1. Radiative TransferVirtually all the exchanges of energy between the earth-atmosphere system and therest of the universe take place by radiative transfer. The earth and its atmosphereare constantly absorbing solar radiation and emitting their own radiation to space.Over a long period of time, the rates of absorption and emission are very nearlyequal, thus the earth-atmosphere system is very nearly in equilibrium with the sun.

Radiative transfer also serves as a mechanism for exchanging energy betweenthe atmosphere and the underlying surface, and among different layers of the at-mosphere. Radiative transfer plays an important role in a number of chemicalreactions in the upper atmosphere and in the formation of photochemical smogs.The transfer properties of visible radiation determine the visibility, the color of thesky and the appearance of clouds. Radiation emitted by the earth and atmosphereand intercepted by satellites is the basis for remote sensing of the atmospherictemperature structure, water vapor amounts, ozone and other trace gases.

2. Spectrum of RadiationElectromagnetic radiation may be viewed as an ensemble of waves propagatingat the speed of light (c∗ = 2.998 × 108m/s through vacum). We characterizeradiation in terms of:

frequency ν = c∗/λ

wavelength λ = c∗/ν

Radiative transfer in planetary atmosphere involves an ensemble of waves witha continuum of wavelengths and frequencies. We partition them into bands :shortwave (λ < 4µm) carries most of the energy associated with solar radia-tion or longwave (λ > 4µm) which refers to the band that encompasses most ofthe terrestrial. The visible region 0.39 − 0.79µm is defined by the range of wave-lengths that the human eye is capable of sensing, and subranges of the visible arediscernible as colors.

3. DefinitionsSolid Angle ω Consider a cone with its vertex at the origin of a concentric spher-

ical surface. The solid angle is defined as the ratio of the area of the sphere

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Figure 1: Ahrens, Chapter 2

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intercepted by the cone to the square of the radius.

ω =A

r2(1)

dω =dA

r2(2)

In spherical coordinates

dA = r2sinθdθdφ (3)

The unit of the solid angle is the steradian. The area cut out of a sphere byone steradian is equal to the square of the radius. Integration over the entirespherical surface then gives ω = 4πsteradians

Flux density F Amount of radiant energy passing through a unit area per unittime. Expressed in [Wm−2]. This flux includes energy contributions fromall wavelengths between some specified limits λ1 and λ2 (the range cannotbe zero). F =

∫ λ2λ1Fλdλ =

∫ ν2ν1Fνdν.

We define monochromatic flux density Fλ as

Fλ = lim∆λ→0F (λ, λ+ ∆λ)

∆λ(4)

If you mark an area on the ground, the amount of daylight falling on itcan be measured in watts per square meter. This is the incident flux ofsolar radiation - and it would decrease with an obstruction, like a cloudor as the evening comes. The flux makes no distinction concerning wherethe radiation is coming from, so in order to completely characterize theradiance field at a given location, we must know not only the flux but alsothe direction from which the radiation is comingthis is the radiant intensity

Intensity I Radiant energy per unit time coming from a specific direction andpassing through a unit area perpendicular to that direction. Total intensityis calculated as the integral over all wavelengths within given limits I =∫ λ2λ1Iλdλ =

∫ ν2ν1Iνdν. The intensity I tells you in detail both the strength and

direction of various sources contributing to the incident flux on a surface. If

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you are looking at the sky, some regions have higher radiant intensity (whenyou look at the sun for example)

The units are [Wm−2sr−1].The monochromatic intensity Iλ and flux den-sity Fλ are related by

Iλ =dF

dωcosθ(5)

We can integrate over the solid angle subtended by a hemisphere to deter-mine the monochromatic flux density coming from all directions

Fλ =

∫ 2πsr

0

Iλcosθdω =

∫ 2π

0

dφ

∫ π/2

0

Iλcosθsinθdφ (6)

4. Blackbody RadiationA blackbody is a surface that absorbs all incident radiation.

The Planck Function Determined experimentally, the intensity of radiation emit-ted by a blackbody is

Bλ =c1λ−5

π(ec2/λT − 1)(7)

where c1 = 3.74 × 10−16Wm2 and c2 = 1.45 × 10−2mK. Theoretical jus-tification of this empirical relationship led to the development of the theoryof quantum physics.

