Research Article Radiation and Heat Transfer in the ...Radiation and Heat Transfer in the...

Hindawi Publishing CorporationInternational Journal of Atmospheric SciencesVolume 2013, Article ID 503727, 26 pageshttp://dx.doi.org/10.1155/2013/503727

Research ArticleRadiation and Heat Transfer in the Atmosphere:A Comprehensive Approach on a Molecular Basis

Hermann Harde

Laser Engineering and Materials Science, Helmut-Schmidt-University Hamburg, Holstenhofweg 85, 22043 Hamburg, Germany

Correspondence should be addressed to Hermann Harde; [email protected]

Received 29 April 2013; Accepted 12 July 2013

Academic Editor: Shaocai Yu

Copyright © 2013 Hermann Harde. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We investigate the interaction of infrared active molecules in the atmosphere with their own thermal background radiation as wellas with radiation from an external blackbody radiator. We show that the background radiation can be well understood only interms of the spontaneous emission of the molecules. The radiation and heat transfer processes in the atmosphere are describedby rate equations which are solved numerically for typical conditions as found in the troposphere and stratosphere, showing theconversion of heat to radiation and vice versa. Consideration of the interaction processes on a molecular scale allows to developa comprehensive theoretical concept for the description of the radiation transfer in the atmosphere. A generalized form of theradiation transfer equation is presented, which covers both limiting cases of thin and dense atmospheres and allows a continuoustransition from low to high densities, controlled by a density dependent parameter. Simulations of the up- and down-wellingradiation and its interaction with the most prominent greenhouse gases water vapour, carbon dioxide, methane, and ozone inthe atmosphere are presented.The radiative forcing at doubled CO

2concentration is found to be 30% smaller than the IPCC-value.

1. Introduction

Radiation processes in the atmosphere play a major role inthe energy and radiation balance of the earth-atmospheresystem. Downwelling radiation causes heating of the earth’ssurface due to direct sunlight absorption and also due tothe back radiation from the atmosphere, which is the sourceterm of the so heavily discussed atmospheric greenhouse oratmospheric heating effect. Upward radiation contributes tocooling and ensures that the absorbed energy from the sunand the terrestrial radiation can be rendered back to spaceand the earth’s temperature can be stabilized.

For all these processes, particularly, the interaction ofradiation with infrared active molecules is of importance.Thesemolecules strongly absorb terrestrial radiation, emittedfrom the earth’s surface, and they can also be excited by someheat transfer in the atmosphere. The absorbed energy is rera-diated uniformly into the full solid angle but to some degreealso re-absorbed in the atmosphere, so that the radiationunderlies a continuous interaction and modification processover the propagation distance.

Although the basic relations for this interaction ofradiation with molecules are already well known since thebeginning of the previous century, up to now the correctapplication of these relations, their importance, and theirconsequences for the atmospheric system are discussed quitecontradictorily in the community of climate sciences.

Therefore, it seems necessary and worthwhile to give abrief review of the main physical relations and to present onthis basis a new approach for the description of the radiationtransfer in the atmosphere.

In Section 2, we start from Einstein’s basic quantum-theoretical considerations of radiation [1] and Planck’s radi-ation law [2] to investigate the interaction of molecules withtheir own thermal background radiation under the influenceof molecular collisions and at thermodynamic equilibrium[3, 4]. We show that the thermal radiation of a gas can bewell understood only in terms of the spontaneous emissionof the molecules. This is valid at low pressures with only fewmolecular collisions as well as at higher pressures and highcollision rates.

2 International Journal of Atmospheric Sciences

In Section 3, also the influence of radiation from anexternal blackbody radiator and an additional excitation bya heat source is studied. The radiation and heat transferprocesses originating from the sun and/or the earth’s surfaceare described by rate equations, which are solved numericallyfor typical conditions as they exist in the troposphere andstratosphere.These examples right away illustrate the conver-sion of heat to radiation and vice versa.

In Section 4, we derive the Schwarzschild equation [5–11] as the fundamental relation for the radiation transferin the atmosphere. This equation is deduced from pureconsiderations on a molecular basis, describing the thermalradiation of a gas as spontaneous emission of the molecules.This equation is investigated under conditions of only fewintermolecular collisions as found in the upper mesosphereor mesopause as well as at high collision rates as observed inthe troposphere. Following some modified considerations ofMilne [12], a generalized form of the radiation transfer equa-tion is presented, which covers both limiting cases of thinand dense atmospheres and allows a continuous transitionfrom low to high densities, controlled by a density dependentparameter. This equation is derived for the spectral radianceas well as for the spectral flux density (spectral intensity) asthe solid angular integral of the radiance.

In Section 5, the generalized radiation transfer equationis applied to simulate the up- and downwelling radiationand its interaction with the most prominent greenhousegases water vapour, carbon dioxide, methane, and ozone inthe atmosphere. From these calculations, a detailed energyand radiation balance can be derived, reflecting the differentcontributions of these gases under quite realistic conditionsin the atmosphere. In particular, they show the dominantinfluence of water vapour over the full infrared spectrum, andthey explain why a further increase in the CO

2concentration

only gives marginal corrections in the radiation budget.It is not the objective of this paper to explain the

fundamentals of the atmospheric greenhouse effect or toprove its existence within this framework. Nevertheless, thebasic considerations and derived relations for the molecularinteraction with radiation have some direct significance forthe understanding and interpretation of this effect, and theygive the theoretical background for its general calculation.

2. Interaction of Molecules with Thermal Bath

When a gas is in thermodynamic equilibrium with itsenvironment it can be described by an average temperature𝑇𝐺. Like anymatter at a given temperature, which is in unison

with its surrounding, it is also a source of gray or blackbodyradiation as part of the environmental thermal bath. At thesame time, this gas is interacting with its own radiation,causing some kind of self-excitation of the molecules whichfinally results in a population of themolecular states, given byBoltzmann’s distribution.

Such interaction first considered by Einstein [1] is repli-cated in the first part of this section with some smallermodifications, but following themain thoughts. In the secondpart of this section, also collisions between the molecules are

Em

En

ΔE = h�mn = hc/𝜆mn

Figure 1: Two-level system with transition between states𝑚 and 𝑛.

included and some basic consequences for the description ofthe thermal bath are derived.

2.1. Einstein’s Derivation ofThermal Radiation. Themoleculesare characterized by a transition between the energy states𝐸

𝑚

and 𝐸𝑛with the transition energy

Δ𝐸 = 𝐸𝑚− 𝐸

𝑛= ℎ]

𝑚𝑛=

ℎ𝑐

𝜆𝑚𝑛

, (1)

where ℎ is Planck’s constant, 𝑐 the vacuum speed of light, ]𝑚𝑛

the transition frequency, and 𝜆𝑚𝑛

is the transitionwavelength(see Figure 1).

Planckian Radiation. The cavity radiation of a black bodyat temperature 𝑇

𝐺can be represented by its spectral energy

density 𝑢𝜆(units: J/m3/𝜇m), which obeys Planck’s radiation

law [2] with

𝑢𝜆=8𝜋𝑛

4ℎ𝑐

𝜆5

1

𝑒ℎ𝑐/𝑘𝑇𝐺𝜆 − 1, (2)

or as a function of frequency assumes the form

𝑢] = 𝑢𝜆𝜆2

𝑐𝑛=8𝜋𝑛

3ℎ]3

𝑐3

1

𝑒ℎ]/𝑘𝑇𝐺 − 1. (3)

This distribution is shown in Figure 2 for three differenttemperatures as a function of wavelength. 𝑛 is the refractiveindex of the gas.

