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# 10 Radiative–Convective Equilibrium Temperature Profiles

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transmission of the various band systems of importance in the atmosphere, insertion

of these into a computational analog of the radiative transfer equation, and iteration

to obtain a steady balance solution. Such models are a much more sophisticated version of the simple radiative equilibrium model discussed in Section 3.8.

The variables that determine the fluxes of radiant energy in the atmosphere include the atmospheric gaseous composition, the aerosol and cloud characteristics,

the surface albedo, and the insolation. Since horizontal transport of energy by atmospheric and oceanic motion affects the local climate, it is of most interest to calculate the radiative equilibrium for conditions averaged over the globe. In a globalmean model the temperature and all other variables depend only on altitude, and the

globally averaged insolation and solar zenith angle are appropriate. To understand

the basic radiative energy balance of Earth, we need to specify the following:

1. H 2 0 : Water vapor is the most important gas for the transfer of radiation in the

atmosphere. Its distribution is highly variable. The sources and sinks (evaporation

and condensation) are determined by the climate itself, and they are fast compared to

the rate at which the atmosphere’s motion mixes moist and dry air together. Water

vapor has a vibration-rotation band near 6.3 pm and a rotation continuum at wavelengths longer than about 12 pm. It is also the principal absorber of solar radiation in

the troposphere.

2. COz: The mixing ratio of carbon dioxide is increasing about 0.4% per year

primarily because of coal and oil combustion. In 1990 the value was about

350 ppmv. Because sources and sinks of C 0 2 are slow compared to the time it takes

the atmosphere to mix thoroughly, its mixing ratio can be assumed constant with latitude and altitude up to about 100 km. The strong vibration-rotation band of C02 at

15 pm is important for longwave radiative transfer. A significant amount of solar radiation is absorbed by carbon dioxide.

3. 0 3 : Ozone has fast sources and sinks in the stratosphere where most of the atmospheric ozone resides. Near the surface ozone is produced in association with

photochemical smog. Its concentration in the middle and upper stratosphere is dependent on temperature, insolation, and a host of photochemically active trace

species. Ozone has a vibration-rotation band near 9.6 pm that is important for longwave energy transfer, and also has a dissociation continuum that absorbs solar radiation between 200 and 300 nm. Absorption of solar radiation by ozone heats the middle atmosphere and causes the temperature increase with height that defines the

stratosphere and tropopause.

4. Aerosols: Atmospheric aerosols of various types affect the transmission of

both solar and terrestrial radiation. A layer of sulfuric acid aerosols exists near 25 km

in the midstratosphere. Sulfate aerosols in the troposphere are also radiatively important and seem to be increasing as a result of human activity, primarily fossil fuel

combustion.

5. Surface albedo: The surface albedo is highly variable from location to location in land areas, depending on the type and condition of the surface material and

68

3

vegetation. Over open ocean it is mostly a function of solar zenith angle, but it also

depends on sea state. When the surface is snow covered, its albedo is generally much

higher than when surface ice is not present.

6 . Clouds: Clouds vary considerably in amount and type over the globe. They

have very important effects on longwave and solar energy transfer in the atmosphere. The distribution in time and space and the optical properties of clouds are important for climate. For a global-mean radiative equilibrium calculation, cloud radiative properties must be specified. The simplest approach is to assume plane-parallel

clouds and specify their distribution in the vertical. The optical properties of the

clouds must also be specified. For solar radiation one must specify the fractions of a

beam of radiation that are absorbed and reflected by clouds, which can be called the

absorptivity and rejectivity, respectively. Normally, water clouds have relatively

weak solar absorption, but they effectively scatter solar radiation back toward space

and so have high reflectivities. Thick clouds can be assumed to be black bodies for

longwave radiation, so that they absorb all incident longwave radiation and emit like

a blackbody with the temperature of the atmosphere at the same level as the cloud. A

simple approach is to specify the properties of three types of clouds, as in Table 3.2.

The albedo values are based on old estimates and are not necessarily the most representative set, but they are the values used in the calculations shown here.

