Potter T.D., Colman B.R. (co-chief editors). The handbook of weather, climate, and water: dynamics, climate physical meteorology, weather systems, and measurements

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the radius vector of Earth, or ðr

=rÞ

, which is an eccentricity correction factor of

Earth’s orbit. A comm on approximation is



¼ 1:000110 þ 0:034221 cos G þ 0:001280 sin G þ 0:000719 cos 2G

þ 0:000077 sin 2G ð1Þ

where G ¼ 2pðd

 1Þ=365 and d

is the day of the year, which ranges from 1 on

January 1 to 365 (366 on a leap year) on December 31.

To determine the solar ﬂux within Earth’s atmospher e requires deﬁning some

fundamental parameters that describe the position of the sun relative to a location

on Earth. The ecliptic plane is the plane of revolution of Earth around the sun. Earth

rotates around its polar axis, which is inclined at approximately 23:5



from the

normal to the ecliptic plane. The angle between the polar axis and the normal to

the ecliptic plane remains unchanged as does the angle between Earth’s celestial

equator plane and the ecliptic plane (Fig. 1). The angle between the line joining the

centers of the sun and Earth to the equatorial plane changes and is known as the solar

declination, d. It is zero at the vernal and autumnal equinoxes and approximately

23:5



at the summer solstice and 23:5



at the winter solstice. A useful approximate

equation to express the declination in degrees is

d ¼ 0:006918 0:399912 cos G þ 0:070257 sin G  0:006758 cos 2G

þ 0:000907 sin 2G ð2Þ

Figure 3 Deﬁnition of Earth’s azimuth and zenith angles. See ftp site for color image.

346

RADIATION IN THE ATMOSPHERE: OBSERVATIONS AND APPLICATIONS

The solar zenith angle y

is the angle between the local zenith and the line join ing

the observer and the sun. The solar zenith angle ranges between 0



and 90



and is

determined from

cos y

¼ sin d sin f þ cos d cos f cos o ð3Þ

where f is the geographic latitude. The hour angle o is measured at the celestial

pole between the observer’s longitude and the solar longitude. The hour angle is zero

at noon and positive in the morning, changing 15



per hour.

The solar azimuth c is the angle at the local zenith between the plane of the

observer’s meridian and the plane of a great circle passing through the zenith and the

sun. It is zero at the south and measured positive to the east and varies between

180



and is calculated from

cos C ¼

ðcos y

sin f  sin dÞ

sin

1

½cos y

cos f

ð4Þ

and 0



 C  90



when cos C  0 and 90



 C  180



when cos C  0.

Incoming Solar Radiation at the Top of the Atmosphere

Earth’s rotation causes daily changes in the incoming solar radiation while the

position of its axis relative to the sun causes the seasonal changes.

The Solar Constant

The ﬂux at the top of Earth’s atmosphere on a horizontal surface is

F ¼ S



cos y

ð5Þ

where S

is the solar constant, the rate of total solar energy at all wavelengths

incident on a unit area exposed normally to rays of the sun at 1 AU.

The solar constant can be measured from satellites or derived from the conserva-

tion of energy. The power emitted by the sun is approximately 3:9  10

W. The

radius of the sun is approximately 7  10

m, so the ﬂux at the surface of the sun is

sun

 6:3  10

W=m

2

. What is the irradiance reaching Earth ðS

Þ? Assuming the

power from the sun is constant, by conservation of energy the total power cross ing a

sphere at a radius equivalent to the Earth–sun distance must be equal to the power

emitted at the sun’s surface, so

ð4pR

Þ¼F

sun

ð4pR

Þð6Þ

¼ F

sun



 1368 W=m

ð7Þ

2 BOUNDARY CONDITIONS 347

The amount of solar energy reaching Earth is not constant, but a function of

distance from the sun, and therefore time of year, as indicated by Eq. (5).

The spectral distribution of the solar ﬂux at the top of the atmosphere is shown in

Figure 4. While the sun is often considered to have a radiative temperature of

approximately 5777 K, Figure 4 demonstrates that the sun is not a perfect blackbody.