Boltzmann’s Relation. Due to Boltzmann’s principle, the rela-tive population of the states 𝑚 and 𝑛 in thermal equilibriumis [3, 4]

𝑁𝑚

𝑁𝑛

=𝑔𝑚

𝑔𝑛

𝑒−ℎ]𝑚𝑛/𝑘𝑇𝐺 , (4)

with𝑁𝑚and𝑁

𝑛as the population densities of the upper and

lower state, 𝑔𝑚and 𝑔

𝑛the statistical weights representing the

degeneracy of these states, 𝑘 as the Boltzmann constant, and𝑇𝐺as the temperature of the gas.For the moment, neglecting any collisions of the

molecules, between these states three different transitions cantake place.

Spontaneous Emission. The spontaneous emission occursindependent of any external field from 𝑚 → 𝑛 and ischaracterized by emitting statistically a photon of energyℎ]

𝑚𝑛into the solid angle 4𝜋with the probability 𝑑𝑊𝑠

𝑚𝑛within

the time interval 𝑑𝑡

𝑑𝑊𝑠

𝑚𝑛= 𝐴

𝑚𝑛𝑑𝑡. (5)

International Journal of Atmospheric Sciences 3

0 2 4 6 8 10Wavelength 𝜆 (𝜇m)

6

5

4

3

2

1

0

×10−3

Spec

tral

ener

gy d

ensit

yu𝜆

(J/m

3/𝜇

m)

T = 2000K T = 1500K T = 1000K

Figure 2: Spectral energy density of blackbody radiation at differenttemperatures.

𝐴𝑚𝑛

is the Einstein coefficient of spontaneous emission,sometimes also called the spontaneous emission probability(units: s−1).

Induced Absorption. With the molecules subjected to anelectromagnetic field, the energy of themolecules can changein that way, that due to a resonant interaction with theradiation the molecules can be excited or de-excited. When amolecule changes from 𝑛 → 𝑚, it absorbs a photon of energyℎ]

𝑚𝑛and increases its internal energy by this amount, while

the radiation energy is decreasing by the same amount.The probability for this process is found by integrating

over all frequency components within the interval Δ], con-tributing to an interaction with the molecules:

𝑑𝑊𝑖

𝑛𝑚= 𝐵

𝑛𝑚∫Δ]𝑢]𝑔 (]) 𝑑] ⋅ 𝑑𝑡. (6)

This process is known as induced absorptionwith𝐵𝑛𝑚

as Ein-stein’s coefficient of induced absorption (units: m3⋅Hz/J/s),𝑢] as the spectral energy density of the radiation (units:J/m3/Hz) and 𝑔(]) as a normalized lineshape function whichdescribes the frequency dependent interaction of the radia-tion with the molecules and generally satisfies the relation:

∫

∞

0

𝑔 (]) 𝑑] = 1. (7)

Since 𝑢] is much broader than 𝑔(]), it can be assumed tobe constant over the linewidth, and with (7), the integral in(6) can be replaced by the spectral energy density on thetransition frequency:

∫Δ]𝑢]𝑔 (]) 𝑑] = 𝑢]𝑚𝑛 . (8)

Then the probability for induced absorption processes simplybecomes

𝑑𝑊𝑖

𝑛𝑚= 𝐵

𝑛𝑚𝑢]𝑚𝑛

𝑑𝑡. (9)

Induced Emission. A transition from 𝑚 → 𝑛 caused by theradiation is called the induced or stimulated emission. Theprobability for this transition is

𝑑𝑊𝑖

𝑚𝑛= 𝐵

𝑚𝑛∫Δ]𝑢]𝑔 (]) 𝑑] ⋅ 𝑑𝑡 = 𝐵𝑚𝑛𝑢]𝑚𝑛𝑑𝑡 (10)

with 𝐵𝑚𝑛

as the Einstein coefficient of induced emission.

Total Transition Rates. Under thermodynamic equilibrium,the total number of absorbing transitions must be the sameas the number of emissions. These numbers depend on thepopulation of a state and the probabilities for a transition tothe other state. According to (5)–(10), this can be expressedby

𝐵𝑛𝑚𝑢]𝑚𝑛

𝑁𝑛= (𝐵

𝑚𝑛𝑢]𝑚𝑛

+ 𝐴𝑚𝑛)𝑁

𝑚, (11)

or more universally described by rate equations:

𝑑𝑁𝑚

𝑑𝑡= +𝐵


𝑁𝑛− (𝐵


+ 𝐴𝑚𝑛)𝑁

𝑚,

𝑑𝑁𝑛

𝑑𝑡= −𝐵


𝑁𝑛+ (𝐵


+ 𝐴𝑚𝑛)𝑁

𝑚.

(12)

For 𝑑𝑁𝑚/𝑑𝑡 = 𝑑𝑁

𝑛/𝑑𝑡 = 0, (12) gets identical to (11). Using

(4) in (11) gives

𝑔𝑛𝐵𝑛𝑚𝑢]𝑚𝑛

= 𝑔𝑚(𝐵


+ 𝐴𝑚𝑛) 𝑒

−ℎ]𝑚𝑛/𝑘𝑇𝐺 . (13)

Assuming that with 𝑇𝐺also 𝑢]𝑚𝑛 gets infinite, 𝐵𝑚𝑛 and 𝐵𝑛𝑚

must satisfy the relation:

𝑔𝑛𝐵𝑛𝑚

= 𝑔𝑚𝐵𝑚𝑛. (14)

Then (13) becomes

𝐵𝑚𝑛𝑢]𝑚𝑛

𝑒ℎ]𝑚𝑛/𝑘𝑇𝐺 = 𝐵


+ 𝐴𝑚𝑛, (15)

or resolving to 𝑢]𝑚𝑛 gives

𝑢]𝑚𝑛=𝐴𝑚𝑛

𝐵𝑚𝑛

1

𝑒ℎ]𝑚𝑛/𝑘𝑇𝐺 − 1. (16)

This expression in Einstein’s consideration is of the same typeas the Planck distribution for the spectral energy density.Therefore, comparison of (3) and (16) at ]

𝑚𝑛gives for 𝐵

𝑚𝑛:

𝐵𝑚𝑛

=𝑔𝑛

𝑔𝑚

𝐵𝑛𝑚

=𝐴𝑚𝑛𝑐3

8𝜋𝑛3ℎ]3𝑚𝑛

=𝑐

𝑛

1

ℎ]𝑚𝑛

𝐴𝑚𝑛𝜆2

𝑚𝑛

8𝜋𝑛2, (17)

showing that the induced transition probabilities are alsoproportional to the spontaneous emission rate 𝐴

𝑚𝑛, and in

units of the photon energy, ℎ]𝑚𝑛, are scaling with 𝜆2.

2.2. Relationship to Other Spectroscopic Quantities. The Ein-stein coefficients for induced absorption and emission aredirectly related to some other well-established quantities inspectroscopy, the cross sections for induced transitions, andthe absorption and gain coefficient of a sample.


2.2.1. Cross Section and Absorption Coefficient. Radiationpropagating in 𝑟-direction through an absorbing sample isattenuated due to the interaction with the molecules. Thedecay of the spectral energy obeys Lambert-Beer’s law, heregiven in its differential form:

𝑑𝑢]

𝑑𝑟= −𝛼

𝑛𝑚(]) 𝑢] = −𝜎𝑛𝑚 (])𝑁𝑛𝑢], (18)

where 𝛼𝑛𝑚(]) is the absorption coefficient (units: cm−1)

and 𝜎𝑛𝑚(]) is the cross section (units: cm2) for induced

absorption.For a more general analysis, however, also emission

processes have to be considered, which partly or completelycompensate for the absorption losses. Then two cases haveto be distinguished, the situation we discuss in this section,where the molecules are part of an environmental thermalbath, and on the other hand, the case where a directedexternal radiation prevails upon a gas cloud, which will beconsidered in the next section.

