handbook of atmospheric science || atmospheric energy and the structure of the atmosphere

2.1 INTRODUCTION

The Sun provides almost all of the energy input to the Earth, its oceans and atmosphere, a massive5 ¥ 1024 Jyr-1; in contrast the internal energy of theEarth generates only around 1021 Jyr-1. The solarinput is responsible for driving the atmosphericcirculation, maintaining the temperature struc-ture of the atmosphere and evaporating water intothe atmosphere to initiate the hydrological cycle.In addition, it initiates many chemical processesin the atmosphere and is central to photosynthe-sis, the process by which the biosphere reducescarbon dioxide (CO2) to carbohydrates. It is clearlyimportant to understand the radiation balance of the Earth and its atmosphere since the transferof solar radiation underpins so many of the impor-tant processes in the atmosphere, biosphere, andoceans.

The average solar flux reaching the top of theEarth’s atmosphere and the average temperature ofthe Earth vary by only fractions of a percent fromone year to the next. This indicates that, althoughthe Earth receives a huge amount of energy eachyear, it does not retain it and loses the sameamount to space. The system is in balance and wemust be able to understand the way this balance ismaintained if we are to predict the effect futurechanges to parts of the system will have on thewhole. The Earth’s atmosphere is not totally trans-parent to either incoming or outgoing radiationand the interaction between the atmosphere andradiation controls the surface radiation budget,

the physics of the atmosphere, and its chemicalcomposition.

2.2 THE STRUCTURE OF THEEARTH’S ATMOSPHERE

Before we discuss atmospheric radiation in moredetail we will give a brief overview of the verticalstructure of the atmosphere. Figure 2.1 shows thechange in temperature and density with height.There is of course considerable variability fromday to day, seasonally and from one location to an-other. However, the main features in the profile aretypical of the vertical structure of the atmospherein general. The pressure profile of the atmospherecan be calculated from the change in pressure, dp,experienced in a small change of height, dz:

(2.1)

where g is the acceleration due to gravity and r isthe density of air. The acceleration due to gravitycan be treated as constant as the atmosphere is thinwith respect to the radius of the Earth but the den-sity clearly varies with altitude. However, air be-haves more or less as an ideal gas, so:

(2.2)

where M is the molar mass of air, R is the gas con-stant (8.314JK-1 mol-1) and T is the temperature inKelvin. This gives:

r =MpRT

d dp g z= - r

2 Atmospheric Energy and theStructure of the Atmosphere

HUGH COE AND ANN R. WEBB

Handbook of Atmospheric Science: Principles and ApplicationsEdited by C.N. Hewitt, Andrea V. Jackson

Copyright © 2003 by Blackwell Publishing Ltd

36 hugh coe and ann r. webb

(2.3)

which can be integrated from the surface (z = 0,p = p0):

(2.4)

This is known as the hydrostatic equation. The de-nominator, RT/gM, has units of length and is oftenreferred to as the scale height. The scale height isthe vertical distance over which the pressure fallsto 1/e of its initial value.

As can be seen from Fig. 2.1, the temperaturedoes not remain constant throughout the atmos-phere and so the scale height also changes withheight. This analysis assumes that the atmosphereis composed of a gas of single molar mass, M. In reality the atmosphere is composed of severalgases and so we might expect a different scale

p pz

RT gM

z

=-

( )Ú00

expd

dd

pp

gMRT

z= -

height for each gas at any altitude. If this was thecase then the composition of the atmosphere in theabsence of sources and sinks would vary withheight as the heavier molecules have smaller scaleheights. This is not observed below 100km as themean free path of molecules, or molecular mixinglength, is much smaller than the macroscopic mix-ing length resulting from turbulence and convec-tion. As a result macroscopic mixing processes actequally on all molecules and dominate the mole-cules’ specific diffusion processes. Above 100kmthis is not the case and molecular separation withheight is observed. The mean molar mass of air inthe lower part of the atmosphere is determined bythe mix of molecular nitrogen and oxygen and is28.8g.

The atmosphere can be subdivided into layersbased on its thermal structure. Closest to the sur-face the temperature reduces with height to a min-imum at around 10km, known as the tropopause.The region closest to the surface is known as thetroposphere (tropos —Greek for “turning”). At thesurface the temperature varies from minima ofaround -50°C at the wintertime poles to maximaof 40°C over the continents close to the Equator.The temperature in the troposphere decreases by,on average, 6.5Kkm-1 from the surface to thetropopause. The stratosphere (stratus —Greek for“layered”) is a region between 10 and 50km inwhich the temperature profile of the atmosphereincreases to a maximum at the stratopause. Thisinversion is caused, as we shall see below, by ab-sorption of solar ultraviolet radiation by a layer ofozone (O3). Increasing temperature with heightsuppresses vertical mixing through the strato-sphere and so causes its layered structure.

Above 50km warming by ultraviolet absorp-tion can no longer compete with the coolingprocesses and temperatures begin to decrease. Thisregion is called the mesosphere and extends toaround 90km when the atmospheric temperaturereaches a second minimum, the mesopause. How-ever, unlike the troposphere where the rate of de-crease of temperature with height, the lapse rate, issufficient for convection to occur, the mesosphericlapse rate is only around 2.75Kkm-1 and the layerremains stable. Above the mesopause the temper-

Molecular density (cm–3)

1e+13 1e+14 1e+15 1e+16 1e+17 1e+18 1e+19 1e+200.0001

0.001

0.01

0.1

10

100

1000

1

180 200 220 240 260 280 3000

20

40

60

80

100

Temperature (K)

Pres

sure

(m

bar

)

Alt

itude

(km

)

Fig. 2.1 The average vertical temperature andmolecular density structure of the Earth’s atmosphere.The data are taken from the US standard atmosphereand represent a time and spatial average. A localinstantaneous sounding will vary considerably from thisprofile.

Atmospheric Energy and Structure 37

ature again increases through the so-called ther-mosphere. At these altitudes the air becomes sothin that molecular collisions become infrequent.Thus atomic species with high translational ener-gy cannot redistribute that energy to highly excit-ed vibrational and rotational states in molecularspecies. The high temperatures in the thermos-phere reflect not a large energy source but the in-ability of the thin atmosphere at these altitudes tolose energy via radiative transfer.

2.3 SOLAR AND TERRESTRIALRADIATION

2.3.1 Solar radiation

The Sun is a middle-aged, medium-sized star witha composition of approximately 75% hydrogenand 25% helium. The Sun’s energy is derived fromthe fusion of hydrogen into helium nuclei and isthen transferred to the surface of the Sun via short-wave electromagnetic radiation. Although the Sunhas a radius of 7.0 ¥ 105 km, virtually all the energyreceived by the Earth is emitted by the outer 500km, known as the photosphere. The Solar pho-tosphere emits light across the entire electromag-netic spectrum, from gamma rays to radio waves.However, most of the radiative power incident atthe top of the Earth’s atmosphere is due to light ofwavelength between 200nm in the ultraviolet and4mm in the infrared, with a peak intensity at about490nm in the green part of the visible region of thespectrum.

The irradiance of the Sun as a function of wave-length is shown in Fig. 2.2. The Sun’s photospherehas a temperature of approximately 5800K and canbe thought of as a blackbody emitter of this tem-perature. A blackbody emits the maximumamount of radiation possible at each wavelength ata given temperature. Planck related the emissivepower, or intensity, of a blackbody B(l, T) at a givenwavelength, l, to the temperature, T, of theemitter by

(2.5)B Thc

hc k Tl

l l,

exp( ) = ( ) -

21

2

5

where k is the Boltzmann constant (1.381 ¥10-23 JK-1), c is the speed of light in vacuum (2.998¥ 108 ms-1), and h is Planck’s constant (6.626 ¥1034 J s). The blackbody curve for an emitter at atemperature of 5777K is also shown in Fig. 2.2. Thetotal flux emitted by a blackbody radiator, FB, andthe total emitted intensity B, can be found by inte-grating the Planck blackbody function (2.1) over allwavelengths

(2.6)

where s is the Stefan–Boltzmann constant (5.671¥ 10-8 Wm-2 K-4).

Solar radiation has an average intensity of ap-proximately 1370Wm-2 at the distance of theEarth from the Sun. This value is often referred toas the solar constant, S, although it varies withtime on a wide variety of scales. The Sun rotateswith a period of 27 days and both active, brighterregions known as faculae and less active, darker regions known as sunspots face the Earth duringeach rotation. The output from these different re-gions of the Sun varies by between 0.1 and 0.3% ofthe total flux. The number of sunspots on the surface of the Sun varies with a cycle of 11 years,

F B B T TB = = ( ) =•

Úp p l l s, d 4

0

0 0.4 0.8 1.2 1.6 2.0Wavelength (mm)

Sola

r sp

ectr

al irr

adia

nce

,(W

m–2 m

m–1)

500

1000

1500

2000

2500

Fig. 2.2 The spectral irradiance from the Sun comparedto that of a blackbody at 5777K. (From Iqbal 1983.)


causing variations in radiative flux at the top of theEarth’s atmosphere of the order of 1%. Lower fre-quency variations in the solar flux, again of theorder of 1–2%, have also been inferred from iso-topic abundance measurements.

