A THEORY OF ANNUAL TEMPERATURE VARIATION

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Gleeson, Thomas Alexander, 1920a theory of annual temperature variations. New York, 19^0* xiii,77 typewritten leaves, diagr 29cm.

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SEW 1081 0NIV5R6ITY milVlRSITI HEIGHTS

A THEORY OF ANNUAL TEMPERATURE VARIATIONS By

\&+'

THOMAS A. GLEESON

April 1, 1950

A dissertation in the department of meteorology submitted in partial ful­ fillment of the requirements for the degree of Doctor of Philosophy at New York University.

ACKNOWLEDGMENTS Professors B. Baurwitz, J. E. Miller and Dr. E. W. Hewson have offered useful comments on this work, which are appreciated.

The author is particularly grateful to Pro­

fessor H. A. Panofsky for very many valuable suggestions and criticisms, and to Dr. W. A. Baum for providing facilities and suggestions that were exceedingly helpful. Massey very kindly typed the manuscript.

-ii-

Mrs, J.

PHELIMIHART STATEMENT OP OBJECTIVES

The primary purpose of this thesis is to obtain a theoretical representation of the annual variation of temperature at a geographic location in terms of important components of the vertical energy bal­ ance for the atmosphere and earth.

To this end, the procedure outlined

below is followed. Three models are studied consecutively: Model 1. The troposphere is considered to be composed of two lay­ ers, (l) a surface layer of constant mean density wherein a vertical heat exchange coefficient is assumed to increase linearly with eleva­ tion, and (2) an upper layer wherein the exchange coefficient is assumed to be constant but the decrease of density with elevation is recognized. The surface layer of the earth is a third layer within which the soil density and coefficient of heat conduction are assumed to be constant. Differential equations of vertical heat transport are chosen to represent each of these three layers.

Solutions to these equations are

obtained and inter-related by suitable conditions at intervening bound­ aries.

In particular, one of the two boundary conditions at the earth's

surface consists of a theoretical energy balance composed of factors that influence the temperature.

When the boundary conditions are satisfied,

the annual temperature variation may be computed by means of values for these factors. Data exist for all factors but the exchange coefficient in air. Evaluation of this coefficient is then necessary.

By trial and error the

annual temperature variation at a chosen station is found to be determined -iii-

accurately when this coefficient is assigned a particular value which is approximately the same at all elevations. Model 2. The above result then suggests that a surface layer of the troposphere wherein the exchange coefficient varies linearly with elevation is not necessary.

This surface layer is no longer postulated;

the solution for the upper layer is extended to include the total depth of the troposphere.

The surface boundary conditions and the solution to

the differential equation for the surface layer of the earth are retained from the first model. Unsatisfactory theoretical descriptions of the annual temperature variation at the same station are obtained by means of this second model. Inherent in this model is an unreal!stically large decrease of vertical heat transport upward from the surface. Model *3. The troposphere is then considered to be one layer charac­ terized by a constant mean density for which there exists a constant mean exchange coefficient.

A differential equation of heat transport is chosen

to represent this layer.

The solution to this equation is related to the

previous solution of the differential equation for the surfacelayer

of the

earth by means of previous boundary conditions at the surface. This third model is simpler to use than the first model and yields better results than the second model when applied to annual temperature variations at the same station.

Therefore the first two models are neg­

lected entirely, while the third model is applied to temperature varia­ tions at variouslevels at three additional stations, and is used to com­ pute the vertical heat exchange coefficient in the ocean. Neglect of certain physical processes, particularly advection,

limits the regions wherein the theoretical representation of annual temperature variations can he successful.

CONTENTS ACKNOWLEDGMENTS........................................ ii PRELIMINARY STATEMENT OP OBJECTIVES..................... iii TABLE OP SYMBOLS....................................... ix INTRODUCTI ON............................................ 1 PAST I.

THEORETICAL CONSIDERATIONS OP HEAT VARIATIONS IN THE LOWER TROPOSPHERE.........................A

PART II.

THEORETICAL CONSIDERATIONS OF THE HEAT VARIATIONS IN THE GROUND....................... 11

PART III.

BOUNDARY CONDITIONS AT THE EARTH*S SURFACE..................................... 13

PART IV.

THEORETICAL CONSIDERATIONS OF HEAT VARIATIONS IN THE UPPER TROPOSPHERE....................... 25

PART V.

CONSIDERATIONS OP CERTAIN QUANTITIES............. 30

PART VI.

AN ALTERNATIVE SOLUTION FOR THETROPOSPHERE.......Al

PART VII.

APPLICATIONS OP MODEL THREE................... A7

PART VIII.

SECONDARY RESULTS OBTAINED BY MEANS OP MODEL THREE................................. 66

PART IX.

CONCLUSIONS...................................73

BIBLIOGRAPHY........................................... 76 TABLES TABLE 1, Amplitudes of the annual temperature variations for one year and computed values of Kc at Seahrook Farms, New Jersey.................................. .8 TABLE 2. Ratios of average evaporation to average precipitation for various types of "bare soil. ........................... 18 TABLE 3.

Mean monthly temperatures (°C) for 7 A.M. and 2 P.M., local time, at 8 mm and 2.2 m, Potsdam, Germany..........................33

TABLE A.

Data for Ely, Nevada...................... 35

-vi-

TABLE 5. Data for Hebron, Labrador................. 48 TABLE 6. Data

for Ciudad Lerdo, Mexico.............. 51

TABLE 7» Data

for Chicago, Illinois...........

TABLE 8. Data

for the North PacificOcean.............67

TABLE 9. Eddy conductivities at various depths in the Bay of Biscay and the Kuroshio Area.........

53

70

FIGURES

FIGURE 1,

Computed and observed annual variations of temperature at 2 m above the surface, Ely, Nevada..............................,36

FIGURE 2.

Computed and observed annual variations of temperature at 2 m above the surface, Hebron, Labrador .....................

FIGTJFE 3.

50

Computed and observed annual variations of temperature at 2 m above the surface, Ciudad Lerdo, Mexico...............’...... 52

FIGURE 4, Computed and observed annual variations of temperature at 2 m above the surface, Chicago, Illinois........................ 5^ FIGURE 5.

Maximum amplitudes and phase retardations of the annual variations of temperature in the ground, computed from Hebron data and observed at Konigsberg................ 55

FIGURE 6.

