Water-Loss Investigations: 

Lake Hefner Studies, 

Technical Report

GEOLOGICAL SURVEY PROFESSIONAL PAPER 269

Prepared in collaboration with the U. S. 

Department of the Navy, Bureau of Ships, Navy 

Electronics Laboratory; U. S. Department of the 

Interior, Bureau of Reclamation; U. S. 

Department of Commerce, W^eather Bureau.

A reprint of Geological Survey Circular 229, which was arranged and reproduced by Navy Electronics 

Laboratory in 1952, with part of the edition issued as N. E. L. Report 327, San Diego 52, Calif.

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1954
UNITED STATES DEPARTMENT OF THE INTERIOR 

Douglas McKay, Secretary

GEOLOGICAL SURVEY 

W. E. Wrather, Director

For sale by the Superintendent of Documents, U. S. Government Printing Office 

Washington 25, D. C. - Price $1.75 (paper cover)
r\\ IS v. \ x> I /-\ 

METEOROLOGICAL STATION

WATER STAGE RECORDER

CREST STAGE GAGE 

NONRECORDING RAIN GAGE

ft EVAPORATION PAN STATION RECORDING RAIN GAGE

CONTOUR INTERVAL 20 FEET DATUM IS MEAN SEA LEVEL

Q SEEPAGE MEASUREMENT STATION

£ THERMAL SURVEY STATION

0 10 20 30 40 50

SCAL£ ^ H|JNOREDS Q(:

N

I a

Map of Lake Hefner, Showing Equipment Installations, Adapted from Britton (SE) 

Quadrangle, U.S. Geological Survey, 1951.

Map of State of Oklahoma showing location of Lake Hefner.

i ^Slpg^'.fcliC.'SS

ii
FOREWORD

Water and its wise management and use are 

vital to the economic stability and growth of the 

western United States. Thus, every "lost river" must be 

explored to determine and, if possible, prevent 

water losses.

Evaporation is one of these "lost rivers." Loss 

through evaporation must be considered in the operation of any reservoir but, in the case of large reservoirs with extensive surface areas, it becomes a 

major factor in planning and management.

Methods of estimating evaporation which have 

proved adequate for small reservoirs may not suffice 

for a large reservoir in a region where water is 

scarce. A case in point is Lake Mead, the largest manmade body of water in the world, created by the 

construction of Hoover Dam on the lower Colorado 

River between Nevada and Arizona.

Lake Mead is the key reservoir for the distribution of lower Colorado River water to Arizona, southern California, and portions of Nevada, Utah, and 

New Mexico. Without water this area would be almost 

uninhabited, and the bulk of the supply comes from 

the lower Colorado. Every drop is precious; the 

demand far exceeds the supply.

For this reason, the Bureau of Reclamation is 

obligated to remove every possible source of error 

in its handling of the water in Lake Mead. The evaporation loss cannot be prevented, but accurate knowledge of evaporation permits more efficient use of 

the remaining water, increases the security of existing 

developments, and allows better planning of future 

developments.

To obtain the best knowledge of evaporation 

it was necessary to bring together experts in many 

fields. Ideas, theories, and methods for estimating 

evaporation have been proposed by engineers, meteorologists, hydrologists, oceanographers, physicists, 

and many others, but there has been no sound basis 

for agreement as to their relative merits. Furthermore, some of the theories and ideas required a precision of measurement of natural factors unattainable 

with existing instruments. By cooperc'ive work involving agencies of the Department of the Navy, the Geological Survey and Bureau of Reclamation of the 

Department of the Interior, and the Weather Bureau 

of the Department of Commerce, it was possible to 

assemble the necessary assortment of skills.

Lake Mead presents a large and complex problem and, therefore, was undesirable as a "proving 

ground" for the various ideas and theories. A smaller, 

simpler reservoir was required for the first studies 

to test the procedures effectively. Lake Hefner was 

selected for this purpose and this report describes 

the work done there and the results obtained. The 

results were satisfactory and the work is being continued at Lake Mead.

The aim of the studies was not only the development of methods for use on large reservoirs, but also 

the development of new methods, or the improvement of existing methods, of potential use on all reservoirs, even the smallest, and an enlargement of 

knowledge of evaporation in general. The results 

as well as having a direct practical application, may 

be a major contribution toward an understanding of 

evaporation and related subjects.

Commissioner of Reclamation

PREFACE

This report was originally published in 1952, 

jointly by the Geological Survey and the Navy 

Electronics Laboratory, as Geological Survey 

Circular 229 and as N. E. L. Report 327; it was 

arranged and reproduced by the Navy Electronics 

Laboratory. The continuing demand for the report 

exhausted the first printing and made a reprinting necessary. In reprinting by the Geological 

Survey it was considered desirable to do so in a 

form that would permit wider distribution to reference libraries. The new format does not involve changes in the body of the report except 

for corrections indicated on the errata slip that

accompanied the first printing and three additional corrections (the exponent in the denominator of the equation for a, p. 60, was changed 

from 1-n to 2-2n, the Gatewood reference, p. 69, 

remains unchanged except for Lower Safford Valley, and the Richardson reference, p. 119, now 

reads American Society of Civil Engineers). Reprinting was accomplished by use of reproducible 

material generously provided by the Navy Electronics Laboratory.

Observational data, on which the report is 

based, are presented in a second report published as "Water-Loss Investigations: Lake Hefner Studies, Base Data Report," Geological Survey Professional Paper 270 and Navy. Electronics Laboratory Report 328.

CONTENTS

AL Abstract.............................................................................................................................. x

Introduction................................................................................................................................... 1

Personnel............................................................................................... >t>-m-- 3

Acknowl edgments.......................................................................................................................... 4

General description of Lake Hefner, by G. Earl Harbeck, Jr........................................................... 5

Basis for choice of Lake Hefner.................................................................................................. 5

Physical and climatological characteristics.............................................................................. 6

References....................................................................................................................................... 8

Geology and ground-water hydrology of the Lake Hefner area, by P. E. Dennis......................... 10

Scope of the study.......................................................................................................................... 10

Geology and water bearing properties of the rocks................................................................... 10

Ground water hydrology................................................................................................................. 12

Conclusions on geology and hydrology........................................................................................ 16

References....................................................................................................................................... 16

The water budget control, by G. Earl Harbeck, Jr., and Frank W. Kennon................................... 17

Instrumentation and methods.......................................................................................................... 17

Results.............................................................................................................................................. 31

Conclusions on the water budget.................................................................................................. 34

References........................................................................................................................................ 34

Instrumentation for mass transfer and energy-budget studies, by Lloyd J. Anderson................ 35

General description of equipment.................................................................................................. 35

Mass transfer studies.................................................................................................................... 36

Energy-budget studies......................t............................................................................................ 38

Overall performance and maintenance.......................................................................................... 41

Conclusions on instrumentation.................................................................................................... 44

References.................................... A.............................................. ................................................... 44

Mass-transfer studies, by J. J. Marciano and G. Earl Harbeck, Jr................................................. 46

Theory.......................................................................................................................... .............. 45

Mass-transfer equations................................................................................................................ 54

Comparison of computed and measured evaporation................................................................... 61

Conclusions on mass-transfer studies and applications.......................................................... 68

References........................................................................................................................................ 69

Energy-budget studies, by Ernest R. Anderson................................................................................. 71

Introduction...................................................................................................................................... 71

Solar radiation.................................................................................................................................. 74

Reflection of short wave radiation.............................................................................................. 78

Effective back radiation................................................................................................................. 88

Advected energy.............................................................................................................................. 98

Change in energy storage............................................................................................................... 100

Ratio of energy used for conduction ot sensible heat to that used for evaporation.............. 104

Energy conducted as sensible heat by the atmosphere.............................................................. 110

Energy utilized by evaporative processes................................................................................... 110

Conclusions from energy budget studies.................................................................................... 117

References........................................................................................................................................ 118

The Cummings radiation integrator, by G. Earl Harbeck, Jr............................................................. 120

Theory........................................................................^^ 120

Instruments and methods................................................................................................................ 121

Results.............................................................................................................................................. 124

Conclusions on Cummings radiation integrator........................................................................... 126

References........................................................................................................................................ 126

Lake and pan evaporation, by Max A. Kohler...................................................................................... 127

Background and objectives............................................................................................................ 127

Instrumentation and operational procedure.................................................................................. 127

Pan evaporation and meteorological factors................................................................................ 132

Lake evaporation and meteorological factors........................................................................ 136

	v»
Page

Lake and pan evaporation Continued

Pan vs lake evaporation.................................................................................................................. 139

Relative evaporation from pans of various types........................................................................ 144

Conclusions on lake and pan evaporation.................................................................................... 146

References........................................................................................................................................ 148

Summary and recommendations............................................................................................................... 149

Summary of results........................................................................................................................... 149

Primary recommendations................................................................................................................ 151

Secondary recommendations............................................................................................................ 151

Appendix: Symbols and dimensions....................................................................................................... 153

................................................._ 157

ILLUSTRATIONS

Page 

Map of Lake Hefner, showing equipment installations.............................................................. Frontispiece

Figure 1. Monthly rainfall at Lake Hefner.......................................................................................... 7

2. Frequency of occurrence of various wind speeds at Lake Hefner................................. 7

3. Monthly variation in wind speed at Lake Hefner.............................................................. 7

4. Frequency of occurrence of winds from various directions at Lake Hefner................. 7

5. Average monthly air and water surface temperatures at Lake Hefner.......................... 8

6. Average monthly vapor pressures and saturation vapor pressures at Lake Hefner..... 8

7. Comparison of meteorological data at Oklahoma City during period of Lake

Hefner investigations with 1934 1951 averages....................................................... 9

8. Piezometric map of Lake Hefner area showing water table contours and other

hydrologic features........................................................................................................... 13

9. Cross sections along lines shown in figure 8.................................................................. 15

10. Error in computed evaporation resulting from inconsistencies in measurements of

change in stage of Lake Hefner...................................................................................... 18

11. Isohyetal map for the storm of 24 July 1951 at Lake Hefner.......................................... 22

12. Variability of storm rainfall................................................................................................ 23

13. Pipe well and recording-gage shelter; concrete control with V-notch weir above

Lake Hefner (location C)................................................................................................. 23

14. Rating curve for gaging station on Bluff Creek above Lake Hefner.............................. 23

15. Hydrograph of daily flow of Bluff Creek above Lake Hefner.......................................... 24

16. Culvert, weirs, and crest-stage gage at location J........................................................ 25

17. Estimated stage hydrograph at gage H resulting from storm of 2 June 1950................ 25

18. Rating curve for gaging station on Lake Hefner Canal................................................... 27

19. Relation between Venturi meter coefficient and differential head................................ 28

20. Venturi meter calibration curve.......................................................................................... 29

21. Effect of evapotranspiration losses on measured shallow seepage from Lake

Hefner................................................................................................................................. 29

22. Water budget computation for the week ending 9 September 1950.................................. 32

23. Barge station......................................................................................................................... 38

24. South station.......................................................................................................................... 38

25. Northeast station.................................................................................................................. 39

26. Intake tower station.............................................................................................................. 39

27. Lake Thermocouple float at barge...................................................................................... 39

28. Lower level instruments on barge station......................................................................... 39

29. Radiation instrument shelter on barge.... .......................................................,..^... n........... 40

30. TPR installation on boat..................................................................................................... 40

31. Data yield (days per month) of meteorological and radiation equipment...................... 42

32. Meteorological instrument shelter on barge...................................................................... 43

33. Icing on barge........................................................................................................................ 45

34. Growth of a boundary layer along one side of a flat plate.............................................. 46

35. The roughness constant C as a function of (u *fs/v)..................................................... 47

36. Laminar, smooth, and rough flow........................................................................................ 48

viii
Page

Figure 37. Wave height as a function of wind speed and fetch........................................................ 50

38. The stability parameter a.................................................................................................... 52

39. Relation between Richardson's number, /3 and S............................................................ 53

40. Momentum exchange in a boundary layer over a flat plate............................................. 55

41. Analogy between momentum boundary layer and the vapor blanket, for 8 = 8W.... ...... 55

42. The relation between Xr and ¥r /Xr .................................................................................. 57

43. Growth of the vapor blanket downwind.............................................................................. 57

44. Variation in unit evaporation downwind............................................................................ 58

45. Experimental test of the Thornthwaite-Holzman evaporation equation....................... 63

46. Experimental test of Sverdrup's (1946) evaporation equation........................................ 65

47. Experimental test of Sverdrup's (1937) evaporation equation........................................ 66

48. Experimental test of Button's evaporation equation........................................................ 66

49. Comparison of Sverdrup's (1937) equation, Sutton's equation, and the Lake

Hefner empirical equation............................................................................................... 67

50. Average monthly solar radiation for Lake Hefner from May 1950 to August 1951....... 75

51. Standard error of estimate of solar radiation computed from Kennedy's method as

a function of cloud cover................................................................................................. 77

52. Standard error of estimate of solar radiation computed from Kennedy's method as

a function of length of time interval.............................................................................. 77

53. Theoretical radiation coming from the sun and sky for Mt. Wilson and Washington,

D. C., according to King................................................................................................. 79

54. Variation of reflectivity of direct solar radiation with altitude of the sun for

pure water as obtained from Fresnel's formula............................................................ 81

55. Theoretical total reflectivity for completely diffuse radiation...................................... 81

56. Effect of the variation of air-mass turbidity on the reflectivity of a natural

water surface..................................................................................................................... 81

57. Theoretical variation of reflected energy as a function of sun altitude for Mt.

Wilson and Washington, D. C.......................................................................................... 82

58. Theoretical variation of reflected energy at the latitude of Lake Hefner 

(35°30'N) as a function of season and time of day using turbidity data 

from Mt. Wilson and Washington, D. C........................................................................... 83

59. Observations on the reflectivity of a natural water surface under polar Canadian 

and tropical Gulf air masses as a function of the wind speed at the 8-meter 

level................................................................................................................................... 84

60. Median reflectivity of a water surface as a function of sun altitude for polar

Canadian (0) and tropical Gulf (+) air masses under clear skies.............................. 85

61. Reflectivity of a natural water surface under clear skies.............................................. 85

62. Reflectivity of a natural water surface under low clouds.............................................. 86

63. Reflectivity of a natural water surface under high clouds............................................. 87

64. Comparison of reflected solar radiation observed at Lake Hefner with that

computed from empirical equations................................................................................ 88

65. Average monthly reflected solar radiation for Lake Hefner from 1 May 1950 to

1 September 1951.............................................................................................................. 89

66. Average monthly atmospheric radiation for Lake Hefner from 1 May 1950 to

1 September 1951................................................ ..................................................... 91

67. Plot of atmospheric radiation equations of the Angstrom and modified Angstrom

type for the Lake Hefner clear sky observations........................................................ 92

68. Plot of atmospheric radiation equations of the Brunt type and the least-square

straight line for the Lake Hefner clear sky observations.......................................... 93

69. Atmospheric radiation as a function of cloud height, cloud amount, and vapor

pressure............................................................................................................................. 95

70. Long-wave radiation from Lake Hefner from 1 May 1950 to 1 September 1951.......... 99

71. Reflected atmospheric radiation from Lake Hefner from 1 May 1950 to

1 September 1951............................................................................................................. 99

72. Daily advected energy for Lake Hefner from 1 May 1950 to 1 September

1951. Base temperature 0°C......................................................................................... 100

73. Comparison of energy storage computed by averaging by depth with average

of 16 stations (A) and single station (B).................................................................... 102

74. Energy storage (A) and variation in lake elevation, area, and volume (B) for

Lake Hefner from 15 April 1950 to 1 September 1951................................................ 103

ix
Page 

Figure 75. Diurnal variation of energy storage for Lake Hefner................................................... 104

76. Daily Bowen ratios for Lake Hefner from 1 May 1950 to 1 September 1951............. 107

77. Bowen ratios for 5 , 10 , and 20 day periods for Lake Hefner from 1 May 1950

to 1 September 1951...................................................................................................... 107

78. Composite hypothetical effect on evaporation of neglecting radiative diffusivity, 

variations in equality of diffusivity coefficients for water vapor and heat, and 

spray in computing Bowen ratios................................................................................ 109

79. Relationship between wind speed and differences between water-budget evaporation and energy budget evaporation....................................................................... 109

80. Average monthly amount of energy conducted to (+) or from (-) Lake Hefner from

1 May 1950 to 1 September 1951................................................................................. Ill

81. Comparison of daily and two day evaporation from Lake Hefner.............................. Ill

82. Comparison of three and four day evaporation from Lake Hefner.......................... Ill

83. Comparison of five and six day evaporation from Lake Hefner.............................. 112

84. Comparison of seven and eight day evaporation from Lake Hefner........................ 112

85. Comparison of nine and ten day evaporation from Lake Hefner............................. 113

86. Accuracy of energy budget evaporation determinations as a function of the

length of computational period for Lake Hefner....................................................... 113

87. Accuracy of energy budget evaporation (A periods only) determinations as a

function of the length of computational period........................................................ 115

88. Accuracy of energy budget evaporation (high evaporation months and A periods

only) determinations as a function of the length of computational period........... 115

89. Evaporation comparison of periods that coincide with thermal surveys................... 116

90. Cross section of Cummings radiation integrator.......................................................... 122

91. Comparison between rainfall and wet bulb temperatures for selected storms........ 123

92. Comparison between computed and observed evaporation.......................................... 125

93. Plan view of pan instaflations at Lake Hefner............................................................ 130

94. Photographs of pan installations at Lake Hefner........................................................ 131

95. Relation for class A pan at south station, Lake Hefner, using water

temperature.................................................................................................................... 133

96. Relation for class A pan at south station, Lake Hefner, using energy balance

approach........................................................................................................................ 135

97. Relation for BPI pan at south station, Lake Hefner, using energy balance

approach......................................................................................................................... 136

98. The effect of heat storage in the BPI pan.................................................................... 137

99. The effect of pan or lake size on rate of evaporation................................................ 143

100. Relation of class A pan coefficient to wind and relative vapor pressure

difference...................................................................................................................... 145

101. Plotting of monthly class A pan coefficient versus ratio of vapor pressure

difference...................................................................................................................... 145

TABLES

Page

Table 1. Monthly evaporation from Lake Hefner............................................................................ 34

2. Percent occurrence of instrument faults, meteorological data^ Lake Hefner

studies...................................................................................;.......................................... 45

3. Adiabatic wind profiles over Lake Hefner...................................................................... 49

4. Boundary-layer thickness at x = 1750 meters................................................................ 51

5. Ratio of average computed to observed evaporation...................................................... 65

6. Number of days per month of complete radiation record................................................. 73

7. Comparison of solar radiation computed from Mosby's formula and that observed

at Lake Hefner................................................................................................................ 76

8. Average turbidity factors for various types of air masses according to Haurwitz.... 80

9. Various values of constants for Brunt's atmospheric radiation equation................... 93

10. Various values of constants for Angstrom's atmospheric radiation equation............ 93
Page 

Table 11. Computed vs observed atmospheric radiation ratios..........]............................................. $4

12. Computed vs observed atmospheric radiation for selected days.................................... 95

13. Reflectivity of various water samples as a function of source temperature and

water temperature.............................................................................................................. 97

14. Reflectivity of contaminated water samples...................................................................... 97

15. Monthly averages of the advected water volumes at Lake Hefner in acre feet

per day................................................................................................................................ 99

16. Change in energy storage over yearly and monthly time intervals................................. 103

17. Change in energy storage during short time intervals when air and surface water

temperature range is small and lake volume constant.................................................. 103

18. Pertinent data for A water budget days when Bowen ratio is approximately 1.00... 107

19. Comparison of energy budget and water budget evaporation for periods 12 to 30

days in length.................................................................................................................... 114

20. Pertinent data for evaporation periods coinciding with thermal cruises....................... 116

21. Comparison between average incoming radiation measured by Cummings radiation

integrator and by pyrheliometers and flat plate radiometer....................................... 124

22. Comparison between computed and observed evaporation from Lake Hefner for

periods of various lengths............................................................................................... 125

23. Comparison between values of indicated radiation items for Cummings radiation

integrator and for Lake Hefner........................................................................................ 125

24. Summary of monthly and annual pan to lake coefficients............................................. 128

25. Summary of pan to pan ratios............................................................................................ '29

26. Evaporation relations [E = (e0 - ea ) (a + bu)] for pans at south station..................... 133

27. Equations for lake evaporations, E = (en - ea ) (a 4- i»u)................................................ 137

28. Summary of monthly pan and lake evaporation and monthly and annual pan to lake

coefficients (Lake Hefner experiment)........................................................................- 140

29. Pan-to-pan ratios (Lake Hefner experiment).........,........................................................ 147

XI
WATER-LOSS INVESTIGATIONS

LAKE HEFNER STUDIES, TECHNICAL REPORT

ABSTRACT

STATEMENT OF PROBLEM

Develop an improved method or methods for the 

determination, and if possible the prediction 

from climatological and limnological data, of 

water losses by evaporation using mass-transfer 

and energy-budget theory. Develop improved 

techniques for converting evaporation-pan data 

to estimated lake evaporation. Test the proposed 

methods and techniques at Lake Hefner, Oklahoma, using evaporation computed from the water budget of the lake as the control.

CONCLUSIONS

1. .The water-budget control met the requirement 

that errors in the water budget should not exceed 5 per cent of the monthly evaporation. In 

addition, 62 per cent of the determinations of 

daily evaporation-were sufficiently accurate to 

provide a check against estimates of daily evaporation by mass-transfer and energy-budget 

theories.

2. Only two (0. G. Sutton's and Sverdrup's 1937 

form) of the many mass-transfer equations tested 

proved adequate for predicting daily evaporation 

with sufficient accuracy to be generally useful, 

employing equipment now available.

3. A simple empirical equation was developed 

from the water-budget data, using wind speeds 

and vapor-pressure differences measured at Lake 

Hefner. An operational version of this equation 

proved satisfactory for computing daily evaporation at Lake Hefner using standard meteorological observations from a nearby weather station, 

plus the surface-water temperature of the lake.

4. The energy-budget equation proved satisfactory for computing evaporation for periods of 7 

days or longer provided the water budget can be 

approximated closely enough to estimate advected energy. The primary limitation on the accuracy of the method, for short periods of time,

is the evaluation of change in energy storage.

5. The Cummings Radiation Integrator (CRI) 

promises to provide a satisfactory substitute for 

the expensive radiation-measuring equipment 

used at Lake Hefner, thus simplifying the computation of evaporation by the energy-budget 

method.

6. Study of evaporation from four types of evaporation pans confirmed previous studies in showing that pan evaporation is generally higher than 

lake evaporation and that pan-to-lake coefficients have a pronounced seasonal variation. 

Annual pan-to-lake coefficients are reasonably 

consistent with previous values, but further 

studies of such coefficients under different conditions of climate, lake area, and lake depth 

will be required for more accurate interpretation 

of available evaporation-pan records in terms of 

lake evaporation.

RECOMMENDATIONS

1. Conduct further studies of the mass-transfer 

equations (Sutton's, Sverdrup's, and the new, 

simplified, empirical one), the energy-budget 

equation, and pan evaporation on a comparative 

basis at Lake Mead.

2. Test the proposed simplified method for obtaining continuous records of evaporation for 

any reservoir at Lake Mead and at a number of 

additional reservoirs with different conditions of 

terrain and climate.

3. Study the mass-transfer and energy-budget 

techniques as methods for estimating water 

losses by evapotranspiration.

4. Expand the present program for the collection 

of water-temperature data in lakes, reservoirs, 

and streams throughout the country and obtain 

wind data at the sites of selected existing and 

proposed reservoirs. These data, together with 

available climatological information, would assist in predicting evaporation prior to the establishment of a new reservoir. They would also be 

useful in interpreting existing records of pan 

evaporation in terms of lake evaporation.

XII
INTRODUCTION

Net water loss by evaporation, and by transpiration from vegetation along water courses, in the 

17 western states has been roughly estimated to be 

over 30 million acre-feet per year, more than twice 

the annual flow of the Colorado River. This would 

be enough water to irrigate 15,000 square miles of 

desert, an area nearly half that of the state of New 

York or Pennsylvania.

Over half this loss is by evaporation from reservoirs, lakes, and streams, approximately 15 million acre-feet being from reservoirs alone. Each new 

reservoir adds to this loss, yet reservoirs must be built 

to salvage water which otherwise would be lost and 

make it available at the time it is most needed.

There is no practicable known method for preventing evaporation losses, but it is possible to reduce 

them by proper planning of reservoir location and by 

establishing suitable operating procedures. Evaporation loss is only one of many factors involved in 

reservoir location and operation, but it is important 

that its magnitude be accurately determined so that 

it can be properly weighted against the other factors. 

Average yearly evaporation figures are not sufficient 

even if fairly accurately known; the monthly variation must be determined, especially when a chain of 

reservoirs of different areas and depths are operated 

as an integrated group, as is the case on the lower 

Colorado River.

Water losses by transpiration from phreatophytes 

(deep-rooted plants, mostly nonbeneficial, which use 

a great deal of water, for example, salt cedar) are 

estimated to be as great or greater than losses by 

evaporation. These losses may be greatly reduced 

by removal of the phreatophytes, but this procedure 

is uncertain and expensive. Again, accurate estimates 

of water losses by transpiration are needed to determine whether a removal program is worthwhile.

The Lake Hefner study represents an integrated 

attack on the water-loss problem. It comprised investigations of both old and relatively new methods 

of determining evaporation. It is an excellent example 

of a successful cooperative effort by specialists in 

different fields; the collaborating scientists and engineers were from five fields of science (geology, 

hydrology, meteorology, oceanography, and physics) 

and from five agencies in three departments of the 

federal government (Bureau of Reclamation and Geological Survey of the Interior Department, Bureau

of Ships and Navy Electronics Laboratory of the 

Navy Department, and Weather Bureau of the Department of Commerce).

Historical Review

The Lake Hefner study was an outgrowth of the 

cooperative investigations at Lake Mead, in Arizona 

and Nevada, undertaken in late 1947, 1948, and 

early 1949 by representatives of the Departments 

named above. The Navy Electronics Laboratory contributed to the Lake Mead investigations by applying 

oceanographic techniques to the determination of 

seasonal patterns of circulation in the lake,- the 

temperature data obtained were also used, with meteorological data, for estimating the thermal energy 

budget of the lake.

The latter study resulted in preliminary estimates 

of monthly evaporation. Discussion of the energybudget technique by representatives of the agencies 

concerned indicated that a wider application of this 

approach might be worthwhile. The application of 

mass-transfer theory, which attempts to explain evaporation in terms of the factors affecting the transport 

of water vapor, to the evaporation problem was also 

discussed. Numerous investigators had studied the 

mass-transfer problem, but the practical instrumentation required for measuring the meteorological factors 

in the field had not been developed. The Navy Electronics Laboratory was in the process of developing 

instruments for measurement of meteorological factors 

affecting radio-wave propagation; these instruments 

appeared to be adaptable to the mass-transfer measurements.

At a joint conference on 13-15 December 1948 

in Boulder City, Nevada, it was decided to investigate 

these relatively untested techniques, and a program 

was outlined. Subsequent conferences modified the 

details, but the basic program remained unchanged 

except for the addition of investigations by the 

Weather Bureau of evaporation from standard and 

experimental evaporation pans a necessary part of 

any comprehensive study of methods for determining evaporation.

The Problem

At present there are four basic methods for 

determining evaporation from water bodies:

1. The wafer-budgef method is the most direct; 

evaporation is determined by difference from measurements of inflow, outflow, and changes in storage.
Unfortunately this method is not generally practicable 

because errors in measuring inflow and outflow are 

often large compared to evaporation, and seepage 

and bank storage are uncertain and often unmeasurable terms.

2. Evaporation pans have been used for many 

years as an index to evaporation from lakes and 

reservoirs; over 4000 station-years of record have 

been collected at points throughout the country. In 

spite of many investigations, however, the general 

validity of "pan coefficients" (determined under specific experimental conditions) used for converting pan 

evaporation to lake evaporation has always been 

subject to question.

3. The thermal energy-buc/gef method has been 

used for some time to estimate evaporation from the 

oceans and has been applied to restricted studies of 

evaporation from inland water bodies. It is based 

upon assessment of all the sources of incoming and 

outgoing thermal energy plus changes in energy storage, the difference being the energy utilized in evaporation. General application of this technique has 

awaited the development of adequate instruments.

4. Modern mass-transfer theory is the basis for 

the fourth method measurement of the factors 

affecting the actual removal of water vapor from a 

lake by processes of turbulent diffusion and transport.

The last two methods had never been adequately 

tested at natural scale; for such a test an acuurate independent determination of evaporation by the waterbudget method was essential. Tests of evaporation 

pans against independent measurements of evaporation from a large natural water body, with concurrent

measurement of pertinent meteorological factors, were 

also inadequate.

Lake Mead was not suitable for the initial test; 

the water budget could not be determined with sufficient accuracy and its large size, irregular shape, and 

mountainous terrain introduced too many complicating factors. The first step in the program, therefore, 

was the search for a lake or reservoir where an accurate water budget could be established as a control. 

Requirements for testing the energy-budget and masstransfer approaches introduced further restrictions, so 

that more than a hundred lakes and reservoirs in the 

West had to be considered to find one whose characteristics approached the stringent requirements.

After considering the advantages and disadvantages of all of these lakes and reservoirs, Lake Hefner, 

Oklahoma City, was chosen at a joint conference of 

the cooperating agencies in Boulder City on 17-19 

October 1949. Installation of the equipment at Lake 

Hefner was completed in April 1950 and, after a preliminary study of meteorologic and hydrologic conditions, routine measurements began the latter part 

of April and continued through August 1951.

This volume reports the results of this investigation the most comprehensive study of its type yet 

undertaken. Several million measurements were recorded, read, tabulated, and analyzed. These will 

be summarized and published in a subsequent volume 

(Volume II, Data Report) of this series. The results 

appear to have justified the effort; in addition to increasing the value of records obtained by traditional 

methods, the new techniques have proved both scientifically sound and operationally practicable.
PERSONNEL >

The following list of personnel contributing to 

this project includes only those assigned by the collaborating agencies for appreciable periods of time; 

contributors not directly assigned are listed under 

"acknowledgments."

General Supervision

Bureau of Reclamation. J. R. Riter, Chief, Hydrology Branch, Project Planning Division, was in 

general charge of the work for this division of the 

Bureau, with technical direction of the work under 

the immediate supervision of W. U. Garstka. Washington representative of the Division was M. D. Dubrow, succeeded by W. D. Romig. C. P. Vetter, Chief, 

Office of River Control, Region III, was in charge 

for that office.

Geological Survey. R. W. Davenport, Chief, Technical Coordination Branch, Water Resources Division, 

was responsible for general supervision of the work 

of the Survey, with technical direction by W. B. 

Langbein, Chief, Research Section, Technical Coordination Branch. G. E. Harbeck, Jr., was responsible 

for the water-budget work and for field liaison. W. O. 

Smith acted as the Geological Survey representative 

during preliminary phases of the work.

Bureau of Ships. Throughout most of the project, G. B. Cummings, Chief Civilian Assistant, Sonar 

Branch, Electronics Division, was responsible for the 

Bureau's participation. His personal interest and unusual ability to coordinate diverse activities contributed greatly to the establishment and success of 

the project.

Navy Electronics Laboratory. R. Dana Russell, 

Senior Consultant (Geophysics), was in general charge 

of the project for the Laboratory, with L. J. Anderson 

responsible for the instrumentation, E. R. Anderson for 

the energy-budget approach, and J. J. Marciano for 

the mass-transfer work.

Weafher Bureau. W. E. Hiatt, Chief, Hydrologic 

Services Division, was in general charge for the Bureau, succeeding the late Merrill Bernard, who with 

R. K. Linsley, now at Stanford University, was responsible for the initiation of the Bureau's phase of work.

Field Personnel

S. K. Jackson, District Engineer (SW), Water Resources Division, Geological Survey, served as local 

coordinator for the field work at Oklahoma City. 

The construction and operation of the stream-gaging 

and water-level installations at Lake Hefner, together with other observations required for the waterbudget control, were directed locally by F. W. Kennon, Hydraulic Engineer, Geological Survey. He was 

assisted by H. O. Wires and, for part of the time, by 

G. E. Koberg. This group also constructed and operated the Cummings Radiation Integrator, mapped 

the shore of the lake, and provided horizontal control 

for the hydrographic survey. During part of the time 

they were also responsible for the maintenance and 

operation of one of the shore meteorological stations.

Installation, operation, and maintenance of the 

meteorological equipment at Lake Hefner were directed locally by LT C. C. McCall, USN, assisted by 

J. D. M. Freitas, MNC, USN, and H. E. Knudsen, MNC, 

USN, assigned to the project by the Bureau of Ships. 

This group was also responsible for taking the bathymetric data required for the hydrographic survey 

of the lake. In addition, they were responsible for 

the thermal surveys of the lake (observations of temperature-depth profiles at 16 stations distributed over 

the lake at intervals of 7 to 10 days, plus a daily 

profile at one station).

The field work of the Weather Bureau was coordinated by R. J. MacConnell, South Central Area 

* Hydrologic Engineer. The observational program was 

under the direction of W. E. Maughan, Meteorologist 

in Charge, Oklahoma City Office. E. L. Smith served 

as observer until April 1951, when he was relieved 

by B. J. Stringer.

The success of the entire project was largely due 

to the conscientious and untiring efforts of these field 

personnel, who regularly obtained observations under 

a variety of weather conditions, frequently unfavorable to the operation of experimental equipment.

Office Personnel

Processing of the data for the mass-transfer 

and energy-budget studies required the maintenance 

of a staff of six to seven people at the Navy Electronics Laboratory. Though these individuals initially 

were Laboratory personnel, the Laboratory had to 

concentrate its effort on urgent defense work soon 

after the beginning of hostilities in Korea, and personnel thereafter were furnished by the Geological 

Survey.

Processing of the meteorological data was at 

first under the supervision of J. J. Marciano, Navy 

Electronics Laboratory, and later under G. E. Harbeck, Jr., Geological Survey, with G. E. Koberg, also 

of the Survey, in immediate charge of the work. 

Machine computations of the voluminous data (on 

about 200,000 IBM punch cards) were made by the
Institute of Numerical Analysis, National Bureau of 

Standards, under a contract with the Geological Survey. The data were analyzed by J. J. Marciano and 

G. E. Harbeck, Jr., assisted by G. E. Koberg.

Processing and analysis of the energy-budget 

data were under the supervision of E. R. Anderson, 

Navy Electronics Laboratory; O. E. Leppanen of the 

Geological Survey assisted with the statistical analyses.

Analyses for the Weather Bureau part of the 

program were performed in the Hydrologic Investigations Section (T. J. Nordensen, Chief), with W. E. 

Fox and E. S. Thompson responsible for most of the 

analytical work. Planning of the analyses and preparation of the Weather Bureau's section of the report 

were under the direct supervision of M. A. Kohler, 

Chief Research Hydrologist.

Final editing of the complete report and preparation for publication were done by the Technical 

Information Division of the Navy Electronics Laboratory. R. Dana Russell acted as editor for the 

collaborating agencies and prepared the sections for 

which no author is listed.

ACKNOWLEDGMENTS

In a large and complex cooperative effort such 

as the study reported here, it is almost impossible to 

acknowledge the assistance of all the people who 

contributed, directly or indirectly, to the successful 

completion of the work. A great many individuals 

in the field offices of the Bureau of Reclamation and 

the Geological Survey contributed data and assistance in the search for a suitable lake; their help has 

been acknowledged in the report on that phase of 

the study. It is hoped that most of the other major 

contributors not directly assigned to the work have 

been included below.

The active cooperation of officials of the Oklahoma City Water Department is gratefully acknowledged, particularly the assistance of M. B. Cunningham, Superintendent and Engineer, F. S. Taylor, then 

in charge of the Lake Hefner Filter Plant, S. K. Bean, 

Civil Engineer, and Frank Herrmann, Office Engineer.

t Calibration of the Venturi meter at the Filter 

Plant was performed by D. M. Lancaster, R. H. Kuemmich, and R. B. Dexter of the Hydraulic Laboratory 

Section, Research and Geology Division, Bureau of 

Reclamation, Denver, Colorado. The description in 

this report of the technique and equipment used was 

taken from their unpublished report prepared upon 

completion of the calibration.

D. W. Pritchard, formerly of the Navy Electronics Laboratory and now Director, Chesapeake Bay 

Institute, The Johns Hopkins University, contributed to 

the preliminary planning and some of the theoretical 

analysis, and also made suggestions for the adaptation of the energy-budget technique to the Cummings 

Radiation Integrator. R. L. Denton, Navy Electronics 

Laboratory, designed the galvanometer amplifier 

used in the recording equipment for the meteorological measurements, and F. R. Bellaire, also of the 

Laboratory, was responsible for the construction of 

much of the meteorological instrumentation and for 

many of the basic instrumentation ideas. E. E. Gossard of the Laboratory assisted in the review and 

analysis of mass-transfer theories. Chemical analyses 

of the samples of water from Lake Hefner and Lake 

Mead used in the emissivity measurements were made 

by T. B. Dover and C. S. Howard of the Quality of 

Water Branch, Water Resources Division, Geological Survey.

Critical review and comments on the development of the energy-budget equation were contributed 

by Dr. H. U. Sverdrup, Director, Norsk Polarinstitutt, 

Oslo, Norway, and by Dr. R. B. Montgomery, Brown 

University, and on the mass-transfer approach by 

O. G. Sutton, Military College of Science, Shrivenham, England.

Rough drafts of this report have been critically 

reviewed by a number of people whose suggestions 

have been incorporated in the text without specific 

acknowledgment. Review and comments by J. T. Gier 

and R. V. Dunkle, University of California, E. C. Buffington, Navy Electronics Laboratory, and H. C. S. 

Thorn, C. S. Gilman, F. D. White, L. P. Harrison, J. L. H. 

Paulhus, and T. H. MacDonald, Weather Bureau, have 

been particularly helpful.
General Description of Lake Hefner

by G. Earl Harbeck, Jr.*

BASIS FOR CHOICE 

OF LAKE HEFNER

The selection of a reservoir or lake suitable for 

a comprehensive study of the exchange of energy 

between the atmosphere and a water surface is not 

a simple matter. It was planned to determine evaporation by three methods: (1) water budget, (2) energy budget, and (3) mass transfer. Since the water 

budget was to be the control for evaluating the other 

two methods, it was imperative that the water budget 

be accurate. This was the paramount consideration 

in selecting the lake (Harbeck et a/., 1951).

The energy-budget and mass-transfer methods 

also imposed restrictions on the choice of the lake. 

It was desired, for example, to minimize the effects 

of mechanically induced turbulence on the wind structure, and for that reason canyon reservoirs were 

deemed unsatisfactory. A deep lake was desired to 

avoid the problem connected with the penetration of 

solar radiation to the lake bottom.

The requirements established for the lake were 

as follows, in order of importance:

1. Water Budget

a. The error in the monthly difference between 

total inflow and outflow, including both surface and 

subsurface flows, and allowing for changes in storage 

(including bank storage), must be less than 5 per cent 

of the best existing estimate of the mean monthly 

evaporation loss. Inflow and outflow preferably should 

be as small as possible during the period of high 

evaporation.

b. Irtfrequent short periods of storm inflow, during which the water budget cannot be determined 

with the accuracy required under la, can be tolerated.

c. Subsurface inflow and outflow must be negligible compared with evaporation, unless it is known 

that they can be measured accurately.

d. Substantial bank storage is undesirable.

e. Transpiration losses must be small.

f. An accurate area-capacity curve is needed, 

but if not available, a hydrographic survey can 

be made.

* Hydraulic Engineer, Water Resources Division, U. S. 

Geological Survey.

2. Size

a. The minimum desirable dimensions are 3 miles 

in width and 5 miles in length, or the area should not 

be less than 10 square miles if the water body is 

nearly circular or if the longer dimension is in the 

direction of the prevailing summer winds.

b. The maximum desirable size is 50 square 

miles if the lake is nearly circular, and otherwise 30 

square miles.

3. Shape

a. A circular shape is ideal. A very irregular 

shore line is unsatisfactory; an irregular shore line 

downwind is not so objectionable as one upwind. 

A long, narrow lake is not satisfactory.

b. An unobstructed expanse of water (few or no 

islands) extending 5 miles in the direction of the prevailing summer winds is desirable.

4. Depth

a. At least 80 per cent of the lake should be 

more than 5 feet deep (and. preferably more than 10 

feet). A deep lake is desirable.

b. Playa lakes should be considered if they 

meet all requirements except depth.

5. Topographic Setting

a. Low relief is preferable; lakes in canyons are 

not satisfactory.

b. The drainage area preferably should be small.

6. Location and Climate

a. An arid region is preferable.

b. There should be fairly long periods of no 

rainfall during the season of high evaporation, but 

infrequent storms are not objectionable.

c. It is preferable, but not essential, that the 

lake remain unfrozen in the winter.

The search for a lake suitable for the study has 

already been discussed (Harbeck ef a/., 1951). The 

Geological Survey district offices in the western states 

and the Bureau of Reclamation regional offices were 

furnished copies of the specifications and were asked 

for recommendations. Their comments were also solicited on the suitability of lakes that appeared, on the 

maps available, to approach the desired size and

30Z520 O - 54 - Z
shape. Those that appeared to satisfy the requirements in most respects were visited, and field examinations were made.

The final choice of Lake Hefner near Oklahoma 

City involved no compromise as to the accuracy of 

the water budget, although in certain other respects, 

such as size and climate, it does not meet the specifications completely. These departures from the specifications were unimportant and, in practice, the 

smaller size actually proved advantageous.

It was believed that satisfactory water budgets 

could be obtained for certain other lakes, such as 

Big Sage Reservoir in northern California and Pyramid 

and Walker Lakes in western Nevada, but these lakes 

were rejected because Lake Hefner appeared to approach the ideal lake more closely in most other 

respects.

PHYSICAL AND CLIMATOLOGICAL 

CHARACTERISTICS

The Reservoir Area and The Dam

Lake Hefner is a water-supply reservoir, owned 

by the City of Oklahoma City. The lake is fairly 

regular in shape (see map, frontispiece) and is formed 

by a long horseshoe-shaped dam on Bluff Creek in 

the southeastern part of T. 13 N., R. 4 W., approximately 8 miles northwest of the center of the city. 

The topography surrounding Lake Hefner is flat to 

gently rolling. The dam itself is the only significant 

obstruction to smooth wind flow in the area. The land 

slopes generally toward the northwest, roughly 30 to 

40 feet per mile. The recent hydrographic survey of 

the lake shows its capacity at full pool (elevation 

1199 feet) to be 75,355 acre-feet and its surface area 

2587 acres. During the period 26 April 1950 to 31 

August 1951, however, the stage was maintained between 1190.8 and 1195.3 feet; contents ranged from 

about 55,900 to 66,100 acre-feet and surface areas 

from 2148 to 2386 acres.

Although Lake Hefner lies in the Cimarron River 

Basin, it is supplied principally by a canal from the 

North Canadian River. In this area the elevation of 

the North Canadian River is several hundred feet 

higher than that of the Cimarron River, and only 

a short diversion canal was needed to permit gravity 

flow of selected water (chemical quality being the 

criterion) from the North Canadian River into Lake 

Hefner. Under the general operational plan, diversions are infrequent and of only a few days' duration, 

but they may be of considerable magnitude since 

the canal capacity is 1500 cfs.

Natural inflow to Lake Hefner is of little consequence so far as reservoir replenishment is concerned. The natural drainage area above the lake 

is only about 30 per cent larger than the area of 

the lake at full pool, so that storm runoff into Lake 

Hefner is usually much less than rainfall on the 

lake surface.

Vegetation is sparse except along the water 

courses. The soil is generally thin, and native grass 

is the predominant cover. The lake shore is relatively 

clean. There are a few marshy areas north and west 

of the dam, supplied by seepage through the dam. 

The average size of the marshy areas is considerably 

less than an acre, and the water consumed by the 

vegetation is accounted for in the water budget.

The Lake Hefner area has been developed for 

recreational purposes by the City of Oklahoma City. 

On the south shore, concrete boat pens have been 

provided in the boat harbor and a golf course has 

been constructed. A good road encircles the lake.

The dam and dike are of rolled earth fill with 

a clay core. The maximum height of the dam from 

thalweg to the road on top of the embankment is 

approximately 105 feet, but the average height is 

considerably less. The total length of the dam and 

dike is 3'/2 miles. Flow through an ungated spillway 

at the east end of the dam would begin if the stage 

reached 1201 feet. Tile drains and catch basins were 

installed in the main part of the dam to collect seepage through the dam and rainfall on the downstream 

face. The water thus collected is drained to the Bluff 

Creek channel below the dam.

Climatology

The climate at Lake Hefner has been classed as 

subhumid by Thornthwaite (1931). For the purposes 

of the water-loss study, a site having an arid climate 

would have been preferred, but of all the requirements governing the selection of a site, this was considered to be of least importance. Theorefically, at 

least, it would be desirable to choose a site where 

there is no rainfall, inflow, or outflow, and where 

evaporation rates are high. Evaporation rates at Lake 

Hefner were considered to be satisfactorily high, and 

the only effect of rainfall was to decrease the number 

of days for which an accurate water budget could 

be obtained.

Normal annual rainfall at Oklahoma City is approximately 31 inches. During the last 12 months of 

the observation period, rainfall at Lake Hefner totalled 27.1 inches. Rainfall at Oklahoma City is 

usually greatest during the April-June period, as
was observed in 1951. During the 4-month period 

May to August 1950, however, rainfall at Lake Hefner totalled approximately 18 inches, with abnormally high rainfall in May and July. The monthly 

variation in rainfall is illustrated in figure 1.

Wind speeds measured at the 8-meter level at 

the barge station averaged 11.5 knots* during the 

16-month period ot observation. The relative frequency of occurrence of various wind speeds is illustrated in figure 2. Wind speeds were in the range 

4 to 14 knots 59 per cent of the time and were less 

than 4 knots only 8 per cent of the time. The modal 

wind speed was approximately 7 knots.

The monthly variation in wind speed at Lake 

. Hefner is shown in figure 3. (Data obtained during 

May and June 1950, the instrument "shakedown" 

period, were not used.) Although the month-to-month 

variation is much more erratic than for other climatic 

factors, such as temperature, strong winds predominate during the winter and spring months; even in 

summer months, the winds can hardly be classed 

as light.

Wind directions at the barge were tabulated 

using a special code, as shown in figure 4. in which 

the figures in the sectors are the code numbers used. 

This code was designed to facilitate selection, by the 

computing machines, of data from the station considered to be most representative of the air upwind 

from the barge. The considerations underlying construction of the code were as follows: (1) normal 

wind-direction distribution, as shown by a wind rose 

furnished by the Oklahoma City Weather Bureau 

Office; (2) number and location of the shore measuring stations, each location being determined by terrain and other practical considerations as well as by 

dominant wind directions; (3) terrain and other obstructions affecting air flow; (4) the fact that the 

equipment recorded wind direction only by the eight 

cardinal and intercardinal points of the compass. The 

arbitrary code shown in figure 4 repres* nts a compromise between these various factors for sectors 2, 

3, and 4, the intent being that measurements at 

stations 2, 3, and 4 be representative of their respective sectors. The remaining area was arbitrarily 

divided into three equal parts (sectors 1, 5, and 6). 

It was later found that the winds at station 4 (the 

tower station) were affected by the dike, particularly 

below 16 meters, and that station 3 was more rep-

* The Navy anemometers used in this study registered 

wind speed in knots. A knot is one nautical mile, or 1.15 

statute miles, per hour.

ilMss is 123 5 ilM

1990 1751

Figure 1. Monthly rainfall at Lake Hefner.

I'5

8

o 5

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 

WIND SPEED IN KNOTS

Figure 2. Frequency of occurrence of various wind speeds at 

Lake Hefner.

Figure 3. Monthly variation in wind speed at Lake Hefner.

Figure 4. Frequency of occurrence of winds from various directions at Lake Hefner.
resentative of both sectors 3 and 4; these sectors 

were therefore combined in the analysis and data 

from station 3 used in the case of winds from 

both sectors.

The results of an analysis of the frequency of 

winds from various directions are also illustrated 

in figure 4. During the entire period of observation, 

there was no 3-hour period in which the wind direction was coded as 1, 5, or 6. Winds from these directions undoubtedly occurred for short periods of time, 

but sustained westerly winds were not observed. The 

code 2 winds (SSW to ESE) reached a maximum frequency of 70 to 80 per cent in late spring and early 

summer. There appeared to be no well-defined seasonal variations of frequency of code 3 winds (ESE 

to NNE). The seasonal variation in code 4 winds (NNE 

to NNW) was apparent; northerly winds prevailed 

15 to 20 per cent of the time in March and April, but 

less than 3 per cent of the time during the summer 

months.

The seasonal variation in air and water-surface 

temperatures is illustrated in figure 5. The usual seasonal lag is readily apparent, particularly during the 

fall, winter, and spring.

The monthly variation in evaporativity as indicated by vapor pressures is illustrated in figure 6. 

For constant wind, evaporation is directly proportional 

to the so-called Dalton Difference, i.e., the saturation 

vapor pressure at the temperature of the water 

surface minus the vapor pressure of the air. The vapor 

pressure difference is, of course, greatest in summer 

and least in winter, and during the 16-month period 

of observation at Lake Hefner the monthly average 

ranged from 1.1 mb in February 1951 to 14.4 mb 

in August 1951.

The observations at Lake Hefner did not in themselves provide any information as to whether evaporation during this period was representative of average 

conditions, since there were no previous records at 

the lake available for the determination of normals. 

However, it was possible to compare climatological 

records obtained at the Weather Bureau city and 

airport stations during the observational period with 

long-term averages at those locations. The results 

are shown in figure 7, prepared by the Weather 

Bureau.

During the period July-September 1950, it is 

probable that evaporation from Lake Hefner was 

below normal; wind speeds, and vapor pressure differences as indicated by air and dew-point temperatures, were below normal. As might be expected, 

sunshine was also below normal.

During the period June-August 1951, wind speeds 

and air temperatures were near normal. Dew-point 

temperatures were above normal, however, and evaporation from Lake Hefner was probably slightly below 

normal during this period also.

Monthly precipitation at Oklahoma City was 

substantially above normal during May and July 

1950, dnd May 1951, but was well below normal 

during the period September-December 1950.

For the entire 16-month period of observation, 

it is believed that evaporation may have been slightly 

below average because of the two below-normal 

summers but that the period, in general, was reasonably representative.

REFERENCES

Harbeck, G. E., ef a/., "Utility of Selected Western 

Lakes and Reservoirs for Water-Loss Studies," U. S. 

Geological Survey, Circular, no. 103, March 1951.

Thornthwaite, C. W., "The Climates of North America 

According to a New Classification," Geographical 

Review, vol. 21, 1931, pp. 633-655.

AIR TEMPERATURE, 8-M LEVEL 

 .. WATER-SURFACE TEMPERATURE

TEMPERATURE IN ° C 1til 0 UM

.- 

S^ ^

\

\

V** <*

*

S

/

4

*J 

^

r

**

M » eg S 1 » I 1 i 1 M

1950 "31

Figure 5. Average monthly air and water-surface temperatures 

at Lake Hefner.

VAPOR PRESSURE OF AIR AT 8-M LEVEL

. SATURATION VAPOR PRESSURE AT THE 

TEMPERATURE OF THE WATER SURFACE

Si* IB; Hi M

Figure 6. Average monthly vapor pressures and saturation 

vapor pressures at Lake Hefner.
15

«io

UJ

1 5 

n

1950 1951 

MAY. JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG

 -

"**'*' q

* -

NORMAN, OKLA., PAN EVAPORATION (1937-1951)

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,-.-- 

*->'**

^

1 

5 80

on 

S 60

40

DC

° 20 

ff 10

t/) 

UJ

80

Li.

ffi 60

UJ

§40 

20

,'X

PER CENT POSSIBLE SUNSHINE (AIRPORT STATION SINCE MAY 1951)

/'X

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"*» _s f

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^ ^

, - ' ---.-' '*

WIND VELOCITY (AIRPORT STATION)

v~-

-:-,,

f~~~ ~~-

DEW-POINT TEMPERATURE (AIRPORT STATION)

..^

 <r*

.---

4

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S*

3

60

40

15 

10 

5

n

-' '-- 

AIR TEMPERATURE (CITY OFFICE)

~\

\

,--' ** 

^**~

.-'"

\ \

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/\

/:*" * «S^ftM

PRECIPITATION (CITY OFFICE)

- . -^ .'----.

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X

,'* *

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~-

MAY JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG 

1950 1951

AVERAGE FOR PERIOD 1934-1951

 OBSERVED (OR ESTIMATED) DURING PERIOD OF LAKE HEFNER 

INVESTIGATIONS

Figure 7. Comparison of meteorological data at Oklahoma City during period of Lake Hefner 

investigations with 1934-1951 averages.
Geology and Ground-Water Hydrology of the Lake Hefner Area

by P. E. Dennis*

SCOPE OF STUDY

It has been generally assumed that seepage to 

Lake Hefner is negligible, and that seepage from the 

lake is largely measured in surface-water drains.

The study of the geology and ground-water hydrology was designed to check the validity of these 

assumptions. Specifically, it was undertaken to provide 

the best (1) estimate of maximum seepage out of the 

lake, (2) estimate of seepage into the lake, and (3) 

evaluation of the present surface measurements of 

outward seepage through the dam.

The results are based on field work done during 

June 1950. A complete inventory was made of farm, 

domestic, and unused wells in the immediate vicinity 

of the lake; static water-level measurements were obtained in twenty-five of the wells. Records of wells 

in the area obtained by the WPA in 1936 were examined, including seven wells subsequently covered 

by the lake. Forty test holes ranging from 10 to 50 

feet in depth were drilled by a rotary machine and 

logged from ditch samples. Three hand-auger holes 

were put down to the water table. In all, 68 

points were obtained on the water surface and water 

levels were measured at least twice in most of them. 

Levels were run to the wells and test holes. Pumping tests were attempted on about a dozen test holes; 

reasonably satisfactory results were obtained on 

three. Exposures of the Hennessey shale in the area 

were examined and described. The contact between 

soil-covered Hennessey and exposed Hennessey was 

mapped in part of the area.

GEOLOGY AND WATER-BEARING 

PROPERTIES OF THE ROCKS

Lake Hefner rests on the flat-lying red shale and 

siltstone beds of the Hennessey shale. Most of the 

alluvial cover in this area has been stripped off by 

Bluff Creek and its tributaries, leaving only 1 to 3 

feet of clayey soil along the flood plains of larger 

streams and on some of the broader divides. Older 

alluvial deposits are found only on the divide between the North Canadian and Cimarron Rivers. The 

Hennessey shale is more than 250 feet thick and is 

underlain by the Garber sandstone and Wellington 

formations, whose sandstone beds yield soft artesian

* Ground-water Geologist, Water Resources Division, U. S. 

Geological Survey.

water to deep wells. The Hennessey, Garber, and 

Wellington formations are of Permian age, and are 

the only formations which appear related to the 

ground-water hydrology of the Lake Hefner area.

Soil and Recent Deposits

The regimen of Bluff Creek is in marked contrast 

to that of the North Canadian River, especially with 

regard to soil and Recent deposits. The Bluff Creek 

drainage has stripped most of the soil and alluvial 

mantle from the bed rock, but the more sluggish 

North Canadian drainage has added Recent floodplain deposits to older alluvial materials and soils.

If appreciable soil and alluvial deposits were 

present beneath and adjacent to the lake, their effect 

on the area hydrology might be considerable. Accordingly, the west branch of Bluff Creek, comparable 

in most respects to the east branch where the lake is 

impounded, was mapped in detail. Surface exposures 

of the Hennessey shale were delineated from the 

areas covered by soil. Some soil was found to be 

generally present on the broader divides and in the 

vegetated parts of the larger stream courses. This soil, 

however, is thin and clayey. Bedrock is exposed almost everywhere on the steeper slopes, along road 

cuts, and in the stream bottoms. The total area of 

outcrop is about equal to that covered by soil.

Thirty of the forty test holes drilled around the 

margin of the take entered bedrock immediately; no 

appreciable amount of soil cover was found. The 

thickness of soil penetrated, if present, was commonly only about 1 foot, and the maximum was 

about 3 feet.

Logs of wells in the area show that none are 

developed in alluvial material. Practically all shallow 

wells (less than 100 feet) draw water from red shale. 

This obtained also for the area later covered by the 

lake, as indicated by WPA records of seven shallow 

wells in that area.

Pleistocene Terrace Deposits

Alluvium, consisting of gray-to-buff-colored sand, 

gravel, and clay, underlies the broad terrace-like divide separating the Bluff Creek and North Canadian 

River drainage basin. Jacobsen and Reed (1949) have 

described these materials as Pleistocene terrace deposits. Some beds consist of well-sorted sand and 

others of unsorted clay, sand, and gravel. Part of the

10
The regional dip of the beds in the Lake Hefner 

area is west at about 40 feet per mile. Bruntoncompass measurements of outcrops show strikes ranging between N. and N. 20° E., and dips ranging 

between 1 ° and 7° W. A few minor folds and dips 

to the east and south were noted. Much of this surface 

structure has probably resulted from slumping and 

soil creep.

Composite Section of Hennessey Shale in the 

Vicinity of Lake Hefner

Description of Material Thickness in Feet

terrace area is covered by windblown sand. Much 

precipitation is absorbed by these terrace materials, 

and sand beds near the base constitute an important 

aquifer in the Bethany and Warr-Acres areas.

The canal connecting the North Canadian River 

with Lake Hefner is cut through these terrace materials for 3 miles of its 5-mile length and acts as a 

drainage ditch when it is not being used to transport 

surface water. Drainage water entering the lake 

through the canal comes from this source.

Much of the low-flow surface water that enters 

the lake through the tributaries of Bluff Creek also 

originates in these terrace materials. All springs and

seeps examined in the Lake Hefner drainage area ___ ____ 

issued at the contact between these alluvial materials "

i ,. it. i i .1 ii . , Soil, shaly, reddish brown. ..................... 2

and the underlying shale. Thus all ground water from sha|e bright fed> fi$si|e ..................... 4

the terrace materials that enters the lake can be Sandstone, red, fine-grained; lenses out east and

measured in various channels. west of section ............................. 2

The terrace materials probably recharge the up- Shale, with interbedded siltstone................ 3

per part of the Hennessey shale also. Joints and other

openings in the shale are small, however, and they Shale, red, fissile............................. 2

accept recharge slowly. Where saturated terrace sand Shale and liltstone' thin beds interbedded 3

. . . . .1111 . . , Shale, red, fissile............................. 7

is in contact with the shale, a continuous supply of

water is available; direct precipitation on the shale _. , .

... . ,, _, , , , Shale, red, silty.............................. 2

would Ic. 0 3ly run off. The rate of this recharge proc- siltstone, red, shaly............................ 5

ess was not determined, but it is probably slow; the shale, silty .................................. 3

recharge would affect only the shale that underlies Siltstone with intraformational conglomerate, shaly

Or is Very near the alluvial cover. partings (seeps issue from this member)*...... V4

Siltstone ..................................... 2

Hennessey Shale Shale, silty .................................. 3

The Hennessey shale controls seepage conditions _, ...

, . riii > , £>ay, red, shaly............................... 5

in the vicinity of the lake. The formation consists of Sna |e Ted si |t 8

bright red interbedded silty shale and siltstone. Most Shale and siltstone, thin beds interbedded........ 13

of the more massive beds are siltstones, but a few

beds of fine-grained sandstone also occur. The beds Shale, red, fissile.............................. 9

were revealed to be lenticular by tracing some of Shale with thin beds of siltstone................. 5

them along their strike. The lithology of the beds Shale, red, fissile.............................. 15

underlying the lake can be approximated from the

........ , ... Shale, red, fissile ............................. 4

composite section in the adjoining column compiled siltstone red 73

from descriptions of exposures in the lake area. Shale, red, fissile.............................. 3

Breaks in the section are indicated by spacing; the Shale, red, fissile.............................. 6

composite nature of this section may involve un- Shale and siltstone, interbedded. ................ 5

avoidable minor duplications or omissions. Sandstone, fine-grained, cross-bedded............ 1

A .1 . A -i i r i r Siltstone and fine-grained sandstone, mostly thin- Another composite section compiled from logs of bedded 3 

the test holes is similar to the above except that the siltstone, shaly and with shale partings.......... 8

massive Siltstone and Sandstone beds at the base of Shale, fissile, mostly red but containing 1- to 3-inch

the outcrop section were not penetrated by the drill. layer of greenish-gray shale (seeps)t.......... 5

The weathered Hennessey is a stiff, plastic clay, Sandstone, fine-grained ....................... i

and wetted cuttings from test holes showed no perii TL L i j L xi_ » * Seeps issue from shaly partings and from vertical joints meable openings. The shale exposed above the water ... '

LI i i i '" Sllt$tonetable, however, had numerous joints some of which t Seeps i$soe from bedding.plane joint$/ or sheor$/ in

were probably produced by shrinkage on drying. shale above underlying sandstone.

1 1
The Hennessey Shale as an Aquifer

General indications suggest that the Hennessey 

shale is extremely impermeable. However, in the Lake 

Hefner area the Hennessey is an aquifer and furnishes small quantities of water to wells. This is substantiated by the WPA well records, oral reports of 

drillers and farmers in the area, and the inventory of 

wells and test holes pumped during this investigation.

Wells north of the terrace materials can be classified in three groups. About 80 to 90 per cent are 

shallow and derive their water from red shale. The 

water from these is generally satisfactory in quality 

but in many places inadequate in quantity for a 

windmill. A few wells have been drilled to depths of 

100 to 300 feet; these derive their water from sandstone or shale, or both. The water generally is highly 

mineralized and is used only for watering stock. The 

third type of well is 400 to 800 feet in depth and 

derives water from the Garber sandstone and Wellington formation. We are only concerned with the 

first group of wells.

One of the most interesting features of the shallow wells is that the water in them comes from shale 

and not from sandstone beds in the shale. Most of 

these wells supply small quantities required for household use, but not every drilling is successful. The water 

generally comes from depths of 10 to 20 feet, although such wells are usually drilled 40 to 70 feet 

deep to provide storage. If any particular permeable 

bed were the aquifer for these wells, one would expect the wells to become deeper westward because 

of the regional dip. Such is not the case. Also, in wells 

on higher ridges, water comes from shallower depths 

than would be expected if the aquifer were a nearly 

horizontal bed.

In all the forty test holes, water was encountered 

at depths of 2 to 25 feet. When pumped, the open 

test holes yielded a fraction of a gallon to 7 gallons 

a minute, being replenished from the surrounding 

shale. In none of the test holes was water obtained 

from siltstone or sandstone.

This evidence supports the conclusion that the 

near-surface part of the Hennessey shale in the Lake 

Hefner area is an aquifer of low yield. The waterbearing zone is generally within 10 to 20 feet of the 

surface and is not restricted to any particular beds, 

although the shale is more likely to be water-bearing 

than is the siltstone or sandstone. The interstices are 

joints and other fractures (possibly solution cavities, 

also) which do not persist at depth. It seems likely

that the openings may be confined to the zone that 

has been above the water table during some part of 

Recent geologic time.

The Garber Sandstone and 

Wellington Formation

Sandstones of the Garber sandstone and Wellington formation constitute the most important aquifers in the Oklahoma City area (Jacobsen and 

Reed, 1949).

It is evident that the artesian heads in the wells 

in these aquifers are not sufficient to force water 

through the shales into Lake Hefner. According to 

Jacobsen and Reed (1949), the original static head 

probably was about 100 feet below the surface. 

Sufficient head probably never was present to make 

the formations a source of the shallow ground water.

GROUND-WATER HYDROLOGY

Adequacy of the Present Surface 

Measurements of Outward Seepage 

Through the Dam

The dam that contains Lake Hefner is of rolledearth fill and is not completely impermeable. Drains 

have been provided around most of the embankment to channel and remove the outward seepage 

through the dam. Points at which measurements are 

presently being made on the drainage ditches are 

marked AA to FF on the piezometric map (fig. 8).

On the west side of the lake some seepage 

through the dam probably reaches the west branch 

of Bluff Creek without being caught in the measured 

drains. On the other hand, a part of the seepage 

measured at station FF comes from the east and is 

unrelated to seepage through the dam and deep 

seepage from the lake. However, the amount of such 

seepage from the east appears to be small.

All the measured drains have rich growths of 

water-consuming plants along their reaches, and 

evapotranspiration losses during summer months probably are very high. Therefore, only the early spring, 

winter, and late fall measurements are likely to be 

reliable. These were made and are reported in the 

section entitled "Water-Budget Control."

The Piezometric Map

Elevation points on the water surface as determined in 68 shallow wells and test holes around 

the lake are the basis for the piezometric map (fig. 8). 

When the data were being assembled, it was not

12
MEASURED POINTS ON THE WATER 

SURFACE - NUMBERS 1 TO 43 ARE 

TEST HOLES, 50 TO 73 ARE SHALLOW WELLS

WELLS IN THE ARTESIAN AQUIFERS 

OF THE GARBER SANDSTONE AND 

WELLINGTON FORMATION

CONTOUR LINE ON THE WATER 

TABLE

SURFACE-WATER DRAINAGE DIVIDE

APPROXIMATE NORTH MARGIN OF 

PLEISTOCENE TERRACE DEPOSITS

® SEEPAGE MEASUREMENT STATIONS

Figure 8. Piezometric map of Lake Hefner area showing watertable contours and other hydrologic features.