Also shown in Figure 4 is the percentage of solar energy below a given wavelength

(dotted line) conﬁ rming that most of the sun’s energy resides at wavelengths less

than 4 mm.

The daily radiation on a horizontal surface in joules per square meter per day

(J=m

day) is

day

sunset

sunrise

Fdt¼ 2

Fdo

ð8Þ

After converting dt to hour angle,

2p radians

24 hour

Figure 4 Spectral distribution of solar energy at the top of the atmosphere. Solid line is

incoming solar spectral ﬂux, and dashed line is the radiation emitted by a blackbody with a

temperature of 5777 K. See ftp site for color image.

348

RADIATION IN THE ATMOSPHERE: OBSERVATIONS AND APPLICATIONS

we obtain the average daily insolation on a level surface at the top of the

atmosphere as

day

ðr

=rÞ

ðo

sin d sin f þ cos d cos f sin o

ð9Þ

In polar regions during summer, when the sun is always above the surface, o

equals p and the extraterrestrial daily ﬂux is

day

¼ S



sin d sin f

The daily variation of insolation at the top of the atmosphere as a function of latitude

and day of year is depicted in Figure 5. Since Earth is closer to the sun in January,

the Southern Hemisphere maximum insolation during summer is about 7% higher

than the maximum insolation during Northern Hemisphere summer. The insolation

of the polar regions is greater than that near the equator during the summer solstice.

This results from the longer days, despite the high solar zenith angles at these high

Figure 5 Daily insolation at top of atmosphere as function of latitude and day of year. Also

shown is the solar declination angle (thick dashed line). See ftp site for color image.

2 BOUNDARY CONDITIONS 349

latitudes. However, the annual average insolation at the top of the atmosphere at the

poles is less than half the annual average insolation at the equator.

Surface Radiative Properties

The surface albedo r

is deﬁned as the ratio of the radiation reﬂected from the

surface to the radiation incident on the surface. The surface albedo varies from

approximately 5% for calm, deep water to over 90% for fresh snow. The surface

albedo over land depends on the type and condition of the vegetation or bare ground.

Thus, over land, the surface albedo varies from location to location and with time.

Over water, the surface albedo is also a strong function of solar zenith angle.

The surface albedo depends on the wavelength of the incid ent radiation. Figure 6

is an example of spectral reﬂection of various surfaces. Snow is very reﬂective at

visible wavelengths (0.4 to 0:7 mm) and less reﬂective an the near-infrared wave-

lengths. Plants have higher reﬂectances in the near-infrared than in the visible.

Photosynthesis is effective at absorbing visible energy. When plants dry out, their

chlorophyll conten t decreases and the reﬂectance at visible wavelength increases.

In the longwave spectral region we generally speak of surface emissivity, or

emittance, e

¼ 1  r

. It is common in models to assume that the surface emittance

Figure 6 Examples of the spectral albedo of different surfaces. See ftp site for color image.

350

RADIATION IN THE ATMOSPHERE: OBSERVATIONS AND APPLICATIONS

in the infrared ðe

Þ is spectrally independent and equal to 1, even though the infrared

emissivity of most surfaces is between 0.9 and 0.98 (Table 1). Assuming that the

surface emi ssivity is 1 leads to approximately a 5% error in upward ﬂuxes. The

emittance of some surfaces is also wavelength dependent. In particular, sand and

desert surface emissivity varies in the 8- to 12-mm window regime between 0.8

and 0.95.

3 RADIATIVE HEAT ING RATES

This section provides examples of radiative heating proﬁles for clear and cloud

conditions. If more radiative energy enters the layer than leaves the layer, a heating

of the layer results. Since the atmosphere is too cold to emit much radiation below

3 mm, atmospheric solar radiative heating rates must all be positive. Since the atmos-

phere emits and absorbs longwave radiation, longwave radiative heating rates can be

positive (net energy gain) or negative (net energy loss).