2.3.2 Terrestrial radiation

The Earth also acts as a blackbody radiator, but asits global mean surface temperature, Ts, is 288K,most of the irradiance from the Earth is in theinfrared part of the spectrum and peaks at about 10mm. Figure 2.3a shows a blackbody curve for anemitter of temperature 288K compared to one at5777K, representing the solar spectrum. There isvery little overlap between the incoming solarradiation at ultraviolet and visible wavelengthsand the outgoing infrared radiation from theEarth’s surface. Thus, incoming solar radiationand outgoing terrestrial radiation are distinct fromone another, separated by a gap at around 4mm, andare often referred to as shortwave (SW) and long-wave (LW) radiation respectively.

As the mean surface temperature of the Earthchanges little from year to year and has varied byless than 5°C in the past 20,000 years it is clear thatthe system is in equilibrium and the energy inputsmust be balanced by energy losses. The effectivearea of the Earth receiving sunlight at any one timeis given by pR2, where R is the radius of the Earth,yet the total area of the Earth is 4pR2, so the averageradiant flux over the Earth is given by S/4. Howev-er, not all of the incoming radiation is absorbed bythe surface; some is reflected back to space by thesurface, clouds, aerosol particles, or scatteringfrom molecules in the atmosphere. The fractionalreflectance is known as the global mean planetaryreflectance or albedo, A. The average surface albe-do is around 0.15 but the high reflectivity of cloudsleads to an overall planetary albedo of 0.3. Thus the incoming irradiance absorbed by the Earth’ssurface, Fs, is given by

(2.7)

and has a value of 240Wm-2.

F AS

s = -( )14

Wavelength (mm)

0.1 1.0 10.0 100.00

20

40

60

80

100

Abso

rpti

on (

%)

0.1 1.0 10.0 100.0

0.1 1.0 10.0 100.0

0

20

40

60

80

100

Abso

rpti

on (

%)

(c)

(a)

(b)

CO2

O3O3O3O2

CO2CO2

H2OH2OH2OH2O H2O

Fig. 2.3 Panel (a) shows the blackbody curves for 5777Kand 280K, representing the solar photosphere and theEarth’s surface. The Sun emits mainly in the visible,while the Earth emits predominantly in the infrared.The so-called incoming shortwave and outgoinglongwave radiation is separated by a gap at around 4mm.Panels (b) and (c) show the fractional absorption ofradiation from the top of the atmosphere to 10km andthe surface of Earth respectively. The main absorbers ineach wavelength region are indicated in panel (c). Mostabsorption of longwave outgoing radiation occurs in thetroposphere and is due principally to water vapor andCO2. However, note that the strong absorption band ofO3 that occurs at 9.6mm in the center of the so-calledatmospheric window is due mainly to stratosphericabsorption.


Fs must be balanced by the outgoing blackbodyradiation of the Earth given by sTe

4, where Te is theeffective blackbody temperature of the Earth-atmosphere system. Equating incoming and out-going fluxes gives an expression for Te

(2.8)

that yields an equilibrium temperature of 255K,compared to 288K, the average surface temper-ature of the Earth. The fact that the Earth–atmosphere system is 33K warmer than predictedby this simple calculation suggests that otherprocesses act to offset the loss of heat by longwavecooling. Even if the albedo is halved to completelyremove the contribution of clouds to the planetaryalbedo the equilibrium temperature only rises to268K. To understand why the Earth–atmospheresystem is warmer than predicted by the simple cal-culation above we need to examine the interactionbetween trace constituents in the atmosphere andincoming and outgoing radiation.

2.4 ABSORPTION OF RADIATIONBY TRACE GASES

So far we have assumed that the atmosphere actssimply to scatter and reflect incoming shortwaveradiation and does not absorb light. However, thisis not the case. The atmosphere interacts withboth incoming solar radiation and outgoing terres-trial radiation and as we will see the strength of theinteraction as a function of wavelength is responsi-ble for the heating of the lower atmosphere.

Figure 2.3 illustrates the effect of absorption bytrace gases in the atmosphere on the transmissionof incoming shortwave radiation from the Sun andoutgoing longwave radiation from the Earth. Fig-ure 2.3a shows the blackbody curves for emittersat 5777K and 280K respectively, representing thesolar and terrestrial emission spectra. Figure 2.3band c shows the fraction of light entering the top ofthe Earth’s atmosphere that is absorbed beforereaching 10km and sea level respectively, as afunction of wavelength. At 10km, the top of the

TA S

e =-( )Ê

Ëˆ¯

14

14

s

troposphere, virtually all radiation below 290nmhas been absorbed. All radiation below 100nm isabsorbed in the thermosphere above 100km. Mol-ecular oxygen absorbs strongly at wavelengths be-tween 100 and 200nm and also in a weaker bandbetween 200 and 245nm. The oxygen absorptionsappreciably attenuate incoming ultraviolet radia-tion of wavelengths less than 200nm above an alti-tude of 50km. Radiation of wavelengths between200 and 300nm is strongly absorbed in the strato-sphere by the oxygen trimer, ozone, and transmis-sion of radiation of wavelengths less than 290nmis negligible below 10km. Between 300 and 800nm the stratosphere is only weakly absorbingand most of the solar radiation at these wave-lengths is transmitted into the troposphere. Acomparison of Fig. 2.3b and c shows that there islittle further absorption in the troposphere atwavelengths below 600nm but H2O and CO2 ab-sorption bands, whose abundances are dominatedby their tropospheric concentrations, deplete thenear infrared part of the incoming solar flux appre-ciably. As a result the solar irradiance at the surfaceis dominated by visible wavelengths.

The interaction between the outgoing long-wave radiation and the atmosphere can also beseen in Fig. 2.3b and c and compared with an irradi-ance spectrum of a blackbody emitter of tempera-ture 288K, representing the radiation emittedfrom the surface of the Earth in Fig. 2.3a. Severaldifferent molecules are efficient absorbers of in-frared radiation and many of these are most abun-dant in the troposphere. Consequently much ofthe outgoing radiation is absorbed in the lowest 10km of the atmosphere. Much of the outgoingradiation of wavelengths less than 7mm is ab-sorbed by water vapor, with some contributionfrom methane and nitrous oxide, N2O. Radiationof wavelengths longer than 13mm is efficiently ab-sorbed by CO2, whose absorption band is centeredat 15mm. This band is particularly important as itlies close to the maximum of the longwave irradi-ance spectrum. At longer wavelengths water vaporis excited into many rotational states that ef-fectively form an absorption continuum beyond 25mm. Minor absorbers between the CO2 andwater bands are mainly N2O and CH4. The only


fraction of the outgoing radiation that is transmit-ted through the troposphere without undergoingappreciable absorption lies in the so-called atmos-pheric window between 7 and 13mm.

The only significant absorptions of infrared ra-diation in the stratosphere are due to ozone. The9.6mm band of ozone happens to lie in the middleof the atmospheric window and as a result meansthat stratospheric ozone plays a significant role inthe outgoing longwave radiation budget of theEarth.

The absorption of radiation by gases in the at-mosphere is complex and mainly involves severaltrace gas species rather than major constituents.These interactions are key to the chemical compo-sition, thermal structure, and radiative balance ofour atmosphere.

2.5 SOLAR RADIATION, OZONE,AND THE STRATOSPHERIC

TEMPERATURE PROFILE

We have already seen that ozone is a very efficientabsorber of solar radiation between 200 and 300nm. Its detailed chemistry is discussed inChapter 7. However, it is worth briefly mentioninghere as its formation and presence controls the stratospheric temperature profile. Figure 2.4shows an average vertical profile of ozone throughthe atmosphere compared to an average tempera-ture profile. The main layer of ozone in theatmosphere is situated between 15 and 30km andreaches a maximum concentration of around 5 ¥1012 molecules cm-3 at 22km. The maximum tem-perature at the top of the stratosphere occurs ataround 50km, well above the main ozone layer. Tounderstand the effect of stratospheric ozone on thetemperature profile we need to understand the wayozone is created and destroyed in the mesosphereand stratosphere.

