Salby M.L. Fundamentals of Atmospheric Physics

Подождите немного. Документ загружается.

50 1 A Global

View

itudes have a deficit of net radiation (cooling). During northern winter, the

entire Northern Hemisphere poleward of 15 ~ absorbs less energy in the form

of SW radiation than it emits as LW radiation. At the same time, most of the

Southern Hemisphere absorbs more energy in the form of SW radiation than

it emits as LW radiation. Only over Antarctica, where permanent snow cover

produces a high albedo, is the net radiation negative. Net radiation is posi-

tive and large almost uniformly across the tropics and in the subtropics of the

summer hemisphere, where strong insolation and cloud-free conditions prevail.

Even though, individually, albedo and OLR exhibit large geographical varia-

tions across the tropics, a cancellation between these components in regions

of convective activity nearly eliminates those anomalies from the distribution

of net radiation. As a result, net radiative forcing of the earth-atmosphere

system exhibits a good measure of zonal symmetry.

The optical properties described above lead to nonuniform heating of the

earth-atmosphere system. Low latitudes are heated radiatively, whereas middle

and high latitudes are cooled radiatively. To preserve thermal equilibrium, the

surplus of radiative energy arriving at low latitudes must be transferred to

middle and high latitudes, where it offsets a deficit of radiative energy. About

60% of this meridional transfer of energy is accomplished by the general

circulation of the atmosphere.

1.5 The General Circulation

The term

general circulation

usually refers to the aggregate of motions con-

trolling transfers of heat, momentum, and constituents. Broadly speaking, the

general circulation also includes interactions with the earth's surface (e.g.,

transfers of latent heat and moisture from the oceans) because atmospheric

behavior, especially that operating on timescales longer than a month, is in-

fluenced importantly by such interactions. For this reason, properties of the

lower boundary, like sea surface temperature (SST), figure importantly in the

general circulation of the atmosphere.

The general circulation is maintained against frictional dissipation by a

conversion of potential energy, which is associated with the distribution of

atmospheric mass, to kinetic energy, which is associated with the motion of

air. Radiative heating acts to expand the atmospheric column at low latitudes,

according to hydrostatic equilibrium (1.17), and to raise its center of mass. By

contrast, radiative cooling acts to compress the atmospheric column at middle

and high latitudes and lower its center of mass there. The uneven distribution

of mass that results introduces an imbalance of pressure forces that drives a

meridional overturning, with air rising at low latitudes and sinking at middle

and high latitudes.

The simple meridional circulation implied above is modified importantly

by the earth's rotation. As is evident from the instantaneous circulation at

1.5

The General Circulation

500 mb (Fig. 1.9a), the large-scale circulation remains nearly parallel to con-

tours of isobaric height. Net radiative heating in Fig. 1.29c tends to establish

a time-mean thermal structure in which isotherms and contours of isobaric

height are oriented parallel to latitude circles. Consequently, the time-mean

circulation at 500 mb (Fig. 1.10b) is nearly circumpolar at middle and high

latitudes. Characterized by a nearly zonal jet stream, the time-mean circula-

tion possesses only a small meridional component to transfer heat between

the equator and poles. A similar conclusion applies to the stratosphere, where

time-mean motion is strongly zonal (Fig. 1.10b).

For this reason, asymmetries in the instantaneous circulation that deflect

air meridionally play a key role in transferring heat between the equator and

poles. In the troposphere, much of the heat transfer is accomplished by un-

steady synoptic weather systems, which transport heat in sloping convection

that exchanges cold polar air with warm tropical air. Ubiquitous in the tropo-

sphere, those disturbances contain much of the kinetic energy at midlatitudes.

They develop preferentially in the North Pacific and North Atlantic storm

tracks and in the continuous storm track of the Southern Hemisphere. By

rearranging air, synoptic disturbances also control the distributions of water

vapor and other constituents produced at the earth's surface.

In the stratosphere and mesosphere, synoptic disturbances are absent. Plan-

etary waves, which propagate upward from the troposphere (Fig. 1.10), play

a role at these altitudes similar to the one played by synoptic disturbances in

the troposphere. Generated near the earth's surface, these global-scale dis-

turbances force the middle atmosphere mechanically. By deflecting air across

latitude circles, planetary waves transport heat and constituents between low

latitudes and high latitudes. Such transport is behind the largest abundances

of ozone being found at middle and high latitudes (Fig. 1.18), despite its pro-

duction at low latitudes.