Radiatively Active Gases in the Atmosphere

The three major absorbing and emitting gases in the stratosphere and troposphere are

ozone, carbon dioxide, and water vapor. There are also important minor constituents

such as chloroﬂuorocarbons (CFCs) and methane. While N

and O

are the most

abundant gases in the atmosphere, from an atmospheric energetics point of view they

are of small importance. The solar energy below 0:2 mm is absorbed by O, NO, O

and N

before it reaches the stratosphere.

Ozone ðO

Þ primarily absorbs in the ultraviolet and in the 9.6-mm region. Solar

radiation in the Hartley spectral band (0.2 to 0:3 mm) is absorbed in the upper

stratosphere and mesosphere by ozone. Absorption by O

in the Huggins band

(0.3 to 0:36 mm) is not as strong as in the Hartley bands. Ozone absorbs weakly

in the 44- to 0.76-mm region, and strongly around the 9.6-mm region, where radiation

is emitted by the surface.

Carbon dioxide is generally a weak absorber in the solar spectrum with very weak

absorption in the 2.0-, 1.6-, and 1.4-mm bands. The 2.7-mm band is strong enough that

TABLE 1 Infrared Emissivities

of Some Surfaces

Water 0.92–0.96

Ice 0.96

Fresh snow 0.82–0.99

Dry sand 0.89–0.90

Wet sand 0.96

Desert 0.90–0.91

Dry concrete 0.71–0.88

Pine forest 0.90

Grass 0.90

3 RADIATIVE HEATING RATES 351

it should be included in calculations of solar absorption, though it overlaps with H

The 4.3-mm band is important more in the infrared region due to the small amount of

solar energy in this band. This 4-mm band is important for remote sensing atmospheric

temperature proﬁles. CO

absorbs signiﬁcantly in the 15-mm band from about 12.5 to

16:7 mm (600 to 800 cm

1

). It is these differences in the shortwave and infrared

properties of CO

(and atmospheric water vapor) that lead to the greenhouse effect.

Water vapor absorbs in the vibrational and rotational bands (ground-state transi-

tions). In terms of radiative transfer through the atmosphere, the important water

vapor absorption bands in the solar spectrum are centered at 0.94, 1.1, 1.38, 1.87,

2.7, and 3:2 mm. In the infrared, H

O has a strong vibrational-absorption band at

6:3 mm. The rotational band extends from approximately 13 mm to 1 mm. In the

region between these two infrared water vapor bands is the continuum, 8 to

13 mm, known as the atmospheric window. The continuum enhances absorption in

the lower regions of the moist tropical atmosphere.

Radiative Heating Rates under Clear Skies

The previous chapter discussed the procedures to calculate radiative ﬂuxes and

heating rates in the atmosphere. In this secti on we discuss examples of radiative

heating and cooling under clear-sky conditions.

Atmospheric radiative heating rates due to absorption of solar energy are given in

Figure 7 for different solar zenith angles and for tropical conditions. As the solar

zenith angle decreases, the total heating of the atmosphere decreases as the solar

energy incident on a horizontal surface at the top of atm osphere decreases. Figure 8

demonstrates the absorption by the individual gases if they existed in the atmosphere

alone. The large heating in the stratosphere is due to absorption of solar energy by

. A minim um in the heating occurs in the upper troposphere. The increased

heating in the lower troposphere is due to water vapor. CO

contributes little to

the solar heating of the atmosphere.

Infrared heating rates are shown in Figure 9 for standard tropical, midlatitude

summer, and subarctic summer conditions. Negative values indicate a cooling. The

larger cooling rates in the lower troposphere for the tropical conditions aris e due to

the warmer temperatures and larger amounts of water vapor. The contribut ions by

O, CO

, and O

to the cooling in a tropical atmosphere are shown in Figure 10.

In the tropical moist atmosphere, the water vapor continuum (8- to 13-mm region)

makes signiﬁ cant contributions to the cooling to the lowest layers of the atmosphere.

accounts for the large cooling in the stratosphere. The positive radiative heating

rates by O

in the stratosphere arise due to the large amounts of radiation in the

9.6-mm band.

