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Chapter X. Evaporation from Wet Surfaces

# Chapter X. Evaporation from Wet Surfaces

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252

X. E V A P O R A T I O N F R O M W E T S U R F A C E S

To minimize these problems of determination, they can be approached in three ways: as the supply of water to the site where

vaporization takes place, as the removal of vapor from this site, and as

the rate a t which energy is converted at the site into latent heat. Where

does this energy come from?

The Energetics of Evaporation

The simple energy-budget equation is balanced in the form:

+R,

-RE

5

+-H

-E

radiant energy absorbed by the evaporating surface, including both shortwave and long-wave forms; also called "radiant-energy intake" (see

Chdpter XI)

radiant energy emitted by the evaporating surface in accordance with the

Stefan-Boltzmann relation

heat exchange with layers below the evaporating surface

exchange of sensible heat with the air

conversion of energy by the vaporization process into latent heat

0

Here + means energy gained by the evaporating surface and - means

energy lost from it. The fluxes of sensible heat (S and H) might be

directed either toward or away from the evaporating surface under

different conditions of water or air temperature distribution; their

direction is indicated by + or -. We will examine these energy

sources in the order given.

water bodies at the earth's surface; for convenience they are called

"short wave" and "long wave." Solar radiation falls in the short

and clouds in the atmosphere is called long wave (wavelengths longer

than about 4 pm). Part of each flux is reflected by the earth's surface,

although in the case of water surfaces the fractions are generally less

than 0.05; the rest penetrates the surface and is progressively absorbed

with distance traveled.

The average rates at which short-wave and long-wave radiation is

absorbed in water bodies are large. For example, during summer in

central North America they are about 200 W m- of short wave and 350

of long wave. These amounts of energy supply correspond to the daily

vaporization of approximately 7 and 1 2 kg of water rnp2 (depths of 7

and 12 rnm day-'). Both forms of radiation vary through the day, the

familiar on-and-off cycling of solar radiation (600 W mP2or more in

the middle of the day, zero during the night) being particularly

253

D E T E R M I N I N G E V A P O R A T I O N RATES

striking. Long wave is more steady in both the daily and annual

cycles.

Not all of the energy from absorbed radiation can be converted into

evaporation or other nonradiative energy conversions. The price of

existence as a wet surface is to maintain a surface temperature of at

least O"C, and this implies the emission of long-wave radiation at a

rate of 315 W m-' from the surface.

Summing all the gains by absorption of short-wave and long-wave

radiation and subtracting the 315 W m-' just noted, we have the

accompanying tabulation as typical of day and night in summer and

winter in the interior of North America (in W IT-'),

Typical day

Midday in summer

Night in summer

Midday in winter

Night in winter

" Taking

Absorption of

long-wave

atmosphere

600

370

0

330

290

270

130

0

Emission of

surface"

-415

-415

-315

-315

Sum

+555

- 85

+ 105

-45

the surface temperature as 20°C in summer, 0°C in winte1

Clearly the sum of all the fluxes of radiation is a highly variable

source of energy; it is large in the long summer days, with a net loss

every night. While these variations are more or less predictable, except

for the effect of passing cloud decks, they nevertheless impose a

pattern on the other, or nonradiative fluxes in the energy-budget

expression cited earlier. By definition, the sum of the nonradiative

fluxes, S, H, and E , must be equal and opposite to the sums of the

radiative fluxes given in the accompanying tabulation.

Substrate-Heat Flux One alternative to radiant energy as a source is

the substrate, that is, the layers of soil or water underlying the surface.

The soil-heat flux is directed toward the active surface more than half

the time, i.e., during the hours of later afternoon and darkness. In

some cases it supports a small rate of evaporation from the soil itself.

Heat moves upward from the deeper layers of a water body to its

surface during the months of fall and winter. Low intensity of

radiation in these months has little inhibiting effect on evaporation,

which proceeds vigorously until the time when ice forms on the lake

surface, if it does. The enormous amounts of heat stored in deep water

254

X.

E V A P O R A T I O N F R O M WET S U R F A C E S

bodies make i t possible for them to evaporate much more water than

Sensible-Heat Flux The second alternative source of energy for

evaporation is the atmosphere itself, which thus is simultaneously

humidified and cooled by turbulent exchange between it and the

underlying wet surface. In this situation the surface must be cooler

than the air and also moister.

