Salby M.L. Fundamentals of Atmospheric Physics

10

1 A Global View

I00

90

80

7O

E

6O

I,I

a

D 5O

30

20

I0

TEMPERATURE (K)

120 160 200 240 280 520

I%\ i I l ."I i I t I i I

- \ :"

\\ i

\ :.. _

\\ "'....

-

N

..,,, ~

X X "'""....

N ..'"

XN..."" _

;" N ~

:" \ \ "

--- Den-sit;- i...

N~\

.....

9 Temperoture

............ ~\

""..

IIIIIIIII

i ttllml t tlimtl

iiiiiiiii i iiiiiii I IIIIHII IIIIII

I0 -3 I0 -2 I0 -I

I I0 102 103

PRESSURE (mb)

I ~ lllnul I Ittntd i tittt~ ! Iltllld I tltltli t llttntl I Ittnul i ~l

tO -6 tO -5 tO -4 tO -3 tO -2 tO -I I

DENSITY (kg

m -3)

Figure 1.2 Global-mean pressure (solid), density (dashed), and temperature (dotted), as

functions of altitude.

Source:

U.S. Standard Atmosphere (1976).

Diffusion of momentum and heat associated with those properties dissipate

atmospheric motions by destroying gradients of velocity and temperature.

Below 100 km, the mean free path is small enough for turbulent eddies in

the circulation to be only weakly damped by molecular diffusion. At those al-

titudes, bulk transport by turbulent air motions dominates diffusive transport

of atmospheric constituents. Because turbulent air motions stir different gases

with equal efficiency, the mixing ratios of passive constituents are homoge-

neous in this region and the components of air are said to be "well mixed."

Turbulent mixing below 100 km makes the densities of passive constituents

decrease with altitude at the same exponential rate, which gives air a homo-

geneous composition with constant mixing ratios rN2 ~ 0.78, ro2 ~ 0.21 and

the constant gas 9 3

properties

M(t = 28.96 g mo1-1 (1.18.1)

R a

= 287.05 Jkg-lK -1 . (1.18.2)

3properties of dry air are tabulated in Appendix B along with other thermodynamic constants.

1.2 Composition and Structure 11

E

v

LU

s

I--

,_1

<

300

200

100

0

10-10 10-9 10 4

TEMPERATURE (K)

200 400 600 800 1000 1200 1400 1600 1800

,~ i 1, ~ i ~, , ' ~l' , , , i 1 i 1 i

\ \ " "~ ~'\ /TN , .... '/: /

I\ \ ~) ~ ~ j Solar ,' ....

I \ , ~, \ ,~: 9

....

I \ ~'~" ~, f~ ........ TActive

T ~I:~ " ~ 02 ~'-

/" \ ~

!, \\ \%\ ~,

~:~'~

.......

' ' I I I ,I I I I I "~ ~1 "1

10 8 10 7 10 -6 1,0 -5 10 -4 10 -3 10 2 10 1 10 0 101 102 10 3

PRESSURE (rob)

NUMBER DENSITY (rn "3)

1012 1013 10 TM 1015 1016 1017 1018 1019 1020 1021 102 2 1023 102 4 102 5 102 6 102 7 102 8

I ,i ,i , i i , I t, i L i ,i i , i I , i i

12 14 1'6 1'8 20 2'2 2'4 2'6 2'8 ' 30

MOLAR WEIGHT

Figure 1.3 Global-mean pressure (bold), temperature (shaded), mean molar weight (solid),

and number densities of atmospheric constituents, as functions of altitude.

Source:

U.S. Standard

Atmosphere (1976).

The well-mixed region below 100 km is known as the

homosphere

and processes

within it form the primary focus of this book.

Above 100 km, the mean free path quickly becomes larger than turbulent

displacements of air. As a result, turbulent air motions are strongly damped

by diffusion of momentum and heat, and diffusive transport becomes the

dominant mechanism for transferring properties vertically. The transition from

turbulent transport to diffusive transport occurs at the

homopause

(also known

as the

turbopause),

which has an average altitude of about 100 km. Figure 1.4

shows a rocket vapor trail traversing this region. Below 100 km, the trail is

marked by turbulent eddies which are produced in the wake of the rocket and

which homogenize different constituents with equal efficiency. These eddies

12

1 A Global View

are conspicuously absent above 107 km, where molecular diffusion suppresses

turbulent air motions and becomes the prevailing form of vertical transport.