Figure 3.16 shows a calculated temperature profile that is in radiative equilibrium. Atmospheric temperatures in radiative equilibrium decrease rapidly with altitude near the surface. In the troposphere, radiative equilibrium temperature profiles

are hydrostatically unstable in the sense that parcels of air that are elevated slightly

will become buoyant and continue to rise. In the real atmosphere, atmospheric motions move heat away from the surface and mix it through the troposphere. Figure 2.4

indicates that 60% of the energy removal from the surface is done by the transport of

heat and water vapor by atmospheric motions and only 40% by net longwave radiation emission. The global mean temperature profile of Earth's atmosphere is not in

radiative equilibrium, but rather in radiative-convective equilibrium. To obtain a realistic global-mean vertical energy balance, the vertical flux of energy by atmospheric motions must be included.

Table 3.2

Values of Cloud Shortwave Reflectivity and Absorptivity and Fractional Area

Coverage Assumed in Manabe and Strickler (1964)

~

'be

High (cirrus)

Medium (cumulus)

Low (stratus)

~

~

~

SW reflectivity

SW absorptivity

% of area

0.21

0.48

0.69

0.005

0.020

0.035

0.228

0.090

0.313

(Reprinted with permission from the American Meteorological Society.)

69

2.3

10-

n^

E

w

v

fK

3

u)

B

fK

100.

a

A:,/

1000

TEMPERATURE (K)

Fig. 3.16 Calculated temperature profiles for radiative equilibrium, and thermal equilibrium with

lapse rates of 9.8OC km-' and 6SoC km-'. [From Manabe and Strickler (1964). Reprinted with permission from the American MeteorologicalSociety.]

The simplest artifice by which the effect of vertical energy transports by motions

can be included in a global-mean radiative transfer model is a procedure called convective adjustment. Under this constraint the lapse rate is not allowed to exceed a

Where

. radiative processes would make the lapse rate

critical value, say, 6.5 K h-'

greater than the specified maximum value, a nonradiative upward heat transfer is assumed to occur that maintains the specified lapse rate while conserving energy. This

artificial vertical redistribution of energy is intended to represent the effect of atmospheric motions on the vertical temperature profile without explicitly calculating

nonradiative energy fluxes or atmospheric motions. In a global mean model, this

"adjusted" layer extends from the surface to the tropopause.

A temperature profile that is in energy balance when radiative transfer and convective adjustment are taken into account may be called a radiative-convective

equilibrium or thermal equilibrium profile. Thermal equilibrium profiles for assumed maximum lapse rates of 6.5"C h-' and the dry adiabatic lapse rate of

are shown in Fig. 3.16. The thermal equilibrium profile obtained with

93°C h-'

is close to the observed global mean temperature proa lapse rate of 6.5"C h-!

file. No a priori reason exists for choosing a 6.5"C km-' adjustment lapse rate

other than that it corresponds to the observed global-mean value. The maintenance

70

3 Atmospheric Radiative Transfer and Climate

of the lapse rate of the atmosphere is complex and involves many processes and

scales of motion.

One use of a climate model is to understand what factors are most important

and how changes in these factors will affect the climate. In particular, the onedimensional radiative-convective equilibrium model is useful for understanding the

role of trace gases and clouds in determining the temperature profile. Figure 3.17

shows three equilibrium profiles obtained with different gaseous compositions, but

without clouds. With only water vapor present a reasonable approximation to the observed profile is obtained except that the stratosphere is absent. Carbon dioxide with

a mixing ratio of 300 ppm raises the temperature about 10 K above the equilibrium

obtained with only water vapor present. A sharp tropopause and the increase of temperature with height that characterizes the stratosphere appear only when solar absorption by ozone is included in the model.

The contributions of individual gases to the heating rate in radiative-convective

equilibrium are shown in Fig. 3.18. In the stratosphere the lapse rate for radiative

equilibrium is never large and positive, so convective adjustment is not required.