A simple example is the wet-bulb thermometer, in which a cotton

wick keeps the bulb wet while i t is strongly ventilated by being

swung. If the air is quite dry, evaporation from the wick removes heat

from the bulb, which has only a small amount of "substrate" heat

storage in its tiny volume and also is shielded from any excess of

radiant energy. The energy budget of the wet bulb thus contains only

two items, the outward flux of latent heat E and the inward flux of

sensible heat H. These offset each other exactly. The drier the air, the

greater are both fluxes since the bulb temperature sinks to a level

much lower than the air temperature.*

In natural situations the other fluxes at the surface of the earth are

usually not zero, so the sensible-heat flux seldom becomes the sole

source of energy for evaporation.

Combined Sources of Energy for Evaporation The energy sources that

support vaporization differ from place to place and time to time. Each

can act singly, but it is more likely that combinations of them provide

the energy that is converted into latent heat.

In a Milwaukee November we see the grass still green, and deduce

that it is continuing its life processes, including transpiration, although at a low rate, powered by the slanting rays of the low sun. It is

not as obvious, perhaps, that at the same time Lake Michigan also is

vaporizing water and at a much higher rate. Its rapid evaporation is

supported less by solar energy than by the heat put into storage in its

deeper layers a few months earlier and now being drawn upon in fall.

largely on current radiant energy, which is supplied at a slow rate,

proceeds slowly at this season when lake evaporation is continuing a t

a daily level of 3-4 kg m-' as a result of its good source of heat.

When a surface is energized from large nonatmospheric heat

sources, either by absorbing radiant energy R or receiving a large flow

* This difference is utilized as a means of determining the dryness of the air-the

psychrometric method.

D E T E R M I N I N G E V A P O R A T I O N RATES

255

of heat from beneath S, the two heat fluxes from it into the atmsophere

find themselves in competition. Usually the latent-heat flux prevails

over that of sensible heat, if the surface is wet. (If it is dry, there is, of

course, little or no evaporation or latent-heat conversion taking place.)

The ratio between these two competing flows of heat, both transported by turbulent convection in the atmosphere, is called the

”Bowen ratio,”named for the oceanographer I.S. Bowen (1926).Usually

the seflsible-heat flux is much the smaller of the two, and the ratio is

of the order of 0.1-0.3. However, by destabilizing the air this flux

increases its diffusivity for water vapor, and the same vapor flux is

moved at a smaller gradient, which allows a lower temperature at the

evaporating surface. In a study of cooling ponds, those unique

ecosystems built to move heat from thermal power plants into the

environment, Ryan et al. (1974) found that thermal convection made

possible the removal of an added flux of 500 W mp2of dump heat from

a 4-km2 pond, with a rise of surface temperature above the 2-m air

temperature of only 9°C. The rate of evaporation, 14 mm day-’, is one

of the highest from any system at the surface of the earth.

Surface Temperature and Vapor Pressure

Each energy flux, including evaporation, is determined in part by

the temperature of the active surface generating it. Long-wave radiation is emitted in accordance with the fourth power of the absolute

temperature of the surface. The flux of sensible heat moves away from

the surface in proportion to the temperature gradient from the surface

into the adjacent air. The heat exchange between the surface and

substrate depends on the corresponding temperature gradient below

the surface.

The latent-heat flux associated with evaporation also follows a

gradient, that of specific humidity or vapor pressure. The surface

value determining this gradient is a single-valued function of the

surface temperature, expressed as the Clausius-Clapeyron relation (for

two physicists of the nineteenth century) in the accompanying tabulation from the Smithsonian Institution (1966). The accelerating trend of

specific humdity with increasing temperature (shown in the last

column) means that during a rise in surface temperature the gradient

driving the latent-heat flux usually grows faster than do the gradients

driving the heat fluxes into the atmosphere and soil. Therefore the

conversion of heat in evaporation tends to preempt most of the

energy.

256

X.

Surface

temperature

("C)

Vapor

pressure

(mb)

-20

0

10

20

25

30

32

35

1.0

6.1

12.3

23.4

31.7

42.4

47.6

56.2

E V A P O R A T I O N F R O M WET S U R F A C E S

Specific humidity at

sea-level pressure

(gm water kg-' air

or parts per 1000)

0.6

3.7

7.4

14.6

19.8

26.3

29.7

35.2

Rate of change in specific

humidity for a 1" change

in surface temperature

(g kg ' deg-7

0.1

0.3

0.5

0.8

1.2

1 5

1.7

1.9

Linacre (1964) and Priestley (1966) have shown that over a wet

surface this preeminence of the latent-heat flux becomes virtually

complete at a surface temperature of about 34°C. The water surface,

continuing to humidify the air, has no further power to warm it. Thus

34°C (307K) represents a maximum temperature for any wet surface,

Verification of this fact is seen in the absence of ocean areas where

water temperatures exceed 34"C, or where temperatures in oceanic air

exceed 34°C (unless air has moved from hotter surfaces of dry land).