The region above the homopause and below 500 km is known as the

het-

erosphere.

In the heterosphere, air flow is nearly laminar. Because it operates

on gases according to their molar weights, molecular diffusion stratifies con-

stituents so that the heaviest species 02 decreases with altitude more rapidly

than the second heaviest species N2 and so forth (Fig. 1.3). For this reason,

composition changes with altitude in the heterosphere, as evidenced by the

mean molar weight in Fig. 1.3. Although constant below the homopause, M

changes abruptly near 100 km, above which it decreases monotonically with

altitude.

Diffusive separation is primarily responsible for the stratification of con-

stituents in the heterosphere. However, photodissociation also plays a role.

Energetic ultraviolet (UV) radiation incident at the top of the atmosphere

dissociates molecular oxygen, providing an important source of atomic oxygen

at these altitudes. In fact, O becomes the dominant form of oxygen not far

above the homopause. Photodissociation of H20 at lower altitudes by less en-

ergetic UV radiation liberates atomic hydrogen, which is gradually mixed and

diffused to higher altitudes. By dissociating such molecules, energetic radiation

is filtered from the solar spectrum penetrating to lower levels.

6J

106 km

,J

102

km

t I

g. 11m

D 108 km

t

112km

Figure

1.4 Constant-density contours of a chemical vapor trail released by a rocket traversing

the turbopause. Beneath 107 km, the vapor trail is distorted by an array of turbulent eddies that

form in the wake of the rocket. Above 107 km, the vapor trail remains laminar, reflecting the

absence of turbulence, and expands under the action of molecular diffusion. Adapted from Roper

(1977).

1.2

Composition and Structure

13

In the homosphere and heterosphere, atmospheric molecules interact

strongly through frequent collisions. Above an altitude of about 500 km,

which is referred to as the

critical level,

molecular collisions are so rare that

a significant fraction of the molecules passes out of the atmosphere without

sustaining a single collision. Known as the

exosphere,

the region above the

critical level contains molecules that leave the denser atmosphere and move

out into space. As is illustrated in Fig. 1.5, these molecules follow ballis-

tic trajectories that are determined by the molecular velocity at the critical

level and the gravitational attraction of the planet. Most of the molecules

in the exosphere are captured by the earth's gravitational potential. They

return to the denser atmosphere along approximately parabolic trajectories.

However, some molecules have velocities great enough to escape the earth's

gravitational potential entirely and those are lost to deep space.

The escape velocity

Ve

is determined by the kinetic energy adequate to

liberate a molecule from the potential well of the planet's gravitational field.

Since that energy equals the work performed to displace a molecule from the

critical level to infinity,

fa ~ (a)2

1 mv~ -

mg o

dr,

(1 19)

7 "

/ \ \

/

\ ,~ \

/ \

i \

I~

\

/ \

I \

/ \

I

\

/ '4 \

/ \

/

IX \

/

I

\ \

/ / \

/

I \ \\

0

Figure 1.5 Schematic cross section of the atmosphere illustrating the homosphere, hetero-

sphere, and the exosphere, in which molecular trajectories are shown.

14

1

A Global View

where a is the earth's radius, go is the gravitational attraction averaged over

the surface of the earth, and the difference between a and the radial distance

to the critical level is negligible. The escape velocity follows from (1.19) as

V e --

v/2go a.

(1.20)

For earth, v e has a value of about 11 km

S -1.

Independent of molecular weight,

Ve

is the same for all molecules. However, different molecules do not all have

the same distribution of velocities. Because energy is equipartitioned in a

molecular ensemble (e.g., Lee, Sears, and Turcotte, 1973), lighter molecules

have greater velocities than heavier ones. Consequently, lighter atmospheric

constituents escape to space more readily than do heavier constituents.

The critical level of the earth's atmosphere lies near 500 km. At this altitude,

the temperature is about 1000 K under conditions of normal solar activity, but

can reach 2000 K during disturbed conditions (Fig. 1.3). Figure 1.6 shows for

0

10

[-/

1 --1U

" "

Z

O -2

10

<

.--I

23

12.

