Dynamic Meteorology

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DYNAMIC METEOROLOGY

DYNAMIC METEOROLOGY BY

BERNHARD HAURWITZ,

PH.D.

Chairman

of the Department of Meteorology College of Engineering, New York University

FIRST EDITION SIXTH IMPRESSION

McGRAW-HILL BOOK COMPANY, NEW YORK AND LONDON 1941

INC.

DYNAMIC METEOROLOGY COPYRIGHT, 1941, BY THE

McGRAW-HiLL BOOK COMPANY,

INC.

PKINTED IN THE UNITED STATE8 OF AMERICA All rights reserved. This book, or may not be reproduced

parts thereof,

in any form without permission of the publishers.

THE MAPLE PRESS COMPANY, YORK,

PA.

PREFACE The largely

great progress of meteorology in recent years has been due to the application of the laws of thermodynamics and

hydrodynamics to the study of the atmosphere and its motions. It is the aim of this book to give an account of these investigations and their results, with regard to applications to weather forecasting and to research.

No

previous knowledge of meteorology is assumed, although training in general meteorology will facilitate the study of the book. A large number of references to literature

some preliminary

have been given in order to enable the reader to consult the

The material presented has been the subject of original papers. lecture courses on Dynamic Meteorology given at the University Toronto during the past six years as part of the meteoroby the university in cooperation with the Meteorological Service of Canada. The scope of the book is, in the main, a theoretical discussion of the various phenomena, of

logical course offered

without a complete descriptive account of the observed phe-

nomena and of the actual practical applications of the theory. The mathematical technique has been kept as simple as possible. Readers who are sufficiently well versed in advanced mathematical methods will know how to obtain solutions for many of the specific problems discussed here by more elegant mathematical methods. Thus, the derivation of the equations of motion on the rotating earth (Sec. 45) could be shortened greatly by the use of vector analysis. Where more advanced results of thermodynamics or of hydrodynamics are used, they have been explained briefly, but the reader will do well to remember that this book does not deal with these subjects but with dynamic meteorology and that for a thorough study of thermodynamical or hydrodynamical problems, specialized textbooks should be consulted.

The problems are chosen partly to supplement the text with material of secondary importance and partly to indicate the possibilities of practical applications.

PREFACE

vi

The formulas are numbered according to the decimal system. The number before the period refers to the section in which the formula appears, the number after the period indicates the The formula with the position of the formula in the section. smaller

number comes

first.

Thus

(17.21) precedes (17.3),

but

follows (17.2).

The author is indebted to Dr. W. Elsasser for permission to reproduce Fig. 21, to the editors of Nature for permission to reproduce Fig. 24, to Prof. J. Bjerknes for permission to reproduce Figs. 45, 79, 80, 86, 89, to Prof. S. Petterssen for permission to reproduce Figs. 55 to 57, to Sir Napier Shaw and Messrs. Constable and Co. for permission to reproduce Fig. 81, and to Mr. C. M. Penner and the National Research Council of Canada for permission to reproduce Figs. 84 and 85. Owing to the present war, it has been impossible to approach all the authors and publishers concerned for permission to reproduce diagrams which appeared in their publications. The author offers his apologies for this omission and hopes that the permission may be considered as granted, since proper references are made in each case

and

since all these diagrams

have originally appeared

in

scientific journals.

The author wishes to express

his gratitude to Prof. J. Patterson,

controller of the Meteorological Service of Canada, to Mr. A. Thomson, assistant controller of the same service, to Prof. C. F.

Brooks, director of Blue Hill Observatory, and to Prof. Sverre Petterssen, head of the Meteorological Department of the Massachusetts Institute of Technology, for their encouragement during the preparation of this book. Sincere thanks are due to Lt. Haakon Anda of the Royal Norwegian Air Force for reading the manuscript, and to Mrs. Haurwitz for her great assistance in preparing the manuscript for publication.

BERNHARD HAURWITZ. CAMBRIDGE, MASSACHUSETTS, Augtist, 1941.

CONTENTS PAGE

PREFACE

v

CHAPTER I.

1.

2.

Units of Pressure, Temperature, and Density

3.

The Composition of the Atmosphere The Gas Equation for Dry Atmospheric Atmospheric Water Vapor

4. 5.

II.

THE EQUATION OF STATE FOR DRY AND MOIST AIR The Earth and Its Gravitational Field

THE EARTH.

1

1

3 5 6

Air

7

ATMOSPHERIC STATICS. ADIABATIC CHANGES OF DRY AIR. 6. The Decrease of the Pressure with Elevation 7. Height Computation of Aerological Ascents 8. Adiabatic Changes of Dry Air

.

.

11

11

13 17

Potential Temperature. The Dry-adiabatic Lapse Hate. Vertical Stability of Dry Air v 10. The Influence of Vertical Motion on the Temperature Lapse Rate and on the Stability of Dry Air 9.