The earth's rotation exerts a smaller influence on air motions at low lati-

tudes. Kinetic energy there is associated primarily with

thermally direct circula-

tions,

in which air rises in regions of heating and sinks in a regions of cooling.

Thermally direct circulations in the tropics are forced by the geographical dis-

tribution of heating (e.g., as is implied by time-mean cloud cover in Fig. 1.25b).

Latent heat release inside the ITCZ drives a meridional

Hadley circulation,

which air rises near the equator and sinks at subtropical latitudes. Subsiding

air in the descending branch of the Hadley circulation maintains deserts that

prevail at subtropical latitudes; compare Fig. 1.29b.

Nonuniform heating also drives zonal overturning, known as a

Walker cir-

culation,

in which air rises at longitudes of heating and sinks at longitudes

of cooling. The nonuniform distribution of land and sea and asymmetries in

radiative, conductive, and latent heating that accompany it lead to Walker

circulations along the equator. The concentration of latent heating over In-

donesia (Fig. 1.29b) forces the Pacific Walker circulation, which is illustrated

in Fig. 1.30. This circulation reinforces easterly trade winds across the equa-

1 A Global

View

... m .... ~ /

Upper

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

Tropospheric

I ~ I,/ I [J I-,f / \ JI. ~ L--

Lower

4_ iq [ .J-

P=P,

_/. ~' 1% ~--I"-, _ j.1 w.

Tropospheric

k"/._~&L.~LX

" ~-~.= ~ J ~ ~

JL =~"~,~='" J

Surface

.........

i ......

/ i:i. ........ """'::';::'=

,nd/ion i:~::':''::'~iii:iil;" ,'i:;" I Pacific ~'""i...!("""i'itlantic N N

0 90E 180 90W 0

Figure 1.30 Schematic of the equatorial Walker circulation. Levels indicated correspond to

isobaric surfaces in the upper and lower troposphere. Adapted from Webster (1983).

torial Pacific, and maintains the arid climate that typifies the eastern Pacific

(compare Figs. 1.29a,b).

Latent heat release also excites unsteady wave motions. Another form of

asymmetry in the circulation, these disturbances propagate the influence of

their source region, which is subsequently deposited in regions where they are

absorbed. Wave motions are also excited mechanically at the earth's surface. By

displacing atmospheric mass vertically, orographic features like the/kips, the

Himalayas, and the Rocky Mountains excite planetary waves and gravity waves

that radiate away from those source regions. Like wave activity generated

by unsteady heating, those disturbances can propagate the influence of their

forcing far from regions where they are excited.

Kinetic energy associated with the general circulation is damped by fric-

tional dissipation, which involves turbulent air motion that operates on a wide

range of scales. About half of the large-scale kinetic energy is dissipated in

the lowest kilometer of the atmosphere. Inside the

planetary boundary layer,

small-scale turbulence extracts energy from large scales and cascades it to

small dimensions, where it is dissipated by molecular diffusion. The remaining

large-scale kinetic energy is dissipated in the free atmosphere by convective

motions and dynamical instability, which generate turbulence from large-scale

organized motion. Large-scale circulations are also damped by thermal dis-

sipation (e.g., by IR cooling to space), which acts on motion indirectly by

destroying the accompanying temperature anomalies.

The description of the general circulation completes this overview of the

atmosphere, which has been presented in terms of a wide range of behavior.

Dynamical behavior comprising the general circulation follows from the strat-

ification of mass and thermal structure, which in turn are shaped by energy

transfer in the atmosphere. Because air motion controls transfers of sensible

and latent heat as well as distributions of radiatively active species, the pro-

cesses responsible for atmospheric behavior are interconnected. With these

phenomenological ingredients as motivation, we now proceed to develop the

fundamental principles of thermodynamics, hydrostatics, radiative transfer,

and atmospheric dynamics that form cornerstones of atmospheric behavior.

Problems

Suggested Reading

The Boltzmann distribution is developed along with other components of

molecular kinetics in

Statistical Thermodynamics

(1973) by Lee, Sears, and

Turcotte.

Aeronomy of the Middle Atmosphere

(1986) by Brasseur and Solomon includes

a comprehensive treatment of anthropogenic trace gases and impacts they may

have on the atmosphere.