Wien’s Displacement Law Differentiating Equation 7 and setting the derivativeequal to zero, gives the wavelength of peak emission for a blackbody attemperature T (HW6).

λm =2897

T(8)

where T in K and λm in µm. An important consequence of Wien displace-ment law is the fact that solar radiation is concentrated in the visible andnear-infrared parts of the spectrum, while radiation emitted by the planetsand their atmospheres is largely confined to the infrared. The nearly com-plete absence of overlap between the curves justifies dealing with solar andplanetary radiation separately in many problems of radiative transfer.

Stefan-Boltzmann Law The black body flux density obtained by integrating the

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Planck function Bλ over all wavelengths.

F = σT 4 (9)

Where σ is the Stefan-Boltzmann constant equal to 5.67× 10−8Wm−2K−4

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5. Radiative properties of Nonblack materialsUnlike blackbodies, which absorb all incident radiation, nonblack bodies suchas gaseous media can also reflect and transmit radiation. We will give a briefdescription of the radiative processes in nonblack bodies. The fate of radiationdepends on wavelength.

1. Transmitted Radiation passes undisturbed. We define the monochromaticfractional transmissivity as Tλ = Iλ(transmitted)

Iλ(incident)

2. Reflected Radiation. Reflectivity is depicted by albedo. Albedo usuallyrepresents all wavelengths and refers to the earth-atmosphere reflection. Wedefine the monochromatic fractional reflectivity as Rλ = Iλ(reflected)

Iλ(incident)

3. Absorbed Increase in internal energy of the object. We define the monochro-matic fractional absorptivity as αλ = Iλ(absorbed)

Iλ(incident)

(a) Ionization-Dissociation Interactions. In these interactions, an electronis stripped from an atom or molecule, or a molecule is torn apart.These interactions occur primarily at ultraviolet and shorter wave-lengths. All solar radiation shorter than about 0.1 µm in wavelengthis absorbed in the upper atmosphere by ionizing atmospheric gases,particularly atomic oxygen. Between 0.1 and 0.2 µm molecular oxy-gen dissociates into atomic oxygen. Radiation between 0.2 and 0.3µm is absorbed by dissociation of ozone. These bands are importantfor preventing the radiation from reaching the ground and in satellitemeteorology for measuring ozone concentrations.

(b) Electronic Transitions Orbital electron jumps between quantized en-ergy levels. These occur mostly in the UV and visible. Ozone, andmolecular oxygen.

(c) Vibrational transitions A molecule changes vibrational energy states.These transitions occur mostly in the infrared portion of the spectrumand are extremely important for satellite meteorology. The two chiefabsorbers in the infrared region of the spectrum are carbon dioxideand water vapor. Symmetric stretching has neither a static or dynamicelectric dipole moment because the symmetry of the molecule is main-tained. If a molecule has no electric dipole moment, the electric fieldof incident radiation cannot interact with the molecule. (This is why

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N2 and O2, the two most abundant gases in the atmosphere, are trans-parent in the infrared.

(d) Rotational transitions a molecule changes rotational states. These oc-cur in the far infrared and microwave portion of the spectrum. Theycan occur at the same time as vibrational transitions. Figure 4.7.

The three are related by αλ +Rλ + Tλ = 1. For a black body αλ = 1.

4. We can also define the Monochromatic Emissivity ελ as the ratio of themonochromatic intensity of the radiation emitted by the body to the corre-sponding blackbody radiation: ελ = Iλ(emitted)

Bλ(T )

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5a. Kirchoff’s Law

It can be shown that the radiation emitted by a given material is a function of tem-perature and wavelength only. Consider an opaque, hollow enclosure with zerotransmissivity into which is placed a slab of finite thickness. In general, this slabwill reflect, absorb and transmit parts of the incident radiation. In addition, itwill emit radiation itself. We now allow the enclosure and the slab to reach ther-modynamic equilibrium, such that the slab and the enclosure walls are the sametemperature. Under this condition, the flow of energy in all directions must be thesame. In thermodynamic equilibrium, the amount entering the slab must exactlyequal the amount leaving, or there would be a net flow of heat to or from the walls,into or out of the slab. Since the slab and the walls are in thermodynamic equi-librium, this would constitute a violation of the Second Law of Thermodynamics.Therefore, the balance equation is:

