64226 > CD
^ CO >
DYNAMIC METEOROLOGY
DYNAMIC METEOROLOGY BY
BERNHARD HAURWITZ,
PH.D.
Chairman
of the Departmen...
136 downloads
546 Views
14MB Size
Report
This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form
64226 > CD
^ CO >
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
McGRAWHILL BOOK COMPANY, NEW YORK AND LONDON 1941
INC.
DYNAMIC METEOROLOGY COPYRIGHT, 1941, BY THE
McGRAWHiLL 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 Dryadiabatic 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 16 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 Saturatedadiabatic
Changes Chart
to
Atmospheric
Processes.
23 26 29 35 36
36 38
39 41
45
Pseudoad iabatic
Lapse Rate 20. Stability with Respect to Saturated Adiabatic Changes. 19. Saturatedadiabatic
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.
Wetbulb Temperature and Wetbulb Potential Tempera70 75 76 78 84
ture 26. Latent Instability 27. Potential, or Convective, Instability
28.
Therm odyiiamic Charts and Airmass 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 Watervapor 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 912
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 1319 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 2021 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 2628
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 2225
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
Atmosphericenergy 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. Largescale Oscillations of the
Atmosphere
Problems 3031
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 centimetergramsecond (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 3M6M. "Smithsonian Meteorological " Tables," Tables 6468. LINKS, F., Meteorologisches Taschenbuch," I, Tables 2627, 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
metertonsecond 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 watervapor 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 watervapor pressure can rise only up to a certain maximum, the saturation, or maximum, vapor pressure
than
e m,
the existing watervapor pressure e is smaller evaporation from liquidwater 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 watervapor 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 watervapor 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 ClausiusClapeyron for the heat of condensation. Besides the absolute humidity, which is used rarely in meteorological practice, the watervapor 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
(57D (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 dryadiabatic 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
pdp
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 watervapor 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 (Tln 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 Tln 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
virtualtemperature 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 Tln p
Ti,z*
.
= f /ln p
chart. Consider, for instance, the virtualtemperature distribution between 900 and 800 mb in Fig. 2. The integral on the
righthand side of (7.2)
represented by the area enclosed
is
= 900 mb and p% = 800 mb and between 273C (0 abs) and the virtualtemperature
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 virtualtemperature 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 socalled "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