13
known that a simple water-table surface was present 

in the shale. On the contrary, a number of confined 

bodies of water, each with its own particular piezometric surface, was the condition to be expected in 

gently dipping, relatively impermeable beds. However, if the aquifer is a series of interconnected joints 

and other fractures in the upper part of the beds, 

then a water table typical of fracture openings might 

be expected.

Most elevations on the shallow-water surface, 

when plotted on the map, are compatible with watertable conditions. The highest water-surface elevations 

correspond with the points that are highest topographically and the water surface appears to be" a 

subdued replica of the land surface. Only locally 

were artesian conditions or perched water-table conditions encountered. In test hole 21, water was encountered in shale below siltstone at a depth of 18 

feet. When the hole was bailed below that depth, 

water could be heard spouting into it under considerable pressure. The water level in this hole is 6.44 

feet below the land surface and at an elevation at 

least 10 feet higher than the water-table surface at 

that point. Thus small local artesian aquifers do 

occur in the shale. In the three hand-auger holes 

that were put down to the water-table, a thin perched 

water body was penetrated above several inches 

of dry shale before the main water body was encountered.

If the piezometric map is considered as a watertable map in fractured material, several characteristics are noted that appear to have resulted from the 

impounding of the lake. From the intake canal and 

eastward and northward around the south and east 

sides of the lake to the spillway the lake is effluent, 

the movement of the shallow ground water being 

toward the lake. The lake probably has modified the 

water table in this area very little, except possibly 

to decrease the gradient slightly. The gradient for 

about 2 miles at the south end of the lake is approximately 40 to 50 feet per mile. Along the east shore 

of the lake the gradient is much gentler and the 

ground-water divide is within half a mile of the lake.

North of the intake canal on the west and north 

sides of the lake to the spillway, the lake is losing 

water by seepage. The dike that contains the lake 

on the west side was built near the crest of the original drainage divide between the east and west 

branches of Bluff Creek. The water table has been 

raised and a steep gradient away from the lake has 

been created by recharge from the lake. The gradient 

westward and northwestward from the lake in this

area is approximately 80 to 100 feet per mile. A 

trough in the water table is created by the drain at 

the northwest end of the dam, and a part of the 

ground water flowing northwestward along the ridge 

finds its way back into this drain. These relations are 

shown by section A-A' in figure 9.

At the northeast end of the dam a major trench 

sloping westward is created by the drains at the base 

of the dam. Beyond the trench, water levels rise rapidly again. Whether they are higher than they were 

before the impounding of the lake is not known. Relations in this area are shown on section B-B' (fig. 9). 

Too few points of control were obtained to permit 

completing the map at the northwest end of the lake.

Permeability of the Hennessey Shale

Examination of the Hennessey shale in outcrops 

and in well cuttings reveals a clayey, silty fissile red 

shale of apparently very low permeability. On the 

basis of such an examination only, one would conclude that the material is an ideal confining bed. The 

fact remains, however, that in its upper pa'rt it is 

sufficiently permeable to furnish water to most of the 

wells in the area. The specific capacities of several 

of these wells are of the order of 1 to 2 gallons per 

minute per foot of drawdown. Bailing of the test holes 

showed that none would yield as much as 40 gallons 

a minute and some only a fraction of a gallon per 

minute. However, one test hole was pumped at the 

rate of 7 gallons per minute for 1 hour with only 

2 feet of drawdown. All these facts indicate that the 

upper part of the Hennessey shale is an aquifer of 

substantial importance, though of small capacity and 

low permeability.

Pumping tests were attempted on several test 

holes and farm wells. For test holes 1 and 26 and 

farm well 53, recovery curves were plotted from 

which transmissibilities were computed by the Theis 

nonequilibrium formula (Theis, 1935). Test hole 1, 

which is northwest of the lake and very close to the 

base of the dam and near the drain, gave a transmissibility of about 2500 gallons per day per foot. 

Test hole 26, which is at the southeast edge of the 

lake, showed a transmissibility of 125. These are 

thought to represent approximately the extremes of 

transmissibility of the shale. Farm well 53, southwest 

of the lake, showed a transmissibility of about 1600, 

which is thought to be considerably higher than average because the well has a reputation of being unusually "strong."

14
200 400 600 800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800 3000 3200 3400 3600 3800 4000

HUNDREDS OF FEET

Figure 9. Cross-sections along lines shown in Figure 8.

Cross-Sectional Areas

The piezometric map (fig. 8) shows clearly that 

the lake is receiving ground water south of the intake 

canal and around the south and east sides of the lake 

to the spillway, and is losing water to the ground 

north of the intake canal around the west and north 

sides of the lake to the spillway. The effluent and 

influent stretches are approximately equal, each being 

about 4 miles in length.

The proper thicknesses to use in computing crosssectional areas of effluent and influent seepage are 

difficult to determine. Evidence of several kinds 

already has been cited indicating that the chief 

movement of water in the Hennessey shale occurs 

near the surface, principally in joints that hardly 

could exist at depth. Where the water table is near 

the surface, as it is in much of the Lake Hefner area, 

most of the movement of water probably occurs 

within the upper 20 feet of saturated shale. Where 

the water table is deep, the thickness of the zone of 

active movement is likely to be even less.

Computation of Influent and Effluent Seepage

A large amount of detailed work probably would 

be required to make any reasonably accurate computation of the influent and effluent seepage at Lake 

Hefner. This is so chiefly because of the nature of 

the interstices that carry the water in the shale. In 

granular material a rather small number of samples 

may permit reasonably accurate determination of the 

average permeability, but in shale the average permeability is much more difficult to determine because 

the fracture openings may be extremely diverse in 

size and distribution. The following estimates are 

therefore to be considered as giving only the probable order of magnitude of the seepage losses from 

and gains to the lake.

The specific capacities of the better wells in the 

Hennessey shale are of the same order of magnitude 

(1 to 2 gallons per minute per foot of drawdown) as 

the specific capacities of the wells in the Garber sandstone, whose permeability has been determined to be 

40 gallons per day per square foot. Assuming, on the

15
basis of similar specific capacities of wells, that the 

Hennessey shale may have about the same permeability as the Garber sandstone, we have one of the 

three factors necessary for computing the seepage. 

The second factor, the water-table slope, as obtained 

from figure 8, is approximately 40 feet per mile for 

portions of the lake gaining water and 80 feet per 

mile for portions losing water. For the third factor, 

the cross-sectional area, we assume that most of the 

ground-water movement occurs in the upper 20 feet 

of saturated material. On the basis of these assumptions, it would appear that seepage to the lake might 

be of the order of 125,000 gallons per day, and 

that the seepage from the lake might be of the order 

of 250,000 gallons per day.

These figures are based on the assumptions that 

all the seepage moving toward the lake is intercepted 

by the lake and that none of the seepage moving 

out of the lake is intercepted by drains but continues 

on as "deep seepage." Neither of these assumptions 

is correct, and it is difficult to estimate what correction factor should be applied. Certainly some of the 

deep seepage continues beneath the lake. The fact 

that part of the deep seepage from the lake is intercepted by Bluff Creek and the drains at its north end 

is indicated by the large trenches they have made 

in the water table and by the visible seeps from the 

bed rock along their channels. Perhaps the figures 

given above should be reduced by half, or more, to 

take care of this factor.

The amount of seepage to the lake can be no 

greater than the amount of seepage originally contributed to Bluff Creek in the area now covered by 

the lake. Actually this seepage should be less, 

because the formation of the lake has decreased 

the gradient. Engineers and others who were acquainted with the area before and during the period 

of lake construction say that all the visible seeps and 

springs feeding this branch of Bluff Creek entered 

above the area subsequently covered by the lake, 

issuing chiefly at the base of the terrace sands.

From all the evidence available it therefore 

appears that the amount of unmeasured seepage 

into the lake is small, perhaps of the order of 50,000 

to 100,000 gallons per day (0.15 to 0.3 acre-foot per 

day), and that the unmeasured deep seepage escaping from the lake is at least twice as much, perhaps 

of the order of 100,000 to 200,000 gallons per day 

(0.3 to 0.6 acre-foot per day).*

CONCLUSIONS ON GEOLOGY 

AND HYDROLOGY

The Hennessey shale underlying Lake Hefner is 

considered to be relatively impermeable. The net 

deep seepage loss from the reservoir is estimated to 

be 0.2 acre-foot per day. There is insufficient evidence 

to warrant any attempt to determine seasonal 

changes, if any, in influent and effluent seepage.

Measurements of shallow seepage losses through 

the dam and dike appear to be adequate if made 

during late fall, winter, and early spring, in order 

to minimize evapotranspiration losses.

REFERENCES

Jacobsen, C. L, and Reed, E. W., "Ground-Water 

Supplies in the Oklahoma City Area, Oklahoma," 

Oklahoma Geological Survey Mineral Report, no. 

20, 1949.

Theis, C. V., "The Relation Between the Lowering of 

the Piezometric Surface and the Rate and Duration 

of Discharge of a Well Using Ground-Water Storage," 

American Geophysical Union, Transactions, vol .16, 

part 2, 1935, pp. 519-524.

* Subsequent to the field examination and preparation 

of this section of the report, sporadic measurements of water 

levels in the forty test holes were made during the period 

June 1950 to September 1951. Unfortunately, many of the 

holes caved in or were otherwise destroyed during that time. 

Ten measurements were made at each of the following test 

holes: 1, 2, 5, 7, and 8, along the west side of the lake; 12, 20, 

and 21, along the south side; 26, and 33, near the east shore; 

and 38, 39, and 40, on the north side of the lake.

The magnitude of water-level fluctuation in the various 

wells varied greatly, but a general seasonal trend was evident. 

Water levels were generally lowest during the period February

to April 1951, but rose rapidly thereafter in response to the 

heavy rains of May, and were at their highest in June 1951. 

Water levels at the end of the period were in general not 

more than 2 or 3 feet higher or lower than at the beginning 

of the period.

Since the seasonal trend in water levels in the area 

near the lake resulted from precipitation rather than changes 

in lake levels, it appeared reasonable to assume that water 

levels over the entire surrounding area followed a similar trend. 

There was insufficient evidence of any change in hydraulic 

gradients of sufficient magnitude to warrant recomputation 

of influent and effluent deep seepage.

16
The Water-Budget Control

by G. Earl Harbeck, Jr., and Frank W. Kennon*

INSTRUMENTATION AND METHODS

Since daily evaporation, as determined by the 

water budget, was the control for the entire project, 

it was imperative that every effort be made to measure each item of the budget to a precision consistent 

with its significance in the water budget and to evaluate the errors inherent in the measurements. Daily 

evaporation was computed using the familiar formula

£ = / O S, (1)

in which E r= evaporation,

S =. change in reservoir contents, 

/ = inflow, and 

O = outflow.

The acre-foot was used as the unit of volume 

in all water-budget computations. Daily evaporation 

was computed in acre-feet and then converted to 

equivalent depths in inches and centimeters.

Changes in Reservoir Contents

An accurate area-capacity curve was considered essential for the determination of daily changes 

in reservoir contents. An area-capacity curve had 

been prepared on the basis of a preconstruction 

transit-stadia survey by the consulting engineer employed by Oklahoma City to design the dam and 

reservoir. It is believed that some material was removed from the reservoir area and used in the construction of the dike, but the exact quantity was 

unknown. The amount of reservoir silting since the 

reservoir was constructed in 1944 was also unknown, 

although it was believed to be small. A resurvey of 

the reservoir was therefore considered necessary.

A plane-table survey was made of the shore 

areas between elevations 1192 and 1200 feet. A 

hydrographic survey of the lake was made, using a 

U. S. Navy plane-personnel boat equipped with a 

Bludworth NK-2 echo-sounder and outboard transceiver to obtain continuous bottom profiles along 

selected courses. A new area-capacity curve was developed on the basis of the resurvey. The curve 

based on the resurvey checks the original curve within 

0.4 per cent at full pool, indicating that the net effects 

of reservoir silting and other changes have been small.

* Hydraulic Engineers, Water Resources Division, U. S. 

Geological Survey.

Since it was planned to determine daily changes 

in the reservoir contents by multiplying the area of 

the lake by the change in stage, accurate stage 

measurements were needed. The principal reference 

gage for all stage measurements was installed at 

the intake tower (see map). It was set to read the 

same as the City Water Department gage, which is 

also fastened to the intake tower, and which is 

graduated from 1180 to 1200 feet.**

Changes in stage were recorded on a Stevens 

type A-35 continuous recorder with a 10:12 gageheight ratio and a 4.8-inches-per-day time scale, 

placed in a wooden shelter over an 18-inch watertight corrugated-steel pipe well. The intake opening 

was made as small as practicable to eliminate surge 

in the well. Seiches were not eliminated, however, 

although it was possible that their amplitudes were 

damped. Standard stage-measurement procedures, as 

described by Corbett (1943), were employed.

The south lake gage was located in the small 

boat harbor at boat pen 498 (see map). Prior to 

23 May 1950, the south lake gage was located at 

a site 1200 feet southwest, but this site was soon 

found to be unsuitable under certain wind conditions 

because of pile-up in the constricted boat harbor. 

The station equipment was the same as at the intake 

tower station.

The first two lake gages installed were placed 

at the north and south ends of Lake Hefner because 

of the overwhelming preponderance of northerly and 

southerly winds, but it was soon evident that additional gages were required to evaluate stage changes 

with sufficient accuracy.

Another lake gage was installed on 30 May 

1950, on the east shore of Lake Hefner at the west 

end of the Oklahoma City Yacht Club dock (see map). 

The station' equipment was like that at the intake 

tower and south gages except that the recorder gageheight scale was 1:6.

A fourth lake gage was installed on the southwest shore of the lake on 27 June 1950. Station 

equipment was similar to that at the other stations, 

except that a Stevens weekly type-F recorder with

** To convert to USC and GS datum, subtract 0.10 foot; 

based on levels from USC and GS traverse station No. 23-A-14 

TT, elevation 1203.44 feet, datum of 1929.

17
a 1:2 gage-height scale was used. From 27 June until 

10 July, the record was unusable because of faulty 

intake action. The intake was enlarged on 18 July, 

and the records obtained thereafter were good except during periods of strong northerly winds. The lake 

was shallow at this location and the bottom sloped 

so gently that considerable pile-up resulted from 

northerly winds. On 11 December 1950, the gage 

was moved to a new location on the west shore of 

the lake (see map), and no further trouble from 

pile-up occurred.

The intake tower gage was used as the reference gage. Daily comparisons of recorded stages 

during periods of light winds were made to detect 

possible changes in datum of the other gages, relative 

to the intake tower gage. Since the other three gages 

were used only to record daily changes in stage, 

it was not necessary that they be set to the same 

datum as the intake tower gage, but it was essential 

that any abrupt changes in datum be detected. For 

this purpose water-level comparisons were deemed 

much more reliable than precise leveling. The east 

and south gages proved to be quite stable. For the 

period 28 May 1950 to 31 August 1951, the net 

change in stage indicated by these two gages and 

the intake tower gage was as follows: intake tower 

gage, -fO.005 foot; south gage, 0.002 foot; east 

gage, 0.006 foot. The west gage was not nearly as 

stable, however, mainly owing to the fact that there 

was no permanent structure to which it could be 

affixed. Corrections for changes in datum because 

of settlement were made as indicated by comparison 

with the intake tower gage during relatively calm 

periods.

To evaluate the effects of errors in recorded 

stage changes (needed for computation of daily 

changes in reservoir contents) the range in indicated 

values of daily change in stage proved useful, for 

it was deemed impracticable to compute the standard 

error of each daily mean change by the usual statistical methods. The variance in daily stage changes 

indicated by the various water-level recorders is a 

measure of the error in this mean. The range in 

indicated stage change is an easily determined measure of the variance. Tippett's mean values of the ratio 

of the range to the standard deviation for various 

sample sizes have been tabulated (Snedecor, 1946). 

The resultant error in the computed change in reservoir contents, and therefore also in evaporation, is 

shown in figure 10.

It has been suggested that the variance resulting from the nonagreement of indicated stage changes

(SURFACE AREA = 2300 ACRES) 

n = NUMBER OF LAKE 

STAGE RECORDERS

ERROR IN COMPUTED EVAPORATION IN ACRE-FEET

Figure 10. Error in computed evaporation resulting from inconsistencies in measurements of change in stage of Lake Hefner.

is not entirely error variance, but also includes systematic variance. A better estimate of the error variance may be obtained from

(fc8 _ h4)2 + (fi4 - M2] (2)

in which S =. standard error of the mean, and 

h = indicated stage change.

The subscripts refer to the gages, which are numbered 

consecutively around the lake.

The standard error of the mean has been computed using equation (2) for 4 days selected at 

random. The standard error of the mean was also 

determined using the range as a measure of the 

standard deviation (see fig. 10), and the results were 

as follows:

Standard Error of the Mean in Acre-Feet

Date

28 June 1950

31 July 1950

22 October 1950

25 May 1950

From Equation (2)

21.2

12.0

30.1

2.5

From Figure 10

22.8

13.4

31.2

3.3

As expected, the standard error of the mean computed using equation (2) is consistently less than that 

obtained using figure 10, for the effect of systematic 

variance has been reduced if not eliminated.

The standard error of the mean was determined 

only for use in classifying daily figures of waterbudget evaporation as to their relative accuracy, as 

discussed in more detail below. The refinement made 

possible by using equation (2), instead of the much

18
simpler method based on the range, was not believed warranted in view of the additional computational work involved. Every effort was made to 

devise an accuracy classification scheme that was 

completely objective, although necessarily somewhat 

qualitative because the time that could be allotted 

justifiably to this analysis was limited. It should be 

noted that the error introduced by using the shortcut method is on the conservative side; some daily 

evaporation figures might actually deserve a higher 

accuracy rating than they were given.

Errors in computed evaporation resulting from errors in measuring changes in stage are not cumulative 

and are therefore independent of the length of 

period. The relative effect of stage errors therefore 

decreases as the length of the period increases.

A study was made of the relation between 

length of period and the magnitude of errors resulting 

from nonagreement of gages. In order to eliminate 

bias in their selection, periods were chosen to begin 

and end on days when the number of lake stage 

recorders in operation changed. For example, in 

1950 during the 15-day period 20 May to 3 June, 

only two recorders were operating. During the 24-day 

period 4 June to 27 June, three recorders were in 

operation, and during the 11-day period 28 June 

to 8 July, all four recorders were in operation. Then 

followed a 10-day period in which three recorders 

were in operation. Thus, in effect, the periods were 

chosen at random, and they varied in length from 

1 to 112 days. The total change in stage for each 

recorder was computed for each of the 32 periods, 

and the error in evaporation determined from figure 

10. It would have been possible to select periods for 

which the agreement among the various gages was 

considerably better, merely by having each period 

begin and end during a near-calm. The procedure 

adopted, however, eliminates any possible bias.

The standard error of the mean change in stage 

was computed for each of the 32 periods. The median 

value of the standard error was 0.0036 foot and, as 

expected, there was no discernible correlation with 

the length of period, since stage errors are not cumulative. Thus, for any length of period, the standard 

error of the indicated stage change is approximately 

0.004 foot, which is equivalent to an error of 9.6 

acre-feet in evaporation, assuming the area to be 

2300 acres.

The effect of thermal expansion of water in the 

reservoir on the computed figures of water-budget 

evaporation was also studied. In most computations 

of this nature the effect of thermal expansion is

ignored, possibly because of the difficulty in evaluating it correctly. It is generally inconsequential in shallow bodies of water.

If the datum to which stage measurements were 

referred remained constant, the error in evaporation 

computed by the water-budget method would depend 

only on the thermal expansion of the water in the 

lake. At Lake Hefner the basic reference point to 

which all stages were referred was a point on the 

walkway surrounding the intake tower a few feet 

above the water. Thus the problem resolved itself to 

the determination of the difference in thermal expansion between the water in the lake and the reinforced 

concrete intake tower. If, for a given temperature 

change, the expansion of the water was exactly the 

same as the expansion of the tower, there would be 

no errors in the water-budget evaporation figures.

The intake tower is a reinforced concrete structure, circular in cross-section. The bottom of its footing 

rests on the Hennessey shale at an elevation of 1142 

feet, according to the construction drawing of the 

City Water Department. The top surface of the walkway is at an elevation of 1199 feet. Thus, the effective height of the tower for computing thermal 

expansion is 57 feet. The commonly accepted coefficient of thermal expansion for reinforced concrete is 

0.0000117 per °C. The thermal expansion of the 

tower is, therefore, 0.000667 foot per °C temperature 

change. The thermal expansion of the water in the 

lake can be most easily determined from tables showing the volume of water at a given temperature 

relative to the volume at 0°C (Chemical Rubber Publishing Co., 1947).

A sample computation for the month of November 1950 is included to illustrate the relative magnitude of the various items. Based on thermal surveys 

of the lake, the average water temperature at the 

beginning of the month was 17.9 °C and at the end 

of the month 7.9 °C. The variation in the thermal 

expansion coefficient with temperature is not linear, 

but since Lake Hefner remained nearly isothermal at 

all seasons, little error was introduced by using an 

average temperature for the entire lake. Average 

content during the month was 62,232 acre-feet, and 

the average area 2298 acres. From the previously 

mentioned tables, the ratio of the volume at 17.9 °C 

to the volume at 0 °C is 1.001186 and the ratio of 

the volume at 7.9 °C to the volume at 0 °C is 

0.999989. Therefore, the ratio of the volume at 

17.9 °C to the volume at 7.9° is 1.001197. The contraction in volume during the month is (1.001197 1) 

X 62,232 or 74.5 acre feet, which is equivalent to

19
a depth of 0.389 inch. During the same period, the 

tower contracted 0.000667 X 12 X (17.9 7.9) or 

0.080 inch. Thus, during November, the lake contracted 0.389 inch and the tower only 0.080 inch, 

a difference of 0.309 inch. Indicated water-budget 

evaporation during November was 6.298 inches, but 

since part of the apparent reduction in volume resulted from thermal contraction rather than evaporation, the indicated evaporation is too large, and the 

corrected volume is 5.989 inches. In this instance, the 

correction was approximately 5 per cent, but it 

would be proportionately greater for a spring month 

when evaporation is less.

Unfortunately, daily evaporation figures could 

not be readily adjusted because day-to-day variations 

in lake temperatures cannot be evaluated with sufficient accuracy. Thermal surveys of the lake were 

not made each day, so that no direct measurements 

of daily temperature changes are available. Average 

daily lake temperatures can be estimated on the basis 

of observed water-surface temperatures, but the standard error of estimate of approximately 0.6°C is 

unduly large in comparison with the usual daily 

change in lake temperature.

If the average lake temperatures on each of 

two successive days can be estimated with a standard 

error of 0.6°C, the standard error of the difference 

between the two estimated temperatures is

V(0.6)2 + (0.6)2 or 0.85°C.

If the lake temperature on the first day is assumed 

to be 20°C and the error in the indicated 24-hour 

temperature change is 0.85°C, the resultant error in 

the adjustment for thermal expansion would then be 

0.13cm, which is 32 per cent of the average daily 

evaporation during the period of observations. Under 

average conditions a week is the minimum period for 

which the error in the adjustment for thermal expansion can be assumed to be less than 5 per cent, if 

average lake temperatures are estimated from watersurface temperatures.

Although it was deemed impracticable to adjust 

daily evaporation figures for the effect of thermal 

expansion, evaporation figures for some longer periods could be corrected. Since the only requirement 

was that the average lake temperature at the beginning and end of the period be measured or estimated 

with sufficient accuracy, evaporation for periods determined by thermal surveys could be adjusted for 

thermal expansion, as could certain other periods 

for which it was believed that temperatures were 

known with sufficient accuracy.

Rainfall on the Lake Surface

Rainfall on the lake surface was frequently a 

major item in the Lake Hefner water budget. Rainfall 

of 0.05 inch or greater was recorded on 86 days 

during the 16-month period of observation. Nearly 

60 per cent of the water withdrawn from Lake Hefner 

for municipal use during the entire period of observation was replaced by precipitation falling directly on 

the lake surface.

Tipping-bucket recording rain gages were located at each of the meteorological stations, including the barge, and 18 Weather Bureau nonrecording 

gages were located on the periphery of the lake 

(see map).

The recording gages operated without loss of 

record except for the barge station. It was apparent 

that the gage on the barge caught spray a few times. 

Also, the motion of the barge was sufficient to cause 

the rain-gage bucket to tip occasionally under certain 

wave conditions. Records for the nonrecording gages 

were complete except for two periods of a day or 

two when a gage was stolen.

Mean rainfall on the lake was taken as the 

simple average of the catches of the 22 gages. For 

a storm that extended over more than one calendar 

day, the amounts credited to each day were computed on the basis of the recording gage records 

by simple proportion.

For five selected storms a comparison was made 

of computed values of mean precipitation on the lake 

as given by (1) the simple average, (2) the Thiessen 

method, and (3) an isohyetal map. The results are 

as follows:

Date of

Storm

5 May 1950

4 July 1950

25. July 1950

9 May 1951

27 May 1951

Mean

Average

0.073

0.750

1.077

1.511

1.849

Rainfall in

Thiessen

Method

0.070

0.785

1.013

1.497

1.864

Inches

Isohyetal

Map

0.066

0.795

1.012

1.486

1.895

It will be noted that results obtained using the 

Thiessen and isohyetal-map methods agree with each 

other better than with the arithmetic mean. This may 

in part result from the much greater weight given the 

record at the barge when computing Thiessen factors 

or preparing an isohyetal map. Since the other gages 

were more or less uniformly distributed around the 

periphery of the lake, the results would be practically 

the same for all methods if the records at the barge

20
were not used. Because the record at the barge was 

less reliable than the others, owing to the possibility 

of mechanically induced bucket tipping or spray, it 

appeared advisable not to weight it more heavily 

than the others. Moreover, the preparation of an 

isohyetal map for each storm would have been an 

arduous task and would have been open to the criticism that the procedure is somewhat subjective. Another advantage of using the arithmetic mean is that 

the standard error of the mean is easily determined. 

The simple average was used as the mean rainfall 

on the lake in the water-budget computations.

Of considerable interest, because of its tremendous area! variability, is the storm of 24 July 

1951, for which an isohyetal map was prepared 

(fig. 11). At the south station, where the recording 

gage showed the heaviest rainfall, precipitation began 

at 2132 and ended at 2244. The total catch was 

1.76 inches, of which 1.48 inches fell between 2200 

and 2230. At rain gage No. 11, 0.45 mile southwest 

of the south station, 1.99 inches fell. In contrast, the 

storm rainfall at gage No. 2, which is 2.40 miles 

northwest of the south station and directly across the 

lake from it, was only 0.02 inch. The average for the 

22 gages was 0.534 inch. The standard error of the 

computed mean rainfall was 0.125 inch, or 23 per 

cent of the mean.

To study the areal variability of storm rainfall, 

the standard error of the mean and the coefficient 

of variation were computed for each storm. The 

standard error of the mean was computed by dividing the standard deviation of the individual rainfall 

amounts by the square root of the number of gages; 

it is thereby assumed that the systematic variance is 

negligible in comparison with the error variance. The 

results, which are illustrated in figure 12, indicate 

that rainfall during large storms tends to be more 

uniformly distributed than during light showers, as 

might intuitively be expected. The coefficient of variation is in general 10 per cent or less for storm rainfalls 

of 0.1 to 1 inch, and about 5 per cent or less for 

larger storms. The standard error of the mean rainfall 

for each storm, expressed in acre-feet, was used in 

determining the relative accuracy of computed waterbudget evaporation.

Surface Inflow

Surface inflow to Lake Hefner results from precipitation on the natural drainage area above the 

lake and from diversion from the North Canadian 

River. Natural inflow usually is a small item in the

water budget because of the relatively small size 

of the drainage basin.

The measurement of natural inflow to Lake 

Hefner was simplified to a considerable degree by 

the "dike" effect of the road encircling the lake, 

which served to confine runoff to relatively few channels where it could be measured. The total drainage 

area tributary to Lake Hefner above the encircling 

road is 2484 acres.

The principal stream, Bluff Creek, and its tributaries have cut back into a high terrace of the North 

Canadian River about a mile south of Lake Hefner. 

This terrace was once a flood plain of the North 

Canadian River, and is now being dissected by tributaries of both the North Canadian and Cimarron 

Rivers.

The deposits underlying the terrace contain relatively large quantities of water. Bluff Creek is cut 

into these water-bearing deposits and is therefore 

seldom dry, in contrast to other small streams in the 

area whose drainage basins are underlain by the 

Hennessey shale. Diurnal fluctuation in flow of Bluff 

Creek is caused by operation of a gravel plant upstream. A gaging station was established on 8 March 

1950 at the bridge on the reservoir road (location 

C, see map).

A Stevens A-35 continuous recorder was housed 

in a wooden shelter over a 24-inch-diameter corrugated-steel pipe well fastened to the upstream left 

wing wall of the bridge (fig. 13). The zero of the 

reference gage was at elevation 1199.86 feet (based 

on an elevation of 1203.44 for U.S.C. & G.S. traverse 

station No. 23-A-14TT, datum of 1929). The control 

for all stages was a sharp-edged steel 90° V-notch 

weir set in the center of a shallow concrete V weir 

6 inches thick and 30 feet long at the upstream side 

of the bridge.

Records of stream flow were obtained using a 

standard stream gaging procedure described by Corbett (1943). The rating curve (fig. 14), was well defined between 0.2 and 40 cfs by 13 discharge measurements made by wading near the gage. Below 

a stage of 1 foot, the rating was based on a standard 

weir rating (U.S. Bureau of Reclamation, 1946). It 

was extended to a stage of 2.6 feet on the basis of 

known hydraulic characteristics of similar controls. 

The momentary peak discharge was 931 cfs on 27 

May 1951. The peak rate of flow was maintained for 

only a very short period of time, as is shown by the 

fact that average discharge on that day was only 

39.6 cfs. However, total inflow to Lake Hefner on this 

and certain other days was so great as to preclude

30Z5ZO O - 54 - 3 21
 7V//'//

22

Figure 11. Isohyetol map for the storm of 24 July 1951 at 

Lake Hefner.
lOOr

0.001

Figure 12. Variability of storm rainfall.

MEAN STORM RAINFALL IN INCHES

Figure 13. Pipe well and recording-gage 

shelter; concrete control with V-notch weir 

above Lake Hefner (location C).

DISCHARGE IN CFS

Figure 14. Rating curve for gaging station on Bluff Creek above Lake Hefner.

23
the computation of evaporation with the requisite 

accuracy; therefore, evaporation was not computed 

for 9, 13, 18, 19 May 1950 and 27 May 1951.

Figure 15 shows the daily variation in inflow of 

Bluff Creek during the period of observation. Daily 

flows are given in acre-feet to permit comparison 

with other items in the water budget. For the 485 

days for which evaporation was computed, the average flow of Bluff Creek was 1.3 acre-feet; the maximum daily flow of Bluff Creek was 39.3 acre-feet, and 

95 per cent of the daily flows were less than 4.6 

acre-feet.

Natural inflow to Lake Hefner was also measured at eight other points (stations A, B, D, E, F, G, 

H, J, see map). Drainage areas above these sites 

range in size from 17 acres at point G to 427 acres 

at point F, and the total of all eight drainage areas 

is 1377 acres. Flow occurred at these gaging stations 

only as a result of storm rainfall. Farm ponds, constructed in upstream drains, have sufficient storage 

capacity to reduce substantially both peak and total 

flows. Except after severe storms, runoff generally 

occurred at only three or four of the stations because 

empty or partly empty farm ponds intercepted all 

surface runoff.

One or more weirs and a crest-stage gage (fig. 

16) were installed at each of the eight locations. A 

continuous record of stage was not obtained. The 

crest-gage used at Lake Hefner consisted of a piece of 

2-inch pipe mounted vertically just upstream from 

the weir or weirs. It was closed at the bottom and

water was allowed to enter through a series of 

14-inch holes spaced 3 inches apart vertically. A small 

amount of ground cork was placed in the pipe, and 

the maximum stage reached by the water was indicated by a ring of cork particles deposited on a 

measuring stick kept in the pipe, which was capped 

to prevent the stick from floating. After each storm, 

the stick was cleaned and the cork supply replenished 

if necessary.

The drainage areas at the various locations and 

the measuring equipment used, in addition to a creststage gage, were as follows:

Location A. Drainage area 288 acres. A 90° 

V-notch weir was installed in a natural drain just upstream from the highway. Stages were obtained by 

measuring down to the water surface from a fixed 

reference point.

Location B. Drainage area 280 acres. Weirs were 

installed in six 18-by-24-inch openings in the rectangular-concrete-box inlet structure at the upstream 

end of three 36-inch concrete pipe culverts. Three 

were 90° V-notch weirs and three were rectangular 

weirs. The crests of the rectangular weirs were 0.7 

foot higher than the point of zero flow of the V-notch 

weirs, so that flows of 27 cfs or less were carried 

entirely by the V-notch weirs.

Location D-J, incl. 90° V-notch weirs were concreted in place at the upstream ends of concrete pipe 

culverts. The drainage area and the number and 

diameter of pipe culverts at each location were 

as follows:

Q36

3 28

20

Figure 15. Hydrograph of daily flow of Bluff Creek above Lake Hefner.

24
Figure 16. Culvert, weirs, and crest-stage 

gage at location j.

Drainage Area Number of Diameter of Culverts 

Location in Acres Culverts in Inches 0.06

COMPUTED DAILY FLOW IN ACRE-FEET

0.12 0.58 0.05 0.05

D

E

F

G

H

J

35

74

427

17

96

160

1

1

3

1

2

4

36

36

60

36

36

36

Total flow at all eight gages, with an aggregate drainage area of 1377 acres, averaged 1.1 acre-feet per 

day, as compared with 1.3 acre-feet per day for 

Bluff Creek, which has a drainage area of 1037 

acres. The difference in yield is caused by the fact 

that some of the Bluff Creek tributaries tap the 

water-bearing deposits of the Bethany terrace, while 

the other small streams do not.

The general procedure was to visit each of the 

gages after storms to obtain one or more observations 

of stage on the recession limb of the hydrograph. 

These observations, together with the crest stage, recorded rainfall intensities, and the Bluff Creek discharge hydrograph, were used to estimate the hydrograph for each gage at which flow occurred, as illustrated in figure 17. The Bluff Creek stage graph 

shows pronounced diurnal fluctuations, caused by 

operation of a gravel plant upstream from the gage, 

but this does not preclude its use as an indicator of 

the general shape of a flood hydrograph. During the 

storm of 2 June, precipitation at the south station 

began at 2149 and ceased at 2342; the total amount 

recorded was 0.44 inch. Flow at gage H on 1 June 

was 0.06 acre-foot as a result of the storm on 28 May, 

and gage H was read on the afternoons of 1 and 2 

June to define the recession from the previous peak. 

The continuous record at Bluff Creek gage showed 

that Bluff Creek began to rise at about 2230 on 

2 June and peaked at 0130 on 3 June. Since the 

drainage area at gage H is 96 acres as compared

Figure 17. Estimated stage hydrograph at gage H resulting 

from storm of 2 June 1950.

with 1037 acres at the Bluff Creek gage, it is assumed that the peak at gage H occurred earlier 

than at Bluff Creek. The heaviest half-hour rainfall 

(0.19 inch) occurred from 2200 to 2230, which coincides with the beginning of the rise in Bluff Creek. 

It seems, therefore, that the rise started at gage H at 

approximately 2230 also, and reached its peak at 

midnight or shortly thereafter. The gage at H was 

visited on the morning of 3 June, the previous peak 

stage noted, and the stage at the time of the visit 

recorded. Stages were observed once daily, or 

oftener, thereafter until flow ceased. Flow on 3 

June at gage H was computed to be 0.58 acre-foot. 

It is believed that the storm discharge hydrograph at the various culvert gages was estimated 

with sufficient accuracy. Flow of Bluff Creek on 3 June

25
was 4.17 acre-feet, and flow at the other culvert 

gages ranged from 0 to 0.95 acre-foot. Total inflow 

was 13.18 acre-feet.

Surface runoff from a small part of the area 

lying east of the Lake Hefner road in the southern 

part of Section 36 (see map) enters Lake Hefner 

through the culverts at-location J, but flow from the 

remainder of Section 36 and from the south half of 

Section 25, that normally would drain into Lake Hefner, does not reach the lake. An intercepting drainage ditch along the east side of the road conducts 

surface runoff from this area in a northerly direction 

to the spillway, whence it eventually flows into Bluff 

Creek several miles below Lake Hefner.

Runoff from the area lying between Lake Hefner 

and the encircling road was not measured. This area 

yaries with reservoir stage, and ranges from 1224 

acres at a water surface elevation of 1191 feet to 

1012 acres at an elevation of 1195 feet. Runoff from 

the ungaged area was estimated by comparison with 

culvert flow or Bluff Creek flow, depending on the 

area! distribution of precipitation and the amount of 

water stored in ponds upstream from the culverts. 

Often a substantial part of the runoff from a particular storm was stored in these ponds and did not 

reach the culverts.

Although it is likely that estimates of ruVioff 

during storm periods may be somewhat in error, it 

must be remembered that the resulting errors in evaporation are small compared with errors in determining 

rainfall on the lake surface. For example, consider a 

storm for which the average rainfall on the lake was 

computed to be 2 inches, or about 380 acre-feet 

on the average area of the lake. From figure 12 it 

is estimated that the coefficient of variation for a 

storm rainfall of 2 inches is 4 per cent, from which 

it follows that the standard error of the mean rainfall on the lake is 15 acre-feet. Storm runoff is rarely 

greater than 25 per cent of the rainfall, and is usually 

much less. Using the 25-per-cent figure, the runoff 

from the ungaged area of 1090 acres would be 23 

acre-feet. An error of 20 per cent in the estimate 

of runoff from the ungaged area would result in an 

error of less than 5 acre-feet, only one-third of the 

error in mean rainfall. Moreover, it is considered that 

because of the wealth of hydrologic data on which 

to base estimates of runoff from the ungaged area, 

the error in these estimates is probably considerably 

less than 20 per cent.

Lake Hefner is replenished by infrequent diversions of water from the North Canadian River through 

the intake canal (see map). The rated capacity of the

canal is 1500 cfs. Records of canal inflow were obtained at a site approximately half a mile above Lake 

Hefner at the point of emergence of an inverted 

siphon under the Northwest Highway. A short concrete transitional section just below the siphon mouth 

is terminated by a low concrete weir. The top of the 

weir is 12 inches above the paved approach section 

and is practically level for its entire length. It has a 

rounded downstream face. In order to measure flows 

with the desired accuracy, a 4-foot rectangular steel 

weir was installed near the left end of the concrete 

weir; its crest was set at 0.35 foot below the crest 

of the concrete weir.

Provisions were made for accurate measurement 

of low flows because of ground-water seepage into 

the intake canal. The canal cuts through the waterbearing deposits of the Bethany terrace, and most 

of the ground and surface water flow it collects is 

drained to a tributary that enters Bluff Creek several 

miles below the dam. Some of the perennial groundwater flow does reach Lake Hefner through the intake 

canal, however, and is occasionally augmented by 

natural runoff from an area of approximately 200 

acres.

A Stevens type-F weekly water-stage recorder 

having a 1:2 gage-height ratio was housed in a small 

wooden shelter over a 12-inch corrugated-steel well 

fastened to the left wall of the concrete transitional 

section just upstream from the weir. The zero of the 

, gage was at elevation 1200.96 feet, datum of 1929. 

Discharge measurements were made by wading or 

boat at a section approximately 700 feet below 

the gage.

The rating curve for the intake canal (fig. 18) 

is based on 34 discharge measurements and is well 

defined between 0 and 1500 cfs. Below a stage of 

0.35 foot (the top of the rectangular weir) a standard 

weir rating was used. Current-meter and portable-Vnotch-weir measurements at low stages check the theoretical rating fairly well. The concrete weir was 

rather insensitive because of its length; at discharges 

of from 500 to 1000 cfs, a change of 0.01 foot in 

stage results in a change of approximately 2 per 

cent in discharge. Twenty-two measurements were 

made at stages above 0.35 foot. Their average deviation from the rating without regard to sign was 

2.7 per cent.

Water was diverted from the North Canadian

River to Lake Hefner on ten occasions during the

. 16-month observation period. During two of these

times natural inflow, including rainfall, was also of

considerable magnitude, and no attempt was made

26
100 

DISCHARGE IN CFS

Figure 18. Rating curve for gaging station on Lake Heftier canal.

to compute evaporation. Diversions ranged in size 

from approximately 800 to 5500 acre-feet in periods 

of 2 to 4 days.

Despite the fact that the records of canal flow 

were considered to be of excellent accuracy by 

usual stream-gaging standards, the diversions were 

of such magnitude that even small percentage errors, 

when expressed in acre-feet, were large compared 

with evaporation. Computed daily evaporation during 

periods when diversions were being made is therefore of doubtful accuracy.

Excluding periods during which diversions were 

being made, the average daily canal inflow was approximately 1 acre-foot. Except during periods of 

reservoir replenishment by diversion from the North 

Canadian River, surface inflow was generally a minor 

item in the water budget. A statistical study to determine the standard error of each daily figure was 

not necessary since wide experience with gagingsration rating curves and stage hydrographs indicates 

the standard error of these operations to be 5 per 

cent. For the 485 days for which evaporation was 

computed, average inflow (excluding canal diversion) 

was 4.4 acre-feet per day as compared with average 

filter plant withdrawals of 30.6 acre-feet per day.

Withdrawals For Municipal Use

Filter plant withdrawals were measured by a 

Venturi meter in the 48-inch raw water line leading 

from the intake tower to the filter plant. Although the 

manufacturer's calibration of the Venturi meter was 

probably sufficiently accurate for the needs of the 

City Water Department, filter plant withdrawals were 

known to be the largest single outflow item in the 

water budget, and precise calibration of the Venturi 

meter was therefore deemed essential. The traditional 

formula for Venturi meter flow (King, 1939) is

Q = (3) l-(d,/dl) 4

where Q = discharge in cfs,

C = Venturi meter coefficient,

g = acceleration of gravity in feet per

second per second,

d! diameter of Venturi meter entrance in feet, 

d2 = diameter of Venturi meter throat

in feet,

hi head of water at Venturi meter entrance in feet of water, and 

h2 = head of water at Venturi meter

throat in feet of water.

The salt-velocity method of calibration was used. 

The major items of equipment were a brine pressure 

tank, a pop valve, two sets of electrodes, an oscillograph, and a water-tube manometer. Brine was injected through the quick-acting pop valve into the 

raw water line at an unbalanced pressure of approximately 80 pounds per square inch. The pop 

valve and each set of electrodes were connected into 

separate oscillograph circuits so as to produce a deflection of the galvanometer when the brine decreased 

the resistance in the circuit. Each set of electrodes was 

connected to form one leg of a bridge in its respective circuit. This arrangement made the oscillograph 

galvanometers very sensitive to any unbalance of the 

bridge, caused by brine passing the electrodes, thus 

permitting the use of low-potential circuits, and also 

of small quantities of brine, thus minimizing density 

currents in the raw water line. The electrodes were 

placed in the 48-inch raw water line at distances of 

100.08 and 211.10 feet downstream from the pop 

valve. The water-tube manometer was installed with 

one tube connected to a manifold at the Venturi 

meter throat and the other tube connected to a mani27
fold at the upstream end of the meter to measure 

the differential head of the meter. The water-tube 

manometer remained in place during the entire period 

of observation at Lake Hefner and was the basic reference gage for the Venturi meter. Dimensions of the 

pipe and Venturi meter were measured to the nearest 0.001 foot.

Brine was injected into the 48-inch line by opening and closing a hand-operated quick-acting gate 

valve as rapidly as possible, which allowed approximately !/2 gallon of brine to flow from the pressure 

tank through the pop valve. The oscillograph was 

started just before the brine was injected and remained in operation until the salt cloud had passed 

the second set of electrodes. The differential head at 

the Venturi meter was observed at 1-minute intervals 

during each test run. Readings of the totalizing Venturi meter recorder in the filter plant were made at 

frequent intervals during each run. Water temperatures were also recorded.

Twelve runs were made for rates of flow ranging 

from approximately 14 to 70 cfs. .Values of C, the 

Venturi meter coefficient, were plotted against the 

differential head (hi h2 ) as shown in figure 19. 

A value of 0.988 was chosen as the upper limit on 

the basis of LeDoux calibration tests of many other 

Venturi meters of different sizes (King, 1939). Fewer 

observations were made at the lower end of the 

curve because of the infrequency of low flows. Results for two runs were discarded because the magnitude of the computed values of C were unreasonable. 

Data from another run were discarded because two 

injections of brine were made, and the salt clouds 

overlapped at the second set of electrodes.

Using values of C taken from figure 19, the 

curve shown in figure 20 was computed. The results 

of the individual runs are also shown for comparison. 

The maximum deviation of any run is 0.5 per cent.

The Venturi meter calibration curve is well defined throughout the range of 14 to 65 acre-feet per 

day. During the period of observation, 95 per cent

of daily withdrawals were in the range 14.1 to 48.4 

acre-feet per day. For the 485 days for which evaporation was computed, the average filter plant withdrawal was 30.6 acre-feet.

The minimum was 4.0 and the maximum 63.2 

acre-feet per day. It should be noted, however, that 

even though the percentage error at extremely low 

flow may be high, the absolute error is small. Although 

the daily withdrawal rate was not generally constant, 

sustained low rates of flow were uncommon, and 

small daily flows generally occurred when the filter 

plant was shut down completely for part of a day.

Since it appeared impracticable to obtain a 

continuous record of differential head as measured by 

the sensitive water-tube manometer at the Venturi 

meter, it was decided to rely as much as possible 

on the record of raw water flow as indicated by the 

Bristol recorder in the filter plant, applying such 

adjustments as might be indicated by simultaneous 

readings of the water-tube manometer and the Bristol 

recorder. Generally the procedure was to record the 

total flow shown by the Bristol meter totalizator over 

a period of 10 minutes to an hour or more, the length 

of the period depending on the rate of flow. The 

water-tube manometer was read frequently during 

the period and the flow, as determined from figure 

20, was compared with that indicated by the Bristol 

meter to determine the adjustment to be applied to 

the Bristol meter readings.

Unfortunately, the relation between flow as indicated by the Bristol meter and flow as indicated 

by the water-tube manometer did not remain constant over the entire period of operation. The Bristol 

meter was adjusted frequently by the City Water Department, and each time it was adjusted the relation 

changed. As far as is known, the Bristol meter was 

not adjusted during the period 25 April to 2 September 1950, but it was adjusted '21 times during 

the period 2 September 1950 to 31 August 1951. It 

was necessary, therefore, to make frequent comparisons between the flow as indicated by the Bristol meter

Figure 19. Relation between Venturi 

meter coefficient and differential head.

.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 

DIFFERENTIAL HEAD AT VENTURI METER IN FEET OF WATER

2.6 2.8 3.0

28
3.0

2.0

1.0

a

3

10.5

0.2

0.1

10 20 30 40 50 60 70 

VENTURI METER FLOW IN ACRE-FEET PER DAY

Figure 20. Venturi meter calibration curve.

Figure 21. Effect of evapotranspiration losses on measured 

shallow seepage from Lake Hefner.

and the flow as indicated by the water-tube manometer. Altogether, 680 such comparisons were made. 

A correction curve was prepared for each period between Bristol meter adjustments. During the period 

June to August 1951, a test was made to verify the 

accuracy of the computed filter plant withdrawals. 

During most of this period the water-tube manometer 

was read each time the rate of flow was changed, 

and daily flow was computed from these readings. 

Daily flow was also computed using the correction 

curves previously described and the results were compared. The standard error of the difference in daily 

flow computed by the two methods during the three 

month period was 2 per cent of the daily mean flow. 

The difference between monthly flows computed by 

the two methods was less than 0.5 per cent.

As a further check of the accuracy of computed 

figures of filter plant withdrawals, a study was made 

of the amounts of water consumed in the plant and 

pumped to the city mains. The average daily flow 

computed from pumping records, with allowance for 

water consumed in the plant, averaged 1.6 per cent 

greater than that computed from the raw water Venturi meter records. It is believed that the major part 

of this difference results from error in estimating the 

quantities of water consumed in the filter plant. Although the water used to wash filters is metered, that 

used to blow sludge from the flocculators is not. 

Records of daily withdrawals computed from pumping records were used only during periods when records from the raw water meter were lacking or 

questionable, as follows: 7-9, 18, 19 June, 19-29 September, 22-25 November, 12-14 January, and 14-20 

February.

For the purpose of classifying the accuracy of 

the daily water-budget figures, an estimate of the 

standard error of all figures of filter plant withdrawals was required. A detailed study made during 

the months of June, July, and August 1951 indicated 

that the standard error of computed withdrawals, 

based on the filter plant record, is 2 per cent, but 

this assumes no error in the relation between watertube manometer readings and actual flow. This error 

is probably much less than 1 per cent, as indicated 

by the range in Venturi meter coefficients shown in 

figure 19. Also, records for a total of 30 days were 

based on pumping records rather than Venturi meter 

records, and their standard error is probably slightly 

greater than 2 per cent. Assuming that the errors 

follow the normal law, it is estimated that the standard error in computed daily figures of filter plant 

withdrawals is 3 per cent.

29
During the spring and summer of 1950, a golf 

course was established on the south shore of Lake 

Hefner, just west of the boat harbor. The greens and 

fairways were irrigated with water from the lake. 

Withdrawals were measured by a totalizing Sparling 

meter. During the entire period of observation the 

amount of water withdrawn from the lake for irrigation rarely exceeded 0.1 acre-foot per day and never 

exceeded 0.2 acre-foot per day, but was, nevertheless, taken into account in computing daily evaporation figures.

Shallow Seepage

Seepage into or out of Lake Hefner may be 

c'assified as shallow and deep seepage; these will 

be discussed separately. Shallow seepage includes 

inseepage from the thin soil mantle overlying the 

Hennessey shale on the south and east sides of the 

lake and outseepage through the dam and dike on 

the west and north sides of the lake, most of which 

is collected and drained to Bluff Creek below the 

dam. Deep seepage, on the other hand, is considered to be the flow in the near-surface part of the 

Hennessey shale, an aquifer of low transmissibility. 

The geologic examination of the Lake Hefner area 

indicated a net deep-seepage loss (see preceding 

section) of 0.2 acre-foot per day, and this figure was 

used in computing daily evaporation for the entire period.

Shallow seepage into Lake Hefner occurs along 

the east and south side shores where the terrain 

slopes gently toward the lake. The reddish-brown, 

shaly soil is quite thin, averaging perhaps 1 foot in 

thickness. The maximum thickness of soil penetrated 

in any of the holes drilled in connection with the 

geologic investigation was 3 feet, and in most instances there was no appreciable soil cover. The 

movement of water through the thin or nonexistent 

soil cover into Lake Hefner was, therefore, considered 

to be negligible.

The possibility of return flow from golf-course 

irrigation during the spring and summer of 1950 was 

considered. The quantity of water applied was generally less than 0.1 acre-foot per day and never 

exceeded 0.2 acre-foot per day. The amounts applied 

were absorbed by the soil and were less than the 

evaporative and transpirative requirements of the 

vegetative cover. Hence, it was considered improbable that any return flow from irrigation reached Lake 

Hefner at any time.

The earth dam and dike on the north and west 

sides of Lake Hefner is of rolled-earth fill with a clay

core, and is not impermeable. A seepage-collection 

system was provided in the main part of the dam and 

drainage ditches were constructed to carry the collected seepage away from the dam. Seepage measurements were made during dry periods when there 

was no natural runoff, as evidenced by complete 

cessation of flow in nearby water courses known to 

be unaffected by seepage from Lake Hefner. Most 

measurements were made with a portable aluminum 

90° V-notch weir, but a few were made with a 

Price pygmy current meter. The variation in lake stage 

throughout the project was only a few feet, so the 

seepage head remained practically constant.

At the time of the detailed geologic and hydrologic examination of Lake Hefner, made shortly 

after observations were begun, it was suggested that 

the computation of seepage inflow be based on measurements made during periods when evapotranspiration losses were at a minimum. Measurements were 

made throughout the period of operations at locations AA, BB, CC, DD, EE, and FF (see map). Seepage 

measurements during the winter were made on days 

of above-freezing temperature to eliminate the possibility of flow being arrested by ice formation. The 

effect of evapotranspiration losses in the seepage 

areas and in the collection channels between the 

dam and points of measurement is apparent (fig. 21). 

Evapotranspiration losses in summer were approximately 0.6 acre-foot per day. The size of the area 

covered by vegetation supplied from seepage was 

not determined.

At AA, BB, and CC, measurements were made of 

the collected flow from a number of small seeps that 

arise in the area between the dike and the northsouth road just west of Lake Hefner. Most of the 

seepage flow at DD comes from a marshy area just 

west of the road in Section 28.

It was recognized that some seepage from Lake 

Hefner may reach the west branch of Bluff Creek 

without first appearing in any of the surface drains. 

On 28 January 1950, current-meter measurements 

were made to determine the increase in flow of the 

west branch of Bluff Creek owing to seepage from 

the west side of Lake Hefner. The reach selected was 

from a point approximately 1700 feet north of the 

south line of Section 33 to a point on the north line 

of Section 28. There was no flow in any of the natural 

drains entering the stream from the west, so the increase in flow in the reach was considered to be 

seepage from Lake Hefner. All ponds in the reach 

were full and spilling except Silver Lake. A temporary staff gage was installed on the shore of Silver

30
Lake and the change in contents was computed on 

the basis of the observed change in stage and the 

area of the lake. The difference between inflow and 

outflow, allowing for the change in contents of Silver 

Lake, was computed to be 0.10 acre-foot per day. 

Although admittedly not of high accuracy, this result 

is of the same order of magnitude as those obtained 

the next year. The average of five measurements of 

seeps AA-DD inclusive, during the late fall, winter, 

and early spring of 1950-51 was 0.22 acre-foot per 

day. Thus, it is probable that the amount of unmeasured seepage reaching the west branch of Bluff 

Creek in the area west of Lake Hefner is small.

All the seepage collected by the drains along 

the north and west sides of the dam is conducted into 

Bluff Creek. In addition, small seeps in Sections 22, 

23, and 27, which are undoubtedly supplied from 

the reservoir, also drain into Bluff Creek. Several 

small uncontrolled farm ponds have been constructed 

to take advantage of the never-failing supply from 

these seeps. These remain full and overflow continuously. All seepage from the northwest side of the 

lake reaches Bluff Creek and was measured at EE.

The determination of seepage at EE was complicated to some extent by waste water from the filter 

plant, which is discharged to the sludge pond (see 

map). Normally the sludge pond is full, and overflows through an uncontrolled morning-glory spillway. 

It can be drawn down if desired, however. Measured 

flow at EE during the period January to August 1950 

may have included some waste water from the filter 

plant. A 90° V-notch weir was installed at a point 

approximately 300 feet below the sludge pond (location JJ). When measurements were made at EE 

on 28 April, 16 June, 18 June, and 15 August 1950, 

the flow at JJ was also measured. The flow at JJ, as 

thus determined, included both the waste water from 

the filter plant and seepage from the reservoir. In 

order to separate the seepage flow from the wastewater flow, it was decided to draw down the sludge 

pond prior to making seepage measurements. The 

flow at JJ was then only seepage since the sludge 

pond was not spilling. The results of six measurements made at JJ with the sludge pond not spilling 

indicated that average seepage at JJ was 0.60 acrefoot per day. It was then possible to adjust the measurements made at EE during the period April to 

August 1950 for filter-plant waste water.

Seepage measured at FF included flow from 

several seeps in the west half of Section 24 and from 

several others in the area below the spillway in 

Section 25. It was obvious that these seeps were supplied from the reservoir, for their flow was perennial, 

in contrast to those farther east. Although there is 

a remote possibility that the flow at FF includes some 

seepage from sources other than the reservoir, no 

such flow was observed during dry periods and, if 

present at all, it must necessarily be small compared 

with the total flow at FF, which is itself an extremely 

small item in the water budget.

Five sets of seepage measurements were made 

during the period 25 October 1950 to 26 March 

1951. The results were as follows:

Daily Flow in Acre-Feet

Date

25 Oet 1950

19 Dec 1950

21 Mar 1951

21 Mar 1951

26 Mar 1951

AA-DD, inc.

0.24

.27

.20

.20

.19

EE

1.60

2.10

1.79

1.57

1.66

FF

0.23

.35

.38

.38

.35

Total

2.07

2.72

2.37

2.15

2.20

Average 0.22 1.74 0.34 2.30

Total seepage during periods of minimum evapotranspiration losses averaged 2.3 acre-feet per day, 

as shown in the above table. This figure was used 

in computing daily evaporation throughout the period 

of observation at Lake Hefner.

RESULTS

Daily evaporation from Lake Hefner was computed for all but 8 days of the 493-day period of 

observations. On those 8 days, inflow volumes were 

so large as to preclude computation of evaporation 

with reasonable accuracy.

A weekly reporting form was designed to facilitate orderly tabulation of the various items comprising the water budget. .Sample computations for the 

week ending 9 September 1950 are shown in figure 

22. Daily values of the various items are believed to 

be reasonably representative of average conditions. 

Daily inflow was small except on Monday, 4 September, and on that day rainfall on the lake surface 

was the largest single item in the water budget. 

Surface inflow resulting from that storm, it will be 

noted, was relatively small. Withdrawals for municipal 

use were the major outflow item, ranging from 20.7 

to 33.8 acre-feet per day. Indicated daily stage 

changes were in reasonably good agreement throughout the week; the maximum daily difference between 

any two recorders was 0.010 foot on 4 September,

31
LAKE HEFNER WATER-LOSS STUDIES 

WATER BUDGET FOR WEEK ENDING 12 P.M. SATURDAY, 9 SEP 1950

SUNDAY MONDAY TUESDAY WEDN'SD'Y THURSDAY FRIDAY SATURDAY

EVAPORATION

Intake tower stage change (ft)

Boat harbor stage change (ft)

East shore stage change (ft)

West shore stage change (ft)

Avg. stage change (ft)

Avg. area during day (acres)

24-hour change in contents

Inflow

Outflow

Evaporation (acre-feet)

Evaporation (inches)

Evaporation (cm)

Avg. contents during day

Mean lake stage

 0.021

 .024

 .027

 .028

 .025

2365

 59.1

6.7

23.2

42.6

.216

.549

65,170

1194.886

4-0.059

4- .059

4- -055

+ .065

4- .060

2365

+ 141.9

165.7

28.5

 4.7

 .024

 .061

65,180

1194.890

 0.031

 .029

 .030

 .036

 .032

2367

 75.7

4.6

27.2

53.1

.269

.684

65,250

1194.920

 0.035

 .038

 .035

 .034

 .036

2365

 85.1

1.9

30.1

56.9

.289

.733

65,170

1194.886

 0.026

 .025

 .020

 .027

 .024 .

2363

 56.7

1.7

34.0

24.4

.124

.315

65,100

1194.856

 0.036

 .036

 .030

 .035

 .034

2362

 80.3

1.4

36.3

45.4

.231

.586

65,030

1194.826

 0.025

 .029

 .028

 .030

 .028

2360

 66.1

1.4

36.3

31.2

.159

.403

64,960

1194.794

INFLOW

Rainfall on lake

Intake canal

Runoff at A

Runoff at B

Runoff at C

Runoff at D

Runoff at E

Runoff at F

Runoff at G

Runoff at H

Runoff at J

Ungaged area

Total inflow

5.12

1.45

0

0

.12

0

0

0

0

0

0

0

6.69

160.23

1.86

0

.30

1.21

0

0

.81

.02

.26

0

1.0

165.69

0

1.59

0

.36

.99

0

0

.61

0

.22

0

.8

4.57

0

1.23

0

.08

.36

0

0

.06

0

0

0

.2

1.93

0

1.23

0

.02

.28

0

0

.04

0

0

0

.1

1.67

0

1.23

0

0

.22

0

0

0

0

0

0

0

1.45

0

1.23

0

0

.22

0

0

0

0

0

0

0

1.45

OUTFLOW

Seepage:

AA + BB 4- CC 4- DD

EE

FF

Deep seepage

Total seepage

Irrigation pumpage

Venturi-meter flow

Total outflow

0.22

1.74

.34

.2

2.5

0

20.7

23.2

0.22

1.74

.34

.2

2.5

0

26.0

28.5

0.22

1.74

.34

.2

2.5

0

24.7

27.2

0.22

1.74

.34

.2

2.5

< 0

27.6

30.1

0.22

1.74

.34

.2

2.5

0

31.5

34.0

0.22

1.74

.34

.2

2.5

0

33.8

36.3

0.22

1.74

.34

.2

2.5

0

33.8

36.3

RAINFALL ON LAKE

Recorder no. 1 (inches)

Recorder no. 2 (inches)

Recorder no. 3 (inches)

Recorder no. 4 (inches)

Non-recording gage no. 1

Non-recording gage no. 2

Non-recording gage no. 3

Non-recording gage no. 4

Non-recording gage no. 5

Non-recording gage no. 6

Non-recording gage no. 7

Non-recording gage no. 8

Non-recording gage no. 9

Non-recording gage no. 10

Non-recording gage no. 1 1

Non-recording gage no. 12

Non-recording gage no. 13

Non-recording gage no. 14

Non-recording gage no. 15

Non-recording gage no. 16

Non-recording gage no. 17

Non-recording gage no. 18

Sum

Average

Area of lake (acres)

Rainfall (acre-feet)

0.01

.01

.02

.02

.02

.02

.04

.04

.03

.03

.03

.03

.02

.02

,03

.03

.03

.03

.03

.03

.03

.02

.57

.026

2365

5.12

1.02

1.26

1.08

.74

.56

.44

.44

.48

.56

.56

.75

.79

.99

.96

1.09

.72

.67

1.08

1.07

.99

.88

.75

17.88

.813

2365

160.23 0 0 0

.

. 0 0

32

Figure 22. Water budget computation for the week ending 9 September 1950.
when the east-shore gage indicated a rise of 0.055 

foot and the west-shore gage a rise of 0.065 foot.

A study of the sample data presented in figure 

22 would lead to the intuitive conclusion that daily 

evaporation figures are probably not of equal accuracy. For example, computed evaporation on 4 

September is probably not as accurate as for other 

days in the week because of the large volume of 

inflow and because of the somewhat larger than 

usual variation in indicated stage changes on that day.

It was, therefore, considered desirable to classify 

the computed daily values of evaporation as to accuracy. The approximate magnitudes of the errors in 

measuring each item in the water budget have already 

been discussed. The error due to nonagreement of in-, 

dicated stage changes was determined for each day 

using the range as a measure of the standard error. 

The error resulting from area! variability of rainfall 

was taken to be the standard error of the mean of 

the catches in the 22 rain gages. The standard error 

of measuring filter plant withdrawals was taken as 

3 per cent, and the standard error of other inflow 

and outflow items as 5 per cent. The individual errors 

were then combined in the usual manner by adding 

the individual variances to obtain the total variance.

When Lake Hefner was selected as the site for 

the study, the accuracy criterion was that the error 

in mean monthly evaporation computed from the 

water budget should not exceed 5 per cent. Average 

monthly evaporation was originally estimated to be 

approximately 1190 acre-feet (Harbeck, 1951). On 

this basis, the error in monthly evaporation should 

not exceed 60 acre-feet. Assuming the standard errors 

in daily evaporation to be normally distributed, the 

standard error in daily evaporation equivalent to a 

monthly error of 60 acre-feet would be 60/\/30, or 

10.9 acre-feet. The following limits for the various 

accuracy classes of daily evaporation were therefore 

selected, using 10 acre-feet as the reference level.

Class

Number of Daily Figures of 

Evaporation in Indicated Class

Class

Standard Error of Computed 

Daily Evaporation in Acre-feet

0- 4.9

5- 9.9

10-19.9

20 and over

Each daily figure of evaporation was classified 

on this basis. The eight days for which evaporation 

was not computed because of extremely large inflow 

volumes were included in class D. The distribution 

was as follows:

A

B

C

D

142

165

108

78

Total 493

Daily evaporation figures for 62 per cent, or 

307 days, were classified as either A or B. It should 

be remembered that this scheme is not infallible; 

errors may be larger or smaller than indicated. Comparisons between daily water-budget evaporation, 

and evaporation computed by one or another of the 

various equations, indicate that the accuracy classification scheme is satisfactory for selecting the most 

reliable figures of daily evaporation. The fact that 

62 per cent of the daily figures were classed as A 

or B indicates that the original requirement as to 

accuracy of the water budget was met.

The system of accuracy classification was also 

used to indicate the relative accuracy of water-budget 

evaporation for selected periods of any number of 

days. Class limits for such periods were computed so 

as to be statistically consistent with the class limits 

for daily evaporation. In general, it was found that 

the accuracy classification improved as the length of 

the period increased. This might be expected since, as 

previously mentioned, the stage measurement error 

was independent of the length of period and its relative effect therefore decreased as the length of period 

increased.

Monthly figures of evaporation from Lake Hefner 

as determined by the water-budget method, including 

adjustments for thermal expansion, are shown in table 

1. Daily figures will be published in Volume II (Data 

Report) of this series.

Of particular interest in table 1 is the variation 

in monthly evaporation. The maximum monthly evaporation of 8.828 inches occurred in August 1951, and 

the minimum of 0.638 inch in February 1951. Evaporation for the year September 1950 to August 1951 

totalled 54.30 inches. Daily evaporation rates of 0.5 

inch or greater were frequently observed. Although 

generally small in amount, condensation occurred on 

56 out of 483 days, most of which were in winter 

or spring, with 15 days in February alone. The frequency of occurrence of condensation was surprisingly large, but not inconsistent with what might have 

been predicted from a study of the meteorological data.