Dry surfaces, on the other hand, are easily heated to high temperatures under strong radiation loading, and in turn heat the overlying

air to temperatures higher than 34°C. The difference between a hot dry

desert surface and the relatively cool foliage of an irrigated field in an

oasis demonstrates that evaporation can remove large amounts of heat.

Surface temperature is a t the same time a cause and a consequence

of energy fluxes. It varies in response to changes in the rate at which

the surface takes in energy and the ease with which i t can get rid of

this energy load. Corresponding variations in the surface vapor

pressure produce changes in the rate of evaporation.

The central problem in talking about evaporation is the scarcity of

direct measurement. Only one method can be classed as direct

measurement; all others depend (a) on statistical correlations of other

fluxes to the vapor flux, (b) on sets of measurements in the energybudget framework just described, (c) on sets of measurements in the

water-budget framework, or (d) Dalton's evaporation law.

Statistical correlations between the vertical flux of vapor and the

fluxes of momentum, heat, carbon dioxide or other substances are not

E V A P O R A T I O N F R O M DEEP WATER B O D I E S

257

precise because of our ignorance of the nature of turbulent transport in

the atmosphere. The mechanisms of this transport do not necessarily

work the same for different physical quantities, i.e., for heat as for

water vapor.

In conditions of neutral thermal stability in the lower layers of the

atmosphere 2nd strong air movement, the turbulent transport mechanisms are most alike. In stable conditions marked by inversion, or in

very unstable conditions this similarity cannot be counted on.

Measurements of energy fluxes to the evaporating surface of a lake

are most accurate over long time intervals, for over a short time the

flow of heat out of the lower layers of water is impossible to

determine. The energy-budget method is most often applied to periods of a month or so, over which the cooling of the lower layers can

be measured.

The water-budget framework also is most useful over long periods

of time, since the movement of water into or out of the lake banks

cannot be evaluated over short times. Over any length of time,

however, the water-budget method is limited by the lack of measurements of precipitation falling into the lake. Because the lake energy

budget often changes atmospheric stability over the lake from the

stability conditions over land and because stability affects precipi tation processes, it is risky to consider that lake rainfall is correctly

represented by rain gages on nearby land sites. (See Chapter V.)

Evaporation from water surfaces is never limited by a shortage of

the evaporating substance, therefore variations are associated with

energy supply or with the vapor gradient from surface to air. Factors

in evaporation can be considered from their connection with either

supplying energy or removing the vapor molecules.

The case of evaporation from snow, which we examined in an

earlier chapter, illustrates this circumstance. In general, evaporation

from snow is limited by a lack of energy supply, as in the Arctic night.

Often, in situations of apparent high energy level, as in advection of

warm marine air over a snow cover, the vapor gradient is toward

rather than away from the snow cover, and no evaporation occurs. Let

us now proceed to examine evaporation from water substance in a

quite different physical state, i.e., bulk water in lake basins and other

aquatic ecosystems.

EVAPORATION FROM DEEP WATER BODIES

All water bodies except the most shallow are distinguished from

other ecosystems by an extremely large capacity to accept heat into

258

X.

E V A P O R A T I O N FROM WET SURFACES

storage and subsequently to give it back. The availability of heat at the

water surface to support evaporation depends less on short-time

fluctuations in radiation than on the stirring of the water or the air.

We need to look at this factor of heat storage in the water.

The Role of Heat Storage

Lake Ontario This lake has received considerable study, some of it

connected with the importance of its level to the operation of the St.

Lawrence Seaway, in hydroelectric production, and as a hazard to

riparian property. In one study of its water budget, a heat budget was

developed (Table I) that illustrates the annual cycle. Like the mountain

snow covers described earlier, evaporation is absent in spring. Even a

contribution of sensible heat from air warmed over surrounding lands

in April and May does little to speed evaporation. Most of the heat

from the warm air, like that from absorbed solar radiation at this

season, simply goes to warm the deep water of the lake. The surface

warms up slowly, and generates little outward movement of water

vapor.

During the cooling phase of the annual cycle, which lasts from

August until February or March, the situation is reversed as heat

stored in the depths of the lake in earlier months now comes to the

surface. Here it makes a large addition to the waning surplus in the

radiation account. The total heat available to go into the atmosphere is

now very great, and supports rapid evaporation during the dark

winter. In fact, evaporation continues through nine months of the

year, and is large in six months (September-February). Its energy

support in September and early October comes chiefly from radiation,

the rest of the time it comes from subsurface heat, which also buffers

the thermal environment of the fish species in this aquatic ecosystem.