Computed and observed maximum amplitudes and phase retardations of annual varia­ tions of potential temperature at the sur­ face, and at 3» 6, and 10 Ion above mean sea level, Ely, Nevada................... 58

FIGURE 7. Computed and observed maximum amplitudes and phase retardations of annual varia­ tions of potential temperature at the surface, and at 3 and 6 km above mean sea level, Hebron, Labrador...................59 FIGURE 8.

Computed and observed maximum amplitudes and phase retardations of annual varia­ tions of potential temperature at the surface, and 3» 6* and 10 km above mean sea level, Ciudad Lerdo, Mexico........... 60

-vii-

FIGURE 9. Computed and observed maximum amplitudes and phase retardations of annual varia­ tions of potential temperature at the surface, and 3» 6« and 10 km above mean sea level, Chicago, Illinois.............. 6l FIGURE 10. Computed and observed annual variations of temperature at the surface, North Pacific Ocean........................... 68 FIGURE 11. Theoretical energy balance in the ver­ tical direction at Ely, Nevada............ 71

-vlii-

TABLE OF SYMBOLS

(Unless otherwise specified, a zero subscript refers to the value of a term at the level 2 = 0; the subscript h refers to the value of term at the level .)

mean annual absorption of the incoming solar radiation by the atmosphere, A1

maximum departure from A.

a

phase retardation of A +

®1

empirical constant, 0.60.

B

albedo,

B

mean annual albedo.

*1

from t = 0.

maximum departure from B.

b

phase retardation of B + B^ from t = 0.

h

empirical constant, O.Oij-2.

c

cloud cover in tenths of sky.

c*

specific heat of soil (or of ocean water), specific heat of air at constant pressure.

c

square root of mean annual water vapor pressure. square root of maximum departure from mean annual water.vapor pressure.

c

phase retardation of C + Cp from t =• 0.

B

mean annual heat loss due to evaporation at the surface.

*1

maximum departure from D,

d

phase retardation of D + D]_ from t — 0.

E

intensity of incoming solar radiation.

E

mean annual intensity of incoming solar radiation.

Ep

:maximum departure from E.

e

:water vapor pressure.

F

:three dimensional heat flux in theatmosphere.

Fc

:eddy transport of heat in the vertical direction in air. • radiative transport of heat in the vertical direction in air.

F^

:vertical heat transport in air.

P

:mean annual cloud cover in tenths of sky.

Fp

:maximum departure from P.

f

:phase retardation of F + Fp from t = 0.

f(t)

: unspecified function of time.

f”(2 )

: unspecified function of elevation.

f'(z) : unspecified function of elevation. 2 g :acceleration due to gravity, g(z) h

i unspecified function of elevation.

: elevation of "boundary "between upper and surface layers in the troposphere,

h* K

:heat content per unit mass of air. :

vertical

heat exchange coefficient inthe atmosphere, Kc+Pr .

Ec :

vertical

eddy dlffusivity in air.

Kr :

vertical

radiative diffusivity in air.

Kq : vertical hea.t exchange coefficient in the atmosphere at the level Z = 0.

E* : coefficient of heat conduction in the ground, (or in the ocean). E” :

vertical

heat exchange coefficient inthe upper troposphere.

jr« . r

vertical

radiative diffusivity in theupper troposphere,

-x-

kei y ker y

series that may be evaluated "by tables

kei y 1

given values of y and y'.

M

. [ i s ] , for

ker y*

L

: latent heat of evaporation.

M]_.

. .Hi|

'■ constant amplitude terms for the annual variations of potential temperature in the lower troposphere.

M£.

. .M^

: constant amplitude terms for the annual variations of temperature in the ground.

I!” 1

. ,MP 4

: constant amplitude terms for the annual variations of .

potential temperature in the upper troposphere. U, u

itg

: constant amplitude terms for the annual variations of potential temperature in the troposphere.

P

: atmospheric pressure.

^1* • *^6 o

: Parameters defined "by equations (39)-

: arc-tan kex■%y

.

q* : arc-tan . u kei y 1 R

: +

H» : +

j

(kei y)2 + (ker y) 2

f(

’2 kei y ‘)2 + (ker y»)

r^

•gas constant for dry air.

R^

:intensity of heat reradiation from water vaporto the ground.

R^

:intensity of heat reradiation fromclouds to the ground.

Tj

:amplitude of the annual variation level Z i »for i = 1,2. -3Ei-

of potential temperature at

solex constant.

parameters defined "by equations (^+3).

same as

except that

appeal's in -place of V

in S' and S' . 1 2

temperature in air. mean annual temperature at level

2—

0.

temperature in the ground or ocean, temperature in the upper troposphere, time. three dimensional wind vector. departure of © from the mean annual

6

at level ?

.

departure of T 1 from mean annual T' at depth-? . departure of Q " from mean annual g" at level a? . • .Wjj, ; parameters defined "by equations (38).

“J"

2/(Z0 + A ?

)

vertical distance. ”

(

mean

qTT J

• a constant >

0.

, averaged over vertical distance and time.

: dry adiabatic lapse rate. mean mean

AT1 -Si

, averaged over vertical distance and time. , averaged over vertical distance and time.

declination of the sun zenith distance of the sun. -xil-

9

:potential temperature.

6

at level Z = 0.

(5Q

:mean annual

Qn

:potential temperature in the uppertroposphere only.

0h

:mean annual 0" at level Z = h.

y\

tconstant and positive rate of change of the vertical heat exchange coefficient with elevation in the surface layer of the troposphere,

y

: frequency of annual variation,

^

:density of air.

g*

:density of soil or ocean water.

—

•

: density of air in the upper troposphere only. by means of (9) and (ll),

lead to

(12) cLP gg.

after

terms cancel each other.

The specific heat, Cp, may be treat-

Jp ed as constant, whereupon this term drops out.

where

V

and V 0

The expanded form of

,/

are the three dimensional wind vector and gradient of

potential temperature respectively, may be substituted into (12); thus

'efrZp] .

(13)

As were the horizontal heat fluxes, the advection of potential temp­ erature will be neglected since both the adequate representation of this advection by means of observational data, and the mathematical operations

—> entailed should possible.

V ’VQ be retained, would be quite difficult if not im­

If now an exchange coefficient, K, is defined by

K s= Kc + Kr •

(1*0

(13) becomes ^

^72 -

^ ^r^AoJ.

(15)

For short period eddies of the order of one day or less, Kc» or *ke analogous coefficient of turbulent mass exchange, Kc

, is commonly in­

ferred to increase with elevation within a surface layer of the troposphere -6-

(for example^ C I] »

.).

likewise for long period eddies of the

order of one year or less, it can "be shown theoretically that Kc or KcC J increases with elevation near the surface.