Ozone is formed from the photo-dissociation ofmolecular oxygen but is itself removed by photo-dissociation. Photo-dissociation is the fragmenta-tion of a molecule as a result of its absorption of aphoton that is energetic enough to break its molec-

ular bonds. Both O2 and O3 absorb ultraviolet lightvery strongly and prevent highly energetic radia-tion penetrating to lower altitudes. Figure 2.5shows the extent to which ultraviolet lightpenetrates the Earth’s atmosphere as a function ofwavelength and indicates the gas species responsi-ble for its absorption. Figure 2.6 shows the absorp-tion cross-section of molecular oxygen. The peakin the absorption cross-section of O2 occurs in theSchumann–Runge continuum at around 145nm.For wavelengths less than 175nm, O2 is dissociat-ed into two oxygen atoms, one of which is elec-tronically excited. This strong absorption preventssunlight of wavelengths below 175nm from pene-trating below around 70km (Fig. 2.5). The oxygenatoms formed as a result of O2 photolysis reactwith other molecules of O2 to form ozone. Figure2.7 shows the strong absorption cross-sections ofozone occurring between 240 and 300nm with a

0

0 2 4 6 8 10

0

20

40

60180 200 220 240 260 280 300

1e+12 2e+12 3e+12 4e+12 5e+12 6e+12

O3 (p.p.m.)

O3 (molecules cm–3)

Alt

itude

(km

)

Temperature (K)

Mixing ratio

Temperature

Concentration

Fig. 2.4 The average vertical profile of ozone andtemperature through the atmosphere. The ozone profileis represented as both a mixing ratio and a molecularconcentration. Data from the AFGL standard ozone andtemperature profiles.


maximum value of 1.1 ¥ 10-17 cm2 at 255nm. As aresult, above 60km ozone is photolysed very effi-ciently back to O2 and atomic oxygen, reducing itsconcentration and favoring the existence of atom-ic oxygen.

The Herzberg continuum between 200 and 240nm is responsible for photolysis of O2 below

60km because the shorter wavelengths have al-ready been removed (Fig. 2.5), while radiation ofthese wavelengths penetrates down to around 20km.

High up in the atmosphere little O3 is producedas the air density is low and there is little O2 to bephotolysed or to subsequently react with theatomic oxygen formed by its photolysis. As we de-scend through the atmosphere the density increas-es, favoring ozone formation via the combinationof atomic and molecular oxygen, and the concen-tration of ozone increases. A maximum concentra-tion in O3 is observed at around 20–25km. Lowerin the stratosphere the overhead ozone column is now significant and absorbs much of the radia-tion between 200 and 290nm, thus limiting pho-tolysis of oxygen and slowing the rate of ozoneformation. The concentration of ozone reducesand reaches a minimum by the tropopause whereradiation of less than 290nm is almost completelyremoved.

The absorption of ultraviolet radiation by bothoxygen and ozone leads to their photolysis, and theenergy involved in these sunlight-induced reac-tions produces local warming. The temperature ata particular altitude will then be a combination ofthe rates of photolysis of the two oxygen species, inparticular ozone, and the air density. The rates ofphotolysis will depend on the local incidence of

0 50 100 150 200 250 300

Wavelength (nm)

50

100

150

200

Penet

rati

on a

ltit

ude

(km

)

N2OO2

O3

O2

Lyman a

Air N+ NO+

N+ O+2

O+2

Fig. 2.5 The extent to which ultraviolet solar radiationpenetrates through the atmosphere as a function ofwavelength. The penetration altitude is the altitude atwhich the initial intensity at any wavelength isattenuated to e-1 of its original intensity. (From Salby1996.)

Wavelength (nm)

180 190 200 210 220 230 240

O2 a

bso

rpti

on c

ross

-sec

tion (

cm–

2)

0

2

4

6

8

10 ¥ 10–24

Fig. 2.6 The absorption cross-section of molecularoxygen as a function of wavelength. (Data from DeMoreet al. 1997.)

Wavelength (nm)150

Abso

rpti

on c

ross

-sec

tion o

f O

3 (

cm2)

200 250 300 350 400

10–22

10–20

10–18

10–16

Fig. 2.7 The absorption cross-section of ozone at 273Kas a function of wavelength between 170 and 360nm.(From Seinfeld & Pandis 1998.)


radiation and thus on the optical density of theatmosphere in the column above at a givenwavelength. This in turn will be dependent on theoverhead concentration profile of O2 and O3 them-selves. As the air density increases any products ofphotochemical processes that remain energetical-ly excited are deactivated more rapidly via an in-creased chance of collisions, leading to an increasein temperature. Although the temperature profileis strongly linked to that of ozone, its maximumoccurs not at the maximum ozone concentration,but above it and close to the region where the pho-tolytic formation and loss processes of ozone aremost rapid.

2.6 TRAPPING OFLONGWAVE RADIATION

Incoming visible and ultraviolet radiation fromthe Sun is energetic enough to excite electronswithin certain optically active molecules. We haveseen that in the cases of ozone and molecular oxygen the photon energy is sufficient to fragmentthe molecule and cause its photolysis. Lessenergetic outgoing photons of infrared wave-lengths induce vibrational and rotational excita-tions of molecules. These excitations do not causechemical changes in the absorbing molecule; in-stead the excited molecule, below 100km at least,is rapidly deactivated by collisions and the energyabsorbed from the original photon is distributedthermally.

We can imagine the effect on a layer of atmos-phere as a result of these interactions. Some frac-tion of the outgoing longwave radiation enteringthe base of the layer is absorbed by molecules suchas CO2, H2O, and CH4 in the layer (see Fig. 2.3c).The absorbed energy is transferred to kinetic ener-gy by collisions between the absorbing moleculesand others in the layer. The layer will itself act as a blackbody and re-radiate infrared radiation;however, the layer will radiate uniformly in alldirections and so act to increase the longwave fluxthrough the lower layers of the atmosphere. Thisprocess raises the local temperature in the lowerlayers of the atmosphere above that predicted from

a straightforward surface budget calculation of thekind described in Section 2.3.

2.7 A SIMPLE MODEL OFRADIATION TRANSFER

Several gases in the atmosphere absorb strongly inthe infrared. As we shall see, each gas has a com-plex absorption pattern made up of many differentindividual vibrational and rotational transitions.The way these different absorptions interact is notstraightforward and should be accounted for in adetailed description of radiative transfer throughthe atmosphere. We discuss some of these effectsand considerations in the next section. However,first we provide a general picture of the processestaking place in the atmosphere by deriving a simple model of radiative transfer based on an at-mosphere that is transparent to incoming short-wave radiation and includes only one trace gas thatabsorbs uniformily at all infrared wavelengths.This model atmosphere is known as a gray atmos-phere. Our model is further simplified by the re-moval of scattering and by assuming that radiationis either emitted or absorbed only in the vertical direction. Lastly, we also assume that each level of the atmosphere is in local thermodynamic equilibrium.

First, we need to describe the absorption of lightby an absorbing species in the atmosphere. The in-tensity of light of wavelength l, I(l), which passesthrough a depth dz of an absorber with numberconcentration n, is reduced by an amount dI(l)given by:

(2.9)

where s(l) is the absorption cross-section at wave-length l and is constant for any given species, and c is the optical depth. We can obtain the intensityof light transmitted a distance z through theabsorber, Iz(l), by integrating eqn 2.9:

(2.10)I I n zz

z

l l s l( ) = ( ) - ( )ÏÌÔ

ÓÔ

¸˝ÔÔÚ0

0

exp d

d d dI I n z Il l s l l c( ) = - ( ) ( ) = ( )


where I0(l) is the initial intensity of light of wave-length l. In cases where the concentration of theabsorber is independent of the depth of the absorb-ing slab the above relation becomes the Beer–Lambert Law:

(2.11)

This is not the case for a vertical slice through theatmosphere.

In our simplified model the single speciesabsorbs uniformly over all wavelengths so we cansimplify eqn 2.10 to give:

(2.12)

where c0 is by convention the optical depth at thebase of the atmosphere.

So far we have only considered the absorption of light. However, we know that the layer will reemit radiation as a blackbody in a similar way so we must also include the intensity of emittedradiation, B, and assuming Kirchoff’s Law:

(2.13)

Furthermore, in any slice of the atmosphere theremay be some downwelling longwave radiationarising from blackbody emission of the layersabove, so we should treat both the upwelling and downwelling radiative fluxes, F≠ and FØ,separately:

(2.14)

The net flux through a layer is given by F = F≠ - FØ, the difference between the upwellingand downwelling radiation. As we have assumedthat the atmosphere is in local thermodynamicequilibrium the flux must not change with height and is therefore constant throughout thedepth of the atmosphere. By summation and sub-traction of the upward and downward fluxes we ob-tain:

dd

anddd

FF B

FF B

≠≠

ØØ= - - = -

cp

cp

d d d dI In z Bn z I B= - + = -( )s s c

I I n z Iz

z

= -ÏÌÔ

ÓÔ

¸˝ÔÔ

= -ÏÌÔ

ÓÔ

¸˝ÔÔ

Ú Ú00

0

0

0

exp exp ,s cc

d d

I I n zz = ( ) - ( ){ }0 l s lexp

(2.15)

where = F≠ + FØ is the total flux leaving one layer.As F is constant these expressions are easily

integrated to give:

(2.16)

The blackbody emission flux of the outermostlayer of the Earth’s atmosphere is given by pB0, andthis must be equal to half of the total flux from thislayer, . As there are no overlying layers to supplya contribution to FØ, = F and so:

(2.17)

In this model the blackbody emission decreaseslinearly with height from the surface to the top ofthe atmosphere and there is a constant differencebetween the up and downwelling fluxes, i.e. F = F≠

- FØ is constant. Futhermore, as there is no heatgained or lost by the atmosphere, the upwellinglongwave radiation leaving the top of the atmos-phere must be equal to the solar radiation absorbedat the Earth’s surface, Fs (eqn 2.7), so at the top ofthe atmosphere

(2.18)

We now consider the boundary conditions at thesurface. We must balance the upward flux ofradiation emitted by the Earth at a temperature Ts,pB(Ts), with the downwelling short and longwaveradiation.