The Influence of Lake Depth The depth of a water body affects heat

storage in the water because with sufficient stirring by wind-up to a

depth of 50 m or more-a large mass of water is brought into thermal

communication with the evaporating surface. The great heat-storage

capacity keeps the surface cool and evaporation slow in the spring and

summer and maintains a warm surface and rapid evaporation in the

fall and winter. In May, for example, the latent-heat flux (dashed

curve, in Fig. X-la) from a 5-m deep lake or reservoir is 120 W m?, or

about 4 mrn day-', whereas from a 20-m lake it is only 30 W m-', a

quarter as much. The deep lake is still taking more heat into storage in

May than the shallow, and allowing much less to escape upward into

the air. The converse occurs in fall.

TABLE I

Water a n d Energy Budget of Lake Ontaiio"

Param-eter

J

F

Net income of energyfrom

+2

Bxchmge of hcat betwecn lake t250

surface and d e e p water*

6

Nei transport of hoat in inflow

and outflow rivers and snow

meltingb

Energy available at lake

+246

surfaceb

Surface temperature'

Surface vapor pressure'

Atmospheric vapor pressure"

+3

8

5

Evaporation'

-3.2

Latent-heat flux'

-90

Sensible-heat flux as residual'-155

M

+35

+I60

-5

M

J

J

A

S

0

ix

D

J

+2

~ 8 3 -140

t185

+ZOO

+20@

+160

+110

+50

t10

-5

-15

-.I65

-245

--I90

-130

-70

0

+65

+145

+ZOO

+250

3

0

41

0

-I

-'I

0

.-.2

-5

6

-59

+10

+7@

+89

+109

t115

fl.53

20

23

20

17

19

.

+192

+57

2

7

4

-

-2.6

-75

-115

A

L

5

-1.4

-25

0

2

7

8

4

8

10

10

12

14

19

22

20

+0.6

--0.3

-10

+70

-0.1

-5

-5

-1.6

-50

-20

-40

+15

-25

+10

15

-2.6

-3.6

--75 -105

-15

-5

11

13

12

-3.0

+190

+246

I

5

10

9

3

8

5

-

-3.0

-85

-85

-30

-70

3

-2.9

-3.2

-85

-90

-105

-155

Units: energy fluxes i n W m .',water Fluxes in m m d ~ y - l ,vapor pressure i n nib, tempcraturcs in "C. Note: 'I'hc olcIer data, c!spc~ially,

are subject to change ds findings of the International .Fie!.d Year on the C~-eatLakes are publishecl. Figul-es dre roundcd. Evaporation E is

calculated by t h e Llalton q u a t i o n E = KVic, .- P A , in wiiich V is wind speed over the Iakc, c,, is vapor pressure over the lakc: c , is vapor

pressure at lak+surfacc ternpcrabi:e. and K is a coefficient (Rich;irds and Irbe, ICjbS).

Bruce and Iiodgers (1962).

Webb (1970).

Richards and Fortin (1962).

Richards and Irbe (1969).

260

X.

200

E V A P O R A T I O N F R O M WET SURFACES

-

150 -

-

N

' E 100

3

.

-

5*

-

50-

t-

2

LLI

o-

z

w

-50

2

=

-100

-150

-

-

-

5

15

10

(a 1

I

I

20

5

10

DEPTH OF RESERVOIR ( m \

I

I

15

20

(b)

Fig. X-1. Energy fluxes in (a) May and (b) October in lakes and reservoirs of different

mean depth (m). The net surplus of radiation of all wavelengths (solid line) is large in

May at the surfaces of all the lakes. The change in heat content of the water ( - X - X ) is

greatest in the deep lakes. In May, the larger lakes extract sensible heat (--.--.) from the

air, while in October they give it up. Evaporation (- - -) decreases in deep lakes in May,

increases in October (Nesina, 1967). From the standpoint of the water surface rather

than the water body itself, the sign for change in heat content would be reversed, as it

represents a removal of energy from the surface in May. The sign for latent heat flux

would also be reversed.

The depth of the lake basin itself is, of course, not a s critical as the

depth of the layer in thermal communication with the lake surface. In

a stratified water body under winter or summer conditions, the depth

of the mixed layer above the thermocline is important. This is related

to the fetch of wind on the lake. Arai (1966) shows that lake

evaporation in spring and summer decreases with greater fetches F,

which mix more cold water to the surface, the coefficient of diffusivity

in the water being proportional to F"' (Arai, 1972). He finds winter

evaporation from deep mid-latitude lakes to increase proportionally to

F1'3.