33
TABLE 1. Monthly Evaporation from Lake Hefner, 

Oklahoma.

Indicated Monthly 

Evaporation 

Month (inches)

May 1950

Jun

Jul

Aog

Sep

Oct

Nov

Dec

Jan 1951

Feb

Mar

Apr

May

Jun

Jul

Aug

3.457*

6.059

6.519

7.410

5.839

6.741

6.298

2.848

2.448

 0.617

3.369

2.533

3.319t

5.502

7.029

8.896

Adjustment 

for Thermal 

Expansion 

(inches)

+ 0.283

+ .292

 .021

 .020

 .196

 .228

 .309

 .006

+ .018

 .021

+ .021

+ .197

+ .170

+ .370

+ .148

 .068

Monthly Evaporation Adjusted for 

Thermal Expansion 

(inches) (cm)

3.740

6.351

6.498

7.390

5.643

6.513

5.989

2.842

2.466

 0.638

3.390

2.730

3.489

5.872

7.177

8.828

9.50

16.13

16.50

18.77

14.33

16.54

15.21

7.22

6.26

-1.62

8.61

6.93

8.86

14.91

18.23

22.42

* Estimated from 24-day total of 2.677 inches 

t Estimated from 30-day total of 3.212 inches

CONCLUSIONS ON THE 

WATER BUDGET

Although it is theoretically possible to use the 

water-budget method for the determination of evaporation from any lake or reservoir, it is usually impracticable to do so because of the effects of errors 

in measuring the various items. Evaporation as determined by this method is a residual, and therefore 

may be subject to considerable error if it is small 

relative to other items. At Lake Hefner, however, 

evaporation was a major item in the daily water 

budget; during the 16-month period of observation, 

outflow was only 10 per cent greater than evaporation. Inflow was generally much less because of the 

sporadic nature of reservoir replenishment.

The requirement that the error in monthly evaporation computed from the water budget should not 

exceed 5 per cent was met. Daily figures are, of 

course, subject to somewhat larger percentage errors, 

particularly during the spring and fall months when 

the effect of thermal expansion of the water in the 

reservoir results in a more or less systematic, but 

unavoidable, error in the daily figures.

REFERENCES

Chemical Rubber Publishing Company, Cleveland, 

Handbook of Chemistry and Physics, 30th ed., Cleveland, 1947, p. 1695.

Corbett, D. M., et a/., "Stream-Gaging Procedure," 

U. S. Geological Survey, Water Supply Paper, no. 

888, 1943.

Harbeck, G. E., et a/., "Utility of Selected Western 

Lakes and Reservoirs for Water-Loss Studies," U. S. 

Geological Survey, Circular, no. 103, March 1951.

King, H. W., Hand-book of Hydraulics for the Solution of Hydraulic Problems, 3d ed., McGraw-Hill, 

1939, p. 474.

Snedecor, G. W., Statistical Methods Applied to Experiments in Agriculture and Biology, 4th ed.. Iowa 

State College Press, 1946, p. 98.

U. S. Bureau of Reclamation, Manual for Measurement of Irrigation Water, 1946.

34
Instrumentation for Mass-Transfer and Energy-Budget Studies

by Lloyd J. Anderson*

To provide the necessary meteorological data for 

computing lake evaporation by the mass-transfer and 

energy-budget methods, it was necessary to develop 

special equipment in some cases and to modify 

standard equipment in others to meet the particular 

requirements of the problem.

The meteorological variables to be measured 

were: air temperature, humidity, and wind speed, at 

2, 4, 8, and 16 meters above the lake surface; lake 

surface temperature; wind direction; and rainfall. 

These were measured and recorded at each of four 

stations. In addition, it was necessary to measure, 

at one station, the incoming solar and terrestrial 

radiation and the solar energy reflected from the 

lake surface.

GENERAL DESCRIPTION 

OF EQUIPMENT

In the design of the equipment system, four 

factors were considered essential:

1. The data must be recorded as accurately as 

possible.

2. The equipment must maintain its calibration 

over extended periods of time.

3. It must operate unattended for as long 

periods as possible.

4. It must use a minimum of electrical power.

Wind speed and direction were measured, respectively, with standard 3-cup Robinson-type contact 

anemometers and a conventional wind vane with 

contacts at cardinal compass points. Rainfall was 

measured with a tipping-bucket rain gage. The only 

adaptation needed for this equipment was the provision of a low-power recording system. An operationaltype Esterline-Angus recorder with a spring-wound 

paper feed was used in conjunction with a capacitor 

discharge circuit for operating the recorder relays.

Temperatures and humidities were measured with 

wet and dry thermocouples. In this way all elements, 

including the lake temperature element, could be 

made up having identical calibrations and, using No. 

30 wires for the wet elements, the minimum ventilation required for attaining the true wet-bulb depression was reduced to 0.5 mph. This eliminated the need 

for forced ventilation, since out of doors the wind 

speed is rarely below this minimum.

To use thermocouples, as well as thermopile 

radiation-measuring equipment, it was necessary to 

develop a low-power amplifier which could amplify 

voltages a thousandfold in order to drive the 1-mil 

Esterline-Angus recorder and which would maintain 

its amplification factor constant within a fraction of 

1 per cent, regardless of normal changes in battery 

voltages and tube characteristics. The amplifier used 

for this purpose was a negative-feedback galvanometer type and fulfilled the requirements with a power 

consumption of 4 watts. It could thus operate for 

periods of two to three weeks on a set of batteries 

small enough to be easily transportable to the station 

sites. Further details on the amplifier, as well as on the 

thermocouple psychrometer, may be obtained by referring to Anderson, Anderson, and Marciano (1950), 

Bellaire and Anderson (1951), and Denton (1951).

Careful comparison was made out of doors 

between the wet and dry thermocouple traces and 

calibrated mercury thermometer readings. In this comparison, the thermocouples were mounted in a radiation shield designed to protect them from solar 

radiation and were ventilated by the wind, which 

varied from 4 knots to nearly calm. The thermometers 

were shaded and ventilated at 8 knots by a blower. 

A plot of thermometer readings taken at 10-second 

intervals showed agreement with the thermocouple 

trace within -|-0.1 0 C over the 30-minute test period, 

except during a period of flat calm, when thermocouple readings rose as much as 0.5°C above the 

thermometer values.

As a result of the above test, and careful spot 

checks in the field, it is believed that wet- and drybulb temperatures were obtained with errors of 

±0.1 °C. Although the Esterline-Angus Company does 

not guarantee its recorders to better than 2 per cent 

of full scale, or 1.0°C in our case, this accuracy was 

considerably improved by calibrating all recorders 

against a standard milliammeter and adjusting them 

so that all were within ±0.1 °C at five points over 

the scale. By loading the pens as lightly as practicable, it was possible to reproduce the same trace 

for a given temperature within ±0.1° when approached from above or below.

* Physicist, Research Division, U. S. Navy Electronics Laboratory, San Diego, California.

35
Numerous attempts to calibrate the cup anemometers resulted in closer agreement between individual units than with the standard. Since the 

anemometers were to be used in the field, they were 

calibrated out of doors against a hot-wire anemometer. The rapid response of the hot wire and the slow 

response of the cup anemometers made it difficult 

to compare results. Because of the complicated hotwire traces obtained, and the logarithmic response to 

wind, it was not possible to planimeter the trace for 

accurate average winds. Installing the anemometers 

on a jeep and driving up- and down-wind also gave 

inconclusive results because of variable ambient winds, 

even under conditions as calm as can be expected.

The results of these tests led to the conclusion 

that, whatever the calibration errors, all units were 

consistent to ±0.5 knot for winds above 3 knots. 

Since the anemometers were maintained in good condition and the cups were new, it is reasonable to 

assume that the absolute calibration was within the 

limits given by the manufacturer.

The radiation measurements consisted of incoming and lake-reflected solar flux measured with an 

Epply pyrheliometer, and total incoming radiation 

measured with a Gier and Dunkle flat-plate radiometer.

The pyrheliometer is a flat circular plate mounted 

horizontally inside a lime-glass bulb. The plate is 

divided into a central white spot, a black ring, and 

an outer white ring. A 10-junction thermopile measures the temperature difference between the black 

and white areas, which is proportional to the radiation flux penetrating the glass bulb.

The radiometer consists of a flat 2-inch square 

plate, mounted horizontally in the blast of a small 

blower. The plate is a sandwich with a blackened 

aluminum upper surface and a polished aluminum 

lower surface; between is a thermopile measuring the 

vertical temperature gradient across an insulating' 

sheet forming the center layer of the sandwich. The 

thermopile voltage is thus proportional to the heat 

flow down through the plate, which in turn is proportional to received energy at the blackened surface 

after deduction of the black-body radiation. To obtain the latter correction, a separate thermocouple is 

used to measure the black-surface temperature. The 

function of the blower blast is to eliminate the effect of the wind on the calibration coefficient of the 

device.

Pyrheliometer and radiometer voltages were recorded using an amplifier and recorder system similar 

to that used for temperature and humidity.

For obtaining temperature-structure data within 

the lake itself, a Western Electric thermistor bead was 

used. It was lowered from a boat on a conducting 

cable, and water temperatures were recorded on a 

1-mil Esterline-Angus recorder in the boat. Temperature changes of the bead caused its resistance to 

change, and this in turn produced an off-balance 

voltage in a Wheatstone-bridge circuit. This voltage 

was amplified and recorded as lake temperature. 

Calibration showed the instrument capable of an 

accuracy of ±0.1 °C, when corrected by bucket 

checks.

After reconnaissance of the Lake Hefner area 

and a study of the wind rose data, it was decided 

to place the four meteorological stations as indicated 

on the map. The primary station, which included the 

radiation equipment for the energy-budget studies, 

was installed on a barge anchored near the center 

of the lake. The most important land station was 

located on the southeast shore of the lake, and served 

to measure the properties of the air before reaching 

the lake surface. The prevailing winds, particularly 

during the summer, were such that this location was 

representative of the up-wind air most of the time. 

During the winter the wind direction is more variable 

and, in order to utilize northerly winds, a station was 

located on the northeast shore of the lake to act 

as the up-wind station for winds between north and 

east. The fourth station was installed on the intake 

tower on the north shore of the lake, about 50 yards 

south of the dike. This station was to collect downwind data for southerly winds. It was not felt necessary to provide for westerly winds, since these 

occurred only a very small percentage of the time.

A more detailed discussion of the development 

of the equipment described above, and the details of 

its construction, are given in Anderson, Anderson, and 

Marciano (1950).

MASS-TRANSFER STUDIES

Initial Measurements

Before the routine evaporation data could be 

collected, it was necessary to evaluate certain parameters such as the variation of wave height with wind 

speed and fetch, as well as the effect of stability on 

the wind profiles. The requirements for this study included the measurement of very small differences 

between winds and temperatures at 8 and 2 meters 

above the lake surface and of wave height as a func36
tion of distance down-wind. The basic equipment for 

these purposes has been described by Anderson, Anderson, and Marciano (1950). It was installed at all 

stations except the northeast.

The wind equipment consisted of two hot-wire 

anemometers connected on opposite sides of a Wheatstone bridge. Due to the logarithmic response of the 

hot wires to wind speed, the off-balance bridge voltage was a measure of the ratio of the winds at 2 

and 8 meters. It was found that this ratio could be 

measured to ±1 per cent, which was far better than 

using two conventional cup anemometers.

The temperature-difference equipment utilized a

5-junction thermopile installed in the radiation shields 

at 2 and 8 meters. The accuracy of measurement was 

±:0.02°C. The wind ratios and temperature differences, as well as the lake temperature, were alternately recorded through the same amplifier and Esterline-Angus recorder system, using the 20-point 

programming switch.

The wave-height equipment consisted of three vertically-floating staff gages marked with alternate

6-inch-wide black and white bands. One gage was 

anchored near the first barge location, the second 

near the middle of the lake, and the third near the 

north shore. In addition to visual and motion-picture 

observations of the staff gages from shore, using a 

10-inch reflecting telescope, close-up observations of 

wave height were made from a small boat. The three 

methods yielded essentially identical data. Six sets 

of such measurements were made, covering wind 

speeds up to 30 miles an hour.

During the first three-week period, the barge 

was located about midway between the south station 

and the center of the lake, and for the remaining 

three weeks the barge was relocated at its final position near the center of the lake. The major equipment 

modification found necessary was damping of the 

wind-ratio recorder to obtain a more readable trace. 

The time constant was increased from 2 to 6 seconds, 

resulting in a trace from which average wind ratios 

(u§/U2) could be much more easily obtained. The 

5-junction thermopiles, used to measure small temperature differences between 2 and 8 meters, performed as expected and no modifications were 

necessary.

Routine Measurements

After the initial measurements, the installation 

of the equipment for routine operation was completed at all four stations, and routine measurements 

were begun, on 15 April 1950. The equipment, as

designed and installed at the model station in San 

Diego, was found to perform satisfactorily. A few 

modifications were needed, most of which resulted 

from maintenance problems and which will be discussed under that heading. The photographs in figures 

23 through 26 show over-all views of the four stations. 

The south station (fig. 24) was located about 100 

yards from the shore of the lake, in order to be above 

the high-water level. The northeast station (fig. 25) 

was mounted on a platform supported by 4-foot pilings for the same purpose. To mount the equipment 

on the intake tower, it was necessary to replace the 

original conical roof by the flat one shown in figure 

26, and to hang the lower-level instruments on 10-foot 

booms in order to avoid, as far as possible, the influence of the tower itself. A provision was made at 

this station for raising or lowering the instrument 

levels to follow the lake stage. At this particular station commercial power was available, and the recording equipment could be located inside the tower. 

Battery power was required for the other three stations. An additional complication was the need for 

providing aircraft warning lights for the northeast 

and barge stations. This required additional batteries 

and clock switches for turning off the lights during 

the daytime. Barge lights were mounted in such a 

way that they had no effect on the radiation measurements.

The lake-temperature thermocouples were the 

only elements requiring appreciable modification for 

satisfactory operation. It was soon found that a galvanic voltage was being generated between the 

thermocouple and float assembly (fig. 27) and the 

steel anchoring cable. During wet weather, partial 

shorting by exposed wet surfaces caused this voltage 

to appear across the amplifier terminals. To eliminate 

this effect, the thermocouple junction was electrically 

insulated from the lake by encasing it in a transparent plastic rod. It was considered that no loss of 

accuracy would result by this procedure, since any 

solar energy absorbed by the plastic would be quickly 

carried away by the lake water. The constant wave 

motion prevented any pile-up of warm water at the 

junction. Since it was necessary to measure the lake 

temperature as close to the surface as possible, the 

junctions were adjusted to float about VA inch below 

the surface. The lake-surface temperature thus measured is an average of the top Va inch of water under 

quiet conditions and probably the top 6 inches of 

water and lower 3 inches of air under rougher conditions. Under virtually all conditions except those 

when still warm air overlies a flat still water surface,

30E5EO O - 54 - 4 37
the wave action tends to remove thermal gradients in 

the top few inches of water. Thus the temperatures as 

recorded should be representative of the interface. 

The temperatures will be even more representative of 

the interface as the wind increases and wave mixing 

becomes stronger, in which case one would also 

expect thermal equilibrium between the air and water 

within the first few inches above and below the interface. Since the wind was above 4 knots 92 per cent 

of the time at Lake Hefner, it is believed that steep 

thermal gradients in the top fraction of an inch occurred very rarely.

ENERGY-BUDGET STUDIES

Radiation Measurements

Figure 23, showing the barge station, gives an 

over-all picture of the radiation equipment installation. The flat-plate radiometer and the Eppley pyrheliometer are on top of the central mast. The two 

downward-looking Eppleys for measuring reflected 

solar energy are mounted on 4-foot arms near the 

tops of the two shorter masts. To avoid radiation 

pickup from the barge, two downward-looking Eppleys 

were used, one covering the south half of the lake 

and the other covering the north half. The bulbs were 

painted to prevent pickup from overlapping sections 

of the lake and from the barge. The outputs of both 

Eppleys were added in series and recorded as a 

single voltage; this was possible because the calibration constants of the two instruments were essentially 

equal. The instrument shelter for the recording equipment is the smaller one nearest the central mast. 

Figure 28 shows a close-up of the southernmost mast 

with a downward Eppley shown near the top. Figure 

29 shows the interior of the instrument shelter, with 

the amplifier and recorder on the top shelf and 

the batteries and accessory equipment on the bottom 

shelf.

The flat-plate radiometer installed on top of the 

central mast performed well under all conditions. It 

was found during analysis of the data, however, that 

the long-wave radiation, which was constant during 

the night, began to decrease at sunrise and reached 

a minimum of about half the nighttime value at 

midday. It began rising again during the late afternoon and by sunset was up to the nighttime value. 

Since this quantity was derived by subtracting the 

incoming solar radiation, as measured by the Eppley, 

from the total incoming radiation, as measured by 

the flat-plate radiometer, there are several possible

Figure 23. bqrge station.

Figure 24. South station.

38
Figure 25. Northeast station.

t 

i

Figure 27. Lake thermocouple float at barge.

Figure 26. intake tower station. Figure 28. Lower level instruments on barge station.

39
causes for this apparent daytime decrease. It does 

not seem reasonable that it should be a real effect. 

Either the Eppley calibration was too high or the 

flat-plate calibration too low or, more probably, the 

reflection coefficient of the flat-plate for visible 

energy was not constant with angle of incidence. To 

check this point, the Eppley readings were compared 

with those of nearby Weather Bureau installations at 

Oklahoma City and subsequently at Las Vegas, Nevada. Comparison showed that the Oklahoma City 

values were 2.6 per cent higher and the Las Vegas 

values 2 per cent lower than the readings of the 

Eppley pyrheliometer used in these studies. Since the 

probable error of calibration is of the order of ±2 

per cent, it seems evident that the instrument in question lies about in the middle of this range.

Accurate terrestrial radiation values might be 

obtained by using a glass-covered flat plate for obtaining solar-energy values. If the covered and uncovered flat plates were connected in series with 

opposite polarity, the resulting recorded value should 

approximate the terrestrial energy flux. Such a scheme 

may be an oversimplification of the problem, but 

should be investigated.

To prevent interaction between the radiation recorder and the temperature recorder, it was necessary 

to use separate ice baths for the reference junctions 

of the flat-plate temperature thermocouple and the 

other thermocouples. When a common ice bath was 

used, interaction was noticed between the two circuits 

during rainy weather, presumably caused by partial 

shorting via the wet surfaces of the mast and barge. 

Similar shorting difficulties were encountered when 

attempting to use a common storage battery for the 

flat-place blower motor and the filament supply of 

the amplifier. Both difficulties were eliminated by suitable isolation.

The flat-plate blower was driven by a small 

6-volt Haydon motor which performed satisfactorily 

for 6 months of almost continuous operation. This performance far exceeded expectation. The motor was 

replaced during a general overhaul of the equipment but was still in good operating condition.

The three Eppley pyrheliometers performed quite 

satisfactorily for the entire 16 months of operation 

and are now in use at Lake Mead.

Temperature Profile Recorder (TPR)

This equipment was developed at NEL to obtain 

more accurate profiles of temperature within the lake. 

It was described io detail by Anderson and Burke 

(1951).

Figure 30 shows the equipment mounted in place 

near the stern of the boat in which it was used for 

the lake-temperature surveys described in a later 

section of this report. It was covered, when not in use,

Figure 29. Radiation instrument shelter on barge.

Figure 30. TPR installation on boat.

40
by the metal box shown in the left background. Amplifier power was supplied by the boat storage battery. Performance was satisfactory except for the last 

2 months of the Lake Hefner studies, when the vibrator became faulty. Similar difficulties had occurred 

on a few previous occasions, but were remedied by 

replacing the vibrator. In June 1951, however, all 

the vibrators became faulty and the equipment was 

sent to NEL for overhaul. It was also recalibrated at 

this time and, after 14 months of operation, the 

thermistor beads were found unchanged in calibration. Tests showed that the entire amplifier had 

maintained its calibration over this length of time. 

The only component which affected the calibration 

of the equipment was the vibrator. Substituting various 

vibrator units shifted the entire temperature scale up 

or down a few divisions, because of the different noise 

levels inherent in the individual units. Correction for 

this shift, however, brought all scales back to the 

original calibration. In practice this effect offers no 

difficulty, since frequent checks were made by comparing the TPR readings with bucket temperatures as 

measured with a calibrated thermometer.

OVER-ALL PERFORMANCE 

AND MAINTENANCE

Accuracy Checks

To maintain the accuracy required for evaporation computations, it was necessary to make periodic 

checks of the equipment in the field. The temperature 

and humidity data were checked weekly at all stations by means of a sling psychrometer. The observed 

wet- and dry-bulb readings were entered on the recorder traces and compared with recorded values 

during the data analysis. The sling-psychrometer 

values were occasionally several tenths of a degree 

higher than the recorded values, although the wetbulb depression always agreed within 0.1 °C. It is 

believed that insufficient shielding of the psychrometer 

from the sun is responsible for these minor discrepancies.

No suitable method was found for checking 

anemometer accuracy in the field. All units were replaced at intervals of 30 days or less. The units which 

had been in use were completely overhauled, the 

overhaul including removal of the shaft and bearings, 

washing out of the lubricant, and filling with fresh 

lubricant. During this time the contact points were 

also checked, and the shaft friction was checked by 

the following procedure. The anemometer body with

cups attached was arranged with its axis horizontal, 

and a small weight was placed on one of the cups 

which was held at the same level as the axis. The 

cup was released, and the weight caused it to rotate. 

At the bottom of the arc the weight dropped off, 

and the cup continued to rotate until eventually 

stopped by shaft friction. It was found that anemometer shafts in good condition made about nine 

revolutions before stopping. Any units which did not 

meet this criterion were relubricated.

The reference bath for the nine thermocouples at 

each station was contained in a 1-gallon Dewar flask. 

Since all temperatures were referred to the bath 

temperature, it was essential that 0°C be maintained 

in the bath at all times. A mercury thermometer was 

permanently mounted in the flask and checked at 

2-day intervals. No appreciable discrepancies were 

noted. 

Usable Data Yield

Table 2* shows the percentage of occurrence 

during each month of operation of the commonly occuring difficulties in collection of data. Some of these 

faults are minor, but others result in unusable data. 

A gradual decrease in the occurrence of most of these 

difficulties is evident throughout the summer and fall 

months of 1950. During the winter, however, wet bulbs 

dried out very frequently, because of freezing of the 

reservoirs. This effect generally occurred several hours 

after the air temperature dropped below 0°C, and 

disappeared abruptly at all levels about 2 hours after 

the air temperature rose above 0°C. On several occasions the temperature dropped a few degrees below 

freezing without the wicks drying out, because presumably, the water in the reservoirs did not freeze 

and thus continued to feed water to the wicks. By 

March, the occurrence of frozen reservoirs had decreased to only 11 per cent and no further difficulty 

was encountered after 21 March.

Missing anemometer pips occurred frequently 

during the first five months. This defect was eliminated 

by periodic checking of the contacts during anemometer overhaul. Fortunately, this defect was not 

serious, because the pips were missing in such a way 

that they could be inserted in the traces during analysis of the data.

Figure 31 shows the yield of usable data during 

the operating period. The radiation equipment, after 

a rather poor start, settled down to an average yield 

of 24 usable days per month during the last year of 

operation. Since the barge meteorological data were 

crucial for all evaporation computations, the yield 

* See page 45.

41
-___.

»_».

.___

USABLE

1"!

PERFE CT

ALL MET. STATIONS

JUN JUL

____

AUG SEP 

1950

OCT

 VB

NOV DEC JAN FEB

USABL

\

- ».

MAR

E

APR.

"T"

PERFE

MAY 

1951

CT

JUN JUL

-

AUG

Figure 31. Data yield (days per month) 

of meteorological and radiation equipment.

of this station is shown separately. The average was 

20 usable days per month, with 14 days per month 

of perfect data. The bottom row of the figure shows 

the average for all four meteorological stations, 21 

days per month of usable data and 16 days of 

perfect data.

To qualify a day as "usable," there could be no 

more than two 3-hour periods containing unusable 

data. Both the temperature and wind recorders had 

to produce usable data simultaneously at all four 

levels (2, 4, 8, and 16 meters). The "perfect" days, 

of course, were those in which no data-recording 

failures occurred during the entire 24-hour period.

Maintenance Problems

The above data yields were the result of careful 

and conscientious attention to maintenance detail by 

the resident maintenance crews. After the initial installation by the equipment designers, it was the duty 

of the maintenance crews to solve the everyday problems that arose, and to solve them quickly to keep 

the data loss as small as possible. Table 2 and figure 

31 give a graphic record of the maintenance crews' 

performance.

As mentioned previously, all the equipment except that at the intake tower was battery-operated. 

Originally, the 180-volt power required by the thermocouple amplifiers was supplied by dry batteries, 

and 6-volt power by storage batteries. Because of the 

persistent wind at Lake Hefner, it was possible to 

install a wind-driven propeller-type battery charger 

at each station. This greatly reduced the labor of 

transferring storage batteries and made it practicable

to use a vibrator-type power supply, operating from 

a storage battery, to supply the 180-volt amplifier. 

In this way all the electrical requirements of the stations were fulfilled by the wind chargers. The cost of 

the charger and vibrator units was saved in a few 

months of operation by eliminating the expense of 

dry batteries. The procedure used with a wind charger 

was to connect it to one set of batteries at the station 

while a second set was in use on the equipment. It 

was not possible to connect the wind charger to the 

batteries while they were in use, because this would 

have necessitated connecting all storage batteries 

together, which was not permissible in the amplifier circuit.

The thermocouple amplifier, which represents the 

heart of each station, performed quite satisfactorily 

with periodic maintenance checks and the replacement of tubes and exciter lamps. Table 2 shows that 

the temperature trace was off scale a fair percentage 

of the time during the first 4 months of operation. 

This was due to sticking of the galvanometer coil 

against its stops whenever it happened to go off 

scale. The problem was solved by modifying the stops 

so that the coil form contacted only a fine point on 

the stop, thus virtually eliminating the cohesive force 

that seemed responsible for the sticking. Very little 

difficulty was experienced after making this modification, in early September.

A routine maintenance check on the input impedance of the amplifier was found necessary to 

prevent inaccuracies resulting from too much current 

being drawn from the external thermocouple circuits. 

It was accomplished by substituting a fixed 500-ohm

42
resistor in place of the thermocouple circuit and adjusting the mechanical zero of the galvanometer until 

zero input voltage was indicated on the recorder. 

Such readjustment was found to be required about 

once a month. In figures 29 and 32, the amplifiers 

are shown resting on a pad of foam rubber. This 

was necessary to eliminate high-frequency vibration 

of the galvanometer suspension. Such vibration was 

particularly troublesome at the northeast station (fig. 

25), where the entire platform would be set in vibration by winds above 15 miles per hour. With a foamrubber pad under the amplifier the trace remained 

unperturbed until the wind exceeded 30 miles per 

hour.

The Esterline-Angus recorders used to record all 

the data were clock-driven to conserve electrical 

power. They performed satisfactorily except during 

conditions of high humidity in the summer. Table 2 

shows the increase in clock stoppage during these 

periods. The difficulty was partially eliminated by installing small light bulbs to dry the inside of the recorder case. The bulbs were powered by worn-out 

amplifier batteries.

Figure 32. Meteorological instrument shelter on barge.

A 20-point switch was used to present, in turn, 

each of the nine thermocouple voltages to the amplifier. In this way all temperatures could be recorded 

on a single tape. The switch was driven by an auxiliary shaft on the recorder and made one revolution 

per hour. Since it was desirable to record each temperature twice an hour, each thermocouple was connected to two contacts. The two extra contacts were 

used to short the amplifier at half-hour intervals. This 

served to identify the beginning and end of each 

sequence and also to indicate the 0°C reading to 

which all temperatures were referred. For the first 

four months, faulty switch operation occurred about 

8 per cent of the time, caused by accumulation of 

dust and oxidized metal on the wiper contact, such 

that when one contact was broken, the next was not 

immediately made. Thus the amplifier was opencircuited for a few seconds, resulting in an off-scale 

trace. Because of its short duration, this did not affect 

the data except when the galvanometer coil stuck 

off-scale, as mentioned above. Routine polishing of 

the wiper and stationary contacts with crocus cloth at 

weekly intervals eliminated this difficulty. A light film 

of instrument oil seemed to reduce further the occurrence by retarding oxidation of the metallic surfaces.

The reference baths for the thermocouples required refilling with ice at weekly intervals during 

the summer and monthly intervals during the winter. 

Several of the Dewar flasks imploded during the Lake 

Hefner observations, in spite of each being mounted 

inside a wooden case filled with rock wool. Even very 

careful handling did not prevent such occurrences. 

Perhaps smaller flasks refilled at more frequent intervals would have suffered fewer casualties, but the 

convenience of the larger size was a very desirable 

feature in field operation.

The wet-bulb reservoirs required filling at 2- to 

4-day periods, depending primarily upon the wind 

speed. After the first 2 months, no appreciable data 

were lost because of dry reservoirs. As mentioned 

previously, frozen wet bulbs during the winter caused 

considerable difficulty, but this was unavoidable and 

represents the major disadvantage of the psychrometric technique. It was believed from the beginning, 

however, that the advantages of the technique far 

outweighed the disadvantages and that heavy loss 

of data during the winter months was less undesirable than the disadvantages of other methods of 

measuring humidity.

Each time the reservoirs were filled, the accumulated dust was removed from the wet thermocouple 

wick. This served to extend the life of the wick as well

43
as to insure attaining of true wet-bulb temperatures. 

The copper wire used on one side of the junctions 

gradually corroded, and several wires had to be replaced toward the end of the operating period. The 

four spare thermocouple assemblies enabled this replacement to be accomplished with virtually no loss 

of data. It was found necessary to install insect 

screens around the psychrometer radiation shields to 

prevent wasps from building nests in the air passages. 

These screens also served to reduce wind speeds in 

the passages and to reduce excessive evaporation 

from the wicks during high winds.

Figure 27 shows a lake thermocouple float. The 

thermocouple is barely discernible inside the triangle. 

As mentioned previously, it was found necessary to 

insulate the junction of this thermocouple from the 

lake water by encasing it in a lucite rod. The thermocouple cable, from the plastic rod to the instrument 

shelter, was also encased in heavy-wall rubber hose, 

both to prevent electrical leakage and to protect the 

cable from mechanical abrasion. Table 2 shows that 

lake-temperature difficulties were essentially eliminated by this treatment.

The radiation equipment required little maintenance except for occasional dusting of the pyrheliometer bulbs and radiometer plate. The latter was 

given a coat of flat black enamel when necessary to 

preserve its absorption properties. The enamel used 

was the same as used by the manufacturer.

The Temperature Profile Recorder required little 

maintenance except for occasional polishing of the 

slip-ring system used for connecting the cable on the 

drum to the amplifier. Accumulation of dust in the 

system resulted in an erratic trace. Repolishing of 

the slip rings with crocus cloth eliminated the difficulty 

for several weeks.

Figure 33 shows a further maintenance problem 

occasionally encountered during the winter. The view 

shown is of the same mast as shown in figure 28. 

It was necessary on this occasion to chip away a great 

deal of ice to keep the barge floating level. Pictures 

such as this make one doubly appreciative of the 

efforts of the maintenance crew.

CONCLUSIONS ON INSTRUMENTATION

The equipment developed for the Lake Hefner 

studies proved to be reliable and practicable. The 

stations had to be visited at 2- to 4-day intervals for 

filling wet-bulb reservoirs, and at weekly intervals for 

other maintenance requirements. At windy locations, 

all power needs were met by wind chargers and 

storage batteries.

The thermocouple psychrometer was found suitable for obtaining accurate humidity data in regions 

where freezing temperatures are not prevalent for 

extended periods. The radiation recording system, 

comprising the Eppley pyrheliometer, flat-plate radiometer, galvanometer amplifier, and Esterline-Angus 

recorder, performed satisfactorily. Indicated daytime 

terrestrial radiation values were too low, possibly due 

to variable reflection of solar radiation by the flatplate surface.

The Temperature Profile Recorder performed well 

and yielded much more accurate profiles than those 

provided by the conventional bathythermograph. 

When surface checks were made against a calibrated 

thermometer the profiles were accurate to 0.1 °C.

REFERENCES

Anderson, E. R., Anderson, L. J., and Marciano, J. J., 

A Review of Evaporation Theory and Development of 

Instrumentation, Lake Mead Water Loss Investigations: 

Interim Report, Navy Electronics Laboratory, Report 

No. 159, 1 February 1950.

Anderson, E. R., and Burke, A. T., "Notes on the 

Development of a Thermistor Temperature Profile Recorder (TPR)," Journal of Marine Research, vol. 10, 

no. 2, 30 November 1951, pp. 168-179.

Bellaire, F. R., and Anderson, L. J., "Thermocouple 

Psychrometer for Field Measurements" American Meteorological Society, Bulletin, vol. 32, no. 6, June 

1951, pp. 217-220.

Denton, R. L., "A Thermocouple Voltage Amplifier 

for Temperature Recording," American Mefeoro/ogica/ 

Society, Bulletin, vol. 32, no. 6, June 1951, pp. 

214-216.

44
TABLE 2. Per Cent Occurrence

Studies

Trace Off-scale

All Temp. Data Unusable

Lake Temp. Unusable

Dry Wet Bulb

Faulty Switch

Clock Stopped

Dry Pens

Anemo. Pips Missing

May

12

7

5

3

8

1

0

48

Jun

14

3

15

3

7

2

0

56

of Instrument Faults, Meteorological Data, Lake Hefner

Jul

15

3

5

1

6

2

2

15

Aug

16

0

0

0

10

2

1

26

Sep

3

0

1

0

1

3

1

13

Oct

3

2

0

3

2

2

0

4

Nov

2

5

3

17

0

1

0

6

Dec

4

2

0

23

0

0

3

0

Jan

5

4

3

28

0

0

5

0

Feb

0

8

0

27

0

4

0

0

Mar

0

5

0

11

0

0

2

0

Apr

0

1

0

0

0

0

0

0

May

1

3

0

0

0

0

0

0

Jun

3

2

0

0

0

0

0

0

Jul

4

4

0

0

0

0

0

0

Aug

2

0

0

0

0

0

0

0

Figure 33. Icing on barge.
Mass-Transfer Studies

by J. J. Marciano* and G. Earl Harbeck, Jr.**

The foundation used for obtaining practical 

results from mass-transfer theory was NEL Report 

159 (Anderson, Anderson, and Marciano, 1950). That 

report contains a detailed review of theory, based 

on the discontinuous- and continuous-mixing concepts 

applied to mass transfer in the boundary layer. As a 

product of the review, a selection was made of a 

series of evaporation equations judged suitable for 

testing the two concepts. Although both concepts had 

some experimental support, they had not been tested 

to the extent or with the precise controls contemplated in the Lake Hefner program.

Several important questions were left unanswered by Anderson, Anderson, and Marciano. First 

was the question of the existence of a critical wind 

speed for air-water boundary-layer processes. The 

answer is tied closely to another question, namely, 

the value of the roughness parameter of the lake 

surface. Evaporation equations are strongly influenced by answers to these two questions. A third 

unknown was the degree of coupling between wind 

and temperature fields in the boundary layer; this 

problem involves a search for a general wind equation for the boundary layer. Finally, it was necessary 

to ascertain whether theory could predict the height 

of wind-generated waves on a relatively small lake.

In the following pages, these problems are again 

discussed in connection with the theory of the boundary layer. Evaporation equations are again reviewed, with the addition of the case of evaporation 

in zero wind. The remaining discussion presents the 

results of tests of several evaporation equations 

against a reliable control.

Few concepts have had more influence on aerodynamics and fluid mechanics than Prandtl's description of the boundary layer in 1904. The theory of the 

boundary layer forms the basis for treating evaporation as a mass-transfer problem.

Consider a fluid flowing with constant velocity 

U past a fixed solid boundary. The velocity will increase from zero at the boundary and reach the 

value U a small distance away. Tangential stresses 

are set up, retarding the flow at distances farther 

and farther from the boundary, so that the width of 

the layer of retarded fluid will increase downstream. 

By definition, the boundary layer is that layer of 

fluid retarded by the boundary. Figure 34 illustrates the process involved.

We can thus divide the fluid into two portions:

1. The relatively thin boundary layer where 

the velocity gradient is large and, consequently, 

where shearing stresses cannot be neglected.

2. The region outside the boundary layer where 

the velocity gradient is small and shearing stresses 

negligible. In the second region the flow closely 

resembles that of the classical nonviscous fluid, with 

the motion determined by pressure forces and the 

flow described with sufficient accuracy by Euler's 

classical equation. Our main concern is the boundary 

layer, particularly the velocity distribution within it.

In nature, the boundary layer is formed as 

shown in figure 34. If the air has been flowing for 

some distance over a reasonably uniform surface, the 

thickness of the layer will be essentially constant 

with respect to downstream distance, changing height 

only insofar as the over-all degree of turbulence may 

warrant. If a new surface is suddenly encountered,

Figure 34. Growth of a boundary layer along one side of a 

flat plate.

* Meteorologist, Research Division, U. S. Navy Electronics 

Laboratory, San Diego 52, California.

** Hydraulic Engineer, Water Resources Division, U. S. 

Geological Survey, San Diego, California.

46
such as would occur at the edge of a lake, a new 

boundary layer will develop, its rate of growth depending on distance as well as the properties of the 

air and the new surface.

Scalar fields, such as temperature and vapor 

pressure, may also be thought of as having boundary layers, although it does not follow that the 

boundary layers for different properties are of equal 

thickness. Boundary-layer theory was first developed 

with momentum as the important property. Since 

then, the concept of "similarity" has been used to 

connect the momentum and heat boundary layers 

(Dryden, 1934; Goldstein, 1938). The same approach 

has been used to connect heat transfer and evaporation (Schmidt, 1929), and momentum transfer and 

evaporation (Millar, 1937). It is now generally accepted that turbulent transports of momentum and 

water vapor are the same, that is, their coefficients 

of eddy transport are interchangeable. Evaporation 

.equations discussed later were developed with that 

principle in mind. The similarity of momentum and 

vapor transfer was shown in some excellent photographs by Albertson (1948) whose work confirmed 

that evaporation is a boundary-layer phenomenon.

The character of the flow and the nature of 

the boundary are important. In experimental work 

it is possible to set up conditions of laminar and 

turbulent flow, but in nature laminar flow is rarely 

observed. The present discussion is based on the 

postulate of turbulent flow at all times. The underlying surface may be "smooth" or "rough" in the 

aerodynamic sense. There exists at the boundary a 

very thin layer, of the order of a fraction of a centimeter in thickness, in which the velocity gradient is 

constant, that is, velocity is proportional to distance 

from the boundary. Such flow is laminar and the 

thin layer is called the "laminar layer." If the protuberances and irregularities of the surface project 

above the laminar layer, they affect the velocity 

distribution, resulting in turbulent flow over a rough 

surface. If the laminar layer completely encloses the 

roughness elements, the flow above is smooth.

Boundary-Layer Velocity Distributions

Considerable evidence exists that for turbulent 

flow without density gradients the velocity in a fully 

established boundary layer over a plane surface 

varies with the logarithm of height. Goldstein (1938, 

pp. 376-382) gave the following equations, based on 

the work of Prandtl and Schlichting:

-H- = C + 5.75 log i. (4)

(a) Smooth flow: C = 5.5 + 5.75 log

V

u.

< 7.08.

(b) Transitional: C = 9.58; 7.08 ^ ^-^ ^ 1^

v

(c) Transitional: C = 11.50 1.62 log ^-^;

v 

(d) Rough flow: C = 8.48; 70.8 ^ .

V

The variation of the roughness constant C is shown 

by figure 35*, taken from Schlichting (1949). In general, C must be determined experimentally. The 

values in equation (4) are for sand roughness of the 

type used by Nikuradse (1933). The actual roughness 

e and the corresponding roughness constant C are 

related to the equivalent sand roughness es, for rough 

flow, by

 =r 8.48 + 5.75 log = C + 5.75 log . (5)

U. fa 

* All symbols and units used are explained in the Appendix.

ROUGHNESS CONSTANT C

O> VI 00«O C

t

1

/

/

/

/ 

*

/* '

^

x

X

^

LIMITS FOR SMOOTH AND ROUGH FLOW 

SMOOTH ROUGH 

NIKURADSE u.t./v < 10 o.e.A > 100 

SCHLICHTING o.e.A < 3.9 u.«./v > 75 

GOLDSTEIN u.«,A < 7.08 U.../K > 70.8

0.5 1.5 2.0 

LOG10 (U.C./K)

2.5 3.0

Figure 35. The roughness constant C as a function of (u. f /'') 

47
In many cases, particularly for flow over water, the 

roughness c cannot be precisely specified; such a 

specification could be made only if the roughness 

elements were of uniform height, shape, and spacing. 

Therefore, equation (4) is used, with the introduction 

of a "roughness parameter" z0/ which is related to 

cs by z0 = fs/30 (Nikuradse, 1933). Substitution for 

cs in equation (4) gives

 = 8.48 + 5.75 log = 

u* 30 ZD

^2.5 In , (6)

zo

which is the equation generally quoted as applicable 

for rough flow in the atmospheric boundary layer. 

The use of 5.75 and 2.5 requires the use of Ic0 = 0.40 

for von Karman's constant (in equation (6), 2.5 = 

1/ko). Although various values of fco appear in the 

literature (Millar, 1937, Montgomery, 1943), the value 

chosen here is consistent with equation (4)) the most 

generally accepted wind equation for the type of 

flow treated, and also with the assumption that the 

friction velocity u* is constant with height in the 

boundary layer. This assumption is part of the discontinuous-mixing (mixing-length) approach to mass 

transfer. In reality u« does vary slightly with height, 

but apparently the small errors so introduced are of 

equal and opposite magnitude to those introduced by 

the assumption of a linear mixing length (Montgomery, ibid. p. 94). Near a boundary, that is, within

a boundary layer, any value of k0 between 0.31 and 

0.45 seems appropriate (Montgomery, ibid. figs. 1, 2). 

The value of 0.40 appears to be the most generally 

accepted value and was therefore used in the computations.

It is somewhat misleading, therefore, to say that 

equation (4) "applies." The wind generates waves on 

a lake surface, and the waves change in height, 

spacing, steepness, and speed downwind. Whether 

the waves or some other surface characteristic represent the roughness elements, it would be difficult to 

describe the entire surface by a single roughness c. 

Therefore equation (6), involving the equivalent sand 

roughness es, must be used for rough flow. Only 

under controlled (laboratory) conditions, could one 

use the right side of equation (5).

Figure 36 illustrates laminar, smooth, and rough 

flow, based on equation (4). In analyzing wind data 

from Lake Hefner, the criteria for the various types 

of flow are those given with equation (4). Using 

e8 1= 30 ZG, these become:

Smooth flows .236;

Transitional: 0.236 2.36; (7)

Rough flow: > 2.36.

Figure 36. Laminar, smooth, 

and rough flow.
In using these criteria, z0 is determined from equation 

(6). Colder (1949) quoted slightly different limits for 

the criteria, based on limits of 4 and 75 for the 

parameter u. «8/v. Lake Hefner wind data do not 

require a closer definition of these limits.

Although equation (4) is based on considerable 

experimental evidence, the expressions for smooth 

and rough flow have also been connected with the 

mixing-length theory of turbulent flow. Since the 

theory may have to be modified, wind measurements 

at Lake Hefner were analyzed in the light of this 

equation. It should be noted that equation (4) does 

not apply very near the boundary since it does not 

describe the laminar flow which exists there. According to evidence summarized by Goldstein (1938), 

equation (4) should not be used when u* z/v is less 

than 30. The lowest height at which wind measurements were made at Lake Hefner (2 meters) meets 

this requirement.

Analysis of some 23,000 individual wind profiles 

measured between 2 and 8 meters over the center of 

Lake Hefner yielded the averages of significant parameters given in table 3, based on 98 cases (3-hour 

averages) in which the lapse rate between the surface 

and 8 meters was essentially adiabatic.* Because of 

the limited number of adiabatic cases, an alternate 

procedure was used to check results. A multipleregression analysis was made, using UH/UU, (T» TO), 

and us as the variables. It was then possible to 

use all the data and to separate the effects of temperature and wind, thus obtaining values of UB/UZ 

(and therefore also z0) for adiabatic conditions.

Two somewhat unexpected conclusions can be 

drawn from table 3: (1) the lake surface is aerodynamically rough at all times; (2) the roughness parameter z0 and the resistance coefficient (u./ug)2 

increase with wind speed. Previous consideration of 

the same problem led to the suggestion, here discarded, that there is a critical smooth-to-rough velocity of the order of 7 meters per second at 8 

meters, and that in the rough range z0 and (u./us)2 

are independent of wind speed. Even if us equals 100, 

for example, the profiles give a value of 5.5 for the 

friction velocity u., whereas with assumed smooth 

flow and the same u8/ u* would assume the value 3.5. 

The friction velocity is a particularly reliable parameter since, for a given height range, it is uniquely determined by the wind profile without regard to any 

assumption of smooth or rough flow. Although the 

ratio us/u2 was used to simplify the analysis, the

* An adiabatic lapse rate would give (T8 TO) = 0.1 °C. 

Differences of 0.2, 0.1, and 0°C were considered adiabatic.

profiles were also examined on semilog plots. When 

the measurements at 2, 4, and 8 meters were considered, no significant deviation from the logarithmic 

equation could be detected. It was therefore assumed 

that logarithmic profiles prevailed down to the 

heights z0 given in table 3.

There is still some doubt about the character 

of the flow below a u$ of about 4 meters per second, 

since light winds were infrequent at Lake Hefner. 

If smooth flow is assumed for light winds, and ug/uo 

plotted against u$, the curve for smooth flow based 

on equation (4a) meets the curve represented by the 

data in table 3 at a value of u8 of approximately 

20 cm per sec. Moreover, it will be seen later that 

even for very low wind speeds two evaporation 

equations based exclusively on rough flow give satisfactory results.

The roughness parameter of a natural water 

surface is a critical factor because the computed 

evaporation depends so much upon it. Values of ZG 

for open water, generally the open ocean, have 

variously been quoted between 0.02 and 0.6 cm. 

Even if there were agreement as to value, it would 

not follow that the same value would apply to a 

small lake. Furthermore, there have been suggestions 

of a more or less abrupt transition between smooth 

and rough flow at a velocity of about 7 meters per 

second, measured at a height of about 8 meters.

A separate investigation of this question was 

carried out, but the necessary adiabatic conditions 

were so rare during the 2-week observation period 

in April 1951 that final results had to await the 

analysis described above. Although z<> was found to 

vary from 0.55 to 1.15 cm, the results can be considered in fair agreement with the value of 0.6 cm 

given by Rossby (1935) for the open ocean under 

equilibrium conditions. That no transitional velocity 

existed was at first surprising, but was confirmed by 

application of every known criterion.

Another test was made to decide whether equilibrium conditions prevailed over the lake surface

TABLE 3. Adiabatic Wind Profiles over Lake Hefner

"8 

(em see" 1 )

100

200

300

400

600

1000

1200

1500

u. 

u8/u2 (cm see" 1 ]

1.239

1.241

1.243

1.245

1.250

1.259

1.265

1.269

5.5

11.3

17.1

22.8

34.7

59.2

72.8

91.9

'0 

1 (em)

0.58

.63

.66

.69

.77

.94

1.05

1.15

<«

(em)

17.4

18.9

19.8

20.7

23.1

28.2

31.6

34.6

«. *o/v

21.4

47.5

75.2

105.2

178.2

371.0

510.0

704.0

(u./«8) 2

0.00306

.00320

.00325

.00326

.00335

.00350

.00366

.00375

Surface

Rough

Rough

Rough

Rough

Rough

Rough

Rough

Rough

49
which is always a wave-generating area. The Sverdrup-Munk (1947) wave-height theory was tested 

with the aid of instruments already described. If 

equilibrium conditions prevailed, then fetch rather 

than wind duration would control wave height. Such 

was found to be the case (fig. 37). The curve for 

measurements at Abbotts Lagoon, reported by Johnson (1950), is shown in the figure as further verification.

Although the values of z0 quoted above are 

based only on measurements near the center of the 

lake, they have been considered applicable to the 

entire surface and used accordingly. There is certainly a possibility that ZQ varies downwind, since 

all the conventional surface characteristics (wave 

height, etc.) do so. On the other hand, one can not 

precisely define, with present knowledge, the roughness elements of a surface which is changing shape 

in a complex fashion downwind.

GROWTH OF THE BOUNDARY LAYER. Schlichting (1949) gives a derivation of an expression for 

the thickness of a growing boundary layer over a 

smooth plate. A similar analysis can be performed for 

a rough plate. The drag W of the water surface can 

be expressed as

W = fr (x) dx = fpu (U u) dz, (8)

0 0

dVY/dx = T (x). (9)

so that

Equation (8) can be rearranged to give the drag in 

terms of the momentum integral J,

(10)

10-' 

8

10-' 

B

 SVERDRUP-MUNK (1947)

 ABBOTTS lAGOON (1950) 

O LAKE HEFNER

Based on table 3, u can be replaced by (see equation (37), Anderson, Anderson, and Marciano, 1950)

u = qu. (z/z0)a 01)

where q and a vary slightly over the velocity range 

observed. In evaluating J we use

u/U = (z/S)tt (12)

with the result that J is a slowly varying number equal 

to a/[(a + 1) (2a + 1)]. The drag is then

8 (x) (13)

and, using equation (9), 

dW

dx

dl' <U)

The shear stress r is eual to pu*2 so, using equation (11),

q2 (S/z0)2a ' 

Now from equations (14) and (15),

a dS 1

(a-f l)(2a+l) dx q2 (8/z0)2a 

and the solution of equation (16) is

_ i

T(n J_ 1W1 J- 9,^2 ,J2a v 1 l 

8-

(15)

(16)

z0 a x

(17)

The thickness of the boundary layer at a downwind 

point representing the average value of x at Lake 

Hefner is given in table 4.

Figure 37. Wave height as a function of 

wind speed and fetch.

8 10s

50
TABLE 4. Boundary-Layer Thickness at x =. 1750 

Meters.

The Rossby-Montgomery equation is

(meters per second) 1 10 15

5 (meters) 41.0 42.9 43.2 47.4

These values of 8 seem high, and in view of the 

uncertainty (Prandtl, 1934) as to boundary-layer 

thickness, they may be considered approximations 

only. The variation with height of various properties 

in the upper part of a boundary layer is so gradual 

that it is difficult to define the exact height at which 

the influence of the boundary ends. During the course 

of the measurements, upwind vapor-pressure profiles 

were compared occasionally with those over the 

middle of the lake. Little or no modification was detectable at 16 meters, the highest level of measurement. Considering the possibility that under stable 

conditions the boundary layer would be even thinner, 

it was decided to use the next lower level, 8 meters, 

as the upper limit of integration in the various computations.

In the development of "point" evaporation equations (for example, Sverdrup's equations, where 

E = F times area) the upper limit of integration can 

be any height within the boundary layer, including 

the value z = 8. For "area" evaporation equations 

(for example, Miller's or Button's), the upper limit of 

integration must be outside the boundary layer, with 

z = 8 the lowest value permissible. The choice of 

z = 8 = 8 meters for both approaches, therefore, 

satisfies these requirements and insures that if the 

barge is properly located, the point solution will be 

merely a special case of the general vapor-blanket 

approach.

GENERAL WIND EQUATIONS. At the close of 

the first phase of the project, it was concluded that 

the search for a general wind equation was far from 

successful. Anderson, Anderson, and Marciano (1950) 

pointed out some of the objections to the RossbyMontgomery (1935) approach. Holzman (1943) made 

an intuitive transformation of Rossby's and Montgomery's results, with little actual improvement. Sverdrup's (1936) wind equation removed some of the 

objections to the first approach, but retained much 

of its mathematical complexity. All these attempts 

at generalization were restricted to assessing the influence of stability, although Sverdrup (1938) later 

suggested that his results also applied to the unstable case.

3z k0 z V ^ 

while Holzman's transformation is

(18)

(19)

The derivation requires that a be constant with respect 

to stability. Sverdrup (1936) found that his measurements, mostly in the stable range, indicated a = 11 

and he later (1938) suggested that the same value 

applies for instability. Deacon (1949) argued that, 

with observed values of Ri = 0.3 and consistent 

wind profiles, a = 11 would result in (aRi) < 1 and 

give imaginary values of 8u/8z when equation (18) 

was used. While a 11 is satisfactory in the stable 

range, the requirement <r = constant thus casts doubt 

on Sverdrup's extension of his results to the unstable 

case. Insofar as equation (19) is concerned, Deacon's 

conclusion that a = 7 has the same drawback the 

equation would break down for Ri > 1/7, giving 

imaginary values of 8u/8z and, therefore, imaginary 

values of evaporation. To confuse the issue further, 

Pasquill (1949) concluded that the Rossby-Montgomery equation (18) provides "reasonable agreement 

with the results in stable conditions if a- 12, while 

in unstable conditions the Holzman formula (19) 

gives corresponding agreement using the same value 

of a." This complete and arbitrary change of basic 

equations with changing stability can hardly be considered satisfactory when one is seeking a general 

wind equation.

A series of unusually detailed wind and temperature profiles was obtained to explore this question 

further. Temperature differences were measured with 

a 5-junction thermocouple accurate to ±0.02°C. 

Wind profiles were obtained with hot-wire anemometers accurate to ±2 per cent. After grouping the 

observations, it was found that, at the 4-meter level, 

the most stable group gave an average value of 

Ri = 0.225, while the most unstable gave Ri = 1.7. 

Neither a rr 11 nor a = 7 would appear to be 

proper. In fact, it appears from the data that a, 

which is the reciprocal of a proportionality constant 

originally used in the Rossby-Montgomery approach, 

must vary with height in order to prevent the two 

proposed wind equations from breaking down.

Figure 38 shows the values of a based on these 

measurements, the notation across the top indicating 

the number of observations entering into each group 

average. Since one cannot use an equation to test

51
NO. OF OBSERVATIONS AVERAGED 

. 10 [ 11 402-J20J

IN EACH S-INTERVAL 

I2-I 21 U

1

1 

I

CURVES BASED ON EQUATIONS (24) AND (25)

 FOR INTERVAL z, 8M 

_ _ _ FOR INTERVAL z» 4M

"*"" «= i . .

 ^"'

-1 -^

1 

1

/I

'

/

 

1

\

'

\ 

\ 

\

Li ' ^

\

\

i

\ rs

V

^\

^^

^- -

 1.0 .8 .6 .4 .2 0 +.2

S X 104 = (T8 rrf/o,2) X 10*

itself, use was made of Deacon's (1949) empirical 

wind equation:

3u/3z = az-0. (20)

This is very similar to Sverdrup's (1936) approach, 

with the added advantage that it can be applied to 

instability as well. In equation (20)

ft > 1 for instability,

ft 1 for the adiabatic case, and (21)

ft < 1 for stability.

Using the definition

(22)

with equation (20) and Deacon's (1949) approximation*

(23)

it can be shown that

T 1 

(24)

The constant a in equation (20) is determined 

from the boundary condition that, as the surface is 

approached, the effects of stability are suppressed, so

* The validity of the assumption that the index /3 is the 

same for wind and temperature was examined using Pasquill's 

(1949) measurements and found to be approximately correct. 

However, the Lake Hefner data do not substantiate this assumption since the wind profiles between 2 and 8 meters were 

essentially logarithmic regardless of stability. The validity of 

Deacon's assumption is therefore questionable, but it has been 

tentatively accepted in order to test the entire theory.

Figure 38. The stability parameter a.

+ 4 +.6 +.8

that the wind approaches that given by the logarithmic law. This results in a = u»/fc0 zo1 """. The constant 

b in equation (23) is determined from the condition 

that as z approaches z0, T approaches T0. The latter 

is not exact but simplifies the computations and makes 

no essential difference in the results. Deacon's equation can be combined with equation (19) above to 

give (Anderson, Anderson, and Marciano, 1950, p. 36)

Ri

(25)

When the value of Ri given by equation (24) is substituted in equation (25), a is obtained and figure 38 

is based on this analysis. The abscissa is based on 

the last term in equation (24), that is,

c _

T8 -Tn (26)

in order to compare results with Deacon's. The relation between S, Ri, and (3 for Lake Hefner is shown 

in figure 39, and since this does not differ appreciably from Deacon's results for a much smaller 

roughness parameter, one must conclude that the 

relationship is unique and that ft is a parameter 

governed only by the Richardson number. It should 

be noted that the great mass of the routine measurements at Lake Hefner over a 16-month period verify, 

in figure 39, conclusions based on the relatively few 

preliminary measurements. It can be concluded, 

therefore, that the curves shown in figure 38, which 

depend on the relationship between S, Ri, and ft, 

would also be further substantiated by using additional data.

52
 1.50

Figure 39. Relation between 

Richardson's number, /9 and S.

 .70 .22

sPo

.30 .55

BASED ON PRELIMINARY MEASUREMENTS 

CLASSIFIED IN FIG. 38 

BASED ON AVERAGES FROM 3000 ROUTINE 

OBSERVATIONS, GROUPED BY (T, Ta) 

AND u.

In summary, there is no basis for the assumption that a is a constant. The measurements indicate 

that a decreases markedly with height and, at any 

given height, varies with stability. Any choice of a 

single reasonable value for a would cause equations 

(18) and (19) to break down numerically, entirely 

aside from the fact that the theoretical basis for awould break down unless it were constant with respect to stability. The alternative, the use of the 

actual variation found in a, would be only an arbitrary adjustment of an empirical number.

A final attempt was made to obtain workable 

results with equation (19). It was found that a and Ri 

vary with height in such a way that their product is 

quasi-constant with height. An evaporation equation 

based on this finding produced such poor results in 

all comparisons that the approach was abandoned.

The possibilities of the above approaches apparently having been exhausted, it was decided to 

consider further Deacon's (1949) empirical wind 

equation, based on equation (20) above:

ft)

(27)

The relation between S and ft shown in figure 39 

is based on this equation. With observed values of 

TS, TO and us, ft can be obtained from figure 39. 

This value of ft, with the observed u$ and z0 already 

known, leads to a value for u*, using equation (27). 

Then, with the usual assumption that u* does not 

vary with height, and accepting Deacon's (1949) 

statement that ft does not vary appreciably with 

height, equation (27) can be used to reconstruct the 

wind profile. It was found that this technique enabled 

measured wind profiles to be reproduced with reasono 

sxio4

+.2 +.4 +.6 +.8 + 1.0

able accuracy. Doubts as to the validity of equation 

(27) are raised below but, even apart from these, 

it would not necessarily follow that a relationship 

suitable as an empirical wind equation would yield 

an evaporation equation which would successfully 

account for the effects of changing stability. Nevertheless, an evaporation equation based on equation 

(27) is developed and tested below.

There is an apparent paradox in connection 

with this analysis which, when resolved, leads to 

some interesting speculations. First, the measured 

wind profiles indicate no appreciable stability effects 

between the 2- and 8-meter levels. With respect to 

equation (27) and the associated analysis, this should 

lead to the conclusion that ft does not vary appreciably from unity, the adiabatic case. On the other 

hand, large variations in stability S were observed, 

leading to large variations in ft. Furthermore, the 

relationship between S and ft is based on a large 

number of observations and verifies similar measurements (Deacon, 1949) over a land surface. Finally, 

it was found that the measured wind profiles could 

be reproduced with reasonable accuracy by the 

method described above. Considering equation (27) 

and the possible variation (that is, either variable 

or constant) with height of each of the three parameters ko, u* and ft, which must appear together, 

there are eight possible combinations which could 

reconcile the apparently contradictory results. One 

of these possible combinations, the one actually used, 

has all three parameters constant with height. This 

combination leads to the conclusion, in order to 

reconcile the results, that the effects of stability are 

not discernible above 2 meters. On the other hand,

30Z5ZO O - 54 - 5 53
it is well known that the local lapse rate 

usually changes with height, so that a combination 

having ft changing with height would also appear 

attractive. Therefore, the departure of ft from unity 

should be increasingly large as the surface is approached, should decrease with height from the 

surface to 2 meters, and then remain essentially 

constant beyond 2 meters. This is in accord with 

common knowledge that the largest lapse rates, both 

stable and unstable, occur just above the surface.

In contrast to this model, Deacon (1949, p. 97) 

states that "it is found that the values of /3 obtained 

from the 4:1 m velocity ratios . . . are generally 

somewhat closer to unity than those giving the best 

fit to the whole velocity profile between 0.5 and 8 

metres. It is considered that this is due to ft not 

being exactly constant with height; the discrepancy 

would be explained if the departure of ft from unity 

increased somewhat with height." Since the boundary 

conditions for equation (27) require ft to approach 1 

as z approaches 0, this requires that the departure 

of ft from unity increase with height, which is exactly 

opposite to the model proposed above.

The contradiction between these models probably arises from the neglect of the slight, but nevertheless real, variation of u» with height. The first 

model appears to violate the boundary condition 

imposed on ft near the surface in deriving equation 

(27). However, from a physical point of view, it is 

preferable to relate ft to the local temperature 

gradient OT/^z), which the first model does, so 

again the inclusion of the real variation of u* with 

height would probably act to reproduce the proper 

value of u.

These considerations lead to the belief that an 

evaporation equation based on equation (27) will 

probably be unsuccessful in assessing the effects of 

varying stability since equation (27),although useful 

as an empirical wind equation, cannot unambiguously 

couple the local wind shear $u/c)z to the local temperature gradient ^T/^z.

A new approach to the turbulence and masstransfer problems has recently been made by Lettau 

(1949). Lettau's boundary layer is not the same as 

that discussed here and is equivalent to the ordinary 

"friction layer." The top of Lettau's layer is the 

level at which the wind vector is first normal to the 

pressure-gradient force. Near the ground, Lettau 

uses a "surface layer," in which turbulence increases, 

and an upper layer in which turbulence decreases, 

with height, presumably reaching some residual value 

at the geostrophic level.

If we arbitrarily designate the 4-meter level 

as the top of Lettau's surface layer, in which the 

vapor flux is constant with height, then Lettau's evaporation equation would be

0.623 PfcoU4(eo-e4) _. (2g)

zo

In 'd'f/'dz, J replaces 6 (potential temperature); Tm 

is the mean temperature in the surface layer; and 

Ou/c)z)0 is the adiabatic wind shear. The main 

problem would be to evaluate the term QT/Qz, which 

varies with height. Equation (23) may be a reasonable first approximation.

For the adiabatic case, the term in brackets in 

the denominator reduces to unity and the above 

equation becomes similar to Sverdrup's (1946), except 

for the smaller interval of integration. Although no 

results are presented below, it is obvious that Lettau's 

equation corrects Sverdrup's in the right direction. 

The ratio of Lettau's evaporation to Sverdrup's works 

out to

FL __ 1.11 (e0 e4 ) ,20.

Fs (e0 e8)

In order to give close agreement between FL and 

observed values, the ratio of vapor-pressure differences in the above equation must have a value of 

about 0.5. The true ratio probably is close to that 

value, and if the exact height of Lettau's surface 

layer were known this point could be examined. But 

Lettau's surface layer is closely related to time 

changes in vapor flux OF/^t), so it is likely that 

over relatively short periods the height of this layer 

would vary within such wide limits that the choice of 

one average value, as in the usual boundary-layer 

approach, would not be proper. While Lettau's theory 

is in several ways an improvement over existing approaches, this study was not designed to test his 

equations. Much more precise measurements and 

more elaborate calculations than those used here 

would be needed to verify his theory.

MASS-TRANSFER EQUATIONS

Two basic approaches were used in the development of the various evaporation equations tested. 

One group of equations is based on the concept of 

discontinuous mixing as developed in Prandtl's mixinglength theory. One of the equations tested is based 

on the concept of continuous mixing first expressed 

quantitatively by Taylor (1922). A physical and mathematical review of the two approaches has been 

given by Anderson, Anderson, and Marciano, (1950).

54
Discontinuous Mixing

EXISTING METHODS. The most general of the 

equations involving the discontinuous-mixing concept 

is that of Millar (1937). His approach is based on the 

growth of a "vapor blanket" or a "moisture boundary 

layer," analogous to the growth of the momentum 

boundary layer illustrated in figure 34. This approach 

will be considered in more detail, with only the case 

of rough flow treated.

Rearranging Millar's symbols, his parameter t, 

the "potential of vapor concentration," can be rewritten as

. 0.622 p ,

r = - (eo e~). (30)

In the discussion below, f is regarded as a transferable property in the same sense that momentum is 

transferred by eddy processes.

Millar's approach is based on the skin friction 

of a flat plate as discussed by Prandtl (1934) and 

von Ka'rman (1934). The underlying principle is that 

friction between a fluid and the underlying surface 

is accompanied by a corresponding change of the 

momentum carried by the fluid.

Compare the momentum carried by a fluid 

through cross section (1) upstream from the edge 

of a plate and through cross section (2) at a distance 

x (fig. 40). The mass passing per unit time per unit 

area through (2) is puA. This mass had momentum U 

per unit mass in (1) and u per unit mass passing 

through (2). The loss of momentum is then pu (U u) 

dA, and the total loss of momentum in unit time is 

then Jpu (U u) dA, the integral being taken over 

the whole of section (2). This is equal to T, the total 

frictional force* acting on the plate from the leading 

edge to x, that is,

T = J> (U u) dz (31)

for a plate of unit width. The value of (U u) is 

negligible for z > 8, the height of the boundary 

layer. In fact, the height where this term becomes 

negligible is taken as the definition of the (momentum) boundary layer. Therefore,

T s u) dz. (32)

The friction per unit area, or the shearing stress r, 

is then dT/dx. Therefore,

* T is the tame as W of equation (8), a different symbol 

being used here to maintain Millar's notation.