This variation is indicated

in Table 1 which contains the annual temperature amplitudes at three ele­ vations at Seabrook Farms, ITew Jersey

during one year, and orders

of magnitude for Kc , computed from these data by means of

,rz_ — -— W -

where )/- yjgr-

JLn

^

Bz-Z,

,

[, 12

p. 223

]

* 'fi. - amplitude in Centigrade degrees at elevation

(This egression is a consequence of (15) if Kc and Kr are treated as constants.) It follows that a fairly reasonable mathematical assumption is that Kc increases at a constant rate in the surface layer. variations of 6 and

~T

, and Kr may be small compared to those of Kc TT"’ . It will be assumed then that 6 isconstant. q — 1, and

0

Ailc

,

In this layer,

is a constant less than K =

. Let ( 16)

I 0 + A Z ,

where K0 =

E at Z =■

0, / = constant

>

0. By substitution (15)

becomes J# =

$§, + A g

-

(iV)

Here, since primary interest is in the annual variation of transformation of variables will be made.

0 - (&c + X 2 )}

v =

departure of 0

Let (18)

where

= mean annual Jf =: mean

6

,a

The transformation will prove

to be advantageous in later boundary conditions. v =

0

from the mean annual 8

at level

at j£ — 0,

averaged over vertical distance and time.

,

Table 1. Amplitudes of the annual temperature variations for one year and computed values of Kc at Seabrook Farms, New Jersey.

Elevation

Amplitude

15.2 cm

14.5°C

Kc

5.3 cm 60.0

14.0 lo3 to

152.0

14.0

-8-

sec

By combination of (l?) and (18),

To facilitate a solution of (19) one may represent v by

p > - ( r - & & ) *

where

,

(2o)

f(t) and g(z) are unspecified functions of time and elevation re­

spectively.

By substitution of (20) into (19)

(21) Equation (21) may be solved by methods Haurwltz has usee! [l3j. Two solutions thus obtained are

R C n ,c o s (x t-f)+

Vi *

and

LMi CaS(ZV*~

v2 = ^

SINC***- ]

0

]

where R ”

± y (kei y)2 + (ker y)2

v ^2 y r -T-

A

y

( K0

+ A*

)_ ,

kei y and ker y are series that may be evaluated by means of tables (see

or

q s arc-tan jg|-Z y_ ' " year

,

’

! R ' and q/ are the same as R and q respectively except that the former terms contain 2 y

in place of V

,

M-^. . .M^ are constant amplitude terms to be determined. The second solution, V£ , is obtained for the special case where

Y-

Ob

which is assumed for mathematical simplicity.

The sum

of the two solutions, v-^ + v2> provides an expression

for v that will "be useful later, v - R ^

cos (X t - q) + H2 sin (/ t - q)]

+ r'[m3

cos

(2/ t - q') +

(22)

sin (2/t - q.')]-(if-

By means of this equation the annual variation of potential tempera^ture at a given elevation in the surface layer of the troposphere may he represented. (a)

The following major steps are involved in the derivation.

/

Internal heat sources or sinks, (for examples, condensation

and evaporation processes aloft) are neglected.

Representations for some

of these phenomena will he attempted in a later houndary condition. (h)

Horizontal transport of heat is not considered.

This inposes

restrictions that will hecome more apparent later. (c)

Vertical heat transport is due to conduction near the surface,

convection, and radiation. (d)

In the surface air layer,

constant and less than

=

0 =

constant, ^ = 1,

jg*'

is

constant > 0.

Expressions for the amplitude terms, M^. . .M^, will he derived later. However it is first necessary to consider the temperature variations in the ground layer of this model, then introduce houndary conditions at the sur­ face, after which solutions for these amplitude terms may he obtained.

•10.

PART II THEORETICAL CONSIDERATIONS OP THE HEAT VARIATION IN THE GROUND The main objective of this section is to obtain a representation of the annual temperature variation at a given depth in the surface layer of the earth.

As in the previous section, this representation

will be composed of solutions to a differential equation.

The equa­

tion to be employed may be written directly: thus n where T- =

Tz%

(23)

temperature in the ground,

=

heat conduction in the ground, t = time, 2 s downward.

constant coefficient of depth, measured negative

This is a simplified form of the classical differential equa­

tion of heat conduction; it being assumed here that soil density is con­ stant. A transformation of variables will prove to be useful later.

{2h)

T '=t ' - H W z ) , where v 1 =

departure of T-/ from the mean annual T; at depth Z

mean annual T

I

at 2 = 0,

Let

, T0 isthe

i \T ^ )S - mean averaged over depth and time;

then by substitution, jur'_

.

(25)

Two, distinct, periodic solutions may be obtained for (3*0 by the method of separation of variables: thus, =. e ® *

[

M~cos (/ t + (5-?

and

-11-

) + Mg7 sin

{y t +

z ),

▼2 = e where

iFz

*

[ Kj

COS

( 2 / t +1$Z

)+

sin ( 2 / t + I ^ Z

)],

. . .M^ are constant amplitude terms to he determined. The sum,

+ v^ , provides an expression for v/that will "be useful

later:

e ^ *'Z [ cos (V t ) + Mg sin (/ t + |)&.Z )] XZii£ + & [ ** C08 (2 y t + |F& Z ) + m' sin (2V t +jf&£ )] .

v' =

(26)

By means of this equation the annual variation of temperature at a given depth in the surface layer of the earth may "be represented.

The

following major steps are involved in its derivation. (a)

The coefficient of heat conduction in the ground and the soil

density are assumed to be constant. (b)

No internal heat sources or sinks, or horizontal heat transports

are considered. . (c)

The choice of the positive signs before the radical terms in

(35) insures that, for the maximum amplitude of vj the amplitude will de­ crease and the phase retardation will advance in time with increasing dis­ tance below the surface, ( Z

is negative below the surface).

These var­

iations appear to be physically reasonable. Consideration will now be given to boundary conditions at the surface so that expressions for the amplitude terms, M * . . .M'/

1

(26) may be obtained.

Hr

-12-

. . . M/j, in (22), and

PART III BOUNDARY CONDITIONS AT THE EARTH*S SURFACE In this section two 'boundary conditions at the earth's surface will "be postulated to relate the solutions previously obtained that represent potential temperature and temperature variations in the surface layers of the troposphere and earth, respectively,

A second purpose of these

conditions is to introduce representations for various factors that in­ fluence the temperature. The first condition is that the potential temperature and the ground temperature are equal at any time at Z = 0,

Thus

Tq' ,

eQ ~

(27)

or from (18), (22), (2A), and (26), + vo r

$

So=

R0^Mi

cos‘Iq ”M2sin

Mg

- Rq(M^ sinqQ+ M2 cos q^),

M1 3

* R ;{M cosa1o 3 "o

K * Eo(M3 sln Thus q^

^o^*

/

are known.