In the actual case, (18) has to be expanded by two termsrepresenting the induced and also the spontaneous emission.Quite similar to the absorption, the induced emission isgiven by the cross section of induced emission 𝜎

𝑚𝑛(]), the

population of the upper state 𝑁𝑚, and the spectral radiation

density 𝑢]. The product 𝜎𝑚𝑛(]) ⋅ 𝑁𝑚 now describes anamplification of 𝑢] and is known as gain coefficient.

Additionally, spontaneously emitted photons within aconsidered volume element and time interval 𝑑𝑡 = 𝑑𝑟 ⋅𝑛/𝑐 contribute to the spectral energy density of the thermalbackground radiation with

𝑑𝑢] = ℎ]𝑚𝑛𝐴𝑚𝑛𝑑𝑡𝑁𝑚𝑔 (]) =ℎ]

𝑚𝑛

𝑐/𝑛𝐴𝑚𝑛𝑑𝑟𝑁

𝑚𝑔 (]) . (19)

Then altogether this gives

𝑑𝑢]

𝑑𝑟= −𝜎

𝑛𝑚(])𝑁

𝑛𝑢] + 𝜎𝑚𝑛 (])𝑁𝑚𝑢] +

ℎ]𝑚𝑛

𝑐/𝑛𝐴𝑚𝑛𝑁𝑚𝑔 (]) .

(20)

As we will see in Section 2.5 and later in Section 3.3 orSection 4, (19) is the source term of the thermal backgroundradiation in a gas, and (20) already represents the theoreticalbasis for calculating the radiation transfer of thermal radia-tion in the atmosphere.

The frequency dependence of 𝜎𝑛𝑚(]) and 𝜎

𝑚𝑛(]), and

thus the resonant interaction of radiation with a moleculartransition can explicitly be expressed by the normalizedlineshape function 𝑔(]) as

𝜎𝑚𝑛(]) = 𝜎0

𝑚𝑛𝑔 (]) , 𝜎

𝑛𝑚(]) = 𝜎0

𝑛𝑚𝑔 (]) . (21)

Equation (20) may be transformed into the time domain by𝑑𝑟 = 𝑐/𝑛 ⋅ 𝑑𝑡 and additionally integrated over the lineshape𝑔(]). When 𝑢] can be assumed to be broad compared to 𝑔(]),

the energy density𝑢 as the integral over the lineshape ofwidthΔ] becomes

∫Δ]

𝑑𝑢]

𝑑𝑡𝑑] =

𝑑𝑢]𝑚𝑛

𝑑𝑡Δ] =

𝑑𝑢

𝑑𝑡

= −𝑐

𝑛(𝜎

0

𝑛𝑚𝑁𝑛− 𝜎

0

𝑚𝑛𝑁𝑚) 𝑢]𝑚𝑛

+ ℎ]𝑚𝑛𝐴𝑚𝑛𝑁𝑚.

(22)

This energy density of the thermal radiation 𝑢 (units: J/m3)can also be expressed in terms of a photon density 𝜌

𝐺[m−3]

in the gas, multiplied with the photon energy ℎ ⋅ ]𝑚𝑛

with

𝑢 = 𝑢]𝑚𝑛Δ] = 𝜌

𝐺⋅ ℎ]

𝑚𝑛. (23)

Since each absorption of a photon reduces the population ofstate 𝑛 and increases𝑚 by the same amount—for an emissionit is just opposite—this yields

𝑑𝜌𝐺

𝑑𝑡=𝑑𝑁

𝑛

𝑑𝑡= −

𝑑𝑁𝑚

𝑑𝑡

= −𝑐/𝑛

ℎ]𝑚𝑛

(𝜎0

𝑛𝑚𝑁𝑛− 𝜎

0

𝑚𝑛𝑁𝑚) 𝑢]𝑚𝑛

+ 𝐴𝑚𝑛𝑁𝑚

(24)

which is identical with the balance in (12). Comparison ofthe first terms on the right side and applying (17) gives theidentity

𝐵𝑛𝑚𝑢]𝑚𝑛

𝑁𝑛=𝑔𝑚

𝑔𝑛

𝑐

𝑛

1

ℎ]𝑚𝑛

𝐴𝑚𝑛𝜆2

𝑚𝑛

8𝜋𝑛2𝑢]𝑚𝑛

𝑁𝑛

=𝑐

𝑛

1

ℎ]𝑚𝑛

𝜎0

𝑛𝑚𝑁𝑛𝑢]𝑚𝑛

,

(25)

and therefore

𝜎0

𝑛𝑚=𝑔𝑚

𝑔𝑛

𝐴𝑚𝑛𝜆2

𝑚𝑛

8𝜋𝑛2. (26)

Comparing the second terms in (12) and (24) results in

𝜎0

𝑚𝑛=𝐴𝑚𝑛𝜆2

𝑚𝑛

8𝜋𝑛2=𝑔𝑛

𝑔𝑚

𝜎0

𝑛𝑚. (27)

So, together with (21) and (27), we derive as the finalexpressions for 𝜎

𝑛𝑚(]) and 𝜎

𝑚𝑛(]):

𝜎𝑚𝑛(]) =

𝑔𝑛

𝑔𝑚

𝜎𝑛𝑚(]) =

𝐴𝑚𝑛𝜆2

𝑚𝑛

8𝜋𝑛2𝑔 (]) ,

𝜎𝑚𝑛(]) =

ℎ]𝑚𝑛

𝑐/𝑛𝐵𝑚𝑛𝑔 (]) , 𝜎

𝑛𝑚(]) =

ℎ]𝑚𝑛

𝑐/𝑛𝐵𝑛𝑚𝑔 (]) .

(28)

2.2.2. Effective Cross Section and Spectral Line Intensity.Often the first two terms on the right side of (20) are unifiedand represented by an effective cross section 𝜎

𝑛𝑚(]). Further


Cros

s sec

tion

(cm

2)

Snm

Frequency

Δ�mn

�mn

Figure 3: For explanation of the spectral line intensity.

relating the interaction to the total number density 𝑁 of themolecules, it applies

𝜎𝑛𝑚(])𝑁 = 𝜎

𝑛𝑚(])𝑁

𝑛− 𝜎

𝑚𝑛(])𝑁

𝑚

= 𝜎𝑛𝑚(])𝑁

𝑛(1 −

𝑔𝑛

𝑔𝑚

𝑁𝑚

𝑁𝑛

)

=ℎ]

𝑚𝑛

𝑐/𝑛𝐵𝑛𝑚𝑔 (])𝑁

𝑛(1 −

𝑔𝑛

𝑔𝑚

𝑁𝑚

𝑁𝑛

) ,

(29)

and 𝜎𝑛𝑚(]) becomes

𝜎𝑛𝑚(]) =

ℎ]𝑚𝑛

𝑐/𝑛𝐵𝑛𝑚

𝑁𝑛

𝑁(1 −

𝑔𝑛

𝑔𝑚

𝑁𝑚

𝑁𝑛

)𝑔 (]) . (30)

Integration of (30) over the linewidth gives the spectral lineintensity 𝑆

𝑛𝑚of a transition (Figure 3):

𝑆𝑛𝑚

= ∫Δ]𝜎𝑛𝑚(]) 𝑑] =

ℎ]𝑚𝑛

𝑐/𝑛𝐵𝑛𝑚

𝑁𝑛

𝑁(1 −

𝑔𝑛

𝑔𝑚

𝑁𝑚

𝑁𝑛

) , (31)

as it is used and tabulated in data bases [13, 14] to characterizethe absorption strength on a transition.