.

The Relation between

Pressure and Temperature Variations 12. Computation of the Ad vection at Great Heights Problems 1-6 11.

III.

ADIABATIC CHANGES OF MOIST AIR

CONDENSATION. 13. 14.

15.

16. 17. 18.

Adiabatic Changes of Moist Air in the Unsaturated State Minimum Inversion Variation of the Dew Point with the Altitude. Condensation Level The Role of the Condensation Nuclei Adiabatic Changes in the Saturated State The Application of the Equations for Saturated-adiabatic

Changes Chart

to

Atmospheric

Processes.

23 26 29 35 36

36 38

39 41

45

Pseudo-ad iabatic

Lapse Rate 20. Stability with Respect to Saturated Adiabatic Changes. 19. Saturated-adiabatic

49 54 56

Conditional Instability iV.

20

FURTHER APPLICATIONS OF THERMODYNAMICS TO THE ATMOSPHERE 21. The Energy

58 of

Thermodynamic

Processes.

The Carnot

Cycle 22. 23. 24.

Entropy Energy Released by the Adiabatic Ascent of Air Equivalent Potential Temperature and Equivalent Temper-

58 62 65

67

ature vii

CONTENTS

viii

PAGE

CHAPTER 25.

Wet-bulb Temperature and Wet-bulb Potential Tempera70 75 76 78 84

ture 26. Latent Instability 27. Potential, or Convective, Instability

28.

Therm odyiiamic Charts and Air-mass Charts Problem 7

V. RADIATION 29.

30.

85

The Laws of Radiation The Solar Radiation The Geographical and Seasonal

85 86

Distribution of the Solar Radiation in the Absence of the Atmosphere 32. The Depletion of the Solar Radiation in the Earth's 31.

Atmosphere 33.

The Albedo

34.

Absorption of Terrestrial Radiation Effect of the Line Structure of the Water-vapor Spectrum on the Atmospheric Emission and Absorption ... General Survey of the Terrestrial Heat Balance The Geographical Distribution of the Outgoing Radiation Computation of the Radiation Currents in the Atmosphere Nocturnal Radiation and the Cooling of the Surface Layers The Differential Equations of Atmospheric Radiation Radiation and the Stratosphere Problem 8

35.

36. 37. 38. 39. 40.

41.

VI.

of the

Earth

43.

44. 45.

46.

47. 48.

.

.

The Hydrodynamic Equations The Physical Equation. Piezotropy

and

54.

Ill

113

114 115 119 121

124 127 131

133 .

.

.

.134 135 138 144

ATMOSPHERIC MOTIONS

145

The

Circulation

Vorticity

Problems 9-12

53.

101

104 106 110

Theorems

51. Circulation

VII. SIMPLE

96 99

114

Plane Motion in Polar Coordinates The Motion on a Rotating Globe The Conservation of Angular Momentum Introduction of a Cartesian Rectangular Coordinate System The Coriolis, or Deflecting, Force of the Earth's Rotation

49. Barotropic and Baroclinic Stratification 50. Streamlines. Divergence and Velocity Potential 52.

89 93 94

The

THE EQUATIONS OF MOTION OF THE ATMOSPHERE 42.

88

The Geostrophic Wind The Inclination of Isobaric

145 147

Surfaces

Temperature Gradients and Geostrophic 148 Motions 150 Steady Motion Along Circular Isobars ... 155 Accelerated Motion and a Changing Pressure Field

55. Horizontal

56.

57.

58. Divergence,

.

Convergence, and Pressure Variation

....

159

CONTENTS

ix

PAQB 163 166

CHAPTIB 59. Pressure Distribution in

a Moving Cyclone

Problems 13-19 VIII. SURFACES OF DISCONTINUITY

167

General Expression for Surfaces of Discontinuity 61. The Pressure Distribution at Fronts 62. Surfaces of Discontinuity in a Geostrophic Wind Field 63. Accelerations at Frontal Surfaces 60.

64.

.

.

.

.167

.

.

Zones of Transition

65. Fronts

and Pressure Tendencies

Problems 20-21 IX. KINEMATICAL ANALYSIS OF THE PRESSURE FIELD 66. 67. 68.

69.

70.

The The The The The

180

Motion of Characteristic Curves Motion of Isobars and Isallobars Motion of Troughs, Wedges, and Pressure Centers Motion of Fronts Application of the Kinematic Formulas to Forecasting .

.

X. ATMOSPHERIC TURBULENCE 72. 73.

Turbulent Motion

74. Prandtl's 75. 76. 77. 78. 79.

80.

Theory

of

Momentum

Transfer .

.

.

XI. TURBULENT

.

.

.