T _dT__d_ T "~ dx ~" dx (33)

We now make the transition to the property f. 

In cross section (1) of figure 41 the evaporation per 

unit area per unit time is Uf0 while in section (2) it 

is uf. If in section (2) the velocity is taken at z = 8, 

then we can use u for both sections. Then the total 

mass of vapor evaporated per unit time, from the 

leading edge to the distance x, is

M (x) = oft, - f) dz. (34)

The evaporation per unit area at a point is dM/dx 

or

 

dx

 

dx S

u (f0 - f) dz (35)

-o

Equation (34) is analogous to equation (32), and 

equation (35) to equation (33). The term in brackets 

in equation (33) is the momentum integral, while the 

corresponding term in equation (35) may be called 

the "vapor integral."

Millar points out that f should be measured at 

the top of the vapor blanket, z Sw, while wind 

speed can be measured at any level. We here make 

the assumption that 8 = 8W and measure both properties at the same level. It was pointed out above, 

in the discussion on the growth of the momentum 

boundary layer, that 8 meters has been chosen for 

this reference level. Values of 8W will be calculated 

below for comparison with 8.

(i) (2)

Figure 40. Momentum exchange in a boundary layer over a 

flat plate.

'"'T r

 " ' u /' ' 1 " Wl

s ' / '

f 1 _/ 1

J / L^/ ,

f, x = 0

r 

f/

/

t

s.1

(1) (2)

Figure 41. Analogy between momentum boundary layer and 

the vapor blanket, for 5 = 8

55
It is necessary at this point to consider the 

several approaches made by Millar. For evaporation 

from "an infinitely extensive, free water surface 

without waves" (Millar, 1937, Section 7), he considers 

a smooth surface, including the laminar layer. For 

evaporation from "an infinitely extensive surface, 

roughened by waves" (Millar, 1937, Section 8), he 

uses a three-layer model (Anderson, Anderson, and 

Marciano, 1950, p. 27), accepting the Rossby 

(1936) distribution of shearing stress in the vicinity 

of the waves. In both these derivations there is no 

mention of a boundary layer or vapor blanket, although the wind equations used apply only in a 

boundary layer. Obviously, Millar was implying that 

8 becomes constant as x approaches infinity. Therefore, equilibrium having been established, it is necessary only to perform the various integrations up to 

the proper height. Evaporation per unit area per 

unit time would not be a function of x, and multiplication by an area would give total evaporation. 

These are essentially the assumptions involved in 

equations referred to in Anderson, Anderson, and 

Marciano (1950) as "point" equations. Sverdrup's 

evaporation equations, for example, are of this type 

and somewhat similar to Millar's. For moderate values 

of x, F is assumed to be computed at the particular 

value of x which yields a true mean for the entire 

surface.

On the other hand, the vapor-blanket analogy 

to the skin-friction plate was applied to "evaporation from finite pans" (Millar, 1937, Section 9), and 

the mathematical treatment indicates the assumption 

of a smooth water surface. However, Millar found 

that "the predicted evaporation was only about half 

as great as was actually observed. Many minute 

ripples were to be seen on the surface of the water 

during the experiments, which gave it a rapidly 

fluctuating, dimpled appearance." He accordingly 

changed an empirical constant, which he had originally derived from Nikuradse's experiments in a 

smooth-walled wind tunnel. It is not clear that the 

mere changing of one constant makes the entire 

derivation, based on smooth flow, applicable to rough 

flow. It must also be noted that Millar chose a value 

for this constant which "best agreed with the observations."

For evaporation from a lake of finite length, with 

the surface roughened by waves, it would be more 

appropriate to combine the vapor-blanket approach 

with the three-layer model used by Millar. It is desirable to determine first the accuracy of Millar's 

original approach.

Millar uses a general wind equation of the form*

" -n

?-. (36)

u. k0

For a rough surface ft = 1, and rj Z/ZQ. In equation (34) above, the variables are transformed so 

that dz = z0 drj. For the lower limit of integration, 

at z = z0, r) = 1. At the upper limit, z = SWI rf = 771 

= 8«,/zo.

Therefore equation (34) is rewritten, using Millar's equation (9b),

M (x) = f u. 4>f0 (l - ^-

J \ <f>l

- (37)

fr fa)-

This is analogous to Millar's equation (9d) which 

applies to a smooth surface. Since ^ = rj\ (8W) = 

771 (x), equation (37) gives total evaporation from a 

strip of unit width and length x. By a similar procedure based on the transformation of variables, Millar's equation (9c) becomes, for the rough case,

"i ii

^ = (Rx)r = Xr fa) = C4*L (V dr). (38) 

Zo \J <pi /

ft ft

The mean evaporation per unit area from a strip of 

length x is M (x)/x, or

F = fo u. (*/X)r. (39)

This is of the same form as Millar's equation (9e), 

but with a different value of F arising from differences 

in u* and (^/X).

Millar describes the evaluation of * and X, and 

the relation between Xr and (*/X)r can be determined with the aid of tables given in Jahnke and 

Emde (1945). This relationship is shown in figure 42.

A corollary result of this analysis is figure 43, 

which shows the growth of the vapor blanket downwind over the lake. It will be seen, by comparing the 

values given in the table in figure 43 with those given 

in table 4, that 8 = 8W. This is gratifying, since it 

tends to verify one of the basic assumptions of the 

mass-transfer approach: complete similarity of eddy 

transport of momentum and water vapor in the atmospheric boundary layer.

* Here Millar's /3 is not to be confused with Deacon's, 

there being no connection.

56
Figure 42. The relation between Xf and

.16

345

LOGlO Xr (X. = «/« )

10'

VALUES OF 8. FOR x = 1750 M 

(TO BE COMPARED WITH TABLE 4) 

u, (MPS) 14 10 15 

8. (M) 44 45.5 47.5 51.7

10"

10s

10* 10' 10s 10'

Figure 43. Growth of the vapor blanket downwind.

57
The above analysis can be used to develop 

evaporation equations by two methods. The exact 

method is to extend equation (37), which gives total 

evaporation from a strip of unit width and length x, 

to a circular area. Curve-fitting figure 42 with the 

relation */X 0.275 (x/z0)-° 135, equation (37) can 

be used with polar coordinates to give total evaporation from a circular area as

E = 0.275 f0 u. z0°- 135 r1M*f(cos 0) 1 - 865 d0. (40)

(41)

After integration and simplification, with 8 meters 

as the reference level,

E _ 0.535^0 us (ep-ejr1 - 865

"~ P [In (800/z0;] 

A more direct but less accurate approach is simply 

to multiply equation (39) by area. But since M (x) 

given by equation (37) is not linear in x, then M (x)/x 

given by equation (39) is only an approximation to 

the true value of mean unit evaporation. However, if 

equation (39) is multiplied by area and use is made 

of the same curve-fit for figure 42, we obtain 

_

0.475 Pko u8 (e0 e{) r1 - 865

~~ (42)

P [In (800/z0;j 

The more exact form, equation (41), gives values 

of E about 12 per cent greater than those obtained 

with equation (42). It will also be found, for Lake 

Hefner, that equation (41) gives values of E within 

2 per cent of those given by Sverdrup's (1946) equation (44) below.

This agreement between the rigorous solution, 

equation (41), and the point equation, equation (44), 

is important. The results verify the belief that if one 

is forced to use a point-evaporation equation, requiring an installation on the lake, such as a barge, the 

computation of total evaporation can be made with 

the equation E = F times area if the equipment is 

properly located. The fact that neither equation (41) 

nor equation (44) gives satisfactory results is of no 

concern the significant point is that when F is computed for the properly chosen downwind point, the

simplified equation gives practically the same results 

as the rigorous solution if the same flow model is used 

in both cases. It should be noted that the relation 

between equations (41) and (44) remains fixed under 

these conditions. The constant 0.535 in equation (41) 

arises from the curve-fit for figure 42, and the values 

quoted above are applicable to Lake Hefner. For 

another lake a different portion of the curve would 

be used and the constant would change accordingly.

The most desirable location of an on-lake installation can be obtained from a graph such as figure 

44, which shows the decrease of unit evaporation (F) 

downwind at Lake Hefner. The "best location" gives 

the true mean value of F. Anchoring problems dictated a location farther downwind but still close 

enough to the "best" location to give a reliable value 

of F. It will be seen that only for relatively small 

lakes would the location of an on-lake installation 

have a major effect on the computed evaporation.

It will be shown below that a laminar-turbulent 

two-layer model produces satisfactory results with a 

point-evaporation equation. It therefore follows that 

the same model would produce about the same results as the rigorous Millar approach described above. 

The latter requires no measurements on or above the 

lake except water-surface temperature. In support of 

this conclusion, it will also be shown below that a 

simplified area-type equation will give results comparable to those given by the best point-type 

equation.

Preliminary calculations were necessary to decide whether the use of a three-layer model, based 

on a discontinuity in u. at the wave crests, would 

improve equation (41). Since equation (41) was found 

to give essentially the same results as Sverdrup's 

(1946) equation given below, and since this threelayer model is known to increase computed evaporation, as compared to Sverdrup's values, by about 10 

per cent, attention was shifted to Sverdrup's equation. 

Only if Sverdrup's equation gave results which were 

too low, would any such modification of equation (41)

.30

V IDEAL LOCATION BARGE BASED ON EQUATION (34) 

M(x) =g0.623pAe)/P] u- zo 

dM/dx = cx~° IM Ae

Figure 44. Variation in unit evaporation downwind.

1000 2000 3000 

DISTANCE FROM SOUTH SHORE (METERS)

58
be worthwhile. It turns out, as discussed later, that 

there is no reason for considering equation (41) 

further. The decision is made on strictly numerical 

grounds, but even the three-layer model (Anderson, 

Anderson, and Marciano, 1950, pp. 27-29) is open 

to question. It sems odd that a model having a 

smooth wind profile below the wave crests should 

yield more evaporation than one in which the rough 

flow extends to the surface.

Several of the equations reviewed during the 

course of the study can be considered special cases 

of general-boundary-layer or vapor-blanket approach. With these equations, one obtains vapor 

flux through unit area in unit time and multiplies by 

the surface area to obtain total evaporation. Most of 

these equations have already been reviewed in detail 

and are only given below because they were tested 

against the water-budget control. Only equation (48) 

is intended to be "general" with respect to effects 

of atmospheric stability; the others are limited to the 

adiabatic case.

Three of the point equations already reviewed 

were tested in the following form: 

C. W. Thornthwaite and B. Holzman (1939)

0.623 pfep2 (u8 u2) (e2 es) A t . (4J)

P [In (800/200)] 2 

H. U. Sverdrup (1946)

0.623 pJc02 us (e0 es) A E = . . . (44)

R. Morris (1948)

P [In (800/zo)] 2

where

0.623 pkp2 u8 (e0 es ) A 

P [In (800/zo)] [In800/aizo)]' 

1.76u»8 u.8 In

(45)

In ai =

u. ln(u.8 //0.017) 

Anderson, Anderson, and Marciano, 1950 (equation (4), p. 64) quote an evaporation equation based 

on Deacon's wind equations (20), (21), and (27) 

above. This equation was not listed among those to 

be tested. Since then, in view of the unsatisfactory 

status of the problem of generalizing the logarithmic 

wind law, it appeared that this relatively new wind 

equation should be investigated. It has been shown 

above that Deacon's equation appears to be satisfactory as a wind equation. Given T8/ ^o/ and u8, 

with ZQ known, one can construct wind profiles agreeing very closely with those observed between 2 and 

8 meters. It remains to be seen whether the parameter ft successfully shows the effect of stability on 

evaporation. Although preliminary indications were 

that an evaporation equation based on these wind

equations might not be satisfactory, the equation 

was developed and tested in the hope that more 

could be learned about the stability problem. 

Point evaporation is

(46)

Using equation (27), K= or

K = k0 u. z0 (47)

Combining equations (46) and (47) and integrating 

as usual to the 8-meter reference level:

_ 0.623pfco2 u8 (1-/?)2 (e0 -e8)A

P [(800/zo) 1 -^ -I]2 

For /? approaching 1, the adiabatic case, equation 

(48) reduces to equation (44). For (3= 1, equation

(47) gives K ICQ u» z which, with equation (46), 

yields equation (44). Values computed from equation

(48) compared with the water-budget values of £ 

are discussed below.

Although specific comparisons are made in detail 

in another section, a general idea of the results is 

necessary at this point. Neither Millar's vapor-blanket 

approach, equation (41), nor any of the point equations (43), (44), (45), and (48) yielded satisfactory 

results. If some modification of theory or some correction of possible errors could not be made, then 

the entire discontinuous-mixing approach would have 

to be abandoned. But the approach has been so 

fruitful over the years that such a step would seem 

drastic. There is still the possibility that the flow 

models used are not complete. After some comparison of approaches it is found that all equations, 

except equation (43), ignore the laminar layer. Sverdrup (1946) recognized the laminar layer in formulating his expression for the eddy coefficient, but 

actually extended the turbulent layer to the surface 

in his integrations. The logical next step would be to 

consider a two-layer model.

Sverdrup (1937) investigated a laminar-turbulent 

two-layer model. This was not listed for testing because it was thought that his more recent equation 

would be more satisfactory. His earlier evaporation 

equation (Sverdrup, 1937, equation 14) is, in our 

notation*

* In an earlier paper Sverdrup (1936) developed essentially the same equation, but for its application discussed values 

of ZQ increasing with wind speed and having values ranging 

up to about 11 cm. These values are much too large and, in his 

1937 paper, he stated that they are applicable only to rapidly 

changing conditions. For steady conditions he recommended 

Z0 = 0.6 cm, and this is the value used in equation (49).

59
0.623 pkp u. (e0 e8)

8. +

. tU'1 

D J

(49) The evaporation equation resulting from Button's 

latest work is similar to the one he developed for 

smooth flow. From the equations given by Button 

(1949), evaporation from a circular lake of radius

Sverdrup defined the thickness of the laminar 

layer, for fully rough flow, as 81 = 27.5v/u., based 

on only a few measurements. This is close to the limiting value for laminar flow (S; = 30v/u.) proposed 

by von Karma'n (1934) on the basis of extensive

r is

where

4 + n

E = 0.623P G' u '2 + n r 2+n (e0 

P

(51)

z n

/2 + n\ 2 +"2 + n . / 2ir [ ! I ! sm[ 

f+f r _V2- _

n; 2,r V2 + n

-) z -4-n2 2

T)

measurements. Equation (49) was tested using the 

definition 8( = 30v/u*.

Continuous Mixing

The concept of continuous mixing, first advanced 

by Taylor (1922), has been applied by Sutton (1932, 

1934, 1949) to the diffusion problem. The latest of 

Button's papers may be considered a generalization 

of his earlier work applying only to flow over smooth 

surfaces.

Sutton (1949) introduces the "macroviscosity" 

N u. z0, presumed to play a part in exchange 

processes over rough surfaces similar to that of kinematic viscosity in smooth flow. For the adiabatic 

wind equation Sutton uses

u 1 . )" u« z 1 .,_. 

 = In . (50) 

u. Jco LN + (v/9)J

For smooth flow ZQ 0, so N 0 and equation (50) 

reduces to equation (4a). For rough flow, N > v 

and equation (50) reduces to equation (6). As in his 

previous work, the more exact logarithmic wind equation is replaced by a power-law approximation in 

order to solve the conventional diffusion equations. 

The macroviscosity also appears in Button's new 

form for the correlation coefficient fy, where

For smooth flow N = 0, and this equation reduces to 

the expression previously used (Sutton, 1934). The 

data selected by Sutton (1949) for illustration appear 

to verify the above equation as a satisfactory approximation.

3n

and, in G', a is given by

n2 n

O ~^^ '

(2-n)1 -"n1 -"z 2 -n (v-fu.z0)" 

(1 - n) (2 - 2n)2 ~2"

This is to be compared with equations (24) through 

(27) in Anderson, Anderson, and Marciano (1950), to 

which the above would reduce if the surface were 

assumed to be smooth (z0 = 0).

It should be noted that the n used by Sutton 

is connected to the a used in equation (11) above by 

the relation a = n/(2 n). Over the range of velocities observed at Lake Hefner, a varies between 0.150 

and 0.175, which is equivalent to a variation of 0.26 

to 0.30 in n. Sutton indicates that typical values of n, 

presumably for a typical range of stabilities, are from 

0.10 to 0.30. This illustrates one of the difficulties involved in using the power law (a variation of equation (12)) between two levels as an approximation 

to the logarithmic law. The variation of n quoted 

for Lake Hefner is for the observed adiabafic wind 

profiles, and it is presumed that the observed variation in ZQ is the cause of this variation. Approximately 

a twofold variation in z0 at Lake Hefner is accompanied by a very small change in n. On the other 

hand, the threefold variation in n, which Sutton implies is typical, can be accounted for only by additional effects of stability. But one cannot be sure of 

the relative importance of stability and dynamic 

effects on the parameter n. From the evidence at 

hand, one could say that n is principally a stability 

parameter, increasing with increasing stability. However, Sutton uses the power law, apparently with 

no intention of accounting for effects of stability. 

Further, it can be calculated from the exact logarithmic law that n must also increase slightly

60
with height to allow equation (51) to fit the entire 

profile closely. This can be illustrated by the data 

in Sutton's (1949) table II, where his power law 

closely represents the logarithmic profile only between 1 and 3 meters. This is perhaps the most serious 

drawback to a wind equation of this type. The power 

law is made approximately correct by fitting, and 

this can be done only for small height ranges. Unless 

the necessary variation of n with height is accounted 

for, a power law assumed to apply over a greater 

interval than 2 or 3 meters introduces a fictitiously 

large wind shear in the expressions for eddy diffusivity. This fictitiously large wind shear would prevail 

at higher levels, so one might say that for high winds 

(large evaporation) the computed evaporation would 

be too small. At low levels the power law would give 

a fictitiously small wind shear, so by similar reasoning, computed evaporation would be expected to be 

too large for low winds.

Another aspect of Sutton's equations has been 

mentioned in Anderson, Anderson, and Marciano 

(1950). Although originally based on Taylor's concept 

of continuous mixing, in the course of his derivations 

Sutton used results obtained from a mixing-length 

(discontinuous-mixing) theory. It would be proper to 

refer to the results as a method, rather than a theory.

Although these remarks have been offered to 

clarify the background to this approach, it is recognized that they give the impression that Sutton's 

evaporation equation might not give satisfactory 

results. Such was not the case, as will be shown 

below. In fact, Sutton's equation gave better results 

than any of the equations originally proposed for test.

EVAPORATION IN THE ABSENCE OF WIND. 

Many of the familiar empirical equations relating 

evaporation to meteorological parameters are of the 

general form

E = Ci(l+Ca u)(eo-e.), (52)

where Ci and C2 are experimental constants. This 

implies that evaporation takes place in the absence 

of wind. Equations of this type are generally obtained by a statistical analysis of data from evaporation-pan records.

Evaporation in the complete absence of wind 

occurs only by molecular diffusion, an extremely slow 

process. Point evaporation is given by

F = PD Oq/^z) (53)

which on integration and substitution yields 

F _ 0.623 PD (e0 eg) 

Pz

For z = 8 meters,

F = 2.52 X 10~6 (eo e*) cm/3 hours. (55)

This serves to give a first approximation. Using boundary-layer concepts, one finds that as u approaches 

zero, the thickness of the laminar layer approaches 

infinity. Therefore, for z approaching infinity in equation (54), F approaches zero. Thus, in comparison 

with evaporation under turbulent conditions, evaporation by molecular diffusion is negligible, unless a 

strong temperature gradient exists so that free convection cannot be ignored.

If substantial temperature gradients do exist, 

evaporation into still air is somewhat greater than 

that given by equation (54). Although stating that 

the experimental verification of his theory had not 

yet been completed, Yamamoto (1950) gave an equation for evaporation into still air, which on substitution of the units and symbols used in this report, is 

as follows:

C -_

0.327P

vD

T \T0.25

 J(eo- e,), (56)

in which yt is the coefficient of expansion of air. 

Using x = 3.45XlQ5 cm for Lake Hefner and 

TO Tz = 5°C, equation (56) reduces to

E-2.05X 10- 4 (e0 -ez) (57)

where eo and ez are in mb and E in cm/3 hours. 

Thus for a given vapor pressure difference, evaporation into still air is approximately the same as evaporation computed using equation (49) with u = 0.32 

knot.

There is sufficient reason, therefore, for ignoring 

the case of zero wind in the experimental tests, aside 

from the fact that this case was never observed. 

Those interested in this aspect of the evaporation 

problem may refer to Hickox (1946), who presents 

the results of a series of excellent experiments.

COMPARISON OF COMPUTED 

AND MEASURED EVAPORATION

Data Processing and Computing

The computation of evaporation from Lake 

Hefner based on the mass-transfer theory required the 

processing of a tremendous number of data. Observations were begun on 26 April 1950 and were 

terminated on 31 August 1951, a period of 493 days. 

At each of the four meteorological stations, wet- and 

dry-bulb temperatures at the 2-, 4-, 8-, and 16-meter

61
levels and water-surface temperatures were measured 

once each half hour and recorded on an EsterlineAngus milliammeter recorder. Each knot of wind 

movement at the same four levels, wind direction, and 

rainfall depths were recorded on an Esterline-Angus 

recorder. Excluding rainfall, a total of 14 items were 

recorded during each half-hour period at each 

station. For all four stations over the entire period 

of observations, the total number of items recorded 

was about 1,325,000. It was apparent that manual 

tabulation of the data and computation of meteorological parameters was impracticable, and the basic 

data were therefore placed on punch cards.

For each half-hour period the value of each of 

the items, as read from the trace, was entered directly on the recorder tape by one person and 

checked by another. The time of observation was 

>aken as the half-hour at which the cycle of observations was completed. Temperatures were read to the 

nearest 0.1 °C, and wind speed to tenths of a knot. 

The wind direction was taken to be that which was 

recorded most frequently during the half-hour cycle. 

For all stations except the barge, wind direction was 

coded to eight points of the compass, with code 

figure 1 representing NE and proceeding therefrom 

in a clockwise direction. For the barge station a different code was adopted, as described in the section 

"General Description of Lake Hefner" and shown 

in figure 4. For certain of the mass-transfer equations 

it was necessary to use .the vapor pressure of the 

unmodified air, and it was belived that this would be 

best accomplished by using dry- and wet-bulb temperatures at the 16-meter level at the station upwind 

from the barge. If the wind at the barge station was 

coded as 2, for example, humidity data at the south 

station were used; if the wind was coded as 3, data 

at the northeast station were used, and for code 4, 

data at the intake tower station. Wind directions 

coded as 1, 5, or 6 were not observed over any 

3-hour period during the studies.

Two sets of punch cards were prepared and 

corresponding cards checked by visual comparison. 

Thus it was possible to correct almost all the punching errors before the two sets of cards containing the 

half-hourly data were sent to the Institute for Numerical Analysis of the National Bureau of Standards. 

The two sets of cards were then compared by machine. Very few discrepancies were found as a result 

of the second comparison, and most of these could 

be resolved quickly. If there was any doubt, the observation in question was deleted before the halfhourly data were tabulated.

Three-hourly averages were computed for each 

item except wind direction. For the barge station the 

most frequent wind direction during the 3-hour 

period was taken as the mean and used in the computations. The results were punched on another set 

of cards and then tabulated.

The set of cards containing the 3-hour average 

data was then used for all subsequent computations. 

Although the evaporation equations tested generally 

appear quite complicated, it was found that they 

could be reduced to but a few variables and parameters. The parameters were given in graphical form. 

Since a digital rather than an analog type of computer was used, it was necessary to prepare a card 

deck for each graph. Evaporation was then computed 

for each 3-hour period for all the equations being 

tested. The results were tabulated and daily Totals 

obtained.

Computed and Observed Evaporation

In considering the results of the mass-transfer 

studies, A and B water-budget days were used for 

all equations except Sverdrup's (1946) form, for 

which only A days were used.* The results are shown 

in figures 45 through 48.

Three different symbols are used in figures 45 

through 48 to indicate atmospheric stability. The 

quantity (T8 70) was computed for each 3-hour 

period. If this difference was found to be positive for 

six or more of the eight 3-hour periods, the day was 

classified stable. If the difference was negative for 

six or more of the eight 3-hour periods, the day was 

classified as unstable. All other days were classified average.

The following values of physical constants were 

used in all computations:

ko = 0.4, 

p = 1000mb, 

P =1.2X 10- 3 gmcm- 3, 

r = 1.725 X 105 cm, 

v = 0.15 cm2 sec" 1 , 

D = 0.25 cm2 sec- 1 .

The Thornthwaite-Holzman and Sverdrup (1946) 

equations can be discussed together since they evolve 

from basically the same approach. Figures 45 and 

46, however, show a striking difference. Sverdrup's

* When results were first assembled it was soon evident 

that Sverdrup's (1946) equation would give values of £ which 

were consistently too large. The pattern having been definitely 

established, it was decided to use only A days so that attention could be gvien to more promising equations.

62
!2 

O

g = 

|

ir

0 0

N = 180 (

STABLE AVERAGE UNSTABLE

S) O

DAILY WATER-BUDGET EVAPORATION IN CM 

^ b> e b

«*>
(1946) equation (fig. 46), which essentially ignores 

the laminar layer in extending the turbulent layer 

to the very surface, yields results that are much too 

large. The discrepancy, furthermore, appears to be 

consistent, suggesting some fundamental inconsistency 

in the approach. It is possible that Sverdrup's (1946) 

equation gives high results because his limits of integration for the wind profile are from z = ZQ to z = z, 

but for the humidity profile they are z =. 0 to z =. z.

The Thornthwaite-Holzman results (fig. 45), while 

they are little better from the point of accuracy desired, at least show a random scatter about the 

perfect correlation line. The Thornthwaite-Holzman 

approach has in its favor the fact that it is much 

simpler than any of the others. The complicated 

aspects of flow near the surface are avoided by 

selecting two reference levels entirely within the turbulent layer. However, the problem arises of measuring small differences in wind speeds and vapor 

pressures, particularly the latter. Vapor-pressure differences between 2 and 8 meters could not be measured with sufficient accuracy with the equipment used 

at Lake Hefner. Moreover, so far as is known, equipment which will measure small vapor-pressure differences with the required accuracy and which is also 

suitable for unattended operation under field conditions is not yet available.

The results obtained with Sverdrup's (1937) 

equation are shown in figure 47. These are believed 

to be the best results obtainable with Sverdrup's 

two-layer model. If, for example, the "average" 

thickness of the laminar layer (8j 11.5v/u») is used, 

the agreement between measured and computed 

evaporation worsens. It should be remembered that 

the individual values shown in figure 47 are daily 

observations, and seldom in practice is a knowledge 

of daily evaporation necessary. As is true for the 

results shown for the other equations tested, the effect 

of atmospheric stability is apparently not significant, 

at least for daily figures of evaporation. It was considered impracticable to attempt to determine evaporation by the water-budget method for periods of 

less than 24 hours, so comparisons for shorter periods 

could not be made.

A graph illustrating the results obtained using 

Millar's vapor-blanket approach was not prepared 

because it was found that it would yield results 

closely approximating those obtained using the Sverdrup (1946) equation. In spite of the difference in 

methods, the two equations, when put in the form 

used for computing, were so nearly alike that it was 

considered unnecessary to use Millar's equation for

detailed computations. Figure 46, which gives the 

results obtained from Sverdrup's (1946) equation, 

can be considered as showing the results to be expected from Millar's equation.

The same argument can be used with Morris' 

(1948) equation, based on a three-layer model 

examined by Millar (1937). Morris modified the 

Sverdrup (1946) approach to yield even more evaporation, and Sverdrup's (1946) equation has already 

been shown to give values that are much too large.

Holzman's (1943) evaporation equation, which 

contains the term In (1 aRi), must be discarded 

completely on the basis of figures 38 and 39. Some 

observed values of (aRi) cause this term to become 

imaginary and computed evaporation therefore also 

becomes imaginary. Moreover, temperatures at the 

reference levels would have to be measured with 

extreme precision, particularly when winds are light. 

For example, if u§ u% 50 cm sec" 1 and Tg TO 

= 1°C, computed evaporation is ten times as great 

at when T8 = TODeacon's evaporation equation (48) was also 

tested for a brief period. It was quickly found that, 

although his wind equation is suitable as an empirical 

wind law, the evaporation equation derived therefrom breaks down. The effect of the stability parameter ft is to over-correct at both ends of the stability 

range. For increasing stability the equation yields 

values which are much too low, while at the other 

end of the stability scale it yields values which are 

much too high.

Frost's (1946) evaporation equation was not 

tested. He assumed a form for the mixing length 

which causes that factor to depend greatly on the 

roughness of the surface, whereas it should be primarily a property of the air. This evaporation equation is otherwise similar in development to Button's 

and, in view of the above discrepancy, was dropped.

Sutton's equation gives essentially as good results (fig. 48) as Sverdrup's (1937) form. It has been 

pointed out above that Sutton's equation was evolved 

from concepts derived from two fundamentally opposed theories of turbulent mixing. Nevertheless, if 

one is seeking a satisfactory method, this equation 

must be considered. However, any modifications of 

Sutton's approach which would increase the computed 

values, such as discussed by Anderson, Anderson, 

and Marciano (1950, pp. 11-12) should not be 

considered.

Table 5 compares the average results that can 

be expected using the several equations discussed 

above. Water-budget evaporation is taken as unity.

64
TABLE 5. Ratios of Average Computed to Observed Evaporation.

Equation Number 

Source in This Report

Water Budget (observed)

Sverdrup (1937)

Sverdrup (1946)

Millar (1937)

Morris (1948)

Sutton (1949)

(1)

(49)

(44)

(41)

(45)

(51)

Evaporation

1.00

1.12

2.10

2.13

2.20

1.07

It is evident that Sverdrup's (1937) equation and 

Button's equation are the only ones that give satisfatctory results. The standard errors of the computed 

figures of daily evaporation (about the line of best 

fit for the points shown in figures 47 and 48) are 

0.109 for Sverdrup's (1937) equation and 0.118 for 

Sutton's. Assuming that the errors are random and

normally distributed, the standard error of a computed figure of weekly evaporation for both equations may be expected to be about 10 per cent, and 

for monthly evaporation about 5 per cent. Because 

both equations give computed values that are too 

high, these computed values must be divided by the 

factors shown in table 5 to give average figures of 

evaporation in agreement with the water-budget data. 

These factors are not constant, but vary with wind 

speed. Computed values are particularly high at high 

wind speeds, especially with Sutton's equation. These 

relationships are shown in figure 49, where these 

two equations are compared with the empirical equation of best fit for the Lake Hefner data, computed 

from water-budget evaporation, wind speed, and 

vapor-pressure difference. This equation is

10-4 u8 (e0 -e8 ). (58)

3

s

5 1t0

2

s o

$

! 

il

.4

 ) /

 /

/6»

/

/ *

' e»«

0

*

/

9

 » i

,

e

e*

 o

/

»

w

«

>. 

5

(

W

_ O

^ 

0

/

1^

* e 

 c

_ 

0

/

° 9*

io$

/

e

o

0

»

N = 95 (A DAYS ONLY) 

STABLE O

AVERAGE 

UNSTABLE S>

4

O

D

U

O

O

DAILY EVAPORATION IN CM COMPUTED USING SVERDRUP (1946) EQUATION

Figure 46. Experimental test of Sverdrup's (1946) evaporation equation.

65
DAILY EVAPORATION IN CM COMPUTED WITH SVERDRUP'S (1937) EQUATION

Figure 47. Experimental test of Sverdrup's (1937) evaporation equation.

®o

N=195 (A + B DAYS)

STABLE O

AVERAGE 

UNSTABLE ©

1.0 1.2 1.4 1.6

DAILY EVAPORATION IN CM COMPUTED USING SUTTON EQUATION

Figure 48. Experimental test of Surton's evaporation equation.

66
MRS.

SVERDRUP (1937)- 

SUTTONBEST FIT FOR 

LAKE HEFNER DATA 

E=6.25X lO-'Me,, e,,)

0 * 6 12 16 20

u, KNOTS

The deviations of computed evaporation from 

the observed water-budget figures are believed to 

result from imperfections in the theoretical equations 

arising from our incomplete knowledge of boundarylayer wind structure, and are therefore not peculiar 

to Lake Hefner.

Notwithstanding the unsuccessful result of the 

search for a general wind equation, the most striking 

feature of the study is that evaporation can be accurately predicted by neglecting stability, that is, by 

using adiabatic equations exclusively. The principal 

effect of stability is to change the thickness of the 

boundary layer (vapor blanket). This effect can be 

taken into account by the proper choice of a reference level.

The effect of the size of lake on the growth of 

the boundary layer has already been discussed. For 

large lakes, the choice of the reference level for wind 

and humidity measurements can be approximated 

using equation 17, and can be determined more accurately by a few field measurements of the height 

to which modification extends since the only requirement is that the reference level be at or above the 

top of the boundary layer. Other evidence is availFigure 49. Comparison of Sverdrup's (1937) 

equation, Button's equation, and the Lake 

Hefner empirical equation.

able to demonstrate that the 8-meter level chosen 

for Lake Hefner was satisfactory and is also suitable 

for smaller lakes.

During the course of developing an empirical 

equation that could be used by the Oklahoma City 

Water Department for day-to-day computations of 

evaporation from Lake Hefner (Harbeck, 1952), a 

study was made using wind and humidity data observed at the Weather Bureau station at the Will 

Rogers Airport, located 13 miles south of Lake 

Hefner. Water-surface temperatures were taken 

as the average of the daily maximum and minimum recorded at the barge station. It was believed 

that this procedure gave a record equivalent to one 

that could have been obtained from a maximumminimum thermometer floating in deep water. Using 

the airport wind and humidity data and the observed 

water-surface temperature, the following empirical 

equation for daily evaporation was obtained as best 

fitting the data: E= 0.00177 u (e0 ea), in which £ 

is in inches, u in miles per hour, and (eo ea) in mb. 

Converting to cgs units, E = 6.47 X 10~4 u (eo ea) 

cm/3 hours, which is in close agreement with the 

empirical equation (58) shown in figure 49.

67
Because of this close agreement, it appears reasonable to assume that the thickness of the modified 

layer at the barge on Lake Hefner was less than 8 

meters, thus substantiating the choice of this level. 

For smaller lakes the modified layer would be thinner, 

and the level selected for Lake Hefner could therefore be used.

CONCLUSIONS ON MASS-TRANSFER 

STUDIES AND APPLICATIONS

General

Although boundary-layer processes are still imperfectly understood, a complete and detailed knowledge of them does not appear necessary for the 

determination of evaporation for periods of a day or 

more.

It was concluded that the wind profile over 

most lakes between the 2- and 8-meter levels can be 

satisfactorily approximated using the logarithmic 

wind law. The effects of changes in atmospheric stability are not clearly understood, but appear to be 

of little practical significance so far as the determination of daily evaporation is concerned.

An evaporation equation based on Sverdrup's 

(1937) two-layer model gives results that are in 

better agreement with observations than those obtainable from any other equation based on the 

mixing-length theory. Button's equation gives about 

the same results. It is realized that all mass-transfer 

equations are, in a sense, partly empirical, for not 

until our understanding of boundary-layer processes 

is complete can a truly theoretical equation be developed. A rigorous wind law is still not available, 

for example, nor are the laws governing the variation 

of humidity with height. The empirical equation (58) 

developed from the water-budget data is, on the 

other hand, not entirely empirical, but embodies some 

of the principles of mass-transfer theory. Though further tests of this equation should be made, it is now 

believed possible, despite gaps in our knowledge, 

to compute daily evaporation from a water surface 

with reasonable accuracy, and in that respect one of 

the chief objectives of the study has been realized. 

It is therefore considered feasible to apply the results 

from Lake Hefner to the next phase of the study the 

determination of evaporation from Lake Mead.

The use of meteorological data from a nearby 

Weather Bureau station offers tremendous advantages in economy of operation. If it can be demonstrated, for any particular reservoir, that meteorological data obtained at some distance from the site are 

reasonably representative of unmodified air conditions prevailing at the reservoir, the determination of 

evaporation for operational purposes becomes relatively simple. The only other physical measurement 

needed, the water-surface temperature, is not difficult 

to obtain.

Of considerable practical significance to those 

who may have occasion to use a mass-transfer equation is the fact that, for routine computations, these 

equations are not nearly so complicated as they 

appear to be at first glance. Although employing 

such mathematical devices as gamma functions and 

integral logarithms, for example, the equations can 

readily be simplified for routine use. Any of the 

equations can be reduced to a simple product of 

several variables, of which one or two may be obtained graphically with little loss in accuracy.

The use of the mass-transfer technique also offers 

considerable promise as a means of determining 

evapotranspiration losses. The magnitude of such 

losses has been approximated in the past by inflowoutflow determinations and other methods (Gatewood 

efa/.7 1950), and by the use of the mass-transfer 

theory (Thornthwaite and Holzman, 1939). In the first 

of these studies, a water-budget control was provided, but the meteorological observations necessary 

for the computation of evapotranspiration losses using 

the mass-transfer theory were not obtained. In the 

second of these studies, good meteorological data 

were secured and evapotranspiration computed, but 

a water-budget control was lacking. Both of these 

omissions need to be made good for a conclusive 

test of the theory. Once the theory has been tested, 

including the determination of appropriate roughness 

parameters and wind and humidity profiles, the technique can be used with confidence for the determination of evapotranspiration losses from different types 

of vegetation under conditions of minimum, maximum, and optimum water supplies.

Specific Findings

THEORY AND BACKGROUND. The results of 

the mass-transfer studies, from the point of view of 

general boundary-layer theory, may be summarized 

as follows:

1. Wind speed near the center of Lake Hefner 

varied logarithmically with height between 2 and 

8 meters, within the limits ascribable to instrumental 

error, regardless of stability variations.

2. The water surface was found to be aerodynamically rough at all times, with no evidence of a

68
critical wind speed. The resistance coefficient increased with wind speed, varying from 3.06 X 10~3 

to 3.75 X 10~ 3; the roughness parameter varied from 

0.58 to 1.15 cm.

3. No completely satisfactory general wind 

equation could be found. For the empirical wind 

equations reviewed, it was found that if wind shear 

is proportional to the height raised to a power, that 

power must vary with Richardson's number and therefore with height.

PRACTICAL RESULTS. From the point of view 

of obtaining an evaporation equation suitable for 

field use, the findings can be summarized as follows:

1. Two of the theoretical equations, Sverdrup's 

(1937) form and Sutton's, give good results.

2. It is believed that the Thornthwaite-Holzman 

equation would give satisfactory results with proper 

instrumentation, but the instrument requirements are 

exacting.

3. All other theoretical evaporation equations 

based on existing models and methods were found 

to be unsatisfactory.

4. A simple empirical equation was developed 

from the water-budget evaporation data, using wind 

speed and vapor-pressure differences measured at 

Lake Hefner. An operational version of this equation 

uses standard data from a nearby U. S. Weather 

Bureau station and the water-surface temperature of 

the lake. No other measurement at the lake is required.

REFERENCES

Albertson, M. L., Analysis of Evaporation as a Boundary-Layer Phenomenon, University of Iowa, Doctoral 

Dissertation, 1948 (see also discussion in Goldstein, 

1938).

Anderson, E. R., Anderson, L. J., and Marciano, J. J., 

"A Review of Evaporation Theory and Development 

of Instrumentation, Lake Mead Water Loss Investigations; Interim Report," Navy Electronics Laboratory, 

Report no. 159, 1 February 1950.

Colder, K. L., "Eddy Diffusion and Evaporation in 

Flow Over Aerodynamically Smooth and Rough Surfaces," Quarterly Journal of Mechanics and Applied 

Mathematics, vol. 2, part 2, June 1949, pp. 153-176.

Deacon, E. L., "Vertical Diffusion in the Lowest Layers 

of the Atmosphere," Royal Meteorological Society, 

Quarterly Journal, vol. 75, no. 323, January 1949, 

pp. 89-103.

Dryden, H. L., "The Aerodynamics of Cooling" (In: 

Durand, W. F., ed., Aerodynamic Theory) Durand 

Reprinting Committee, vol. 6, Div. T, 1934.

Frost, R., "Turbulence and Diffusion in the Lower Atmosphere," Royal Society of London, Proceedings, ser. 

A, vol. 186, no. 1004, June 1946, pp. 20-35.

Gatewood, J. S., et a/., Use of Water by Bottom-land 

Vegetation in Lower Safford Valley, Arizona, U. S. 

Geological Survey, Water-Supply Paper, no. 1103 

1950.

Goldstein, S., ed., Modern Developments in Fluid 

Dynamics, vol. 2, Chapter 15, 1938, Oxford, pp.. 

646-675.

Harbeck, G. Earl, Jr., "Evaporation Research at Lake 

Hefner," American Water Works Association, Journal, 

vol. 44, no. 8, August 1952, pp. 701-706.

Hickox, G. H., "Evaporation from a Free Water Surface," American Society of Civil Engineers, Transactions, vol. Ill, paper 2266, 1946, pp. 1-33.

Holzman, B., "The Influence of Stability on Evaporation," New York Academy of Sciences, Annals, 

vol. 44, Art. 1, May 1943, pp. 13-18.

Jahnke, E., and Emde, F., Tables of Functions with 

Formulae and Curves, 4th ed., Dover,. 1945.

Johnson, J. W., "Relationships Between Wind and 

Waves, Abbotts Lagoon, California," American Geophysical Union, Transactions, vol. 31, no. 3, June 

1950, pp. 386-392.

van Kdrma'n, Th., "Turbulence and Skin Friction," 

Journal of the Aeronautical Sciences, vol. 1, January 

1934, pp. 1-20.

Lettau, H., "Isotropic and Non-isotropic Turbulence 

in the Atmospheric Surface Layer," Cambridge Research Laboratories, Geophysical Research Paper no. 

1, December 1949.

Millar, F. G., "Evaporation From Free Water Surfaces," Canada, Department of Transport, Division 

of Meteorological Services, Canadian Meteorological 

Memoirs, vol. 1, no. 2, 1937, pp. 39-65.

Montgomery, R. B., "Generalization for Cylinders of 

Prandtl's Linear Assumption for Mixing Length," Annals of New York Academy of Sciences, vol. XLIV, 

article I, 1943, pp. 89-103.

Nikuradse, J., "Stromungsgesetze in Rauhen Rohren" 

Forschung auf dem Geb/efe des /ngen/eurwesens, 

Beilage, Heft 361, Berlin 1933.

3025ZO O - 54 - 6 69
Morris, R., "Evaporation from Extensive Surfaces of 

Water Roughened by Waves," Royal Meteorological 

Society, Quarterly Journal, vol. 74, no. 319. January 1948, pp. 1-12.

Pasquill, F., "Eddy Diffusion of Water Vapour and 

Heat Near the Ground," Royal Society of London, 

Proceedings, Ser. A, vol. 198, no. 1052, July 1949, 

pp. 116-140.

Prandtl, L, "On Fluid Motions with Very Small Friction," Proceedings of the Third Mathematical Congress, Heidelberg, 7904, Leipsig, 1905, pp. 484-491.

Prandtl, L., and Tietjens, O. G., Applied Hydro-and 

Aeromechanics, McGraw-Hill, 1934, pp. 64-65.

Prandtl, L., "The Mechanics of Viscous Fluids," (In: 

Durand, W. F., ed., Aerodynamic Theory) Durand Reprinting Committee, vol. 3, Div. G, 1934.

Rossby, C. G., and Montgomery, R. B., "The Layer 

of Frictional Influence in Wind and Ocean Currents," 

Papers in Physical Oceanography and Meteorology, 

vol. 3, no. 3, 1935.

Rossby, C. G., and Montgomery, R. B., "On the Momentum Transfer at the Sea Surface," Papers in Physical Oceanography and Meteorolgy, vol. 4, no. 3, 

1936.

Schlichting, H., "Lecture Series: Boundary Layer Theory, Part II Turbulent Flows," National Advisory 

Committee for Aeronautics Technical Memorandum 

no. 1218, April 1949, pp. 30-32.

Schmidt, E., "Verdunstung und Warmeubergang," 

Gesundheits-lngenieur, vol. 52, 1929, p. 526.

Sutton, O. G., "A Theory of Eddy Diffusion in the 

Atmosphere," Royal Society of London, Proceedings, 

Ser. A, vol. 135, no. 826, February 1932, pp. 143-165.

Sutton, O. G., "Wind Structure and Evaporation in 

a Turbulent Atmosphere," Royal Society of London,

Proceedings, Ser. A, vol. 146, no. 858, October 1934, 

pp. 701-722.

Sutton, O. G., "The Application to Micrometeorolgy 

of the Theory of Turbulent Flow Over Rough Surfaces," Royal Meteorological Society, Quarterly Journal, vol. 75, no. 326, October 1949, pp. 335-350.

Sverdrup, H. U., "Das Maritime Verdunstungsproblem," Anna/en der Hydrographie und Maritimen 

Meteorologie, vol. 64, 1936, pp. 41-47.

Sverdrup, H. U., "The Eddy Conductivity of the Air 

Over a Smooth Snow Field," Geofysiske Publikasjoner, vol. 11, no. 7, 1936, pp. 1-69.

Sverdrup, H. U., "On the Evaporation From the 

Oceans," Journal Marine Research, vol. 1, no. 1, 

1937-38, pp. 3-14.

Sverdrup, H. U., "On the Influence of Stability and 

Instability on the Wind Profile and the Eddy Conductivity Near the Ground," (In: International Congress for Applied Mechanics, 5th, Cambridge, Massachusetts, 1938, Proceedings) Wiley, 1938, pp 369- 

372.

Sverdrup, H. U., "The Humidity Gradient Over the 

Sea Surface," Journal of Meteorology, vol. 3, no. 1, 

March 1946, pp. 1-8.

Sverdrup, H. U., and Munk, W. H., Wind, Sea and 

Swell: Theory of Relations for Forecasting, Hydrographic Office Publication no. 601, Government 

Printing Office, 1947.

Taylor, G. I., "Diffusion by Continuous Movements," 

London Mathematical Society, Proceedings, 2nd Ser., 

vol. 20, June 1922, pp. 196-212."

Thornthwaite, C. W., and Holzman, B., "The Determination of Evaporation From Land and Water Surfaces," Monthly Weather Review, vol. 67, 1939, pp. 

4-11.

Yamamoto, G., "Investigations of Evaporation From 

Pans," American Geophysical Union, Transactions, 

vol. 31, no. 3, June 1950, pp. 349-356.

70
Energy-Budget Studies

by Ernest R. Anderson*

INTRODUCTION

Since 1915, when Schmidt first attempted to 

utilize the (thermal) energy budget to obtain estimates of the annual evaporation from the oceans, 

applications of the energy budget have been restricted because of the difficulty of evaluating certain 

terms, such as reflected solar radiation, atmospheric 

radiation, reflected atmospheric radiation, long-wave 

radiation from the body of water, advected energy, 

energy conducted to or from the body of water as 

sensible heat, and the change of energy storage in 

the body of water.

Schmidt (1915), for instance, neglected the 

change in energy storage by computing evaporation 

from the oceans over a yearly interval. Richardson 

(1931) investigated evaporation from California lakes 

during time intervals when the change in air temperature was small and assumed a corresponding negligible change in energy storage. Sverdrup (1940) 

investigated the energy budget of two particular 

regions of the oceans: (1) a region off the Bay of 

Biscay which was without distinct currents so that 

advected energy could be considered negligible, and 

(2) a portion of the Kuroshio current where the advected energy was assumed constant throughout the 

year and was determined from the energy budget by 

first assuming that evaporation was negligible during 

the early summer. During recent years, Holzman 

(1941) and Penman (1947) both commented on the 

difficulty of evaluating atmospheric radiation, longwave radiation from the body of water, and energy 

storage.

In addition, difficulty has been experienced in 

evaluating the conduction of sensible heat to or 

from the body of water. From the energy-budget 

equation it is possible to obtain the sum of energy 

conducted as sensible heat and energy utilized by 

evaporation. During the early use of the energy 

budget for obtaining evaporation rates, the ratio of 

these two terms was estimated to be around 0.1 or 

0.2. Later investigations showed that the ratio varied 

considerably. Bowen (1926) theoretically examined 

the ratio and developed an expression for it in terms

* Oceanographer, Research Division, U. S. Navy Electronics 

Laboratory, San Diego 52, California.

of easily measured quantities. Considerable controversy, however, still exists over the meaning and 

validity of Bowen's ratio (R).

Thus, past investigations have been confined to 

specific cases and large-scale features to minimize the 

effect of terms that could not be evaluated. However, 

the increasing attention on evaporation from lakes 

and reservoirs (Anderson, Anderson, and Marciano, 

1950), on the cooling action of lakes as applied to 

industrial problems (Throne, 1951), and on the forecasting of thermal conditions in proposed reservoirs 

requires that methods be devised to evaluate each 

of the terms of the energy budget for any specified 

season, locale, and time interval.

The principal objective of the energy-budget investigation at Lake Hefner was to determine the utility 

of the energy budget as a method for computing 

evaporation from natural bodies of water. This made 

it necessary to improve our understanding of certain 

terms in the energy budget, for example, reflected 

solar radiation, change in energy storage, the Bowen 

ratio, and effective back radiation.

The Energy-Budget Equation

The energy budget for the ocean, and as applied in the past to lakes and storage reservoirs, is 

generally expressed as

Q8 Qr Qb Qh Qe + Q«' = QS/ (59) 

where Q8 is the solar radiation incident to the water 

surface, Qr reflected solar radiation, Q& the net 

energy lost by the body of water through the exchange of long-wave radiation between the atmosphere and the body of water, Q/,, energy conducted 

from the body of water to the atmosphere as sensible 

heat, Qe energy utilized for evaporation, Qv > net 

energy advected into the body of water, and Qs the 

increase in energy stored in the body of water. Conduction of energy through the bottom, heating due to 

chemical and biological processes, and the transformation of kinetic energy into thermal energy are 

generally neglected because of their small magnitude.

From equation (59) it follows that the evaporation, £, can be expressed as

_ QB Qr Qft ~f~ QI/ ^3

PL (1 + R)

(60)

71
where L is the latent heat of vaporization and R, generally referred to as the Bowen ratio, is the ratio of 

energy conducted to or from the air as sensible heat 

to the energy lost through evaporation.

The ratio of conduction of sensible heat to evaporation as given by Bowen (1926) is

0-61P TQ 

1000

/ TQ Ta \ I I /

V e0 ea /

.... (o I ;

where P is the atmospheric pressure in mb, T0 is the 

water-surface temperature and Ta the air temperature 

in °C, and e0 is the saturated vapor pressure at the 

surface-water temperature and ea the vapor pressure 

of the air in mb. 

Generally,

Q & = Qa (62)

where Qa is the atmospheric radiation (principally 

from water vapor, cloud droplets, carbon dioxide, 

and ozone), TO the water-surface temperature in °K*, 

cr the Stefan-Boltzmann constant, and e the emissivity 

of the water surface. The emissivity of water has 

variously been taken to be 0.900 to 1.000. In addition, the reflection of atmospheric radiation has generally been neglected.

Saur and Anderson (paper in preparation) discuss in detail the relations existing between advected 

energy, change in energy storage, and change in 

volume of a body of water. These relations will be 

briefly reviewed.

When considering the energy content of a substance it is necessary to refer the energy content to 

some arbitrarily selected base temperature. The 

energy content of a unit mass of a liquid substance 

may be written

c (T T6 ) -f- constant, (63)

where T is the temperature in °C, the unknown constant depends on the base temperature T& and the 

character of the substance, and c is the specific heat. 

Thus, in calculations of a change in energy content 

of a fixed mass of water, the constant and base 

temperature cancel out. However, if the mass of water 

is not fixed, the change in energy content is not independent of the base temperature and must be considered with the advected energy.

Let Vi and V2 be the volumes of the body of 

water at the beginning and end of the period under 

consideration, N and M the total volumes of inflow 

and outflow, and pi, p2, PN, pM the densities of the

* °K represents degrees Kelvin (the absolute centigrade 

scale).'

respective volumes of water; then, the equation of 

continuity is

Pi PN N. (64)

Let

, ana

 p/ n/ + pp np -\- . . .

(65)

M = -f- iii ps ms -f- pe me -f- . . . ,

where po mo is the mass of surface outflow, ps ms the 

mass of out seepage, pe me the mass of evaporated 

water, p/ n/ the mass of surface inflow, and pp np the 

mass of precipitated water. Then, using equations 

(64) and (65)

pi Vi p2 V2 = po f- PS 1^8 H~ Pe ^e ~h 

pi n/ pp rip . . .. (66)

From equations (63) and (66) it follows that the quantity (Qv'-Q^) can be expressed as

c [Vi pi (Ti Tb ) V2 pz (72 7"&) po m0 (To Tb )

 & Pe M

p np (Tp - T6 )

P! nl l ~

(67)

where Tl7 T2, T/, T0, Tp, Ts, and Te are the temperatures of the respective volumes of water. Since equation (66) holds, it is clear that equation (67) is independent of the base temperature, and that the base 

temperature can be arbitrarily selected. In fact, T6 

could be omitted from equation (67) but, as will be 

seen, it is convenient to retain it for the purpose of 

computation. Equation (67) further shows that those 

terms in the energy budget associated with a term 

in the water budget cannot be considered individually, but are independent of the computational 

method only when the sum is considered. The constant in equation (63), being a function of only the 

base temperature and the nature of the substance, 

cancels out as long as equation (66) holds. It is noted 

that, by neglecting differences in density (thermal 

expansion) and using an average value, little error 

is introduced and terms can be computed on the basis 

of volume. The results obtained from equation (67) 

are in units of energy per lake per interval of time. 

This result must be divided by. the area of the lake to 

obtain calories per unit area per interval of time in 

order to be consistent with the units used to measure 

the radiative terms.

In addition to the energy lost because of the 

latent heat of vaporization, the mass of evaporated 

water represents an advected loss from the body 

of water, as indicated by equation (65), solely by the

72
loss of a mass of water at some temperature. Thus 

equation (59) should be written:

= Q« Q, - Oft 

(68)

where Qw is the energy advected out of the body of 

water by the mass of evaporated water, and Qv contains all other advected volumes. Dividing equation 

(68) by QO and rearranging:

(1 + «) + Qw = Q8 - Or - Q6 - Q 5 -+

(69)

Also,

Qo = ELp,, and Qw = cpe E (Je - T6), (70)

where T0 is the temperature at which evaporation 

takes place usually taken as the water-surface 

temperature. Substituting from equation (70) in 

equation (69):

E [p0L (

and

E: 

where

cpe (Te -

Q8 Qr Q6 Q 3- -f Q,,

Po [L (1 + ft) + c (T. - T6 )] '

= Q 0 ,

(71)

(72)

(73)

Qar being the amount of reflected atmospheric radiation. The term (Te Tb ) in the denominator of 

equation (72) is of the nature of a correction factor 

and will vary the evaporation about 5 per cent, depending upon the magnitude of the other terms in 

the equation. Obviously, the nearer the base temperature is to the evaporation temperature, the 

smaller will be the correction.

Intelligent selection of the base temperature can 

materially increase the accuracy of the computed 

evaporation obtained from equation (72). Referring 

to equation (67), generally it is difficult to evaluate 

np and ma . Since the loss of energy through the 

volume of evaporated water is corrected for in equation (72), the base temperature should be selected 

near the temperature of the seepage and/or the 

precipitation temperature, since seepage and precipitation volumes are generally difficult to evaluate. 

By selecting the base temperature in this manner 

the effect of errors due to the errors in volume determination are minimized. In general, the base temperature should be selected as the best temperature 

estimate of the largest unknown advected volume.

If evaporation could be determined directly from 

the water-budget equation, there would be no necessity for using the energy budget. However, except in 

extremely well-controlled investigations, inaccuracies 

exist in the determination of the water budget which 

can be of the same order of magnitude as the evaporation. Since the latent heat of vaporization appears 

in the denominator of equation (72) for computing 

evaporation from the energy budget, a real advantage is obtained by use of the latter method. To utilize 

equation (72) it is necessary to have some knowledge 

of the water budget, but not to the degree of precision necessary to determine evaporation from the 

water budget.

Observational Program at Lake Hefner

The location of the instruments used for the 

energy-budget investigation is shown on the map.

Table 6, showing the number of days of each 

month for which complete radiation data were recorded, indicates in a general way the amount of 

data obtained. Considerable data covering all months 

and seasons were available for the study of energy 

exchange.

During October 1950, difficulty was experienced 

because the thermocouple ice jug rose above 0°C, 

with the result that the flat-plate temperature was 

in error during much of the month. An attempt was 

made to correct the flat-plate temperatures, but it

TABLE 6. Number of Days Per Month of Complete 

Radiation Record.

Month

May 1950

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan 1951

Feb

Mar

Apr

May

Jun

Jul

Aug

Number of 

Days of Data

9

11

11

13

20

27

30

22

29

20

30

29

19

20

22

30

Per Cent of 

Total No. of Days

29

37

35

42

67

87

100

71

94

71

97

97

61

67

71

97

TOTAL 342 70

73
was difficult to judge how far back in time to carry 

the correction. This may have resulted in too high an 

effective back radiation and consequently too low 

an evaporation.

Ice formed on the lake during the month of 

February 1951, making it necessary to beach the 

barge from 9 February to 23 February. During this 

period all energy-budget parameters except the reflected solar radiation were measured at the beach 

position, on the south side of the lake. In addition, it 

was impossible to obtain vertical temperature profiles of the lake from 26 January to 26 February. 

Accurate data for evaluating the change of energy 

storage were therefore lacking during most of 

February.

In late June 1951, difficulty was encountered 

with the Temperature Profile Recorder (TPR), so that 

for the last two months of the observational program 

it was necessary to utilize a bathythermograph to 

obtain water-temperature data.

During months other than those mentioned above 

the instrumentation operated with only minor or local 

difficulties which influenced the observations for only 

short periods of time. For the months discussed above, 

the results, even the monthly average values, may 

have been affected by the instrumental difficulties.

Data Processing

The radiation data were recorded on an Esterline-Angus 1-mil recorder using a 15-point programming switch. The following elements were recorded: 

solar radiation, reflected solar radiation, flat-plate 

radiometer reading, and the temperature of the flat 

plate plus a reference. This recording procedure 

resulted in a 4-minute record of each element spaced 

at 20-minute intervals.

The data were processed by constructing a continuous diurnal curve for each element and then reading off values of each element at half-hourly intervals. 

These 48 readings were averaged to obtain a daily 

average value, and then converted into a daily total. 

The totals were then utilized to establish an energy 

budget from which evaporation could be computed.

The TPR data were processed by reading and recording the mean water temperature by 1-meter 

intervals. From the mean temperatures and mean 

areas for each 1-meter-thick layer, the areas being 

obtained from the area-capacity table for the lake, 

the energy content in each layer was computed. The 

energy contents of the layers were then summed to 

obtain the total energy content, above an arbitrary 

base temperature, of the lake.

SOLAR RADIATION

The short-wave radiation (0.3 /A to approximately 

4.0 ju,) incident to the outside of the earth's atmosphere comes primarily from the sun. This radiation 

in passing through the earth's atmosphere undergoes 

depletion through scattering, reflection, and absorption by gases of the air, water vapor, clouds, and 

dust. As a result of these complex processes, the 

short-wave radiation arrives at the earth's surface 

partly as direct radiation and partly as diffuse radiation.

Two approaches are available for evaluating the 

solar radiation incident to the earth's .surface: (1) 

indirect evaluation in terms of easily observable or 

measurable quantities, and (2) direct measurement 

by suitable instrumentation.

Knowing the amount of radiation incident to the 

outside of the earth's atmosphere, many investigators, 

for example, King (1913), Mosby (1936), Hewson 

(1943), Haurwitz (1948), Klein (1948), and Kennedy 

(1949), examined, both theoretically and empirically, 

the modification of solar radiation by the earth's 

atmosphere, with a view to obtaining an indirect 

method for evaluating this term. Because of the complexity of the processes involved, complete theoretical 

treatment was not possible; hence various assumptions, simplifications, and restrictions were necessary. 

For example, King (1913) examined theoretically the 

scattering of solar radiation by gaseous media and 

obtained complex expressions for the intensity of 

direct and diffuse solar radiation. He considered only 

the effect of molecules of the atmosphere and small 

particles, neglecting the effect of clouds, and assumed 

the earth was a smooth plane and the density of air 

a function of height only; he ignored reflection from 

the earth's surface and refraction by the atmosphere. 

Mosby (1936) developed the following empirical formula for computing the incoming radiation on a horizontal surface, in terms of the average altitude of 

the sun and the average cloudiness:

Q8 = k (1 0.071 C) h cal cm~2 min" 1 , (74)

where Q« is the solar radiation, k a constant that is 

a function of latitude, C the average cloud cover in 

tenths of sky covered, and JJ the average altitude of 

the sun in degrees. This equation is not valid for sun 

altitudes greater than 60 degrees, but gives correct 

results if they are replaced by reduced altitudes. 

Hewson (1943) considered theoretically the effect of 

cloud thickness, water content, and drop size on the 

transmission of solar radiation. Haurwitz (1948) examined empirically the relation between total solar

74
radiation reaching the earth's surface and the amount 

of cloudiness, cloud density, and cloud type. Klein 

(1948), utilizing the results of previous theoretical and 

empirical investigations, presented a method for computing direct and diffuse radiation received on the 

earth's surface. He considered the effects of terrain 

reflection and of depletion by dry air, water vapor, 

dust, and clouds. He summed up: "However, the 

computations described in succeeding sections of this 

report all involve approximations from empirical relations which are true only in the mean and subject 

to large deviations in individual cases. Therefore, 

they should be applied only to average values from 

climatological records taken over as long a period 

of time as possible." Kennedy (1949) presented a 

method for computing daily insolation. The method 

makes use of a pyrheliometer station, similar in latitude .and altitude to the place where the radiation 

data are desired. Data are extrapolated from the 

station to the place in question. The method involves 

obtaining a solution to the following equation:

Q8 = /o a"', (75)

where Qfl is the solar radiation, /0 the solar radiation 

received on a horizontal surface at the exterior of 

the earth's atmosphere, a the atmospheric transmission coefficient, and m the solar air mass or the 

ratio of the length of the actual path of the solar 

beam to the path through zenith. Using known values 

of Q8, /o, and m at a pyrheliometer station, the daily 

atmospheric transmission coefficient a may be computed. The daily value is then plotted against cloudiness in tenths, and a smooth curve fitted to the points. 

From this plot the value of the atmospheric transmission coefficient for the unknown station may be obtained for different degrees of cloudiness. Utilizing 

the proper value of a and tables giving /0 and m as 

functions of latitude and the declination of the sun, 

the solar radiation is computed from equation (75).

In this study, solar radiation was measured 

directly by means of an Eppley pyrheliometer. The 

observations are summarized in figure 50 in terms 

of average monthly values. The numerals indicate the 

number of complete days of solar-radiation observations obtained for each month. In addition, the maximum and minimum daily radiations during the month 

are shown. As expected, there is a roughly sinusoidal 

variation throughout the year with a variation from 

about 250 cal cm~2 day" 1 in December to 625 cal 

cm~2 day" 1 in June. A comparison of July 1950 with 

July 1951 indicates the variation that may occur from 

year to year for any given month. July 1950 had an 

average monthly value of about 175 (or about 25 

per cent) lower than July 1951. According to the U. S. 

Weather Bureau, July 1950 was one of the coldest 

and rainiest on record and this is reflected in the 

low monthly radiation.

The monthly range in maximum and minimum 

varies from about 250 cal cm~2 day" 1 in December 

to 450 cal cm" 2 day" 1 in June.

In past investigations, solar radiation has usually 

been evaluated by one of the indirect methods mentioned above. The direct solar measurements made 

at Lake Hefner over a period of 16 months provided 

an opportunity for evaluating some of these indirect 

methods. Two were selected for evaluation: (1) 

Mosby's formula, equation (74), which represents an 

empirical method designed to provide average 

monthly or annual values 'of solar radiation, and 

(2) Kennedy's method, which represents an extrapolation designed to provide daily solar-radiation 

values. These methods were selected because both 

have been widely used for oceanographic and limnologic studies.

The monthly solar radiation for Lake Hefner from 

July 1950 to August 1951 was computed from Mosby's 

formula. According to Mosby the constant k is a 

function of latitude, and for the latitude of Lake

700

600

. 500 

°400

200

100

0

Figure 50. Average monthly solar radiation for Lake Hefner 

from May 1950 to August 1951.

MAY JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG 

1950 1951

75
Hefner its value is 0.024. The average cloud cover 

was computed from the hourly cloud cover as reported by the U. S. Weather Bureau station located 

13 miles south of Lake Hefner at Will Rogers Airport. 

The average monthly sun altitude was computed by 

means of a method described by Bowditch (1939).

The results obtained are summarized in table 7. 

As indicated, the solar radiation values obtained from 

Mosby's formula are in general too low.

TABLE 7. Comparison of Solar Radiation Computed 

from Mosby's Formula and that Observed at Lake 

Hefner.