M. sin q'), 4

\ /

(29)

cos *o' K

may be determined when

Henceforth

pi

. . .M^, R0. R q , q^, and

. . .Mjj^ will be written in terms of M-^. . .M^,

respectively in equations where the former appear. The second condition is that the flux of energy through the surface

-1>

£ = 0 will "be continuous at any time.

This condition may 1)6 repre­

sented in the form of a postulated vertical energy-balance:

(Intensity of incoming solar radiation) - (absorption of solar radia­ tion by atmosphere) - (albedo) - (heat loss due to evaporation at the

I

surface) - (heat radiation upward from the surface) + (reradiation of heat to surface from water vapor) + (reradiation of heat to surface from clouds) - (heat conducted and convected to the atmosphere from the ground) =

(heat conducted into the ground).

This relationship will be symbolized as follows: £ 1-(A+A^ cos(V t-a)) -(B+B^ cos(

V t-b))] [ E + E^ cos / t^

-(D+D1 cos(y t-d) ) - e T ^ f 1-0.6 -O.Q^f2(C+C1 cos (Xt-c) ) -0.A5 (i-wr-L cos (Xt-f) )]

(30)

c 'e 'K 'ijii'l .

+

In (30), a capital letter with a bar represents an annual mean; with a unit subscript, the maximum departure from that mean.

The phase re­

tardation of a maximum value, measured from t = 0 at the summer solstice, is represented by a small letter within a cosine term.

Units employed

for each of the five, major additive terms are cal cm ‘"sec

.

The mean­

ings to be attached to the separate symbols are discussed below.

1,

E + E f cosy t. This is the intensity of incoming solar radiation, which, for sta­

tions situated poleward of latitude 23.5 degrees, may be evaluated by means of

-Ik -

E = 8.23 x 1(T3

[ cos $ • cos S * sin Xe + % • sin sinSj* (31)

where E = average intensity of incoming solar radiation at a given in­ stant; the number of calories received by a given horizontal square centi­ meter at latitude seconds in

$ during the daylight period divided by the number of

k hours, S = declination of the sun, found for any time of

2

the year from a nautical almanac,

X. ~ hour angle of the sun at sunrise

or sunset; obtained from cos

't 0 =

- tan S •

tan

(f> .

The coefficient 8.23 x 10“3 contains the solar constant,

, 9 k cal cm-^min~^, a transmission factor to be discussed below, o and a conversion factor for reduction to cal cm”“sec . 1

2.

A + A^ cos ( / t - a). These terms symbolize the percentage absorption by the atmosphere

of incoming solar radiation.

Suitable observations of this absorption

are lacking so that an estimated mean value will be used. As quoted in

£f2j , Milankovitch computed the daily insolation on

/

the assumption of 30$ absorption by the atmosphere, irrespective of clouds. According to

C*3 , Kimball computed the scattering by dry air., the ab-

sorption, scattering, and diffuse reflection by solid particles, and the

^

This equation is derived from the cosine law for solar radiation, E

-

S cos ^

where E = intensity of solar radiation incident on a horizontal surface at latitude 0 , S = solar constant, l\ = zenith distance of the sun at hour angle X . It may be verified by spherical trigonometry that cos

l\ -

cos

X • cos S *

cos $

+ sin S

• sin

.

The expression for E, in which cos l\ has been replaced by means of this identity, is integrated over the daylight period and divided by the diurnal period.

-15-

/

absorption and scattering by water vapor.

The net extinction due to

these factors varied from 11$ in summer at Mount Whitney, California, to 37$ in winter at Samoa where water vapor was responsible for more than one half the latter value. per

Powle

[ z j , Kimball [_2j , and Hoel-

determined from measurements the absorption of solar radiation

by water vapor in the atmosphere.

Their values vary with the mass of

water vapor, but are smaller than 20$.

Somewhat arbitrarily, then, a

constant absorption value of 20$ will be assumed for use throughout this study.

This is 5 or 6$ more than is generally allotted to mean

absorption in the terrestrial heat balance.

However, the increase is

more than compensated in this balance by a recent downward revision of the mean albedo discussed below. A = 0.2. in

3.

Thus it is assumed that A^ s

0,

(The factor, 0.2, is included in the coefficient 8.23 x 10“3

(31).)

B + Bx cos (X t - b). These terms represent the albedo for which an adequate body of

cliraatological data is lacking.

Angstrom, according to£3j£/2.J, suggested

a linear relationship between the albedo and cloud cover, B - 0.17 + 0.53C, where B = albedo, C - cloud cover in tenths, 0.17 - sum of 0.08 and

0.09, representing an estimated reflection and scatter from the atmos­ phere, respectively.

If a total sky cover by clouds, C = 1.0, is assumed,

the albedo of the clouds is 70$, from this formula. Fritz

JjJ discussed the premises underlying Angstrom's formula,

and also the work of Danjon who used observations of the "earthshine” to calculate a mean albedo for the earth.

-16-

Two conclusions arrived at

in the discussion are that the mean albedo of clouds was probably closer to

50

% than to 70$. and that the mean albedo for the earth with an assumed

mean cloud cover of 5**$ was probably near work.

35% during the years of Danjon's

In the present work, Angstrom's formula has been adopted and modi­

fied to take into account these late estimates, becoming then B = 0.17 + O.33C.

h.

D + D-l cos (y t - d). This expression denotes the energy loss due to evaporation at the

surface, which likewise must be estimated indirectly.

Few observations

of evaporation from the ground are available, insofar as present require­ ments are concerned.

Meinzer

published data on evaporation in terms

of precipitation amounts, which appear in Table 2.

Evaporation is also

a function of surface humidity, temperature, and wind speed, of course, but no simple relation appears to exist involving these factors.

More­

over, evaporation depends primarily upon the availability of moisture. On the basis of Meinzer's data the assumption is made that the average precipitation is .69 times the average precipitation. The mean monthly heat loss due to evaporation in cal cm

2sec-1 ,

is ^ x where L =

evaporation per month per scuare centimeter number of seconds in a month

'

latent heat of evaporation; 595 - 0.5 z Centigrade temperature.