The first-order balance is between heating produced by solar absorption by ozone

2.3

10-

5E

v

W

a:

3

m

cn

W

n 100-

1000

TEMPERATURE (K)

Fig. 3.17 Thermal equilibrium profiles for three cloudless atmospheres obtained with a critical lapse

rate of 6.5 K km-’. One atmosphere has water vapor only; one includes water vapor and carbon dioxide;

and the third contains water vapor, carbon dioxide, and ozone. [From Manabe and Strickler (1964).

Reprinted with permission from the American Meteorological Society.]

71

I

w

9

0

I

c

3

-( 201-1

m

5

v

RATE O F TEMPERATURE CHANGE ("C/day)

Fig. 3.18 Radiative heating rate profiles for a clear atmosphere. LH20, LC02, and LO3 show the

heating rates associated with longwave cooling by water vapor, carbon dioxide, and ozone, respectively.

The S prefix indicates the heating rate associated with solar absorption by each of these gases. NET is the

sum of the solar and longwave radiative heating rates contributed by all gases. [From Manabe and Strickler (1964). Reprinted with permission from the American Meteorological Society.]

and cooling produced by longwave emission from carbon dioxide. The troposphere

is balanced by convective heat transfer from the surface, where a positive net radiative imbalance exists. In a clear atmosphere this net longwave cooling is closely approximated by the cooling from water vapor emission. In the troposphere, longwave

cooling from carbon dioxide is approximately balanced by solar absorption by water

vapor. From the results of radiative-convective equilibrium calculations presented

in Figs. 3.17 and 3.18, we conclude that water vapor is by far the preeminent greenhouse gas in the natural atmosphere.

Radiative-convective equilibrium models can also be used to examine the effect

of simple clouds on the temperature profile. Figure 3.19 shows the effect of inserting

clouds with various characteristics. Low clouds greatly reduce the temperature at

the surface and in the troposphere, whereas the addition of high clouds can cause the

surface temperature to exceed the value obtained for cloud-free conditions. The

albedos assumed for the lower clouds are higher, so that the greater reflection of

solar radiation from these clouds explains a good part of their stronger cooling effect

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3 Atmospheric Radiative Transfer and Climate

170

210

250

290

330

TEMPERATURE (K)

Fig. 3.19 Thermal equilibrium temperature profiles for atmospheres with various cloud distributions. The cloud heights corresponding to each type of cloud are shown on the right (L = low,

M = medium, and H = high cloud). The heavy dashed line shows the equilibrium profile for clear skies.

[From Manabe and Strickler (1964). Reprinted with permission from the American Meteorological

Society.]

(Table 3.2). Lower clouds have a weaker effect on the escaping longwave radiation,

however, since their top temperatures are warmer, and this also explains part of the

greater cooling effect of low clouds in these calculations.

3.11 A Simple Model for the Net Radiative Effect of Cloudiness

Clouds are potentially very important for the sensitivity of climate, since they can affect both solar and longwave radiative transfer in the atmosphere. Clouds of sufficient thickness are typically almost perfect absorbers of terrestrial radiation and are

at the same time excellent reflectors of solar radiation. These two properties of

clouds produce opposite effects on the radiation balance. The reflection of solar radiation tends to cool Earth. Because of the decrease in temperature with altitude in

the atmosphere, clouds reduce the outgoing terrestrial radiative flux at the top of the

atmosphere, which tends to warm the climate.

We can illustrate the relative roles of the reflection of solar radiation and trapping

of longwave radiation by clouds with a very simple model of their effect on the

global energy balance at the top of the atmosphere. The energy balance at the top of

3.11 A Simple Model for the Net Radiative Effect of Cloudiness

73

the atmosphere is the difference between the absorbed solar radiation and the outgoing longwave radiation (OLR).

where So is the solar constant and apis the albedo, so that (S0/4)(1 - ap)= Qabs is

We wish to calculate the difference in the net radiation that results from adfling a

cloud layer with specified properties to a clear atmosphere.

m~~~= Rcloudy - Rclear = AQabs - LWt (w)

(3.56)

Suppose that we can specify the albedo for both clear and cloudy cbnditions, so

that the difference in absorbed solar radiation is

(3.57)