0

13_ -3

10

_J

<

z

0

I-

0 0. 4

< 1

tr

LL

" H

X

\

X

\

X

\

N

10 -5

-6 L

10

0

2 4 6 8 10 12 14 16 18

Molecular Velocity (km s -1)

Figure

1.6 Boltzmann distribution of velocities for a molecular ensemble of oxygen atoms

and hydrogen atoms. Escape velocity

Ve

for earth also indicated.

1.2 Composition and Structure

15

O and H the Boltzmann distributions of a molecular ensemble as functions of

molecular velocity

dn-4v2 [ (-~o)21

n -- ~/~ v03 exp -

dv, (1.21)

where

dn/n

represents the fractional number of molecules having velocities in

the range

(v, v + dr),

g/ 2k T

v 0 - -- (1.22)

m

is the most probably velocity, m is the mass of the molecules, and k is the

Boltzmann constant. The fraction of molecules with velocities exceeding the

escape velocity is

E I O)21

n e --Vu exp

dv

[ e21

(:ot exp

forv0

<< V e.

For atomic oxygen, the most probable velocity is v 0 = 1.02 km s -1. The

fraction of O molecules having velocities greater than v e is

only

about 10 -45. A

lower bound on the time to deplete all O molecules initially at the critical level

(e.g., neglecting production of O molecules locally by photodissociation and

diffusive transport from below) is given by the mean time between collisions

divided by the fraction of molecules moving upward with v >

v e.

Near 500 km,

the mean time between collisions is of order 10 s. Hence, the time for all O

molecules initially at the critical level to escape the earth's gravitational field

is greater than

10 46

smfar greater than the four billion years the planet has

existed. Heavier species are captured by the earth's gravitational field even

more effectively.

The situation for hydrogen differs sharply. The population of H is dis-

tributed over much higher velocities, so many more molecules exceed the

escape velocity

l) e.

The most probable velocity is 4.08 km

S -1,

whereas the

fraction of molecules having velocities greater than

Ve

is about 10 -4. By pre-

vious reasoning, a significant fraction of H molecules initially at the critical

level is lost to deep space after only 105 s or 1 day. During conditions of dis-

turbed solar activity, temperatures at the critical level are substantially higher,

so hydrogen molecules are boiled off of the atmosphere even faster. The rapid

escape of atomic hydrogen from the planet's gravitational field explains why

H is found only in very small abundances in the earth's atmosphere, despite

its continual production by photodissociation of H20.

16 1 A Global

View

1.2.3 Thermal and Dynamical Structure

The atmosphere is categorized according to its thermal structure, which de-

termines the dynamical properties of individual regions. The simplest picture

of the atmosphere's thermal structure is provided by the vertical profile of

global-mean temperature in Fig. 1.2. From the surface up to about 10 km,

temperature decreases with altitude at a nearly constant

lapse rate,

which is

defined as the rate of "decrease" of temperature with altitude. This layer im-

mediately above the earth's surface is known as the

troposphere,

which means

"turning sphere" and symbolizes the convective overturning that characterizes

this region. Having a global-mean lapse rate of about 6.5 K km -1, the tropo-

sphere contains most of what is known as weather and is driven ultimately by

surface heating. The upper boundary of the troposphere or "tropopause" lies

at an altitude of about 10 km (100 mb) and is marked by a sharp change of

lapse rate.

The region from the tropopause to an altitude of about 85 km is known as

the

middle atmosphere.

Above the tropopause, temperature first remains nearly

constant and then increases in the

stratosphere,

which means "layered sphere"

and is symbolic of properties at these altitudes. Increasing temperature with

altitude (negative lapse rate) in the stratosphere reflects ozone heating, which

results from the absorption of solar UV. Contrary to the troposphere, the

stratosphere involves only weak vertical motions and is dominated by radiative

processes. The upper boundary of the stratosphere or "stratopause" lies at an

altitude of about 50 km (1 mb), where temperature reaches a maximum.