215

MASS EXCHANGE

81. Transfer of Air Properties by 82. The Differential Equation of 84.

216 Turbulent Mass Exchange Turbulent Mass Exchange

Exchange Mixing and Problems 26-28

88. 89. 90. 91.

.

216 220

226 Its

Study by Isentropic Analysis

THE ENERGY OF ATMOSPHERIC MOTIONS 86. The Amount of Available Energy 87.

.

221 The Daily Temperature Period The Transformation of Air Masses by Turbulent Mass

85. Lateral

XII.

188 190 192 195 197

The Vertical Variation of the Wind in the Surface Layer The Variation of the Wind above the Surface Layer .201 The Effect of the Vertical Variation of the Pressure Gradient 207 The Effect of the Centrifugal Force 209 210 The Variability of the Coefficient of Eddy Viscosity The Diurnal Variation of the Wind Velocity 213 Problems 22-25

83.

180 183 184 186 187

188

The Shearing Stresses in a Viscous Fluid Dynamic Similarity and Model Experiments

71.

169 170 173 176 176 179

The The The The The

Atmospheric-energy Equation

.

.

.231 237

238

238 238

241 Energy of Air Columns 241 Dissipation of Energy 244 Energy Transformations in a Closed System Energy of Air Masses of Different Temperature Lying 247 Side by Side

CONTENTS

X CHAPTER

PAQB

The

92.

Effect of

Water Vapor on the Atmospheric Energy 252 253

Transformations Problem 29 XIII.

THE GENERAL CIRCULATION OF THE ATMOSPHERE

254

Survey of the General Circulation

93.

94. Application of the Circulation

254

Theorem

to the General

258

Circulation

The Meridional Heat Transport 262 The Meridional Heat Transport as a Form of Turbulent 265 Mass Exchange The Cellular Structure of the General Circulation .... 267

95. 96.

97.

XIV. THE PERTURBATION THEORY OF ATMOSPHERIC MOTIONS 98. Disturbed and Undisturbed Motion 99. The Perturbation Equations 100. The Boundary Conditions 101. Wave Motion at the Free Surface of a Single Layer. Gravitational Waves 102. Wave Motion at an Internal Surface of Discontinuity. Shearing Waves .

.

,

103. Billow Clouds

An Example

104.

of Inertia

Waves

105. General Discussion of Inertia

Waves

106. Large-scale Oscillations of the

Atmosphere

Problems 30-31

XV. AIR MASSES, FRONTS, CYCLONES, AND ANTICYCLONES

114.

The Convection Theory The Conditions in the Upper Levels The Theories of the Coupling between the

276 282 287 288 292 295 299

The 312 317 318 320

Variations in the

Higher and the Lower Atmosphere 115. Tropical Cyclones

327 334 337

116. Anticyclones

APPENDIX Table Table

271 272 274

300 302 307

Occlusion Process 111. The Barrier Theory 112.

271

300

Masses 108. Fronts and Their Origin 109. The Wave Theory of Cyclones 110. Further Development of Extratropical Cyclones. 107. Air

113.

.

341 I.

II.

Saturation Pressure of Water Vapor, Millibars Numerical Constants

.

...

341 342

NAME INDEX

343

SUBJECT INDEX

347

SOLUTIONS TO PROBLEMS

353

DYNAMIC METEOROLOGY CHAPTER

I

THE EARTH. THE EQUATION OF STATE FOR DRY AND MOIST AIR 1. The Earth and Its Gravitational Field. The earth is approximately a sphere or, more accurately, a spheroid with an

equatorial radius of 6378.4 km and a polar radius of 6356.9 km. For almost all meteorological problems the deviation of the earth from the spherical form may be disregarded, so that the earth may be assumed as exactly spherical with a radius of

6371 km, approximately. A sphere of this radius has roughly the same area and volume as the earth. The angular velocity of the earth's rotation

2w sidereal

The

=

7.292

day

X

10- 5 sec- 1

.

acceleration of gravity that is observed on the earth conthe actual attraction by the earth diminished by the

sists in

caused by the earth's rotaPoints near the equator move faster than those at higher latitudes owing to the earth's rotation. Therefore, the centrifugal force decreases poleward, and consequently the total acceleration of gravity increases. Moreover, owing to the spheroidal shape of the earth, points at higher latitudes are closer to the center of the earth. This is an additional reason for the increase of the acceleration of gravity poleward, for the gravitational force at a point outside the earth is inversely proportional to the distance effect of the centrifugal acceleration

tion.

The total acceleration can be expressed by center. the following formula for the acceleration of gravity at sea level

from the 0o

and at

latitude go

=

)

cm/sec

2

(1.1)

DYNAMIC METEOROLOGY

2

Because the acceleration of gravity decreases with the square from the center, its value g at an altitude z above

of the distance

sea level

is

given by 9

=

or

g

~0o(l

3.14

X

10~ 7 2)

if

E = 6371 km, the mean radius of On mountains, Eq. (1.2) should tion,

owing to the mass

isostatic

of the

The

compensation.