To calculate the change in OLR we subtract (3.39) from (3.45):

t

t

7

AF (-) =Fcloudy (=)- F ~ l e a r ( ~ )

(3.58)

If the top of the cloud is above most of the gaseous absorber of longwave radiation, which is water vapor, then we may make the approximation

qzct

,-] = 1.0

(3.60)

in which case (3.59) becomes

1

AF'(=J)=O\$~ - ~ ~ ~ { z , , o o )

~ T ( Z dT{z',=}

' ) ~

5-{% ?}

(3.61)

or

AFT(,)

= m4

t

zCt - Fclear

(-1

(3.62)

Inserting (3.57) and (3.62) into (3.56) gives an approximate formula for the change

in net radiation at the top of the atmosphere that is produced by the addition of

clouds to a clear atmosphere.

mTOA =-

SO

7

~

t

a +pFclear

( ~ 1 m2t

-

(3.63)

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3 Atmospheric Radiative Transfer and Climate

If the cloud top is above most of the longwave absorber, then (3.63) indicates that

the change in net radiation produced by the cloud depends on the albedo contrast between clear and cloudy conditions and on the temperature at the cloud top. Since

most of the water vapor is in the first few kilometers of the atmosphere, the approximation (3.60) is qualitatively correct for cases with cloud tops above 4 or 5 km.

From (3.63) it is possible that the albedo contrast and the cloud top temperature can

be such that the cloud produces no change in the net radiation. The condition for this is

obtained by setting AR,

= 0 in (3.63) and solving for the cloud top temperature.

> 114

r

(3.64)

If we assume that the temperature decreases with a lapse rate of r f r o m a surface

value of T,, then the temperature of the cloud top can be related to its altitude.

T, = T, -rZct

(3.65)

-Ct

We can use (3.64) and (3.65) to solve for the cloud top altitude for which the reduction in OLR will just cancel the reduction in absorbed solar radiation associated with

the presence of a cloud. For numeric values we can use a solar flux of 1367 W m-2,

'a clear-sky OLR of 265 W rn-2, a surface temperature of 288 K, and a lapse rate of

6.5 K krr-'. These are all reasonable global-mean values. The resulting curve of

cloud top altitude versus albedo contrast is shown as the heavy curve in Fig. 3.20.

Clouds with albedo contrasts and altitudes that fall along the heavy line have no net

0

0

0.1

0.2

0.3

0.4

0.5

0.6

*aP

Fig. 3.20 Contours of change in net radiation at the top of the atmosphere caused by the insertion of

a cloud into a clear atmosphere, plotted against cloud top altitude and the planetary albedo contrast between cloudy and clear conditions. The net radiation changes are calculated with the approximate model

described in Section 3.8 that is invalid for clouds with tops lower than about 4 km. Contours from -100 to

+150 W m-2 are shown at an interval of 50 W m-2. The zero contour where the cloud has no net effect on

the radiation budget at the top of the atmosphere is bold and negative contours are dashed.

3.12 Observed Role of Clouds in the Energy Balance of Earth

75

effect on the energy balance at the top of the atmosphere. Those that fall below the

line will produce a reduction in net radiation, or a cooling, and those above will produce warming. One can obtain the approximate cloud albedo for global average conditions by adding the clear-sky albedo, which is about 15%.As the cloud top rises,

the albedo contrast between cloudy and clear conditions that will just balance the

OLR change also increases. Clouds with high cold tops and low albedos can cause a

significant positive change in net radiation, while low bright clouds can cause a large

negative change in net radiation at the top of the atmosphere.

3.12 Observed Role of Clouds in the Energy Balance of Earth

It is possible to measure the radiative fluxes of energy entering and leaving Earth

from orbiting satellites. If the spatial resolution of the measurements of the energy

fluxes provided by the instrument on the satellite is great enough, then cloud-free

scenes may be identified. These cloud-free scenes can be averaged together to estimate the clear-sky radiation budget. If these cloud-free scenes are taken to represent

the atmosphere in the absence of clouds, then the difference between the cloud-free

radiation budget and the average of all scenes represents the effect of clouds on the

radiation budget. We can call the effect of clouds on the radiation budget the cloud

radiative forcing of the energy balance.