Number Days Observed 

Month Observation QSQ

Jul 1950

Aug

Sep

Oct

Nov

Dec

Jan 1951

Feb

Mar

Apr

May

Jun

Jul

Aug

13

13

22

27

30

24

29

24

30

29

29

24

28

30

(cal

448

565

417

390

287

233

275

322

426

516

541

591

611

576

Computed

Q8C

Difference

Q«c Qso

Per Cent 

Difference

em~2 day" 1 )

383

550

339

367

226

184

189

239

312

449

478

521

563

599

 65

 15

 78

 23

 61

 49

 86

 83

 114

 67

 66

 70

 48

23

 15

_ 3

 19

 6

 21

 21

 31

 26

 27

 13

 12

 12

_ 8

4

The mean difference is about 70 cal cm 2 day" 1, 

or about 15 per cent of the observed values. The 

values obtained from Mosby's formula are dependent 

to a great extent on the accuracy of the estimate of 

the amount of cloud cover. Since cloud cover is only 

estimated to tenths, the formula is rather inflexible 

and considerable difference in the results may be 

caused by varying the cloud cover by one-tenth. 

Obviously, if cloud cover were estimated with greater 

accuracy, say to twentieths or hundredths, greater 

flexibility would be attained. However, since cloud 

cover is generally observed from the ground and furthermore is visually estimated, the use of a scale in 

greater detail than a tenth is not justified. Under 

certain conditions it is difficult for observers to agree 

even within a tenth as to the amount of cloud cover. 

The negative bias, indicated by table 7, can be 

removed by adjusting the constants in equation (74). 

The constant k and the cloud-cover coefficient were 

adjusted individually and simultaneously. Changing 

(1) k to 0.0276, or (2) the cloud-cover coefficient to 

0.0532, or (3) the constant k to 0.025 and the cloudcover coefficient to 0.059 removes the negative bias.

The standard error of estimate, using the new constants, is ±50, ±38, and ±41 cal cm"2 day- 1, 

respectively. These amount to approximately ±10 

per cent of the mean observed solar radiation. Hence, 

there is some improvement from the adjusted constants. The use of these constants, in preference to 

those originally established by Mosby, must be made 

with caution since they are probably only applicable 

to the Lake Hefner area.

The effect of the differences between computed 

and observed solar radiation on evaporation computations depends upon the magnitude of (Qe -{- Q/,) 

and R. For example, during August 1950, September 

1950, and January 1951, the use of solar radiation 

computed from Mosby's formula, using the original 

constants, introduces differences of 6, 31, and 55 

per cent, respectively, in computed evaporation.

On a mean annual basis, the observed solar 

radiation from 1 September 1950 to 1 September 

1951 was 432 cal cm~2 day" 1 , while that obtained 

from Mosby's formula was 362 cal cm"" 2 day" 1 or 

approximately 15 per cent lower.

In summary, Mosby's formula leads to resultsapproximately 15 to 20 per cent too low for monthly 

or yearly values. Solar radiation values obtained 

from Mosby's formula may therefore lead to large 

errors in evaporation computations, unless the values 

are substantiated by independent data; for other 

purposes, the values may be sufficiently accurate.

Kennedy's method of obtaining daily solar radiation consists basically in extrapolating measured 

data from a radiation station to the place in question, 

the two locations being similar in latitude and elevation. Since the U. S. Weather Bureau maintains about 

sixty radiation stations scattered throughout the United 

States, it is generally possible to find a radiation 

station that meets the latitude and elevation requirements. The radiation station at Fresno,* California, 

most nearly meets the requirements for Lake Hefner.

Solar radiation for Lake Hefner was computed 

for each of the 358 days for which a direct measurement of solar radiation was available. Plots of the 

difference between Kennedy's value and the observed 

value versus the observed value indicated that the 

difference was a function of cloud cover. The data 

were therefore examined in terms of cloud cover 

clear, scattered, broken, or overcast. The results are 

presented in figure 51 in terms of the standard error 

of estimate and the standard error of estimate as a 

percentage of the mean observed solar radiation.

* The Oklahoma City data were not used because they 

would not have provided a test of Kennedy's method.

76
For clear skies the standard error of estimate is less 

than 10 cal cm~2 day" 1 , or only about 2 per cent 

of the mean observed radiation. This error is acceptable for purposes of computing evaporation. However, as the cloud cover increases the standard error 

increases to about 115 cal cm" 2 day" 1, or 25 and 

5d per cent for broken and overcast skies, respectively. Errors of this magnitude cannot be tolerated in 

computing evaporation. The increase in error with 

cloud cover is related to the height of the cloud cover. 

Kennedy's method considers radiation only as a function of amount of cloud and not of cloud height. It is 

apparent, however, that the amount of solar radiation 

penetrating a high overcast will be greater than that 

penetrating a low overcast, all other conditions being 

the same; in other words the average daily atmospheric coefficient for high clouds is different from 

that for low clouds. This dependence was also demonstrated by Haurwitz (1948). Hence, to improve the 

accuracy for cloudy conditions, it is necessary to 

select a radiation station similar not only in latitude 

but also in types of clouds to the location in question. Alternatively, separate atmospheric transmission 

coefficients should perhaps be defined for each of the 

major cloud types low, middle, and high.

In addition, solar radiation values were computed for time intervals greater than one day. The 

number of data did not permit a breakdown with 

respect to cloud cover or height. The results are presented in figure 52. The standard error of estimate 

decreases from a value of 85 cal cm~2 day" 1 for 

daily periods to 45 cal cm"2 day" 1 for a 10-day 

period, and then remains essentially constant with 

increasing length of time interval. Expressed as per 

cent of mean observed solar radiation, the error 

decreases from 20 for single days to about 12 for 

intervals greater than 5 days.

It thus appears that Kennedy's method gives 

excellent results for daily solar radiation with clear 

skies. For sky conditions other than clear, errors up 

to 50 per cent may occur when computing daily 

values. These errors may possibly be reduced by 

taking into account the characteristics of the cloud 

cover. For periods longer than 5 days, an error of 

approximately 10 per cent may be expected.

Summarizing, of several indirect methods for 

evaluating solar radiation, two were studied, namely, 

Mosby's, representative of long time intervals, and 

Kennedy's, representative of short time intervals. The 

results obtained were examined in detail and compared with direct observations made at Lake Hefner, 

Oklahoma. The conclusion was that these indirect

methods of evaluating solar radiation will give the 

necessary accuracy for evaporation computations only 

in selected circumstances. For applications other than 

to evaporation, where accuracies of the order of 

15 per cent are required, they are satisfactory. It is 

necessary to measure solar radiation directly to obtain 

the accuracy required for applying the energy budget 

to evaporation determinations.

3

5 50

STANDARD ERROR 

OF ESTIMATE

STANDARD ERROR OF 20 

ESTIMATE EXPRESSED 

AS PERCENT OF MEAN 

OBSERVED SOLAR 

RADIATION

SCATTERED

AMOUNT OF CLOUDS

Figure 51. Standard error of estimate of solar radiation computed from Kennedy's method as a function of cloud cover.

358 STANDARD ERROR OF ESTIMATE

NO. OF PERIODS

 , ., Qslobs;

STANDARD ERROR OF 

ESTIMATE EXPRESSED 

AS PERCENT OF MEAN 

OBSERVED SOLAR 

RADIATION

^,43.0 / 

7. 7 / 4 !7'°

5 10 15 20 

TIME INTERVAL IN DAYS

Figure 52. Standard error of estimate of solar radiation computed from Kennedy's method as a function of length of time 

interval.

77
REFLECTION OF 

SHORT-WAVE RADIATION

As stated by Neiburger (1948), a "knowledge 

of the reflectivity of the sea surface is of fundamental 

importance for heat-balance studies, both of the 

atmosphere and the oceans. In order to evaluate the 

amount of solar energy absorbed by the sea, it is 

necessary to know not only the amount incident on 

the surface, but to know also what portion of that 

is returned, unchanged, by reflection at the surface 

and by scattering from bubbles and suspended particles immediately below it."

The necessity for evaluating each term in the 

energy budget accurately when attempting to determine evaporation has been pointed out in the 

preceding section. For example, an error of 30 to 40* 

cal cm~2 day" 1 in the evaluation of any term in the 

energy budget results in an error of 10 to 20 per cent 

in the computed evaporation.

The reflectivity of short-wave radiation by a 

natural water surface has been studied by relatively 

few investigators. Zenker (1888) made the first theooretical study of the subject, and Schmidt (1915) 

investigated the evaporation for the oceans as a 

whole by utilizing the energy-budget approach. In 

evaluating the incoming solar radiation, Schmidt discussed the theoretical reflectivity of both solar and 

sky radiation and obtained a theoretical value of 

0.17 for the reflectivity of sky (diffuse) radiation.

Kimball and Hand (1930) made measurements 

from an airplane of reflectivity over Chesapeake Bay, 

Potomac River, and Patuxent River, at altitudes ranging from 10 to 1000 feet. They obtained values of 

reflectivity under an overcast sky of 0.06 to 0.10. 

For Chesapeake Bay, they reported a value of 0.097 

for a hydrodynamically smooth surface and 0.034 to 

0.049 when whitecaps were present, suggesting a 

decrease of reflectivity at higher wind speeds.

Powell and Clarke (1936) made 34 measurements of the reflectivity of the sea water at Buzzard's 

Bay in July and August 1935. Measurements were 

made for red and violet light, and a variety of winds, 

sky covers, and sun altitudes. They concluded that: 

(1) the solar radiation is reflected not only from the 

water surface but also from a stratum of relatively 

opaque water (bubbles and suspended material) just 

beneath the surface, and for a reflectivity of 0.09 

cr less, one-half the radiation is returned by the water

* An approximate daily average value of the amount 

of reflected energy at Lake Hefner.

surface itself; (2) reflectivity increases slightly when 

the surface is hydrodynamically rough and in the 

presence of clouds; (3) reflectivity is not more than 

0.09 for sun altitudes greater than 30 degrees in clear 

weather; (4) average reflectivity under an overcast 

sky (diffuse radiation) is 0.08.

Sverdrup (1942) synthesized Schmidt's theoretical 

values for reflectivity of solar radiation and Powell 

and Clarke's observed values for sky radiation, and 

obtained values of reflectivity of 0.25, 0.06, 0.03, and 

0.03 for sun altitudes of 10, 30, 60, and 90 degrees, 

respectively.

Neiburger (1948), using a blimp, studied diffuse 

radiation off the California coast under a stratus 

overcast. He made 117 observations on 45 days 

during the summer of 1945. He concluded that: (1) 

the radiation was not completely diffuse; (2) the 

average reflectivity was 0.105, varying with cloud 

thickness and sun altitude; (3) for completely diffuse 

radiation, Schmidt's value of 0.17 would apply; (4) 

Powell and Clarke's value of 0.08 should be applied 

only for radiation from a high sun passing through 

relatively thin clouds; (5) no correlation exists between wind speed and reflectivity.

As indicated by these studies, there is lack of 

agreement concerning the reflectivity of completely 

diffuse radiation and the effect of wind speed. In 

addition, so far as is known, no one has investigated 

the effect of variable amounts of high, middle, or low 

clouds on the total reflectivity.

The observations made at Lake Hefner provide 

the necessary data for examining the reflectivity of 

a natural water surface in considerable detail. In 

this section the theory of reflectivity of solar and sky 

radiation for an optically flat water surface under 

clear skies will be reviewed; reflectivity will be considered as a function of sun altitude, turbidity* of 

the atmosphere, hydrodynamic nature of the surface, 

and amount and height of cloud cover; and a method 

of determining reflectivity in terms of more easily 

measured parameters will be presented.

As previously mentioned, total incoming and reflected radiation was measured every 20 minutes from 

1 May 1950 to 31 August 1951. Observations were 

made covering variable cloud conditions, winds from 

0 to 40 knots, sun altitudes from 0 to 78 degrees, and 

several different types of air masses.

* Turbidity, unless otherwise noted, is used in this section 

to refer to the turbidity of the atmosphere when no clouds are 

present. As will be pointed out later, the type of air mass is 

related to the turbidity of the air, which in turn is related 

to the reflectivity.

78
Theoretical Reflectivity Under a Clear Sky solar radiation = t\ (S^, C), and (77)

Short-wave radiation reaches any surface on the 

earth partly as direct radiation from the sun (solar 

radiation) and partly as scattered or reflected radiation from the sky (sky radiation). When considering 

the total reflectivity of a surface it is necessary to 

consider the reflectivity of each of these components. 

If a and b are the reflectivities of solar and sky radiation, and m and n are the respective fractions of the 

incoming radiation from the sun and sky, then the 

total reflectivity, Rtl of any horizontal surface may be 

expressed as

Rt am -}- b/r. (76)

The reflectivity of both solar and sky radiation 

from an optically flat water surface under a clear 

sky can be examined by application of Fresnel's formula for the reflection of natural unpolarized light.* 

Theoretical expressions for the amount of short-wave 

radiation coming from the sun and sky were obtained 

by King (1913).

The theoretical examination of each term in 

equation (76) makes it possible to examine the nature 

of the total reflectivity of an optically flat water surface under a clear sky.

King applied the results of Rayleigh on the 

scattering of parallel radiation by molecules and 

small particles to radiation and absorption in the 

earth's atmosphere. He made the following simplifying assumptions: the earth is a smooth plane, the 

density of air is a function of height only, and reflection from the earth's surface and refraction by 

the earth's atmosphere may be neglected. In addition, he considered only the effects of molecules of 

the atmosphere and small particles, neglecting the 

effect of clouds.

By much mathematical manipulation he developed two equations (approximations of a complicated exact equation which is difficult to apply) for 

obtaining the total intensity of sky radiation from 

any direction. One is applicable to large sun altitudes and the other to small altitudes. These equations indicate that both solar and sky radiation are 

functions of the sun's altitude, the coefficient of 

attenuation of scattered radiation, and absorption. 

His attenuation coefficient, C, includes the effect of 

both molecules of gas and dust particles. Hence,

* A complete discussion and mathematical treatment of 

the Fresnel formula is given by Sears (1945) and Houstoun 

(1938).

sky radiation r"2 (S^, C), (78)

where SA is the altitude of the sun.

Using mean coefficients of attenuation for Mt. 

Wilson, California, and Washington, D. C., King computed for these two stations the amount of energy 

coming from the sun and sky as a function of sun's 

altitude. His values are graphically presented in 

figure 53a. The sky radiation is given as a fraction 

of the total short-wave radiation in figure 53b. These

10 20 30 40 5O 60 70 80 »0 

SUN ALTITUDE (DEGREES)

20 30 40 SO 00 70 00 9< 

SUN ALTITUDE (DEGREES)

Figure 53. Theoretical radiation coming from the sun and sky 

for Mt. Wilson and Washington, D. C., according to King.

79
data show the greater amount of sky radiation at 

Washington because of the increased scattering due 

to lower elevation of the station above sea level and 

the greater number of small particles in the atmosphere. The Mt. Wilson data probably reflect scattering caused primarily by molecules, while at Washington there is a greater particulate effect. In each 

place, the sky radiation is nearly independent of 

the altitude of the sun.

King's coefficient of attenuation is similar to 

Linke's (1922) turbidity factor. This factor may be 

written as

(79)

where T is a mean turbidity factor, a a mean extinction coefficient due to molecular scattering, and aw 

and a<j mean coefficients of extinction by water vapor 

(disregarding selective absorption) and dust, respectively.

Haurwitz (1934) demonstrated that the turbidity 

factor varies with respect to the air mass present. 

Table 8 presents the average turbidity factor obTABLE 8. Average Turbidity Factors for Various 

Types of Air Masses According to Haurwitz.

Air Mass Turbidity Factor

Polar Continental 2.42

Modified Polar Pacific 2.65

Modified Polar Continental 2.81

Tropical Gulf plus Tropical Atlantic 3.49

Polar Atlantic plus Modified Polar Atlantic 3.61

tained by Haurwitz for various types of American air 

masses. The effect of the variation in turbidity with 

air mass would be to vary the ratio of sky radiation 

to solar radiation. Thus, for modified Polar Atlantic 

air the relative amount of diffuse radiation would be 

greater than for Polar Continental air.

As previously stated, the reflectivity of an optically flat water surface under a clear sky for direct 

solar radiation can be examined by considering 

Fresnel's formula for the reflectivity of natural unpolarized light. This formula is

tan2 (90 SA r) . sin2 (90 SA 

tan2 (90 SA + r) sin2 (90 SA + r)

where a is the reflectivity of direct solar radiation, 

SA the altitude of the sun above the horizon, r the 

angle of refraction, and sin (90 SA) = p- sin r where 

p. is the index of refraction for water relative to air. 

For sun alititude of 90 degrees, equation (80) reduces to

The index of refraction for pure water relative 

to air at 25°C is 1.33251 (Handbook of Chemistry 

and Physics, 1947) while for sea water of salinity 

34.00 %o* it is 1.33873 at 25°C (Sverdrup, ef a/, 

1942).

Figure 54 shows the variation of reflectivity of 

direct sun radiation with altitude of the sun for pure 

water. In the case of sea water, which has a slightly 

higher index of refraction, the reflectivity is only 

0.000 to 0.003 greater.

In the presence of thick clouds, assuming the 

atmosphere obeys Lambert's cosine law of diffuse 

radiation, the sky light is of equal intensity from all 

directions. Knowing the reflectivity for various sun 

altitudes, and assuming the light is completely diffused, then, as indicated by Schmidt (1915),

M / **

b = j* a (SA) cos SA (82)

where a ($4) is the reflection of direct sun radiation 

as a function of sun altitude.f Equation (82) can be

* Parts per thousand, the standard method of expressing 

the salinity of sea water.

(80)

The area of a surface of revolution around the y-axis is

irR/2 B-/2

ITT J xd*s or in this case ZirRp C cos S4 dSA. The amount of

0 o

ir/2 

reflected radiation is 2irR2 f a (S^) cos S^ e/S^. The mean value

0 

of a function is Cxdt/Cds. In this case the average reflectivity

becomes

ir/2 

fa(

ir/2

2»rR2 j cos S^ dSA 

0

80
evaluated graphically if x = sin S^ and y = a (SA).

i 

Then the integral becomes fydx. The area under

o 

the curve in figure 55 is then the total reflectivity

of completely diffuse sky radiation for an optically 

flat water surface. This evaluation gives 0.173 for 

the reflectivity of sky radiation.

The total reflectivity of an optically flat water 

surface under a clear sky can be evaluated from 

equation (76) using values for a, b, m, and n previously discussed. Rewriting equation (76),

Rt = am -|- bn, where 

1 pan2 (90 SA r) . sin2 (90 SA 

tan2 (90 SA + r)

ir/2

b =

sin2 (90 - SA + r) 

= J a (SA) cos SA d SA,

(76a)

(76b)

o

m =

fi (SAl C)

, and (76c)

n =

, C) (76d)

we obtain

R = } rtan2 

' 2 Ltan2

< 90 - SA - 0 , sin2 (90 - SA - r)

X

tan2 (90 SA + r) sin2 (90 

fi (SA, Q

- SA - r) 1

- SA + r) J

(SA, C)

, C) + f2 (SAl C) h (SA, C) + f2 (SAf C)

7T/2

X fa (SA) cos SA d SA. (83)

From equation (83) it may be concluded that the total 

reflection from an optically flat water surface under 

a clear sky is a function of the altitude of the sun 

and the type of air mass (turbidity, excluding effect 

of clouds).

The effect of increasing turbidity (turbidity data 

from King, 1913) is indicated by figure 56, where 

equation (83) is applied to radiation data for Mt. 

Wilson and Washington. This shows that the effect 

of increasing turbidity at sun altitudes of less than 

20 degrees is to decrease the reflectivity slightly, and 

at sun altitudes greater than 20 degrees to increase 

the reflectivity slightly. For different air masses and 

various amounts of cloud cover, the variations would 

be similar.

The total actual amount of reflected solar and 

sky radiation is

Qr = RtQe, (84)

SUN ALTITUDE (DECREE!)

Figure 54. Variation of reflectivity of direct solar radiation 

with altitude of the sun for pure water as obtained from Fresnel's 

formula.

Figure 55. Theoretical total reflectivity for completely diffuse 

radiation.

SUN ALTITUDE (DEGREES)

Figure 56. Effect of the variation of air-mass turbidity on the 

reflectivity of a natural water surface.

81
where Qr is the total amount of reflected energy and 

Q8 >the total incoming short-wave radiation. Using 

values of reflected energy computed from equation 

(83) and values of total incoming short-wave radiation computed from equations (77) and (78), one can 

examine the variation of the amount of reflected 

energy as a function of sun altitude and time.

If knowledge of the coefficient of attenuation 

for short-wave radiation were adequate, it would be 

possible to obtain a family of curves showing the 

amount of energy reflected from an optically flat 

water surface under a clear sky as a function of sun 

altitude. In addition it would be possible to present 

graphically the amount of reflected energy as a function of latitude and time. The amount of reflected 

energy could then be evaluated for any given location and day.

Since the data are inadequate, the above formulas have been applied to specific cases only. 

Figure 57 shows the variation of reflected energy as 

a function of sun altitude for Mt. Wilson and Washington (King, 1913). Mt. Wilson represents a low 

coefficient of attenuation and Washington a moderately high coefficient. The amount of reflected energy 

increases rapidly at low sun altitudes, reaching a 

maximum at 15 degrees for Mt. Wilson and then 

falling rapidly to virtually a constant value at 40 

degrees. For Washington where the turbidity is 

greater, the amount of reflected energy is less at 

all sun altitudes, the maximum being at 25 degrees 

and less pronounced. As in the case of low turbidity, 

the reflected energy increases rapidly at low sun 

altitudes. The maximum effect of turbidity is for sun 

angles less than 30 degrees. From figure 57 it can 

be seen that the maximum amount of reflected energy 

for relatively transparent air is 0.085 cal cm~2 

min~ 1 and for turbid air 0.06 cal cm~2 min" 1 .

Figure 58 shows the theoretical variation of reflected energy for a given location (Lake Hefner, 35° 

30/ N) as a function of season and time of day. The 

amount of reflected energy increases rapidly to a 

maximum at about two hours after sunrise; it then 

decreases, the amount of the decrease depending 

upon the season of the year, being greater during, 

the summer and less during the winter. By the late 

afternoon, the reflected energy reaches an almost 

constant value. It then increases rapidly until two 

hours before sunset, and then decreases rapidly to 

zero shortly after sunset. A decrease in turbidity increases the reflected energy at all hours of the day 

but exerts its greatest influence just after sunrise 

and before sunset.

SUN ALTITUDE (DEGREES)

Figure 57. Theoretical variation of reflected energy as a function of sun altitude for Mt. Wilson and Washington, D. C.

At this stage, it becomes possible to speculate 

on the effects of cloud cover and air mass. If a uniformly dense cloud cover were introduced, it would 

effectively increase the turbidity of the atmosphere. 

This in turn would theoretically decrease the amount 

of reflected energy for all sun angles, latitudes, and 

times of day. Hence, for any given sun altitude, reflected energy would be maximal on a clear day and 

minimal on an overcast day. Variations in air mass 

should have a similar effect, and the more turbid the 

air mass, the smaller the amount of reflected energy 

for a given sun altitude.

As previously mentioned, Powell and Clarke 

(1936) concluded that the reflected radiation consists 

of radiation reflected both from the water surface 

and from bubbles and suspended matter just beneath the surface. For a total reflectivity of 0.09 cal 

cm~2 min" 1 or less, only one-half is reflected by 

the actual surface. It might be postulated that, with 

increasing winds, the bubbles and suspended matter 

would increase, thereby increasing the total reflected 

energy.

Previous investigations have indicated that a 

water surface changes from hydrodynamically smooth 

to rough at some critical wind speed. For example, 

Munk (1947), using evidence based on observations 

of whitecaps, soaring characteristics of sea gulls, and 

variations in resistance and evaporation coefficients, 

concluded that the critical wind speed was 6 to 7 

meters per second (approximately 12.5 knots). It 

may be speculated that, if other characteristics 

change at this critical wind speed, the reflectivity of 

the water surface would also change. Hence, the wind

82
Figure 58. Theoretical variation of 

reflected energy at the latitude of 

Lake Hefner (35° 30' N) as a function of season and time of day using 

turbidity data from Mt. Wilson and 

Washington, D. C.

--..08

0 .06

5-os

TURBIDITY DATA FOR:

- WASHINGTON D.C.

 MT. WIUSON

21 JUN

21 JUN

12 I 2 

MIDNIGHT

5 6 8 9 10 II 12 I S 9 10 II 12 

MIDNIGHT

in making the water surface hydrodynamically rough 

may also change its reflectivity characteristics.

Recapitulating, the above theoretical examination indicates that the reflectivity of an optically flat 

water surface is primarily a function of the altitude 

of the sun and secondarily a function of the turbidity 

of the atmosphere. The turbidity, in a general sense, 

is in turn a function of the type of air mass present. 

The introduction of clouds into the air mass increases 

the turbidity of the air and thus decreases the amount 

of reflected energy at all sun altitudes. Because radiation may also be reflected by bubbles and particulate material just under the surface, theoretical 

values for reflectivity should be lower than those 

actually observed. The effect of wind is to increase 

the suspended material and bubbles, thereby increasing the reflectivity. An abrupt change in reflectivity 

at wind speeds of 6 to 7 meters per second would 

confirm the concept of a critical wind speed at which 

a surface changes from hydrodynamically smooth 

to rough.

The actual energy reflected, for a given location, is greatest at sun altitudes less than 30 degrees, 

with a nearly constant amount reflected at greater 

sun altitudes. Therefore, on a given day, most energy 

is reflected just after sunrise and just before sunset. 

This characteristic is most pronounced during the summer and least during the winter.

Observations of Reflectivity Under a Clear Sky

At Lake Hefner, 1163 observations of reflectivity 

were made under a clear sky during the year 1 May 

1950 to 30 April 1951. The observations were made 

under a variety of conditions: wind speeds of 0 to 30 

knots; various combinations of air masses, and many 

examples of essentially unadulterated Tropical Gulf 

warm moist air and Polar Canadian cold dry air, 

the former being considerably more turbid than the 

latter; sun altitudes of 0 to 78 degrees. The resulting 

data offer an opportunity to examine in detail the reflectivity of a natural water surface under a clear sky.

Since the type of air mass present is indicative 

of the turbidity of the air, the effect of wind speed, 

when Tropical Gulf or Polar Canadian air only is 

present, will first be examined. A plot of wind speed 

versus reflectivity for a given, small sun-altitude range 

and a given air mass should indicate, by an abrupt 

change in reflectivity at some wind speed, any effect 

of a change in water surface from hydrodynamically 

smooth to hydrodynamically rough. Figure 59 presents 

the observations taken under these conditions. The 

number of observations is indicated by N, and SA 

represents sun altitude. The data clearly indicate the 

absence of any wind effect or critical wind speed 

over the range of wind speeds shown, and the independence of wind speed and reflectivity. The greater 

scatter of individual observations at low sun altitudes,

83
compared to high sun altitudes, is caused by the 

greater percentage error in measuring reflected solar 

and total solar radiation, due to their small magnitude, at the low sun altitudes. At sun altitudes greater 

than 30 degrees, the scatter is small.

Figure 60, a plot of the median reflectivity for 

5-degree sun-altitude intervals, indicates the effect 

of turbidity, as defined by Polar Canadian and Tropical Gulf air masses, on reflectivity. Theoretically, an 

increase in turbidity will increase the reflectivity at 

high sun altitudes and decrease it at low sun altitudes. 

The Lake Hefner data do not show such a systematic 

variation. If it be assumed that the number of data 

are adequate to provide reliable median values of 

reflectivity, the lack of agreement between observation and theory may arise from the turbidity differential between these two air masses being too small 

to demonstrate the expected dependence. Hence, it 

may be concluded that reflectivity under clear skies 

is independent of air-mass turbidity, except for cases 

of extreme turbidity such as might occur in dust-laden 

or smoke-laden air.

Since the reflectivity is apparently independent 

of wind speed and air-mass turbidity, it can be a 

function only of the altitude of the sun. The relationship is shown in figure 61. The numerals indicate 

the number of observations at each altitude. Above 

sun altitudes of 55 degrees, the reflectivity is independent of the sun altitude and its value is approximately 0.05.

POLAR CANADIAN AIR MASS

SA 5-9.9°

3A 10-14.9* , N-79

 *. ' . . ,* » 

 ' * »*...*' % * ".

 . <,

SA 15- 19.9° N-94

SA 20-24.9° N-98

3X25-29.9° N-II5

SA 30-34. 9° _ f t ,N-I02

.05

.05

.05

1- 

>.05

U "

.05

.05

.05

SA'35-39.9° N-94

'.SA 40-44.9° N-5S

.8/45-49.9° " ' N-33

SA 50-54.9° '' ' N-25

SA55-59.9° ' N-40

SA60-64.9° ' N-22

'SA65-69.9° ' N-6

"sA70-74.9° , , , , N . 6

WIND SPEED (KNOTS)

15 20

WIND SPEED (KNOTS) WIND SPEED (KNOTS)

TROPICAL GULF AIR MASS

SA IO-J4.9"

SA 15-19.9° N-24

SA 20-24. 9° N-29

SA25-29.9' N-28

SA30-34.9° ', , , ,N-23

.05 

0 

.05 

0 

.OS

REFLECTIVITY ° > 0<

.OS

.05

.05

SA 35-39.9° ' N-25

SA 40-44.9° ' N - 28

SA45-49.9° ' N -23

SA 50- 54.9° ' N - 19

.-. 

SA 55-59.9° N - IS

SA 60-64.9°' ' ' N - 19

-" . 

SA 65-69.9° ' N - 14

SA 70-74.9° ' N - 7

SA 75-79.9° _ _ " | ' |N .9

0 5 10 15 20 25 30 

WIND SPEED (KNOTS)

WIND SPEED

20 

(f<NOTS) WIND SPEED (KNOTS)

Figure 59. Observations on the reflectivity of a natural water surface under Polar 

Canadian and Tropical Gulf air masses as a function of the wind speed at the 8-meter 

level.

84
of 

«

9 

OtO +

o +

O 4-

o +

.10 .15 .20 

REFLECTIVITY

Figure 60. Median reflectivity of a water surface as a function 

of sun altitude for Polar Canadian (O) and Tropical Gulf (+) 

air masses under clear skies.

.10 .IS .20 

REFLECTIVITY

Comparison of these observations with the semitheoretical reflectivities of Sverdrup (quoted earlier) 

and those in figure 56 for Washington shows that 

for high sun altitudes the Lake Hefner observations 

give higher reflectivities and for low sun altitudes 

lower reflectivities. In general they agree within about 

±0.02. Sverdrup's results would have been closer 

to those obtained at Lake Hefner if he had used 

Schmidt's value of 0.173 for reflectivity of diffuse 

radiation instead of Powell and Clarke's value of 

0.08. The higher observed values of reflectivity obtained at high sun altitudes tend to support Powell 

and Clarke's contention that all reflection does not 

take place directly at the water surface but that 

about "one-half takes place from a stratum of relatively opaque water just beneath the surface." The 

semitheoretical results of Sverdrup and the Washington data do not include this component, while 

the Lake Hefner observations do include it.

The expression (equation 83) for the reflectivity 

of an optically flat water surface under a clear sky 

is complicated and suggests that the expression for 

the reflectivity of a natural water surface under a 

clear sky is also complicated. Nevertheless, the observations presented in figure 61 can be approximated by an equation of the form

Rt = (85)

Figure 61. Reflectivity of a natural water surface under clear 

skies.

where Rt is the reflectivity, SA the sun altitude, and 

a and b constants having values of 1.18 and 0.77 

respectively. This equation is shown by a solid line 

in figure 61.

The reflected energy can be obtained from the 

reflectivity provided the total incoming solar radiation is known. Figure 58 shows the theoretical variation of the amount of reflected energy, from an 

optically flat surface, with season and time of day. 

Days without cloud cover were infrequent at Lake 

Hefner; hence very few observational data are available for comparison. On the occasional clear days, 

the reflected energy rose rapidly after sunrise, remained relatively constant throughout the day, and 

decreased rapidly just before sunset. The secondary 

features shown in figure 58 were unrecognizable. 

The maximum amount of reflected energy observed 

was 0.08 cal cm~2 min" 1 , which agrees very well 

with the predicted maximum of 0.085 cal cm~2 min" 1 

(fig. 57).

Observations of Reflectivity Under a Cloudy Sky

Since for clear skies the reflectivity appears independent of the state of .the water surface and

302520 O - 54 - 7 85
atmospheric turbidity for Polar Canadian and Tropical Gulf air masses, it is safe to assume that under 

cloudy skies the same independence will exist. Enough 

Lake Hefner data are available to examine the reflectivity of a natural water surface under low clouds 

(50 to 6500 feet above the terrain) and high clouds 

(greater than 20,000 feet above the terrain) for three 

cloud amounts, scattered (1/10 to 5/10 sky coverage), broken (6/10 to 9/10 sky coverage), and overcast (10/10 sky coverage). Middle clouds (6500 to 

20,000 feet above the terrain) occur infrequently at 

Lake Hefner, so not enough data are available to 

examine reflectivity under these conditions. The number of observations available for reflectivity study 

under low clouds was 696, distributed as follows: 

scattered clouds, 271; broken clouds, 183; and overcast clouds, 242. These observations were taken from 

1 September 1950 to 31 August 1951.

Figure 62 illustrates the relation between reflectivity and sun altitude for different amounts of low 

clouds. Increased scattering of solar radiation with 

increasing amounts of low clouds is clearly demonstrated. As the amount of cloud increases, the effect 

of sun altitude on reflectivity decreases. The data can 

be closely approximated by empirical equations of 

the same form as for clear skies, giving the constants 

the following values:

consfanf low scattered low broken low overcast

a 2.17 0.78 0.20

b 0.96 0.68 0.30

These equations, using the proper constants, are

plotted as solid lines on figure 62.

Clouds modify the reflectivity of a natural water 

surface the most under a low overcast. As shown in 

figure 62c, the reflectivity is almost independent of 

sun altitude, ranging from 0.05 at 80 degrees to 

0.11 at 5 degrees. Although this range is small, it 

indicates that the solar radiation is not completely 

diffuse and that some direct solar radiation reaches 

the water surface. The average reflectivity over this 

range in sun altitude is about 0.085, which agrees 

well with Powell and Clarke's average reflectivity 

under an overcast sky of 0.08, and is slightly lower 

than the 0.105 obtained by Neiburger under a low 

stratus cloud cover. At low sun altitudes, when solar 

radiation is most diffused, the reflectivity approaches 

the theoretical value of 0.173 for completely diffused 

solar radiation. This indicates that complete diffusion 

of solar radiation is only attained in nature under a 

low overcast cloud cover at very low sun altitudes, 

and that the theoretical value of 0.173, rather than 

0.08 as suggested by Powell and Clarke, is more 

correct. These results agree with Neiburger's conclusion, obtained from observations under California 

stratus.

The Lake Hefner observations under a low 

overcast give values of reflectivity at all sun altitudes 

that are slightly lower than those obtained by Neiburger under stratus. The difference can probably 

be accounted for in terms of the height of the cloud 

cover above the ground. In the case of Lake Hefner 

the observations were made for a period of a year 

under low clouds that varied in height from 50 to 

6500 feet above the ground, with the median reSCATTERED (''|O- 5/ lo)

LOW CLOUDS

BROKEN ("lo-'-'l OVERCAST (I0/| 0 )

In I ' I

.10 .20

REFLECTIVITY REFLECTIVITY REFLECTIVITY

Figure 62. Reflectivity of natural water surface under low clouds.

86
flectivity values probably representing a median cloud 

height of about 3500 feet. Neiburger's observations 

were made under stratus, which has a fairly constant 

height of about 1500 feet. This indicates that, for 

a constant sun altitude, the lower the overcast the 

greater the reflectivity. Thus, for areas where the 

height of the low overcast averaged around 3500 

feet, figure 62c would be useful in obtaining reflectivity but, if the cloud heights were consistently higher 

or lower, modification would be necessary.

The reflectivity of a water surface under variable 

amounts of high clouds is shown in figure 63. Again 

the data can be adequately represented by the empirical formula used for clear skies or low overcast, 

but the constants have the following values: 

constant high scattered high broken high overcast 

a 2.20 1.14 0.51 

b 0.98 0.68 0.58 

A comparison of the reflectivity under high clouds 

with that under low clouds indicates that for scattered-cloud conditions the reflectivities are similar 

in fact, the two essentially coincide. In the case of 

broken and overcast skies, however, the reflectivity 

for high sun altitudes is greater under low clouds, and 

for low sun altitudes is smaller under low clouds. 

This indicates that the scattering or diffusing of solar 

radiation becomes greater as cloud height decreases 

or, what is probably more correct, the lower cloud 

height is associated with thicker clouds and hence 

greater diffusion of solar radiation.

A comparison of the reflectivity under a clear 

sky with that under a low overcast (the two extreme

situations) shows that the reflectivity under low overcast clouds is slightly greater at high sun altitudes and 

less at low sun altitudes, the variation being due to 

the increased turbidity of the atmosphere because of 

the presence of water droplets in the air. The modification of the reflectivity, in these two extreme cases, 

is not great, showing that the effect of turbidity on 

reflectivity is small. Thus, it may be concluded that 

the reflectivity of a natural water surface is primarily 

a function of sun altitude and secondarily a function 

of atmospheric turbidity including the effect of clouds. 

This discussion suggests that the determination 

of the amount of energy reflected from a water surface from empirical relationships, as represented by 

figures 61, 62, and 63, is possible. It is necessary 

to know only the sun altitude, type and amount of 

clouds, and the total incoming solar radiation. The 

type and amount of clouds can usually be obtained 

from Weather Bureau observations and the total incoming solar radiation measured at or near the site 

in question by a suitable pyrheliometer. This eliminates the necessity of attempting the difficult direct 

measurement of reflected solar energy. The amount 

of energy reflected from Lake Hefner on 20 days 

selected at random during the period 1 May 1950 

to 31 August 1950 has been computed from the above 

relationships and compared in figure 64 with the 

observed reflected energy. On the average, the computed reflected energy is smaller than the observed 

by 0.004 cal cm~ 2 min- 1 (5.8 cal cm~ 2 day" 1 ). 

Since the accuracy of the observation is 0.005 cal 

cm~2 min" 1 , the agreement is satisfactory.

SCATTERED ('/|0- S'lo)

HIGH CLOUDS

BROKEN (61o-*/|o) OVERCAST (,IO/| 0 )

.10 .20 

REFLECTIVITY

.10 .20 

REFLECTIVITY

.10 .20 

REFLECTIVITY

Figure 63. Reflectivity of a natural water surface under high clouds.

30Z520 O - 54 -8 87
a

4..

0 IO 2O 30 4O 50 6< 

OBSERVED REFLECTED SOLAR ENERGY^CAl CM"1 DAY" 1 )

Figure 64. Comparison of reflected solar radiation observed at 

Lake Hefner with that computed from empirical equations.

Average Monthly Reflected Solar Radiation 

Observed at Lake Hefner

The average monthly reflected solar radiation 

and the maximum and minimum daily value for each 

month as recorded at Lake Hefner during the 16- 

month observational period are shown in figure 65. 

The reflected radiation was relatively constant, varying from 21 cal cm" 2 day" 1 in February 1951 to 

48 cal cm" 2 day" 1 in June 1950, with an extreme 

daily minimum of less than 1 during February 1951 

and an extreme daily maximum of 55 in June 1950. 

The high solar radiation during the summer is apparently offset by the greater reflectivity at the low sun 

angles during the winter.

Summarizing, the reflectivity of an optically flat 

water surface under clear skies is theoretically a 

function of sun altitude and atmospheric turbidity. 

The Lake Hefner observations indicate that under a 

clear sky the effect of wind and air-mass turbidity 

is negligible, and hence that reflectivity is primarily 

a function of sun altitude. In addition, the observations support previous conclusions reached by Powell 

and Clarke that the energy is reflected partly from 

the water surface and partly from a layer of opaque 

water just beneath the surface. Under cloudy skies 

the reflectivity is uniformly modified by an increase in 

scattering, caused by increased cloud cover and decreased height of cloud cover. Solar radiation is completely diffused only under conditions of low sun

altitude and low overcast clouds, and then approaches the theoretical value.

The variation of observed reflectivity with sun 

altitude, for the various types and amounts of clouds, 

can be approximated by simple, empirical, hyperbolic 

relationships, which can be used to evaluate indirectly the reflected solar radiation. At Lake Hefner, 

the reflected radiation ranged from near zero to 55 

cal cm~2 day" 1 .

EFFECTIVE BACK RADIATION

The effective back radiation has been defined as 

the difference between the long-wave radiation leaving a body of water and that from the atmosphere 

being absorbed by the body of water. The process 

has often been referred to as the "nocturnal" radiation. The terminology is misleading, however, since 

the process takes place during the day as well as 

during the night. In addition, there is evidence that 

its magnitude is greater during the day.

A. Angstrom (1915) presented an excellent review of effective back radiation and of instrumentation for measuring it. The reader is referred to his 

paper for information on the status of the study 

prior to 1915.

All the early instruments had disadvantages. 

Some were directional, that is, measurements had to 

be made at various zenith angles and then integrated 

over the hemisphere to obtain the total atmospheric 

radiation. All were uncompensated for convectional 

effects, and therefore were limited in use to calm 

or nearly calm conditions in order to obtain nonerratic readings. Most could be used only for night 

observations, since the daytime effects of short-wave 

solar radiation could not be eliminated. None were 

able to record continuously and automatically; it was 

necessary to take several individual measurements 

and then average to obtain a single representative 

reading. With the development of the Gier and 

Dunkle flat-plate radiometer in 1948, most of these 

disadvantages were overcome.

Since 1780, when the first primitive measurements of effective back radiation were made, countless observations have been recorded by investigators 

in all parts of the world. A great variety of instruments and techniques were utilized. It is not practicable to review all this work, but certain series of 

measurements are outstanding and deserving of 

attention. Examples are A. Angstrom's (1915) measurements made at Bassour, Algeria, in 1912 and in 

California in 1913; Kimball's (1918) measurements 

made at Washington, D. C, and vicinity in 1914 and

88
Figure 65. Average monthly reflected solar radiation for Lake 

Hefner from 1 May 1950 to 1 September 1951.

MAY JUN JUl AUG SEP OCT NOV DEC JAN FEB MAR 

1930 1951

1915; Dines and Dines (1927) measurements at 

Benson, England, from 1922 to 1926; Robitzsch's 

(1926) measurements at Lindenberg in 1926; Asklof's 

(1920) measurements in Sweden in 1918; Ramanathan 

and Desai's (1932) measurements at Poona, India, in 

1930; Eckel's (1934) measurements made at Vienna 

in 1930 and 1931; Buttner's (1939) measurements 

made in Africa in 1938; Wexler's (1941) measurements made at Fairbanks, Alaska, and Fargo, North 

Dakota, from 1936 to 1938. These observations were 

made with various types of instruments during various 

months of the year.

From these observations several general conclusions regarding atmospheric radiation have been 

formulated, namely, variations of atmospheric radiation at low altitudes are due primarily to variations 

in atmospheric temperature and humidity; the atmospheric radiation increases as the moisture content of 

the air in the lower levels increases; the radiation 

during the daytime seems subject to the same laws 

that hold for the radiation during the night; the effect 

of clouds is variable, with low dense clouds considerably decreasing the effective back radiation and 

a high thin overcast having only a slight effect; there 

is always a positive radiation from earth to sky.

In most cases, effective back radiation has not 

been measured directly. Usually, the atmospheric 

long-wave radiation has been measured, and radiation from the body of water has been computed as 

black-body radiation utilizing Stefan-Boltzmann's law 

corrected by some emissivity coefficient, since water 

radiates as a gray body rather than a black body. 

Therefore, observations have usually been discussed 

in terms of atmospheric radiation, rather than effective back radiation.

The atmospheric radiation does not follow any 

simple law since it is a function of many variables, 

such as the distributions of moisture, temperature, 

ozone, carbon dioxide, and perhaps other atmospheric factors as yet unknown. It was early recognized 

that the total amount of moisture in the air was one

APR MAY JUN JUL AUG

of the primary variables. Since upper-air soundings 

are not always available, early observers attempted 

to correlate the incoming radiation with elements 

easily observed at the ground, such as air temperature and humidity near the place of observation. 

These efforts have led to the establishment of several 

empirical formulas for evaluating atmospheric radiation. These formulas agree only statistically with 

actual observations, since the scatter of any set of 

observations about the line of best fit is very great. 

The following four empirical formulas for computing 

radiation from a clear sky have been proposed, with 

the first two receiving the greatest attention:

Qa

= 0 be Angstrom (86a)

= (oP-bea)/T0 

= a -f- D log ea,

Brunt

Robitzsch

Elsasser

(86b) 

(86c) 

(86d)

where Qa is the atmospheric radiation, a the StefanBoltzmann constant, T0 the absolute temperature of 

the air near the ground, c the Naperian base, ea 

the vapor pressure of the air near the ground, P the 

air pressure, and a, b, and y empirical constants.

Over the usual atmospheric range of vapor pressures, 2 to 30 mb, and using the presently accepted 

values of the constants, the differences between the 

above formulas are small. For example the difference 

between Brunt's and Angstrom's formulas, and between Brunt's and Elsasser's formulas, is only 10 per 

cent. This is not a significant difference, since the 

scatter of the observations is considerably greater. 

These formulas all attempt to relate atmospheric radiation to the local vapor pressure only, whereas it 

should be related to the total vapor content of the 

atmosphere, the carbon dioxide content, ozone content, and perhaps to other unknown factors. Therefore, these formulas can only be considered as 

empirical relationships which will only give average 

evaluations of atmospheric radiation from a clear sky

89
over relatively long periods. This is brought out by 

the variations in the constants, as obtained from 

various series of measurements. For example, twelve 

different sets of observations fitted to Brunt's formula 

gave a value of a from 0.34 to 0.62 and of b from 

0.029 to 0.082; and five different sets of observations 

fitted to Angstrom's formula gave a from 0.71 to 0.81, 

b from 0.24 to 0.33, and y from 0.04 to 0.074.

The above discussion is concerned only with the 

radiation from a clear sky. If clouds are present, the 

atmospheric radiation is increased greatly. Systematic observations under an overcast sky were made 

by Asklof (1920) and Angstrom (1929). They presented their results in the following form:

Qac T04 (1 - A) + AQ0, (87)

where Qac is the atmospheric radiation from an overcast sky, Ta is the temperature of the air in °K, X is an 

empirical coefficient which is a function of cloud 

height, and Qa is the atmospheric radiation from a 

clear sky computed from equation (86a). The constant 

A has been evaluated only for the Asklof and Angstrom observations.

Atmospheric radiation has been treated theoretically by Mugge and Moller (1932), Phillips (1940), 

Elsasser (1940), and more recently by Yamamoto 

(1950) who pointed out that theoretical investigations of atmospheric radiation are not in too 

good agreement with observation. This disagreement is probably due to inability of investigators to 

consider theoretically all the variables contributing to 

the atmospheric radiation because of the complex 

relationships of the variables.

The above discussion may be summarized as 

follows: At the present time it appears possible to 

evaluate from empirical relationships the atmospheric radiation received from a clear sky over a 

long time interval. For shorter time intervals, and for 

times when clouds are present, it is necessary to 

measure it directly.

The long-wave radiation from the body of water 

is evaluated from the Stefan-Boltzmann law for blackbody radiation corrected by a suitable emissivity 

factor. A search of the literature revealed a rather 

meager knowledge of the emissivity of water: Dorsey 

(1940) gave a value of 0.985, Angstrom (1920) 0.94, 

and Richardson (1931) 0.906. Variations of this magnitude in the emissivity factor are important in computing evaporation from the energy budget. For 

example, a change of emissivity from 0.985 to 0.94 

will increase the evaporation by 30 per cent at an 

evaporation rate of 0.2 cm per day and by 20 per

cent at an evaporation rate of 0.4 cm per day. It is 

evident that, for purposes of computing evaporation, 

a more precise evaluation of the emissivity of water 

is essential.

The remainder of this section is devoted to a 

discussion of the observations, pertinent to the longwave radiation exchange, made at Lake Hefner. The 

characteristics of atmospheric radiation are examined 

with respect to the variation of the radiation as a 

function of local vapor pressure and the limitations 

placed upon the use of such relationships. In addition, the emissivity of a natural water surface is 

discussed.

Atmospheric Radiation

At Lake Hefner, the atmospheric radiation was 

measured directly during the night by means of a 

Gier and Dunkle flat-plate radiometer. During the 

day it was evaluated indirectly, using the radiometer 

for measuring both the solar and atmospheric energy 

irradiating the surface of the earth and an Eppley 

pyrheliometer for measuring only the solar radiation; 

the daily atmospheric radiation is the difference between the two measurements.

The diurnal variation of atmospheric radiation as 

obtained from this system of instruments was of an 

unexpected nature. On every one of 340 days of 

available measurements, the atmospheric radiation 

began to decrease at sunrise, reached a minimum at 

solar noon, increased until sunset, and then remained 

relatively constant until sunrise. The magnitude of 

the observed decrease is significant; for example, the 

nighttime value of atmospheric radiation might be 

0.500 cal cm~2 min" 1 , decreasing to 0.250 cal 

cm~2 min"" 1 at solar noon. The magnitude of the 

decrease was greatest under clear skies and least 

under a low overcast.

Intuitively one would expect, not a diurnal variation of this nature, but a relative constancy in the 

atmospheric radiation throughout the day, with perhaps a slight increase during the afternoon as the 

lower atmosphere warms up. Water vapor in the 

atmosphere is a fairly effective absorber of long-wave 

radiation, except for a transparent band from 8.5 

p. to 11.0 ft. If the observed decrease in long-wave 

radiation is a true one, it would imply that some 

other component in the atmosphere radiates very 

strongly at the same wavelengths as the transparent 

band in the water-vapor spectrum and, furthermore, 

has a large diurnal fluctuation. Since this is improbable, the observed daytime values of atmospheric 

radiation appear incorrect, and the observed diurnal

90
variation is therefore probably due to instrumental 

errors. (See "Instrumentation" section for description 

of instruments.)

Some possible instrumental errors are: (1) a high 

calibration constant for the Eppley pyrheliometer, 

(2) a low temperature reading on the flat plate, (3) 

a low calibration constant for the flat-plate radiometer, and (4) an imperfect flat-plate black body, with 

the reflectivity of the flat plate for the solar radiation 

dependent on the sun's altitude.

The U. S. Weather Bureau made solar radiation 

measurements at a site approximately 13 miles south 

of Lake Hefner. The Lake Hefner observations are 

approximately 2.5 per cent higher than those of the 

Weather Bureau. At the present time the same pyrheliometer is in service at Lake Mead. Comparison of 

the measurements being made at Lake Mead with 

those made by the Weather Bureau at Las Vegas, 

Nevada, 30 miles distant, indicates that the Lake 

Mead readings are about 1.5 per cent lower than 

those at Las Vegas. It is therefore concluded that 

these differences are within the limits of accuracy of 

the technique.

A low value for the temperature of the flat plate 

could be obtained if the reference thermocouple 

were not maintained at 0°C. Since ice in the reference 

ice jug was renewed at frequent intervals, any error 

from this source would be present only for short 

periods prior to the refilling of the jug, and could not 

account for the consistent observed variation in atmospheric radiation.

In the absence of a recalibration of the flat-plate 

radiometer, no direct evidence is available to indicate 

that the calibration was incorrect. However, nighttime recorded values agree satisfactorily with previous 

measurements made with other types of instruments.

The last possible instrumental error is concerned

with the reflectivity of the flat plate for solar radiation. Dunkle et a/. (1949) demonstrated in the laboratory that the flat plate is essentially independent 

of the altitude angle of an artificial light source. However, this may not be true when the flat plate is 

irradiated by solar energy, some of which is arriving 

as direct radiation and some as scattered. It has not 

been possible to investigate in detail this possible 

source of instrumental error.

It is tentatively concluded that the dependence 

of the flat-plate reflectivity on sun altitude is the 

primary cause of the observed daily decrease in atmospheric radiation, and that it is necessary to 

develop better instrumentation.

For this study, and in the absence of independent verification, it is assumed that the atmospheric 

radiation is relatively constant throughout the day 

and night, that the night values obtained from the 

Gier and Dunkle flat plate are correct, and that the 

day values are incorrect. An average daily value has 

therefore been obtained from the night measurements 

and this value assumed to apply throughout the 

24-hour period.

Figure 66 indicates the magnitude of the atmospheric radiation as observed at Lake Hefner. The 

average monthly radiation (number indicates the 

number of days of data) and the highest and lowest 

daily value recorded during the month are shown. 

The radiation averages about 800 cal cm~2 day" 1 

during the summer and about 550 cal cm~2 day" 1 

during the winter. These values are approximately 

200 higher than for comparable monthly average 

solar radiation. The greatest daily radiation occurred 

in August 1950, and amounted to 900; the minimum 

occurred in February 1951, and amounted to 375. 

These compare with maximum and minimum solar 

radiations of 750 and 40, respectively.

Figure 66. Average monthly atmospheric radiation for Lake Hefner 

from 1 May 1950 to 1 September 

1951.

900

BOO

f/oo >-

2 

7 «»

5

.500

0.

400

300

/

1

>''NX i i ' i i

:

2 ~"~"il _J.1 13^--^. N X

:__- '"" \ 20\^ ^--^ DAILY MAXIMUM _

\ ,-x 27\ ^^-^.-- "" 

\--'' "^vv N. MONTHL

X\ 30^~ - ^***"

x .

^-^^^ DAILY MINIMUM

^^xs

I 1 1 I 1 I 1 .

MY JUN JUL AUG SEP OCT HOV DEC JAN

950 1951

i i i

*''

-'"^ ^<

Y AVERAGE^^'*29

-20 30

->-,^**"*

xx'""

,.x'x

1 1 1

FEB MAR APR MAY

' , j.- '

/'20 / -»«.

* '

-

-

JUN JUL AUG

91
RADIATION FROM A CLEAR SKY

At Lake Hefner, for the period 1 September 1950 

to 1 September 1951, 1893 observations of the nighttime atmospheric radiation were available for study. 

As previously indicated, the atmospheric radiation is 

primarily a function of the total amount of water 

vapor in the atmosphere. However, because of a lack 

of information concerning the total water-vapor 

content, most investigators have adopted the local 

vapor pressure as an index of the total water-vapor 

content. At the Will Rogers Airport, 13 miles south of 

Lake Hefner, two radiosondes per day were taken 

during the course of the Lake Hefner observations. 

It was originally hoped that these data might be used 

in connection with the study of atmospheric radiation 

at Lake Hefner. It was found, however, that two 

radiosondes per day were not enough to define the 

variation of the total moisture content with the accuracy needed when considering half-hourly radiation 

observations, and that accurate evaluation of total 

water vapor in the atmosphere could not be obtained, 

possibly due to the lack of accuracy or sensitivity of 

the humidity-sensing element and to the fact that the 

soundings did not consistently sample the entire atmospheric column. The use of total water vapor, 

as obtained from the two daily radiosondes, in place 

of the local vapor pressure, did not lead to better 

results. Hence, throughout this study, the vapor pressure at 2 meters has been used as the moisture 

variable. Atmospheric radiation is expressed as the 

ratio of the observed atmospheric radiation to the 

black-body radiation at the temperature of the air 

at 2 meters. Actually, the black-body radiation at 

the mean water-vapor temperature should be used, 

but the determination of this temperature is not feasible. As noted later, the use of the local vapor pressure and the local air temperature, under special 

circumstances, leads to anomalous results.

The 1893 observations were made over a vaporpressure range of 2.9 to 30.3 mb. The data were 

edited to exclude observations made during periods 

of moderate and strong air-mass over-running. This 

editing made the local temperature and vaporpressure observations more representative of the 

air column above the instrument. The ratio of atmospheric radiation to the black-body radiation of 

the atmosphere, defined by the 2-meter air temperature, was plotted against the 2-meter vapor pressure. To simplify plotting, these ratios were grouped 

by 0.5-mb intervals, and median values were obtained for each group. Median values were used 

rather than arithmetic means because the two did 

not differ markedly and the former were easier to 

compute. Figures 67 and 68 show the plots of these 

median ratios versus vapor pressure. The number indicates the number of observations, while the dashed 

lines include two-thirds of the individual observations 

within each 0.5-mb range in vapor pressure. In 

general there is a definite increase in the radiation 

ratio with increasing vapor pressure. However, at low 

vapor pressure (5 to 8 mb) the ratio is higher than 

expected, and also higher than most prior observations. These results are explainable in terms of the 

local meteorology. Observations at these vapor pressures were generally made during the winter, when 

a thin wedge of relatively dry Polar Canadian air 

was present at the ground with over-running of moist 

Tropical Gulf air aloft. Under these circumstances the 

total water vapor in the atmospheric column above 

the station was considerably higher than indicated by 

the local vapor pressure, thereby giving a higher 

atmospheric radiation than expected. In addition, the 

2-meter air temperature was relatively lower than the 

mean water-vapor temperature.

Four empirical equations, plotted on figures 67 

and 68, of the following form were fitted to these 

data:

A A

MODIFIED ANGSTROM 

TYPE EQUATION

/ s' "- "fT

' ^^1^0"" «"« / N~"' 

'"v ~.~£3C^ ' '< I

"v^s^f^x^ /v -"i^r?1." \' \ vx

ANGSTROM TYPE 

EQUATION

b 20

ez '(MILLIBARS)

Figure 67. Plot of atmospheric radiation equations of the Angstrom and modified Angstrom type for the Lake 

Hefner clear sky observations.

92
o.e

0.7

4* /V 2*4 /" '-''" V V

LEAST SQUARES 

STRAIGHT LINE

15 20

ez (MILLIBARS)

25

Figure 68. Plot of atmospheric radiation equations of the Brunt type and the least-square straight line for 

the take Hefner clear sky observations.

Q

= a be ~"

= c +

= (f+ge2) -/!£« », 

and = i + /e2,

Angstrom

Brunt

(88)

Modified Angstrom 

Straight line

where Q0 is the atmospheric radiation, a the StefanBoltzmann constant, T2 2-meter air temperature, e2 

the 2-meter vapor pressure, c the Naperian base, and 

a, b, c, d, f, g, h, i, j, y, and a are constants to be 

determined from the data. The modified Angstrom 

type equation gives a better fit to the data at low 

vapor pressures. With the exception of one point 

at 18.75 mb, determined by only three readings, all 

four curves are within 5 per cent of the median points. 

In addition, the curves stay fairly well within the twothirds band of actual radiation ratios. 

The values of the constants are:

Constant 

a 

b

y

c 

d 

r

9 

h 

a 

i

Value

1.107

0.405

0.022

0.682

0.036

0.752

0.0048

0.542

0.781

0.740

0.0049

Various other determinations of the constants 

in Brunt's and Angstrom's equations have been summarized by Elsasser (1942) and are listed for comparison in tables 9 and 10. The correlation coefficient

indicates the correlation between the local vapor 

pressure and the observed radiation ratio, Qa/aTo4. 

In general, the radiation ratios obtained from the 

Lake Hefner data are higher than those obtained 

by others.

TABLE 9. Various Values of Constants for Brunt's 

Atmospheric-Radiation Equation.

Correlation

Investigator

Dines

Asklof

Angstrom

Angstrom

Boutaric

Kimball

Eckel

Roman

Anderson

TABLE 10.

Place

England

Sweden

Algeria

California

France

Washington, D. C.

Austria

India

Oklahoma

Various Values

c

0.53

0.43

0.48

0.50

0.60

0.44

0.47

0.62

0.68

d

0.065

0.082

0.058

0.032

0.042

0.061

0.063

0.029

0.036

of Constants

Coefficient

0.97

0.83

0.73

0.30

 

0.29

0.89

0.68

0.92

for Angstrom's Atmospheric-Radiation Equation.

Investigator

o lt

Angstrom

Kimball

Eckel

Roman

Anderson

Place

Sweden

Virginia

Austria

India

Oklahoma

a

0.806

0.80

0.71

0.79

1.107

b

0.236

0.326

0.24

0.273

0.405

y

0.115

0.154

0.163

0.112

0.022

There are also considerable variations between determinations. These variations are probably related to 

the implied hypothesis that previous investigators 

have made concerning the relation between the local 

vapor pressure and the total water-vapor content of 

the atmosphere. This relation is probably different for 

different locations, since the composition of the atmosphere differs from one place to another. If it were

93
possible to express the moisture parameter in terms 

of total water-vapor content rather than local vapor 

pressure, the variation would undoubtedly decrease. 

Hence, it is concluded that the use of empirical relations are restricted to locations that have similar airmass structures. To attempt to use them in other 

locations will lead to erroneous results and conclusions.

If the above relations are extrapolated to zero 

vapor pressure (index for an absolutely dry atmosphere), the radiation ratio should express the component of atmospheric radiation related to the other 

constituents in the atmosphere that contribute to atmospheric radiation, namely, carbon dioxide and 

ozone, provided there is no intercbrrelation between 

water vapor and the other constituents. Yamamoto 

(1950) indicates that at a temperature of 280°K (7°C), 

the ratio for carbon dioxide is 0.146 and for ozone is 

0.0086. The zero intercepts for the curves in figures 

67 and 68 are considerably higher than these values. 

The radiating power of the atmosphere is high even 

for exceedingly small moisture contents, owing to the 

very high absorptivity of water vapor for some of the 

long waves. Hence, at very low vapor pressures, the 

ratio should decrease rapidly. Therefore, these empirical relations are only applicable over the range of 

vapor pressures considered, in this case from 3 to 

30 mb.

The four empirical curves were checked against 

four sets of nighttime data not used in obtaining the 

curves themselves, and six sets of nighttime data used 

in obtaining the curves. The latter six sets were chosen 

arbitrarily, except that they were spread over the 

entire vapor-pressure range. The results are given in 

table 11. The empirical values for the four sets of 

data are from 16 per cent higher to 3 per cent lower,

TABLE 11.

Comparison of Computed and Observed Atmospheric-Radiation Ratios

Modified Straight Maximum Minimum

Date Observed Angstrom Angstrom Brunt line Difference Difference

(per cent) (per cent)

Days Not Included In Evaluating Constants

13Jun 1950

15Jun

lOAug

31 Aug

lOOct

4Nov

19 Mar 1951

2 May

/July

28 Aug

0.817

0.778

0.868

0.721

0.802

0.699

0.749

0.728

0.854

0.845

0.864

0.865

0.859

0.832

Days Included

0.804

0.748

0.742

0.789

0.862

0.849

0.864

0.865

0.861

0.837

0.856

0.856

0.846

0.833

0.856

0.857

0.846

0.829

 5.8

 11.2

2.5

 16.1

 4.8

 10.0

0.8

 15.0

in Evaluating Constants

0.817

0.771

0.756

0.806

0.863

0.852

0.812

0.765

0.759

0.801

0.854

0.844

0.806

0.767

0.764

0.795

0.855

0.843

 1.9

 10.3

 2.0

 10.7

 1.1

0.1

 0.2

 7.0

0.9

_8.4

0.0

 0.8

and for the six sets of data from 10 per cent higher 

to 1 per cent lower than observed. Hence it might be 

hazarded that 10-per-cent accuracy might be expected from the empirical curves. Since there is little 

difference between the four empirical relations it 

seems reasonable to consider the straight line as the 

empirical expression for atmospheric radiation. Atmospheric radiation averages about 650 cal cm~2 

day" 1 ; hence, a 10-per-cent error is too large for 

evaporation computations.

RADIATION FROM A CLOUDY SKY

Prior investigations of atmospheric radiation 

were concerned chiefly with the radiation occurring 

with a clear sky. The only investigations dealing with 

the effects of cloud cover are those of Asklof (1920) 

and Angstrom (1929). They treated the atmospheric 

radiation occurring under an overcast sky. The Lake 

Hefner observations include adequate data for considering the atmospheric radiation occurring when 

low, middle, and high clouds in variable amounts 

scattered, broken, or overcast are present.

These data have been examined in a manner 

similar to that used in treating the data concerning 

radiation from a clear sky. A linear relationship has 

been used since it was shown, for the case of clear 

skies, that such a relationship is as adequate as more 

complicated ones in empirically expressing the results 

over the vapor-pressure range generally encountered 

in nature.

Figure 69 summarizes the results, and gives atmospheric radiation as a function of cloud height, 

cloud amount, and local vapor pressure. N indicates 

the number of observations. In each case, the data 

have been treated in the same manner as that employed for clear skies. The heavy dashed lines indicate the least-squares line for each set of data, while 

the light dashed lines indicate the relationship for 

clear-sky condition. An inspection of this figure reveals 

a fairly consistent relation among the three variables. The radiation is least, and the effect of vapor 

pressure is greatest, for clear skies over the entire 

vapor-pressure range; the effect of vapor pressure 

decreases as the cloud amount increases. For any 

given amount of cloud cover, the radiation increases 

over the entire vapor-pressure range as the height 

of the clouds decreases. The modification of the radiation, from the clear-sky case, is least for high 

scattered clouds and greatest for low overcast. For 

low overcast, the radiation is independent of the 

vapor pressure. Since the average height of the low 

overcast is probably around 2500 to 3000 feet above

94
HIGH SCATTERED HIGH BROKEN HIGH OVERCAST

NUMBER OF OBSERVATIONS

 I - » OBS.

 8-10 OBS.

 11-20 OBS. 

> 20 OBS.

MIDDLE SCATTERED MIDDLE BROKEN MIDDLE OVERCAST

.00

.80

.TO

_

' . ___^^^ *

* N -211

, . -Lj^R. ^

^^^J^''' '

N-157

m B» 

 -' '^x^^'--*'. '- ^^^ :;V"""" ;

N-260

LOW SCATTERED LOW BROKEN LOW OVERCAST

2S

TWO-METER VAPOR PRESSURE (mb)

Figure 69. Atmospheric radiation as a function of cloud height, cloud amount, and vapor pressure.

the terrain, this indicates that for low overcast clouds 

below this altitude the radiation is independent of 

vapor pressure and height of clouds.