The Intergovernmental Panel Report on Climate Change

(1990), edited by

Houghton, Jenkins, and Ephraums, provides an overview of the climate prob-

lem, numerical modeling of it, and historical evidence of previous climates.

Atmospheric Science: An Introductory Survey

(1977) by Wallace and Hobbs

contains an excellent introduction to the general circulation and basic concepts

of radiative transfer.

A comprehensive treatment of the global energy budget and its relationship to

the general circulations of the atmosphere and oceans is presented in

Global

Physical Climatology

(1994) by Hartmann.

Problems

1.1. Derive expression (1.15) for the molar fraction of the ith species in terms

of its mass mixing ratio.

1.2. Derive an expression for the volume mixing ratio of the ith species in

terms of its mass mixing ratio.

1.3. Relate the volume mixing ratio of the ith species to the fractional abun-

dance of i molecules present.

1.4. Show that the sum of partial volumes equals the total volume occupied

by a mixture of gases (1.6).

1.5. Demonstrate that 1 atm of pressure is equivalent to that exerted by

(a) a 760-mm column of mercury and (b) a 32-ft column of water. (The

density of Hg at 273 K is 1.36

104

kg m-3.)

1.6. Consider moist air at a pressure of 1 atm and a temperature of 20~

Under these conditions, a relative humidity of 100% corresponds to a

vapor pressure of 23.4 mb. At that vapor pressure, air holds the maxi-

mum abundance of water vapor possible under the foregoing conditions

and is said to be

saturated.

For this mixture, determine (a) the molar

fraction N___H~o, (b) the mean molar weight M, (c) the mean specific gas

constant R, (d) the absolute concentration of vapor PrimO, (e) the mass

mixing ratio of vapor ru~o, and (f) the volume mixing ratio of vapor.

1.7. From the distribution of water vapor mixing ratio shown in Fig. 1.14, plot

the vertical profile of water vapor number density

[HzO](z ) at

(a) the

equator and (b) 60~

1.8. In terms of the vertical profile of water vapor mixing ratio rH20(z ), derive

an expression for the total precipitable water vapor EH~O in millimeters.

54 1 A Global

View

1.9. If

rH~O(4', Z) -- r0(4,)e

-z/h,

where h --- 3 km and r0(4,) is the zonal-mean mixing ratio at the surface

and latitude ~b in Fig. 1.14, use the results of Problem 1.8 to compute

the total precipitable water vapor at (a) the equator and (b) 60~

1.10. From the mixing ratio of ozone ro3 shown in Fig. 1.17, plot the vertical

profile of ozone number density [O3](z) above 100 mb at (a) the equator

and (b) 60~

1.11. As in Problem 10a, but for the contribution to total ozone Eo3(Z) from

100 mb to an altitude z over the equator.

1.12. Clouds contribute much of the Earth's albedo, which has a value of

about 0.30. Calculate the change of equivalent blackbody temperature

corresponding to a 5% reduction of albedo.

1.13. The earth's equivalent blackbody temperature, Te - 255 K, corresponds

to a level in the middle troposphere, above the layer in which water vapor

is concentrated. This also corresponds to the level from which most of the

outgoing LW radiation is emitted to space. Suppose atmospheric water

vapor increases to raise the top of the (optically thick) water vapor

layer by 1 km, while the albedo remains unchanged. (a) What is the

corresponding change of the earth's equivalent blackbody temperature?

(b) If a lapse rate of 6.5 K km -1 is maintained inside the water vapor

layer, by how much will the earth's mean surface temperature change?

1.14. By appealing to the hydrostatic relationship (1.17), contrast the vertical

spacing of isobaric surfaces (p - const) at a horizontal site where air is

cold from one where air is warm.

1.15. Estimate the characteristic timescale for advection (a) in the tropo-

sphere, where synoptic disturbances to the subtropical jet have a char-

acteristic zonal wavenumber of 5 and move slowly relative to the zonal-

mean flow, and (b) in the stratosphere, where planetary wave distur-

bances to the polar-night jet have a zonal wavenumber of approximately

1 and are quasi-stationary.

1.16. The mean pressure observed in a planetary atmosphere varies with

height as

p(z) - 1 + (z/h) 2"

Determine the mean variation of temperature with height.

1.17. Venus has a mean distance from the sun of 0.7 that of Earth and an

albedo of 0.75. Calculate the radiative-equilibrium temperature of Venus

and compare it with the observed surface temperature of 750 K.