Table 3.3 shows estimates of the globally and annually averaged radiation budget

components for average conditions, cloud-free conditions, and the difference between them, which is called cloud forcing. The uncertainty of these estimates is

about 5 W mP2, as indicated by the average net radiation of 5 W m-2, which should

logically be zero. In round numbers the observations indicate that clouds increase

the albedo from 15 to 30%, which results in a reduction of absorbed solar radiation

of 50 W mP2. This cooling is offset somewhat by the greenhouse effect of clouds,

which reduces the OLR by about 30 W m-2. The net cloud forcing of the radiation

budget is thus a loss of about 20 W mP2. The meaning of this number is that, if

clouds could suddenly be removed without changing any other climate variable, then

Table 3.3

Cloud Radiative Forcing as Estimated from Satellite Measurements

OLR

Albedo

~

Average

Cloud-free

Cloud forcing

234

239

+5

266

288

+22

30%

15%

+3 1

-48

-17

+15%

~~

Radiative flux densities are given in W m-2 and albedo in percent. [From Harrison ef d.(1990). 0 American Geophysical Union.]

3.12 Observed Role of Clouds in the Energy Balance of Earth

77

Earth would begin to gain 20 W m-2 in net radiation and consequently begin to

warm up. The size of the temperature increase that might result from such a change

in the radiation balance is the subject of Chapter 9.

The observed distribution of clouds has been estimated in two ways. Surface observers have recorded the type and fractional distribution of clouds and a long record

of such observations has been compiled into a cloud climatology.2 In more recent

years attempts have been made to systematically characterize cloud distributions

from the observations of visible and infrared radiation taken from meteorological

satellite^.^ Each of these data sets has its strengths and weaknesses related to the

viewing geometry (up versus down) and the instrumentation (the human eye versus

a radiometer). Surface observations have a much better view of cloud base, whereas

satellite measurements see the tops of the highest clouds very well and provide a

more direct means of estimating the visible optical depth of the clouds.

Figure 3.21 shows global maps of the fractional area coverage of clouds with tops

at pressures lower than 440 mb (high clouds), clouds with tops at pressures greater

than 680 mb (low clouds) and clouds with tops at any pressure (total cloud amount).

High clouds are concentrated in the convection zones of the tropics over equatorial

South America and Africa, and a major concentration exists over Indonesia and the

adjacent regions of the eastern Indian and western Pacific Oceans. Low clouds are

most prevalent in the subtropical eastern ocean margins and in middle latitudes. The

low cloud concentrations in the eastern subtropical oceans are associated with lower

than average sea surface temperature (SST) (Fig. 7.11) and consist of stratocumulus

clouds trapped below an inversion. Low clouds are heavily concentrated over the

oceanic regions and are less commonly observed over land. The total cloud cover

also shows a preference for oceanic regions, particularly in midlatitudes where the

total cloud cover is greatest. Minima in total cloud cover occur in the subtropics in

desert regions, but regions with low total cloud amounts also occur over the

Caribbean Sea and over the southern subtropical zones of the Pacific, Atlantic, and

Indian oceans.

Estimates of the effect of clouds on the radiative energy budget at the top of the

atmosphere can be derived from satellite measurements of the broadband energy

flux.4 The longwave cloud forcing is the reduction of the OLR by the clouds, and so

ZWarrenet al. (1986, 1988).

3Rossow and Schiffer (1991).

4Hamson etal. (1990).

Fig. 3.21 Annual average cloud fractional area coverage in percent estimated from satellite data

under the International Satellite Cloud Climatology Project [ISCCP, Rossow and Schiffer (1991)J.

(a) Clouds with tops higher than 440 mb, (b) clouds with tops lower than 680 mb, and (c) all clouds. In (a)

and (b) the contour interval is 5%, with values greater than 30% lightly shaded, and greater than 50%

heavily shaded. In (c) the contour interval is also 5%, but light shading is applied for values greater than

50% and heavy shading for values greater than 80%.