2.2.3. Effective Absorption Coefficient. Similar to 𝜎𝑛𝑚(]), with

(30) and (31), an effective absorption coefficient on a transi-tion can be defined as

𝛼𝑛𝑚(]) = 𝜎

𝑛𝑚(])𝑁

=ℎ]

𝑚𝑛

𝑐/𝑛𝐵𝑛𝑚𝑁𝑛(1 −

𝑔𝑛

𝑔𝑚

𝑁𝑚

𝑁𝑛

)𝑔 (])

= 𝑆𝑛𝑚𝑁𝑔 (]) = 𝛼0

𝑛𝑚𝑔 (]) ,

(32)

which after replacing 𝐵𝑛𝑚

from (17) assumes the morecommon form:

𝛼𝑛𝑚(]) =

𝐴𝑚𝑛𝑐2

8𝜋𝑛2]2𝑁𝑛(𝑔𝑚

𝑔𝑛

−𝑁𝑚

𝑁𝑛

)𝑔 (])

=𝐴𝑚𝑛𝜆2

8𝜋𝑛2𝑁𝑛(𝑔𝑚

𝑔𝑛

−𝑁𝑚

𝑁𝑛

)𝑔 (]) .

(33)

Nm

Nn

WGmn Amn

Cmn

WGnm Cnm

Figure 4: Two-level system with transition rates due to stimulated,spontaneous, and collisional processes.

2.3. Collisions. Generally the molecules of a gas underliecollisions, which may perturb the phase of a radiatingmolecule, and additionally cause transitions between themolecular states. The transition rate from 𝑚 → 𝑛 due tode-exciting, nonradiating collisions (superelastic collisions,of 2nd type) may be called 𝐶

𝑚𝑛and that for transitions from

𝑛 → 𝑚 (inelastic collisions, of 1st type) as exciting collisions𝐶𝑛𝑚, respectively (see Figure 4).

2.3.1. Rate Equations. Then, with (25) and the abbreviations𝑊

𝐺

𝑚𝑛and𝑊𝐺

𝑛𝑚as radiation induced transition rates or transi-

tion probabilities (units: s−1)

𝑊𝐺

𝑚𝑛= 𝐵


=𝑐

𝑛

1

ℎ]𝑚𝑛

𝜎0


,

𝑊𝐺

𝑛𝑚= 𝐵


=𝑐

𝑛

1

ℎ]𝑚𝑛

𝜎0


=𝑔𝑚

𝑔𝑛

𝑊𝐺

𝑚𝑛

(34)

the rate equations as generalization of (12) or (24) and addi-tionally supplemented by the balance of the electromagneticenergy density or photon density (see (22)–(24)) assume theform:

𝑑𝑁𝑚

𝑑𝑡= + (𝑊

𝐺

𝑛𝑚+ 𝐶

𝑛𝑚)𝑁

𝑛− (𝑊

𝐺

𝑚𝑛+ 𝐴

𝑚𝑛+ 𝐶

𝑚𝑛)𝑁

𝑚,

𝑑𝑁𝑛

𝑑𝑡= − (𝑊

𝐺

𝑛𝑚+ 𝐶

𝑛𝑚)𝑁

𝑛+ (𝑊

𝐺

𝑚𝑛+ 𝐴

𝑚𝑛+ 𝐶

𝑚𝑛)𝑁

𝑚,

𝑑𝜌𝐺

𝑑𝑡= −𝑊

𝐺

𝑛𝑚𝑁𝑛+𝑊

𝐺

𝑚𝑛𝑁𝑚+ 𝐴

𝑚𝑛𝑁𝑚.

(35)

At thermodynamic equilibrium, the left sides of (35) aregetting zero. Then, also and even particularly in the presenceof collisions the populations of states 𝑚 and 𝑛 will bedetermined by statistical thermodynamics. So, adding thefirst and third equation of (35), together with (4), it is foundsome quite universal relationship for the collision rates

𝐶𝑛𝑚

=𝑔𝑚

𝑔𝑛

𝑒−ℎ]𝑚𝑛/𝑘𝑇𝐺𝐶

𝑚𝑛, (36)

showing that transitions due to inelastic collisions are directlyproportional to those of superelastic collisions with a pro-portionality factor given by Boltzmann’s distribution. From(35), it also results that states, which are not connected by anallowed optical transition, nevertheless will assume the samepopulations as those states with an allowed transition.


2.3.2. Radiation Induced Transition Rates. When replacing𝑢]𝑚𝑛

in (34) by (3), the radiation induced transition rates canbe expressed as

𝑊𝐺

𝑚𝑛=

𝐴𝑚𝑛

𝑒ℎ]𝑚𝑛/𝑘𝑇𝐺 − 1, 𝑊

𝐺

𝑛𝑚=𝑔𝑚

𝑔𝑛

𝐴𝑚𝑛

𝑒ℎ]𝑚𝑛/𝑘𝑇𝐺 − 1. (37)

Inserting some typical numbers into (37), for example, atransition wavelength of 15𝜇m for the prominent CO

2-

absorption band and a temperature of 𝑇𝐺

= 288K, wecalculate a ratio𝑊𝐺

𝑚𝑛/𝐴

𝑚𝑛= 0.037. Assuming that 𝑔

𝑚= 𝑔

𝑛,

almost the same is found for the population ratio (see (4))with 𝑁

𝑚/𝑁

𝑛= 0.036. At spontaneous transition rates of the

order of 𝐴𝑚𝑛

= 1 s−1 for the stronger lines in this CO2-

band then we get a radiation induced transition rate of only𝑊

𝐺

𝑚𝑛= 𝑊

𝐺

𝑛𝑚= 0.03-0.04 s−1.

Under conditions as found in the troposphere with colli-sion rates between molecules of several 109 s−1, any inducedtransition rate due to the thermal background radiation isorders of magnitude smaller, and even up to the stratosphereand mesosphere, most of the transitions are caused bycollisions, so that above all they determine the population ofthe states and in any case ensure a fast adjustment of a localthermodynamic equilibrium in the gas.

Nevertheless, the absolute numbers of induced absorp-tion and emission processes per volume, scaling with thepopulation density of the involved states (see (35)), can bequite significant. So, at a CO

2concentration of 400 ppm,

the population in the lower state 𝑛 is estimated to be about𝑁𝑛∼ 7 × 10

20m−3 (dependent on its energy above theground level). Then more than 1019 absorption processes perm3 are expected, and since such excitations can take placesimultaneously on many independent transitions, this resultsin a strong overall interaction of the molecules with thethermal bath, which according to Einstein’s considerationseven in the absence of collisions leads to the thermodynamicequilibrium.

2.4. Linewidth and Lineshape of a Transition. The linewidthof an optical or infrared transition is determined by differenteffects.

2.4.1. Natural Linewidth and Lorentzian Lineshape. Formolecules in rest and without any collisions and also neglect-ing power broadening due to induced transitions, the spectralwidth of a line only depends on the natural linewidth

Δ]𝑚𝑛

=1

2𝜋𝜏𝑚

+1

2𝜋𝜏𝑛

≈𝐴𝑚𝑛

2𝜋(38)

which for a two-level system and 𝜏𝑛≫ 𝜏

𝑚is essentially

determined by the spontaneous transition rate 𝐴𝑚𝑛

≈ 1/𝜏𝑚.

𝜏𝑛and 𝜏

𝑚are the lifetimes of the lower and upper state.

The lineshape is given by a Lorentzian 𝑔𝐿(]), which in the

normalized form can be written as

𝑔𝐿(]) =

Δ]𝑚𝑛/2𝜋

(] − ]𝑚𝑛)2

+ (Δ]𝑚𝑛/2)

2, ∫

∞

0

𝑔𝐿(]) 𝑑] = 1.

(39)

2.4.2. Collision Broadening. With collisions in a gas, thelinewidth will considerably be broadened due to state andphase changing collisions. Then the width can be approxi-mated by

Δ]𝑚𝑛

≈1

2𝜋(𝐴

𝑚𝑛+ 𝐶

𝑚𝑛+ 𝐶

𝑛𝑚+ 𝐶phase) , (40)

where 𝐶phase is an additional rate for phase changing colli-sions. The lineshape is further represented by a Lorentzian,only with the new homogenous width Δ]

𝑚𝑛(FWHM)

according to (40).