1.18. Calculate the fractional exchanges of energy in the global-mean energy

budget (Fig. 1.27), referenced to the mean incoming flux.

Chapter 2

Thermodynamics of Gases

The link between the circulation and transfers of radiative, sensible, and

latent heat between the earth's surface and the atmosphere is thermodynamics.

Thermodynamics deals with internal transformations of the energy of a system

and exchanges of energy between that system and its environment. Here, we

develop the principles of thermodynamics for a discrete system, namely, an

air parcel moving through the circulation. In Chapter 10, those principles are

generalized to a continuum of such systems, which represents the atmosphere

as a whole.

2.1 Thermodynamic Concepts

A thermodynamic system refers to a specified collection of matter (Fig. 2.1).

Such a system is said to be "closed" if no mass is exchanged with its surround-

ings and "open" otherwise. The air parcel that will serve as our system is,

in principle, closed. In practice, mass can be exchanged with the surround-

ings through entrainment and mixing across the system's boundary, which is

referred to as the control surface. In addition, trace species like water vapor

can be absorbed through diffusion across the control surface. Above the plan-

etary boundary layer, such exchanges are slow compared to other processes

influencing an air parcel, so the system may be regarded as closed.

The thermodynamic state of a system is defined by the various properties

characterizing it. In a strict sense, all of those properties must be specified

to define the system's thermodynamic state. However, that requirement is

simplified for many applications, as is discussed below.

2.1.1 Thermodynamic Properties

Two types of properties characterize the state of a system. A property that does

not depend on the mass of the system is said to be intensive, and extensive oth-

erwise. Intensive and extensive properties are usually denoted with lowercase

and uppercase symbols, respectively. Pressure and temperature are examples

Thermodynamics of Gases

Mean Boundary

Expanded Boundary

Figure 2.1 A specified collection of matter defining an infinitesimal air parcel. The system's

thermodynamic state is characterized by the extensive properties m and Z and by the intensive

properties p, T, and z.

of intensive properties, whereas volume is an extensive property. An intensive

property z may be defined from an extensive property Z, by referencing the

latter to the mass m of the system

z = --, (2.1)

in which case the intensive property is referred to as a

specific property. The

specific volume v =

Vim

is an example. A system is said to be

homogeneous

if its properties do not vary in space, and

heterogeneous

otherwise. Because

an air parcel is of infinitesimal dimension, it is by definition homogeneous so

long as it involves only gas phase. On the other hand, stratification of density

and pressure make the atmosphere as a whole a heterogeneous system.

A system can exchange energy with its surroundings through two fundamen-

tal mechanisms. It can perform work on the surroundings, which represents a

mechanical exchange of energy with its environment. In addition, heat can be

transferred across the control surface, which represents a thermal exchange of

energy between the system and its environment.

2.1.2 Expansion Work

The system relevant to the atmosphere is a compressible gas, perhaps con-

taining an aerosol of liquid and solid particles. For this reason, the primary

mechanical means of exchanging energy with the environment is expansion

work (Fig. 2.1). If a pressure imbalance exists across the control surface, it is

(~pdv

2.1

Thermodynamic Concepts

Figure 2.2 Expansion work performed by

the system during a cyclic process, wherein it is

restored to its initial state.

out of mechanical equilibrium. The system will then expand or contract to re-

lieve the mechanical imbalance. By adjusting to the environmental pressure,

the system performs expansion work or has such work performed on it. The

incremental expansion work

performed by displacing a section of control

surface

dS = dSn,

with unit normal n, perpendicular to itself by

= (pdS) . dn

(2.2)

= pdV,

where

dV = dSdn

is the incremental volume displaced by the section

dS.

Integrating (2.2) between the initial and final volumes yields the net work

performed "by" the system

W12 - pdV.

(2.3)

Dividing both sides by the mass of the system then gives the specific work:

w12 -

pdv.

(2.4.1)

For a cyclic variation, namely, one in which the initial and final states of the

system are identical,

- f pdv.

(2.4.2)

The cyclic work (2.4.2) is represented in the area enclosed by the path of the

system in the

p-v

plane (Fig. 2.2). Although the system is restored to its initial

state, namely, the net change of properties is zero, the same is not true of the

work performed during the cycle. Unless the system returns to its initial state

along the same path it followed out of that state, net work is performed during

a cyclic variation.