The data shown in figure 69 can be represented 

empirically by an equation of the following form:

Qa/c,T24 = a + befl/ (89)

where

a = 0.740 -f 0.025 Ce-°- OB84fc, 

b = 0.00490 0.00054 Ce"0 - 000*,

and 1600^ h ^ oo.

Radiation for cloud heights under 1600 feet must be 

considered equivalent to that occurring when the 

cloud height is 1600 feet, but actually the radiation 

would probably continue to increase as cloud height 

decreases. In the above equation ea is the vapor

pressure in mb, C the cloud amount in tenths of sky* 

covered, c is the Naperian base, and h the cloud 

height in thousands of feet. Equation (89), using appropriate cloud heights and amounts, is shown in 

figure 69 as a solid line. The agreement with the 

least-squares line is good.

Table 12 summarizes some comparisons of observed atmospheric radiation and that obtained from 

equation (89). These data indicate that accuracies 

around 5 to 10 per cent are possible using the empirical equation. Although the percentage errors are 

small, the actual error varies from 15 to 75 cal cm~2 

day" 1 , which is too large for purposes of computing evaporation.

Recapitulating, this analysis indicates that: (1) 

Empirical relationships between atmospheric radiation

TABLE 12. Comparison of Computed and Observed Atmospheric Radiation for Selected Days

Atmospheric Radiation

(cal cm- 2 day-l)

Date

14Jul 1950

6 Aug

14Sep

24Sep

23 Jan 1951

8 Apr

Observed

699.8

905.8

803.5

733.0

560.2

535.7

Computed

743.0

829.4

822.2

710.1

544.3

565.9

Difference

43.2

 76.4

17.7

 22.9

 15.9

30.2

Difference

(per cent)

5.8

 8.4

2.2

 3.2

 2.8

5.6

T2

(°C)

22.8

24.5

18.5

20.6

4.1

8.9

«2

(mb)

18.4

26.4

19.5

14.1

5.2

4.9

C

(tenths)

3

7

10

3

7

1

h

(1000's

of feet*)

37

14

1

32

31

60

* For days with partly cloudy and clear skies, height is arbitrarily set at 100,000 feet for purposes of 

determining an average height for the day.

95
and local vapor pressure may be used if 10-per-cent 

accuracy is acceptable, provided the air-mass structure is similar to that of the area where the original 

observations were recorded. For other areas, with no 

consideration of the air masses involved, the accuracy 

of the relationships is more questionable. (2) A linear 

relationship between atmospheric radiation and local 

vapor pressure (over the vapor-pressure range generally observed in nature) is as adequate as the more 

complicated expressions of Brunt and Angstrom. (3) 

These empirical determinations are inadequate for 

evaluating atmospheric radiation for purposes of 

evaporation determinations. (4) To obtain more accurate methods of determining atmospheric radiation, 

in terms of more easily evaluated parameters, it will 

be necessary to consider the total vapor content of 

the atmosphere as the moisture variable, rather than 

the local vapor pressure.

Radiation from the Body of Water

Long-wave radiation from a body of water can 

be computed from the Stefan-Boltzmann law for 

.black-body radiation, corrected by a suitable emissivity factor for water, since water radiates as a gray 

body rather than a black body. It is necessary, for 

purposes of evaporation determination, to know the 

emissivity of the water surface to two significant 

figures. Since doubt existed as to the value of the 

emissivity of water, the Navy Electronics Laboratory 

contracted with the Physical Standards Laboratory, 

Institute of Engineering Research, University of California, for determinations of the emissivity of four 

water samples distilled water, Lake Hefner water, 

Lake Mead water, and sea water. The salinities of 

the Lake Hefner, Lake Mead, and sea water were 

347 ppm, 572 ppm, and 33.42 °/oo (parts per thousand), respectively. To determine the extremes of any 

possible variation of emissivity, the emissivity was 

determined over a range of water temperatures from 

0 to 30°C and, in addition, for a sample of water 

contaminated with a thin oil film.*

The emissivity of a body of water can be evaluated by direct measurement or indirectly from a 

measurement of the reflectivity of the water surface.

The direct method consists of heating the sample, 

and taking measurements of the surface temperature with a thermocouple and of the radiation from 

the surface with a directional radiometer. There are 

several objections to this method: (1) It requires a 

fairly large temperature difference between the

* This work was done under the supervision of R. V. Dunkle 

and J. T. Gier.

sample and the surroundings. This involves cooling 

the surroundings, since the emissivity was desired over 

a water-temperature range of 0 to 30°C. (2) Since 

water is opaque to long-wave radiation, and since 

most of the emitted energy comes from a layer very 

close to the water surface, it is extremely difficult 

to obtain precise surface-temperature measurements. 

(3) An appreciable error in the direct emissivity measurement may arise because of the absorption of 

radiation by water vapor in the air.

The indirect method consists of measuring the 

reflectivity of the water surface, using calibrated 

energy sources of different temperatures. According 

to Kirchoff's law, when the source temperature is 

equal to the water-surface temperature, the emissivity 

is precisely equal to the absorptivity. In addition, the 

absorptivity is equal to one minus the reflectivity. 

Thus when the water-surface temperature is equal to 

the source temperature, the emissivity is equal to 

one minus the reflectivity. Therefore, determination of 

reflectivity as a function of source temperature and 

extrapolation of the resulting curve to the temperature of the sample enable an indirect determination of 

the emissivity of the water sample. The reflectivity is 

measured by a directional radiometer which is first 

calibrated against a calibrated energy source over 

the path lengths to be used in the experiment. The 

reflected energy is obtained by taking the difference 

between a measurement of the ambient energy 

(energy source shielded from the water sample) and 

a measurement made when the energy source is unshielded. Since the energy source is calibrated, the 

reflection is the ratio of the reflected energy to the 

total energy output of the calibrated energy source. 

The indirect method has several advantages: (1) The 

sample can be maintained at any convenient temperature, the only restriction being that the temperature of the environment remain constant. (2) Since 

the reflectivity of water is low, a large percentage 

error in the measurement of reflectivity results in a 

small error in the emissivity. (3) The problem of 

water-vapor absorption is avoided by calibrating the 

radiometer against the source used and over the 

same path lengths used in making the reflectivity 

measurement. The major disadvantage in this indirect 

method is drift in the measurements caused by changes 

in water-surface temperature. This source of error 

can be minimized by making the two required readings as rapidly as possible.

The Physical Standards Laboratory, therefore, 

used the indirect method of evaluating the emissivity. 

Two different energy sources were utilized. The high96
TABLE 13. Reflectivity of Various Water Samples 

as a Function of Source and Water Temperatures.

Distilled Water

Source Water

Temp. Temp. Reflectivity

4320 48 0.023 

72 0.022

86 0.022

4140 48 0.023

72.5 0.022

87 0.022

3920 48 0.022

73 0.022

86 0.023

315 43 0.030

313 71 0.031

316 86 0.030

Sea Water

(33.42 <>/ 00)

Source Water

Temp. Temp. Reflectivity

4320 51 0.022 

68 0.022

86 0.022

4140 51 0.023

68 0.023

86 0.023

3920 51 0.023

68 0.023

86 0.023

303 46 0.032

316 71 0.030

320 86 0.031

408 72 0.029

420 86 0.030

549 75 0.024

557 75 0.026

204 85 0.030

203 62 0.030

Lake Mead Water

(572 ppm)

Source Water

Temp. Temp. Reflectivity

4320 48 0.023 

73 0.022

86 0.021

4140 49 0.025

73 0.022

86 0.022

3920 50 0.023

73 0.022

86 0.022

311 46 0.029

307 73 0.033

312 86 0.030

315 90 0.033

405 69 0.026

549 72 0.027

551 86 0.024

405 67 0.026

212 65 0.031

221 85 0.030

405 67 0.031

406 85 0.031

Water Analysis (ppm)

Date 2 Oct 1951

Silica 11

Iron .03

Calcium 71

Magnesium 27

Sodium 81

Potassium 3.0

Bicarbonate 137

Sulfate 244

Chloride 66

Fluoride .1

Nitrate 1.0

Lake Hefner Water

(347 ppm)

Source Water

Temp. Temp. Reflectivity

4320 44 0.022 

68 0.023

86 0.022

4140 44 0.023

68 0.023

86 0.023

3920 86 0.023

68 0.023

86 0.022

307 47 0.032

315 72 0.033

317 86 0.031

408 76 0.030

411 86 0.029

551 73 0.028

556 86 0.026

212 69 0.032

207 85 0.030

Water Analysis (ppm)

Date 27 Sep 1951

Silica 2.4

Iron .00

Calcium 46

Magnesium 17

Sodium 53

Potassium 5.8

Bicarbonate 170

Sulfate 75

Chloride 60

Fluoride .3

Nitrate .4

temperature source, a 1000-watt lamp, calibrated 

for total directional radiation and temperature as 

a function of source voltage, was used to produce 

source temperatures around 4000°F. The low-temperature source, a metallic surface coated with lamp 

black and heated electrically, produced source temperatures around 300°F.

Table 13 summarizes the individual reflectivity 

measurements. An inspection of the data in table 

13 indicates that the effect of salinity and watersurface temperature is negligible. Since the data did 

not justify any other curve, a straight line was fitted 

to the plot of reflectivity versus source temperature. 

The equation for this line is

r = 0.0301 3.343 X 1Q- 6 T3I (90)

where r is the reflectivity of the water and T8 the 

source temperature in °C.

The greatest deviations from the straight line of 

best fit for the high-temperature source were +0.002 

and 0.001, with a mean deviation of 0.001. For 

the low-temperature source, the greatest deviations 

were 0.005 and -{-0.003, with a mean deviation 

of 0.002. Extrapolation of this line to temperatures 

(above freezing) normally encountered in natural 

bodies of water gives a reflectivity of 0.030. It is 

concluded that the emissivity of water is 0.970, with 

an estimated accuracy of ±0.005, and that it is independent of water temperature and dissolved solids.

A few data were collected on the reflectivity of 

tap water, muddy tap water, and tap water contaminated by an oil film. These data are summarized 

in table 14. As expected, the tap water is similar 

in reflectivity to the previously discussed samples. 

Sediment suspended in the tap water has no appreciable effect on the reflectivity. The contamination of 

the water surface by an oil film, however, modifies 

the reflectivity significantly, increasing it by about 50 

per cent and decreasing the emissivity from 0.970 to 

0.956. A modification of emissivity of this magnitude 

is significant when considering energy-budget evaporation computations, since it can introduce errors of

TABLE 14. Reflectivity of Contaminated Water Samples

Source

Temp.

(°F)

208

Muddy Tap

Water

Water

Temp. Reflec-

(°F) tivity

64 0.027

Source

Temp.

(°F)

216

219

287

Oil Film on

Tap Water

Water

Temp.

(°F)

67

76

76

Reflectivity

0.045

0.041

0.046

Source

Temp.

(°F)

138

Tap

Water

Water

Temp.

(°F)

66

Reflectivity

0.028

97
10 to 15 per cent at evaporation rates of around 

0.2 cm per day. It is speculated that for certain 

natural bodies of water, where considerable oil contamination may result from excessive plankton growth, 

domestic waste, or industrial waste, the evaporation 

rate may be considerably modified through the influence of the contamination on the emissivity.*

Because of time requirements it was necessary, 

in this study, to utilize a preliminary value of emissivity obtained prior to the completion of the above 

measurements. Preliminary data indicated an emissivity of about 0.967. This value was therefore used 

in computing evaporation. The evaporation figures 

were not recomputed because, over a range of watersurface temperatures of 7 to 32°C, the use of the 

emissivity value of 0.970 would increase the effective 

back radiation by only 2 cal cm" 2 day" 1 . The 

accuracy of the determination of emissivity is ±0.005, 

which amounts to an accuracy of about ±4 cal 

cm" 2 day" 1 in the effective back radiation.

The long-wave radiation from Lake Hefner is 

shown in figure 70. The average monthly, as well as 

the individual maximum and minimum daily values 

for each month, are indicated. The radiation averages around 900 cal cm~2 day" 1 during the summer, 

and 680 cal cm" 2 day" 1 during the winter, with the 

highest daily recorded value being 930 and the 

lowest 660. This is the largest single term in the 

energy budget.

Reflected Atmospheric Radiation

In prior evaporation studies utilizing the energy 

budget, the reflected atmospheric energy has generally been considered negligible. In this section it 

will be demonstrated that the magnitude of this term 

is such that it must be considered.

In the preceding section it has been shown that 

the reflectivity of a water surface, for a source temperature ranging from 0 to 30°C, is 0.030. Since the 

atmosphere is radiating at approximately these temperatures, it follows that the reflectivity of a water 

surface for atmospheric radiation is also about 0.030.

The average monthly reflected atmospheric radiation, as well as the daily maximum and minimum 

for each month, is shown in figure 71. The reflected 

atmospheric energy is nearly constant throughout the 

year, averaging about 26 cal cm"2 day" 1 during 

the summer and about 18 cal cm"2 day" 1 during the 

winter, with a daily recorded maximum of 29 and a

* Oil films, of course, also affect evaporation in other ways, 

but this problem was not studied.

daily recorded minimum of 12. The reflected atmospheric radiation is, on the average, slightly less than 

the reflected solar radiation, and is large enough, if 

neglected, to introduce a definite error into evaporation determinations based on the energy budget.

In summary, the atmospheric-radiation observations at Lake Hefner were examined in an empirical 

manner similar to that employed by previous investigators. However, instead of considering atmospheric 

radiation only from a clear sky and as a function only 

of local vapor pressure, which was done in most 

previous studies, it was possible to take account of 

the effects of cloud amount and height. An empirical 

expression was derived relating atmospheric radiation 

to the local vapor pressure (used as the moisturecontent index), cloud amount, and cloud height. This 

expression allows the computation of atmospheric 

radiation with an accuracy of about 10 per cent 

of the observed value, provided it is used in areas 

that have a relation between local vapor pressure 

and total atmospheric-water content similar to that 

at Lake Hefner.

In connection with long-wave radiation from the 

water surface, considerable work was done to determine the emissivity of water. It was shown that 

the emissivity of water is independent of its temperature and composition and is 0.970 ±0.005.

A consideration of reflected long-wave radiation 

indicated that this energy-budget term cannot be 

neglected.

ADVECTED ENERGY

In evaluating the energy advected into or out 

of a body of water, it is necessary to study the 

climatological, hydrological, and geographical characteristics of the particular location. These characteristics influence the choice of the most suitable method 

for measuring advected energy. It is not possible to 

establish a method that will apply in all cases, but 

it is possible to derive certain general principles, techniques, and cautions.

Advected energy is defined as the net energy 

gained or lost by a body of water through the ingress 

or egress of a volume of water. Advected volumes 

may result from surface inflow, rainfall, seepage, 

bank storage, controlled outflows, evaporation, and 

condensation. In some situations all these must be 

considered, while in other situations some may be 

negligible. The feasibility of applying the energy 

budget to the determination of evaporation at any

98
Figure 70. Long-w a v e radiation 

from Lake Hefner from 1 May 1950 

to 1 September 1951.

1000

>; 900

3

L 800 

O

S700

y 2 '7^<N 22 v 20 3Vx^^

^""^~ ^ '^

1 1 1 1 1 1 1 1 1 1 1 1 MAY JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY 

1950 1951

.. -

DAILY MAXIMUM 

MONTHLY AVERAGE 

-DAILY MINIMUM

i i i

JUN JUL AUG

Figure 71. Reflected atmospheric 

radiation from Lake Hefner from 

1 May 1950 to 1 September 1951.

11111

11 _ jT __ 13 --"^x

9^^".---'"\ ^^^.?° ^x-.^_

 " N ^^^^ 27

\ t *^

\--'~ "^^^ ^

>v^

i i i i i i i i

_.-'"' .,

DAILY MAXIMUM ,'' ' 'K>s',

^^*- ~~ 29^^.^^^ .,* '

^~fr ' MONTHLY AVERAGE ^*^ ^''

' .. __^~-^2b 30 ^,.--'

Xvv..^ DAILY MINIMUM ^-'''

^ , **

sn 30

'

-

-

AUG SEP 

1950

OCT NOV DEC JAN FEB MAR APR MAY 

1951

lake will be determined largely by the ability to 

evaluate adequately the various parameters entering 

into the advected term. It is probable that, for certain 

situations, as in "run of the river" lakes or reservoirs, 

it may be impossible to utilize the energy budget 

because of difficulties encountered in connection with 

estimating advected energy.

Several aspects of advected energy have already 

been discussed in the beginning of this section, for 

example, the relationship between advected energy 

and change in energy storage for a lake of varying 

volume, the energy advected out of the lake by the 

volume of evaporated water, and the selection and 

use of a base temperature.

In the case of Lake Hefner, surface inflow, rainfall directly on the lake surface, seepage, and water 

withdrawals made by the city filter plant, are the 

primary advected volumes of water. The temperatures of these components were computed as follows: 

The temperature of the surface inflow was taken as 

the average of the maximum and minimum 2-meter 

air temperatures at the south station. Rainfall into 

the lake was assumed to be at the same temperature 

as the flat-plate radiometer at the time rainfall took 

place. Seepage temperatures were assumed to be 

the same as that of the deep bottom water and were 

obtained from vertical temperature profiles of the 

deep part of the lake. The temperature of city-filterplant withdrawal was evaluated utilizing the periodic 

TPR cruise temperature profile taken at the intake 

tower and the day-to-day temperature profile as 

recorded in the filter plant; after March 1951 it was 

measured by a thermocouple placed in the intake 

tower at the withdrawal depth.

Due to the simplicity ot the geographic and 

hydrologic situation at Lake Hefner, it was possible 

to evaluate the various advected components with 

extreme accuracy. Table 15 summarizes information 

concerning the magnitude of the four advected components as well as the monthly average of advected 

energy. Surface inflow and seepage were extremely 

small, while filter-plant withdrawals were, on the 

average, equal to evaporation (average daily evaporation was approximately 30 acre-feet per day), and 

rainfall averaged about one-half the evaporation. 

Rain fell on 115 days out of 488, so on about 75 

per cent of the days rainfall was zero.

TABLE 15. Monthly Averages of the Advected 

Water Volumes at Lake Hefner in Acre-feet per Day.

Month

May 1950

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan 1951

Feb

Mar

Apr

May

Jun

Jul

Aug

Surface 

Inflow

8.1

4.0

7.2

6.3

1.7 '

1.1

1.1

0.9

0.9

2.1

2.2

3.8

15.0

8.3

3.5

0.5

3.7

Rainfall

39.0

13.0

43.6

13.8

12.1

5.2

4.4

0.1

4.2

9.6

9.3

25.9

37.3

20.0

17.1

11.7

16.6

Filter-Plant 

Withdrawal

33.3

36.4

35.1

34.8

29.0

24.3

23.2

28.5

25.2

26.3

25.8

26.6

29.0

34.7

41.4

40.7

30.8

OutSeepage

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

2.3

99
In evaluating the advected energy, it is necessary to select a base temperature for each time interval under consideration. Since, in the case of Lake 

Hefner, the magnitudes of the various advected components were both small and accurately measured, a 

constant base temperature of 0°C was selected. 

Figure 72 presents the daily values of advected 

energy. Figures at the top show the values for the 

corresponding days. In general, energy is advected 

out of the lake, and the magnitude is small, being 

about 15 cal cm~ 2 day" 1 during the summer and 

approximately zero during the winter. It is evident 

that very little improvement could be effected by 

using a variable temperature base. Days on which 

the advected energy was positive were days when it 

rained and, generally, the heavier the rain the larger 

the advected term. On 18 May 1951 the advected 

energy amounted to 603.0 cal cm~2 day" 1 . It is obvious that under these circumstances a more suitable 

temperature base than 0°C could be employed.

Anderson and Pritchard (1951) made a study 

of the energy advected into and out of Lake Mead. 

The geographic and hydrologic characteristics of Lake 

Mead are considerably more complex than at Lake 

Hefner, and a different base temperature was used.

The Lake Hefner and Lake Mead studies indicated that the evaluation of advected energy for any 

specific body of water is an individual problem, and 

extreme caution must be exercised in adopting procedures for the evaluation. It is possible that lack 

of adequate consideration of this term could lead to 

seriously erroneous evaporation results.

CHANGE IN ENERGY STORAGE

Early investigators, in attempting to utilize the 

energy budget, had difficulty in evaluating the 

storage of energy in a body of water over any specified time. It was necessary to select time intervals, 

or regions of the globe, where change in energy 

storage could be assumed negligible and therefore 

ignored. These assumptions were in many cases difficult to substantiate.

For example, Scrim id t (1915) computed evaporation from the oceans over a yearly interval. Probably 

his assumption that energy storage was zero over a 

period of one year was valid, provided average 

meteorological conditions prevailed.

In 1931 Richardson discussed the application of 

the energy equation to the determination of evaporation from lakes and reservoirs. He stated, "Storage 

of energy ... for a short period . . ., as determined 

from a study of data collected in California, is negligible." This may be true in certain California regions 

where the annual, seasonal, and daily range in air 

temperature may be small, and hence little energy 

stored in a body of water. However, since most areas 

experience a large air-temperature range, its truth 

is the exception rather than the rule. The application 

of the energy budget to evaporation determinations 

has, in the past, therefore been restricted to time 

intervals of a year or, in the case of shorter time 

intervals, to areas and times of small air-temperature 

range.

Through the development of instruments capable 

of obtaining vertical temperature profiles, and the 

development of the echo sounder for obtaining 

bottom depths, instruments are available for evaluating this energy-budget term. The following study of 

energy storage is based upon temperature data taken 

at Lake Hefner, utilizing primarily the TPR.

Methods of Evaluation

As indicated by equation (67), the change of 

energy storage is directly related to the advection of 

energy. This section is concerned with evaluating that 

part of equation (67), Vi (Ti 76) V2 (72 T6 ), 

which represents the change in energy storage. The 

energy storage can be computed by a numerical 

integration method, provided vertical temperature 

profiles and an area-capacity table for the body 

of water are available. The time rate of change of 

energy storage is defined as the change of energy 

storage, Q>

Figure 72. Daily advected energy 

for Lake Hefner from 1 May 1950 to 

1 September 1951. Base temperatore 0°C.

r so i *

0 30

I 20

8 10

I °

» 10

g 20

5 30

a 40

50 MAY 

1950

88

32

.888

JUN JUL AUG SEP OCT NOV DEC JAN 

1951

FEB MAR APR MAY JUN JUL AUG

100
The energy storage for a column of water one 

square cm in area and extending from the surface 

to the bottom may be expressed as:

«Kcolumn) = /* i ~ & < ' 

<=1

where Q?(Coiumn) is the energy storage, p the density, 

c the specific heat, Tt the average temperature of a 

standard layer of thickness Ah, T6 the base temperature, n the number of standard layers, and C the 

amount of energy, above the base temperature, in a 

layer of nonstandard thickness extending from the 

bottom of the standard layer to the bottom of the 

body of water.

If the temperature structure of the body of water 

is uniform, then the energy storage above a base 

temperature is

water. It is obvious that this procedure is time consuming and tedious, and hence would restrict the use 

of the energy-budget approach.

A second method of evaluating the energy 

storage is to (1) take many individual temperature 

profiles in all regions of the body of water, (2) 

average all the individual profiles regardless of depth 

of water, and (3) compute the energy storage from 

equation (92).

A third method is to take one profile in a deep 

section of the water and consider it representative 

of the entire body of water. This procedure, of course, 

assumes that the temperature structure of the body 

of water is uniform and, if sufficiently accurate, is 

desirable because it eliminates the necessity for 

making many temperature profiles and simplifies the 

computational procedure.

Q*<totai) =

t=l

 Tb) A/i< -f Ac C, (92) Energy Storage at Lake Hefner

where AA/I{ is the mean horizontal area of each 

standard layer and A0 the mean area of the nonstandard layer.

The change of energy storage over a given time 

interval is then the difference in the energy storage 

at the beginning of the time interval and the end 

(primed symbols), or

 T6) Aht

Tb) Ahi + A0 C Ac' C. (93)

It should be reiterated that equation (93) is 

independent of the base temperature, provided the 

volume of the body of water remains constant. If the 

volume varies, the energy budget itself is independent of the base temperature, since it is necessary to 

consider both advected energy and stored energy 

in this case.

It is probable that in any body of water the 

regions of similar energy storage change are defined 

by the depth of water. That is, shallower water 

would probably experience a change of energy storage different from that in deeper water. Therefore 

the most accurate procedure for obtaining the energy 

storage is to (1) divide the body of water into several 

smaller regions representing different depths of water, 

(2) take several temperature profiles of each region 

and average them to obtain a representative temperature profile for the region, (3) compute the 

energy storage for each smaller volume from equation (92), and (4) sum the individual energy storages 

to obtain the total energy storage of the body of

The original observation program for Lake 

Hefner required thermal surveys at 10-day intervals 

at 16 stations located with respect to the depth of 

water, as indicated on the map. The program was 

later changed to complete thermal surveys every 7 

days, with a daily profile being made at either 

station 6 or 2, depending upon weather conditions. 

Because of the high wind speeds experienced in this 

area, it was difficult to adhere rigorously to the 

program. During the 16 months of observations, 50 

complete thermal surveys and 160 daily measurements at station 6 or 2 were made.

Prior to 1 March 1951, the observations were 

made at a time of day convenient for the observers, 

usually the forenoon but frequently the afternoon. 

As the analysis progressed, it was found that the 

time of day was important. If the observation were 

made in the morning at one time and in the afternoon 

at a later time, the afternoon heating of the upper 

layers of the water introduced an apparent change in 

energy storage. To eliminate this transient effect on 

energy-storage computations, after 1 March 1951 the 

observations were made about an hour before dawn 

when thermal conditions were most stable. The best 

times for the observations are obviously the beginning 

and end of the period under consideration. In this 

study evaporation was computed daily from midnight 

to midnight, but it was not practicable to make midnight observations.

From 1 May 1950 to 1 July 1951, the observations were made with the TPR. During July and 

August 1951, a 70- or 180-foot BT was employed, 

because of operational difficulties experienced with

101
the TPR. Evaluations of energy storage based on 

observations made with the TPR are more accurate 

than those with the BT, due to the greater precision 

of the former instrument.

In addition to periodic information regarding 

vertical temperature structure, it is necessary to know 

the volume of water. Prior to the construction of the 

Lake Hefner dam, an area-capacity table had been 

prepared from a topographic survey. Although this 

table was probably accurate at the time it was prepared, no information was available on sedimentation 

that may have occurred or on the possibility that 

earth had been removed from the reservoir basin 

in the construction of the dam. Hence a bathymetric 

survey, using transit shore control, was made on 2-11 

August 1950, with the upper 10 feet of the lake 

plane-tabled on 14-24 March 1950. The results of 

these surveys indicate that the reservoir capacity is 

approximately 0.4 per cent less than indicated by the 

preconstruction survey.

The energy storage for each cruise was computed, using equation (92), by each of the three 

methods outlined above: (1) averaging by depth, (2) 

averaging the 16 stations, and (3) using a single deep 

station. The change in energy storage, expressed in 

cal cm" 2 day" 1 , was then computed for the intervals 

between cruises, using equation (93). The results are 

shown graphically in figures 73a and 73b. The correlation coefficients are 0.99 and 0.98, respectively, 

while the standard errors of estimate are ±16.8 and 

±31.3 cal cm~ 2 day- 1 , respectively. Method (2)' 

gives results close enough to those of method (1) to 

justify its use, but method (3) does not agree with 

method (1) as closely as desired. However, in this 

study it was not practicable to obtain complete 16- 

station data all the time, and hence it was necessary 

to make single-station evaluations.

The temperature measurements were accurate to 

±0.1 °C. If the maximum error were present in a 

temperature profile in the same direction at all depths, 

it would introduce a maximum error of approximately 

70 cal cm" 2 over the time interval considered. Such 

an energy error is considerable over short time intervals, for example, of a few days, but over a period 

of 7 to 10 days is not great.

Figure 74 presents the energy storage for lake 

Hefner from 14 April 1950 to 31 August 1951. The 

numerals indicate the complete cruise numbers and 

the "dots" without numerals a single temperature 

profile. As indicated by equation (67), the energy 

storage is independent of advected energy only if 

the volume of the water remains constant. Figure 74

also shows the variation in the volume of Lake Hefner 

for the above time interval. The volume varied from 

a minimum of 56,000 acre-feet on 8 May 1950 to 

65,000 acre-feet on 25 August 1950, with a mean 

volume of approximately 62,000 acre-feet, or during 

the 16 months of observation it varied by about 15 

per cent of the mean volume.

From these data on energy storage, it is possible 

to examine the assumptions of various investigators 

concerning the magnitude of the change in energy 

storage over both long and short time intervals when 

volume changes are nil or small. Table 16 gives the 

variation of energy storage for monthly and yearly 

intervals. The variation over a yearly cycle is less 

than 10 cal cm"" 2 day 1, while over the months of 

January and August, when the monthly variations 

would probably be the least, it is from 2, to 60 

cal cm" 2 day" 1 . During months other than January

Figure 73. Comparison of energy storage computed by averaging by depth with average of 16 stations (a) and single station (b).

102
_II93 2400 

-2300 

c2000 i!)60000

g

a 35000

tf 30000-

Figure 74. Energy storage (a) and variation in lake elevation, area, and volume (b) for Lake Hefner 

from 15 April 1950 to 1 September 1951.

TABLE 16. Change in Energy Storage Over Yearly and 

Monthly Intervals.

Yearly

Yearly

Monthly

Monthly

Monthly

Time Interval

1

15 May 1950-1 5 May 1951

20Jul 1 950-20 Jul 1951

1 Jan 1951-31 Jan 1951

1 Aug 1950-31 Aug 1950

1 Aug 1951-31 Aug 1951

Change in

Storage

(eal cm~ 2 day" 1 )

+0.7

+7.8

 1.7

 13.5

-58.4

Change in

Volume

(per cent)

 8.5

+ 1.1

 2.0

0.0

-3.9

and August, the change in energy storage is considerable as shown by figure 74. In addition, the variation from year to year for the month of August is 

considerable, being about 15 and 60 cal cm~2 

day" 1 for August 1950 and August 1951, respectively. It may be concluded that the assumption of 

negligible change in energy storage is (1) valid over 

a yearly cycle provided the volume does not change, 

(2) is of dubious validity over monthly intervals during 

January and August, and (3) is definitely invalid for 

other months.

Table 17 indicates the change in energy storage 

over two short time intervals when the range in air 

and water-surface temperature was small and the 

volume constant. The change in energy storage in one 

case was negligible and the other considerable, 

making it impossible to draw general conclusions.

TABLE 17. Change in Energy Storage During Short 

Time Intervals When Range of Air and Surface-Water 

Temperature is Small and Lake Volume Constant

Time Interval

13 Dec 1950-16 Dec 1950 

(4 days)

6 Aug 1950-12 Aug 1950 

(7 days)

Average Daily 

Air-Temperature 

Range (°C)

Average Daily 

Water Temperature 

Range (°C)

Change in 

Energy Storage 

(cal cm"~ 2 day"" 1)

2.9 to 4.9 

(range 2.0)

4.3 to 4.7 

(range 0.4)

+ 1.4

24.5 to 27.5 

(range 3.0)

25.8 to 27.4 

(range 1 .6)

+46.5

On several occasions attempts were made to 

obtain 24-hour series of vertical temperature profiles 

taken at half-hour intervals. Although the change in 

energy storage is a residual term from several components, each varying in some manner during a 

24-hour period, it was believed that perhaps some 

information on the diurnal variation of energy storage 

might be obtained. Only two such attempts were completely successful: one from 1300 CST, 16 November 

1950 to 1130 CST, 17 November 1950; and another 

from 0730 CST, 21 March 1951 to 0730 CST, 22 

March 1951. Figure 75 presents the results from these 

two series of observations. They fail to indicate any

30Z5ZO O - 54 - 9 103
1 1' I' I' I' I ' I' I' I' I' I' I' I' I' I' I' I' I' 111' I' I' I' I'

17 NOV 1950 16 NOV 1950

^o

10 12 14 

TIME (CST)

Figure 75. Diurnal variation of energy storage for Lake Hefner.

consistent diurnal trend the November example indicates a possible linear decrease during the 24-hour 

period, while the March example indicates a possible 

sinusoidal variation, with a maximum energy storage 

during the late afternoon and a minimum during the 

early morning. Of more significance, however, is the 

abrupt decrease and increase in the energy storage 

observed at 2130 CST and 1330 CST on 16 November 1950 and 21 March 1951, respectively. These 

abrupt changes are apparently associated with the 

movement of cold and warm eddies of water through 

the area where the temperature profiles were taken. 

With the passage of the eddy, the energy storage returned to a more normal value. These abrupt changes 

are equivalent to approximately 350 cal cm~2.

These examples indicate the possibility of introducing considerable error into the evaluation when 

single-station profiles are used since, under the above 

conditions, single stations did not yield profiles representative of the entire body of water. This error, 

however, compared to errors of other terms in the 

energy budget, decreases as the length of the time 

interval increases and, for longer time intervals, say 

a week or month, the technique may be used with 

confidence.

Summarizing, analysis of three methods of evaluating the change of energy storage indicates that 

the temperature structure of Lake Hefner is essentially 

vertically and horizontally uniform, suggesting that 

the change in energy storage can be evaluated from 

a single vertical temperature profile or at most from 

an average of several profiles taken in different 

depths of water. Because of limits to the accuracy of 

the temperature measurements, considerable error 

may be present when evaluating the change of

energy storage over time intervals of a few days. For 

time intervals of the order of 7 to 10 days, however, 

the accuracy is probably adequate for evaporation 

determinations. Examination of the magnitude of 

energy-storage change substantiates that over annual 

cycles the change is negligible, but not over shorter 

time intervals, for example, a month or less.

RATIO OF ENERGY USED FOR 

CONDUCTION OF SENSIBLE HEAT 

TO THAT USED FOR EVAPORATION

From the energy budget, it is possible to evaluate 

the sum of the energy utilized by evaporative processes and that conducted to or from the body of 

water by the air as sensible heat. Since neither term 

can be evaluated directly, it is necessary to employ 

a ratio, R, the ratio of conducted heat to energy 

utilized by evaporation. Early workers, utilizing the 

energy equation for obtaining evaporation, have 

assumed various values for this ratio.

Bowen Ratio

In 1926 Bowen attempted theoretically to relate 

the value of R to easily measurable quantities. He 

concluded that the ratio could be expressed as follows:

R = c

e0 e0 / 1000

(94)

where TO and T0 are the temperatures of the air and 

water surface, e0 and ea the saturated vapor pressure 

at the temperature of the water surface and the vapor 

pressure of the air, respectively, and P the atmospheric pressure. The limiting values of c are 0.58 and 

0.66, depending upon the state of the atmosphere.

104
Bowen concluded that c, under normal atmospheric 

conditions, was approximately 0.61.

The fundamental equations from which Bowen 

started apply to the transfer of heat, and water vapor 

by processes of molecular diffusion. It might, therefore, be concluded that the computed Bowen ratio 

is valid only under conditions of laminar flow. However, as Bowen states, ''it can be expected that heat 

losses by evaporation and diffusion and by conduction will follow the same laws and will be affected 

in the same way by convection." This implies that the 

ratio between the energy losses will be independent 

of the state of atmospheric turbulence.

In addition, Pritchard (unpublished notes) derived the same ratio as Bowen from mass-transfer 

concepts, using Sverdrup's (1946) relationships as a 

basis. He found that c varied from 0.57 for a smooth 

surface to 0.66 for a rough surface, which agrees with 

Bowen's limiting values. A similar analysis was made 

by Thorn (1951) using the Thornthwaite-Holzman 

evaporation equation.

Sverdrup (1943) attempted an analysis of the 

general validity of the Bowen ratio. Recent work had 

suggested to him that it might be necessary to consider the effects of radiation on the transfer of heat 

and of spray upon evaporation. These effects might 

make the Bowen ratio dependent on turbulence, and 

it would then have to be modified.

Sverdrup first examined radiation for air in laminar motion or at rest. In a moist atmosphere, a flux 

of radiant energy is directed from regions of higher 

to regions of lower temperature. This flux is dependent upon the selective emission and absorption of 

long-wave radiation by water vapor, and is therefore 

not present in dry air. Its effect can be described by 

means of a coefficient which Brunt (1944) calls "radiative diffusivity." The radiative diffusivity has the 

same dimensions as the diffusion coefficient for heat 

energy and, when problems of diffusion of heat 

energy in moist air are considered, it must be added 

to the heat-energy diffusion coefficient. The ratio, R, 

in this form is greater than in Bowen's form, the 

increase depending upon the relative magnitudes of 

the diffusivity coefficients for heat energy and radiation.

Very little is known about the value of radiative 

diffusivity near a boundary surface. However, making 

certain primitive assumptions concerning radiative 

diffusivity, Sverdrup concluded that the ratio was 

about 1.30 times that obtained from equation (94). 

This conclusion was only semiquantitative. He also 

concluded that the application of Bowen's formula

to atmospheric conditions breaks down because evaporation is increased at high wind speeds by spray. 

Montgomery (1940) suggested that evaporation from 

spray was of great importance and that, as a result, 

the total evaporation would not remain proportional 

to the wind speed as is nearly the case when transfer 

by eddy diffusivity only is considered, but would increase more rapidly at high wind speeds. If this is 

true, the Bowen formula gives values of the ratio 

that are too high at higher wind speeds. N. W. Cummings, in a discussion of Sverdrup's (1943) paper, 

indicated that in his opinion the presence of spray 

does not invalidate the Bowen ratio.

In summary, Sverdrup (1943) concluded that 

Bowen's formula gives only an approximate value, 

the closeness of the approximation depending upon 

the radiative diffusivity, turbulence near the surface, 

and the effect of spray upon evaporation.

The various developments of the ratio of heat 

losses by convection to those by evaporation depend 

fundamentally on the assumption that the eddy diffusivities for these two quantities are identical. There is 

evidence indicating that this assumption is not always 

valid. For example, Pasquill (1949) made an experimental study of the various factors involved in the 

turbulent transfer of water vapor and heat in the 

lowest layers of the atmosphere over open grasslands. From the data he evaluated the vertical components of the eddy diffusivities for water vapor and 

heat. He demonstrated, from evaluation of the turbulent heat flux using heat-balance considerations, that 

the eddy diffusivity for heat is reasonably equal to 

that for water vapor under stable conditions, but is 

substantially greater for unstable conditions, and may 

be twice as great.

Commenting on these observed differences in 

eddy diffusivities for water vapor and heat, Pasquill 

stated: "Although it is not proposed in this paper to 

deal in any detail with theories of turbulent mixing 

it may now be noted that, in qualitative implication 

at any rate, the present results are not entirely unexpected. In a recent treatment of convection near 

the ground Sutton (1948) has assembled evidence for 

the existence of a difference in heat and momentum 

transport. Furthermore, Priestley and Swinbank (1947) 

have recently put forward a modification of the classical theory of turbulent transport of heat, taking into 

account buoyancy effects. The latter treatment possesses implications as regards the transfer of physical 

quantities other than that of heat, and provides 

reasons for expecting different values of the eddy 

diffusivity to apply to the transport of different prop105
erties. As far as is known the present results provide 

the first direct experimental demonstration of the existence of this feature in the cases of matter and heat."

Although the eddy diffusivity for heat is approximately equivalent to that for water vapor under 

stable conditions, it may be twice as great for unstable conditions, and the computed evaporation rates 

33 per cent too high. Hence Pasquill concluded that 

the methods described for computing evaporation 

from the energy budget may only be applied with 

confidence when the atmosphere is stable, and may 

be seriously in error under unstable conditions when 

evaporation is at a maximum.

Since considerable controversy exists over the 

validity and meaning of the Bowen ratio, an attempt 

was made with Lake Hefner data to compute Bowen 

ratios by observing all factors in equation (72) and 

comparing the results with those obtained from meteorological data. However, it was found that the 

various factors had not been measured with the 

requisite accuracy, and inconsistent results were obtained. It is suggested that a wind tunnel experiment 

be designed to investigate the validity of the Bowen 

ratio.

When its value is small, the Bowen ratio occurs 

in equation (72) as a corrective term. Hence, errors 

in its evolution are reflected as much smaller errors 

in evaporation. When the ratio approaches unity, it no 

longer is a small corrective term, and errors in its 

determination are of greater importance.

The remainder of this section covers a study, 

based on Lake Hefner data, of the implications of 

the Bowen ratio and its effect upon the evaluation of 

evaporation from the energy budget.

Bowen Ratios Observed at Lake Hefner

From the data collected at Lake Hefner, the 

Bowen ratio was computed for single days and also 

for periods varying in length up to 30 days. The ratio 

was computed from equation (94), utilizing a value 

of 0.61 for c, the surface water-temperature taken 

at the barge, and the barge 2-meter wet- and 

dry-bulb temperatures. Temperatures over the length 

of period under consideration were averaged. The 

ratios for daily, 5-, 10-, and 20-day periods are presented in figures 76 and 77. The most striking feature 

of these figures is the relatively small variation in the 

ratio, from an extreme of +0.5 to 0.5. The average 

variation is from +0.2 to 0.2 from May through 

October, but from October to May the daily variation 

is much greater, the greatest variation occurring

during February. For the 16-month period, the extreme variation in the ratio was from 20.02 occurring the latter part of April to +31.50 in February. 

As the length of the period increases, the range becomes smaller, until for periods of 20 days the ratio 

only varies throughout the year from +0.25 to 0.25.

Bowen ratios of approximately 1.00, large 

positive ratios, and large negative ratios will next be 

examined with respect to their validity and their effect 

upon evaporation computations.

Referring to equation (71), ratios having a value 

of or close to 1.00 are of interest. Under these conditions the first term in the brackets on the left side 

of the equation assumes very small values (the second 

term is always small), with the result that small variations in (Q8 Qr Q6 Q^+QJ will vary the 

computed evaporation extremely. For the case when 

the ratio is equal to 1.00 and the base temperature equals the surface temperature, the. quantity 

[pL (1 + R) + cpe (Te Tb )j is equal to zero. Then 

the right-hand side of equation (71) must also be 

equal to zero, and the evaporation is indeterminate. 

It must be pointed out that on the right-hand side of 

the equation the quantity (Qu O^) is not independent of the base temperature, since the term Qv does 

not include the energy advected out of the body of 

water by the mass of evaporated water. This latter advected mass of water is accounted for on the left-hand 

side of the equation by the quantity cpe (T6 T&). 

Hence, if the base temperature is changed in the 

situation discussed above, the quantity [pe L (1 + R) 

+ cpe (Te TD)] will have a finite value. The quantity (Q« Qr Qft QS+ Qv) may then assume 

any value depending upon the particular situation.

On a daily basis, there were 14 days when the 

ratio was between 0.70 and 1.30. In all 14 cases 

the lower layer of the atmosphere was stable, with 

an increase in temperature between 0 and 2 meters 

of 1.6 to 8.9°C. The vapor pressure in the lower 2 

meters decreased with height from 1.1 to 4.9 mb. 

Unfortunately, only two of these days had water 

budgets classified as A. Pertinent data for these two 

days are summarized in table 18. In order that the 

computed evaporation equal the observed evaporation, (Q8 Qr Q& Qy+ Qv) should have a 

value of 3 and 1 cal cm~2 day" 1 on 22 December 

and 18 January, respectively. In other words, (Q8 

Qr Q6 Q$ + Qv) would be in error by 35 and 

185 cal cm~2 day" 1, respectively. The only term in 

the energy budget that could vary the amount indicated above is the change in energy storage. On a 

daily basis, errors of the above magnitude in evaluat106
TABLE 18. Pertinent Data for A Water-Budget Days When Bowen Ratio is Approximately 1.00.

Evaporation Evaporation Q Q Q J. (1 _j_ /}) + 7-

Bowen T2 T0 Water Budget Energy Budget e2 eo Q» + Qv

Date Ratio (°C) (em) (cm) (mb) (cal cm" 2 day- 1 )

22 Dec 1950 0.865 1.6 0.043 

18 Jan 1951 1.109 5.1 0.015

-0.376 -1.1

-1.291 2.7

 32 

184

85 

-60

+2.5 

+2.0 

+ 1.5 

+ 1.0

I___I____I____I___I____I I I

83$:=

i in; i__i J___I____I

MAY JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG 

1950 1951

Figure 76. Daily Bowen ratios for 

Lake Hefner from 1 May 1950 to 

1 September 1951.

+ 1.5

+ 1.0

0+0.5

2

z o

5 0.5

 1.0

 1.5

1950 1951 

MAY JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG

12.134

5-DAY

+0.5

0

-0.5

0+0.5 

1

z o

1 -0.5 - 20-DAY

MAY JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG 

1950 1951

Figure 77. Bowen ratios for 5-, 10-, 

and 20-day periods for Lake Hefner from 1 May 1950 to 1 September 1951.

107
ing this term are possible, since temperature-depth 

measurements were not made at the beginning and 

end of the evaporation period. Greater accuracy in 

evaporation determinations will be obtained if the 

energy-storage measurements are made at the proper 

time intervals.

In addition, Bowen ratios around 1.00 occurred 

during eight periods varying in length from 4 to 16 

days. Results similar to those above are indicated. 

Errors in evaluation of (Qs Qr Q& Q$ + QJ, 

of a magnitude attributable to errors in energystorage evaluation, can account for the difference 

between computed and observed evaporation. One 

of these periods, 7 days in length, is of particular 

interest since complete 16-station thermal surveys 

were made both at the beginning and end of the 

period. Under these conditions it was possible to make 

the most accurate evaluation of the change in energy 

storage that the technique employed would permit. 

Pertinent data for this period follow:

Date .................12 Jan to 18 Jan 1951

Bowen ratio ................. 1.126

Water-budget evaporation ..... 0.027 cm

Energy-budget evaporation ..... 0.123 cm

Temperature difference between 0

and 2 meters ................ 3.2°C

Vapor pressure difference between

0 and 2 meters............... 1.5 mb

(Q8 Qr Qb Q5 + Qv) 18 cal cm- 2 day- 1 

L(l + R) + TW .............. -150

If (Q8 Qr Q6 Q* -f Q«) were equal to 4 cal 

cm~2 day" 1 , or 14 cal cm"2 day" 1 smaller than 

observed, then the computed evaporation would 

agree exactly with the observed evaporation. Perhaps this difference is suggestive of the accuracy with 

which the change in energy storage can be evaluated. 

Negative ratios greater than 3.00 occurred 

on 7 days, and positive ratios greater than -j-3.00 

on 8 days. In addition, for 13 periods, varying from 

2 to 8 days in length, ratios greater than ±3.00 

were computed. These all occurred during the winter 

and spring when there was a southerly flow of warm 

moist air over a cold lake, with a resulting temperature inversion between 0 and 2 meters of 2.2 to 

9.3°C, and a difference in vapor pressure between 

the air and air saturated at the temperature of the 

water of 0.1 to 1.0 mb. For large Bowen ratios the 

denominator of equation (72) becomes very large. 

Under these conditions, differences between computed 

and observed evaporations cannot be accounted for 

by errors in evaluating the change of energy storage, since large changes in (Q8 Qr Qft Q$ 

-j- Qv) are necessary to vary the computed evaporation by an appreciable amount. However, small errors 

in evaluating the vapor-pressure difference in equation (94) become extremely important, since these 

small differences are reflected as a large change in 

the Bowen ratio.

From these observations it might be concluded 

that evaporation determinations made when the 

Bowen ratio approximates 1.00, or attains large 

positive or negative values, may be in considerable 

error. In the first case, errors in evaluating the change 

in energy storage are reflected as large changes in 

evaporation while, in the second, small errors in 

vapor-pressure difference cause the same effect. However, only 8.0 per cent of the time did the Bowen ratio 

attain the extreme values discussed above, and only 

during periods of low evaporation or condensation. 

During periods of high evaporation the ratio was 

small, and errors in its evaluation were reflected as 

smaller errors in evaporation due to its entering equation (72) as a corrective term. For periods greater 

than 10 days, the extreme values of the ratio never 

occurred. In a practical sense, errors in the Bowen 

ratio do not greatly affect the accuracy of the computed evaporation, because evaporation determinations for periods of under 10 days are rarely necessary or desired.

As previously stated, doubt has been cast upon 

the validity of using the Bowen ratio due to three 

assumptions: (1) neglect of the effect of radiative 

diffusivity, (2) equality of the coefficients of mixing 

for water vapor and heat over all ranges of stability, 

and (3) neglect of the effect of spray. As previously 

indicated, differences of opinion exist concerning the 

effect of spray on the Bowen ratio. No attempt will 

be made to resolve these differences. In the following 

discussion it is assumed that spray modifies the ratio, 

as suggested by Sverdrup and Montgomery.

Figure 78 has been prepared to indicate hypothetically what the effect would be on the computed 

evaporation provided these three effects were principally functions of wind speed and varied in some 

assumed fashion.

For example, assume, as Pasquill suggests, that 

the eddy diffusivity coefficients for water vapor and 

heat are equal under stable or neutrally stable conditions, and that the diffusivity for heat is greater 

than for water vapor when the atmosphere is unstable. Then, assuming that increasing wind speed is 

indicative of an increasingly unstable atmosphere, the 

presently computed evaporation rates for higher wind

108
speeds would be too large, and it would be necessary 

to apply a negative correction to the computed evaporation as indicated in figure 78a.

Radiative diffusivity is assumed to be independent of wind speed. Hence, there would always be a 

negative correction to apply to the computed evaporation, since the neglect of radiative diffusivity gives 

a Bowen ratio that is too small. This is shown schematically in figure 78b.

Figure 78c indicates the possible error if spray 

is neglected. No error would occur at wind speeds 

lower than the speed necessary to produce whitecaps. 

When this speed is attained, water droplets are blown 

into the air. Evaporation would then occur from these 

water droplets before they fall back into the body of 

water. The computed evaporation would then be too 

small, and a positive correction would have to be 

made to the computed evaporation.

The composite of these three effects would be 

as indicated in figure 78d. At low wind speeds the 

computed evaporation would be too large and at 

high wind speeds it would be too small.

UJ

S 1 11 »

STABILITY EFFECT RADIATIVE-DIFFUSIVITY 

EFFECT

WIND SPEED WIND SPEED

SPRAY EFFECT

WHITE CAPS BEGIN 

TO APPEAR

COMPOSITE EFFECT

WIND SPEED WIND SPEED

Figure 78. Composite hypothetical effect on evaporation of 

neglecting radiative diffusivity, variations in equality of diffusivity 

coefficients for water vapor and heat, and spray in .computing 

Bowen ratios.

An examination of the Lake Hefner data should 

decide whether or not a consistent composite error, 

such as indicated above, is introduced into the evaporation results when using the Bowen ratio in its 

original form. The results of such an analysis are 

given in figure 79 for 142 evaporation periods of 7 

or more days in length. For periods of the same length 

(that is, 7 days) there was no overlapping, but the 

longer periods necessarily included the shorter ones. 

The difference between these water-budget evaporations and energy-budget evaporations are plotted 

versus the average wind speed at 8 meters. The 

longer evaporation periods were selected in order to 

minimize errors in energy-storage evaluations. It is 

evident that the scatter is essentially random, and no 

relation, such as suggested above, is indicated.

Finally, the Lake Hefner data indicate that the 

assumptions made in the development of the Bowen 

ratio concerning spray, radiative diffusivity, and atmospheric stability do not introduce a consistent error 

into evaporation determinations. The Bowen ratio 

appears to be sufficiently accurate for computing 

energy utilized by evaporation for most conditions. In 

exceptional conditions, for example, when evaporation rates are small and the difference in vapor 

pressure of the atmosphere and that of air saturated 

at the surface-water temperature approaches instrumental accuracy, the Bowen ratio is inadequate. The 

latter conditions occur only 8 per cent of the time.

8- /METER WIND SPEED (KNOTS)

Figure 79. Relationship between wind speed and differences 

between water-budget evaporation and energy-budget evaporation.

109
The heat carried away from a water surface 

through a unit area in unit time may be expressed 

as follows:

(95)

where cp is the specific heat of the air at constant 

pressure, A the vertical component of the eddy conductivity, dT/dz the temperature gradient of the air, 

and y the adiabatic lapse rate. The term cpA enters, 

instead of the molecular coefficient of heat conductivity of the air, because the air is nearly always in 

turbulent motion; the degree of turbulence is indicated 

by A, the eddy conductivity. To utilize equation (95), 

it is necessary to evaluate the eddy conductivity.

Observations were made at Lake Hefner of all 

quantities necessary to evaluate eddy conductivity. 

It was hoped to compute eddy conductivity from the 

data, and then find a method for obtaining it from 

more easily measured parameters, thus making it 

possible, in the future, to compute Q^ from equation 

(95). However, it was found that (Qe -}- QJ, the evaporation, and/or dT/dz were not measured with 

enough precision to obtain consistent values of eddy 

conductivity. Hence, the attempt was abandoned.

It was possible to obtain a knowledge of the 

gross magnitude of the conducted heat through use 

of the Bowen ratio. From the preceding discussion of 

this ratio, it was concluded that satisfactory values of 

conducted heat could be obtained in all but the few 

situations where the Bowen ratio leads to erroneous 

results.

The average monthly conducted heat for the 16 

months of observations is shown in figure 80. Most 

energy was conducted to the lake in April 1951, 

when it averaged 84.0 cal cm~2 day" 1 ; most was 

conducted away in November 1950, when it averaged 

75.0 cal cm~2 day" 1 . During most months, the conducted heat is a large enough term to be considered 

in the energy budget.

The ratio of conducted heat to energy utilized 

by evaporation varied, during the last 12 months of 

observations, from 0.324 in February 1951 to 

-fO.252 in November 1950; the annual average ratio 

was 0.030. Hence, on a monthly basis the conducted heat at Lake Hefner must be considered, while 

on an annual basis it may be neglected.

Evaporation was computed, by means of the 

energy budget, for periods of 1,2, 3, 4, 5, 6, 7, 8, 9, 

10, 12, 14, 16, 18, 20, 25, and 30 days, and for 

periods of variable length for which complete (16- 

station) thermal-survey data were available at both 

the beginning and the end of the period. Only periods 

where all terms in the energy budget were measured 

continually were used. No extrapolated data are 

included. The periods are non-overlapping and consecutive in time, and no attempt was made to exert 

judgment as to which days should be included in a 

particular period. The data were edited only to exclude periods where the evaluation of energy storage 

was obviously inadequate for example, no storage 

determinations were made from 26 January 1951 

to 26 February 1951.

Enough data for periods varying from 1 to 10 

days in length and for periods coinciding with the 

thermal cruises were available for statistical treatment. For periods greater than 10 days, the data were 

too few to treat statistically.

The data for the above periods are plotted in 

figures 81 through 85. These figures clearly indicate 

the better agreement between observed and computed evaporation as the length of the time interval 

increases. In nearly every case the number of points 

above the 45-degree line equals the number below, 

while the average-energy-budget and the averagewater-budget evaporations agree within 2.4 to 

-f-7.1 per cent. It may be concluded that the data 

are essentially randomly distributed about the line of 

perfect agreement.

Since the data were scattered normally about 

the 45-degree line, it was possible to compute standard errors of estimate; these are shown as dashed 

lines in figures 81 through 85. One standard error of 

estimate should contain, by definition, approximately 

two-thirds of all the cases. These standard errors of 

estimate, as well as the standard errors of estimate 

expressed as a percentage of the mean energybudget evaporation, are shown as a function of the 

length of period in figure 86. The improvement in 

accuracy of the computations is marked as the length 

of period increases to 4 days, and thence more gradual, until for periods of 10 days the standard error 

of estimate is about ±12 to 15 per cent of the mean 

energy-budget evaporation. The curve appears to 

follow the expected square-root law reasonably well.

110
Figure 80. Average monthly 

amount of energy conducted to (+) 

or from ( ) Lake Hefner from 1 

May 1950 to 1 September 1951.

-80

JUN JUL AUG SEP OCT NOV DEC JAN FEB MAR APR MAY JUN JUL AUG 

1950 1951

DAILY EVAPORATION 

(202 OBSERVATIONS)

TWO-DAY EVAPORATION 

(112 OBSERVATIONS)

A WATER BUDGET DAYS 

B WATER BUDGET DAYS

.2 ' .* .» .e i.o 1.2 1.4 i.a

 > / WATER BUDGET (CM) WATER BUDGET (CM)

Figure 81. Comparison of daily and two-day evaporation from Lake Hefner. Dashed line represents one 

standard error of estimate.

THREE-DAY EVAPORATION 

(71 OBSERVATIONS)

FOUR-DAY EVAPORATION 

(51 OBSERVATIONS)

I,,

-X

Figure 82. Comparison of three- and four-day evaporation from Lake Hefner. Dashed line represents 

one standard error of estimate. 111
FIVE-DAY EVAPORATION 

(37 OBSERVATIONS)

SIX-DAY EVAPORATION 

(37 OBSERVATIONS)

'/ .2 .4 .« .8 1.0 1.2 1.4 1.8 

' WATER BUDGET (CM)

/ .2 .4 .« .6 1.0 1.2 1.4 1.6 

', WATER BUDGET (CM)

Figure 83. Comparison of five- and six-day evaporation from Lake Hefner. Dashed line represents 

one standard error of estimate.

SEVEN-DAY EVAPORATION 

(29 OBSERVATIONS)

EIGHT-DAY EVAPORATION 

(24 OBSERVATIONS)

/

.&/

///

/ .2 .4 .6 .6 I.O 1.2 1.4 1.8 

WATER BUDGET (CM)

-.4 -.2 / / .2 .4 .6 .» 1.0 1.2 1.4 I.e 

WATER BUDGET (CM)

Figure 84. Comparison of seven- and eight-day evaporation from Lake Hefner. Dashed line represents 

one standard error of estimate.

112
NINE-DAY EVAPORATION 

(19 O5SERVATIONS)

TEN-DAY EVAPORATION 

(16 OBSERVATIONS)

1.8 

1.4

1.2

"?

fo

§ 

>.

0

5 .8

Z

.4

.2

/

.4 -..//;

/

-.2

-.4

1.8 

1.4

1.2

?

& 1.0

' u)

/ J 0'

X / X O

xC/V i 

X V^/ 3

' / / uj -6

/'/*/* *

/'*, / * A WATER BUDGET DAYS ' 4 

/ '"/ " B WATER BUDGET DAYS

( //

/ / . ,.,,,.,.,,. . X

/ .2 .4 .8 .8 1.0 1.2 1.4 1.8 -.4 -.2 / 'S 

' WATER BUDGET (CM) // '

-.2 

-.4

/.

f//

/ //

/ /'/ //x"7"

'- /$'

/{

' .2 .4 .8 .8 I.O 1.2 1.4 1.8 

WATER BUDGET (CM)

Figure 85. Comparison of nine- and ten-day evaporation from Lake Hefner. Dashed line represents 

one standard error of estimate.

Figure 86. Accuracy of energy-budget evaporation determinations as 

a function of the length of computational period for Lake Hefner. 

Figure indicates the number of A and B periods included.

123456789 10 

LENGTH OF PERIOD (DAYS)

DC 7

gft ± 4

J___I___I___I___I

I 234 5 0 7 8 9 10 

LENGTH OF PERIOD (DAYS)

113
It must be remembered that these data include both 

A and B water-budget days, and that the control 

itself is subject to certain inaccuracies previously 

discussed.

The individual sets of data for periods longer 

than 10 days, which could not be treated statistically, 

are listed-in table 19. With respect to percentages, 

the accuracies of these data vary considerably. On 

the whole, there does not appear to be any improvement in accuracy as the length of period increases 

beyond 10 days. This may be related to the fact, 

previously indicated, that the standard errors of estimate of computations of energy-storage changes from 

an average of all cruise stations or a single deep 

station and from the more detailed layer-depth 

method are ±17 and ±31 cal cm~2 day" 1 , respectively. The available A and B data, for periods 

when evaporation was greater than 0.5 cm day" 1, 

indicate that an accuracy of ±19 per cent is possible 

for periods longer than 10 days. For shorter periods 

the accuracy is inferior and decreases to about ±70 

per cent for single days.

A better agreement between observed and computed evaporation was obtained for A periods only. 

The errors in the water budget were then minimal, 

and the differences should be representative of the 

energy-budget variations alone. Figure 87 represents 

such an analysis. As expected, the standard error of 

estimate becomes less, having a value of ±0.036 

cm for a 10-day period; expressed as a percentage 

of the mean energy-budget evaporation, the error 

is ±6.4.

At Lake Hefner there is a season of high evaporation and one of low evaporation, occurring during 

the summer and fall, June through November, and 

during the winter and spring, December through May, 

respectively. When considering accuracies in terms 

of percentages of the mean evaporation, the results 

differ for the two seasons. The results for the highevaporation season are shown in figure 88. Comparing figure 88 with figure 87 shows that the 

standard error of estimate for the high-evaporation 

season is about the same as for all A period data, 

but the standard error of estimate expressed as a 

percentage of the mean energy-budget evaporation 

decreases. For periods of 10 days the standard error 

of estimate is approximately ±5 per cent of the mean 

energy-budget evaporation. For the low-evaporation 

season, the data are insufficient for statistical treatment. However, they do indicate that the standard 

error of estimate would be about the same as for the 

high-evaporation season, but the percentage error

considerably larger. For example, for single-day 

periods, the standard error of estimate was ±72 per 

cent of the mean, whereas in the case of the highevaporation season it was ±47 per cent.

It is evident that it is difficult to assign a particular percentage error to the energy-budget-evaporation determinations unless the basis of comparison 

is clearly indicated. It may be concluded that comparisons with A water-budget periods indicate that:

TABLE 19. Comparison of Energy - Budget and 

Water-Budget Evaporation for Periods of 12 to 30 

Days.

Length 

of Period

12 days

14 days

16 days

1 8 days

20 days

25 days

30 days

Classification

A

B

A

B

A

A

A

B

B

A

A

A

B

A

A

A

A

A

A

A

A

A

A

B

B

A

B

A

A

A

A

B

B

A

A

A

A

A

B

A

B

Water 

Budget 

(cm)

0.425

0.614

0.494

0.773

0.392

0.361

0.139

0.108

0.129

0.379

0.606

0.640

0.690

0.699

0.623

0.712

0.368

0.325

0.377

0.363

0.634

0.698

0.578

0.609

0.379

0.314

0.408

0.190

0.704

0.538

0.604

0.392

0.431

0.744

0.518

0.751

0.551

0.573

0.382

0.518

0.338

Energy 

Budget 

(em)

0.421

0.578

0.398

0.654

0.360

0.261

0.249

0.257

0.336

0.493

0.723

0.650

0.758

0.565

0.445

0.628

0.325

0.356

0.378

0.446

0.667

0.670

0.511

0.554

0.358

0.328

0.423

0.327

0.761

0.476

0.509

0.354

0.447

0.804

0.493

0.801

0.482

0.502

0.447

0.482

0.401

Difference 

(cm)

 0.006

 0.036

 0.096

 0.119

 0.032

 0.100

0.110

0.149

0.207

0.114

0.117

0.010

0.068

 0.134

 0.178

 0.084

 0.043

0.031

0.001

0.083

0.033

 0.028

 0.067

 0.055

 0.021

0.014

0.015

0.137

0.047

 0.062

 0.095

 0.038

0.016

0.060

 0.025

0.050

 0.069

 0.071

0.065

 0.036

0.063

Difference 

(per cent)

 0.9

_5.9

 19.5

 15.4

_8.2

 27.6

79.0

138.0

161.0

30.1

19.3

1.6

9.9

 19.2

 28.6

 11.8

 11.7

9.5

0.3

22.8

5.2

_4.0

 11.6

_9.0

_5.5

4.5

3.7

72.0

6.7

 11.5

 15.7

 9.7

3.7

8.1

 4.8

6.7

 12.5

 12.4

17.0

 7.0

18.6

114
I 23450789 IO 

LENGTH OF PERIOD (DAYS)

I I I I

LENGTH OF PERIOD (DAYS)

Figure 87. Accuracy of energy-budget evaporation (A periods 

only) determinations as a function of length of computational 

period. Figure indicates number of A periods included.

(1) the standard error of estimate over a period 

of a year, a high-evaporation period, or a lowevaporation period is about 0.250 cm for daily 

periods, decreasing rapidly to about ±0.140 cm for 

4-day periods, and then less rapidly to about ±0.050 

cm for 10-day periods; (2) very little additional improvement occurs for periods longer than 10 days; (3) 

the standard error of estimate expressed as a percentage of the mean energy-budget evaporation varies, 

depending upon how the mean energy-budget evaporation is selected. For mean energy-budget evaporations obtained for a complete year of data, for the 

high-evaporation season, and for the low-evaporation 

season, the error for daily periods is about ±52, 

±47, and ±72 per cent, respectively; while for 

10-day periods it is about ±7, ±5, and ±40 (estimated) per cent respectively.

The above comparisons of the water-budget and 

energy-budget evaporations are characterized by a 

considerable scattering of the individual points, with 

the scattering decreasing as the length of the period 

increases. An inspection of the energy-budget equaI 23458789 IO 

LENGTH OF PERIOD (DAYS)

±8°

S

LENGTH OF PERIOD (DAYS)

Figure 88. Accuracy of energy-budget evaporation (high-evaporation months and A periods only) determinations as a function 

of the length of computational period. Figure indicates the 

number of periods included.

tion, equation (72), indicates that errors in evaluating 

the Bowen ratio and/or the change in energy storage 

could introduce such scattering.

