Arya S.P.S. Introduction to Micrometeorology

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178 11 Momentum and

Heat

Exchanges

l0c-

010

15m

4.2m

O-w/u. + x

"618.

0 •

10'

-z/L

Fig. 11.9 Normalized standard deviations of vertical velocity and temperature fluctua-

tions as functions of

-z1L, compared with local free-convection similarity relations. [From

Businger (1973); after Monji (1972).]

orological

data

from homogeneous sites. In particular, the empirical for-

mula

1.25[1 -

3(z/L)]1I3

01.32)

obtained

from

a compilation

many aircraft and

tower

data

taken

over

land

and

sea

can

recommended

for unstable conditions (see also

Panofsky

and

Dutton,

1984, Chap. 7).

The

surface

heat

flux in a convective boundary layer (CBL) can also be

estimated from

measurements

(Tu

(Tv

any

convenient height in the

lower

part

the

CBL

where

the empirical similarity relations o; =

(Tv

O.6W*

are

found to be valid (Caughey and Palmer, 1979). Since, the

convective velocity scale

W* is related to the mixed-layer height and the

surface

heat

flux,

the

latter

can

be estimated if the mixed-layer height is

measured

independently (e.g., from

temperature

sounding).

In some applications, the above-mentioned similarity relations between

the turbulent fluxes and variances are used to determine the latter from

indirect estimates

the

former.

For

example, the horizontal and vertical

spread

chimney

plume or a pollutant cloud is intimately related to

velocity variances, which

need

to be indirectly estimated in the absence

turbulence

measurements.

Problems and Exercises 179

11.6 APPLICATIONS

Theory and observations of momentum and heat exchanges between

the atmosphere and the

earth's

surface, presented in this chapter, may

have the following practical applications:

• Systematic ordering of mean wind, temperature, and turbulence data

with stability in the surface layer

• Quantitative description

the surface layer wind and temperature

profiles and their relationship to the surface fluxes of momentum and

heat

• Relating different stability parameters

• Specifying

eddy

diffusivities and bulk transfer coefficients as func-

tions of stability in the parameterizations of fluxes

• Determining surface and

PBL

fluxes from different types of mean

flow and turbulence measurements

• Parameterizing turbulence and diffusion in the atmospheric surface

layer

• Determining the surface energy budget

PROBLEMS

AND

EXERCISES

1. What maximum

error

or uncertainty might be expected in the determi-

nation of the Obukhov length, if the fluxes of momentum and heat have

uncertainties of

±20%

associated with them?

2. Using the definitions of the various scales and parameters and the

basic

Monin-Obukhov

similarity relations [Eq. (11.2)], verify or de-

rive

Eqs.

(11.4) and (11.5).

3. Using the forms of similarity functions obtained in the 1968 Kansas

Field Program, calculate and plot the Richardson number as a function

L in the range - 2

2, and compare it with the graph based

on the simpler relations [Eq. (11.10)]. What conclusions can you draw

from this?

4. Substituting from the

M-O

similarity profile relations in the bulk trans-

fer formulas [Eq. (11.15)]

(a) Derive the expressions given by Eq. (11.16) for

Co and C

functions of

Land

Zoo

(b) Write the same as functions of the bulk Richardson number

RiB

for

stable conditions

(RiB>

0), and plot

CD/CON

as a function of

RiB

for z/zo = 10

and 10

•

What

conclusions

can

you draw from this?

180 11 Momentum and Heat Exchanges

The

following

measurements

mean

wind

and

potential

temperature

were

taken

around

noon

during

the

1968

Kansas

Field Program:

(m)

U (m sec

(K)

5.81

307.20

6.70

306.65

7.49

306.28

8.14

305.88

8.66

305.62

UsingtheM-Osimilarityrelations<Ph

<p~

15(zIL)]-1/2 for

the

unstable

surface

layer

(a)

Calculate

and

plot

Ri as a function

height.

(b)

Determine

the

Obukhov

length from

the

above

plot.

(c)

Calculate

the

surface

shear

stress

TO and the

heat

flux H«, using the

above

data

from

the

lowest

two

and

4 m) heights;

take

p = 1.2

kg m

and

= 10

kg-

K-I.