(L2)

+

[i

z is expressed the earth.

in

meters.

be replaced by another equamountain and the imperfect

consideration of these corrections

far into geodesy and is not of great importance to the meteorologist who finds these figures in tables. 1

would lead too

The

height z of a point above sea level can also be expressed

by the difference between the potential of gravity at sea level and at the altitude z. The potential at the altitude z is numerically equal to the work done when the unit of mass is lifted from It is called the geopotential. The sea level up to this height. the the and between exists relation geopotential ^ following height

z:

+ = $*gdz according to which, with

(1.3)

(1.2),

z<E,

the denominator on the right side of this last equation very nearly unity so that numerically ^ is about 10 times larger than z if the meter is used as the unit of length. In order to obtain approximate numerical equality between the geopotential and the corresponding altitude the former is usually

Because

is

expressed in a unit that

is

10 times smaller than the one following " the "dynamic meter or

from Eq. (1.31). This unit is called "geodynamic meter." It should be the dynamic meter is not an altitude unit mass. The hundredth part of

clearly understood that but rather an energy per the dynamic meter is a dynamic centimeter, 1000 dynamic meters are a dynamic kilo1

"Smithsonian Meteorological Tables," 5th

ed.,

Smithsonian Institution,

Washington, D. C., 1931. BJERKNES, V., "Dynamic Meteorology and " Hydrography, Tables 1M and 2M, Carnegie Institution of Washington, 1910. Washington,

THE EARTH meter, etc. units, it is

3

above sea level is expressed in these dynamic height" to distinguish it from the

If the height

called

"

ordinary geometric height. Obviously the following relation between the dynamic height ^ and the geometric height z:

exists

Y

~ 10

1

+

(1.4)

(z/E)

Because go = 9.8 m/sec 2 [if go is expressed in centimeter-gramsecond (cgs) units the factor Jlo has to be replaced by Koool> With the aid is about 2 per cent smaller numerically than z. of (1.4) and (1.1), dynamic heights and geometric heights may be transformed one into the other. In meteorological practice where 1 speed is essential, tables are used for this transformation. the The practical advantage of over the dynamic height geometric height z is due to the possibility of combining the variations of the acceleration of gravity g with the variable ^ which measures the elevation (see Sec. 6). \l/

\l/

Dynamically, the surfaces of equal potential are more important than the surfaces of equal height because the force of gravity is everywhere normal to the former while it has a component Therefore, a sphere would be in equilibparallel to the latter. rium on a surface of equal potential but would roll toward the equator on a surface of constant height. The surfaces of equal geometric and dynamic height intersect each other, but the inclination is small. The equipotential from surface 20,000 dyn. meters, for instance, descends 107

m

the equator to the pole. 2. Units of Pressure, Temperature, and Density. is defined as the force exerted on the unit area.

system being the dyne, unit of pressure in the cgs system

of force in the cgs

2 Dynes/cm = gm

it

Pressure

The

cm"" 1 sec~ 2

This quantity is too small for practical use in meteorology. " 6 pressure of 10 cgs units has been called 1 bar." 1

bar

=

unit

follows that the

A

10 6 dynes/cm 2

BjERKNBS, op. cit., Tables 3M-6M. "Smithsonian Meteorological " Tables," Tables 64-68. LINKS, F., Meteorologisches Taschenbuch," I, Tables 26-27, Akademische Verlagsgesellschaft, Leipzig, 1931. l

DYNAMIC METEOROLOGY

4

and

in practice the millibar,

in

is

the thousandth part of a bar, in which the atmospheric

i.e.,

most countries used as the unit

pressure

is

expressed

mb =

1

10 3

dynes/cm

2

In addition to the millibar the following expressions are sometimes used 1 1 decibar = 10" bar :

1 1

may

It

centibar

microbar

= =

10~ 2 bar

10~ 6 bar

be noted that the centibar

is

the unit of pressure in the

meter-ton-second system. In practice the atmospheric pressure is most frequently determined by the height of a mercury coluitin exerting the same

Consequently, the pressure observations pressure as the air. Because the are given in units of length, millimeters or inches. is and the of 13.6 acceleration of mercury density gravity at sea

and 45 latitude is 980.6 cm/sec 2 the pressure column of height 1 mm in cgs units is

level

1

of

,

mm Hg =

10- 1

X

13.6

X

980.6

Similarly, the pressure of a

because

1 in.

=

25.4

=

1333 dynes/cm 2

mercury column

a mercury

=

1.333

of height 1 in.

mb is,

mm, 1 in.

