, Ice and Perrnafrost Reaearch Eatablishment,

R e s . P a p . 6 . f 9 5 3 .

Halgtead, M.H., and lff. Covey. Some meteorological aspecte of

e v a p o t r a a s p i r a t i o n . S o i l S c i . S o c . A m . P r o c . 2 f ( 5 ) : 4 6 1 - 4 6 4 . 1 9 5 7 .

Hydrology Subcomrnittee, Aaaoc. Cqrrm. Geod. and Geophya., Nat.

Reg. Coun., Ottawa. Minutes of fifth meeting, March 1960.

Johnsson, H. Termisk-hydrilogiaka Studier i Sjiin Kldmrningen.

(Thermal-hydrological etudiee in the Lake KtSmmingen). Geogr.

A n n . S t o c k h . 2 8 : I - 1 5 4 . 1 9 4 5 .

l l . L i n s l e y , R . K . J r . , M a x A . K o h l e r , a n d J . L . P a u l h u e . H y d r o l o g y f o r E n g r s . M c G r a w - H i l l . f 9 5 8 .

Mansfield, W.tff. The inlluence of monolayere on evaporation frorn

w a t e r a t o r a g e s . A u s t . J . A p p l . S c i . l 0 ( l ) : 6 5 - 7 2 . t 9 5 9 .

Mateer, C.L. A prelirninary eetirnate of the average insolation in

C a n a d a . C a n . J . A g r . S c i . t 5 z 5 7 9 - 5 9 4 . 1 9 5 5 .

t .

z .

3 . 4 . 5 . 6 . 7 . 8 . 9 .

l o .

t 2 . t 3 .

(20)

EVAPORATION FROM WATER SNOW AND ICE 4 ? \ 1 4 .

1 5 .

Mittar, D.H. Snow cover and ctimate in the Sierra Nevada, C a l i f o r n i a . U n i v . C a l i f . P u b . G e o g . I l : l - 2 1 8 . 1 9 5 5 . Pelton, w.L., K.M. King, and C.B. Tanner. An evaluation of the

Thornthwaite and mean temperature methods for deterrnining p o t e n t i a t e v a p o t r a n s p i r a t i o n . A g r o n . J . 5 2 ( 7 1 t t 8 7 - 3 9 5 . 1 9 6 0 . Penman, H.L. Evaporatioa: an introductory survey. Netherlands

J . A g r . S c i . V o l . 4 , N o . l . 1 9 5 6 .

P r l e s t l e y , C . H . B . T u r b u l e n t t r a n s f e r i n t h e l o w e r a t r n o s p h e r e . r J n i v . of Chicago Prese. 1 9 5 9 .

U.S. Departrnent of the Interior. Water-loee investigations: Lake Hefner studies, technical report. Geol. Surv. Prof. Pap. 259.

1 9 5 4 .

1 6 . ,

l ? .

1 8 .

19. U.S. Departrnent of the Interior. Tfater-loes inveetigations: Lake Mead etudiea, Geol. Surv. Prof. Pap. 298. f958.

(21)

z

o @ @ ll

j

Y

>

J Y

z

o

|('

ri

tl

j

=

F F

o

o o ro o o OJ o S S U O S S V

o

9

N O t I V T O V U

(un +z/ urc

os u slruorvc)

(22)

EVAPORATION FROM WATER, SNOW AND ICE 49

+500 +400

rcE suRFAcE-+-

WATER SURFACE

o

=

ct

o

att at, gg tl

o

J

()

E I $ GI E lrl F 3 l!

o

=

()

o.t7

o.34

o.5l

'300

'200

+100

-roo

-200

+200

+loo

o

-200

rloo

o

-too

-200 -300

LONG WAVE RADIATION

l---t---t---)h--tt+ --*p--r(-- -tF--a___*_--r(- - -*---ta

tt -t,-:-*+-t--#f, -;a1-1

RATE OF CHANGE

OF HEAT STORED

WARMING WATER ICE MELTING

CONVECTIVE

COMPONENT

-/..-"---\y--1-l - , . 1 * .