The

following

observations

were

taken

on a

summer

evening during

the

Kansas

Experiment

(with a surface

pressure

1000 mbar):

z (m) 2 4 8 16

U (m

sec-I)

2.84 3.39 4.01 4.85

T (0C) 33.09 33.31 33.57 33.82

(a)

Use

the

graphical

version

the

profile

method

discussed in the

text

determine

and

e*.

(b)

Use

the

gradient

aerodynamic

method

for

the

same

purpose,

using

information

from

the

lowest

two

and

4 m) heights only.

(c)

Compare

the

results

the

above

two

methods

with

the

eddy

cor-

relation

measurements

uw =

-0.044

rn' sec ?

and

-0.023

sec-I

During

the

Minnesota

1973

Experiment,

the

following

mean

tempera-

ture

measurements

were

made

from a

tethered

balloon at 75-min in-

tervals:

z (m)

305

610

914

1219

1217-1332 (CDT)

21.75

19.54

17.16

14.43

11.66

8.99

1332-1447 (CDT)

22.46

20.25

17.86

15.16

12.34

9.61

(a)

Calculate

the

warming

rate

at various levels and

the

average value

for

the

whole

PBL.

Problems and Exercises 18I

(b) Calculate the surface heat flux in the middle of the observation

period when the

PBL

height was 1400 m, assuming a constant

(average) warming rate, independent of height. Compare this with

the direct (eddy correlation) measurement

ofw8

= 0.20 m

sec

daytime, using temperature measurements at only one level and

remote sensing of the

PBL

height from the surface.

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Chapter 12

Evaporation from Homogeneous

Surfaces

12.1 THE PROCESS OF EVAPORATION

The evaporation in the atmosphere and consequent phenomena of con-

densation, cloud formation, and precipitation have fascinated scientists,

philosophers, and common people since time immemorial. Early philoso-

phers and scientists proposed many theories and explanations of the

evaporation process; for a briefreview of the history the reader may refer

to Brutsaert (1982). As early as in the fourth century

Be, Aristotle recog-

nized

that

"wind

is more influential in evaporation than the

sun."

How-

ever, it was not until the nineteenth century that the foundations of the

modern quantitative theories of evaporation and other transport phenom-

ena

were laid by J. Dalton, A. Fick, and O. Reynolds.

Evaporation is the phenomenon by which a substance is converted

from the liquid state into vapor state.

In the atmospheric surface layer

evaporation may take place from a free water surface or a moist soil

surface, as well as from the leaves of living plants and trees (here, we are

ignoring the distinction between evaporation and transpiration). The de-

tails of the physical process of conversion from the liquid to vapor state

may differ in different cases, but the basic mechanisms of transport of

vapor away from any interface are the same. Very close to the interface,

vapor is transferred through molecular exchanges in the same manner as

heat

and momentum are transferred. Most of the transfer takes place

within a few molecular free path lengths of the surface. The vigorous

molecular activity in this region is indicated by the observation that an

estimated 2 to 3 kg of water moves across a free water interface in each

direction

every

second, for each square meter of the interface (Munn,

1966, Chap. 10). The

net

transport of water vapor is only a tiny fraction of

the total

transport

in each direction. According to

Fick's

law, the net flux

a material (e.g.,

water

vapor) in a given direction is proportional to its

concentration (specific humidity) gradient in that direction, the coefficient

183

184 12 Evaporation from Homogeneous Surfaces

of proportionality being defined as the moecular diffusivity. Thus, the

evaporation rate at a horizontal surface is given by

Eo =

-paw(aQlaz)

(12.1)

in which

is the molecular diffusivity of water vapor. The above rela-

tionship is

expected

to be valid only in the laminar or viscous sublayer in

which the molecular exchange remains the primary (perhaps the only)

transport mechanism. The same mechanism is operating at the air-soil

and

air-leaf

interfaces. However, natural surfaces, not being very smooth

and uniform, may not have well-defined molecular sublayers which could

be amenable to direct measurements. Therefore, the practical utility of

Eq. (12.1) for estimating evaporation in the atmosphere remains question-

able. Some attempts have been made by physical chemists and fluid dyna-

mists to study evaporation and

other

transport processes on the molecular

scale. Micrometeorologists are more interested, however, in the ex-

change processes

over

the much larger scales that are encountered in the

atmospheric surface layer and the PBL.