2.4.3. Doppler Broadening. Since the molecules are at anaverage temperature 𝑇

𝐺, they also possess an average kinetic

energy

𝐸kin =3

2𝑘𝑇

𝐺=𝑚𝐺

2V2, (41)

with 𝑚𝐺as the mass and V2 as the velocity of the molecules

squared and averaged. Due to a Doppler shift of the movingparticles, the molecular transition frequency is additionallybroadened and the number of molecules interacting withintheir homogeneous linewidth with the radiation at frequency] is limited. This inhomogeneous broadening is determinedby Maxwell’s velocity distribution and known as Dopplerbroadening. The normalized Doppler lineshape is given by aGaussian function of the form:

𝑔𝐷(]) =

2√ln 2√𝜋Δ]

𝐷

exp(−(] − ]

𝑚𝑛)2

(Δ]𝐷/2)

2ln 2) (42)

with the Doppler linewidth

Δ]𝐷= 2]

𝑚𝑛(2𝑘𝑇

𝐺

𝑚𝐺𝑐2

ln 2)1/2

= 7.16 ⋅ 10−7]

𝑚𝑛√𝑇𝐺

𝑀, (43)

where 𝑀 is the molecular weight in atomic units and 𝑇𝐺

specified in K.

2.4.4. Voigt Profile. For the general case of collision andDoppler broadening, a convolution of 𝑔

𝐷(]) and 𝑔

𝐿(] −

]) gives the universal lineshape 𝑔𝑉(]) representing a Voigt

profile of the form:

𝑔𝑉(]) = ∫

∞

0

𝑔𝐷(]) 𝑔

𝐿(] − ]) 𝑑]

=√ln 2𝜋√𝜋

Δ]𝑚𝑛

Δ]𝐷

× ∫

∞

0

exp(−(] − ]

𝑚𝑛)2

(Δ]𝐷/2)

2ln 2)

⋅𝑑]

(] − ])2

+ (Δ]𝑚𝑛/2)

2.

(44)


2.5. Spontaneous Emission as Thermal Background Radiation.From the rate equations (35), it is clear that also in thepresence of collisions, the absolute number of spontaneouslyemitted photons per time should be the same as that withoutcollisions. This is also a consequence of (36) indicatingthat with an increasing rate 𝐶

𝑚𝑛for de-exciting transitions

(without radiation), also the rate 𝐶𝑛𝑚

of exiting collisions isgrowing and just compensating for any losses, even when thebranching ratio 𝑅 = 𝐴

𝑚𝑛/𝐶

𝑚𝑛of radiating to nonradiating

transitions decreases. In other words, when the moleculeis in state 𝑚, the probability for an individual spontaneousemission act is reduced by the ratio 𝑅, but at the same time,the number of occurring decays per time increases with 𝐶

𝑚𝑛.

Then the spectral power density in the gas due tospontaneous emissions is (see (19))

𝑑𝑢]

𝑑𝑡= ℎ]

𝑚𝑛𝐴𝑚𝑛𝑁𝑚𝑔 (]) , (45)

representing a spectral generation rate of photons of energyℎ]

𝑚𝑛per volume. Photons emerging from a volume element

generally spread out into neighbouring areas, but in the sameway, there is a backflow from the neighbourhood, whichin a homogeneous medium just compensates these losses.Nevertheless, photons have an average lifetime, before theyare annihilated due to an absorption in the gas. With anaverage photon lifetime

𝜏ph = 𝑙ph𝑛

𝑐=

𝑛

𝛼𝑛𝑚𝑐, (46)

where 𝑙ph = 1/𝛼𝑛𝑚 is the mean free path of a photon in thegas before it is absorbed, we can write for the spectral energydensity:

𝑢] = ℎ]𝑚𝑛𝐴𝑚𝑛𝜏ph𝑁𝑚𝑔 (]) =𝑛

𝑐ℎ]

𝑚𝑛

𝐴𝑚𝑛

𝛼𝑛𝑚

𝑁𝑚𝑔 (]) . (47)

The same result is derived when transforming (20) intothe time domain and assuming a local thermodynamicequilibrium with 𝑑𝑢]/𝑑𝑡 = 0:

(𝜎𝑛𝑚(])𝑁

𝑛− 𝜎

𝑚𝑛(])𝑁

𝑚) 𝑢]= 𝛼𝑛𝑚𝑢]

=𝑛

𝑐ℎ]

𝑚𝑛𝐴𝑚𝑛𝑁𝑚𝑔 (]) .

(48)

With 𝛼𝑛𝑚(]) from (33) and Boltzmann’s relation (4), then the

spectral energy density at ]𝑚𝑛

is found to be

𝑢]𝑚𝑛=8𝜋𝑛

3ℎ]3

𝑚𝑛

𝑐3

1

(𝑔𝑚𝑁𝑛/𝑔

𝑛𝑁𝑚− 1)

=8𝜋𝑛

3ℎ]3

𝑚𝑛

𝑐3

1

𝑒ℎ]𝑚𝑛/𝑘𝑇𝐺 − 1.

(49)

This is the well-known Planck formula (3) and shows thatwithout an additional external excitation at thermodynamicequilibrium, the spontaneous emission of molecules can beunderstood as nothing else as the thermal radiation of a gason the transition frequency.

This derivation differs insofar from Einstein’s consid-eration leading to (16), as he concluded that a radiationfield, interacting with the molecules at thermal and radiationequilibrium, just had to be of the type of a Planckianradiator, while here we consider the origin of the thermalradiation in a gas sample, which exclusively is determinedand rightfully defined by the spontaneous emission of themolecules themselves. This is also valid in the presence ofmolecular collisions. Because of this origin, the thermalbackground radiation only exists on discrete frequencies,determined by the transition frequencies and the linewidthsof the molecules, as long as no external radiation is present.But on these frequencies, the radiation strength is the sameas that of a blackbody radiator.

Since this spontaneous radiation is isotropically emittingphotons into the full solid angle 4𝜋, in average half of theradiation is directed upward and half downward.

3. Interaction of Molecules with ThermalRadiation from an External Source

In this section, we consider the interaction of molecules withan additional blackbody radiation emitted by an externalsource like the earth’s surface or adjacent atmospheric layers.We also investigate the transfer of absorbed radiation to heatin the presence of molecular collisions, causing a rise ofthe atmospheric temperature. And vice versa, we study thetransfer of a heat flux to radiation resulting in a cooling of thegas. An appropriate means to describe the mutual interactionof these processes is to express this by coupled rate equationswhich are solved numerically.

3.1. Basic Quantities

3.1.1. Spectral Radiance. The power radiated by a surfaceelement 𝑑𝐴 on the frequency ] in the frequency interval 𝑑]and into the solid angle element 𝑑Ω is also determined byPlanck’s radiation law:

𝐼],Ω (𝑇𝐸) 𝑑𝐴 cos𝛽𝑑]𝑑Ω

=2ℎ]3𝑛2

𝑐2

1

𝑒ℎ]/𝑘𝑇𝐸 − 1𝑑𝐴 cos𝛽𝑑]𝑑Ω,

(50)

with 𝐼],Ω(𝑇𝐸) as the spectral radiance (units: W/m2/

Hz/sterad) and 𝑇𝐸as the temperature of the emitting surface

of the source (e.g., earth’s surface). The cosine term accountsfor the fact that for an emission in a direction given by theazimuthal angle 𝜑 and the polar angle 𝛽, only the projectionof 𝑑𝐴 perpendicular to this direction is efficient as radiatingsurface (Lambertian radiator).