Above the stratopause, temperature again decreases with altitude in the

mesosphere,

where ozone heating diminishes. Convective motions and radiative

processes are both important in the mesosphere. Meteor trails form in this

region of the atmosphere, as do lower layers of the ionosphere during daylight

hours. The "mesopause" lies at an altitude of about 85 km (0.01 mb), where

a second minimum of temperature is reached.

Above the mesopause, temperature increases steadily in the

thermosphere

(compare Fig. 1.3). Unlike lower regions, the thermosphere cannot be treated

as an electrically neutral continuum. Ionization of molecules by energetic solar

radiation produces a plasma of free electrons and ions, each of which interacts

differently with the earth's electric and magnetic fields. As is apparent in Fig.

1.3, this region of the atmosphere is influenced strongly by variations of solar

activity. However, its influence on processes below the mesopause is very

limited.

A more complete picture of the thermal structure of the atmosphere is

provided by the zonal-mean temperature T, where the overbar denotes the

longitudinal average, which is shown in Fig. 1.7 as a function of latitude and al-

titude during northern winter. In the troposphere, temperature decreases with

altitude and latitude. The tropopause, which is characterized by an abrupt

change of lapse rate, is highest in the tropics (-~16 km), where temperatures

1.2 Composition and Structure 17

TEMPERATURE (K)

80-~~~~~________----~~..~-~ ~/''

2201//I

, , I ~0.0[

7o-

1 o.,

E 60 - ~........__.___....~~.. 'u

~260 :::0

~~

9

r'q

50 ~'"--,~ ~ ~ I co

E3 (/)

D C

H

40 ~ ~- ;e

rn

~ ~ 240 ~~ ~ 10 ----

--J 50 ~

~.~ ~< i

3

20 ~ I00

,o

0 IO00

90S 60S 30S 0 30N 60N 90N

LATITUDE

Figure 1.7 Zonal-mean temperature during northern winter as a function of latitude and

altitude. Adapted from Fleming

et al.

(1988).

are quite cold, and lowest in polar regions (~8 km). A sharp change of zonal-

mean lapse rate is not observed at midlatitudes, which is symbolized by a

break in the tropopause. In the stratosphere, temperature increases with alti-

tude. Temperatures are warmest over the summer pole and decrease steadily

to coldest values over the winter pole. In the mesosphere, where temperature

again decreases with altitude, the horizontal temperature gradient is reversed.

Temperatures are actually coldest over the summer pole, which lies in per-

petual daylight, and increase steadily to warmest values over the winter pole,

which lies in perpetual darkness. This peculiarity of the temperature distribu-

tion, which is contrary to radiative considerations, illustrates the importance

of dynamics to establishing observed thermal structure.

The thermal structure in Fig. 1.7 is related closely to the zonal-mean circula-

tion ~, which is shown in Fig. 1.8 at the same time of year. In the troposphere,

the circulation is characterized by subtropical

jet streams,

which strengthen

with altitude up to the tropopause. These jets describe circumpolar motion

that is westerly in each hemisphere/ Above the subtropical jets, the zonal-

mean flow first weakens with altitude and then intensifies with opposite sign

in the two hemispheres. In the winter hemisphere, westerlies intensify above

the tropopause in the

polar-night jet,

which reaches speeds of 60 m s -1 in the

lower mesosphere. In the summer hemisphere, westerly flow weakens above

the tropopause and is then replaced by easterly flow that intensifies up to the

mesosphere. Reaching speeds somewhat stronger than the zonal-mean flow

4In meteorological parlance,

westerly

refers to motion from the west and

easterly

to motion

from the east.

18

80

70

60

E

50

I,I

D 40

2O

I0

0

90S 60S 50S 0

A Global View

ZONAL VELOCITY (m s -1)

-

,/ ,/ z.,.~,~-L.'--],,\\,I

V20

.y

, ,~ rx\,201 '

I' 0.01

/ /./,/,~?~".~l i I~ / 40 ,\~

I I !, ~ t. ~j \

\~,\ x\\\\\ ~

o I

-o

x x\x ,. ,. \ ~.r

m

\ ~\ x x x ~, \ ,., t.h

\ \ \. \

~-~v..y.l

/ Ift'

m

\ \-~ ~'-40'

I I ; ~"

~_.0~

\. "'L'----

)1 )

I0 cr

~ -20//

~o~"~ _~

~ - Ioo

------'OOO

:5ON 60N 90N

LATITUDE

Figure 1.8 As in Fig. 1.7, but for the zonal component of velocity.

in the winter hemisphere, this easterly circulation merges below with weak

easterlies in the tropical troposphere.