Radiative Heating under Cloudy Conditions

Clouds signiﬁcantly alter the radiative heating and cooling of the atmosphere and at

Earth’s surface. Clouds also undergo physical changes (e.g., particle size distribu-

tion, water content, and cloud top and base altitude) as they form, grow, and dissi-

352 RADIATION IN THE ATMOSPHERE: OBSERVATIONS AND APPLICATIONS

pate. These physical changes are capable of affecting the radiative characteristics of

the cloud. Radiative processes have a strong inﬂuence on the convective structure

and water budget of clouds and affect the particle growth rate.

The previous chapter examined the appropriate equations for calculating the

radiative properties of clouds. In this section, we examine the radiative impact of

clouds on the atmospheric heating proﬁle. Figure 11 shows the impact of four

clouds, each 1 km thick, with differing cloud microphysics on the radiative heating

of a midlatitude summer atmosphere with a solar zenith angle of 15



. Three of the

clouds have r

eff

radius of 20 mm with differing ice water content (IWC). Heating

rates are expressed in degrees Celsius per day. The larger the IWC the greater the

energy convergence, expressed as a heating, within the cloud, as more solar radiation

is absorbed. The larger the IWC, the more the heating below cloud base is reduced

because of the reduced amount of solar energy below the cloud.

The presence of the cloud affects the heating proﬁles above the cloud. As the

cloud optical depth increases, more energy is reﬂected upward, enabling absorption

in gaseous absorption bands. The particle size also impacts the heating proﬁle.

Comparing the two clouds with the same IWC, but different r

eff

indicates that the

cloud with the large particle size absorbs less solar energy.

Figure 7 Shortwave radiative heating in degrees celsius per day for standard tropical

conditions and three solar zenith angles. See ftp site for color image.

3 RADIATIVE HEATING RATES 353

Figure 12 depicts the impact of clouds on the longwave heating, for a midlatitude

summer atmosphere. The cloud is 2 km thick with a top at 10 km. The top of the

cloud has a net energy loss, expressed as a cooling. The cloud base can experience a

radiative warming or cooling, depending on the IWC. Increasing the IWC reduces

the cooling below cloud base as more energy is emitted by the cloud into this lower

layer. Increasing the particle size reduces the cloud radiative heating, if the IWC is

ﬁxed.

We will now consider how radiation interacts with a cloudy layer, accounting for

multiple scatteri ng within the cloud. The previous chapter discussed how radiation

interacts with a particle, or distribution of particles, in terms of the single scattering

properties of the par ticles. There are some useful limits of the two stream model to

consider:

lim

d!0

R ¼

ð10Þ

Figure 8 Shortwave radiative heating for standard tropical conditions by H

O, CO

, and O

and a solar zenith angle of 30



. See ftp site for color image.

354

RADIATION IN THE ATMOSPHERE: OBSERVATIONS AND APPLICATIONS

lim

d!0

A ¼

ð1  o

Þð11Þ

lim

d!1

R ¼

ða

þ kg

ð1 þ km

Þðk þ g

ð12Þ

These limits become useful in understanding the relationship between a cloud’s

microphysical properties and its radiative properties. For example, the reﬂectance is

inversely proportional to the cosine of the solar zenith angle. Thu s, as the sun gets

lower in the sky, the reﬂectance of a cloud increases. Cloud albedo increases with

optical depth, or the water p ath, though it does approach a limiting value. Reﬂec-

tance is directly proportional to the single scattering albedo. As the par ticles

composing a cloud get smaller, o

increases, and so does the cloud reﬂectance.

Cloud absorption decreases with increasing solar zenith angle, the opposite of the

case for gases. For optically thin clouds, and thin aerosol layers, the absorption is

proportional to ð1  o

Þ. The variation of absorption with increasing r

is a combi-

nation of two competing factors: Droplet absorption 1  o

increases with increas-

ing drop size, while the extinction, and thus optical depth, decreases with incre asing

Figure 9 Longwave radiative heating for standard tropical, midlatitude summer, and

subarctic summer conditions. See ftp site for color image.

3 RADIATIVE HEATING RATES 355