(2.19)

where cs is the optical depth of the lowest layer ofthe atmosphere. However,

(2.20)

which when evaluated at the surface gives

(2.21)p p cB T B Fs s s( ) = ( ) + 2

F F F B Fs- = = -Ø2 2p

p cB T Fs s s( ) = + ( )ØF

F F Fs≠ = =

BF

BF

= + = +( )2 2

10pc

pc

FF

F B F F c= = +2p cand

F

dd

anddd

FF B

FF

cp

c= - =2 ,


This expression implies that there is a temperaturediscontinuity between the surface and the coolerlowest layer of the atmosphere. Figure 2.8 showsschematically how the radiation fluxes vary withoptical depth through the atmosphere and empha-sizes the discontinuous blackbody emission fluxat the surface.

If we assume that our absorber varies in concen-tration solely as a function of pressure then we canexpress its optical density in the atmosphere as

(2.22)

where z is the height above the surface and Hs is thescale height in the surface layers, approximately 7km. When the atmosphere is optically thin, c < 1,then radiation traverses the level with little inter-action. When c > 1 radiation is absorbed efficientlywithin the layer and successive layers do notstrongly interact. We will therefore choose cs = 1to crudely illustrate how our model works. We canevaluate c and so using eqn 2.17 we can derive B asa function height. However, from eqn 2.6:

c c= -{ }s sz Hexp

and so we can calculate the temperature at anylevel in the atmosphere. The results of such a cal-culation are shown in Fig. 2.9. The discontinuity inthe temperature at the surface is again obvious,reducing from 282K to 255K above the surface.The temperature falls rapidly with height andtends to some limit, the so-called skin tempera-ture, the temperature as z tends to infinity,(Fs/2s)1/4. If the atmosphere did not interact withoutgoing longwave radiation (c = 0) the tempera-ture of the atmosphere would be constant with al-titude and be equal to the skin temperature.

Clearly the temperature in the troposphere doesnot vary in this way. However, the temperatureprofile in the lower stratosphere (Fig. 2.1) is close tothat described above. In Section 2.9 we discussconvection and show that the rate of decrease oftemperature with height, or lapse rate, throughmost of the troposphere is determined by heattransport by convection rather than radiativetransfer. The average lapse rate in the tropopause is

F B B T TB = = ( ) =•

Úp p l l s, d 4

0

FF

Fs

( c)

(T )

F

B

( c)

( c)

Fs

p

Bp s

( c )Bp s

c = 0

c = cs

Fig. 2.8 Schematic of the variation of upwelling anddownwelling radiation and blackbody emission fluxeswith optical depth through the atmosphere. Thedifference between the upwelling and downwellingradiation fluxes is constant with height and the modelpredicts a discontinuity in the blackbody emission fluxat the surface.

210 220 230 240 250 260 270 280

Temperature (K)

Skin temperature

Discontinuity

Radiative equilibrium in a gray atmosphere

Lapse rate 6.5 K km–1

0

10

20

30

Alt

itude

(km

)

Fig. 2.9 The temperature structure of the grayatmosphere. The temperature predicted by a model ofradiative equilibrium in a gray atmosphere tends to theskin temperature at high altitudes. At the surface thereis a marked discontinuity. In reality the warm surface ofthe Earth heats the air immediately above and initiatesconvection, modeled by the average tropospheric lapserate of 6.5Kkm-1.


around 6.5Kkm-1, somewhere between the lapserate of dry air and that of cloudy air. This lapse rateis also shown in Fig. 2.9.

In the model, the air in contact with the groundis heated by the surface, becomes buoyant and ini-tiates convection. In this way convective process-es dominate the heat transfer of the troposphereand the temperature of the overlying layers de-creases with height at the average troposphericlapse rate. The temperature is greater than thatpredicted by the radiative scheme to a height of 8km, above which the radiative scheme predictswarmer temperatures than the lapse rate. So themodel predicts a convective lower atmospherethat is turbulent and well mixed and a transition ataround 8km to a stable atmosphere whose temper-ature structure is controlled by radiative process-es. The transition level in this model is a littlelower than the observed tropopause but neverthe-less this simple scheme predicts the broad temper-ature characteristics of the atmosphere below 20km.

2.8 A BRIEF OVERVIEW OF MORECOMPLEX RADIATIVE TRANSFER

In Section 2.7 we made very simple assumptionsabout the nature of the absorber in our model at-mosphere. In reality of course the situation is con-siderably more complex. The absorption spectrumof each radiatively active gas is composed of manyindividual lines. Figure 2.10 shows part of the 14mm band of CO2, and the complex structure inthe band is immediately evident. At some wave-lengths much of the light is transmitted, while at

other wavelengths the absorption by the atmos-phere is total.

Although the individual absorption lines arisefrom discrete transitions between particularvibrational and rotational energy levels within anabsorbing molecule the individual lines are notinfinitely narrow. The lifetime of the state towhich the molecule has been excited can never bepredicted exactly as there is always some small butfinite uncertainty in the energy of the excitedstate. This uncertainty in the decay time leads to abroadening of the transition line over a range of fre-quencies. However for vibrational and rotationaltransitions in the infrared part of the spectrumother line broadening mechanisms are moreimportant.

The absorbing molecules can collide with othermolecules while in the excited state and this af-fects the re-radiation of light from the molecule.This broadening effect is known as pressure broad-ening. Lastly, Doppler broadening occurs as a re-sult of the absorbing or emitting molecule movingin either the same direction as or the opposite di-rection to the photon of light it emits. This leads toa small frequency shift, observed as line broaden-ing. In the troposphere the frequency of collisionsis high and pressure broadening is the dominantmechanism.

When the atmospheric column of a gas absorbsonly a small fraction of the incident radiation theabsorption is said to be weak at that wavelength. Inthis case the thicker the layer of absorber the lighttraverses the greater the attenuation. This reduc-tion in intensity is described by the Beer–LambertLaw (eqn 2.11). However, as can be seen in Fig.2.10, several gases in the atmosphere absorb very

0.0680 688 696 704 712 720 728 736

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

Wavenumber (cm–1)

Transm

itta

nce

Abso

rpta

nce

Fig. 2.10 The 14mm wavelengthband of carbon dioxide. The complexstructure of absorptions within theband is immediately obvious. (FromSalby 1996.)


strongly and certain individual transitions effec-tively absorb all light of that wavelength throughthe atmospheric column, and the absorptanceapproaches unity. This is known as saturation:radiation at these wavelengths has effectively beenremoved and the only further change in the absorp-tion spectrum of the atmosphere after further pas-sage through the absorbing atmosphere is at theedges of the line where the gas is more weakly ab-sorbing. Saturation will therefore change the shapeof the absorbing band through the depth of the ab-sorbing column. In the case where the spectrum ofan absorbing gas contains many strong absorptionlines close together, saturation may lead to themerging of these lines and the absorption spec-trum may become continuous.

Furthermore, unlike the simple model we dis-cussed above, there are many different absorbingspecies in the real atmosphere, each with its ownspectral characteristics. This can lead to some ofthe absorbing features of different gases overlap-ping and saturation at some wavelengths mayoccur even though the contribution from each of the individual absorbing molecules may beweak.

In theory all of these effects can be treatedexplicitly. However, in practice this is difficult and also computationally very expensive. Radia-tion schemes usually use some parameterizationsof the absorption spectra, or band models that rep-resent the general features of the absorption spec-tra over a range of wavelengths. The absorptionspectrum of CO2 around 14mm (Fig. 2.10) shows aregular pattern of absorption lines and can be mod-eled using an evenly spaced set of lines separatedby mean line spacing with a line strength derivedfrom the mean strength of lines in the band. Ab-sorption lines of other absorbers may be randomlyspaced and this needs to be accounted for in theband model.