Iλ −RλIλ = TλIλ + Eλ (10)

Where Eλ is the emitted radiance in the same direction as Iλ. But TλIλ = Iλ(1 −αλ −Rλ) since αλ +Rλ + Tλ = 1. Therefore,

Iλ(1 −Rλ) = Iλ(1 − αλ −Rλ) + Eλ (11)

Thus, Eλ − αλIλ = 0 or Eλ = αλIλThus, inside of an opaque, hollow enclosure in thermodynamic equilibrium,

the amount emitted by the slab equals the amount absorbed by the slab. We nowimagine our enclosure to be replaced by a different one, constructed from a differ-ent material, and again allow it to come into thermodynamic equilibrium with thesame slab and at the same temperature as before. Consequently, the slab emissionwill be the same as before, since it depends only on temperature and wavelength,neither of which has been changed. Similarly, the slab absorption will not changebecause the slab material is the same. Thus we have:

Eλ = αλI′λ (12)

Where I ′λ is the incident radiation on the slab in the new enclosure, thus itfollows that Iλ = I ′λ.Thus, the radiation within an opaque, hollow enclosure isindependent of the material from which the walls are made. Re-writing the aboveequation we see Eλ

αlambda= Iλb = f(T, λ) only and Iλb is the radiance inside an

opaque hollow enclosure at temperature T and wavelength λ.

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This result is known as Kirchhoff’s Law, which states that “The ratio of theemission to the fractional absorptivity of a slab of any material in a state of ther-modynamic equilibrium and at wavelength λ is equal to a constant”.

We may now define the fractional emissivity ελ as the ratio of the radiationemitted at the wavelength λ to that within a hollow enclosure at the same temper-ature or:

ελ =EλIλb

(13)

From this definition, we see that Eλ = ελIλb. But from Kirchoff’s Law it thenfollows:

ελ = αλ (14)

Or the fractional emissivity equals the fractional absorptivityKirchoff’s Law is fundamental to further development of the subject of radia-

tive transfer, and is frequently applied in a variety of applications. Recalling thatit is strictly valid only under conditions of thermodynamic equilibrium, it is nev-ertheless generally assumed to be valid for atmospheric problems even though theatmosphere is not strictly in thermodynamic equilibrium.

We may now carry this thought experiment one step further. Let’s replacethis slab by an ideal black body such that, by definition, it completely absorbs allradiation falling on it. Inside the hollow enclosure then, the radiation leaving theblack body slab consists entirely of radiation emitted by the slab. The equilibriumcondition becomes

Iλb = Eλ (15)

leading to the important conclusion that the radiation flowing in any directionwithin the hollow enclosure in thermodynamic equilibrium is equal to the energyemitted in the same direction as an ideal black body. Such radiation is calledblack body radiation, and from our earlier arguments is isotropic or equal in alldirections.

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6. Examples1. Prove that the intensity I of solar radiation is independent of distance from

the sun, provided that the distance is large and that radiation emitted fromeach elemental area on the sun is independent of the zenith angle.

2. The average flux density Fe of solar radiation reaching the earth’s orbit is1368Wm−2. Nearly all the radiation is emitted from the outermost visiblelayer of the sun, which has a mean radius of 7×108m. Calculate the equiva-lent blackbody temperature or effective temperature of this layer. The meandistance between the earth and sun is 1.5 × 1011m.

3. Calculate the equivalent blackbody temperature of the earth assuming aplanetary albedo αp = 0.3 where αp is the fraction of the total incidentsolar radiation that is reflected and scattered back to space. Assume that theearth is in radiative equilibrium.

4. A completely gray flat surface on the moon with an absorptivity of 0.9 isexposed to direct overhead solar radiation. What is the radiative equilibriumtemperature of the surface? If the actual temperature is 300K, what is thenet flux density above the surface?