Hg =

33.86

mb

The following scales are used to express temperature: According to the centigrade scale, the freezing and boiling points of " " water at normal atmospheric pressure (760 Hg = 1013 mb)

mm

and 100, respectively. According to the Fahrenheit scale, these two fixed points have the values 32 and 212. The relation between the two scales is therefore have the values

PC = %(tF The Reaumur

32)

(2.1)

scale according to which the freezing point of

and

boiling point 80 is today not used in meteorolto the absolute temperature scale the freezing ogy. According of has the water value 273 1 and the boiling point 373, so point that the absolute temperature T is, in degrees centigrade,

water

is

its

T = 1

This figure

is

t

C

sufficiently accurate for

+ 273 all

meteorological problems.

(2.2)

THE EARTH For a discussion

of the theoretical foundations of the absolute

temperature scale the reader thermodynamics.

is

referred to the textbooks

on

The density p is defined as mass per unit volume. Its unit in The specific volume v is the volume the cgs system is gm/cm3 per unit mass. It is obviously .

v

=

i

(2.3)

p 3.

The Composition

a mixture

of the

of various gases.

Atmosphere.

The two main

Atmospheric

air is

constituents in the

lower layers are nitrogen and oxygen which account for 99 per cent of volume and mass of the air. A critical survey by Paneth 1 shows the composition of the air near the surface to be as given in the following table in abbreviated form :

There are

also small traces of neon, helium, krypton, xenon,

ozone, radon, and perhaps hydrogen present. The table refers to completely dry air. The water vapor of the air is variable, for water may freeze, condense, and evaporate at the temperatures encountered in the atmosphere.

It there-

fore requires separate consideration (Sec. 5). The observations indicate that the composition of the atmos-

phere remains virtually unchanged at least up to 20 km. Ozone becomes more abundant at greater heights, with a maximum between 20 and 30 km. It has great influence upon the emission and absorption of radiation in the upper atmosphere, but its

amount is not sufficient to affect the density of the air directly. At greater altitudes, but probably not below 100 km, lighter 2 For the problems of dynamic gases must become predominant. 1

PANETH, F. A., Quart. J. Roy. Met. Soc., 65, 304, 1939. CHAPMAN, S., and MILNE, E. A., Quart. J. Roy. Met. Soc., 46, 357, 1928. HAUBWITZ, B., The Physical State of the Upper Atmosphere, J. Roy. Astr. Soc. Can., 1937, 1938. CHAPMAN, S., and PRICE, W. C., Report on Progress 2

in Physics, Phys. Soc. London, 3, 42, 1937.

DYNAMIC METEOROLOGY

6

meteorology the state of the high atmosphere

not important,

is

at least according to our present knowledge.

The Gas Equation

In thermofor Dry Atmospheric Air. the relation exists that between shown following dynamics, T and absolute of an ideal gas: temperature pressure p, density p, 4.

it is

10 6 ergs/gm degree = 1.986 cal/gm degree, is the molecular weight of the the universal gas constant, and For actual gases, (4.1) holds as long as they are in a state gas.

Here

72*

=

83.13

X

m

away from condensation. Therefore, the equation can always be used for the atmospheric gases at ordinary temperatures and pressures, with the exception of water vapor. sufficiently far

For a mixture of two or more gases, as, for instance, for atmospheric air, a similar formula holds. To simplify matters a mixture of only two components will be considered. The gases may have the volumes V\ and V*, the masses M\ and M^ the same pressure p, and temperature T. Because Pl

it

=

M

l

,

and

p2

=

M

2

y-

follows from the gas equation (4.1), as long as the gases are

separated in two containers, that

R* p " If is

M

l

miVi

A an d

T

p

R* M* _ T

m 7 2

2

the containers are brought together and the separating wall removed, each gas occupies the whole volume V.

V=

Fi

+F

2

Consequently the sum of the partial pressures of both gases

This relation states Dalton's law, viz., that the sum of the partial pressures is equal to the total pressure of a mixture of gases.

The preceding equation may be written

m

THE EARTH

7

provided that the "molecular weight of the mixture"

M + M, = M !

m

Because Af i

+ Af = M 2

,

i

mi

M

is

defined

by

2

W2

the total mass of the gas mixture

Thus, the gas equation for a mixture of gases is also given by (4.1) provided that a mean molecular weight m is introduced according to (4.2). If the mixture consists of more than two components, its molecular weight is given by

m

^J

'

mi

From the table in Sec. 3 the molecular weight of the air is 28.97 if nitrogen, oxygen, argon, and carbon found to be m dioxide are taken into account. Since the universal gas constant R* appears in the equation mostly divided by the molecular weight m, it will be convenient to introduce the gas constant for (dry) air

R =

=

2.87

X

10 6

cm 2

sec~ 2 (deg)- 1

In addition to the other gases 6. Atmospheric Water Vapor. enumerated in Sec. 3, atmospheric air contains a certain amount As of water vapor which varies widely with time and locality. or is water fusion as condensation no vapor taking place, long may be treated as an ideal gas. If e is the water-vapor pressure,

mw its

molecular weight,

m^ =

18, p w its density,

T

its

tempera-

ture, according to Eq. (4.1)

"

where

m^/m =

0.621.