2.9 CONDUCTION, CONVECTION,AND SENSIBLE AND LATENT HEAT

2.9.1 Introduction

So far we have only considered transfer of heat

through the atmosphere by radiative processes.Certainly radiative transfer of energy is very im-portant in the atmosphere. However, energy mayalso be transferred through conduction and con-vection. The process of conduction occurs by thetransfer of kinetic energy from one molecule to anadjacent one. The process will be most efficientwhen the molecules are tightly constrained insolids and especially when there is a defined structure to the material, such as in a metal. Gases,including air, have low thermal conductivities and so the atmosphere is a poor conductor of heat. Although conduction can be neglected in the atmosphere it is the main mechanism bywhich heat is transferred away from the warm surface through the underlying layers of soil orrock.

Convection occurs much more efficiently thanconduction in fluids as warmer parts of the masscan mix much more rapidly with cooler parts and transfer heat. This transfer of heat on themacroscale is far faster than transfer on themolecular scale and makes this process extremelyimportant when considering heat transfer in the atmosphere. Heat is exchanged between theEarth’s surface, which is radiatively heated, andthe lowest layer of the atmosphere by conductionat the molecular level. The heating of the air caus-es density changes in the fluid and locally the airexpands. This makes the warmed parcel morebuoyant and may in itself cause the parcel to mixthrough the bulk of the air above, a process knownas free convection.

However, the atmosphere is continually stirred by large-scale winds generated by pressuregradients and motion around and over mountainranges. This process forces the heated air close tothe ground to mix through the air above and warmthe whole air mass. Hence, this process is knownas forced convection.

Convection then mixes parcels of warm andcold air together and so changes the temperature ofthe two parcels. The warm parcel loses heat as itcools and the colder parcel gains heat as it warms.Enthalpy, or specific heat, is transferred along thistemperature gradient. The specific heat content ofa parcel of air of unit mass is defined as cpT, where


cp is the specific heat at constant pressure and T isthe temperature of the parcel.

Energy may be transferred indirectly, withoutchanging the temperature of the air parcel, througha change in phase of water in the atmosphere, oth-erwise known as latent heat. A large amount ofheat is required to change liquid water to watervapor and the same amount of energy is releasedwhen water vapor condenses and a cloud forms.Already we can see that cloud formation releasesenergy and so will have an effect on the tempera-ture profile compared to the dry atmosphere. Thelatent heat of vaporization, L, is the energy re-quired to convert 1kg of liquid water to watervapor at the same temperature: at 0°C L = 2.5 ¥106 Jkg-1. The latent heat of melting is the energyrequired to melt 1kg of ice to form liquid water. At0°C this is around 3.3 ¥ 105 Jkg-1. We will now lookin more detail at the effects sensible and latent

heats have on the temperature structure of thetroposphere.

2.9.2 Sensible heat and the temperaturestructure of the dry atmosphere

We will now consider the processes acting on a ris-ing parcel of air in the atmosphere from a ther-modynamic perspective to obtain a verticaltemperature profile assuming that the atmosphereis well mixed. This is shown schematically in Fig.2.11. A rising parcel of air will expand because thepressure exerted on the parcel by the surroundingair reduces with height. The work done by the airparcel in expanding is at the expense of its own in-ternal energy and so the temperature of the parcelfalls. We can consider the effect of a small changein altitude on a dry air parcel of unit mass byconsidering small perturbations to the pressure,

T2

T1

T1 r1,

T2 r2,

Dry air parcel Saturated air parcel

dQ = dw – du dQ = 0 = cpdT – vdp

dQ = –Ldr = cpdT – vdp

dT

dz= – = –

g

cpGd

dT

dz= += – = +

g

cp

dr

dz

L

cp

dr

dz

L

cpGd

g s

Fig. 2.11 Schematic representation of the adiabatic cooling of dry and cloudy air parcels in the atmosphere.


dp, volume, dv, and temperature, dT, of that parcelusing the ideal gas law (pv = RT).

(2.23)

where R is the molar gas constant (8.314Jmol-1

K-1). If we expand and assume that the products ofthe increments are negligible:

(2.24)

The work done by the parcel of gas at pressure p inexpanding by a volume increment dv is

(2.25)

From the first law of thermodynamics we knowthat the quantity of heat, dQ, supplied to a unitmass of a gas is balanced by an increase in the in-ternal energy of the gas, du, and the external workdone by the gas, dw:

(2.26)

We should now introduce the specific heats at con-stant pressure and volume, cp and cv, which are de-fined in the following way:

(2.27)

Let us first consider the case when there is no vol-ume change, dv = 0. In this case

(2.28a)

and so from eqn 2.27:

(2.28b)

In general then:

(2.29)

We may now consider the particular case whenthere is no pressure change, dp = 0.

d d d d d dQ c T p v c T R T v pv v= + = + -

d du c Tv=

d dQ u=

cQT

cQTp

pv

v= Ê

Ëˆ¯ = Ê

Ëˆ¯

dd

anddd

d d d d dQ u w u p v= + = +

d dw p v=

p v v p R Td d d+ =

p p v v R T T+( ) +( ) = +( )d d d

(2.30a)

and so from eqn 2.27

(2.30b)

we have

(2.31)

In most situations in the troposphere vertical mo-tion of air is rapid enough to far outweigh any heattransfer by conduction or radiation. Under thesecircumstances there is no net exchange of heat anddQ = 0, such processes are known as adiabaticchanges and

(2.32a)

or

(2.32b)

On integrating, we can see that

(2.33)

This sole dependence of the temperature of an airparcel on pressure during an adiabatic processmeans that we can find the temperature an air par-cel would have if it was moved from some arbitrarypressure level to 1000mbar adiabatically. Thistemperature is known as the potential tempera-ture, q, where

(2.34)

If q is constant with height then the atmosphere issaid to be in convective equilibrium and the fall oftemperature with height, or lapse rate, is found by substituting the hydrostatic equation into

qk

= ÊËÁ

ˆ¯T

p1000

Tp

Rc

c c

cKp

p v

p= = =

-=constant, k 0 288.

d dTT

R pc pp

=

c T v p RTpppd d

d= = Ê

ËÁˆ¯

d d dQ c T v pp= -

cQTp

p= Ê

Ëˆ¯

dd

d dQ c R Tv= +( )


eqn 2.32. Remembering that we have considered aparcel of unit mass, so vr = 1:

(2.35)

or

(2.36)

Gd is known as the dry adiabatic lapse rate and isthe rate of reduction of temperature with heightthrough the atmosphere assuming that the air isunsaturated and in convective equilibrium. Thatis to say that if a parcel of air is displaced verticallyits temperature at the new pressure will be thesame as the surrounding air at that level. Thoughnot true locally, on average the atmosphere wouldshow a lapse rate close to this value as long as theair is dry and there is no contribution from phasetransitions of water to the energy budget.

2.9.3 Stability of dry air

The measured lapse rate in the atmosphere, g, isthe observed rate of change of temperature withheight. We can compare this value with the dry adi-abatic lapse rate to investigate the likely extent ofvertical motion of an air parcel in that layer. Thetwo environment curves in Fig. 2.12, marked Aand B, have lapse rates that are respectively greaterand less than Gd. An environment curve shows thevertical temperature structure of the real environ-ment. First consider environment curve A. If aparcel of air at point O is displaced upwards itstemperature reduces along the dry adiabatic lapserate and will therefore be higher than that of its sur-roundings. Since the pressure is the same, the den-sity of the parcel must be less than that of itssurroundings. The air parcel will then have posi-tive buoyancy and will be accelerated upwards.Similarly, if the parcel is displaced downwardsthen it becomes cooler and denser than its sur-roundings and sinks further. An atmosphere underthese conditions is said to be unstable.

Conversely, when we consider a parcel of air atpoint O on environment curve B, an upward dis-

dd

K kmTz

gcp

d= - = - = - -G 9 8 1.

c T g zpd d+ = 0

placement of the air parcel along the dry adiabaticlapse rate curve will result in the temperature ofthe parcel displacement being cooler than that ofits surroundings and more dense. Likewise, adownward displacement will result in the parcelbeing warmer and more buoyant than its sur-roundings. In both cases the parcel is subjected to arestoring force that tends to return it back to pointO. Vertical columns of air with temperature pro-files similar to the environment curve B are said tobe stable.

2.9.4 Latent heat and the effect of clouds on the vertical temperature structure

We have so far assumed that the atmosphere is dry:that is to say, no water exists in the atmosphere in the liquid or solid phase. However, water canchange phase readily under atmospheric condi-tions and such changes of state produce large

B

O

650

700

750

800

850

900

950

1000

A

Dry adiabatic lapse rate

Temperature (°C)

–20 –10 0

0

500

1000

1500

2000

2500

3000

3500

10 20

Alt

itude

(m)

Pre

ssure

(m

bar)

Fig. 2.12 The lapse rates of the two environment curvesA and B are respectively greater and less than the dryadiabatic lapse rate of 9.8Kkm-1. If an air parcel at Oascends adiabatically its temperature changes at a rateGd. If the surrounding environment has a lapse rategreater than this, A, then the ascending parcel is warmerthan the air around it, and is buoyant and unstable. If thelapse rate of the surrounding air is less than that of theascending parcel, B, then the displaced parcel is coolerthan its surroundings and the air is stable.


changes in the energy budget of the system. It is itsability to act as an energy store that makes water soimportant in the troposphere.