5. A flat surface is subject to overhead solar radiation as in the previous ex-ample. The absorptivity is 0.1 for solar radiation and 0.8 in the infraredpart of the spectrum, where most of the emission takes place. Compute theradiative equilibrium temperature.

6. Calculate the radiative equilibrium temperature of the earth’s surface andatmosphere assuming that the atmosphere can be regarded as a thin layerwith absorptivity of 0.1 for solar radiation and 0.8 for terrestrial radiation.Assume that the earth’s surface radiates as a black body.

7. Radiative-Convective Equilibrium Temperature Pro-files (from Global Physical Climatology by Hart-man)

One can solve the radiative transfer equation for global mean terrestrial condi-tions. This involves construction of appropriate models for the transmission of

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the various band systems of importance in the atmosphere, insertion of these intoa computational model of the radiative transfer equation and iteration to obtaina steady balance solution. The variables that determine the fluxes of radiant en-ergy in the atmosphere include the atmospheric gaseous components, aerosols andcloud characteristics, surface albedo and insolation. In a global mean model, thetemperature only depends on altitude. We need to specify:

H2O Most important gas for radiative transfer. Water vapor has a rotation bandnear 6.3µm and a rotation continuum at wavelengths longer than 12 µm. Italso absorbs solar in the troposphere.

CO2 Important because of increased concentrations. Mixing ration can be as-sumed constant with latitude and altitude up to 100km. Strong vibration-rotation band of CO2 at 15µm makes it important for long wave radiativetransfer. Also absorbs significant amount of solar.

O3 Ozone has fast sources and sinks in the stratosphere. In the surface it is relatedto photochemical smog. Ozone has a vibration-rotation band near 9.6 µmthat is important for long wave energy, and has a dissociation continuumthat absorbs solar between 200 and 300 nm. Absorption heats the middleatmosphere and causes the temperature increase with height that defines thestratosphere and troposphere.

Aerosols Affect transmission of both solar and terrestrial radiation. Sulfate aerosolsin the troposphere are raditively important.

Surface Albedo Highly variable from location to location depending on typeand condition of surface material and vegetation. When the surface is snowcovered, its albedo is generally much higher.

Clouds Clouds vary considerably in amount and type one the globe. They havetry important effects on long wave and solar energy transfer in the atmo-sphere. The optical properties of the clouds must also be specified. Waterclouds have weak solar absorption, but they seatter solar very effectively.Thick clouds can be assumed to be black bodies for long wave radiation. Asimple approach is to specify the properties of three types of clouds.

Looking at Figure 3.17 we see that with only water vapor present, a reasonableapproximation to the observed profile is obtained except that the stratosphere

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is absent. Carbon dioxide with a mixing ratio of 300 ppm raises the tempera-ture by 10 K above the equilibrium obtained with only water vapor present. Asharp tropopause and the increase of temperature with height that characterizesthe stratosphere appear only when solar absorption by ozone is included in themodel.

Figure 3.16 shows a calculated temperature profile that is in radiative equi-librium. Atmospheric temperatures in radiative equilibrium decrease rapidly withaltitude near the surface. In the troposphere, radiative equilibrium temperatureprofiles are hydrostatically unstable in the sense that parcels of air that are ele-vated 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

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troposphere. 60% by latent and sensible heat and 40% by net long wave radia-tion emission. The global mean temperature profile of the earth is not in radiativeequilibrium, but rather in radiative-convective equilibrium. To obtain a realisticglobal-mean vertical energy balance, the vertical flux of energy by atmosphericmotions must be included.

The simplest artifice by which the effect of vertical energy transports by mo-tions can be included in a global-mean radiative transfer model is a procedurecalled convective adjustment. Under this constraint the lapse rate is not allowedto exceed a critical value (6.5Kkm−1. Where radiative processes would makethe lapse rate greater, a non radiative upward heat transfer is assumed to occurthat maintains the specified lapse rate while conserving energy. This ”adjusted”layer extends from the surface to the tropopause. A temperature profile that is inenergy balance when radiative transfer and convective adjustment are taken intoaccount may be called a radiative-convective equilibrium or thermal equilibriumprofile.The thermal equilibrium profile obtained with a lapse rate of 6.5 Kkm−1

is close to the observed global mean temperature profile.

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