=

~^ RT

>

It is convenient to introduce the gas

The temperature T of the for (dry) air in (5.1). water vapor may be assumed as equal to the temperature of the constant

R

dry air with which it is mixed. Therefore, it is not necessary to denote it by a subscript w. In meteorology the density of water " vapor is frequently called absolute humidity."

DYNAMIC METEOROLOGY

8

The

total density p of the moist air is the sum of the density and of water vapor. The partial pressure of dry

of dry air

e when p p Conseauently

air is

P

=

is

the total pressure of the moist

g

j)

^ 0.621

~~ftf

^j ^

IP

I

"D>P

I

*

"""

"r^i 0.379 - j

air.

1

/ra\ (5.2)

This equation shows that moist air is lighter than dry air of the same temperature and pressure, for the water vapor is lighter

than the air that it replaces. In problems where only the density of the air is important, dry air of somewhat higher temperature may be assumed to be This temperature which the substituted for the actual moist air. to be of the same density as order fictitious dry air should have in the same the actual moist air under pressure is called the ''virtual temperature

"

T*.

According to

T* = 1

The

density of moist air

may

(5.2),

then be written (5.4)

At a given temperature the water-vapor pressure can rise only up to a certain maximum, the saturation, or maximum, vapor pressure

than

e m,

the existing water-vapor pressure e is smaller evaporation from liquid-water surfaces or ice can take em.

If

= e m an equilibrium is reached between the liquid 2 the gaseous state; if e > e m condensation occurs. and (or solid) Below the freezing point, one has to distinguish between the saturation pressure over ice and over water. place;

if

e

,

,

It should be clearly understood that the fact of saturation is independent of the presence of other gases besides water vapor. If water of a certain temperature is brought into a vessel con-

taining no other gas, the water-vapor pressure, by evaporation, will reach the same saturation value as if air or any other gas water vapor condenses and falls out as precipitation, the resultnot be e, for the water vapor by itself is not in hydrostatic equilibrium (see Chap. II, Prob. 1). 2 For modifications of this statement due to the surface tension of water 1

But

if

all

ing pressure

fall will

Sec. 16. droplets and the pressure of dissolved substances in water, see

THE EARTH

9

were present. The maximum water-vapor pressure depends only on the vapor temperature. It is, therefore, not strictly correct to say that the air is saturated with water vapor. Some justification for such a statement may, however, be found in the fact that the atmospheric water vapor has the same temperature as the air of which it forms a part. Because the saturation pressure depends on the temperature, its magnitude is indirectly " the air The

saturated by temperature. expression air" will therefore be used, for its brevity, in the following influenced

discussion.

The

variation of e m with the temperature

is

given in Table

I,

(page 341). Tetens has given an empirical formula for e m based on the laboratory measurements. If e m is the saturation vapor pressure in millibars and t the temperature in degrees centigrade, 1

em

The constants a and Over

=

6.11

X

10 *T*

(5 6 ) '

b are as follows:

ice,

a

=

9.5,

b

=

265.5

a

=

7.5,

b

=

237.3

Over water,

A similar theoretical formula can easily be derived from the 2 equation of Clausius-Clapeyron for the heat of condensation. Besides the absolute humidity, which is used rarely in meteorological practice, the water-vapor content may be expressed by numerous other quantities. The relative humidity f is the ratio of the actual

vapor pressure to the saturation pressure at the

existing temperature,

/

=

fm

(5.6)

or, according to (5.1),

/

The

relative

humidity

may

- -^-

(5.61)

PW max

thus also be defined as the ratio of

the actual absolute humidity to the maximum absolute humidity possible at the existing temperature. 1

2

2d

TETENS, 0., Z. Geophysik, 6, 297, 1930. See, for instance, D. Brunt, "Physical ed., p. 103,

Cambridge University

Press,

and Dynamical Meteorology," London, 1939.

DYNAMIC METEOROLOGY

10

The

humidity q is the ratio of the absolute humidity of water vapor) to the density of the moist air, (density specific

The mixing

ratio

density of dry

w

is

the ratio of the absolute humidity to the

air,

w = -^- =

0.621

~~

'

(5.8)

following relations exist between the specific humidity and the mixing ratio according to their definitions (5.7) and (5.8):

The

^ -2-

(5-7D (5.81)

Because e < p as seen from Table I, which gives the maximum water- vapor pressures at different temperatures, (5.7) and (5.8) can in practice be simplified to q

~ w ~ 0.621 -

(5.82)

Mixing ratio and specific humidity are figures without physical dimensions. Owing to their smallness, it is convenient in practice to express them in grams of water vapor per kilogram of air In Sec. 13, it will be shown that q and w remain (dry or moist). constant for dry-adiabatic changes.