DEC JAITI FEB MAR APR MAY .,UNE.T.LY AIJG SEPT OCT No\' DEC JAN FEB

(23)

o

=

(J

o

at, q t! E

3

()

e#

LONG WAVE

RADIATION

EVAPORATION

at E I sf $l (n

=

o

J F M A M . ' . , , A

LEGEND

SMALL TANK I FT DEEP

SMALL

LAKE 30 FT DEEP

LAKE ONTARIO

Figure 3 - Cornpariaon of energy balance large lake.

a - - - a - - - a I-X-t o-o-o

CONVECTION

(24)

E V A P O R A T I O N F R O M W A T E R S N O W A N D I C E

5 r

EVAPORATION CMS / OAY

O T T A W A

( E X P O S E D

S I T E )

GLACIER ( SHELTERED

SITE )

FT. FRANCES (SHELTERED

FROM WIND EXPOSED

TO

FT. FRANCES

(SHEL

FROM SUN AND WIND)

2 0

t o

o

U' z

9 z o

F

| r l o

ur

at

3 o

l.L

o 3 0

E Lrl

3 z o

f z

to

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5 0

40

3 0

?o

l o

Figure 4 - Hiatograrns of evaporation observations at Ottarra, Glacier, Fort Francea.

(25)

at, E I sr (\l U'

=

()

ul F E z I F G

o

o-u, o lrj (l f to tt,

=

o . 8

o . 7

o ' 6

o . 5

o . 4

o . 3

o . 2

o . r

o r z

o

? 4 6 8 r O 1 ? 1 4 V A P O U R P R E S S U R E D I F F E R E N C E ( M B S ) L E G E N D t 6 NO OF OAILY OBSERVATIONS WATER O a

tr

I SNOW A A X

IO FOOT TANK OTTAWA CLASS.h..TANK OTTAWA PEAT MOSS TANK OTTAWA cLAss "A" TANK NoRMAN wELLs FORT FRANCES G L A C I E R OTTAWA 3 0 3 0 4 4 9 8 90 2A 2 l

Figure 5 - Measured evaporation rate p r e a s u r e d i f f e r e n c e .

cornpared to vapour

(26)

DISCUSSION ON WATER SNOlry AND ICE

Dr. ORVIG pointed out that Mr. Wiltiams, in discussing the energy balance method of measuring and calculating evaporatio'n, wrote the Bowen ratio as the ratio of the temperature difference at two levels to the vapour pressure difference at the same two levels. This assurnes the same value for the exchange coefficiente in heat and water vapour exchange. Measure-rnenta of these coefficients, however, aeern to give as many different values as the nurnber of gites investigated. Therefore, in order to obtain general results, it would eeem preferable to uee the energy balance method.

Penman has found a high correlation between net radiation and urater uee (evapotranspiration) over a grass surface. In Canada there are large areas of peculiar vegetation typee which do not behave as ttuormaltt vegetation. For example, surfacee in the central part of the Labrador-Ungava peninsula have very low rates of evapotranspiration, arrd lo\r correlation between water use and net radiation. Such conditions, of lower-than-expected evapotranapiration and hence high runoff, are also found ia similar areaa in Finland. The low evaporation is cauged by the iaautating effect of the lichen co\rer, which does not act aB a traneporting agent for water from the ground to the atrnosphere.

Mr. WILLIAMS in reply stated that over snow co\rers there are periods when it is most difficult to obtain meaningful values for Bowen's ratio. For exarnple, if warrn air is paasing over a meltiDg anow cover the difference in temperature between the eurface and the air is great, whereas, depending on the vapour pres6ure of the air, the difference between the vapour preasure of the air and the vapour pressure of the surface can be emall, and it can even fluctuate betweea plue and rninus valuea. In thia case, calculated values of Bowenra ratio catr fluctuate between large poeitlve and large negative valuea, and it ie very difficult to use Bowenrs ratio to solve for snow evaporation in the eaergy balance equatioa.