3.1.2. Spectral Flux Density—Spectral Intensity. Integrationover the solid angle Ω gives the spectral flux density. Then,representing 𝑑Ω in spherical coordinates as 𝑑Ω = sin𝛽𝑑𝛽𝑑𝜑


𝜑

𝛽

dA

d𝜑

d𝛽

Figure 5: Radiation from a surface element 𝑑𝐴 into the solid angle𝑑Ω = sin𝛽𝑑𝛽𝑑𝜑.

with 𝑑𝜑 as the azimuthal angle interval and 𝑑𝛽 as the polarinterval (see Figure 5), this results in:

𝐼] = ∫

2𝜋

Ω

𝐼],Ω (𝑇𝐸) cos𝛽𝑑Ω

= ∫

2𝜋

𝜑=0

∫

𝜋/2

𝛽=0

𝐼],Ω (𝑇𝐸) cos𝛽 sin𝛽𝑑𝛽𝑑𝜑

=2𝜋ℎ]3𝑛2

𝑐2

1

𝑒ℎ]/𝑘𝑇𝐸 − 1

(51)

or in wavelengths units

𝐼𝜆=𝑐𝑛

𝜆2𝐼] =

2𝜋ℎ𝑐2𝑛3

𝜆5

1

𝑒ℎ𝑐/𝑘𝑇𝐸𝜆 − 1. (52)

𝐼𝜆and 𝐼], also known as spectral intensities, are specified

in units of W/m2/𝜇m and W/m2/Hz, respectively. Theyrepresent the power flux per frequency or wavelength and persurface area into that hemisphere, which can be seen fromthe radiating surface element. The spectral distribution of aPlanck radiator of 𝑇 = 299K as a function of wavelength isshown in Figure 6.

3.2. Ray Propagation in a Lossy Medium

3.2.1. Spectral Radiance. Radiation passing an absorbingsample generally obeys Lambert-Beer’s law, as already appliedin (18) for the spectral energy density 𝑢]. The same holds forthe spectral radiance with

𝑑𝐼],Ω

𝑑𝑟= −𝛼

𝑛𝑚(]) 𝐼],Ω, (53)

where 𝑑𝑟 is the propagation distance of 𝐼],Ω in the sample.This is valid independent of the chosen coordinate system.The letter 𝑟 is used when not explicitly a propagation

35

30

25

20

15

10

5

00 10 20 30 40 50 60

Wavelength (𝜇m)

Spec

tral

inte

nsity

(Wm

−2𝜇

m−1)

Earth299K

Figure 6: Spectrum of a Planck radiator at 299K temperature.

z

𝛽

Δz

I�,Ω cos 𝛽dΩ

Δz/cos 𝛽

Figure 7: Radiation passing a gas layer.

perpendicular to the earth’s surface or a layer (𝑧-direction)is meant.

3.2.2. Spectral Intensity. For the spectral intensity, which dueto the properties of a Lambertian radiator consists of a bunchof rays with different propagation directions and brightness,some basic deviations have to be recognized. So, because ofthe individual propagation directions spreading over a solidangle of 2𝜋, only for a homogeneously absorbing sphere andin a spherical coordinate system with the radiation source inthe center of this sphere the distances to pass the sample, andthus the individual contributions to the overall absorption,would be the same.

But radiation, emerging from a plane parallel surfaceand passing an absorbing layer of thickness Δ𝑧, is bettercharacterized by its average expansion perpendicular to thelayer surface in 𝑧-direction. This means, that with respect tothis direction an individual ray, propagating under an angle 𝛽to the surface normal covers a distance Δ𝑙 = Δ𝑧/cos𝛽 beforeleaving the layer (see Figure 7). Therefore, such a ray on oneside contributes to a larger relative absorption, and on theother side, it donates this to the spectral intensity only withweight 𝐼],Ωcos𝛽𝑑Ω due to Lambert’s law.

This means that to first order, each individual beamdirection suffers from the same absolute attenuation, andparticularly the weaker rays under larger propagation angles𝛽 waste relatively more of their previous spectral radiance.Thus, especially at higher absorption strengths and longerpropagation lengths, the initial Lambertian distribution willbe more and more modified. A criterion for an almostunaltered distribution may be that for angles 𝛽 ≤ 𝛽max theinequality 𝛼

𝑛𝑚⋅ Δ𝑧/cos𝛽 ≪ 1 is satisfied. The absorption


loss for the spectral intensity as the integral of the spectralradiance 𝐼],Ω (see (51)) then is given by

∫𝑑𝐼],Ω (𝑧) cos𝛽𝑑Ω𝑑𝑧

= −𝛼𝑛𝑚(]) ∫

2𝜋

0

∫

𝜋/2

0

𝐼],Ω

cos𝛽cos𝛽

sin𝛽𝑑𝛽𝑑𝜑.

(54)

The cosine terms under the integral sign just compensate and(54) can be written as

𝑑𝐼] (𝑧, 𝑇𝐸)

𝑑𝑧= −2𝛼

𝑛𝑚(]) 𝐼] (𝑧, 𝑇𝐸) . (55)

This differential equation for the spectral intensity shows thatthe effective absorption coefficient is twice that of the spectralradiance, or in other words, the average propagation length ofthe radiation to pass the layer is twice the layer thickness.Thislast statementmeans thatwe also can assume radiation,whichis absorbed at the regular absorption coefficient 𝛼

𝑛𝑚, but is

propagating as a beam under an angle of 60∘ to the surfacenormal (1/cos 60∘ = 2). In practice, it even might give senseto deviate from an angle of 60∘, to compensate for deviationsof the earth’s or oceanic surface from a Lambertian radiator,and to account for contributions due to Mie and Rayleighscattering.

The absorbed spectral power density 𝐿] (power pervolume and per frequency) with respect to the 𝑧-directionthen is found to be

𝐿] (𝑧) =𝑑𝐼] (𝑧)

𝑑𝑧= −2𝛼

𝑛𝑚(]) 𝐼] (0) 𝑒

−2𝛼𝑛𝑚(])𝑧, (56)

with 𝐼](0) as the initial spectral intensity at 𝑧 = 0 as given by(51).

An additional decrease of 𝐿] with 𝑧 due to a lateralexpansion of the radiation over the hemisphere can beneglected, since any propagation of the radiation is assumedto be small compared to the earth’s radius (consideration ofan extended radiating parallel plane).

Integration of (56) over the lineshape 𝑔(]) within thespectral intervalΔ] gives the absorbed power density𝐿 (units:W/m3), which similar to (23) may also be expressed as lossor annihilation of photons per volume (𝜌

𝐸in m−3) and per

time. Since the average propagation speed of 𝐼] in 𝑧-directionis 𝑐/2𝑛, also the differentials transform as 𝑑𝑧 = 𝑐/2𝑛⋅𝑑𝑡. With𝐼] from (51), this results in

𝐿 (𝑧) =2𝑛

𝑐∫Δ]

𝑑𝐼]

𝑑𝑡𝑑] =

2𝑛

𝑐

𝑑𝐼]𝑚𝑛

𝑑𝑡Δ] = 2

𝑑𝜌𝐸

𝑑𝑡ℎ]

𝑚𝑛

= −𝐴𝑚𝑛

2𝑁𝑛(𝑔𝑚

𝑔𝑛

−𝑁𝑚

𝑁𝑛

)ℎ]

𝑚𝑛

𝑒ℎ]𝑚𝑛/𝑘𝑇𝐸 − 1𝑒−2𝛼𝑛𝑚𝑧,

(57)

or for the photon density 𝜌𝐸in terms of the induced transi-

tion rates𝑊𝐸𝑛𝑚

and𝑊𝐸𝑚𝑛

caused by the external field:

𝑑𝜌𝐸

𝑑𝑡= −𝑊

𝐸

𝑛𝑚𝑁𝑛+𝑊

𝐸

𝑚𝑛𝑁𝑚

= −𝐴𝑚𝑛

4(𝑔𝑚

𝑔𝑛

𝑁𝑛− 𝑁

𝑚)

1

𝑒ℎ]𝑚𝑛/𝑘𝑇𝐸 − 1𝑒−2𝛼𝑛𝑚𝑧,

(58)

with 𝛼𝑛𝑚

= 𝛼0

𝑛𝑚/Δ] = 𝑆

𝑛𝑚𝑁/Δ] as an averaged absorption

coefficient over the linewidth (see also (32)) and with

𝑊𝐸

𝑚𝑛(𝑧) =

𝐴𝑚𝑛𝜆2

𝑚𝑛

8𝜋𝑛2ℎ]𝑚𝑛

𝐼]𝑚𝑛(𝑧) =

𝜎0

𝑚𝑛

ℎ]𝑚𝑛

𝐼]𝑚𝑛(𝑧)

=𝐴𝑚𝑛

4

𝑒−2𝑆𝑛𝑚𝑁𝑧/Δ]

𝑒ℎ]𝑚𝑛/𝑘𝑇𝐸 − 1,

𝑊𝐸

𝑛𝑚=𝑔𝑚

𝑔𝑛

𝑊𝐸

𝑚𝑛.