These quantities are there-

fore useful for the identification of air masses.

The dew point r is the temperature to which the air has to be cooled, at constant pressure, in order to become saturated.

CHAPTER

II

ATMOSPHERIC STATICS, ADIABATIC CHANGES OF DRY AIR 6.

The Decrease

of the Pressure with Elevation,

The atmos-

pheric pressure at any level in the atmosphere represents very accurately the total weight of the air column above the unit area

At greater

at the level of observation.

pressure,

column 1).

altitudes the pressure

is

mass above the observer.

consequently smaller, for there is less To find the rate of decrease of the consider a vertical air

of unit cross section (Fig.

At the

level z the pressure

p; at the level z

+

p

dz, it is

p-dp

P

is

z+cfz

z

dp.

The

pressure difference is equal to the weight of the air column of the

height dz. If dz is chosen sufficiently small so that the density

Fia.

.

Decrease of the pressure with altitude.

and the acceleration of the gravity g may be regarded as constant in the height interval under consideration,

dp

= -gpdz

(6.1)

called the "hydrostatic equation."

sometimes This equation As long as the water-vapor content can be neglected, the density is

(4.1)

which

may

altitude th^e

be substituted in is

it

neglected,

dp

(6.1).

If

the variation of g with

follows that

~

~^ g

(6.11)

~p

and, by integration, that

p

(6.2) 11

DYNAMIC METEOROLOGY

12

where po is the pressure at the earth's surface. If the temperais independent of the altitude, (6.2) may be written

ture

p

=

po e~*T

(6.21)

The assumption

of a constant temperature in the vertical a good approximation to the average temperature In the lower part of the distribution in the stratosphere. atmosphere, the troposphere, the temperature distribution is rep-

direction

is

resented better by a function decreasing linearly with the height,

T= The constant a

is

To

-

az

called the "lapse rate of

temperature" or the

"vertical temperature gradient," even though the latter expresWhen the temperature sion should rather be reserved for dT/dz. is negative and the atmosphere shows an "inversion" of the temperature lapse within an atmospheric layer, the layer is rate; when a. =

increases with the altitude, the lapse rate

"isothermal." If

the temperature

of (6.2) gives the

is

a linear function of the height, integration

equation

_T\

]Ra

Upon (6.1), it

introducing the geopotential

^ according

to (1.3) in

follows that

dp Equation

When

(6.22)

(6.3)

=

-pcty

(6.3)

can be integrated in the same manner as (6.1). is used instead of the geometric height,

the geopotential

the variable acceleration of the gravity no longer appears in the equations. The influence of the atmospheric moisture content on the decrease of the pressure with altitude can be taken into account

by using the

virtual temperature

T*

instead of

!T.

From Eq.

followed that moist air of the temperature T and of the vapor pressure e has the same density as dry air of the temperature (5.3), it

T* = 1

-

-

*i\ (^ ( *' 6)

0.379(e/p)

where T* was the virtual temperature of the

air.

Therefore,

ATMOSPHERIC STATICS

13

T should be replaced in the prethe virtual temperature T*. ceding equations by The baro7. Height Computation of Aerological Ascents. for moist air the

metric formula

temperature

is

used for the solution of a great number of

practical problems as, for instance, for the height computations of aerological ascents. Because the aerological data must be

1000 -10

10

20

4000

Temperature, C 2000

1000

FIG. 2.

July

3,

3000

30

Dyn. meter Height computation of an aerological ascent after V. Bjerknes. Toronto, 1939. (The ordinate is p 2 ^ 8 not in p for reasons given on page 23.) -

,

quickly available for the daily weather analysis, a

number

of

methods have been developed for the computation of the height of any point in the atmosphere for which aerological observations 2 are available. 1 Only the method of V. Bjerknes will be described here.

From

the aerological ascents the pressure

ture T, and the relative humidity / for a 1

STOVE, G.,

"

number

Meteorologisches Taschenbuch,"

II,

p,

the tempera-

of points in the

Akademische Verlags-

gesellschaft, Leipzig, 1933. 2

BJERKNES,

"

V.,

Dynamic Meteorology and Hydrography," Chap.

Carnegie Institution of Washington, Washington, D. C., 1910.