The amount of water vapor in a parcel of air maybe expressed as the mass mixing ratio, r, where r = rv/ra and rv and ra are the densities of watervapor and dry air in the parcel. The mixing ratio isusually expressed in units of grams of water perkilogram of air. The maximum amount of watervapor an air parcel can hold at any given tempera-ture is given by the saturation mixing ratio, rw. Thesaturation mixing ratio can be thought of as theamount of water in the vapor phase above athermally isolated body of pure liquid water atequilibrium. Any further increase in water into the already saturated air parcel will lead to con-densation. The saturation mixing ratio is solely afunction of temperature: as the parcel warms itscapacity to hold water vapor increases and con-versely at colder temperatures air can hold lessmoisture before condensation occurs. The Antarc-tic continent may therefore be thought of as adesert because there is little moisture available inthe air at such low temperatures, reducing thepossibility of significant precipitation.

The adiabatic changes experienced by a cloudyair parcel are shown schematically in Fig. 2.11.Consider an unsaturated air parcel close to theEarth’s surface. As the air parcel rises it will cooland its saturation mixing ratio will decrease. Ifthere is no exchange between the air parcel and itsenvironment the water vapor mixing ratio of theparcel remains constant. If the parcel continues torise it will eventually reach a level at which theambient mixing ratio of the parcel is equal to thesaturation mixing ratio. This point is known as thecondensation level and is often observed by a clear-ly defined base to a layer of clouds. As the air parcelcontinues to rise the water vapor mixing ratio is at,or slightly above, rw and so condensation occurs.During condensation, heat is released into the airparcel. Some of the energy required to expand theparcel as it rises into a lower pressure environmentis met by this energy release and so the parcel nolonger needs to meet all of the work of expansionfrom its own internal energy. The effect of conden-sation within a cloud is therefore to offset some of

the temperature reduction with height of a risingparcel under dry adiabatic conditions by releasingenergy through the phase transition of water.

We can modify our mathematical description ofthe dry atmosphere to account for these changes inthe following way. Unlike the dry atmospherewhere there is no change in heat of the parcel withheight, heat is provided to the parcel by the mass ofwater condensed. If we are considering a parcel ofunit mass the latent heat released will supply achange in heat dQ = -Ldr, where dr is the change inmixing ratio of water vapor in the parcel. We cantherefore modify eqn 2.31 to give:

(2.37)

and by substituting the hydrostatic equation:

(2.38)

As a result we can see that the reduction in temper-ature with height of a cloudy air parcel under adia-batic conditions, the saturated adiabatic lapse rate,gs, is given by:

(2.39)

As the cloudy air parcel ascends the water vapormixing ratio decreases with height as water iscondensed onto cloud droplets so dr/dz is negative.This reduces gs below the dry adiabatic lapse rate,Gd, by an amount that is directly proportional tothe mass of water vapor condensing in the rising airover a fixed height interval.

It is clear that as the amount of water vapor aparcel can hold at saturation is a strong function oftemperature, the saturated adiabatic lapse rate willvary markedly depending on the temperature ofthe air parcel. At high latitudes, or altitudes withvery cold temperatures, an air parcel can hold verylittle water vapor at saturation. As a result most ofthe energy for expansion of a rising cloudy air par-cel must still come from the internal energy of theparcel and the lapse rate is little different from theparcel in dry conditions. However, at low latitudesand altitudes air temperatures are higher and there

g sp p

dp

Tz

gc

Lc

rz

Lc

rz

= - = + = +dd

dd

dd

G

- = +L r c T g zpd d d

d d d dQ L r c T v pp= - = -


is a considerable amount of water vapor held in theparcel at saturation. Under these circumstances,condensation may contribute significantly to theenergy budget of the parcel. At high ambient tem-peratures it may be that gs may be as low as 0.35Gd.

It should also be pointed out that the extremelylow temperatures at the tropopause act as a coldtrap. At these very low temperatures air cansustain very little water in the vapor phase beforereaching saturation. As water vapor is condensedand precipitates out of the parcel the remainingparcel is very dry and little water vapor is trans-ferred into the stratosphere.

Clearly the average lapse rate of the tropospherewill be less than the dry adiabatic lapse rate butwill be considerably more than the 3.5Kkm-1

experienced in the lower levels of equatorial cu-mulonimbus clouds. In fact the average lapse ratein the troposphere is around 6.5Kkm-1. This lapserate was used in the consideration of the tempera-ture structure of the lower atmosphere in Section2.7.

2.9.5 Stability in cloudy air

It follows from the above that the rates of change oftemperature of an ascending air parcel will be dif-ferent in dry and cloudy air. We can imagine thenthat an air parcel will respond differently to smallvertical perturbations in position depending onwhether cloud is present in the air parcel or not.What is certainly true is that if an air parcel is un-stable in dry air (see discussion in Section 2.9.3)then it must be unstable in cloudy conditions. Putanother way, if the ambient temperature reduceswith height more steeply than the dry adiabaticlapse rate it must fall faster than the saturatedlapse rate.

However, the converse clearly does not hold.Consider an air parcel at point O in Fig. 2.13. Theambient fall in temperature with height is lessthan the dry adiabatic lapse rate and in dry air thiswill lead to any vertical motions of a parcel beingsuppressed as the air parcel is stable. However, ifthe temperature structure of the atmosphere is thesame but the air parcel at O has a water vapor mix-ing ratio of 6gkg-1 then the parcel at point O will be

just saturated and so will be at cloudbase. The re-duction in the adiabatic lapse rate caused by heatreleased during condensation is sufficient to re-duce the saturated adiabatic lapse rate to less thanthe ambient temperature profile in the cloudycolumn above point O. Any vertical displacementof an air parcel subjected to these conditions willlead to it becoming more buoyant than the airaround it and hence the parcel will be unstable.This is known as conditional instability and a sta-ble column of air that is forced to rise over orogra-phy or a frontal zone may cool until condensationoccurs. This may release enough latent heat tomake the air column positively buoyant and henceunstable.

Table 2.1 shows the five different stability crite-ria possible from absolutely unstable to absolutelystable. Neutral stability occurs in dry air when anyvertical movement of an air parcel neither increas-es nor decreases its buoyancy relative to the sur-rounding air. The lapse rate under these conditions

–20 –15 –10 –5

Temperature (°C)

Dry adiabatic lapse rate Pre

ssure

(m

bar)

Alt

itude

(m)

0 5

O

10 150

500

1000

1500

2000

2500

3000

3500

environmental lapse ratewet adiabatic lapse rate

1000

950

900

850

800

750

700

650

Mixing ratio = 6 g kg–1

at cloudbase

Fig. 2.13 The effect of moisture on the atmosphericlapse rate. A parcel of air at O has a mixing ratio of 6gkg-1 of water vapor and so is just at saturation. As the airparcel rises adiabatically its temperature will decrease ata rate gs as some of the energy required to expand theparcel is supplied by the condensation of water vapor. Ifthe environment curve is greater than gs and less than Gdthe parcel is stable in dry air but becomes unstable whencloud is present; in other words, conditionally unstable.


will be equal to Gd. Similarly neutral stability canalso occur in cloudy air, though of course the lapserate under cloudy conditions will then be equal to gs.

2.10 THE ENERGY BUDGET FORTHE EARTH’S ATMOSPHERE

We have seen that the input of energy to the Earthand its atmosphere comes from the Sun in the formof shortwave visible and ultraviolet radiation.However, the temperature of the Earth and its at-mosphere is not changing considerably with timeso the system appears to be close to steady stateoverall and energy inputs balance energy losses.We have discussed the re-radiation of longer-wavelength infrared radiation from the Earth andits atmosphere and also the transfer of energythrough the atmosphere through convection andlatent heat release. We should now consider thecontributions these processes make to the overallenergy budget of the Earth atmosphere system.First, we will consider the energy budget averagedover the whole globe and simply consider the rela-tive contributions of the various processes as global means. Although this gives a good indica-tion of which of the processes are most importantit does not give us any idea of the behavior of the at-mosphere at a particular location. We will there-fore consider how the energy budget varies fromone part of the globe to another and discuss the ef-fect of these inhomogeneities on the atmosphereof the Earth.