58 2

Thermodynamics of Gases

On a fluid system, another form of work can be performed, one analogous

to stirring. Paddle work corresponds to a rearrangement of fluid, as would occur

by turning a paddle immersed in the system. The reaction of the fluid against

the paddle must be overcome by a torque, which in turn produces work when

exerted across an angular displacement. Although secondary to expansion

work, paddle work has applications to disspative processes associated with

turbulence (see Fig. 2.1).

2.1.3 Heat Transfer

Energy can also be exchanged thermally via heat transfer Q across the system's

control surface. If it moves into an environment of a higher temperature, an air

parcel will absorb heat from its surroundings, for example, through diffusion

or thermal conduction. If the system is open, a similar process can occur

through the absorption of water vapor from the surroundings, condensation

and the release of latent heat, and precipitation out of the system of the

condensed water. Heat can also be exchanged through radiative transfer. By

communicating radiatively with surroundings at a higher temperature, an air

parcel will absorb more radiant energy than it emits.

For many applications, heat transfer is secondary to processes introduced

through motion, which operates on a timescale of order one day and shorter.

This is especially true above the boundary layer, where turbulent mixing be-

tween bodies of air is relatively weak. In the free atmosphere and away

from clouds, the prevailing form of heat transfer is radiative. Operating on

a timescale of order two weeks, radiative transfer is slow by comparison with

expansion work, which usually influences an air parcel on timescales of only

a day.

If no heat is exchanged between a system and its environment, the control

surface is said to be adiabatic, and diabatic otherwise. Because heat transfer

is slow compared to other processes influencing a parcel, adiabatic behavior

is a good approximation for many applications. Obviously, heat transfer must

be central to processes that operate on long timescales since it is ultimately

responsible for driving the atmosphere. For this reason, diabatic effects prove

to be important for the long-term maintenance of the general circulation, even

though they can be ignored for behavior operating on shorter timescales.

2.1.4 State Variables and Thermodynamic Processes

In general, describing the thermodynamic state of a system requires us to

specify all of its properties. However, that requirement can be relaxed for

gases and other substances at normal temperatures and pressures. A pure

substance is one whose thermodynamic state is uniquely determined by any

two intensive properties, which are then referred to as state variables. From

2.1 Thermodynamic Concepts

any two state variables Zl and z2, a third z 3 can be determined through an

equation of state

of the form

f(Zl, Z2,

Z3) -- 0,

or, equivalently,

z3 = g(zl,

Z2).

In this sense, a pure substance has two

thermodynamic degrees of freedom.

Any two state variables fix the thermodynamic state. An ideal gas is a pure

substance that has as its equation of state the ideal gas law (1.1). Because

no more than two intensive properties are independent, the state of such a

system can be specified in the plane of any two state variables (Za, z2). The

latter represents the

state space

of the system, which defines the collection of

all possible thermodynamic states. For instance, the state space of an ideal gas

may be represented by the v- T plane. Then any third state variable describes

a surface as a function of the two independent state variables, for example,

p = p(v, T),

which is illustrated in Fig. 2.3 for an ideal gas.

A homogeneous system is said to be in

mechanical equilibrium

if there

exists at most an infinitesimal pressure difference between the system and

its surroundings (unless its control surface happens to be rigid). Likewise,

a homogeneous system is said to be in

thermal equilibrium

if there exists

at most an infinitesimal temperature difference between the system and its

surroundings (unless its control surface happens to be adiabatic). If it is in

mechanical equilibrium and in thermal equilibrium, a homogeneous system is

thermodynamic equilibrium.

Thermodynamics addresses how a system evolves from one state to another,

for example, from one position in state space to another position. The trans-

formation of a system between two states describes a path in state space, which

is referred to as a

thermodynamic process.

As depicted in Fig. 2.4, infinitely

many paths connect two thermodynamic states. Thus, a process depends on

the particular path in state space followed by the system. However, the change

of a state variable depends only on the initial and final states of the system,

namely, it is "path independent." As illustrated in Fig. 2.2, a cyclic process can

be decomposed into two legs: a forward leg, from the initial state to an in-

termediate state, and a reverse leg, from that intermediate state back to the

initial state. Because it is path independent, the change of a state variable

z along the reverse leg must equal minus the change along the forward leg.

Therefore, the cyclic integral of a state variable vanishes

f dz - O.

(2.5)