Here the temperature value has only a small effect, so that the mean heat loss is not appreciably dependent upon it.

5«

c^e -jp Ko ( y ! ) 0 . This quantity is the heat energy transport between the ground and

-17-

Table 2. Ratios of average evaporation to average precipitation for various types of bare soil.

Average evaporation x 100 Average precipitation

Type of Bare Soil

Fine Sand

16

Clay

72

Loam

69

Sandy Loam

69

Peat

35

-18-

the atmosphere at Z- 0.

Its evaluation will he discussed later.

the potential temperature increases with elevation,

^*($)0

When is

positive and represents a downward heat transport tending to establish a dry adiabatic lapse rate.

Conversely when the potential temperature

decreases with elevation, this term is negative and represents an upward heat transport to establish a dry adiabatic lapse rate.

6.

6 .,*[

-A0^-2(C+C^ cos (/t - c) -0.25(l’+I’i cos (Vt - f) )] .

This expression represents the net radiation of heat upward from the surface after reradiation of heat to the surface from water vapor and clouds.

Its development will be discussed.

A fairly reasonable assumption is that the earth radiates as a black body in the infrared region of the spectrum.

Then the heat radiated

upward from the ground may be described by the Stephan-Boltzmann law, E0 = S

T0\

(32)

where E0 = intensity of emission,

6

- Boltzmann constant, TQ is the

temperature at Z - 0. Brunt C53

suggested an empirical formula for the flux of radia­

tion from water vapor in the atmosphere to the ground, Hi s where a^

«

(ax + bx |T? ), .

(33)

~ 0 .60, b^ — 0.0h2, e = water vapor pressure in millibars.

For mathematical simplicity this formula is adopted in preference to one suggested by Angstrom, as described in £3] , wherein e appears in an exponent.

Both formulas presumably give results less accurate

than those obtained from Elsasser's radiation chart £3] , but this chart is not particularly advantageous here because of the lack of historical series of reliable data on humidities aloft.

-19-

In Brunt's formula,

ir r

j

is replaced here by a function.

c + Cp cos ( y t - c) , to describe a time variation in the vapor pressure. Brunt i n

describes another empirical formula suggested by Ang­

strom for the portion of heat radiated from the ground that isinter­ cepted and reradiated back tothe ground by clouds; R2 =

0.9 C (C *0k-R,) ,

(3*0

where C is again the cloud cover in tenths of the sky, which is re­ placed here by a function, f + Fp cos ( V t - f)

,

to describe a time variation in the cloud cover. If now (32), (33)t and (3^) are combined, one obtains g T ^ l

- 0.6 - O.OA-2Inr- 0.9

c (1 - 0.6-0.0h2

l/e)J^

(I)

It can be shown that the term 0.9 C (l -0.6 -0.0h2fe) may be sim­ plified to O.25 C with little error, for values of e usually found in the atmosphere.

The error involved appears to be well within the limits

of error obtainable with Brunt's and Angstrom's empirical formulas

[ 3 , pp. 13^ and 1*1*1J.

This simplification is desirable as it shortens

future mathematical operations considerably.

Then, with substitutions

for e and C from above, (I) becomes finally, S T q ^ 1 -0.6 - 0.0h2 (0 + Cp c o s ^ t - c) ) - 0.25 (F + Fp cos(^t - f))J .

7.

c e • k' ( j £ \ . This final function describes the heat conduction into the ground,

where C'=specific heat of the soil, C ,= density of the soil.

-20-

When

the temperature increases with depth, ( - ^ ) is negative ( Z

decreases

downward), and a flow of heat toward the surface, due to the existing gradient, is indicated.

Conversely, when the temperature decreases

with depth, the function is positive and represents a flow of heat away from the surface. It will he noted that most of the above quantities have been de­ scribed as having cyclic variations with a one-year period.

This crude

assumption is for mathematical simplicity, with a consequent sacrifice of some accuracy.

In various instances where observed effects are ap­

proximately constant or otherwise vary from this representation, period­ ic terms will be dropped, and only mean annual values for these effects will be used. In the equation for the second boundary condition (30), T0 may be replaced by T0 + v0 , (T0 = The expanded form of

0O

and v0 = v£

at Z = 0, from (28) ).

& T0^ is then

G (T0 + v0)^ - 6 Tq2* + k « v0Tq3 plus other comparatively small quantities which will be neglected for mathematical simplicity. Substitutions are possible for respectively.

9

Thus from (20), (22), (2*0, and (26),

= (5£)

over vertical distance and time.

Let assumed to be constant for mathematical

-25-

convenience.

By substitution (40) becomes

j g - ’-

r tjk i'iz "

(kz)

It will be assumed for simplicity that factors in the final term

of

(42) represent constant mean values for the layer. Periodic solutions of (42) are possible by the method of separation of variables.

One solution may be of the form f > * ; ,

and, for the special case where

((‘T l f 11- - Ulijh -j-'i£ «

& 2- }

a second solution may be of the form

-i2Kt where f^*1( Z ) and fg1^ * ) 8X6 unspecified functions of elevation^ i - ) } - l . Thus V i m _ ^ 5i2'JM;L«i cos (/t - S2Z

- f x"- 4 r 0 * 6e " 1 L /("re”Tn 0

).

for

on v M is unknown unless the term

is evaluated, which is a topic for

later consideration. The expressions for the annual variations of potential temperature in the upper and surface layers of the troposphere, (22) and (44) re­ spectively, will now be connected by means of a condition at the inter­ vening boundary at £ - h.

This condition is that the potential tempera­

tures in both layers are equal at any time at Z =

-27-

h:

eh

r

0 h" ;

(/J-5)

or, from (18), (22), (41), and (44):

0o + +

=

M1 cos^ * " °-h) + M2 sin ^

Eh » [

f 0 +

e

cos

(2/ t -

q ^ 1)

+

* " Q-h^J

sin (2/t -

q ^ ')]

S,lr[ Mi'’cos (/ t - Sgh) + M2 '• sin (/ t - SgiOj

C !/ [ M3 m

COS (2 y t -S2 ' h) + M ^ "

where a subscript hrefers to values at 2 -

sin

(2^ t -

%

(**)

Sg* h)]

,

h.

This relation can be rearranged to a form wherein and

+

Qa ,

axe additive to terms composed of either cosyt, sin/ t, cos 2/t,

or sin 2yt, with constant coefficients.

In order for the relation to

be generally valid,

& ¥ao rX

C l ~S —~K + e* - e±

and the sum of the coefficients

,

(47)

of a given trigonometricterm

on

one

side must equal the sum of the coefficient of the same term on the other side, as before.