VI,

DYNAMIC METEOROLOGY

14

atmosphere are obtained. They are plotted on a chart whose abscissa is the temperature on a linear scale and whose ordinate As an is the pressure on a logarithmic scale (T-ln p chart). Toronto on the ascent made at airplane July 3, 1939, example is

plotted in Fig. 2 (broken curve). for this ascent are

The data

The height of the Toronto airport To find the height of each point temperature has to be determined tion of this quantity from (5.3)

much obtain

is

187 dyn. meters.

of observation the virtual

first.

would

Because the computarequire, in practice, too on the T-ln p chart to

time, provision has been made The difference between virtual temperit more directly.

ature and temperature

is

approximately

T* - T =

^^ T

6

0.379 f

As long as the relative humidity is 100 per cent, the difference T is a function of pressure and temperature only. T* Therefixed of T* T is the and the value by pressure fore, temperature of each point on the chart. It is indicated by the distance between each two successive short vertical lines on every isobar For instance, when the representing a multiple of 100 mb. the mb is and 700 pressure temperature +10C, the virtual of air be about +12C. When the saturated would temperature T is obtained relative humidity is less than 100 per cent, T* T for saturated air by /. by multiplying the difference T* In the previous example a relative humidity of 50 per cent would In this manner the give a virtual temperature of +HC. 7

virtual-temperature curve can be plotted quite easily

(full

curve

in Fig. 2.)

The

height

may now

be expressed in dynamic meters in order

to eliminate the acceleration of gravity g. Upon substituting the equation of state for moist air (5.4) in (6.3), it follows that

ATMOSPHERIC STATICS

IS

and, by integration

fa

fa

= _

f ln

_R 1

Pl

10

=

T*d(ln p)

\

Jin

1 77^

10

Pl

#^1,2* In

v ^

(7.1)

p2

Here

-

(In pi

Ti, 2

*

is

In p2)7Y2*

a suitably defined

between pi and p 2

r*d(ln p)

(7.2)

mean

virtual temperature in the layer can easily be found on the T-ln p

Ti,z*

.

= f /ln p

chart. Consider, for instance, the virtual-temperature distribution between 900 and 800 mb in Fig. 2. The integral on the

right-hand side of (7.2)

represented by the area enclosed

is

= 900 mb and p% = 800 mb and between 273C (0 abs) and the virtual-temperature

between the isobars pi the isotherm curve.

t

=

Equation

(7.2)

shows that the isotherm representing the

virtual temperature T* must be chosen so that the area enclosed between the isobars 900 and 800 and the iso-

mean

mb

therms

273C and

is

TI,**

mb

equal to the area given

by the

Thus, the shaded triangles in Fig. 2, which are bounded by the virtual-temperature curve, the isotherm In Ti, 2 *, and the isobars 900 mb and 800 mb must be equal.

integral in (7.2).

practice the

mean

virtual temperature of a layer can be deter-

mined quite accurately in this manner even if the virtualtemperature curve is more complicated. The mean virtual temperatures for the ascent at Toronto on July 3, 1939, are given in Fig. 2 under the heading T m *. The dynamic height difference between two pressure levels depends only on the mean virtual temperature of the layer. In practice the height differences between levels whose pressures are multiples of 100 mb, the so-called "standard" isobaric sur-

Tables giving the dynamic height between standard isobaric surfaces for various virtual

faces, are first determined.

differences

1

temperatures are available. If the pressure p 2 is not a standard pressure,

(7.1)

may be

written .

l

BjERKNEs, Taschenbuch,"

=

-

+

a may be considered, i.e., where the stable originally. When Ap > 0, i.e., when the air descends, a! < a, i.e., the lapse rate becomes smaller, and when Ap is sufficiently large, a' may even become zero or is

When the lapse rate is negative, the temperature negative. Thus an inversion may be formed by increases with elevation. sinking and spreading of the

air.

This process occurs frequently

where large inversions are observed which from their origin are called subsidence inversions. 1 Because the stratification is more stable the smaller in the center of stagnant anticyclones

compared with the adiabatic lapse rate, the be stated by saying that descending motion in may an atmosphere with originally stable stratification increases the stability of the air. On the other hand, when the air ascends the lapse rate

(or

is

also

result

when

rate a

its

cross section decreases),

becomes

i.e.,

when Ap
T, downward motions is just the opposite. Downward motion increases the lapse rate; upward motions

the effect of upward and

make

the lapse rate smaller.

When the lapse unchanged.

of finite height, 2 1

2

NAMIAS,

J.,

HAURWITZ,

rate

was

originally adiabatic (r = a) it remains also be extended to air columns

The method can

but the lapse rates resulting from vertical

Harvard Met. Studies, No. B.,

Ann. Hydr.,

2,

69, 22, 1931.

1934.

DYNAMIC METEOROLOGY

26

adiabatic motion of finite air columns are not very different from those obtained from the preceding formula (10.21).

For a graphical determination of the change of the lapse rate in a layer of air that ascends or descends adiabatically, the in Fig. 6 adiabatic chart may be used. The full curve may represent the original pressure and temperature distribution.

AB

subjected to vertical adiabatic motion, each must move along an adiabat (broken curves). If A point of comes to rest at a pressure p