Reservoirs If water from a deep storage reservoir in summer is

taken from its middle or lower layers, which are usually colder than its

E V A P O R A T I O N FROM DEEP WATER B O D I E S

261

surface layer,* the result is that less energy is removed in the discharge

water than is brought into the reservoir by inflowing streams. Energy

accumulates in the reservoir, its surface temperature rises, and evaporation increases. In Roosevelt Lake in Arizona this fashion of withdrawal probably increases annual evaporation 150 mm (Koberg, 1960).

In contrast, smaller reservoirs downstream on the same river, which

are fed by cold water from Roosevelt Lake, experience less evaporation. For the system of reservoirs on the Salt River the total change

was

7.0

X

2.6

X

10'' mR

10" m '

more evaporation from Roosevelt Lake

less evaporation from the lower reservoirs

4.4

X

10' m3

added loss of water by evaporation

If all reservoirs in the system are not under the same administration, a

transfer of income equal to the value of the changed evaporation also

takes place to the benefit of the downstream managers.

In warm dry regions where reservoirs must hold water over periods

generate streams of hot, dry air moving over the reservoirs. The result

is that reservoir regulation of stream flow for irrigation or other

purposes incurs excessive costs of water storage in the form of high

evaporation.

This fact sets limits in physical terms, quite in addition to the

economic costs of construction labor and materials, to the degree of

regulation that is logical on any stream, regardless of the availability of

subsidized low-interest funds. On the Colorado River, reservoir storage of 36 x 10' m3 in 1959 provided an annual regulation of 7.8 X 10'

m3 at a cost of 1.0 x 10' m3 lost by evaporation (Langbein, 1959). Net

gain is 6.8 x lo9 m3 (7.8 less cost of 1.0).

Proposals to build more reservoirs to store a total of 97 x 10' m" of

water would provide 9.8 x 10' m3 of annual regulation, less 2.6 x lo9

m3 lost by evaporation. This is nearly three times the present amount

of evaporation. A large expenditure on dam construction results in a

net annual regulation of 7.2 X 10' m3, not much more than the earlier

6.8 gain. Hydrologically, this gain over the present situation is

insignificant. The toll of evaporation is a major obstacle in the way of

reaching an artificially controlled water supply, in which the yearly

* In some cases, cold water is drawn to preserve downstream habitat for cold-water

species of fish. In others, it is withdrawn for operating convenience and might cause

damage to downstream water users, e.g., rice irrigators.

262

X.

E V A P O R A T I O N F R O M WET S U R F A C E S

regime of natural input would be completely transformed into a yearly

regime fitted to man's requirements.

The Role of Surface and Atmospheric Conditions

In Lake Ontario in fall, heat storage provides a large flow of heat to

the surface, keeping it warm and its vapor pressure high. A large

amount of heat is available for transfer from surface to atmosphere.

The rate of transfer and the partition of the total between sensible heat

and latent heat (as vapor) depends on two gradients from the surface

into the air. These are the gradient of temperature, depending on the

water-to-air temperature difference (3" in fall over Lake Ontario), and

the gradient of vapor pressure, depending on the difference between

surface vapor pressure (a function of surface temperature, as noted

earlier) and atmospheric vapor pressure. The water-to-air difference in

vapor pressure above Lake Ontario runs about 2 mb (Table I).

Estimates of the evaporation from a deep water body can be made

from data on its surface temperature (and hence surface humidity q s ) ,

atmospheric vapor pressure q , and wind speed u . Based on Dalton's

law, a later form of the relation is the Shuleikin-Sverdrup equation

E

= XP49S

-

9)r

in which x is a coefficient and p is air density, the product xp

approximately equal to 2.5 x

g cmP3(or 2.5 g m-") (Budyko, 1971,

Eq. 2.88, p. 108).

Wind speed is important for two reasons: high winds accelerate the

upward turbulent movement of vapor (and sensible heat) from the

surface; and heat stored in deep water is brought to the surface by

vertical mixing within the water body, brought about by wind stress

on its surface.

If the second effect did not exist, the removal of sensible and latent

heat from the surface would chill it, reducing both flows. As the Lake

Ontario data show, however, the removal of as much as 200 W m-'

from the surface in fall reduces its temperature only slowly because the

reservoir of subsurface heat prevents such chilling. Otherwise, cooling

of the surface would result in a lowering of vapor pressure to a point

a t which evaporation would cease.

Wind-generated turbulence brings dry, often warmer air down to

the water surface and carries vapor up from it. Because the vapor

pressure of the surface is maintained reasonably constant, these

variations in atmospheric turbulence produce large short-term variations in evaporation.

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