Pour equations become available from which the follow­

ing relations are obtainable. Ml " =

fa

cos

(Sgh - q.h ) + M2 sin (Sgh - qh )J

M2 M =

K1 sin ^

V'

“3 eos (V h - V> + \ 8l” (S2,h

V

=$

- °-h) + M2

cos

[- M 3 s1” = 0.2. cal gm” 1deg”‘1', (specific heat of soil), K* = 4.7 x 10"^cm^sec“^ , (coefficient of heat conduction in soil). This value

is chosen from the work of Johnson and Davies

who com­

puted the specific conductivity of heat in soil from temperature measure­ ments. r = 1.5 X lO-^cnf1 , ( P=-TV»

43

)

in tlie upper layer

of the troposphere). With regard to the terms ( (

X" -

)

X~

)z

,

, and

{Z--JL) that appear in expressions for the

annual variations of potential temperature and temperature (22), (24), and (44), they will not be considered henceforth for two reasons. first is that there are no apparent methods of evaluating rate of change of the radiative diffusivity with elevation;

-30-

The , the

X‘, the

mean variation of ground temperature with depth,' and Kp11, the mean radiative diffusivity of the upper troposphere.

Secondly^ since the

chief concern here is with annual variations rather than numerical values of potential temperature and temperature at a given time, these terms are not necessary for a comparison of computed and observed varia­ tions at a given level.

Thus, for a given level

2

, a comparison will

be made between the observed annual variation and the theoretical annual variation represented by computed values for v (24), v 1 (26), or v 11 (44) without the terms just discussed. The eddy diffusivity Kc has a wide theoretical range of magnitudes, Shaw £233 published values of KcC computed by several investigators using various methods, which vary from 1 to 1000 gm cm"^sec~^ , dependent upon the physical situation considered and the assumptions made in the compu­ tation.

Such wide variations in K c6 must be attributed primarily to

Kc rather than to the density.

It is difficult then to estimate values

for Kc in the present problem.

This term is included in the exchange co­

efficient, K, which was defined by K = Kc + Kr . The radiative diffusi­ vity, Kr has been estimated by Brunt £33

to be of the order of magnitude

io3 cm^sec">^ , which he considers to be higher than normal values.

Thus

it appears K cannot be estimated satisfactorily for present purposes. Since

and

are vital to the further application of Model 1,

it will be necessary to compute values for these two quantities. Additional information on K0 ,

, and h, the height of the sur­

face layer of the troposphere, is possible if these terms are treated as dependent variables in the equations for the annual variation of poten­ tial temperature. A particular station can be chosen for which there exist sufficient climatological data to permit the calculation of the other variables in these equations, including potential temperature. -31-

j

Then the "behavior of these terms can "be studied. In this procedure a difficulty develops in the satisfaction of the second "boundary condition at

2 - 0, (20). Values for the varia­

tions of potential temperature at

2

~ 0 , are required in this "boundary

equation, "but temperature measurements are normally made at Z = 2 meters approximately; not at

2 - 0.

It appears necessary then to represent the

variation of potential temperature at iation at

2

-

2

meters.

2

- 0 in terms of the observed var­

This will be discussed further.

The mean annual temperature at Z = 0, Tfl , cannot be evaluated in terms of other variables with the present theory. to compute T0 from observed temperatures.

It is then necessary

Lacking climatological data

for temperatures at 2 = 0 , one is practically forced to use the mean annual temperature value at a higher level.

It will be assumed then

that the observed mean annual temperature at

2 - 2 meters is T0 . Gei­

ger [10] published values of the mean monthly temperatures for 2 P.M. local time in summer and 7 A.M. local time in winter, from 8 years of observations at

, . meters and 8 mm at Potsdam, Germany.

2 2

produced in Table 3»

These are re­

The temperature extremes at each level can be

averaged to obtain a crude value for the mean annual temperature at that level:

283.2 A at 2.2 meters, 285.I A at 8 mm.

The closeness of these

values suggests that, for this problem, this assumption will not lead to large numerical errors. If the July and February temperatures are averaged for each level, the annual ranges obtained are lA.l C at 8 mm and 11.3 C at 2.2 m. difference in ranges is then 2.8 C.

The

Presumably this difference would be

smaller if the ranges were computed from mean diurnal temperature values rather than from afternoon temperatures in summer and morning temperatures

-32-

Table 3* Mean monthly temperatures (°C) for 7 A.M. and 2 P.M., local time, at 8 mm and 2.2 m, Potsdam, Germany.

2 P. M. Jun

(8 years of data.)

7 A. M.

Jul

Aug

Dec

Jan

Feb

Mar

8 mm

25.A

26.2

25.2

- 1.3

- 1.8

-1.9

0.2

2.2. m

20.3

21.5

21.3

- 0.3

-0.9

-1.1

0.9

-3>

in winter, since the mean diurnal temperatures at "both levels would "be more nearly equal.

This suggests that in general the difference

between the mean annual ranges at the surface and 2 meters at a given location may not he very large. Climatological data for Ely, Nevada, were used to examine K0 , -yjj

, and h.

Little rainfall is experienced at this station, there­

fore actual evaporation rates there should also "be small.

Thus any

errors due to an inadequate representation of heat loss from evaporation were presumably minimized. Table

k contains these data, source references, and values for

parameters in P-^........ Pg , computed from the data.

The observed

annual temperature curve at ?- 2 meters was fitted quite closely, Fig­ ure 1 , by means of the equation for v-^ (22), and a suitable choice of values for K0 and

which appear in M]_ . .

This choice was de­

termined by trial and error; the equations involved are so complicated that a more systematic solution for KQ and q^ did not seem feasible. Terms containing cos 2 / t and sin

2 / t were small, and were found

to have no noticeable effect on the maximum amplitude or the phase re­ tardation of the computed curve.

These terms did tend to increase the

curvature at the maximum potential temperature in summer and decrease the curvature at the minimum in winter.

There appears to be little

physical significance in these distortions of the otherwise symmetrical sine curve, especially since the curve of the observed variation does not even remotely exhibit similar distortions.

Hereafter terms contain­

ing cos 2 /t and sin 2 / t in the equations for v, v', and v" will be neglected; their initial purpose was to facilitate the solutions. For the computation of the theoretical curve,

g

=

2

-1

-

0.78 cm sec cm -3*i-

Table 4. Data for Ely, Nevada 39.15° N, 114.53° ¥; Elev: 1.9 km Temperature

Cloud Cover

Specific Humidity

Precipitation

Jan

268.7 A

•50

3-3 °/oo

2.03 g

Feb

269-5

•50

4.5

2.29

Mar

271.2.