2.10.1 The average energy budget

Over one year around 5.5 ¥ 1024 J of solar energy isreceived at the top of the Earth’s atmosphere in theform of visible and ultraviolet radiation. If, as inFig. 2.14, we assume this input to be 100 units wecan compare the other parts of the energy budget ofthe Earth–atmosphere system with respect to thistotal. The planetary albedo is approximately 0.3and so 30% of the incoming shortwave radiation is reflected back to space with no interaction. Two-thirds of this reflection is from cloud tops,one-fifth is from molecular scattering in the at-mosphere, and the remainder is from reflection atthe Earth’s surface. Of the remaining 70 units ofenergy, 19 are absorbed by the atmosphere and 51are absorbed by the surface.

We have seen that the Earth itself acts as ablackbody radiator and emits radiation from itssurface, mainly at infrared wavelengths. Overall,the number of units of energy lost from the surfaceas longwave radiation is 117, greater than the ener-gy input from the Sun. However, of this large lossfrom the surface only 6 units escape to space di-rectly, while 111 units are absorbed by the atmos-phere, mainly by water vapor, CO2, and clouds. Inaddition to the absorption of shortwave and long-wave radiation, the atmosphere gains a further 23units from evaporation of surface water into the at-mosphere in the hydrological cycle and also 7 unitsfrom sensible heat transfer by convective process-es. In total, then, the atmosphere gains 160 units ofenergy per year.

Clearly the atmosphere is not undergoing a verylarge warming and this energy gain is balancedoverall by loss of the same amount of energy in theform of longwave radiation. Of the 160 units lostby the atmosphere 96 units are re-radiated back tothe surface and 64 units are radiated out to space.This longwave re-radiation by the atmosphere alsobalances the energy budget at the Earth’s surface,where in total 147 units of energy are received inthe form of shortwave and longwave radiation andlost by longwave radiation, evaporation, and con-vective transfer. Likewise, the incoming 100 unitsof solar energy are balanced by a loss of 100 units

Table 2.1 Stability criteria for a moist air parcel.

1 g < gs Absolutely stable

2 g = gs Saturated neutral

3 gs > g > Gd Conditionally unstable

4 g = Gd Dry neutral

5 g > Gd Absolutely unstable

The ambient reduction in temperature with height, g, may be greater

or less than the change in temperature with height induced by either

a dry or a saturated adiabatic process, Gd or gs.


from reflection and longwave radiative emissionfrom both the Earth and the atmosphere.

2.10.2 Variations in the heat budget across the globe

The above energy budget is averaged over all lati-tudes and seasons. Clearly, the shortwave radia-tion input is not uniform over the entire globe.Low latitudes receive considerably more energyfrom solar radiation than higher latitudes. Further-more, the albedo of the surface determines thefraction of sunlight reflected away from the sur-face and so affects the amount of shortwave radia-tion absorbed. For example, the albedo of fresh

snow is very high (0.8–0.9), while that of the oceanis as low as 0.08. The heating rates therefore varygreatly as a function of latitude, but this does notlead to a concomitant increase in temperaturewith time. One explanation of this is that the out-going longwave radiation flux also varies in a simi-lar way. However, satellite data show that this isnot the case. The average albedo of the Earth’s landsurfaces is higher than that of the oceans and asmost of the land surface is in the northern hemi-sphere and the southern hemisphere is mostlyocean there are differences in the heating rates ofthe two hemispheres and hence in their dynamicalcirculation.

Figure 2.15a shows the calculated shortwave

+96 +51

–117

–64 –96+19 +111 +23 +7

–23 –7

–4–64 –20 –6 –6Sun+100

Net loss to space = –100

Energy lost by theatmosphere = –160

Energy gained by theatmosphere = +160

Energy gained at the surface = +147 Energy lost = –147

Fig. 2.14 The average energy budget of the Earth–atmosphere system. The contributions from shortwave andlongwave radiation to the energy budget are indicated by straight and wavy arrows respectively. One-fifth of the loss ofenergy from the surface is in the form of sensible and latent heat, the remainder is due to longwave radiation.


and longwave energy budgets of the Earth–atmosphere system averaged over time as a func-tion of latitude. There is an excess of incomingshortwave radiation between 35°S and 40°N and adeficit at higher latitudes compared with the out-going longwave radiation budget. If equilibriumwere to be maintained at every latitude the short-wave and longwave radiation should balance local-ly and the two curves in Fig. 2.15a would beidentical. Because they are not, and local meantemperatures close to the Equator are not increas-ing with time and those close to the poles are notdecreasing, heat energy must be transported fromlow latitudes poleward. This is achieved by circu-lation within both the ocean and the atmosphere,transporting heat away from the Equator towardthe pole and maintaining a higher temperature atlatitudes greater than 50° than would be possible

from a system in radiative equilibrium, illustratedby the thin broken curve in Fig. 2.15a.

The difference between the incoming short-wave and outgoing terrestrial longwave radiationis known as the net radiation and is shown as theshaded area in Fig. 2.15a. Net radiation is positiveclose to the Equator indicating energy gain andnegative above 40°. The energy transfer from Equa-tor to pole can be calculated for each hemisphereby integrating the net radiation from the Equatortoward the pole. The result of this calculation isshown in Fig. 2.15b. The energy transfer reaches amaximum around 40°, the latitude below whichthe Earth–atmosphere system on average gains en-ergy. The energy is transported to higher latitudeswhere the system on averages loses more energythan it receives. This heat pump supplies the ener-gy required to drive the ocean circulation and glob-al wind patterns in the atmosphere.

Figure 2.16 shows the spatial variability of netradiation over the globe for February 2002 that hasbeen calculated from the combined informationfrom several different satellites during the EarthRadiation Budget Experiment (ERBE). The strikingfeature of the global variability of net radiation isthe zonal homogeneity. The shortwave radiativeflux is largely dependent on the albedo and thisvaries markedly from one location to another atsimilar latitudes. The atmosphere above theoceans at low latitudes is characterized by clearskies and so has a low albedo of around 0.1, where-as over the equatorial land masses large-scale or-ganized convection leads to large cumulonimbusclouds and hence increased albedos greater than0.3. However, the cloud tops over land are ataround 10–12km and consequently have low tem-peratures. There is therefore significant longwavetrapping over low-latitude land masses and this ap-proximately offsets the change in albedo and leadsto the lack of zonal variability in the net radiationover the globe.

Detailed satellite measurements of net radia-tion can be used to deduce the size of the energytransfer from the Equator to the poles. However,this gives us no indication of the relative size of thecontributions made to this transfer by the circula-tion of the ocean and atmosphere.

90 N 60

(b)

(a)

50 40 40 50 60 90 S30 3020 2010 100

Latitude (degrees)

Mer

idio

nal en

ergy

transf

er (

10

15

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tion (

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DeficitDeficit

Fig. 2.15 (a) The calculated shortwave and longwaveenergy budgets of the Earth–atmosphere systemaveraged over time as a function of latitude. There is anexcess of shortwave solar radiation at low latitudesrelative to the longwave losses in the same region of theplanet. This increase in heat flux is balanced by a netloss of radiation at latitudes above 40°. As thetemperatures of the lower atmosphere in both thetropical and polar regions are approximately constantwith time there must be a poleward transfer of energy,shown in (b). (From Wells 1997, © John Wiley & SonsLimited. Reproduced with permission.)


2.11 ENERGY TRANSFER IN THEATMOSPHERE AND OCEAN

As we have seen, for the Earth’s ocean–atmospheresystem to be in equilibrium large quantities of heatmust be transferred from the Equator toward thepoles. Circulation systems in both the atmosphereand the ocean are significant in achieving thisredistribution of heat. In the atmosphere, the highlevels of net radiation at the Equator warm thesurface and drive large-scale convection. The ris-ing air cools and condensation enhances instabil-ity and fuels further convection by very largereleases of latent heat. Penetration of the rising airinto the stratosphere is greatly suppressed by thelarge temperature inversion at the tropopause andthe air flows poleward. As the air moves away from

the Equator the air cools as it transports heat pole-ward. The air is no longer buoyant and sinks, pro-viding a return flow at low level towards theEquator. This circulation is known as the Hadleycirculation and extends to around 30° latitude,north and south of the Equator. A small fraction ofthe descending air flows poleward and rises in themid-latitudes before a weak upper-level mean flowreturns the air toward the Equator. This secondcell is known as a Ferrel cell.

The Hadley cell has a strong meridional compo-nent as it is driven by convection. In contrast themean circulation in the mid-latitudes is driven bylarge latitudinal pressure differences and the flowis largely westerly with only small mean merid-ional flow. However, this situation is unstable and synoptic-scale transient perturbations to the

Fig. 2.16 A false color map of the spatial variability of net radiation over the globe for February 2002. The data wereobtained from several different satellites and combined during the Earth Radiation Budget Experiment (ERBE). (Datacourtesy of the ERBE and CERES Projects, NASA LaRC: http://www.earthobservatory.nasa.gov)


westerly flow, in the form of cyclonic and anticy-clonic weather systems, very efficiently transportheat poleward.