(59)

Similar to (37), these rates are again proportional to thespontaneous transition probability (rate) 𝐴

𝑚𝑛, but now they

depend on the temperature 𝑇𝐸of the external source and the

propagation depth 𝑧.Due to the fact that radiation from a surface with a

Lambertian distribution and only from one hemisphere isacting on the molecules, different to (37), a factor of 1/4appears in this equation.

At a temperature 𝑇𝐸= 288K and wavelength 𝜆 = 15 𝜇m,

for example, the ratio of induced to spontaneous transitionsdue to the external field at 𝑧 = 0 is

𝑊𝐸

𝑚𝑛

𝐴𝑚𝑛

= 0.0093 ≈ 1%. (60)

For the case of a two-level system, 𝐴𝑚𝑛

can also be expressedby the natural linewidth of the transition with (see (38))

Δ]𝑁𝑚𝑛

=𝐴𝑚𝑛

2𝜋. (61)

Then (59) becomes

𝑊𝐸

𝑚𝑛(𝑧) =

𝜋

2Δ]𝑁

𝑚𝑛

𝑒−2𝑆𝑛𝑚𝑁𝑧/Δ]

𝑒ℎ]𝑚𝑛/𝑘𝑇𝐸 − 1. (62)

With a typical natural width of the order of only 0.1 Hz in the15 𝜇m band of CO

2, then the induced transition rate will be

less than 0.01 s−1. Even on the strongest transitions around4.2 𝜇m, the natural linewidths are only∼100Hz, and thereforethe induced transition rates are about 10 s−1.

Despite of these small rates, the overall absorption of anincident beam can be quite significant. In the atmosphere,the greenhouse gases are absorbing on hundred thousands oftransitions over long propagation lengths and at molecularnumber densities of 1019–1023 per m3. On the strongest linesin the 15𝜇m band of CO

2, the absorption coefficient at the

center of a line even goes up to 1m−1. Then already within adistance of a fewm, the total power will be absorbed on thesefrequencies. So, altogether about 85% of the total IR radiationemerging from the earth’s surface will be absorbed by thesegases.

3.2.3. Alternative Calculation for the Spectral Intensity. Forsome applications, it might be more advantageous, first tosolve the differential equation (53) for the spectral radianceas a function of 𝑧 and also 𝛽, before integrating overΩ.


Nm

Nn

WGmnAmn

Cmn

WGnm

WEmn

WEnmCnm

Figure 8: Two-level system with transition rates including anexternal excitation.

Then an integration in 𝑧-direction over the length 𝑧/cos𝛽with a 𝑧-dependent absorption coefficient gives

𝐼],Ω (𝑧) = 𝐼],Ω (0) 𝑒−(1/ cos𝛽) ∫𝑧

0𝛼𝑛𝑚(𝑧

)𝑑𝑧

, (63)

and a further integration overΩ results in

𝐼] (𝑧) = ∫

2𝜋

0

∫

𝜋/2

0

𝐼],Ω (𝑧) cos𝛽 sin𝛽𝑑𝛽𝑑𝜑

= 𝐼],Ω (0) ∫

2𝜋

0

∫

𝜋/2

0

𝑒−(1/cos𝛽)∫𝑧

0𝛼𝑛𝑚(𝑧

)𝑑𝑧

× cos𝛽 sin𝛽𝑑𝛽𝑑𝜑.

(64)

First integrating over 𝜑 and introducing the optical depth∫𝑧

0𝛼𝑛𝑚(𝑧)𝑑𝑧

= 𝜏 as well as the substitution 𝑢 = 𝜏/cos𝛽,

we can write

𝐼] (𝜏) = 2𝜋𝐼],Ω (0) 𝜏2∫

∞

𝜏

𝑒−𝑢

𝑢3𝑑𝑢

= 𝐼] (0) 𝜏2(𝐸

1(𝜏) + 𝑒

−𝜏(1

𝜏2−1

𝜏))

(65)

with 𝐼](0) = 𝜋𝐼],Ω(0) (see (51)) and the exponential integral

𝐸1(𝜏) = ∫

∞

𝜏

𝑒−𝑢

𝑢𝑑𝑢 = −0.5772 − ln 𝜏 +

∞

∑

𝑘=1

(−1)𝑘+1𝜏𝑘

𝑘 ⋅ 𝑘!. (66)

Since the further considerations in this paper are concen-trating on the radiation transfer in the atmosphere underthe influence of thermal background radiation, it is moreappropriate to describe the interaction of radiation with themolecules by a stepwise propagation through thin layers ofdepth Δ𝑧 as given by (55).

3.3. Rate Equations under Atmospheric Conditions. As afurther generalization of the rate equations (35), in thissubsection, we additionally consider the influence of theexternal radiation, which together with the thermal back-ground radiation is acting on the molecules in the presenceof collisions (see Figure 8).

And different to Section 2, the infrared active moleculesare considered as a trace gas in an open system, the

atmosphere, which has to come into balance with its environ-ment.Thenmolecules radiating due to their temperature andthus loosing part of their energy have to get this energy backfrom the surrounding by IR radiation, by sensitive or latentheat or also by absorption of sunlight. This is a consequenceof energy conservation.

The radiation loss is assumed to be proportional to theactual photon density 𝜌

𝐺and scaling with a flux rate 𝜙.

This loss can be compensated by the absorbed power ofthe incident radiation, for example, the terrestrial radiation,which is further expressed in terms of the photon density𝜌𝐸(see (57)), and it can also be replaced by thermal energy.

Therefore, the initial rate equations have to be supplementedby additional relations for these two processes.

It is evident that both, the incident radiation and theheat flux, will be limited by some genuine interactions. So,the radiation can only contribute to a further excitation, aslong as it is not completely absorbed. At higher moleculardensities, however, the penetration depth of the radiationin the gas is decreasing and therewith also the effectiveexcitation over some longer volume element. Integrating (55)over the linewidth while using the definitions of (31) and (32)and then integrating over 𝑧 results in

𝐼]𝑚𝑛(𝑧) = 𝐼]𝑚𝑛

(0) 𝑒−2𝑆𝑛𝑚𝑁𝑧/Δ] (67)

which due to Lambert-Beer’s law describes the averagedspectral intensity over the linewidth as a function of thepropagation in 𝑧-direction. From this, it is quite obvious todefine the penetration depth as the length 𝐿

𝑃, over which the

initial intensity reduces to 1/𝑒 and thus the exponent in (67)gets unity with

𝐿𝑃=

Δ]

2𝑆𝑛𝑚𝑁𝑆

. (68)

From (68), also the molecular number density𝑁𝑆is found, at

which the initial intensity just drops to 1/𝑒 after an interactionlength 𝐿

𝑃. Since 𝑁

𝑆characterizes the density at which an

excitation of the gas gradually comes to an end and in thissense saturation takes place, it is known as the saturationdensity, where 𝑁