Mr. GOLD added that in determining the energy balance, it ie aesumed that:

Q . = K " u ( e o - e " ) Q s = K . u ( e o - e 3 )

At some stage in the calculation, an assumption equivalent to Bowenrs ratio muat be rnade. In this case, it ig assurnted that R"/K" ie constant. Also, it is not only the change in the character of the energy exchange frorn one area to another which must be considered, but also t}e change with tirne at one area.

Dr. ORVIG replied that in investigating the importauce of radiation, in the energy balance equation, it is found that it changeg markedly from one tirne to another and also with locatim. Radiation is overwhelrning in the ablation of glaciers in temperate latitudes, such as Switzerland, and in the early part of the melting aeason in high

(27)

latitudee. ln locatiooe Dear opea water and, ia general, late io the

ablation Eeason, the importance of eddy llux increasea a6 more *arm air

ie advected over the snow and ice surfacea. The relative importance of

convection and radiation in the heat tranaport to the eurface dependa

therefore on both tirne and location.

Dr. PORTMAN was interested ia Mr. Williamsr statement that

radiorneters tt... are still at the atate of development where they require

practically constant eupervision ...rr for reliable resulte. Hig own

experience in working with ventilated radiometera for nearly a dozen

yeats hae led to an opposite view. In fact, the ability to measure net

radiation is the one bright spot in an otherwise diecouraging overall

rneasuretrrent problem.

Mr. WILLIAMS repl.ied that he could only speak about his own

experience in the meaaurenrent of net radiation. Two commercially

available net radiometera were compared, and it was found that there

w e r e l a r g e d i f f e r e n c e s o n a s h o r t t e r r n b a s i s , b u t t h a t o v e r a p e r i o d o f several days the agreement was reasonable but by no rneans perfect. It is pointed out that it is much more difficult to get a reasonable

lreasurernent of net radiation over snow coversr where the net radiation

terrn can be srnall and of tJle same order of magnitude as the convective and heat atorage terms.

Mr. MUKAMMAL, said that the eddy transfer coefficients of heat

and moisture are different, as has been proved by Priestley, but t}is

difference diminishes as the surface is approached since the stability

effect is at a rninirnurn. It is therefole recornmended that the potential

difference of ternperature and hurnidity should be measurcd as near the

surface as possible for cornputing flowenrs ratio.

Mr. CAVADIAS said that in the paper there is the remark rrPenman has atternpted to elirninate the need for water ternperature

obgervations through eirnultaneous Bolution of an empirical masg-transfer

equation and an energy balance equation.rr In lHydrology for Engineersrt

by Linaley, Paulhus aad Kohler, there is a similar remark that "Penrnan

haa shown that the need for water temperature observationg can be

eliminated.r' Is it not true that Penman has not shown, but sirnply

assurned, that the need for water temperaturea can be elirninated?

Mr. WILLIAMS replied that he had not checked witJ: field data to

see whether or not Penrnanrs method of elirninating the need for water

temperature observations is satisfactory, and he would rather not

comment on this question without actual calculations to back up any comrnents.

(28)

A liet of all publications of the Division of Building Research is available and may be ob-tained from the Publications Section, Division of B u i l d i n g R e s e a r c h , N a t i o n a l R e s e a r c h C o u n c i l , Ottawa, Ganada.

Figure

Figure  I  illustratee  the  differencea  in  the  energy  available  frorn s h o r t - w a v e   r a d i a t i o n   a t   t h e   t w o   s i t e s
Figure  2  shows  the  estimated  monthly  values  of  the  components  of the  energy  balance
Figure  2  indicatee  that  during  the  apring,  about  the  game  arnount  of energy  is  used  in  evaporating  ice  ag  in  rnelting  it
Figure  3  ehows  that  the  rate  of  change  ofheat  atorage  and convection  terma  are  relatively  emau  for  the  tank  and  that  net  radiation ie  approxirnately  equal  to,  and  in  phase  with,  evaporation
+5

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