On individual days, the circulation is more complex and involves a good

deal more variability than is represented in the zonally and time-averaged

distributions in Figs. 1.7 and 1.8. Figure 1.9a shows for an individual day

the instantaneous circulation on the 500-mb isobaric surface: that locus of

points where the pressure equals 500 mb. Contoured over the circulation is

the altitude of the 500-mb isobaric surface. Because hydrostatic equilibrium

(1.17) implies a single-valued relationship between pressure and altitude (Fig.

1.2), the elevation of this isobaric surface may be interpreted analogously to

the pressure on a surface of constant altitude. Because pressure decreases

monotonically with altitude, low elevation of the 500-mb surface corresponds

to low pressure on the surface with a constant altitude of 5 km. Similarly, high

elevation of the 500-mb surface corresponds to high pressure on that constant

altitude surface. Therefore, centers of low elevation that punctuate the height

of the 500-mb surface in Fig. 1.9a imply centers of low pressure on a constant

altitude surface, whereas centers of high elevation imply the reverse.

The 500-mb surface slopes downward toward the pole, in the direction of

decreasing tropospheric temperature (Fig. 1.7). The circulation at 500 mb is

characterized by westerly circumpolar flow, which corresponds to the zonal-

mean subtropical jet in Fig. 1.8. But the jet stream on the individual day shown

is far from zonally symmetric. Half a dozen depressions of the 500-mb sur-

face, which are associated with

synoptic weather systems,

distort the circulation

into a wavy pattern that meanders around the globe. These unsteady distur-

bances deflect the air stream meridionally and migrate from west to east. The

circumpolar flow, although highly disturbed by these features, remains nearly

parallel to contours of isobaric elevation.

1.2 Composition and Structure

19

--- (a) 500 mb: Mar 4, 1984 ~,~;E

.- "-...~.,. ,.

~,., .~ ~.,-" -,-.,

"-.t:---

"--..,...,,,,,

,.-.-- ...........

.. , '~..." ~.'.': ~'

/:"-4\ " "

'

-* i

o" Z;

"-- ~

" i

,,--b':"

"~,~-. "-- 'v-,,: " "-

- .

............... :'.',

............... .-::

"L~ .---""

"; "~

'

~ ,, ......... --....

,.>-~

l

..........

.... . ~.~.f.f"

r176

,.o--'P

',.b / ' . - ,

; ~ :.'

, l/

#

.."

:

5; ~

t

,z! .i' a~l ......... .., -"

\ / .'

9 ,.%. ~,, lVl , ~

,,

X /o

% "~ "" '~'"~ Z

: Z

", - ~,, --, = - ;&-'"- "- X- ........ ......... "-

..... ,, .... -

-~ ....... .% ~..--~, ..-'r .............. ~.~ ..... -..

%,,. ,,,: ..,Y' "-'--

50 m/s

........................

;.-

.......- ......

Figure 1.9 (a) Height (contours) of and horizontal velocity (vectors) on the 500-mb isobaric

surface for March 4, 1984.

(continues)

In addition to synoptic weather systems, disturbances of global dimension

also appear in the 500-mb circulation. Known as

planetary waves,

these distur-

bances are manifested by a displacement of the circumpolar flow out of zonal

symmetry (e.g., at high latitudes) and by a gradual undulation about latitude

circles of the jet stream. Planetary waves are more evident in the time-mean

circulation, shown in Fig. 1.9b, in which unsteady synoptic weather systems

are filtered out. The time-mean circulation exhibits more zonal symmetry and

therefore a greater correspondence to the subtropical jet in Fig. 1.8 than does

the instantaneous circulation in Fig. 1.9a. However, the time-mean flow is

still disturbed, only on larger scales. Steady planetary waves, which are not

removed by time-averaging, displace the circumpolar flow out of zonal sym-

metry to again deflect the air stream across latitude circles. In addition, the

Salby M.L. Fundamentals of Atmospheric Physics

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