The Bowen ratio has already been discussed in 

considerable detail and the conclusion reached that 

the ratio is sufficiently accurate. However, during 

periods when the difference between vapor pressure 

of the air and that of air saturated at the water 

temperature is small, considerable error is introduced 

because of instrumental inaccuracies. Such conditions 

occurred rather infrequently and only during periods 

of low evaporation; hence, they could not be responsible for the above scattering.

By inference, then, the scattering can be explained in terms of the errors in the technique used 

in evaluating the change of energy storage. During 

the Lake Hefner observational program, attempts 

were made to obtain complete thermal surveys every 

7 days, although the average time interval was probably closer to 9 days; single temperature soundings 

were made on as many other days as possible. The 

time of determinations rarely coincided with the evap115
oration periods, since the latter were selected chronologically. In general, any error in the change in 

energy storage would decrease as the number of 

days in the time interval increased, since the error is 

not cumulative. Hence, the agreement between the 

energy-budget and water-budget evaporations will 

improve as the length of the period increases. This 

improvement was observed.

To illustrate the effect of errors in determinations 

of change in energy storage, the energy-budget evaporation was computed for periods that coincided with 

the times of the thermal cruises, minimizing, of course, 

the error attributable to the change in energy 

storage. The periods varied from 7 to 19 days, the 

majority being 8 or 9 days long. It must be remembered that the errors inherent in the technique employed to evaluate this term, as previously pointed 

out, may introduce considerable error for periods 

shorter than 7 days. Hence, periods shorter than 7 

days were eliminated from consideration. Data for 

nine A periods and four B periods were available, 

the pertinent data being given in table 20 and in 

figure 89. Of the A periods, seven occurred during 

the high-evaporation season and two during the lowevaporation season. The standard error of estimate 

for the nine A periods was ±0.044 cm or ±9.1 per 

cent of the mean energy-budget evaporation; for the 

seven A periods during the high-evaporation months, 

the standard error of estimate was ±0.028 cm or 

±5.5 per cent of the mean energy-budget evaporation.

The significance of the above results as related 

to errors in the evaluation of the change in energy

TABLE 20. Pertinent Data for Evaporation Periods Coinciding with 

Thermal Cruises.

Date

1 4-22 Sep 1950

8-17 Oct

17-25Oct

2-20 Nov

3-1 2 Jan 1951

4-10 Mar

10-1 9 Mar

3-1 6 Apr

16-23 Apr

12-19 Jun

2-10 Aug

15-22Aug

22-30 Aug

Length

(dayi)

9

10

9

19

10

7

10

14

8

8

9

8

9

Classification

A

A

A

A

A

B

B

A

B

A

B

A

A

Water

Budget

Evap.

(cm)

0.372

0.533

0.384

0.468

0.371

0.306

0.346

0.417

0.283

0.394

0.746

0.703

0.779

Energy

Budget

Evap.

(cm)

0.372

0.493

0.396

0.496

0.290

0.350

0.406

0.481

0.374

0.349

0.878

0.708

0.780

Difference

(cm)

0.000

 0.040

0.012

 0.028

 0.081

0.044

0.060

0.064

0.091

 0.045

0.132

0.005

0.001

Per Cent

0.0

 7.5

3.1

 6.0

 21.8

14.4

17.3

15.3

32.2

 11.4

17.7

0.7

0.1

storage is obscured by the variation in the length 

of the periods. Of more interest are the results obtained during the five 8- and 9-day A periods. The 

standard error of estimate for these periods was 

±0.023 cm or ±4.4 per cent of the mean energybudget evaporation. In comparison, the standard 

error of estimate obtained for all 8- and 9-day 

periods was ±0.077 and ±0.066 cm, respectively, 

or ±15.5 and ±13.7 per cent of the mean energybudget evaporation. These comparisons indicate a 

definite improvement related to the better evaluation 

of the change in energy storage.

It is concluded that evaluation of the change in 

energy storage is the primary limitation to the accuracy of evaporation determinations utilizing the 

energy-budget technique for periods shorter than a 

year, while the Bowen ratio is a secondary limitation 

of importance only during periods of low evaporation 

and in specific circumstances which occur infrequently. In addition, for periods of greater than 

7 days, optimum accuracies of about ±0.030 cm 

or ±5.8 per cent of the mean evaporation can be 

expected during the high-evaporation months from 

the energy-budget technique.

Summarizing, comparisons of evaporation determinations at Lake Hefner by the energy-budget 

and water-budget methods result in the following 

major conclusions: (1) the energy-budget equation 

must be used with caution for periods of less than

WATER BUDGET (CM)

Figure 89. Evaporation comparison of periods that coincide 

with thermal surveys. Dashed lines represent one standard error 

of estimate.

116
7 days; (2) when applied to periods greater than 7 

days, it will yield an accuracy approaching ±5.0 

per cent of the mean energy-budget evaporation, providing all terms are evaluated with the utmost care, 

particularly the change in energy storage.

CONCLUSIONS FROM 

ENERGY-BUDGET STUDIES

1. The classical energy-budget equation for the 

oceans, and as applied in the past to lakes and storage reservoirs, must be modified to correct for the 

advected loss of the evaporated water. Neglect of 

this term will cause an error in the computed evaporation of the order of 5 per cent, depending upon 

the magnitude of the other terms in the equation.

2. For computing evaporation, indirect methods 

of evaluating sotar radiation, such as those proposed 

by Mosby and Kennedy, are inadequate. It is necessary to evaluate solar radiation by direct measurement to obtain the requisite accuracy.

3. Theoretically, the reflectivity of an optically 

flat water surface under a clear sky, for solar radiation, is a function of sun altitude and atmospheric 

turbidity. In addition, previous evidence (Munk, 1947) 

indicated that wind speed may be a factor. The Lake 

Hefner observations showed that the effects of wind 

and air-mass turbidity are negligible, and hence that 

reflectivity is primarily a function of sun altitude.

4. Under cloudy skies, the reflectivity is uniformly modified by an increase in scattering, caused 

by an increasing amount and a decreasing height of 

the cloud cover.

5. Solar radiation is completely diffused only 

under conditions of low sun altitude and low overcast 

clouds. Under these conditions the theoretical reflectivity for completely diffuse radiation is approached.

6. Simple, empirical, hyperbolic relationships 

can be used to express the variation of reflectivity 

with sun altitude for any given cloud condition. These 

relationships can be used to evaluate indirectly the 

reflected solar radiation with enough accuracy for 

purposes of evaporation computation.

7. A linear function expresses the relation between atmospheric radiation and local vapor pressure 

(over the vapor-pressure range generally observed in 

nature) just as adequately as the more complicated 

expressions of Brunt and Angstrom.

8. Empirical relations between atmospheric radiation and local vapor pressure may be used if 

10-per-cent accuracy in computed atmospheric radiation is acceptable, provided they are used in areas

of air-mass structure similar to that of the area where 

the original observations were recorded. If they are 

extended to other areas with no consideration of the 

air-masses involved, the accuracy of the result is 

questionable. For purposes of evaporation determination, empirical evaluations are inadequate for evaluating atmospheric radiation.

9. To obtain a general empirical method for 

evaluating atmospheric radiation it is necessary to 

consider the total vapor content of the atmosphere 

as the moisture variable, rather than the local vapor 

pressure (that is, as determined by standard observations).

10. The emissivity of a natural water surface is 

independent of water temperature and the composition of the water. The emissivity has been found to 

be 0.970 ±0.005.

11. The evaluation of advected energy for any 

particular body of water is an individual problem. It 

is possible for certain situations, as in "run of the 

river" lakes and reservoirs, that inability to evaluate 

this term adequately will prohibit the use of the 

energy-budget approach to the determination of 

evaporation.

12. Errors in evaluating advected energy may 

be minimized through proper selection of the base 

temperature.

13. The Lake Hefner data indicate that the 

energy storage can be evaluated from an average of 

several profiles taken in different depths of water.

14. The Lake Hefner data indicate that the 

Bowen ratio is valid and gives consistent results except 

when the difference between the vapor pressure of 

the atmosphere and that of air saturated at the 

water-surface temperature is small.

15. No error seems to be introduced into evaporation computations by neglect of the effects of 

radiative diffusivity, stability of the air, and spray.

16. Because of limits in the accuracy of the 

temperature measurements, considerable error may 

occur in evaluating the change of energy storage 

over time intervals of a few days. However, for time 

intervals of the order of 7 to 10 days, the accuracy 

is probably adequate for evaporation determinations. 

Over annual cycles the change in energy storage is 

negligible.

17. The energy-budget equation, when applied 

to periods greater than 7 days, will result in a maximum accuracy approaching ±5 per cent of the mean 

energy-budget evaporation, providing all terms in 

the energy budget have been evaluated with the utmost accuracy, particularly change in energy storage.

117
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118
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119
Cummings Radiation Integrator

by G. Earl Harbeck, Jr.*

Application of the energy-budget technique to 

the determination of evaporation from lakes and reservoirs was advocated for many years by the late 

N. W. Cummings, of San Bernardino Valley Junior 

College, San Bernardino, California. He suggested the 

use of a thermally insulated pan to measure certain 

net radiation items, and employed such a pan to 

compute evaporation from Bear Lake in Utah and 

Idaho in the summer of 1937 (Cummings, 1940). A 

water-budget control for this experiment was lacking, 

however, and Cummings could only compare his 

results with those obtained from a nearby Weather 

Bureau Class A pan.**

Shortly before Cummings' death, he outlined in 

a personal communication to the writer his suggested 

design of an insulated pan to be constructed and 

operated as a part of the Lake Hefner project. Cummings' proposed design was studied and the basic 

features incorporated in the instrument constructed 

at Lake Hefner.

THEORY

The energy-budget for a lake or reservoir is generally expressed as

Q8 - Qr - Q6 - Qh - Qe + Qv - Qw = Q«» (59)

in which Q8 = solar radiation incident to the wafer

surface,

Qr = reflected solar radiation, 

Qb net energy lost by the body of water

through the exchange of long-wave

radiation between the atmosphere

and body of water, 

Qh energy conducted from the body of

water by the atmosphere as sensible

heat,

Qe = energy utilized by evaporation, 

Qv = net energy advected into the body of

water, 

Qw energy advected by the evaporated

water, and 

Q$= increase in energy stored in the body

of water.

* Hydraulic Engineer, Water Resources Division, U. S. 

Geologic Survey.

** See next section for discussion of Class A pans.

The effective back radiation, Q6 , is considered to be 

the net exchange of long-wave radiation between the 

atmosphere and the body of water, and for the 

purposes of this discussion can be separated into its 

various components as follows:

Q6 = Qbs Qa + Qor, (96)

in which Qbs = long-wave radiation emitted by the

body of water, 

Qa incoming long-wave radiation from

% the atmosphere, and 

Qar reflected long-wave radiation. 

Substituting in equation (59), we obtain

Q8 Qr + Qa Qar 

(97)

Consider, then, a pan placed near the shore of 

the lake. Since the pan is in effect merely a small 

body of water, it will also have an energy budget 

which may be expressed as

Q/ - Q/ + Qa' - Qa/ - Q6/

- <V - Q/ + Q/ - Qw' = Qs' (98)

in which the primed symbols refer to the pan.

The basic assumption is that the net sum of certain radiation terms is the same for the lake as for 

a pan on the shore of the lake, as follows:

Qs Qr' + Qa' Qa/

= Q8 - Qr + Qa - Qar. (99)

Over a short period of time, such as an hour, it is 

probably not true that long-wave and short-wave incoming radiations (Qs + Qa) are the same for the 

pan and the lake because of transient cloud effects, 

but for longer periods, such as a day or week, the 

assumption appears reasonable. Since the amount 

of reflected long-wave radiation, Qor, is dependent 

only on the emissivity, or rather the absorptivity, of 

the water, there appears to be no reason to question 

the validity of the assumption as far as this item 

is concerned, providing lake water is used in the pan 

and does not become contaminated with substances 

that might change its emissivity. The validity of the 

assumption that reflected solar radiation is the same 

for the lake as for the pan is not as obvious. In the 

discussion of reflectivity of a natural water surface 

(see preceding section entitled "Energy-Budget

120
Studies") it was concluded that reflectivity was independent of wind speed and, therefore, also of surface 

roughness. Therefore, the amount of solar energy reflected from the relatively smooth surface of the water 

in the pan should not be materially different from 

that reflected from the usually rougher lake surface. 

The possibility that some solar radiation may be reflected by the bottom of the pan has not been taken 

into account. However, it should be remembered that 

reflected solar radiation is a relatively small item 

compared with other items in the energy budget. 

During the period of observation at Lake Hefner, 

daily reflected solar radiation ranged between approximately 1 and 55 cal cm~2 day" 1 , as compared 

with a range of approximately 50 to 750 cal cm~ 2 

day" 1 for incoming solar radiation.

Equation (98) may then be written as follows:

+ Q,/ + (V - Q.' + Cb' + (l 00)

Using the familiar relations R' = GV/QC', Qe 

pff L', QA' = PF' R' L', and Qw' = pcF (T0 T&) and 

using a base temperature, T6, of 0°C, equation (100) 

becomes

Q.' - Q/ + Qa' - Q«/ = pF' [L' (1 + RO + cT0'] 

+ Q*/ - Q.' + <V. 001)

Combining equations (97), (99), and (101), and 

using the relations Qe = pFL, Qh = pFRL, Qw = pcF 

(To T6), T6 = 0°C and the close approximations

F' \L' (1 + If) + To'} + (Q»/ - Q; + <V) - (Q». - Qu + Qa) (102) 

L (1 + R) + To

INSTRUMENTS AND METHODS

The Cummings Radiation Integrator (CRI) was 

constructed as indicated in figure 90. Several features 

of its construction are worthy of note. The insulation was designed to limit the heat loss or gain by 

conduction to 3 per cent of the energy required to 

evaporate 0.1 inch of water per day, assuming a 

temperature gradient of 10°F through the insulation. 

Actually the average daily heat loss through the 

insulation did not exceed 0.5 per cent of the energy 

used for evaporation, principally because the observed temperature gradients were much less than 

had been assumed. A wooden platform covered with 

building paper was laid on a carefully leveled site. 

A block of Perlite concrete 24 inches thick was constructed on this platform as a base for the pan. It

would have been preferable to use a more efficient 

insulating material for the base, such as rock wool, 

but it was obvious that the insulating effect of rock 

wool would be destroyed by the packing down resulting from the load of more than 1300 pounds of 

water alone. Perlite concrete appeared to be an 

acceptable compromise, for its structural strength 

was adequate and its thermal conductivity fairly low. 

After the base had been poured and the forms removed, it was covered with building paper, waterproofed with a bituminous coating, and then painted 

with aluminum paint. Because Perlite concrete is 

porous and its thermal efficiency seriously impaired 

by moisture, it was considered essential that the base 

block be placed above ground and waterproofed. 

When the pan was dismantled at the completion of 

the project, the Perlite block was dry, indicating that 

its insulation properties had been unimpaired.

The galvanized sheet-metal pan was 48 inches 

in diameter and 24 inches deep. It was surrounded 

by an 88-inch-diameter sheet-metal ring, and the 

20-inch space between the pan and ring filled with 

rock wool. The rock wool was covered by an annular 

sheet-metal roof or rim which sloped toward the 

outside to shed rain. The roof was fastened to the 

outside ring, but was separated from the pan to 

prevent thermal conduction.

The rim projected inward over the pan for a 

distance of 2 inches, and was provided with a '/2-inch 

lip to prevent driving rain from entering the air gap 

between the pan and the rim. The purpose of the 

overhanging rim was to maintain the area of the 

water surface exposed to solar radiation as nearly 

constant as possible. Obviously, for small sun angles, 

some of the incoming radiation was reflected from 

the side of the pan but, at least during the middle of 

the day when solar radiation was greatest, the watersurface area remained constant.

A fixed-point gage in a small stilling well was 

affixed to the side of the pan. The point of the gage 

was placed approximately 4 inches below the rim 

to allow for storm rainfall; overflow was to be 

guarded against to prevent wetting of the rock-wool 

insulation.

It was originally contemplated that the pan be 

serviced at intervals of approximately one week. The 

weekly schedule was adhered to for a time, but later 

it became evident that pan servicings should be timed 

to coincide with thermal surveys of the lake, which 

were made at somewhat irregular intervals because 

of weather conditions. When the pan was serviced, 

the water was stirred thoroughly with a stick and its

302520 O - 54 - 10 121
SLOPE W PER FT

20» H

20-GA. GALVANIZED 

SHEET STEEL

ROCK WOOL

Figure 90. Cross-section of Cummings Radiation Integrator.

temperature then measured with a calibrated thermometer. Enough water was then added to bring the 

level to the point of the gage; both the quantity and 

temperature of water added were carefully measured. The water in the pan was stirred again and its 

temperature measured. On some occasions, it was 

necessary to remove storm rainfall to bring the water 

level to the point of the gage, but the procedure 

was similar.

Using the average temperature as found and 

the volume and temperature of the water added, the 

theoretical temperature of the full pan was computed 

and compared with the temperature measured after 

the pan had been filled and stirred. In no instance 

did the theoretical and observed temperatures differ 

by more than 0.3°C, and the difference was usually 

less than 0.1 °C.

The CRI was located in the enclosure at the 

south meteorological station. It was placed in operation on 2 November 1950, and weekly observations 

were made until 30 November, when it was deemed 

advisable to remove the water from the pan to 

prevent possible damage from freezing. The pan was 

refilled on 7 March 1951 and observations were 

made until 31 August 1951. After a preliminary 

analysis of the data obtained in November, March, 

and April, it was evident that certain additional information was needed. Using equipment similar to 

that employed for temperature measurements at the 

meteorological station, thermocouples were installed 

to measure (1) the water-surface temperature in the 

pan, (2) the temperature of the underside of the 

overhanging rim, (3) the temperature of the soil under 

the pan, and (4) rainfall temperature. A reduced 

chart speed was used and each temperature was 

measured once each half hour.

The thermocouple for water-surface temperature 

was suspended from a small float in the center of the 

pan and was adjusted to remain just under the surface 

of the water. The rim thermocouple was placed 

against the underside of the steel rim but, because 

of the plastic insulation on the thermocouple junction, 

it was not in direct contact with the steel. The thermocouple for measuring soil temperature was buried Va 

inch in the soil under the pan.

The instrument used for measuring rainfall temperature was essentially a small rain gage with a 

thermocouple in the storage receptacle. A plastic 

funnel having a maximum diameter of 4-13/16 inches 

and a throat diameter of 7/16 inch was used. A 

small plastic cup, % inch in diameter and Vi inch in 

height, was placed about 3 inches below the funnel. 

A thermocouple was placed in the cup. It is believed 

that during rainless periods the temperature indicated 

by the thermocouple would be, on a daily basis at 

least, not greatly different from the temperature of 

the ambient air. When precipitation occurred, however, it was reasoned that the funnel and cup would 

soon reach the temperature of the rain. The cup was 

purposely made very small so that it would soon fill 

and run over. It was placed 3 inches below the funnel 

in order that the falling water would splash in the 

cup, thus assuring thorough mixing. Even a rainfall 

of only 0.02 inch was sufficient to cause the cup to 

overflow, but little importance was attached to record 

temperatures of light rains because of the possibility 

that equilibrium had not been reached.

The analysis of the data was made using in 

general the methods described in the preceding section. The method used in computing certain items of 

the energy budget for the CRI are believed to be of 

interest, however.

122
Although evaporation takes, place over the 

entire water surface of the CRI, the effective area 

for incoming radiation is only the area encompassed 

by the rim. The sum of the four radiation items in 

equation (100) (which is the basic equation for the 

energy budget of the CRI) is needed for the solution 

of equation (97). These must be computed in terms of 

the area encompassed by the rim, whereas each of 

the terms on the right side of equation (100), Q/ -{- 

QA' + Q&a' Q/ + Qp> + Qw, must be evaluated 

on the basis of the entire water surface area. The 

sum of these last items must then be divided by the 

effective area for incoming radiation to determine the 

sum of Q/ Q/ + Qa' Qo/ on a unit-area basis.

The computation of Q6/, the outgoing longwave radiation from the water surface in the CRI, was 

complicated by the effect of the overhanging rim. 

Temperature records obtained during the period May 

to August indicate that the rim temperature was 

generally 1° to 5°C higher than the water-surface 

temperature, and the interchange of energy between 

the water and the rim could not be neglected. The 

underside of the rim was painted black, using the 

same paint employed to blacken the flat-plate radiometer, and its long-wave emissivity was taken as 

0.90. The preliminary value of 0.967 was used for 

emissivity of water. The amounts of energy being 

received, emitted, absorbed, and reflected by both 

the water and the rim were computed by successive 

approximations. In general, the result was a net gain 

for the water. The rim temperature was higher and 

its emissivity lower, but its reflectivity was also higher. 

The net Q&/ for the entire pan was then computed 

as the difference between that being emitted by the 

water and that being received from the rim.

The computation of Q/, the advected energy, 

was simple, since energy could be brought in only

by rainfall or through the sides and bottom of the 

CRI. The volume of advected rainfall was determined 

by multiplying the depth of rainfall recorded at the 

south meteorological station by the area encompassed 

by the rim. The temperature of the rainfall was measured using the device described above. The results 

obtained for two selected storms are shown in figure 

91. After the rainfall had cooled the small receiving 

cup, the indicated rainfall temperature remained 

fairly constant. Wet-bulb temperatures recorded at 

the 8-meter level are also shown in figure 91, and 

it is apparent that they agree closely with rainfall 

temperatures. Other investigators have previously 

obtained similar results so it appears unnecessary to 

discuss the matter at length. As far as the CRI is 

concerned, energy brought in by rainfall is usually 

a relatively small item in the energy budget, and 

rainfall-temperature measurements were made only 

to substantiate the generally accepted conclusion that 

rainfall and wet-bulb temperatures are approximately 

equal.

The heat exchange through the insulated sides 

and bottom of the pan was also a minor item. The 

thermal conductivity of the Perlite concrete was taken 

as the figure supplied by the manufacturer for the 

particular mix used, and the temperature gradient 

was taken as the difference between the water temperature and the temperature of the soil under the 

pan. Calculations of the heat exchange through the 

rock wool were made in similar fashion, the temperature gradient being taken as the difference between 

the water temperature and the dry-bulb temperature.

The change in energy storage in the CRI, Q^, 

was computed from volumes and temperatures measured each time the pan was serviced. Compared to 

a similar determination for Lake Hefner, it was a 

simple procedure.

STORM OF 4

RAIK

I 1

*-

JULY 1951

FALL .65" 

 1 

TEMPERATURE >-L

WET-BULB TEA 

AT 8-M. LEVE

OF RAINFALL

,O .ft i

APERATURE

L

0300 0600 

TIME IN HOURS

STORM OF 2 

RAINF>

a

\

4 JULY 1951

kLL 1.75"

"

TEMPERATURE

WET-BULB TEJ 

AT 8-M. LEVE

OF RAINFALL

P 

APERATURE

.

00 2200 2300 24

Figure 91. Comparison between rainfall 

and wet-bulb temperatures for selected 

storms.

123
The Bowen ratio for the lake, R, was computed 

using dry- and wet-bulb temperatures measured at 

the 2-meter level at the barge. Water-surface temperatures were also taken from records obtained at 

the barge. The Bowen ratio for the pan, K, was computed using the same wet- and dry-bulb temperatures, and the temperature of the water surface in 

the pan. Wet-bulb, dry-bulb, and water-surface temperatures were averaged for each period. Vapor 

pressures were computed using these average temperatures. Although the relation between vapor pressure and temperature is not linear, for a week 

selected at random the vapor-pressure difference 

computed using average temperatures differed by 

only 2 per cent from the average of the vaporpressure differences computed using the 3-hourly 

data.

Thermocouples for measuring water-surface and 

soil temperatures were installed in the CRI on 1 May 

1951 and those for measuring rim and rainfall temperatures were installed 25 May 1951. Of these, the 

water-surface temperature measurement was by far 

the most important. The amount of long-wave radiation emitted by the water in the pan, which depends 

only on the water-surface temperature, is a major 

item in the energy budget for the pan. Its importance 

relative to that of other items was not recognized at 

the time observation began. An attempt was made 

to estimate water temperatures, during the period 

before measurements were begun, but it was considered that the possible resultant error might be so 

large as to cause the results to be questionable. 

Rim temperatures during part of May were taken as 

equal to the air temperature at the 2-meter level at 

the south station.

RESULTS

Since the CRI was suggested as a possible replacement for the more expensive and complicated 

radiation instruments, it is necessary to show whether 

it measures the total radiation received, Q* Qr 

-f- Q0 Oar/ with sufficient accuracy. Using all periods for which radiation data were sufficiently complete, the sum of the four radiation items was 

computed from the CRI records and from the radiation records. The results are shown in table 21. 

Although the general agreement is excellent, it should 

be remembered that a relatively small error in measuring radiation may affect computed evaporation 

considerably. For example, from the data in table 21, 

the standard error of measuring net radiation using

TABLE 21. Comparison Between Average Net Incoming Radiation Measured by Cummings Radiation 

Integrator and by Pyrheliometers and Flat-plate Radiometer

Net Incoming Radiation

Period

12-19 Jun

28Jun-9Jul

17-24Jul

24 Jul-2 Aug

2-10 Aug

15-22 Aog

22-29 Aug

Weighted

average

Computed from

CRI Records

(cat cm~2 day" 1)

1310

1296

1434

1281

1357

1282

1306

1321

Measured Using

Radiation Instruments

(cal cm~2 day" 1)

1349

1295

1418

1366

1364

1295

1307

1340

the CRI is 36 cal cm~2 day" 1 , which is only about 

3 per cent of the mean. An error of this mggnitude 

would result in an error of about 9 per cent in evaporation if it is assumed that the radiation instruments 

are without error.

Because of possible large errors resulting from 

the lack of measurements of the temperature of the 

water surface in the pan, evaporation was not computed for periods in November, March, and April. An 

attempt was made to compute evaporation for each 

period between CRI servicings, which were made approximately once each week. After data for the 

first five months had been analyzed, it was discovered that CRI servicings should have been timed 

to coincide with thermal surveys of the lake. The 

change in energy storage, Q$ was found to be of 

considerable importance particularly for short periods. The day-to-day change in storage in the lake 

was obviously erratic, and values interpolated between thermal surveys were therefore questionable. 

After this had been recognized, CRI servicings were 

made immediately after a successful thermal survey 

of the lake had been completed.

It was deemed advisable to group certain of the 

CRI servicing periods to obtain a smaller number of 

longer periods for which the change in energy storage 

in the lake was sufficiently well defined by thermal 

surveys. Commencing in June 1951, however, the 

coincident periods averaged slightly less than 8 

days in length.

Evaporation was computed for 13 periods ranging in length from 5 to 20 days. The results are 

shown in table 22 and figure 92. For all periods, the 

mean daily evaporation as determined using the CRI

124
TABLE 22. Comparison Between Computed and Observed Evaporation from 

Lake Hefner for Periods of Various Length

Period

3-23 May

23 May-5 Jun

5-12 Jun

12-19 Jun

19-28 Jun

28 Jun-9 Jul

9-1 7 Jul

17-24 Jul

24 Jul-2 Aug

2-10 Aug

10-15 Aug

15-22 Aug

22-29 Aug

Weighted average

Length of

Period

(day.)

20

13

7

7

9

11

8

7

9

8

5

7

7

Computed

Evaporation

(cm day-1)

0.249

.441

.242

.371

.696

.593

.680

.835

.439

.911

.693

.733

.762

.544

Observed

Evaporation

(cm day *)

0.242

.415

.247

.371

.566

.601

.531

.712

.516

.808

.631

.712

.766

.509

Error

(per cent)

+2.9

+6.3

 1.0

0.0

+23.0

 1.3

+28.1

+17.3

 17.5

+ 12.7

+9.8

+2.9

 0.5

+6.9

Accuracy

Clauificatlon

D

D

A

A

A

A

A

A

B

B

B

B

B

WATER-BUDGET DAILYEVAPORATION AVERAGE CMIN

'-'iouA-uib-Wao-oa

/

y

X

fy

 9.

/ I 3

7

/ 

94 

8«

%

/

 7

/

 8

FIGURES BESIDE PLOTTED POINTS INDICATE 

NUMBER OF DAYS IN PERIOD

.2 .3 .4 .5 .6 .7 .8 .9 

COMPUTED AVERAGE DAILY EVAPORATION IN CM

1.0

Figure 92. Comparison between computed and observed evaporation.

was 6.5 per cent greater than the corresponding 

water-budget evaporation, and the standard error of 

estimate was 0.72 cm, or 14.1 per cent of the mean. 

The computed results for periods of high evaporation 

appear to be consistently greater than those observed. No explanation can be given for this trend, 

if it be real, but it is suggested that it may result 

from incomplete knowledge of the interchange of 

energy between the water surface and the underside 

of the rim. The rim effect is not directly associated 

with the amount of evaporation, as might be inferred 

from figure 92, but rather with rim temperature, 

which in turn is correlated with solar radiation and 

thus with evaporation.

Similar computations were made using meteorological data obtained at the Weather Bureau 

station of the Will Rogers Airport, 13 miles 

south of Lake Hefner, for the determination of R and 

R'. There was no significant difference in the computed evaporation figures, which is not surprising, 

since Ta and ea in the expression for the Bowen ratio 

are specified as the temperature and vapor pressure 

of the air before modification by the lake has occurred. It is believed that data obtained at the airport 

are representative of average conditions over periods 

of a week or longer. Moreover, this procedure indicated that it is not necessary to measure air temperature and humidity at the lake, which simplifies the instrumental requirements considerably. It should not be 

assumed that this procedure can be followed at any 

lake; but a series of sling psychrometer checks should 

be adequate to show whether this assumption is valid.

Table 23 illustrates the relative magnitudes of 

certain of the terms in equation (102). The differences 

between L' (1 + R') and L (1 -f R) are highly correlated with the corresponding differences between 

7"</ and r0 because the difference between R and R' 

depends largely on these two water-surface temperatures. The differences between (Q6/ Q/ -f Q^') 

and (Q6s Qv -f Q$) reflect to a large extent the 

change in energy storage in the lake. During the 

period 5-12 June, Q$ was -|-235 cal cm" 1 day" 1 . 

If this had been neglected, computed evaporation for 

the period would have been 0.614 cm instead of 

0.242 cm. An assumption that changes in energy storage in the lake are negligible cannot be justified, 

except possibly on an annual basis.

TABLE 23. Comparison Between Values of Indicated Radiation Items for 

Cummings Radiation Integrator and for Lake Hefner

Period

3-23 May

23 May-5 Jun

5-12 Jun

12-19 Jun

19-28 Jun

28 Jun-9 Jul

9-1 7 Jul

17-24 Jul

24 Jul-2 Aug

2-10 Aug

10-15 Aug

15-22 Aug

22-29 Aug

1* (1 + RO 1(1+ R) rc' 

(ealcm- 1 ) ('

541

601

636

617

545

549

562

520

574

456

526

524

436

572

609

610

609

550

591

581

555

623

557

602

599

541

17.4

20.5

22.0

24.2

25.0

23.9

25.8

26.3

26.7

26.0

25.1

24.8

24.3

T0 Q( 

'C)

17.8

20.7

21.5

24.0

25.1

24.8

26.3

27.9

28.4

28.4

27.2

26.9

26.0

(cal cm~2 day" 1)

803

835

876

876

866

854

875

865

910

839

878

836

868

950

897

1076

1075

981

931

984

947

995

824

879

823

874

125
CONCLUSIONS ON CUMMINGS 

RADIATION INTEGRATOR

The conclusions presented here are based on 

only a relatively short period of observation. Further 

experimental work is desirable, and is believed warranted on the basis of the findings at Lake Hefner.

1. The Cummings Radiation Integrator offers 

considerable promise as an instrument for measuring 

certain net radiation items. For purposes of computing evaporation, using the energy-budget technique, the instrument appears to be a satisfactory 

substitute for the flat-plate radiometer and pyrheliometers. It should be emphasized that no empirical 

pan coefficient is involved if the CRI is used for this 

purpose. Only certain physical constants, whose 

values can be or have been established in the laboratory, are needed. It is estimated that the error 

in computed weekly evaporation during the summer 

months will ordinarily be 10 to 15 per cent or less, 

if measurements of energy storage in the lake are 

also made at weekly intervals; on a monthly basis, 

the error is estimated as 5 to 10 per cent. There is 

no apparent reason why the CRI cannot be operated 

in winter using a saline solution or some other liquid 

whose emissivity and other properties are known.

2. The problem of the interchange of energy 

between the water surface and the underside of the 

rim is as yet imperfectly understood. Elimination of 

the rim would be desirable if some means could be 

found of maintaining a completely filled pan or of 

satisfactorily accounting for the effect of a vertical

rim. The rim effect could be minimized by using a 

larger pan.

3. The energy conducted to or from the CRI 

as sensible heat was determined by the use of the 

Bowen ratio. A comprehensive study of the validity 

of the Bowen ratio, particularly as used in connection 

with the CRI, would be of considerable value. The 

use:of a wind tunnel for this study may be practical 

if the necessary physical measurements can be made 

with the requisite accuracy.

4. CRI or other radiation records obtained at a 

reservoir site prior to construction are desirable for 

making preconstruction estimates of evaporation using 

the energy-budget technique.

5. The use of the CRI in connection with studies 

of evaporation losses from vegetation should be investigated. No data are yet available as to the areal 

distribution of incoming long-wave radiation. Pyrheliometer records of incoming solar radiation at selected 

locations are available, but these are insufficient because long-wave radiation is excluded. Although the 

energy budget of a plant is admittedly more complex 

than that of a lake, the relation between total radiation received and evapotranspiration losses should 

be studied in relation to available moisture.

REFERENCE

Cummings, N. W., "The Evaporation-Energy Equations 

and their Practical Application/' American Geophysical Union, Transactions, vol. 21, part 2, 1940, pp. 

512-522.

126
Lake and Pan Evaporation

by Max A. Kohler*

BACKGROUND AND OBJECTIVES

The Weather Bureau has for many years collected evaporation data as observed in a standard 

4-foot pan (commonly known as the Class A pan), 

and many additional stations equipped with pans of 

various types have been operated by other Federal, 

state, and local agencies. Admittedly, the data cannot be considered to represent evapotranspiration 

from adjacent land areas or evaporation from a 

nearby lake, but they do constitute an index to the 

evaporative power of the atmosphere, and thereby 

serve a real purpose in depicting geographical variations in climate.

Pan-evaporation data are of direct value in 

several fields, such as planning facilities for evapoative-cooling air conditioning, and have been widely 

used in an indirect manner for estimating evaporation 

from lakes and evapotranspiration. Estimates of lake 

evaporation are commonly derived by applying a 

"pan coefficient" to observed pan evaporation. This 

section is primarily devoted to a discussion of the 

Lake Hefner results relative to this latter application 

of pan data. Specifically, the objectives of the part 

of the Lake Hefner program devoted to pan studies 

were as follows:

1. Obtain further information on the relative 

evaporation from various types of pans.

2. Investigate the reliability of the pan-coefficient 

approach for estimating monthly and annual lake 

evaporation.

3. Derive a more reliable procedure for estimating lake evaporation from pan evaporation and 

related meteorological data normally collected by 

the Weather Bureau in its established observational 

programs.

To provide ready reference for comparative purposes, table 24 presents a summary of pan coefficients 

derived in numerous past studies. The investigator, 

geographical location, and reference publication are 

given in each case. Attention is called to the fact 

that in some cases a 12-foot sunken pan was assumed 

to be equivalent to a lake. Considering variations in

* Chief Research Hydrologist, Hydrologic Services Division, U. S. Weather Bureau.

climate and lake dimensions and the probable reliability of computed lake evaporation, the coefficients 

are reasonably consistent. Table 25, also presented 

for comparative purposes, summarizes results of simultaneous observations from various types of pans as 

ratios. The periods of record indicated in the footnotes of this table are only approximate. For example, 

"Apr-Oct 1940-45" would mean that observations 

during the 6-year period were, on the average, 

started by 1 April and discontinued at the end of 

October. Tables 24 and 25 contain all readily accessible comparative data; however, further search 

may reveal data which have not been included.

In the analyses in this chapter, particular attention has been directed to the Class A pan since it is 

the official Weather Bureau instrument. This is not 

to be construed as meaning that the other pans are 

not equally susceptible to such detailed analysis or 

that the Class A pan is believed to be the "best" of 

the group. Time was simply not available for complete analysis prior to publication of this report.

INSTRUMENTATION AND 

OPERATIONAL PROCEDURE

Three separate installations were mainatined 

around Lake Hefner to provide data for the evaporation-pan studies (see map). A plan view of each 

installation is shown in figure 93. The photographs 

in figure 94 show various aspects of the installations.

Observations at the pan stations consisted of 

pan evaporation, precipitation, pan-water temperature, humidity, air temperature, wind, and, at the 

south station only, soil moisture and temperature. 

Observations began on 24 April 1950, and terminated 31 August 1951, except that the BPI pan was 

not installed until 19 May 1950. Observations were 

initially made at about 1500 because of the heavy 

morning work load at the Oklahoma City Weather 

Bureau Office. Because of the inherent difficulties experienced in interpreting data from the maximum and 

minimum thermometers, and to terminate the day 

more nearly at the time of minimum evaporation, 

observation time was shifted to about 0900 on 11 

June 1950.

127
TABLE 24. Summary of Monthly and Annual Pan-to-Lake Coefficients

Investigator

Lippincott, J. B.

O'Neill, C.M.

Rohwer, Carl

Sleight, R. B.

Sleight, R. B.

White, W. N.

Young, A. A.

Young, A. A.

1. B. and W. C., United

States and Mexico

Special Committee on4

Irrigation Hydraulics

Computed Values5

Computed Values6

Rowher, Carl

Young, A. A.

Reference Location

10 Lake Hodges, Calif.

12 Newell Reservoir, Canada

14 Ft. Collins, Colo. (85-ft reservoir)

15 Denver, Colo. (12-ft pan, 3 ft deep)

15 Denver, Colo. (12-ft pan, 3 ft deep)

1 8 Milford Exp. Stn., Utah (pan, 1 2 ft x 3 ft)

21 Fullerton, Calif, (pan, 12 ft x 3 ft)

21 Lake Elsinore, Calif.

24 Ft. Mclntosh, Tex. (pan, 12 ft x 3 ft)

22 (recommendation of Subc. on Evap.)

8, 23 Red Bluff Reservoir, Texas

8 Lake Okeechobee, Florida

14 Ft. Collins, Colo. (85-ft reservoir)

21 Lake Elsinore, Calif.

Years of Basis of 

Record Coefficient

1919-21 Jun-Octl

1919-25 Approx. May-Sep

1926-28 Apr-Nov2

1915-16 7 Mean Annual

1916 Jun-Oct

1926-27 May-Oct

1936-39 Mean Annual

1939-41 Mean Annual

1950 Feb-Dec

Mean Annual

1939-47 Mean Annual

1940-46 Mean Annual

Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Dec

1927-28 Apr

May

Jun

Jul

Aug

1926-28 Sep

Oct

Nov

1939-41 Jan

Feb

Mar

Apr

May

Jun

Jul

Aug

Sept

Oct

Nov

Dec

BPI 

Class A (Sunken)

0.95

0.70

0.67

0.94

0.67

0.77 0.94

0.77

0.72

0.70

0.68

0.81

0.77

0.69

0.73

0.84

0.82

0.85

0.91

0.91

0.85

0.76

0.71

0.83

0.60

0.63

0.69

0.69

0.71

0.82

0.72

0.77

0.82

0.63

0.68

0.66

0.68

0.77

0.74

0.78

0.87

0.93

0.97

0.95

Colorado 

(Sunken)

0.96

0.79

0.89

0.78

0.98

0.87

0.83

0.89

1.00

0.97

1.08

1.19

1.20

1.05

0.84

0.77

0.90

0.75

0.76

0.77

0.76

0.75

0.86

0.82

0.99

Screened 

(Sunken)

0.98

0.98

0.89

0.96

0.77

0.92

0.87

0.90

0.99

0.95

1.02

1.10

1.12

1.12

1.03

1 Fragmentary record; periods of data total 6 months; 2-inch chicken wire over pan.

2 No data Apr-Aug 1926.

3 Record broken for 2'/2 months in winter.

4 Not an investigation, but recommendations based on study of all previous investigations.

5 Computed from evaporation data on reservoirs and pans, respectively, from references (8) and (23); "Years of record" refers to pan data.

6 From reference (8) and WB files; Class A pan at Belle Glade; Colorado (sunken) pan at Moore Haven. 

f Nov-Nov.

128
TABLE 25. Summary of Pan-to-Pan Ratios.

_PonX

PanY

1"

Z

°"

Class A

BPI

(Sunken)

Colorado

(Sunken)

Screened

(Sunken)

PAN "X"

Class A

1.30 Lake Kickapoo, Tex. (2)

1.20 Buchanan Dam, Tex. (3)

1.33 Denver, Colo. (4)

1.25 Balmorhea, Tex. (6)

1.28 Pardee Res., Cal. (8)

1.45 Ft. Assinniboine, Mont. (9)

1.29 Yuma Fid. Sta., Cal. (10)

1.21 Fullerton, Cal. (10)

1.46 Hays, Kans. (14)

1.31 Average

1.13 Ft. Collins, Colo. (5)

1.22 Belle Glade, Fla. (7)

1.18 Pardee Res., Cal. (8)

1.15 Fullerton, Cal. (10)

1.06 Henshaw Res., Cal. (13)

1.15 Average

1.10 Mansfield Dam, Tex. (1)

1.27 Lake Kickapoo, Tex. (2)

1.20 Buchanan Dam, Tex. (3)

1.36 Dryden, Tex. (3)

1.25 Ft. Mclntosh, Tex. (3)

1.31 Yuma Fid. Sta., Cal. (10)

1.25 Fullerton, Cal. (10)

1.26 Lake Elsinore, Cal. (11)

1.34 San Jacinto, Cal. (12)

1.26 Average

BPI (Sunken)

0.77 Lake Kickapoo, Tex. (2)

0.83 Buchanan Dam, Tex. (3)

0.75 Denver, Colo. (4)

0.80 Balmorhea, Tex. (6)

0.78 Pardee Res., Cal. (8)

0.69 Ft. Assinniboine, Mont. (9)

0.78 Yuma Field St., Cal. (10)

0.82 Fullerton, Cal. (10)

0.68 Hays, Kans. (14)

0.77 Average

0.92 San Pablo Res., Cal. (8)

0.97 Upper San Leandro

Res., Cal. (8)

0.93 Pardee Res., Cal. (8)

0.95 Fullerton, Cal. (10)

0.94 Average

0.98 Lake Kickapoo, Tex. (2)

1.00 Buchanan Dam, Tex. (3)

1.02 Yuma Fid. Sta., Cal. (10)

1.04 Fullerton, Cal. (10)

1.01 Average

Colorado (Sunken)

0.89 Ft. Collins, Colo. (5)

0.82 Belle Glade, Fla. (7)

0.84 Pardee Res., Cal. (8)

0.87 Fullerton, Cal. (10)

0.95 Henshaw Res., Cal. (13)

0.87 Average

1.09 San Pablo Res., Cal. (8)

1.03 Upper San Leandro

Res., Cal. (8)

1.08 Pardee Res., Cal. (8)

1.05 Fullerton, Cal. (10)

1.06 Average

1.09 Fullerton, Cal. (10)

Screened (Sunken)

0.91 Mansfield Dam, Tex. (1)

0.79 Lake Kickapoo, Tex. (2)

0.83 Buchanan Dam, Tex. (3)

0.73 Dryden, Tex. (3)

0.80 Ft. Mclntosh, Tex. (3)

0.77 Yuma Fid. Sta., Co . (10)

0.80 Fullerton, Cal. (10)

0.79 Lake Elsinore, Cal. (11)

0.75 San Jacinto, Cal. (12)

0.80 Average

1.02 Lake Kickapoo, Tex. (2)

1.00 Buchanan Dam, Tex. (3)

.98 Yuma Fid. St., Cal. (10)

.97 Fullerton, Cal. (10)

0.99 Average

0.92 Fullerton, Cal. (10)

(1) Jan 1950-Decl951

(2) Apr-Oct 1950-51

(3) Jan-Dec 1950

(4) Jun-Octl916

(5) Sep1926-Novl928

(6) Apr 1941-Dec 1948

(7) Jun 1941-Dec 1948

(8) Jan 1930-Dec 1944

(9) Apr-Sep 1949-51

(10) Jan 1937-Decl939

(11) Jul 1939-Dec1943

(12) Jul 1939-Moy 1946

(13) Jan 1942-Decl943

(14) Apr-Sep 1937-1950

129
Figure 93. Plan view of pan Installations 

at Lake Hefner.

1 INSTRUMENT SHELTER

2 CLASS A EVAPORATION PAN

3 BPI EVAPORATION PAN (SUNKEN)

4 COLORADO EVAPORATION PAN (SUNKEN)

5 SCREENED EVAPORATION PAN (SUNKEN)

6 STANDARD 8-INCH RAIN GAGE

7 RECORDING RAIN GAGE

8 ANEMOMETER, MOUNTED ON CLASS A PLATFORM

9 SOIL MOISTURE AND TEMPERATURE INSTALLATION

10 GOTHAM GAS-BULB THERMOGRAPH

11 CUMMINGS RADIATION INTEGRATOR

12 HIGH MAST (WIND, TEMPERATURE, AND HUMIDITY)

13 LOW MAST (WIND, TEMPERATURE, AND HUMIDITY)

14 HOUSING FOR NAVY RECORDER EQUIPMENT

15 STORAGE TANKS

^

NORTHEAST 

STATION NO. 2

^

-29.^(2)

m 6°0

L

NORTHEAST 

STATION NO.

SOUTH STATION

o V

7

ooo 151515

Bo

5 0

Evaporation Pans

Class A pans were installed at each site to indicate areal variations, and additional pans of various 

types were installed at the south station for comparative purposes. Two Class A pans were maintained at 

the northeast site, one in a standard enclosure and 

the other more openly exposed, to test the effect of 

obstructions (shelter, rain gage, etc.) on observed 

evaporation.

The Class A pans were fabricated to standard 

specifications (4 ft in diameter and 10 in. deep) and 

mounted on standard wooden platforms. These pans 

were generally filled to within about 2 in. of the rim 

and refilled when the level had dropped approximately 1 in. The sunken screened pan was 2 ft in 

diameter and 3 ft deep, covered with Vi-in.-mesh 

screen; the Colorado sunken pan was 3 ft square and 

18 in. deep; and the BPI pan was 6 ft in diameter 

and 2 ft deep. Each of the sunken pans was installed 

with the rim about 2 in. above the ground surface, 

and the water level was maintained at or slightly 

below ground level.

Water level in the pans was observed with hook 

gages calibrated to thousandths of an inch. Since the 

project was to be of limited duration, every effort 

was made to maintain observations as complete as 

possible, even during the winter. The pans were not 

emptied during cold weather as is customary. To 

avoid damage by freezing, small holes were drilled 

through the ice, thus relieving the pressure.

During September and early October 1950, 

windblown vegetation tended to collect on the windward side of the enclosures and in the sunke'n pans, 

particularly. Although an effort was made to keep 

the pans and fences free of vegetation, wind and 

pan evaporation were at times reduced somewhat. 

Visual observation and analysis of the data demonstrate that observed pan evaporation was occasionally in error due to blowing snow and to splash-out 

or spill-over during periods of heavy rain and high 

wind. This deficiency of the pan is discussed in considerable detail later in this section.

Water Temperatures

The Class A pan at the south station was 

equipped with a Gotham gas-bulb thermograph for 

recording surface-water temperature. The element 

was shielded from the sun and floated as near the 

surface as was feasible. It is believed the temperatures represent an average through about the top 

%-in. of depth. All other pans were equipped with 

Sixes maximum-minimum thermometers, floated at a 

depth of approximately Vi-in. below the surface and 

shielded from the sun. The thermometers were removed from the pans during freezing temperatures 

to avoid damage.

Precipitation

The network of precipitation gages utilized for 

the project is discussed in the section of this report 

on "Instrumentation." It will suffice to state here that 

standard 8-in. gages were placed at each of the 

three pan installations to assist in computing pan 

evaporation.

130
a. South station, looking east. Pans, in order of increasing distance from camera, are (1) Colorado, (2) 

screened, (3) Class A, and (4) Bureau of Plant Industry 

(BPI). Soil moisture-temperature shelter surmounted by 

Bouyoucos bridge in foreground.

c. Showing float mounting of Sixes maximum-minimum 

thermometer in BPI can.

e. I'-.ui ,'i'. -.. .! MMullation, looking west. Station no. 1 in 

foreground, station no. 2 in background, and recording 

rain gage approximately midway between the two enclosures.

b. South station, looking west across BPI pan toward 

Class A pan.

d. Northeast station no. 1, looking north.

f. Northeast installation, looking e^it. It will be noted 

that the pan at station no. 2 (foreground) is relatively 

unobstructed.

g. Southwest station, looking east. h. Southwest station, looking west-southwest.

Figure 94. Photographs of pan installations at Luke Hefner.

131
Wind Movement

Anemometers were mounted on the supports of 

all Class A pans in standard fashion, with the plane 

of the cups 6 in. above the rim of the pans. The 

anemometers used were 3-cup totalizing (odometer) 

Bendix-Friez Model 349-N. Twenty-four-hour wind 

movement in statute miles was observed and recorded 

at each observation.

Air Temperature and Humidity

As is standard practice, maximum and minimum 

air temperatures were observed at each of the three 

sites. In addition, wet- and dry-bulb temperatures 

were read at each observation and each site was 

equipped with a hygrothermograph, although these 

data were not used in the analyses, since more reliable psychrometric data were available from the 

mass-transfer stations. These instruments were exposed in standard cotton region shelters (5 ft above 

the ground surface).

Soil Moisture and Temperature

The experimental work conducted at Lake Hefner 

did not require observations of soil moisture and 

temperature. However, it was envisioned that the 

project data might later be used for related studies 

in which such information would be required. Accordingly, thermistors and Bouyoucos blocks were 

installed at the south station at depths of 10, 20, 

30, 50 and 100 cm. Satisfactory soil temperature 

records were obtained, but the soil-moisture data 

appear somewhat erratic, although the data have 

not been analyzed critically.

PAN EVAPORATION AND 

METEOROLOGICAL FACTORS

Numerous attempts have been made in the past 

to derive reliable relations between pan evaporation 

and meteorological factors (Bigelow, 1910; Norton, 

1917; Meyer, 1942; Rohwer, 1931; Theis, 1931). Obvious purposes to be served by such relationships are:

1. Extrapolation of short-period and seasonal 

records.

2. Derivation of estimated data for meteorolgical stations at which pan observations are not made.

3. Tests of the reliability and representativeness 

of observed data.

4. Aids in studying lake-pan relations.

Over much of the country, pan observations are 

not made during the winter months because of subfreezing temperatures. This represents a serious deficiency in the data, particularly when studying 

pan-to-pan and pan-to-lake coefficients. Thus, a relation that could be used to estimate winter evaporation would be of practical value. Concerning estimates of evaporation at meteorological stations, 

continued and numerous requests for additional evaporation stations point up the need for more complete 

geographical coverage of evaporation data. Regarding the third purpose, it should be pointed out,that 

Weather Bureau evaporation stations are attended 

by cooperative observers and, under the circumstances, it is not always possible to select a truly 

representative site or to maintain observational procedures on a fully standard basis. Some observers, 

for instance, add water frequently and maintain 

proper water level, while others do not refill the pan 

until the water level has dropped far too low. Such 

variations in observational procedures are known to 

affect pan evaporation and, therefore, tend to produce fictitious variations from station to station.

Two types of pan relations were developed, the 

first being based on observed water-temperature data 

and the second utilizing an energy-balance approach.

Water-Temperature Approach

Most empirically derived equations for evaporation are similar in form to that set forth by Dalton 

well over a century ago, namely,

= (e0 ea) f (u), (103)

where £ is the evaporation in unit time, eo and ea 

are the vapor pressures of the evaporating surface 

and the atmosphere, respectively, and f (u) is a function of horizontal wind speed. Although some equations include a pressure term, variations in this factor 

have no significant effect on evaporation at a particular site. The form of f (u) is usually assumed to 

be such that equation (103) becomes either

or

= (e0 ea) (a + bu)

£ = c (e0 e0) un,

(104)

(105)

132
where a, b, c and n are constants. Although these 

two equations appear quite different, they fit observed pan data almost equally well because of the 

limited range of the wind data. Equation (104) was 

fitted (least squares) to the pan data for the south 

station with the results shown in table 26 and figure 

95. The derived constants (a and b) yield 24-hour 

evaporation in inches where u is the wind movement 

at the Class A pan in miles per day, and the atmospheric vapor pressure (ea in inches of mercury) is 

observed at the 2-meter level (mass-transfer station). 

Values of ea were obtained from mean 3-hourly wetand dry-bulb temperatures and then averaged for 

the day. The vapor pressure of the water surface 

(ey) was based on the mean 24-hour temperature 

(average of maximum and minimum). The derived 

constants are strictly applicable only when the vaporpressure difference is computed as stated. The substitution of 6-hourly observations for 3-hourly in 

computing ea should have little effect on the results. 

However, computation of the mean daily water temperature by any other method may produce a slight 

bias in computed evaporation.

Days with rain were excluded from the correlation because of the poor reliability of the evaporation data during such periods. The records prior to 

12 June 1950 were excluded because of the change 

in time of observation, and periods when the pans 

were frozen could not be used because only accumulated evaporation was observed and water temperatures were not available. The fact that the Class A 

pan was occasionally frozen over when the sunken 

pans were ice-free accounts for the difference in 

number of observations used.

It will be noted from the table that the wind 

factor (b) displays but little variation from pan to 

pan as compared with the other constant (a). Water 

temperatures in the different pans, as indicated by 

values of (eo ea ), are consistent for the over-all

periods used. However, with respect to comparative 

analysis of the data presented in table 26, it should 

be pointed out that the data used for the various 

pans were not altogether concurrent.

A pan relation based on water temperature is 

limited in application because water-temperature

VAPOR PRESSURE DIFFERENCE IN INCHES OF MERCURY 

0 0.20 0.40 0.60 0.80 1.00 1.20 1.40

Figure 95. Relation for Class A pan at south station. Lake 

Hefner, using water temperature.

TABLE 26. Evaporation Relations [£ = (e0 ea) (a -f- bu)] for Pans at South Station.

Pan

Type

Class A

BPI

Colorado

Screened

Days

of

Record

266

288

285

292

Standard Error

Correlation

a

0.417

0.253

0.212

0.104

b

0.0040

0.0040

0.0044

0.0044

Index

0.92

0.91

0.89

0.86

of Estimate

inches

0.053

0.040

0.050

0.049

% of avg.

18

19

22

24

Daily Average*

Eva p.

(in.)

0.294

0.211

0.229

0.203

eO-«a

(in. Hg)

0.324

0.295

0.320

0.337

Wind

(mi/day)

122.4

125.0

126.0

125.8

* For number of days shown in second column.

133
data are seldom available and, when this element is 

observed, pan evaporation is naturally observed as 

well. Nevertheless, such a relation is of value in panto-pan and pan-to-lake studies, as is demonstrated 

subsequently, and it is reasonably applicable for extended periods (e. g., monthly) if the air temperature 

is assumed to equal the water temperature in the 

pan (Linsley, et a/., 1949).

Energy-Balance Approach

Pan-water temperature obviously depends on a 

number of factors which affect the heat exchange 

between the sun, atmosphere, earth, and pan. From 

an energy approach, water temperature should correlate highly with air temperature, solar radiation, 

evaporation, and wind, all of which are readily 

observed. Thus, by assuming numerous functional relations, application of least-squares analysis should 

facilitate selection of a reasonably reliable regression. The required functional relation was known to 

be complex and of a form unsuited to least-squares 

analysis with water temperature as the dependent 

variable. Accordingly, resort was made to an approximate equation of energy balance to derive a 

suitable regression relation.

The concept of combining the aerodynamic and 

energy approaches for treatment of the evaporation 

problem was used by Jacobs (1942) in considering 

ocean evaporation, and by Penman (1948) for small 

water bodies. Basically, the two equations involving 

water temperature and evaporation are combined 

to eliminate the need for water-temperature observations. Assuming the change in heat storage of the 

water body to be negligible, Penman derived the 

equation

E 1

A + y

(Qn A + yEa) (106)

where A is the slope of the saturation-vapor-pressure 

vs temperature curve (des/dJ) at the air temperature 

Ta; Ea is the evaporation given by equation (104) 

or (105), assuming water temperature equal to air 

temperature; Qn is the net radiant energy expressed 

in the same units as those of E; and y is defined by 

the equation

R = (107)

in which R is Bowen's dimensionless ratio (see section 

of report on "Energy-Budget Studies"). If evaporation 

and convective transfer of sensible heat are restricted

to equivalent, identical surfaces, it can be shown that 

y = 0.010 (P/Po) inches of mercury per °F, where 

P and P0 are actual and standard atmospheric pressures, respectively.

Penman used an equation in which

Qn = f (Qs, a, C) (108)

where Q8 is short-wave radiation from sun and sky, 

Rt is the reflection coefficient for the surface, and C 

is the cloud cover in tenths.

Equations (106) and (108), with the coefficients 

presented by Penman, are admittedly not strictly applicable to a Class A pan, but assuming the form of 

equation (106) to be applicable, pan data were 

correlated with meteorological elements by the coaxial, graphical technique (Linsley et a/., 1949) illustrated in figure 96. First assuming the standard value 

(0.01) of y, curves were constructed in quadrants 

II and III (light, dashed lines) directly from equation 

(106) so that the entering axis of II represents Qn A 

and the final axis of III is E. Since A is a function of 

Ta only, and this factor in conjunction with solar 

radiation Qs constitutes a reasonably reliable index 

of Qn, the data were plotted in quadrant I to define 

the relation between Qn A, Q8, and Ta. That is, the 

chart sequence was entered in reverse order with 

E, Ta, and Ea [thus solving Qn A = EA + y (E Ea)] ; 

values of Qn A were plotted against Ta in quadrant I; 

the points were labeled with values of Qs; and a family of curves (light, dashed lines) was drawn to fit the 

plotted data. Values of Ea are those obtained from 

the inset relation in the upper left-hand quadrant 

which, incidentally, is the Class A relation of figure 

95 with Ta substituted for TOSince evaporation can occur only from the water 

surface while convective transfer of sensible heat 

takes place at the sides and bottom of the Class A 

pan as well, y would be expected to have a value 

greater than that originally assumed (0.01). Accordingly, values of Qn A were computed from quadrant I 

for each day and, using these data, the values of y 

required to satisfy equation (106) were determined. 

The derived values of y appeared to be independent 

of wind, averaging about 0.025. Curves in quadrants 

II and III were then revised to make y = 0.025 

(heavy, solid curves); data were again plotted in 

quadrant I; and the radiation curves were revised for 

the larger value of y (heavy, solid curves). The final 

relation, based on data for 246 days without rainfall, 

has a correlation index of 0.96, and the standard 

error of estimate is 0.038 in. (13 per cent of the

134
NOTE: The solid lines fetcept for the upper 

left-hand quadrant) art based on V -0.015 

the light dashed lines on "r *O.OIO.

0.20 0.30 0.40 O.SO 

DAILY PAN EVAPORATION IN INCHES

Figure 96. Relation for Class A pan at south station, Lake 

Hefner, using energy balance approach.

average evaporation for those days used in the 

analysis).

How well the relation will fit the data for other 

localities is not yet known, but data from eight firstorder Weather Bureau stations equipped with Class 

A pans are being assembled for this purpose. Admittedly, y is a function of pressure; and vapor 

pressure and cloud cover (equation 108) are neglected so far as Qn is concerned. However, it is believed that these omissions will not materially affect 

the general applicability of the relation for low-level 

stations. In mountainous regions, y will undoubtedly 

require adjustment for reduced pressure.

Figure 97, which is based on evaporation from 

the BPI pan, was derived in the same manner as 

figure 96. As would be expected, the value of y for 

this pan is less than for the Class A pan (0.015 as 

compared to 0.025). It should be realized, however, 

that the value of y, among other things, depends 

to some extent upon the type of material used in 

fabricating the pan because of the resulting effect 

on conduction.

The relation of figure 97, based on 288 days of 

record, has a correlation index of 0.86 and a standard error of estimate of 0.049 in. While the energybalance approach yields a higher correlation for the

135
0.004 0.006 0.012 0.016 0.020 0.02<J 0.028

Figure 97. Relation for BPI pan at south station, Lake Hefner, using energy balance approach.

Class A pan than was obtained when using water 

temperature, the correlation index for the BPI pan 

dropped from 0.91 to 0.86. Realizing that the reduced correlation index was at least partially the 

result of heat storage in the relatively deeper, sunken 

BPI pan, the errors in computed daily evaporation 

were plotted against the changes in pan-watersurface temperature as shown in figure 98. The degree of correlation indicates that changes in heat 

storage cannot be assumed zero for the BPI pan 

when considering daily evaporation.

There is little doubt that similar relations could 

be developed for other types of pans. Moreover, 

it may be possible to apply a similar approach directly to a lake if some means can be found for 

extrapolating changes in heat storage in a fashion 

somewhat like that employed for water storage in 

streamflow routing.

LAKE EVAPORATION AND 

METEOROLOGICAL FACTORS

In the previous section it was shown that daily 

pan evaporation could be estimated from water 

temperature, vapor pressure of the air, and wind 

movement. It was visualized that if a similar relation 

could be developed for lake evaporation, the two 

could be compared for the purpose of analyzing 

the differences between pan and lake. Evaluation of 

similar equations derived by Rohwer (1931), Meyer 

(1942), Folse (1929), Hickox (1946), Norton (1917), 

and Sleight (1927) from less detailed information 

would also become possible.

With these objectives in mind, correlations were 

made with several combinations of data and assumed 

wind functions. The results of this analysis are presented in table 27.

136
The regression constants yield evaporation in 

inches per day with u in miles per day and vapor 

pressures in inches of mercury. Water temperature 

and 4-meter wind at the barge station were used 

in all six correlations, and dewpoint was taken at the 

2-meter level for the barge or upwind station as indicated. Since the rate of evaporation at any instant 

is proportional to the corresponding instantaneous 

value of (eo ea) f(u), best results should be expected when daily evaporation is related to the 

average value of (e0 en ) f(u) rather than daily 

averages of (eo en) and u. Accordingly, 3-hourly 

means of (eo e<J and (eo ea) u were averaged 

for the day and used in solving for regression constants. To determine the relative accuracy of the 

shorter method, the data used for line 5 of table 27 

were reanalyzed using daily means of TO (to obtain 

GO)» ea, and u. The resulting values of a and b were 

0.064 and 0.00250, respectively, and the correlation 

index was 0.952. Thus, it is seen that the simpler 

approach is essentially as reliable, although the regression constants are slightly different. This conclusion cannot be considered necessarily of general 

application, however, since it is dependent upon the 

amplitude of diurnal fluctuations in (eo ea) and u, 

as well as any difference in phase.

The selection of 2-meter dewpoint and 4-meter 

wind in the correlations of table 27 may appear 

somewhat arbitrary. It was believed that the 2-meter 

dewpoint (lowest level of observation) should constitute a reliable index to the vapor-pressure gradient 

near the water surface. Moreover, dewpoint observations taken in a standard Weather Bureau shelter 

(ground exposure) are at approximately the 2-meter 

level. The barge relations were developed first and, 

since it seemed possible that the barge itself might 

affect the wind observed at the 2-meter level, it 

appeared advisable to use the 4-meter wind. Actually, the levels selected are rather immaterial, 

since the resulting equations can readily be converted through analysis of the observations at different levels.

It will be noted that some correlations were made 

in which the coefficient a was assumed to be zero 

(lines 3 and 6 of the table), and others in which it 

was assumed to have a value other than zero. The 

latter are in the form of equation (104), while the 

former correspond to equation (105) with n equal 

to unity. Statistically, the derived values of a are not 

significantly different from zero and, therefore, may 

partially result from the tendency for imperfect correlation to reduce the slope of regression. Correlao

at

g 0.10

1o

i 0.05

=>

S -ro

1Ua -

o

10

| 0.05

z

z

o

z

0 0.15

QC

n sn

,

 77i

.

'

 T

. . /

. . 4

**,'<

 ..«***

. f «fl

%f \

!

.

»

L

^ /

i* 

r.«> 

^ ^: * 

,. » .

C : r ; >

'.-:/x .

, : ' :

 t

. . . .

i

 . 

*

15 10 5 0 + 5 10

CHANGE IN MINIMUM WATER TEMPERATURE FROM YESTERDAY IN F '

Figure 98. The effect of heat storage in the BPI pan.

tions using the same data as for line 2, assuming 

various values of n, and solving for corresponding 

values of c in equation (105), did not yield a correlation index greater than that obtained for equation 

(104). Thus, the Lake Hefner data provide little to 

support preference for either equation.

By assuming equilibrium conditions and substituting reasonable values in the standard diffusion

TABLE 27. Equations for Lake Evaporation, E = (eo ea) (a -f- bu)

15

1

2

3

4

5

6

Dayt

of

Record

141

115

115

141

115

115

Period

of

Rocord

Apr 1950-Aug 1951

Sop 1950-Aug 1951

S«p 1950-Aug 1951

Apr 1950-Aug 1951

Sep 1950-Aug 1951

Sep 1950-Aug 1951

Standard Error

Dewpoint

Station

Barge

Barge

Barge

Upwind

Upwind

Upwind

a

0.069

0.051

0.00*

0.082

0.068

0.00*

b

.00268

.00287

.00304

.00234

.00246

.00270

Correlation

Index

0.942

0.957

0.956

0.941

0.954

0.952

of Ettimi

Inehei %

0.039

0.035

0.035

0.039

0.036

0.037

ite

Axg.