In the lower

part

(surface layer) of the atmospheric boundary layer the

air motion is almost always turbulent. Here, the water vapor is efficiently

transported away from the interfacial molecular sublayer by turbulent

eddies. Because transport and

the

intensity of turbulent mixing depend on

the surface roughness, wind

shear

or friction velocity, and thermal stratifi-

cation, the evaporation rate also depends on the above factors, as well as

on the average specific humidity gradient. The frequently used gradient-

transport relation for the same is

E =

-pKwcaQlaz)

(12.2)

in which the eddy diffusivity

replaces the molecular diffusivity in Eq.

(12.1).

While

Fick's

law or Eq. (12.1) has a firm theoretical (e.g., the kinetic

theory of gases) and experimental basis,

the

analogous relation [Eq.

(12.2)] for a turbulent flow is based on a phenomenological gradient-

transport theory whose limitations have already been pointed out in

Chapter 9.

Better

flux-gradient relations obtained on the basis of the

Monin-Obukhov

similarity theory and carefully conducted micromete-

orological experiments will be discussed in a later section.

12.2

POTENTIAL

EVAPORATION

AND

EVAPOTRANSPIRATION

Over a bare land surface with no standing water, the soil moisture is the

only source of

water

for evaporation. Therefore, the rate of evaporation

12.2 Potential Evaporation-Evapotranspiration

185

must depend on the moisture content of the topmost layer of the soil.

When the soil surface is fully saturated and the

soil moisture content is

not a

limiting factor in evaporation, this maximum rate of evaporation for

the given surface weather conditions is called potential evaporation (E

) .

When the soil becomes drier, the rate of evaporation for the given atmo-

spheric conditions (particularly the surface temperature, near-surface

wind speed, specific humidity, and stability) also depends on the moisture

content of

the

topmost

soil layer which, in turn, depends on the soil

moisture flow through the soil. The relationship between the rate of evap-

oration

(Eo)

and

the soil moisture flow rate

(M)

is given by the instantane-

ous

water

balance equation at the surface

Eo = M«

or the

water

balance equation for the subsurface layer,

Eo = M

t1M

(12.3)

(12.4)

in which

t1M is the rate of storage of soil moisture in the layer per unit

area of the surface, and

is the moisture flow rate at the bottom of the

layer.

The

above simplified water budget equations would be valid only

during the periods of no precipitation, irrigation, or runoff at the surface.

The vertical moisture flow rate, or the flux of soil moisture, is related to

the vertical gradient of the soil moisture content

(S) through Darcy's law

(12.5)

where k

is the hydraulic conductivity through the soil medium. Note that

Eq. (12.5)

(Darcy's

law) is analogous to Eq. (4.1) (Fourier's law) for heat

conduction. Similarly, following the derivation of Fourier's heat conduc-

tion equation [Eq. (4.2) or (4.3)), one

can

obtain

(a/at)(pS) = - aM/az

or, after substituting from Eq. (12.5),

as/at

(alaZ)[O'm(aS/aZ)]

(12.6)

(12.7)

where O'm is the diffusivity of soil moisture.

For

surfaces with live vegetation, a significant portion of water vapor

transfer to the atmosphere is by transpiration from the vegetation, most of

which takes place through stomata of leaves. Transpiration may be con-

sidered an important by-product of the process of photosynthesis and

respiration, in which plants take CO

during daytime and give out the

same at nighttime. Transpiration may also be a physiological necessity for

plants to draw up dissolved nutrients from the soil through their roots.