.44

5.0

2.54

Apr

275.7

.47

5.6

2.29

May

283.1

.39

9.1

2.29

Jun.

286.9

•30

11.6

2.03

Jul

292.7

.24

14.7

1.2.7

Aug

290.3

.21

14.5

1.57

Sep

285.5

.19

9.8

I .27

Oct

281.4

.25

6.4

2.03

Nov

272.2

.43

4.0

2.03

Dec

269.3

.49

3-6

3.30

9 years

[

2

h O O u>

a u »o U) o

? uy J

? «

z M P

3

z

in

£

S

in

5 a




C = 2 .62 mb

Cf= 0.239 cal

E

z 12

e




this figure was obtained when the assumed value for the eddy conductivi­ ty in the ocean, K* €

', is 0.222 gm cnT^sec"*^.

This value may he compared with values for the eddy conductivity computed from annuel temperature observations in the Bay of Biscay and the Kuroshio are of the Pacific Ocean

, appearing in Table 9»

These values have a fairly wide range so that perhaps 0.222 is not for­ biddingly small.

Of course, the latter value is not strictly comparable

with the above, since it represents an average over depth.

It might be

expected that an average that included the small values of the eddy con­ ductivity to be expected in comparatively stable lower layers would be small.

However, in view of the assumptions made in its derivation, the

value 0.222 is probably not very reliable. (2)

A visual representation of the annual variations of the var­

ious considered factors appearing in the heat balance at Ely is shown in Figure 11.

These curves were computed from values in Table 4 by

means of the energy balance equation (20).

Included at the top of the

diagram for comparison purposes is the curve of the theoretical variation of temperature at two meters. The remaining curves represent; (b) the albedo, (c) the mean diur­ nal intensity of incoming solar radiation not absorbed in the atmosphere, (d) heat radiation upward from the surface, (e) conductive and convective heat transport from the surface to the atmosphere, (f) heat radiation from water vapor to the surface, (g) heat radiation from clouds to the surface, (h) mean annual heat loss due to evaporation at the surface, (i) conductive heat transport from the surface into the earth.

With

regard to curves (e) and (i), positive values indicate heat transports away from the surface, while negative values Indicate heat transports to the surface.

-69-

Table 9. Eddy conductivities at various depths in the Bay of Biscay and the Kuroshio Area

Kuroshio Area

16.A g

O

Bay of Biscay

\ wT CD o1

Eddy Conductivity

Depth 0 m

3-2

25

2.1

50

3.8

100

(a)

(*)

78

58

0

34

43

25

23

39

50

22

32

100

-70-

Figure 11: Theoretical energy balance in the vertical direction at Ely, HevacLa. (a) theoretical temperature at 2 m, (b) albedo, (c) incoming solar radiation, (d) heat radiation upward from the surface, (e) conductive and convective heat transport from the surface to the atmosphere, (f) heat radiation from water vapor to the surface, (g) heat radiation from clouds to the surface, (h) mean annual heat loss due to evaporation at the surface, (i) conductive heat transport from the surface into the earth. With regard to curves (e) and (i), negative values indicate heat transport toward the surface.

201 /S

(o) h o-

3a 25

20

87-

PER

6

5-

IO~3 CAL

C M 2 5EC

albedo

35

3-

fe) o* -t.

MAY

JUNE

AUG

SEPT 25

NOV S

DEC 22

TIME

Figure 11* -7/-

FEB

MAR

MAY

It will "be noted that the most dominant influences in the the­ oretical energy "balance are solar radiation, infra-red radiation from the surface to space, and vertical transport of heat "between the surface and the atmosphere.

Interestingly enough, the vertical transports in

air and ground are depicted as "being greatest at the solstices, not at the occurrences of extreme temperature values.

The physical validity of this

result of speculative. An estimate of the ratio of the vertical heat transports in ground and air may "be obtained "by numerical substitutions for terms in This ratio is found to be

XX* i.

_

ZKK[% .

; a value which is approximately rsgrc—

sentative for the four continental stations considered. value for this ratio may be somewhat different.

In reality the

One important reason

for this is that the solution (51) describing temperature variations in the atmosphere applies to a model wherein the atmosphere eventually sends back to the surface all heat it has received from the surface (see Figure 11).

This does not happen in reality; "rich of the heat is radiated from

the atmosphere to space.

Indirectly the above value may be affected ly

thi s di sere oancy.

-72-

PART IX CONCLUSIONS Concerning the mathematical representation for annual variations of potential temperature in air and temperature in the ground, herein developed and applied, we offer the following conclusions. 1.

The theoretical representation of the annual variations of po­

tential temperature is quite accurate at two meters above the surface at the land stations considered.

Presumably equal accuracy would be obtained

at other land stations similarly removed from marked maritime Influences. 2.

A qualitative comparison showed close agreement between observed

and theoretical annual temperature variations at various depths in the surface layer of the earth.

As the computed variations depended upon a

value for the coefficient of heat conduction in soil that was previously determined in a study of soil temperatures

, close agreement was

perhaps to be expected. 3.

At upper levels, theoretical annual variations of potential

temperature showed poorer agreement with observed variations than near the earth*s surface.

Neglect of horizontal heat transport and local ther­

mal effects (such as condensation) which are present above the surface probably account for part of this lack of agreement aloft.

Above 3 hm,

however, these neglected effects appear to have reduced significance; so that a general solution to a differential equation for vertical heat transport in the troposphere can be used with some success to approximate annual variations at upper levels in terms of observed variations at 3 km» The vertical heat exchange coefficient in air, K, may be assumed to increase linearly with elevation, but accurate descriptions of annual

-73-

potential temperature variations at two meters are possible if VK

■jjg — 0.

The constancy of Z with elevation may be a result of

the assumption that Z has merely a linear variation.

5.

In the present study, the value of the exchange coefficient K

depends upon the assumed mean values of the density and potential temperature

0

vertical distance and time. somewhat arbitrary.

6

, temperature T,

, of air, which represent averages over The vertical distance to be considered is

However, once mean values for these quantities are

chosen, a value for K is determined. accurately the annual

It is then

possible to represent

variations of potential temperature

near the surface

at the four land stations considered by means of values of Z within the range (2.5 ±0.2) x 10^ cm^ sec~\ when it is assumed that

~r = 1. J gm cm—3 J , -q

—3

10

€

is

It appears reasonable that Z would vary from one

geographic location to another; but it cannot be concluded that values in this range are closely indicative of this horizontal variation since crude assumptions regarding other factors that determine the temperature affect the numerical results obtained in this study. 6.