22

20

20

22

21

22

' Auumed to be zero.

. 30Z5EO O - 54 - 11 137
equation, it can be shown that evaporation should 

essentially cease when there is no wind (see section 

on "Mass-Transfer Studies"). Nevertheless, the possibility that a is significantly greater than zero cannot be dismissed by showing that evaporation from 

a large water body is insignificant under equilibrium 

conditions when u = 0. Other facts to be considered 

are:

1. The anemometers in use, although probably 

the best commercially available, have a starting 

speed of approximately 1.5 mph in laboratory tests. 

Thus, on days with light winds the true speed may 

be greater than observed a systematic error which 

tends to increase the statistically derived value of a.

2. Because of insufficient ventilation, the true 

wet-bulb temperature is lower than the observed 

during relatively calm periods by an amount depending upon wind speed (see fig. 10 of reference 

1 and section on "Instrumentation"). Since the error 

is negligible for wind speeds in excess of 0.5 mph, 

this effect is probably not appreciable, but would 

tend to increase the constant a.

3. During daylight hours when solar radiation 

is heating the water, evaporation into still air would 

occur primarily not by diffusion through a deep layer 

of air, but by natural convection created by (a) 

the temperature difference between the water and 

the air, and (b) the difference in the densities of the 

evaporated water vapor and the surrounding air.

It should be pointed out that the derived constant a is much larger for the pans (except the 

screened) than for the lake. This is to be expected 

because of the lip effect and because items (1) and 

(3) above become much more important in the case 

of the pans. With respect to item (1), the pan anemometer is at a lower level where the velocities are 

smaller, and with respect to item (3), some natural 

convection can occur directly over a pan without 

registering on the anemometer.

Referring again to table 27, it should be noted 

that the data are identical for correlations (1) and 

(2), and for correlations (4) and (5), except for the 

record periods. Analyses were first made using all 

days with A water-budget data. From examination 

of the resultant errors, it appeared that the records 

of surface-water temperature of the first few months 

were not as reliable as those of the later months 

and, accordingly, the regression equations were recomputed using only the last full year of data. The 

correlation indexes based on the last full year of 

record are significantly higher than those for the 

total period.

Correlations (4), (5), and (6) are equivalent to 

(1), (2), and (3), respectively, except that the dewpoint of the upwind station was substituted for that 

at the barge station. To the extent that evaporation 

at the barge can be considered to equal the average 

over the lake, the first three regressions represent 

point evaporation formulas. The three based on upwind dewpoint are in every sense geared to a lake 

the size of Hefner, and the regression constants may 

therefore require adjustment if the lake dimensions 

change appreciably.

Conclusions expressed in the literature vary considerably as to the effect of lake size, or downwind 

dimension, on the average rate of evaporation per 

unit area. Rohwer (1931) concluded that size has 

no appreciable effect for diameters in excess of 12 

feet; Hickox (1946) derived an equation in which 

evaporation varies as the 0.25 power of diameter; 

Albertson, in a discussion of Hickox's paper, stresses 

the effect of assumed boundary conditions; and 

Sutton's equation (Anderson, Anderson, and Marciano, 1950) gives 0.11 power of diameter when 

the lapse rate is adiabatic. Data reported by Young 

(1947) on sunken pans follow 0.11 power for 

diameters of 2 to 12 feet, while for similar data 

collected by Sleight (1917) the power is more nearly 

 0.14. Figure 99 shows the data of Young, Sleight, 

and Rohwer, as well as several points for lakes. In 

those cases where 12-foot pan data were not available for comparison with lake data (such as the 

Hefner case), the points are based on data from 

other pans as explained in the notes.

Figure 99 indicates (within the range of data) 

that the evaporation from the relatively large lake 

is approximately equivalent to that from the 12-foot 

sunken pan. Moreover, examination of the figure will 

show that the only manner in which lake size could 

have any appreciable effect would require that evaporation from a 12-foot natural lake be considerably 

greater than that from the 12-foot pan. Since pans 

in general overestimate lake evaporation because 

of lip effects and heat conduction characteristics, it 

seems reasonable to conclude that lake size (within 

the range of interest) has no appreciable effect on 

the rate of evaporation. It should be noted, however, 

that this conclusion does not agree with fig. 44, based 

on Millar's (1937) theory.

If it is assumed that the rate of evaporation is 

essentially independent of lake size within the range 

of interest, the upwind regression for Lake Hefner 

may be reasonably reliable for other lakes providing 

representative observations of water temperature,

138
4-meter wind, and 2-meter dewpoint are available. 

Moreover, there is frequently a high degree of correlation of dewpoint and wind as observed at different sites within a limited area, so that a relatively 

small number of observations of these elements at 

the lake, when related to data from a nearby meteorological station, may be sufficient for estimating 

average annual or monthly evaporation. The reliability of such an approach can, of course, not be 

determined until verification data are available for 

numerous additional lakes. As a matter of interest, 

mean daily Lake Hefner observations were correlated 

with data for Will Rogers Airport (WBAS) with the 

following results:

4-meter barge wind = 0.82 WBAS wind; 

2-meter upwind vapor pressure = 0.96 WBAS vapor

pressure.

The correlation coefficient for the wind relation is 

0.935, and that for vapor pressure is 0.994; the 

standard error of estimate for the wind relation is 

1.8 mph, and for vapor pressure 0.025 in. of mercury. Both analyses were based on 479 days of 

record.

It will be noted that equation (58) in the "Mass 

Transfer" section is similar in form to the empirical 

equations shown in lines (3) and (6) of table 27. The 

question naturally arises as to how nearly these 

equations agree when reduced to the same units 

and levels of observation. To facilitate such comparison, the following relations were derived from 

approximately 40 days of data selected at random: 

(e0 ea) for 2-meter level = 0.91 (e0 ea) for 

8-meter level, and

u4 = 0.90 u8.

Using these equations and converting to proper units, 

the value of b in line (3) of table 27 becomes 

6.5 X 10~4 compared to the value of 6.25 X 10~4 

given in equation (58). This converted value of 

6.5 X 10~ 4 .is based on the last 12 months of 

record. Basing the computations on the full period 

of record (b = 0.00292) yields a converted value of 

6.2 X 10- 4 .

PAN VS LAKE EVAPORATION

The practice of applying a coefficient to pan 

evaporation to estimate that occurring from an adjacent lake is of long standing, even though the reliability of this approach has always been subject to 

question. Coefficients derived and recommended by 

various investigators are summarized in table 24,

and results for the Lake Hefner experiment are presented in table 28.

There are several ways in which pan-to-lake 

coefficients can be computed for an experiment such 

as that at Lake Hefner, all of which merit consideration. The first approach is to use all observed data; 

second, to use only those data which are considered 

reliable; and third, to adjust all data which are 

believed to be in error.

If the errors in observed daily evaporation were 

randomly distributed, and if each weather event had 

an independent effect on evaporation, then all three 

approaches should give reliable and comparable 

results. However, this is not the case with respect 

to lake and pan evaporation. For example, the errors 

in pan evaporation brought about by water splashed 

out during periods of rain and high wind are always 

in the same direction. Similarly, during periods of 

heavy snowfall accompanied by high wind, the 

precipitation gage underestimates the amount of 

snowfall and this, in conjunction with the blowing 

snow trapped by the lake, results in an underestimate 

of lake evaporation as computed by the water 

budget. In the case of a pan, the effect of a particular weather event on evaporation is relatively 

independent of antecedent weather, largely because 

of its limited heat capacity. In the case of the lake, 

however, the evaporation occurring on one day is not 

independent of antecedent weather because the temperature of the lake is a function of the energy 

exchange over a considerable period of time. This 

carry-over effect results from the larger heat capacity 

of the lake. The large difference in heat capacities of 

the pan and lake produces marked variations in the 

daily pan coefficients during periods of changing 

weather. The elimination of questionable data would 

probably yield reliable annual coefficients except for 

the fact that the correlation between weather events 

and reliability of evaporation data produced biased 

results. In view of these facts, it would appear that 

pan coefficients should be based on continuous 

records adjusted where necessary for apparent discrepancies which would create a bias. Accordingly 

daily pan and lake evaporation were computed by 

applying the empirical relations described previously 

in this section. The computed values were substituted 

for observed data when the differences were large, 

provided the source of the errors could be accounted for.

Examination of the Lake Hefner pan evaporation 

data indicates that unreasonably high evaporation 

was frequently observed for the Class A pans during

139
TABLE 28. Summary of Monthly Pan and Lake Evaporation and Monthly and Annual Pan-to-Lake Coefficients.

(Lake Hefner Experiment)

Evaporation (inches]

Class A Pan

Mo. Year

May 1950

Jun

Jul

Aug

Sep

Oct

Nov

Dec

Jan 1951

Feb

Mar

Apr

May

Jun

Jul

Aug

Annual Totals

Jun-May

Jul-Jun

Aug-Jul

Sep-Aug

NE Sta.

No. 1

9.04

10.72

7.77

8.42

6.13

7.32

4.53

2.56

3.06

3.34

6.48

9.02

8.66

9.06

11.63

12.58

NE Sta.

No. 2

8.10

10.28

7.61

8.05

5.97

7.11

4.36

2.45

3.04

3.22

6.26

8.66

8.13

8.70

11.05

12.11

SW S

Sta. Sta. Avg**

8.53 9.13 8.90

10.24 10.98 10.65

7.52 7.54 7.61

8.18 8.41 8.34

6.1 1 5.96 6.07

7.00 7.32 7.21

4.49 4.56 4.53

2.77 2.70 2.68

3.36 3.34 3.25

3.39 3.38 3.37

6.98 6.64 6.70

9.05 8.72 8.93

8.19 8.37 8.41

8.74 9.37 9.06

10.55 11.29 11.16

12.09 12.25 12.31

77.92 77.75

76.31 76.16

80.06 79.71

83.90 83.68

BPI

8.38

5.64

6.62

4.76

5.38

3.83

2.08

2.61

2.02

5.13

6.44

6.18

6.95

8.45

9.18

59.07

57.64

60.45

63.01

*

Colorado

6.60

8.56

5.93

6.65

4.81

5.94

4.16

2.21

3.24

2.15

5.72

7.53

7.05

7.87

9.39

10.08

63.95

63.26

66.72

70.15

Screened

5.66

7.08

5.00

5.19

4.32

5.51

4.59

2.81

3.15

2.31

5.67

6.87

6.13

6.43

7.88

8.90

58.63

57.98

60.86

64.57

Lake

3.14t

6.35

5.43

6.83

5.65

6.51

5.99

2.84

2.47

0.44

3.39

3.44

4.40 1

5.87

7.18

8.83

53.75

53.27

55.01

57.01

Pan Coefficients

Class A Pan

S

Sta. Avg**

0.34 0.35

0.58 0.60

0.72 0.71

0.81 0.82

0.95 0.93

0.89 0.90

1.31 1.32

1.05 1.06

0.74 0.76

0.13 0.13

0.51 0.51

0.39 0.39

0.53 0.52

0.63 0.65

0.64 0.64

0.72 0.72

0.69 0.69

0.70 0.70

0.69 0.69

0.68 0.68

BPI

0.76

0.96

1.03

1.19

1.21

1.56

1.37

0.95

0.22

0.66

0.53

0.71

0.84

0.85

0.96

0.91

0.92

0.91

0.90

Colorado

0.48

0.74

0.92

1.03

1.17

1.10

1.44

1.29

0.76

0.20

0.59

0.46

0.62

0.75

0.76

0.88

0.84

0.84

0.82

0.81

Screened

0.55

0.90

.09

.32

.31

.18

.31

.01

0.78

0.19

0.60

0.50

0.72

0.91

0.91

0.99

0.92

0.92

0.90

0.88

Evaporation amounts and coefficients based on observed data, eliminating all days (approx. 14%) requiring 

Annual Totals adjustment for either pan or lake evaporation.

Jun-May

Jul-Jun

Aug-Jul

Sep-Aug

68.92

65.64

70.55

74.21

68.50

65.30

70.09

73.82

51.66

49.11

52.49

54.90

56.23

54.17

58.23

61.33

51.58

50.10

53.47

56.78

47.95

46.82

48.96

50.34

0.70

0.71

0.69

0.68

0.70

0.72

0.70

0.68

0.93

0.95

0.93

0.92

0.85

0.86

0.84

0.82

0.93

0.93

0.92

0.89

Annual Totals Evaporation amounts and coefficients based on observed water-budget and pan data for full period.

Jun-May

Jul-Jun

Aug-Jul

Sep-Aug

80.59

79.48

83.61

87.65

81.01

79.76

83.46

87.76

58.81

57.74

60.86

63.58

64.36

64.00

67.73

71.20

58.40

57.52

60.29

64.03

52.67

52.19

52.87

54.31

0.65

0.66

0.63

0.62

0.65

0.65

0.63

0.62

0.90

0.90

0.87

0.85

0.82

0.82

0.78

0.76

0.90

0.91

0.88

0.85

Partial Yr. Totals Evaporation amounts and coefficients -based on adjusted water-budget and pan data.

Jun-Oct

Nov-May

May-Oct

40.21

37.71

49.34

39.88

37.87

48.78

30.78

28.29

31.89

32.06

38.49

27.10

31.53

32.76

30.77

22.97

33.91

0.76

0.61

0.69

0.77

0.61

0.70

1.00

0.81

0.96

0.72

0.88

1.14

0.73

1.04

* Except as noted otherwise, the pan evaporation listed in this table has been corrected for splash-out, and the lake evaporation has

been adjusted during periods of high inflow or blowing snow as explained in the text. 

** Average of all Class A pans except NE No. 2, which was installed for comparative purposes, 

t Records missing for 7 days. Monthly total obtained by adding computed values for missing days and adjusting "D" days adjacent

to the missing days. 

t Computed value used for one day of missing record. Further adjustments given in text.

140
periods of rain, particularly when accompanied by 

high winds. Raindrop splash and wave action, as 

well as pan overflow, are the probable causes. Accordingly, evaporation was computed for all pans 

from relations of the type shown in figure 95 for 

each storm period. The computed values were then 

compared with the observed data and substituted 

therefor when discrepancies were apparent. All 

monthly data in table 28 were adjusted in this 

manner.

The total observed and adjusted evaporations in

inches at the south station for the 16-month period 

are as follows:

Observed Adjusted

Class A ............124.48 119.96

BPI ................ 83.92 83.66

Colorado ........... 99.18 97.87

Screened ........... 86.34 87.50

Records for the Class A pan required adjustment on 

48 days; the adjustment was in excess of 0.20 in. on 

nine days; and for one exceptionally windy day when 

the pan was nearly full, the adjustment was 0.47. 

Apparently, about as much water (and silt) splashed 

into the sunken pans by raindrop impact as was 

splashed out. Since only the screened pan required 

an upward adjustment, it is possible that the screen 

reduced the splash-out of rain falling into the pan, 

but has essentially no effect on splash from the 

ground surface into the pan.

The computed lake evaporation was substituted 

for the observed only for those few periods when the 

reliability of the water budget could be questioned 

because of (1) heavy precipitation accompanied by 

considerable local runoff, (2) high inflow from the 

intake canal, and (3) heavy snowfall with high winds. 

Most of those water-budget days which were designated as C or D were so classfied because of the 

error in determining mean daily change in lake level 

brought about by variations in the wind pattern over 

the lake. Generally speaking, errors of this type are 

compensating and, therefore, need not be considered 

when evaluating monthly and annual evaporation. 

This is not necessarily true when such days precede 

or follow a period for which adjustments are made, 

since the compensating day (or days) may fall within 

the adjusted period. In each case, adjustments were 

made for periods between days classified as A or B. 

Following is a list of the periods for which adjustments were made and the reasons for the apparent 

errors:

Period

7-14 May 1950 

17-19 May 1950 

17-21 July 1950

24-26 July 1950 

18-20 August 1950 

12-16 February 1951 

19-21 February 1951 

29 April-1 May 1951 

8-11 May 1951 

17-22 May 1951 

27 May 1951

through intake 

through intake

Remarks

Missing water-budget data. 

Missing water-budget data. 

High inflow through intake 

canal.

High inflow 

canal.

High inflow 

canal.

Snow accompanied by high 

wind and low temperature. 

High inflow through intake 

canal and heavy rainfall. 

Heavy rainfall culvert weirs 

overtopped.

Heavy rainfall and high inflow 

through intake canal. 

High inflow through intake 

canal and heavy local runoff. 

Missing water-budget data.

Comparison of the adjusted monthly lake evaporation values in table 28 with the observed waterbudget amounts listed in table 1 shows substantial 

differences for individual months, although the 16- 

month totals agree reasonably well (78.76 in. for the 

adjusted data, and 78.30 in. for the observed waterbudget data). The adjusted value of 0.44 in. for 

February 1951 is of particular interest, since the 

water budget shows a negative evaporation (condensation) of 0.638 in. for the month. Thus, the 

adjustments for two brief periods (12-16 and 19-21 

February) account for a change of 1.08 in. in the 

month's total. Since the total evaporation for the 

remaining 20 days of the month computed from the 

lake relation is within 0.1 in. of the observed waterbudget data, it is apparent that there is no pronounced bias during this season of the year.

All monthly data and coefficients in table 28 

are based on adjusted pan and lake evaporation as 

described in the preceding paragraphs. The moving 

annual coefficients immediately below the monthly 

data and the seasonal coefficients at the bottom of 

the table are also based on adjusted pan and lake 

evaporation. The other two sets of annual coefficients, 

based on (1) only those days which did not require 

pan or lake adjustment, and (2) observed pan and 

lake data for the full period, are given in order that 

a comparison may be made of the three approaches. 

It will be noted that the first two sets of annual 

coefficients agree rather closely. However, this is not 

the case on a monthly basis; for example, the ad141
justed data give a June Class A coefficient of 0.65, 

whereas the selected data (23 days) give a value 

of 0.76.

Table 28 shows that the range in monthly coefficients is greater for all Lake Hefner pans than is the 

case for Lake Elsinore, Lake Okeechobee, and the 

85-ft reservoir at Ft. Collins, Colo. This great range 

is undoubtedly caused partially by the sequence of 

weather during the Lake Hefner experiment. Nevertheless, there is little doubt that pan coefficients for 

Lake Hefner are normally much higher in late fall 

than in early spring because of the lag between pan 

(or air) and lake temperature. The limited data available for relatively few lakes indicate that the range 

is greatest for deep lakes in localities with large 

annual range in temperature.

Examination of tables 24 and 28 reveals variation in annual and warm-season coefficients of appreciable magnitude. The obvious question arises as to 

whether or not this variation is indicative of the reliability of the approach. There is, of course, the 

possibility that the variations can be explained and 

used to advantage. Among the possible causes of the 

indicated variations are:

1. Errors in lake evaporation (water budget).

2. Variations in observational practice at pan 

stations.

3. Variations from standard enclosure.

4. Variations in pan exposure, particularly with 

respect to the lake.

5. Differences in lake size and depth.

6. Geographical variations in climate.

7. Difference between part-year and annual coefficients.

Sufficient, reliable pan-vs-lake evaporation data 

should provide a means of evaluating these various 

factors, although in the case of climatic variations it 

may be necessary to resort to geographical differentiation. To be sure, little would be gained by the 

coefficient approach if the required water-budget 

data were sufficient of themselves to estimate evaporation reliably from all other existing and proposed 

lakes, but this is believed not to be the case. For 

example, if the Class A coefficient were assumed to 

be 0.70 for those lakes shown in tables 24 and 28, 

the extreme indicated error in mean annual lake 

evaporation would be about 12 per cent and, if this 

could be reduced by half through consideration of 

one or more of the factors listed, the coefficient approach would undoubtedly provide the required 

accuracy in estimating mean annual evaporation 

from existing and proposed lakes and reservoirs.

Discussing the seven items in the order listed, 

the accuracy of water-budget data from natural lakes 

is often in question because the magnitude of inflow 

and outflow seepage is not known. Although the 

water-budget data for Lake Hefner are exceptionally 

accurate, the data represented in table 24 were 

derived in a somewhat less rigorous manner. It is 

obvious, therefore, that a considerable portion of 

the variation in pan coefficients may be the result of 

incorrect lake evaporation. A detailed discussion of 

the effects of variations in observational practice at 

pan stations is not presented here, since they have 

been described in several previous publications 

(Hickox, 1946; Rohwer, 1931; Young, 1947; "Evaporation from Water Surfaces," 1934).

The effect of variations from standard exposure 

of the pan and related instruments is amply shown 

by the differences in evaporation and wind observed 

at the two northeast Lake Hefner stations. Quite the 

opposite of what might be expected, the site with 

least obstructions consistently recorded lower wind 

movement at pan height and, as a consequence, less 

evaporation by 4.3 per cent. Anemometers were interchanged to check possible instrumental deficiencies 

and observations made upwind from the enclosures 

showed no appreciable difference between the sites. 

Apparently, the obstructions at the site with standard 

layout create turbulence with a net transport of 

momentum downward to anemometer height.

Wind is probably the most important factor to 

be considered in selecting a pan site representative of 

a particular lake, since it has an appreciable effect 

upon pan evaporation, and it can be extremely 

variable over a relatively small area. If wind at the 

pan site is appreciably less than over the comparative lake, the true pan coefficient would necessarily 

be relatively high.

Figure 99 indicates that lake area has little 

effect on the rate of evaporation (depth per unit of 

time), although the data are admittedly insufficient 

to be conclusive. Depth of water, on the other hand, 

quite definitely influences the monthly coefficients 

the greater the depth, the greater the range. Thus, 

part-year, or seasonal, coefficients must be related to 

lake depth. Although mean annual coefficients are 

not greatly affected by depth, it appears that they 

decrease as depth increases and as the annual air 

temperature range increases.

There is, of course, the possibility that the pan 

coefficients vary from region to region because of 

variations in climatic factors and this is frequently 

advanced as a reason why pan coefficients cannot be

142
85-ft dram, reservoir Fort Collins, Colorado (Rohwer, Ref. 14) 

Colorado pan 26.6% greater than 85-ft reservoir 

Class A Pan'42.7% greater than 85-ft reservoir 

Colorado pan 25.9% greater than 12-ft pan « Sleight as quoted 

Class A pan 42.1% greater than 12-ft pan « Sleight as quoted 

Ratio 85-ft reservoir to 12-ft pan based on Colorado, 0.994 

Ratio 85-ft reservoir to 12-ft pan based on Class A, 0.996 

Mean = 0.995

Lake 

Ratio 

Ratio 

Ratio 

Ratio 

Ratio 

Ratio 

Ratio 

Ratio 

Ratio

by Rohwer 

by Rohwer

Hefner, Oklahoma 

Lake to Class A pan = 0.69 

Lake to BPI pan = 0.91 

Lake to Colorado pan = 0.83 

Class A to 12-ft pan = 1.421; Sleight (Ref. 15} 

BPI to 12-ft pan = 1.066; Sleight 

Colorado to 12-ft pan = 1.259; Sleight 

Lake to 12-ft pan based on Class A, 0.980 

Lake to 12-ft pan based on BPI, 0.970 

Lake to 12-ft pan based on Colorado, 1.045 

Mean = 0.998

Lake Elsinore, Calif., by Young (Ref. 21)

Period 1939-1941: Lake evap. = 56.24 in.; Class A 

Fullerton experiments Ratio 12-ft pan to Screened 

Fullerton experiments Ratio 12-ft pan to Class A. = 

Ratio Lake to 12-ft pan based on Class A, 1.003 

Ratio Lake to 12-ft pan based on Screened, 1.007 

Mean = 1.005

§ 2.00 

5 1.80 

oe 1.60, 

O 1 .40

"; 1.20

< 1.00

8 8°

£ - 6° 

in 

Z 40

2

ee 

O

IU

^ .20

u.

O 

O

i <>

\ \ \

V^_^_

 

L_ x_ ^^ c

 .H

N*"""T *^x

-X.

**

/ -

\

\

Vx

^^** ^^i^

LEGEND

o Ref. 21 

Circular sunken pans 3-ft deep

X Ref. 15 

Circular sunken pans 3 ft deep

^^

^%.

Tneor

 *-*

etica

4

cu ^ 'or d;0ft

~^r t7^-c

y

\

\

\

 ^^

\

'

Buena 

Period 

Ratio 1

Ratio L

s~

\ J

~*~~^

Vista Lake, 

Jul-Nov: Ra 

2-ft pan to 

ake to 12-ft

[ ^^

Calif. - 

tio Lak 

sunken 

pan, (

 ^,

-Har 

e to 

3 ft 

.921

=

d

s

1 2 4 6 8 10 20 40 60 80100 200 400 600 800(000 2QOO 40QO 600o800

DIAMETER OF PAN OR LAKE - FEET

Figure 99. The effect of pan or lake size on rate of evaporation.

6*
relied upon. If, however, the effect is not pronounced 

 and the rather minor variations observed in pan 

coefficients indicate this to be the case rather 

limited pan-vs-lake data should provide the means 

for showing the variation geographically by isolines 

on a map.

Finally, it should be pointed out that coefficients 

based on part-year data are apt to be quite erratic 

because of the seasonal variation brought about by 

heat storage in the lake. That they should not be 

used as annual coefficients is demonstrated by the 

last three lines of table 28. Except for the close 

agreement of the annual and the May-October coefficient for the Class A pan, the part-year coefficients 

for all pans are considerably different from the 

annual values.

The preceding discussion has been directed 

toward evaluation of the pan-coefficient approach to 

estimating annual or mean annual lake evaporation. 

Observations demonstrate conclusively that the coefficients vary throughout the year and that the mean 

annual coefficient cannot be used to estimate monthly 

lake evaporation. To provide a ready basis for distributing annual evaporation computed from annual 

pan coefficients, the Class A equation in table 26 

and the fifth equation in table 27 were combined 

as shown in figure 100. Since the wind data used in 

developing the two relations were different, it was 

necessary to relate wind at the pan to that at the 

barge (4-meter level). It will be seen that while the 

pan coefficient is a nonlinear function of wind, the 

effect is not great within the range of winds normally 

experienced, particularly for periods as long as a 

month. It should be pointed out that the relation of 

figure 100 is based on daily data and, because of 

the nonlinear effect of wind, application to monthly 

average data will result in an overestimate of the 

annual evaporation.

It is evident from table 28 that the coefficient 

for a particular month may vary considerably from 

year to year, even though the annual coefficient is 

reasonably stable. Variations in weather, coupled 

with the large heat capacity of the lake, can produce 

an appreciable seasonal shift in the monthly coefficients. Consequently, the monthly coefficients derived 

for the sixteen months of record at Lake Hefner may 

actually deviate considerably from the normal values. 

Moreover, even if normal coefficients were available 

for Lake Hefner, they would not necessarily be applicable to other lakes. To provide a means for distributing estimated annual evaporation, figure 101 

was developed assuming that the seasonal march in

coefficients is wholly the result of temperature lag 

between the pan and the lake. While figure 101 

is quite similar to figure 100, the data plotted were 

computed on a mean monthly basis (evaporation, 

wind, water temperature, and dewpoint). It will be 

noted that the data display a moderate seasonal 

trend, but that the effect of wind is unimportant 

within the small range of speeds observed.

Figure 101 provides a means of estimating the 

monthly pan coefficients from observed pan and lake 

water-temperatures and dewpoint. Pan water-temperatures have not normally been observed at Class 

A evaporation stations, but can be computed by 

entering a relation of the type shown in figure 95 

with observed pan evaporation, wind, and dewpoint. 

In most cases, it will be necessary to use dewpoint 

data from a nearby meteorological station. Lack of 

lake surface-water-temperature data constitutes the 

major obstacle to the application of figure 101. 

However, most approaches to the evaporation problem require surface-water-temperature data and, 

therefore, it would seem that a program for initiating 

nation-wide observations would be justified.

RELATIVE EVAPORATION FROM 

PANS OF VARIOUS TYPES

Because several types of evaporation pans are 

rather widely used in this country, numerous investigations have been conducted (Rohwer, 1931; Sleight, 

1917; Young, 1947) to determine the relative evaporation from each. The results of many of these experiments are listed in table 25 in the form of ratios. 

In addition to ratios taken from references at the end 

of this chapter, values have been derived from published and unpublished data in Weather Bureau files.

It will be noted from the footnotes that some 

of the values are based on part-year data and, in 

most cases, the record periods are short. Considering 

these facts, the ratios are amazingly consistent six 

determinations of BPI to Class A ratio in Texas and 

California average 0.797 with a maximum departure 

of 4 per cent. The three lowest values of 0.69, 0.75, 

and 0.77 are all based on part-year data, while all 

larger values are computed from full-year data. 

Whether the low ratios for Denver (0.75) and for Ft. 

Assinniboine (0.69) are entirely the result of climatic factors or, to some extent, of the use of partyear data cannot be determined with the limited 

records available.

Monthly and annual pan-to-pan ratios for the 

Lake Hefner observations are listed in table 29. It

144
Q. -n

5» «"

RATIO OF LAKE HEFNER EVAPORATION TO CLASS A PAN EVAPORATION

\

\

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3TO

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\

<4 ^

\

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e?cT

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O OL 3

og-i

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RATIO OF LAKE HEFNER EVAPORATION 

ro o ro !

o « 

-, <D _,

2 3) o

w n O c 5' '

01

5 Iff

a 3 a 

w Q o

^-n 

5" t>

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will be noted that two sets of ratios are listed; the 

first column in each block is based on data corrected 

for splash-out as described earlier in the section; the 

second column is based on incomplete data, eliminating those days when splash-out occurred. While the 

paired monthly values show minor discrepancies, the 

annual values are essentially equivalent.

Minor seasonal fluctuations may exist, but they 

are masked by other variations, except in the case 

of the screened pan. Ratios of the screened pan to 

each of the other pans show a definite seasonal 

march, with the maximum occurring in December and 

the minimum in June. Apparently, the screen affects 

the exchange of radiant energy (and convective 

transfer). It might be expected that the Class A pan 

with its vertical sides exposed to a maximum of 

winter solar radiation would display the opposite 

trend, but this is not borne out by the data for Lake 

Hefner.

Relations of the type shown in figures 96 and 

97 should be of considerable value in studying the 

relative evaporation from different .types of pans 

under specified conditions, particularly if they can 

be shown to be generally applicable. For example, 

the difference in y (pages 134 and 135) for the Class 

A and the BPI pans apparently indicates that climatic 

variations in wind would result in corresponding 

variations in the pan-to-pan ratio. However, taking 

an air temperature of 80°F, a dewpoint of 50°F, and 

radiation of 500 cal cm"" 2 , figures 96 and 97 yield 

a Class A to BPI ratio of 1.48 for u 50 miles per 

day, and 1.46 for u = 200 miles per day, indicating 

little wind effect on the pan ratio. A rather thorough 

analysis of this type is planned for the near future.

Comparing the Lake Hefner ratios of table 29 

with the average ratios of table 25, it will be seen 

that the values are reasonably consistent. The greatest 

differences are in the ratios for the Class A to Colorado and Class A to screened. Much of the discrepancy in these values results from the determinations 

at Henshaw Reservoir and Mansfield Dam.

CONCLUSIONS ON LAKE 

AND PAN EVAPORATION

Before discussing the conclusions to be reached 

from the Lake Hefner observations and the analyses 

presented in this chapter, it seems appropriate to 

emphasize the "inconclusive" nature of a single evaporation experiment, at least with respect to the pan 

approach. The Lake Hefner experiment has undoubtedly provided the most reliable data available in 

an over-all sense, and has gone far in providing 

substantiating evidence. Nevertheless, additional similar experiments must be performed under other climatic regimes before the reliability of the empirical 

approaches described can be accurately determined. 

Nor can any conclusions be reached at this time as 

to the relative reliability of the several empirical 

approaches discussed. Possibly all should be considered if the problem at hand warrants.

The following is a summary of the more important conclusions drawn from the material in this 

section.

1. Measured pan evaporation is frequently 

greater than true pan evaporation because of splashout and blow-out during periods of heavy rain and 

high winds.

2. Differences in wind movement and pan evaporation at the two northeast stations indicate that 

obstructions can increase wind at pan height and 

thus increase the rate of evaporation by a few 

per cent.

3. Comparative observations on the four types 

of pans at Lake Hefner agree quite closely with data 

obtained from other studies (tables 25 and 29).

4. Daily pan evaporation at Lake Hefner can 

be accurately estimated from air temperature, solar 

radiation, dewpoint, and wind using an energybalance approach in conjunction with equation (104).

5. No appreciable seasonal variation was found 

in the pan-to-pan ratios for the Class A, BPI, and 

Colorado pans at Lake Hefner. The ratio of the 

screened pan to each of the other three, however, 

does display a minimum near June and a maximum 

in about December.

6. Annual pan coefficients based on Lake Hefner 

data are reasonably consistent with those derived 

from previous studies. The coefficients derived for 

Lake Hefner from adjusted pan and lake data are:

Class A .......0.69

BPI ...........0.91

Colorado ......0.83

Screened ......0.91

To indicate the magnitude of the splash-out adjustment, the coefficients were computed using observed 

pan data and adjusted lake data; the results were 

as follows: the Class A coefficient became 0.66, the 

Colorado 0.82, the BPI 0.91, and the screened 0.91.

146
TABLE 29. Pan-to-Pan Ratios (Lake Hefner Experiment)

_ ., Pan X Ratio = 

PanY

1-

c

s.

Class A

(S. Station)

BPI

(Sunken)

Colorado

(Sunken)

Screened

(Sunken)

Pan "X"

Class A

(South Station)

Jan .28 1.28

Feb .67 1.79

Mar .29 1.35

Apr .35 1.36

May .35 1.39

Mun .33 1.35

*Jul .34 1.35

*Aug .31 1.33

Sep .25 1.24

Oct .36 1 .36

Nov .19 1.26

Dec .30 1.30

Ann. 1.33 1.35

Jan 1.03 1.03

Feb 1.57 1.59

Mar 1.16 1.22

Apr 1.16 1.15

*May 1.28 1.30

*Jun 1.24 1.25

*Jul 1.24 1.24

*Aug 1.24 1.26

Sep 1.24 1.24

Oct 1.23 1.23

Nov 1.10 1.15

Dec 1.22 1.22

Ann. 1.22 1.23

Jan 1.06 1.06

Feb 1.46 1.56

Mar 1.17 1.25

Apr 1.27 1.26

*May 1.48 1.51

Mun 1.51 1.51

Mul 1.46 1.46

*Aug 1.47 1.49

Sep 1.38 1.39

Oct 1.33 1.33

Nov 0.99 1.03

Dec 0.96 0.96

Ann. 1.32 1.34

BPI

(Sunken)

Jan 0.78 0.78

Feb 0.60 0.56

Mar 0.77 0.74

Apr 0.74 0.77

*May 0.74 0.72

*Jun 0.75 0.74

*Jul 0.75 0.74

*Aug 0.76 0.75

Sep 0.80 0.80

Oct 0.73 0.73

Nov 0.84 0.79

Dec 0.77 0.77

Ann. 0.75 0.74

Jan 0.81 0.81

Feb 0.94 0.88

Mar 0.90 0.90

Apr 0.86 0.86

*May 0.88 0.87

*Jun 0.93 0.93

*Jul 0.93 0.91

*Aug 0.94 0.95

Sep 0.99 1.00

Oct 0.91 0.91

Nov 0.92 0.91

Dec 0.94 0.94

Ann. 0.91 0.91

Jan 0.83 0.83

Feb 0.87 0.87

Mar 0.90 0.93

Apr 0.94 0.93

*May .01 .00

Mun .13 .12

Mul .09 .08

*Aug .12 .13

Sep .10 .12

Sep 0.98 0.98

Nov 0.83 0.82

Dec 0.74 0.74

Ann. 0.99 0.99

Colorado

(Sunken)

Jan 0.97 0.97

Feb 0.64 0.63

Mar 0.86 0.82

Apr 0.86 0.87

*May 0.78 0.77

Mun 0.81 0.80

Mul 0.81 0.81

*Aug 0.81 0.80

Sep 0.81 0.80

Oct 0.81 0.81

Nov 0.91 0.87

Dec 0.82 0.82

Ann. 0.82 0.81

Jan .24 .24

Feb .06 .13

Mar .11 .11

Apr .17 .19

*May .14 .15

Mun .07 .08

Mul .08 .10

*Aug .06 .06

Sep .01 .00

Oct .10 .10

Nov 1.09 .10

Dec 1.06 1.06

Ann. 1.10 1.10

Jan 'l.03 1.03

Feb 0.93 0.99

Mar .01 .03

Apr .10 .10

*May .16 .16

Mun .22 .21

Mul .18 .18

*Aug .19 .19

Sep .11 .12

Sep .08 .08

Nov 0.91 0.90

Dec 0.79 0.79

Ann. 1.09 1.09

Screened

(Sunken)

Jan 0.94 0.94

Feb 0.68 0.64

Mar 0.85 0.80

Apr 0.79 0.79

*May 0.67 0.66

Mun 0.66 0.66

Mul 0.68 0.68

*Aug 0.68 0.67

Sep 0.72 0.72

Oct 0.75 0.75

Nov 1.01 0.97

Dec 1.04 1.04

Ann. 0.75 0.74

Jan 1.21 .21

Jan 1.14 .15

Mar 1.11 .08

Apr 1.07 .08

*May 0.99 .00

Mun 0.88 0.89

Mul 0.91 0.93

*Aug 0.89 0.89

Sep 0.91 0.90

Oct 1.02 1.02

Nov 1.20 1.22

Dec 1.35 1.35

Ann. 1.01 1.01

Jan 0.97 0.97

Feb 1.07 1.01

Mar 0.99 0.97

Apr 0.91 0.91

*May 0.87 0.86

Mun 0.82 0.83

Mul 0.85 0.85

*Aug 0.84 0.84

Sep 0.90 0.90

Oct 0.93 0.93

Nov 1.10 1.11

Dec 1.27 1.27

Ann. 0.92 0.92

Note: The first column in each block is based on data corrected for splash-out as described in the text; 

the second column is based on incomplete data, eliminating those days when splash-out occurred. 

'Average of two years of data.

147
7. Monthly pan coefficients display a pronounced seasonal march which is principally the result of temperature lag between the lake and pan 

brought about by the difference in heat capacity 

of the two water bodies.

8. Annual lake evaporation can probably be 

estimated within 10-15 per cent (on the average) by 

applying an annual coefficient to pan evaporation, 

provided lake depth and climatic regime are taken 

into account in selecting the coefficient. Available 

data are admittedly insufficient to evaluate properly 

the effects of lake depth and climatic regime on pan 

coefficients. However, it is hoped that current Lake 

Mead observations, and possibly those from other 

anticipated projects, will shed some light on this 

phase of the problem.

9. Although equation (105) appears to be more 

in agreement with present theory, comparative analysis indicates that equation (104) fits the Lake Hefner 

data equally well.

10. Analyses presented apparently substantiate 

the conclusions of Rohwer and others that the rate 

of lake evaporation (depth per unit of time) is not 

greatly influenced by the area of the water surface 

in the range considered. This conclusion disagrees, 

however, with the material presented in the section

on "Mass Transfer."

Although the Lake Hefner experiment has appreciably advanced our knowledge of evaporation, 

it is not yet possible to reach unalterable conclusions 

as to which of the pans tested provides the most 

reliable means of estimating lake evaporation. The 

physical characteristics of the BPI (sunken) pan seem 

to be most nearly representative of those of a natural 

body of water and, therefore, this pan merits high 

consideration on a theoretical basis. The Class A 

pan, on the other hand, is subject to convective and 

radiant heat transfer to and from the sides and 

bottom. One effect of this characteristic is to increase 

y beyond Bowen's theoretical value, and even beyond 

that for the BPI pan, as shown in figures 96 and 97. 

This dissimilarity of the Class A pan is not necessarily 

serious, provided a coefficient appropriate to the 

given geophysical conditions is used. The relatively 

few determinations which have been made to date 

certainly indicate that climatic variations in the Class 

A pan coefficient are rather minor and, if all its 

operational advantages are viewed in the light of 

the many hundreds of station-years of record now 

available, any decision to replace the Class A pan 

in favor of another will merit considerable deliberation.

REFERENCES

1. Anderson, E. R., Anderson, L. J., and Marciano, 

J. J., "A Review of Evaporation Theory and Development of Instrumentation, Lake Mead Water Loss Investigations; Interim Report," Navy Electronics Laboratory, Report no. 159, 1 February 1950.

2. Bigelow, F. H., "Studies of the Phenomena of 

Evaporation of Water Over Lakes and Reservoirs," 

Monthly Weather Review, vol. 35, 1907, pp. 311-316; 

vol. 36, 1908, pp. 24-39, 437-445; vol. 38, 1910, 

pp. 307-313.

3. Folse, J. A., "A New Method of Estimating StreamFlow, Based Upon a New Evaporation Formula," 

Carnegie Institution of Washington, Publication no. 

400, 1929.

4. Hording, S. T., "Evaporation from Large WaterSurfaces Based on Records in California and Nevada," American Geophysical Union, Transactions, 

vol. 16, Pt. 2, 1935, pp. 507-511.

5. Hickox, G. H., "Evaporation from a Free Water 

Surface," American Society of Civil Engineers, Transactions, vol. Ill, paper 2266, 1946, pp. 1-33.

6. Norton, R. E., "A New Evaporation Formula Developed," Engineering News-Record, vol. 78, April 

26, 1917, pp. 196-199.

7. Jacobs, W. C, "On the Energy Exchange Between 

Sea and Atmosphere," Journal of Marine Research, 

vol. 5, no. 1, 1942, pp. 37-66.

8. Langbein, W. B., "Research on Evaporation from 

Lake and Reservoirs," Paper presented at Brussels 

Assembly, 1951, pp. 9-11.

9. Linsley, R. K., er af.. Applied Hydrology, McGrawHill, 1949.

10. Lippincott, J. B., "Evaporation on United States 

Reclamation Projects," Discussion, American Society 

of Civil Engineers, Transactions, vol. 90, 1927, pp. 

351-363.

11. Meyer, A. F., "Evaporation from Lakes and Reservoirs," Minnesota Resources Commission, Bulletin, 

June 1942.

12. O'Neill, CM., "Evaporation on United States 

Reclamation Projects," Discussion, American Society 

of Civil Engineers, Transactions, vol. 90, 1927, pp. 

343-351.

13. Penman, H. L., "Natural Evaporation from Open 

Water, Bare Soil and Grass," Royal Society of London, Proceedings, Series A, vol. 193, no. 1032, April 

1948, pp. 120-145.

14. Rohwer, C., "Evaporation From Free Water Surfaces," U. S. Deportment of Agriculture, Technical 

Bulletin no. 271, December 1931.

15. Sleight, R. B., "Evaporation from Surfaces of 

Water and River-Bed Materials," Journal of Agricultural Research, vol. 10, no. 5, 1917.

16. Sleight, R. B., "Evaporation on United States 

Reclamation Projects," Discussion, American Soc/efy 

of Civil Engineers, Transactions, vol. 90, 1927, pp. 

303-316.

17. Theis, C. V., "Some Recent Studies in Evaporation," American Geophysical Union, Transactions, vol. 

12, 1931, pp. 220-224.

18. White, W.N., "A Method of Estimating GroundWater Supplies Based on Discharge by Plants and 

Evaporation from Soil," U. S. Geological Survey, 

Wafer-Supp/y Paper, no. 659a, 1932, pp. 1-105.

19. Whitney, H. A., "Evaporation from Lake Conches, 

Mexico," Discussion, American Society of Civil Engineers, Transactions, vol. 80, 1916 pp. 1983-91.

20. Young, A. A., "Evaporation from Water Surfaces 

in California," A Summary of Pan Records and Coefficients, 1881 to 1946, California Dept. of Public 

Works, Division of Water Resources, Bulletin, no. 54, 

1947.

21. Young, A. A., "Some Recent Evaporation Investigations," American Geophysical Union, Transactions, 

vol. 28, no. 2, April 1947, pp. 279-284.

22. "Evaporation from Water Surfaces: a Symposium," American Society of Civil Engineers, Transactions, vol. 99, 1934, pp. 671-747.

23. "Mean Monthly and Annual Evaporation from 

the Free Water Service for United States, Alaska, 

Hawaii and Puerto Rico," U. S. Weather Bureau, 

Technical Papers, no. 13, 1951, pp. 1-10.

24. "Flow of Rio Grande and Related Data from 

Elephant Butte Dam, New Mexico, to Gulf of Mexico, 

1950," U. S. International Boundary and Water Commission, United States and Mexico, Water Bulletin, 

no. 20, 1952, pp. 1-108.

148
Summary and Recommendations-Lake Hefner Studies

SUMMARY OF RESULTS

1. Daily evaporation from Lake Hefner was determined by the water-budget method, that is, by 

measuring inflow, outflow, and changes in storage. 

Although this method is impracticable for most reservoirs, good results were obtained at Lake Hefner 

because: (1) natural surface inflow was generally 

negligible; (2) filter-plant withdrawals, the major 

outflow item, could be measured accurately; (3) 

changes in storage could be evaluated with sufficient 

accuracy; (4) deep seepage was almost negligible; 

(5) evaporation, although a residual by this method 

of determination, was a major item in the water 

budget, thus minimizing the effects of errors in measuring the other items. The water-budget control met 

the fundamental requirement that errors in the water 

budget not exceed 5 per cent of the monthly evaporation. An objective scheme of classifying daily waterbudget evaporation figures as to their relative accuracy into four categories, A, B, C, and D, was 

used to select the most reliable daily figures for 

purposes of comparison with results obtained using 

the mass-transfer and energy-budget techniques. 

Sixty-two per cent of the daily figures were A or 

B, and only these values were used for the comparison.

A

149
2. The equipment system developed for measuring and recording the factors needed in the masstransfer and energy-budget methods proved reliable 

and practicable. The average data yield during the 

16-month operating period was 70 per cent, in spite 

of such difficulties as frozen wet-bulb reservoirs during 

the winter.

3. In the mass-transfer studies, both mixinglength and continuous-mixing concepts were reviewed 

in detail and the most promising evaporation equations derived therefrom were tested against the waterbudget figures. The results of these studies were as 

follows:

a. No deviation from the logarithmic wind law 

could be detected between 2 and 8 meters, regardless of stability conditions, over periods of 3 hours 

or longer.

b. The lake surface was aerodynamically rough 

at all times, with no evidence of a critical wind speed 

for water-air boundary processes. The roughness 

length varied from 0.55 to 1.15 cm, increasing with 

wind speed. Taken as a whole, the evidence indicates 

that flow conditions very near the surface are critically important in predicting evaporation.

c. Of the mass-transfer equations tested, only 

O. G. Sutton's and Sverdrup's (1937) form proved satisfactory for use with available field instruments.

d. A new quasi-empirical equation was developed from the water-budget data, using wind 

speeds and vapor-pressure differences measured at 

Lake Hefner. A field version of this equation, using 

standard meteorological data from a Weather Bureau installation 13 miles from the lake and requiring 

only a measurement of the water-surface temperature 

at the lake, also gave good results. General application of this simplified technique depends upon 

whether the standard meteorological data are generally representative of the up-wind air approaching 

the lake.

4. The energy-budget equation was analyzed 

in detail to gain a more complete understanding of 

the physical processes involved, and an improved 

form of the equation was developed which makes it 

more generally useful. Evaporation determined with 

this equation, compared with results given by the 

water-budget control, showed that:

a. For periods of less than seven days, errors 

are likely to be excessive, primarily because changes 

in energy storage cannot be measured with sufficient 

accuracy.

b. For periods of seven days or more, evaporation can be determined with an accuracy approaching ±5 per cent if all terms in the energy budget are 

carefully evaluated, particularly changes in energy 

storage.

5. The Cummings Radiation Integrator, essentially an efficiently insulated pan, was used as an 

instrument for measuring the net sum of certain 

radiation items. Although some aspects of this technique need further study, indications are that, for 

purposes of determining evaporation, the CRI will 

prove to be a satisfactory substitute for the complicated and expensive radiation instruments required 

for direct measurement of the individual items.

6. A plan for obtaining continuous records of 

evaporation from any reservoir has been developed 

and tested at Lake Hefner. This plan utilizes the simplified mass-transfer equation mentioned above, with 

a calibration period of sufficient length to determine 

whether data from the nearest first-order Weather 

Bureau station are sufficiently representative. Calibration would be by water budget if practicable and, 

if not, by energy-budget or by more detailed masstransfer measurements at the lake. If the Weather 

Bureau data prove not to be sufficiently representative, a skeleton meteorological installation at the reservoir would be necessary. Since the only other observations required would be water-surface temperatures, the field observations and computations would 

be relatively simple.

7. Studies of evaporation pans at Lake Hefner 

included observations from four Weather Bureau 

Class A pans, one Colorado pan, one Bureau of 

Plant Industry pan, and one screened pan; the last 

three were sunken types. The following results 

were obtained from this study:

a. Pronounced seasonal and short-period fluctuations in the ratio of lake evaporation to that 

from all types of pans tested result principally from 

differences in surface-water temperature, augmented 

to some extent by other factors, such as variations 

in wind and the sun's altitude.

b. Annual coefficients between pan and Lake 

Hefner evaporation are as follows:

Class A pan.............. 0.69

BPI pan .................0.91

Colorado pan ............0.83

Screened pan ............0.91

While it is noted that the Class A pan coefficient is 

the lowest of the group, its range in monthly values

150
was less (though not significantly so) than that of any 

other for the period of observation.

c. A high degree of correlation exists between 

pan coefficients and dewpoint, wind, and lake and 

pan temperatures. This relation explains much of the 

seasonal variation in pan coefficients.

d. Statistical relations were derived which show 

that daily pan evaporation at Lake Hefner can be 

reliably estimated from observations of surface-water 

temperature, dewpoint, and wind. Daily lake evaporation is also dependent on these factors, but may 

be appreciably affected by other factors as well, 

such as size and shape of lake, terrain in the vicinity, 

and possibly stability of the air.-The true effects of 

these factors cannot be determined without additional observations at other lakes.

PRIMARY RECOMMENDATIONS

1. The techniques and methods tested at Lake 

Hefner should be given further tests at another lake 

with different terrain and climatic conditions. Although a water-budget control would be desirable 

for this additional test, it is considered that the results 

at Lake Hefner are sufficiently satisfactory and that 

enough has been learned of the physical processes 

involved to justify making the tests on a comparative 

basis without an accurate water budget. The following methods should be-employed simultaneously and 

the results compared: (a) the mass-transfer method 

using Button's and Sverdrup's (1937) equations; (b) 

the quasi-empirical mass-transfer equation, including 

a comparison of measurements from the meteorological installation with data from the nearest firstorder Weather Bureau station; (c) the energy-budget 

method; (d) the Cummings Radiation Integrator, comparing results with radiation measurements for the 

energy budget; (e) pan evaporation. In view of the 

importance of evaporation in the operation of reservoirs on the lower Colorado River, and in accordance 

with previous plans, these further tests are being 

conducted at Lake Mead.

2. The simplified field method for obtaining continuous records of evaporation described above (paragraph 6 of "Summary of Results") should be tried 

at a number of additional reservoirs under conditions 

of different terrain and climate.

3. Both the mass-transfer and energy-budget 

approaches appear promising as methods for determining water losses by evapotranspiration. It is 

recommended that these applications be further 

studied.

4. Both the mass-transfer and energy-budget 

techniques also offer promise for the prediction of 

evaporation from available climatological data, prior 

to the establishment of a lake or reservoir. The chief 

item lacking is information on both surface temperatures and energy storage in lakes of varied sizes and 

shapes and in different climates. Data on surfacewater temperatures and wind movement are also 

needed for better interpretation of existing records 

of pan evaporation. It is therefore recommended that 

the present program for measuring water temperatures in lakes, reservoirs, and streams be considerably 

expanded, and that wind data be taken at selected 

lakes and reservoirs.

SECONDARY RECOMMENDATIONS

5. The experimental instrumentation for the 

mass-transfer and energy-budget studies at Lake 

Hefner proved very satisfactory and is recommended 

for use in other applications requiring measurements 

of temperature, humidity, wind, or radiation, provided 

that humidity measurements are not required during 

extended periods of below-freezing weather. For 

general application of mass-transfer or energy-budget 

techniques, however, further development should be 

undertaken with a view to producing commercial 

equipment requiring less servicing and maintenance.

6. Further research on mass-transfer processes 

is needed, with more refined measurements of wind, 

humidity, and temperature immediately above the 

water surface, especially just above, near, and below 

wave crests. Attention should also be focussed on 

evaluation of von Karman's constant under field conditions.

7. Further research along the following lines is 

recommended to facilitate more general use of the 

energy-budget technique:

a. Additional observations on the reflectivity of 

a water surface under variable amounts of middle 

clouds.

b. Additional measurements of atmospheric radiation in conjunction with measurements of total 

atmospheric water-vapor content.

c. Development of better instruments and evaluation methods for measuring changes in energy 

storage.

d. Further studies of the Bowen ratio. Experiments in a suitable wind tunnel would probably yield 

useful results. The use of the Bowen ratio in connection with the Cummings Radiation Integrator might 

also be studied in this way.

151

Appendix: Symbols and Dimensions

The numbers in parentheses refer to the equations where the symbol first appears or 

where additional clarification may be obtained. There is some duplication of symbols 

because of the desire to preserve the notation used in the original papers.

Symbol aaaaaad

Ow

Ol bbbcc

CPdeeea

6i es ez e0 f

fo9hhhhk

kommme m0

m, nnn

Dimensions Description

LT 1 A function of u., k , and z (20).

Atmospheric transmission coefficient (75).

Reflectivity of solar radiation from a water surface (76).

Mean extinction coefficient due to molecular scattering (79).

Empirical constant (85, 86, 104).

Coefficient of extinction due to dust (79).

Coefficient of extinction due to water vapor (79).

A frictional parameter (45).

An empirical constant (23).

Reflectivity of sky radiation from a water surface (76).

Empirical constant (85, 86, 104).

|_2 "T 2 Q l Specific heat of water (63, 101).

Empirical constant (94, 105).

L^ T 2 8 1 Specific heat of air at constant pressure (95).

|_ Venturi-meter throat diameter (3).

ML~ 1 T 2 Vapor pressure (numerical subscript indicates height in meters) (28).

Emissivity of a water surface (62).

ML" 1 T 2 Vapor pressure of the air (52).

ML 1 T 2 Vapor pressure of unmodified air.

ML~ 1 T 2 Saturation vapor pressure at T . a

ML~ * T 2 Vapor pressure at a height z (30).

ML" 1 T 2 Vapor pressure of saturated air at the temperature of the water surface (28).

ML 3 Potential of vapor concentration (30).

ML~ 3 Value of f outside the vapor blanket (34).

LT 2 Acceleration of gravity (3).

L Indicated stage change (2).

L Venturi-meter head (3).

L Cloud height (89).

Average altitude of the sun in degrees (74).

Constant that is a function of latitude (74).

von Karman's constant (18).

Solar air mass; the ratio of the length of th* actual path of the solar beam to the 

path through zenith (75).

That part of the total incoming short-wave radiation that comes from the sun (76).

L3 Volume of evaporated water (65). L3 Volume of surface outflow (65). L 3 Volume of out-seepage (65).

An empirical constant (51).

That part of the total incoming short-wave radiation that comes from the sky (76).

Empirical coefficient or constant (105).

Symbol

n/ np qrrru

Uz

u»

U«8

v/ Xz

zoAAAc

AAA4

CCCCDEEFFLFB11

/oJKLMMNN0PQ

QaQfi

Dimensions L3 L3L

LT1

LT1

LT1

LT1

LT1 LLLL2

ML^T1 L2L2

ML2 T~2

L2T -i

L3T -i

L3T -i

ML 2 T x

ML2 T~1

ML~2 T1

L3T -i

L

MT~3

L2T-i

L2T-2

ML^T1 L3L2 T -i L3L3 T-i

ML- 1 T2 L3 T-i

ML2 T2

ML2 T2

Description

Volume of surface inflow (65).

Volume of precipitation (65).

Empirical constant (11); also specific humidity (53).

Radius of a circular evaporating surface (40).

Angle of refraction (80).

Reflectivity of a water surface (90).

Average wind speed in x-direction (numerical subscript indicates 

meters) (4).

Average wind speed at a height z (24).

Friction velocity, equal to V r/p (4).

Friction velocity over a smooth surface (45).

Vertical component of fluctuating velocity

Distance along horizontal coordinate axis (8).

Distance along the vertical coordinate axis (4).

Roughness parameter (6).

Area of lake surface (43).

Vertical component of eddy conductivity (95).

Mean horizontal area of a standard layer (92).

Experimental constant (4).

Attenuation coefficient (77).

The amount of energy in a nonstandard layer (91).

Average cloud cover in tenths of sky covered (74).

Molecular vapor diffusivity (49).

Volume of evaporated water in unit time (1).

Evaporation given by equation (104) or (105), assuming e. = e .

Evaporation computed from Lettau's equation (29).

Evaporation computed from Sverdrup's equation (38), (29).

Volume of inflow in unit time (1).

Thickness of an intermediate layer (45).

Solar radiation received on a horizontal surface at the exterior of 

atmosphere (75).

Momentum integral (10).

Eddy diffusivity (46).

Latent heat of vaporization (60).

Total evaporation from a strip of unit width and length x (34).

Total volume of outflow (64).

The macroviscosity (50).

Total volume of inflow (64).

Volume of outflow in unit time (1).

Atmospheric pressure (28).

Venturi-meter discharge (3).

Atmospheric radiation (long wave) (62).

The net energy lost by the body of water through the exchange of 

radiation between the atmosphere and the body of water (59).

height in

the earth's

long-wave

153

154
Symbol

Qe

Qh

Q»

Or

Qe

Q.'

Qv

Q»8

Q«

Q*

Qac

Qar

R

Dimensions

ML2 T-2

ML2 T-2

ML2 T-2

ML2 T- 2

ML2 T-2

ML2 T-2

ML2 T-2

ML2 T-2

ML2 T-2

ML2 T-2

ML2 T-2

ML2 T-2

Description

Energy utilized by evaporation (59).

Energy conducted from the body of water by the atmosphere as sensible heat (59).

Net radiant energy defined by equation (108).

Reflected solar radiation (59).

Solar radiation incident to the water surface (59).

Net energy advected into the body of water by all water volumes entering or 

leaving the body of water (59).

Net energy advected into the body of water by all volumes entering or leaving 

the body of water, except that volume leaving as evaporated water (68).

Long-wave radiation emitted by the body of water (96).

Energy advected out of the body of water by the mass of evaporated water (68).

The increase in energy stored in the body of water (59).

Atmospheric radiation from an overcast sky (87).

Reflected atmospheric radiation (73).

Bowen ratio or ratio of Q to Q (60). 

A e

Rt Turbulent correlation coefficient

Ri Richardson's Number (18).

Rt Total reflectivity of solar and sky radiation from a water surface (76).

R<e Ratio x/zfl (38).

S

S

S

L3 T-i

n-2 e

Change in reservoir contents in unit time (1).

Standard error of mean (2).

Stability parameter (26).

5^ Altitude of the sun, in degrees, above the horizon (77).

r

r

r

T«

TO

T6

Te

T/

T«

To

Tp

T8

T8

Tz

Ti

T2

U

y

y

W

e

e

e

e

e

e

e

e

e

e

e

e

e

e

e

IT- 1

L3

L3

LT-2

Absolute temperature (22).

Temperature (numerical subscript indicates height in meters) (24).

Mean turbidity factor (79).

Temperature of the air (61).

Absolute temperature of the air near the ground (86).

Arbitrary base temperature (63).

Average temperature of evaporated water (67).

Average temperature of surface inflow (67).

Average temperature of a standard layer of water (91).

Average temperature of surface outflow (67).

Average temperature of precipitation (67).

Average temperature of out-seepage (67).

Source temperature (90).

Temperature at a height z (24).

Average temperature of a body of water at the beginning of a period (67).

Average temperature of a body of water at the end of a period (67).

Horizontal wind speed outside the boundary layer (8).

Volume of a body of water at the beginning of a time interval (64).

Volume of a body of water at the end of a time interval (64).

Frictional drag of water surface (8).

155

Symbol Dimensions Description

a

ft

An empirical constant (11).

An empirical constant (20).

Q An empirical number (36).

y

y

y

8

8e

8W

6

«.

L- 1 ^

L

L

L

L

L

A number defined by equation (107).

Empirical constant (86).

Adiabatic lapse rate (95).

Thickness of the momentum boundary layer (10).

Thickness of the laminar film (49).

Thickness of the vapor blanket (34).

Mean diameter of roughness elements (5).

Naperian base (86).

Equivalent sand roughness (4).

 The ratio Z/ZD (36).

 - The value of ij at z = 6 (37).

A

pV

I

P

P

PB

po

PP

pa

PI

PM

P*r

pi

P2

<r

(T

T

L2 T -1

T

ML- 3

ML- 3

ML- 3

ML-3

ML-3

ML- 3

ML- 3

ML- 3

ML- 3

ML- 3

ML- 3

MT~3 e~4

ML- 1 !-2

Empirical coefficient (87).

Index of refraction for pure water relative to air (80).

Kinematic viscosity of the air (4).

Time required for R, to decrease from 1 to 0

Density of the air (8).

Density of water (60).

Average density of evaporated water (65).

Average density of surface outflow (65).

Average density of the precipitation (65).

Average density of out-seepage (65).

Average density of the surface inflow (65).

Average density of the volume of outflow (64).

Average density of the volume of inflow (64).

Average density of the body of water at the beginning of a time interval (64).

Average density of the body of water at the end of a time interval (64).

An empirical number (18).

Stefan-Boltzmann constant for black-body radiation (62).

Eddy shearing stress (8).

A The ratio u/u, (36).

A, The value of (p at z = 8 (37).

"ri y)

r

A

T

M!-2

The gamma function (51).

Slope of saturation vapor pressure vs temperature curve (106).

Total f fictional force (31).

X- The ratio x/z. (38).

U

tyr The integral defined in (37).

156
INDEX

A Page 

Acknowledgments.......................................... 4

Anderson, Ernest R....................................... 71

Andersen, Lloyd J......................................... 35

Appendix: Symbols and dimensions.................. 153-156

Atmospheric radiation.................................... 90-91

B

Bowen ratio................................................104-117

Geology and ground-water hydrology of Lake Page

Hefner area Continued 

geology and water-bearing properties............. 10-12

Pleistocene ten ace deposits....................... 12

scope of study.......................................... 10

soil and Recent deposits............................ 10

Harbeck, G. Earl, Jr.

Hennessey shale.....i

as an aquifer.......<

composite section, 

permeability of.....

H

 5, 17, 46, 120

..... 10-12

12

11

14

Climatology (Lake Hefner area)....................... 6-7

Cummings, N. W........................................... 120

Cu mm ings Radiation Integrator........................120-126

conclusions............................................. 126

instruments and methods................. 121-124, fig. 90

results. ...................................................124-125

theory of................................................. 120-121.

Dennis, P. E............................................... 10-16

Energy-budget studies................................... 7 1- 119

advected energy........................................ 98-100

change in energy storage............................ 100-104

conclusions....... .».« «» . « ««««« » 117

effective back-radiation.............................. 88-98

energy conducted as sensible heat by the

atmosphere........................................ 110

energy utilized by evaporative processes....... 110-117

introduction............................................. 71-74

ratio of energy used for conduction of sensible heat to that used for evaporation.... 104-109

reflection of short-wave radiation................. 78-88

solar radiation.........»«»«.«. *«.««» »*« «» « . 74-77

Evaporation. (See Lake and pan evaporation.).*.. 1-2, 

31-34, 61-68, fig. 22, table 1

Garber sandstone and Wellington formation...!

Geology and ground-water hydrology of Lake

Hefner area...................................

conclusions....... .».. *.« «« » .........<

10-

12

16

16

Instrumentation for mass-transfer and energybudget studies.................................... 35*45

conclusions.............*............................... 44

energy-budget studies...................38-41, figs. 23-30

equipment, general description of*...»..».....«... 35-36

mass-transfer studies................................. 36-38

over-all performance and maintenance............ 41-44

Introduction................................................ 1-3

Kennon, Frank W. 

Koehler, Max A»«.<

17

127

Lake and pan evaporation..............................127-148

background and objectives.......................... 127

conclusions*................................146-148, table 29

instrumentation and operational procedure.*... 127-132, 

tables 24-26, figs. 93-97

lake evaporation and meteorological studies... 136-139 

pan evaporation and meteorological factors.....132-136

pan versus lake evaporation............139-144, table 28

relative evaporation from pans of different

types................................................144-146

Lake Hefner:

basis for choice of.................................... 5-6

computation of influent and effluent seepage

of...............*.*.................................. 15-16

cross-sectional area of.............................15, fig. 9

general description of........» *» .....* .......... 5-9

historical review of.......»..«*....... ....« *. »»» 1

physical and climafological characteristics of. 6-8 

piezometric map of.................................... 12

157
M Page Page

Marciano, J. J............................................. 46

Mass-transfer studies.................................... 46-70

comparison of computed and measured

evaporation...................4.................... 61-68

conclusions............................................. 68-69

mass-transfer equations.............................. 54-61

theory..................................................... 46-54

Personnel) 3-4

Radiation. See Energy-budget studies* 

Rainfall* See Water-budget control* 

Reservoir area and dam, changes in content of. 

description of........................................«

Storms, comparison of precipitation, 

gaging procedures.........*...........

isohyetal map of.......................

17

6

........... 20-21

........... 25-27

........... fig. 11

variability of rainfall*........* * « »* fig. 12

158

U. S. GOVERNMENT PRINTING OFFICE : 1954 O 302520