Moisture transfer through roots, stands, and leaves is considerably more

efficient than that through the soil pores, because the former is forced by a

186 12 Evaporation from Homogeneous Surfaces

strong osmotic pressure gradient. Still, evaporation from the bare soil

surface in

between

the plants is by no means negligible. Because evapora-

tion

and

transpiration

occur

simultaneously and it is not easy to distin-

guish between the vapor transferred by the two processes, the term

evapotranspiration is sometimes used to describe the total water vapor

transfer to the atmosphere. More often, though, meteorologists use evap-

oration as a substitute or synonym for evapotranspiration. The distinction

between the two disappears, anyway, when one considers the vegetative

surface as a composite of soil and

leaf

surfaces, which contribute to the

total

water

vapor

transfer (evaporation) to the atmosphere per unit hori-

zontal

area

of the composite surface,

per

unit time. In this gross sense,

details of the vegetation canopy (e.g., leaves and branches) and small-

scale transfer processes around the individual surface elements are essen-

tially glossed

over

and

the canopy is considered as an idealized, homoge-

neous material layer which has a finite thickness and mass, and which can

store or release heat and water vapor. In particular, the water budget for a

vegetative surface is often too complicated to be used as a practical tool

for estimating evaporation.

The

concept

of potential evaporation (E

)

can also be extended to any

vegetative surface for which

represents the maximum evapotranspira-

tion likely from the vegetative surface for a given set of surface weather

conditions. The potential evapotranspiration is generally less than the free

water

surface evaporation under the same weather conditions, especially

in humid regions.

The

former

can

exceed the latter, however, in certain

arid regions and under certain weather conditions (see, e.g., Rosenberg

al., 1983,

Chapter

7).

Actual evaporation usually differs from the potential evaporation, be-

cause

the

surface may not be saturated and the plants may not be drawing

water

from the soil and transpiring at their maximum rate. Still, the con-

cept

of potential evaporation is quite useful in agricultural and hydrologi-

cal applications.

12.3 MODIFIED

MONIN-OBUKHOV

SIMILARITY RELATIONS

The original

Monin-Obukhov

similarity hypothesis and subsequent re-

lations based on the same, as discussed in the preceding chapter, are

strictly valid when buoyancy effects of water vapor can be ignored. When

substantial evaporation occurs and it affects the density stratificaiton in

the surface layer, modified

M-O

similarity relations incorporating the

buoyancy effects of water vapor would be more appropriate.

12.3 Monin-Obukhov Similarity Relations 187

Some buoyancy effects of water vapor have already been discussed in

Chapter

5, where the concepts of virtual temperature and virtual potential

temperature were introduced. A similar concept is that of the virtual heat

flux

Hs , which may be interpreted as the flux of virtual temperature

(H;

pCpw(}v).

The modified

Monin-Obukhov

similarity hypothesis

states that, in a homogeneous and stationary atmospheric surface layer,

the mean gradients and turbulence structure depend only on four indepen-

dent

variables: z, U*,

g/T

vo,

and Hvo/pc

•

As mentioned in Chapter 5, the

difference between the virtual temperature and actual temperature is no

more than 7 K and frequently less than 2 K, so that the buoyancy parame-

ter

g/T

does

not

differ much from gl

T«.

However, the virtual heat flux

may differ significantly from the actual heat flux, and the two are approxi-

mately related as

(12.8)

This follows from the approximate relationship between the fluctuations

of virtual temperature, actual temperature, and specific humidity

(}v =

(}

0.610q

(12.9)

which, in turn,

can

be derived from the relationship between

and

El,

noting

that

= 0

(}vand

= 0 +

(}.

These derivations are left as an

exercise for the reader.

The relation given by Eq. (12.8) can also be expressed in terms of the

latent

heat

flux H

or the Bowen ratio B, as

(12.10)

(12.11)

in which

= 0.61c

p®/L

is a dimensionless coefficient. Since

is only

weakly

dependent

on the actual temperature in "C, a constant value of

= 0.07, corresponding to ® = 280 K, is frequently used in the above

relations.

From

Eq. (12.10) it is obvious that the suggested modification to

the

M-O

hypothesis is necessary only when

IBI

< 1, i.e., when the latent

heat

flux exceeds the sensible heat flux. This is usually the case over the

oceans,

but

not so common

over

the land areas.

Note

that in the modified similarity hypothesis both the sensible heat

flux and

water

vapor flux are considered together in an appropriate combi-

nation (the virtual

heat

flux) and

not

separately. Therefore, a modified

Obukhov length is defined as

L =

-uV

-.!L

)

pCp

However, the scales of temperature, virtual temperature, and specific

humidity have to be defined from their respective fluxes, i.e.,