Brunt £3] estimated that 1,3 1 10-^ cm^ sec ^ was rather higher

than the normal values of the radiative diffusivity. is quite small compared to 2.5 x 10

6

cm

Since this value

2 - 1 sec , which represents the sum

of the eddy and radiative diffusivities, it must be concluded that the latter value refers almost entirely to the eddy diffusivity, in this study. 7. 10

6

The high order of magnitude of the exchange coefficient,

2 - 1 sec

cm

, seems to be fairly reasonable because it satisfies two

independent requirements.

Firstly, this magnitude is necessary for the

accurate representation of annual temperature variations near the surface.

Secondly, the representation of annual variations aloft, once the surface variations are known, will yield the correct order of magni­ tude for the amplitudes of the upper variations only if K has this order of magnitude as a lower limit. 8.

Theannual variation of K is seemingly unimportant in the the­

oretical determination of annual potential temperature variations at two meters. 9.

The eddy conductivity in the North Pacific Ocean was computed

to he 0.222

orT'sec^.

Presumably this value is inaccurate due to neg­

lect of horizontal heat transport, as well as other thermal effects below the ocean surface. 10. For each of the continental stations studied, the indicated ratio of heat conducted into the ground to heat transported tc the air by conduction, convection, and radiation from the surface is

^

^

This result is somewhat misleading insofar as the solution for the atmosphere is predicated upon a model wherein all heat delivered to the atmosphere from the surface eventually is returned to the surface.

-75-

BIBLIOGBAPHY 1.

Beers, N.B., 19^-4: Temperature and turbulence in the lower atmosphere. J. Meteor., vol. 1, 7&*88.

2.

Berry, F.A. , Bollay, E., and Beers, N.B., 19^55 Handbook of Meteorology. Hew York, McGraw Hill Book Co. Inc., 1068 pp. (see pp. 292-296.)

3.

Brunt, D., 19^1: Physical and Dynamical Meteorology. London, Cambridge University Press, h28 pp.

h.

Dwight, H.B., 19^7J Tables of Integrals and Other Mathematical Data. New York, Macmillan Co., 250 pp.

5.

Elsasser, W., 19^2: Heat ti'ansfer "by infrared radiation in the atmosphere. Harvard Meteor. Studies, no. 6 , 107 pp.

6.

Fleagle, B., 1950: Badiation theory of local temperature differences, (unpublished).

7.

Fritz, S., 192+9*. The albedo of the planet earth and of clouds. J. Meteor., vol. 6 , 277-282.

8.

Frost, E., 1948: Calculation of night minimum temperatures. Met. Office. London. Prof. Notes, no. 95* 5 PP*

9.

Geiger, E . , 1942: Das Elima der bodennahen Luftschlcht: Ein Lehrbuch der Mikrokllmatologie. Braunschweig, F. Vieweg, *05 PP* (see p. 3 * 0

10.

11.

12.

13.

. 19271 The Climate of the Layer of Air Near the Ground. Translated by J. Lelghly, 1942, for U.S. Dept, of Agric., 15b pp. (see pp. 13, 35*) Groen, P., 19’'?: Note on the theory of nocturnal radiational cooling of the earth’s surface. J. Meteor., vol. 4, 63-66. Haurwitz, B., 1941: Dynamic Meteorology. New York, McGraw Hill Book Co. Inc., 365 PP* ____________, 1936: The daily temperature period for a linear variation of the austausch coefficient. Trans. Boy. Soc. Canada. 3rd Series, Sec III, vol. 30.

14. McGrawHill 15.

. and Austin, J.M. , 1944: Climatology. New York, Book Co. Inc., 410 pp. (see p. 158 and Plate III.)

Jahnke, E., and Emde, F., I9A3 : Tables of Functions with Formulas and Curves. New York, Dover Publications, 303 plus 76 pp. (see pp. 272-254.)

-76-

Johnson, IT., and Davies, E., 1927: Some measurements of temperature near the surface in various kinds of soils. Quart. Jour. Boy. Met. Soc., vol. 53. 45-59. Kendrew, W., 1941: The Climates of the Continents. Oxford University Press, 473 PP*

London,

Koppen, W., und Geiger, B., 1930: Handbuch der Klimatologie. Berlin, Getruder Borntrager, vol. 1, A. (see Milankovitch.) ______________________ . 1936: Bandbuch der Kllmatoloaie. Berlin, Getruder Borntrager, vol. 2, J. Leighly, J., 1949: Climatology since the year 1800, American C-eophysical Union, vol. 30, 658-672.

Trans.

Melnzer, 0., 1942: Hydrology. New York, McGraw Hill Book Co.Inc., 712 pp. (see p. 292.) Shaw, IT. , 1942: Manual of Meteorology. Lond.on, Cambridge University Press, vol. 2, 472 pp. ■ 1942: Manual of Meteorology. London, Cambridge University Press, vol 4, 359 PP* Sverd-rup, H. , 1942: Oceanography for Meteorologists. New York, Prentice Hall Inc., 243 PP« (see p. 232.) ___________ , Johnson, M. , and Fleming, B . , 1942: The Oceans. New York, Prentice Hall Inc., IO87 pp. Thornthwaite, C.W., 1948: Micrometeorology of the Surface Layer of the Atmosphere. Baltimore, Johns Hopkins University, Beport no. 3 , 32 pp. U.S. Dept, of Agric., 193^: Atlas of American Agriculture. Washington, U.S. Gov. Print. Off., 217 pp. . 1941: Climate and Man: Yearbook of Agricul­ ture. Washington, U.S. Gov. Print. Off., 1248 pp. U.S. Wea. Bur., 1939, 1940, 1941, 1944-1948: Mon. Wea. Bev.. Washington, vol. 67-69, 72-76, selected pages. _____________ , 1942: Normal Temperature and Pressure Charts. Northern Hemisphere. Washington, 72 pp. U. of Chicago, Inst, of Met., 1942: A Table of Potential Temperatures. Chicago, U. of Chicago Press, 30 PP* Wexler, H., 1944: Determination of the normal regions of heating and cooling in the atmosphere by means of aerological data. J. Meteor.. vol. 1, 23-28.

-77LIBBAR2 OP SEW lOHi ONIVXReiTf nfllflRSITY HEKWT9