Radiation transfers more heat to the ocean thanto the land because the albedo of the sea surface is generally less than that of the land and the seasurface temperatures at low latitudes, where theradiative input is highest, are lower than the landsurface temperatures at the same latitude, reduc-ing longwave cooling. The largest net loss of heatfrom the ocean is by evaporation rather than sensi-ble heat loss. The heat capacity of water is large andso, unlike the atmosphere, the ocean can store verysignificant quantities of heat. The oceans accumu-late large amounts of heat in the mid-latitudes inthe summer, which is released back to the atmos-phere during the winter, giving rise to increasedcloud and precipitation at these latitudes at thistime of year. However, the wintertime heat lossfrom the mid-latitude oceans is larger than theirheat gain in summer and a poleward flux of heat isrequired to maintain the equilibrium.

Figure 2.17 shows the contributions made bythe atmosphere and ocean to the total net energytransport in the northern hemisphere as a functionof latitude. To more clearly see the contributionsof the processes described above the energy trans-port of the atmosphere is split into its mean andtransient components. At low latitudes heat istransported approximately equally by the oceansand the atmosphere, whose contribution is almostentirely due to the Hadley circulation. The oceanheat flux dominates over the atmosphere between15 and 30°N. This is the region south of the mid-latitude circulation where the Hadley circulationis descending. The maximum heat flux occurs ataround 30°N and at higher latitudes the atmos-pheric transport is once more larger than that inthe ocean. However, unlike the equatorial regionsthe main transport mechanism is not the mean cir-culation but the transient synoptic-scale perturba-tions to the mean zonal flow.

2.12 SOLAR RADIATION ANDTHE BIOSPHERE

In addition to providing the energy to our Earth-

atmosphere system and providing a climatology inwhich life can flourish, solar radiation has manydirect influences on the biosphere (the narrowband of the atmosphere–Earth–ocean systemwhere living organisms are found). There are alsoindirect effects and complex feedback systemsbetween the atmosphere, biosphere, and radiativetransfer, especially when man’s activity is consid-ered as part of the biosphere: for instance, anthro-pogenic emissions of carbon dioxide and methaneand ozone depletion. Here we will concentrate onthe more direct effects of solar radiation on lifeforms.

In many respects the behavior of living organ-isms in sunlight is not an effect, in the sense of anexisting organism responding to an externally im-posed stimulus, but an example of the evolution-ary adaptation of the organism to utilize availableresources. Many of the so-called effects of radia-tion (usually detrimental) come from an upsettingof the balance between the radiation climate inwhich the organism evolved and that to which it iscurrently exposed; this change might be consid-ered as an external stimulus.

The radiation balance of the Earth–atmospheresystem discussed in earlier sections is only con-cerned with the total energy in the complete solarwaveband (0.3–4mm). However, many of the pho-toreactions initiated by sunlight are wavelength

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Fig. 2.17 The total poleward longitudinally averagednet transport of heat as a function of latitude. The totalhas been subdivided into the atmospheric componentand that due to the ocean circulation (shaded area). Theatmospheric contribution has been divided into itsmean and transient components (lower and upper partsof the non shaded area respectively) to separate the fluxdue to general circulation and that arising from synopticscale mixing processes. (From Gill 1982.)


dependent and different parts of the solar spectrumhave to be considered for different reactions. In ad-dition, the simple physical energy contained in awaveband is not always a good indicator of its po-tential to induce the desired reaction; the actionspectrum is also required. The action spectrum, orresponse spectrum, for a given reaction describesthe wavelength-dependent sensitivity of the reac-tion or target body to the incident radiation, oftennormalized to unity at the wavelength of maxi-mum response. Thus if the incident solar spectralintensity is I(l) and the action spectrum of interestis denoted by R(l) then the intensity of biologicallyeffective radiation (i.e. the physical energy weight-ed with its effectiveness in producing the specifiedreaction) will be:

(2.40)

Many of the important biological action spectraare in the ultraviolet and visible portions of thesolar spectrum, where the individual photons havemost energy. Approximately half of the total solarenergy is in the visible part of the spectrum, with asmall amount in the ultraviolet and the rest at in-frared wavelengths (see Fig. 2.3).

Two fundamental uses of solar radiation by in-habitants of the biosphere are vision and photosyn-thesis. Respectively they allow the majority ofmobile creatures to see, and plants to convert solarenergy into a usable form (sugar), initially forthemselves and then, as plant matter, for other lev-els of the food chain. Our optical system (the eye)responds to visible radiation (wavelengths be-tween 400 and 700nm), and the photopic responsepeaks in the middle of this range (green light). It isno coincidence that the solar spectrum, both ex-traterrestrially and at the surface, is a maximum inthe same wavelength region. Understanding the il-luminance (visually effective radiation with R(l)equal to the photopic response in eqn 2.40) is im-portant in, for example, building design and ismeasured with a luxmeter that has a responsespectrum very similar to the photopic response ofthe eye.

Systems that photosynthesize also make gooduse of the same waveband. Photosynthesis bychlorophyll-containing plants is the process by

I Rl l l( ) ( )Ú d

which water and atmospheric CO2 are convertedinto simple sugars (and thence more complex com-pounds), oxygen, and water. The absorptance oftypical green leaves exceeds 90% at blue and redwavelengths, but decreases to less than 80% in thegreen waveband, where reflectance and transmit-tance increase (hence the observed green color).Absorptance drops precipitously at the longwaveend of the visible spectrum and is less than 5% be-tween 0.7 and 1mm, thereafter increasing again.The reflection and transmission of infrared radia-tion helps to prevent the plant from overheating.Given the comparatively constant spectral com-position to photosynthetically active radiation(PAR), the photosynthesis rate will depend on theintensity of the radiation, plus water availability,carbon dioxide concentration, and the presence or lack of other environmental stresses (e.g. tem-perature). In the absence of other limiting factorsthe photosynthesis rate increases almost linearlywith increasing incident radiation up to a limitingvalue. At this point the plant is light saturated andfurther increases in radiation are not beneficial.The light saturation point occurs at different irra-diance and photosynthesis rates depending on thetype of plant: both are low for shade-loving plantsand increase until, for some plants, it is difficult toreach light saturation in sunlight.

At ultraviolet wavelengths photon energies be-come sufficient to cause damage and it is the detri-mental effects of ultraviolet that are most oftencited, although there are beneficial effects as well.Prominence has been given to the UVB wavebandas it is radiation in this waveband that is most af-fected by changes in stratospheric ozone. Ozonedepletion leads to increased UVB at the surface and a shift of the short-wavelength limit of thesolar spectrum to shorter (more damaging) wave-lengths. In humans and animals UVB radiation is necessary for skeletal health as it initiates thecutaneous synthesis of vitamin D, but it alsodamages DNA, produces sunburn/tanning, affectsskin-mediated immunosuppression, and causesdamage to the eye. Sunburn is probably the bestknown detrimental effect, with an action spec-trum that includes both UVB and UVA radiation. Itis also associated with increased risk of skin can-cer, particularly the most fatal variety, malignant


melanoma. Skin cancers, like chronic eye damageresulting in cataracts, are diseases whose risks in-crease with accumulated lifetime exposure to ul-traviolet, moderated for skin cancers by the skin’snatural sensitivity to ultraviolet radiation. Fairskinned people are at greater risk than those withnaturally high levels of pigmentation. The relationbetween skin color and ultraviolet effects is a goodexample of an evolutionary balancing act. Peopleoriginating from high latitudes (low levels of sun-light and ultraviolet) have fair, sun-sensitive skinswith little melanin (a competing absorber forultraviolet photons and responsible for color intanned, brown, or black skin), enabling them totake advantage of available ultraviolet for vitaminD synthesis. Movement to higher radiationenvironments (equatorwards) increases the risksof sunburn and skin cancer. Conversely, highlypigmented peoples from low latitudes have naturalprotection against high levels of ultraviolet there,but if they move polewards where ultraviolet is

reduced they can become susceptible to vitamin Ddeficiency.

REFERENCES

De More, W.B., Sander, S.P., Golden, D.M. et al. (1997)Chemical Kinetics and Photochemical Data for Usein Stratospheric Modeling. Evaluation No. 12. JetPropulsion Laboratory, Pasadena, CA.

Gill, A.E. (1982) Atmosphere–Ocean Dynamics.Academic, London.

Iqbal, M. (1983) An Introduction to Solar Radiation.Academic Press, Toronto.

Salby, M.L. (1996) Fundamentals of AtmosphericPhysics. Academic Press, San Diego.

Seinfeld, J.H. & Pandis, S.N. (1998) Atmospheric Chem-istry and Physics: From Air Pollution to ClimateChange. Wiley, New York.

Wells, N. (1997) The Atmosphere and Ocean: A PhysicalIntroduction, 2nd edn. Wiley, New York.

handbook of atmospheric science || atmospheric energy and the structure of the atmosphere

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