𝑆and also 𝑆

𝑛𝑚refer to the total number

density𝑁 of the gas.Since at constant pressure, the number density in the

gas is changing with the actual temperature due to Gay-Lussac’s law and the molecules are distributed over hundredsof states and substates, the molecular density, contributing toan interaction with the radiation on the transition 𝑛 → 𝑚, isgiven by [15]

𝑁𝑛= 𝑁

0

𝑇0

𝑇𝐺

𝑔𝑛𝑒−𝐸𝑛/𝑘𝑇𝐺

𝑃 (𝑇𝐺), (69)

where 𝑁0is the molecular number density at an initial

temperature 𝑇0, 𝐸

𝑛is the energy of the lower level 𝑛 above


the ground state, and𝑃(𝑇𝐺) is the total internal partition sum,

defined as [15, 16]

𝑃 (𝑇𝐺) = ∑

𝑖

𝑔𝑖𝑒−𝐸𝑖/𝑘𝑇𝐺

. (70)

In the atmosphere typical propagation, lengths are of theorder of several km. So, for a stronger CO

2transition in the

15 𝜇m band with a spontaneous rate 𝐴 = 1 s−1, 𝑁𝑛/𝑁 =

𝑁𝑛/𝑁

0= 0.07, a spectral line intensity 𝑆

𝑛𝑚or integrated cross

section of

𝑆𝑛𝑚

= 𝜎0

𝑛𝑚=𝐴𝑚𝑛𝜆2

8𝜋𝑛2

𝑁𝑛

𝑁(1 −

𝑔𝑛

𝑔𝑚

𝑁𝑚

𝑁𝑛

)

= 6 × 10−13m2Hz = 2 × 10−19 cm−1/ (molecules⋅cm−2)

(71)

and a spectral width at ground pressure of Δ] = 5GHz,the saturation density over a typical length of 𝐿

𝑃= 1 km

assumes a value of 𝑁𝑆≈ 4 × 10

18molecules/m3. Since thenumber density of the air at 1013 hPa and 288K is 𝑁air =2.55 × 10

25m−3, saturation on the considered transitionalready occurs at a concentration of less than one ppm. Itshould also be noticed that at higher altitudes and thus lowerdensities, also the linewidth Δ] due to pressure broadeningreduces. This means that to first order, also the saturationdensity decreases with Δ], while the gas concentration inthe atmosphere, at which saturation appears, almost remainsconstant. The same applies for the penetration depth.

For the further considerations, it is adequate to introducean average spectral intensity 𝐼]𝑚𝑛 as

𝐼]𝑚𝑛=

1

𝐿𝑃

∫

𝐿𝑃

0

𝐼]𝑚𝑛(𝑧) 𝑑𝑧

=Δ]

2𝑆𝑛𝑚𝑁𝐿

𝑃

𝐼]𝑚𝑛(0) ⋅ (1 − 𝑒

−2𝑆𝑛𝑚𝑁𝐿𝑃/Δ])

(72)

or equivalently an average photon density

𝜌𝐸=𝑛

𝑐

Δ]

ℎ]𝑚𝑛

𝐼]𝑚𝑛=

Δ]

2𝑆𝑛𝑚𝑁𝐿

𝑃

𝜌𝐸(0) ⋅ (1 − 𝑒

−2𝑆𝑛𝑚𝑁𝐿𝑃/Δ])

(73)

which characterizes the incident radiation with respect to itsaverage excitation over 𝐿

𝑃in the rate equations. In general,

the incident flux 𝜌𝐸(0) onto an atmospheric layer consists of

two terms, the up- and the downwelling radiation.Any heat flux, supplied to the gas volume, contributes

to a volume expansion, and via collisions and excitation ofmolecules, this can also be transferred to radiation energy.Simultaneously absorbed radiation can be released as heat

in the gas. Therefore, the rate equations are additionallysupplemented by a balance for the heat energy density of theair𝑄 [J/m3], which under isobaric conditions andmaking useof the ideal gas equation in the form 𝑝air = 𝑁air ⋅ 𝑘 ⋅ 𝑇𝐺 can bewritten as

Δ𝑄 = 𝑐𝑝𝜌airΔ𝑇𝐺 = (

𝑓

2+ 1)

𝑅𝐺

𝑀air

𝑀air𝑁𝐴

𝑝air𝑘𝑇

𝐺

Δ𝑇𝐺

=7

2

𝑝air𝑇𝐺

Δ𝑇𝐺,

(74)

or after integration

𝑄 (𝑇𝐺) =

7

2𝑝air ∫

𝑇𝐺

𝑇0

1

𝑇

𝐺

𝑑𝑇

𝐺=7

2𝑝air ln(

𝑇𝐺

𝑇0

) . (75)

Here 𝑐𝑝represents the specific heat capacity of the air at

constant pressure with 𝑐𝑝= (𝑓/2 + 1) ⋅ 𝑅

𝐺/𝑀air, where 𝑓 are

the degrees of freedom of a molecule (for N2and O

2: 𝑓 = 5)

and 𝑅𝐺= 𝑘 ⋅ 𝑁

𝐴is the universal gas constant (at room

temperature: 𝑐𝑝= 1.01 kJ/kg/K). 𝜌air is the specific weight

of the air (at room temperature and ground pressure: 𝜌air =1.29 kg/m3), which can be expressed as 𝜌air = 𝑚air ⋅ 𝑁air =𝑀air/𝑁𝐴 ⋅𝑁air = 𝑀air/𝑁𝐴 ⋅ 𝑝air/(𝑘𝑇𝐺)with𝑚air as the mass ofan air molecule,𝑀air the mol weight,𝑁𝐴 Avogadro’s number,𝑁air the number density, and 𝑝air the pressure of the air.

The thermal energy can be supplied to a volume elementby different means. So, thermal convection and conductivityin the gas contribute to a heat flux j

𝐻[J/(s⋅m2)], causing

a temporal change of 𝑄 proportional to div j𝐻, or for one

direction proportional to 𝜕𝑗𝐻𝑧/𝜕𝑧. j

𝐻is a vector and points

from hot to cold. Since this flux is strongly dominated byconvection, it can be well approximated by 𝑗

𝐻𝑧= ℎ

𝐶⋅ (𝑇

𝐸−

𝑇𝐺) with ℎ

𝐶as the heat transfer coefficient, typically of the

order of ℎ𝐶≈ 10–15W/m2/K. At some more or less uniform

distribution of the incident heat flux over the troposphere(altitude 𝐻

𝑇≈ 12 km), the temporal change of 𝑄 due to

convection may be expressed as 𝜕𝑗𝐻𝑧/𝜕𝑧 = ℎ

𝐶/𝐻

𝑇⋅ (𝑇

𝐸−𝑇

𝐺)

with ℎ𝐶/𝐻

𝑇≈ 1mW/m3/K.

Another contribution to the thermal energy originatesfrom the absorbed sunlight, which in the presence of col-lisions is released as kinetic or rotational energy of themolecules. Similarly, latent heat can be set free in the air. Inthe rate equations, both contributions are represented by asource term 𝑞SL [J/(s⋅m

3)].Finally, the heat balance is determined by exciting and de-

exciting collisions, changing the population of the states 𝑛 and𝑚, and reducing or increasing the energy by ℎ ⋅ ]

𝑚𝑛.

Altogether, this results in a set of coupled differentialequations, which describe the simultaneous interaction ofmolecules with their self-generated thermal backgroundradiation as well as with radiation from the earth’s surface


and/or a neighbouring layer, this all in the presence ofcollisions and under the influence of heat transfer processes:

𝑑𝑁𝑚

𝑑𝑡= +(

ℎ]𝑚𝑛

Δ]𝐵𝑛𝑚(𝜌𝐺+ 𝜌

𝐸) + 𝐶

𝑛𝑚)𝑁

𝑛

− (ℎ]

𝑚𝑛

Δ]𝐵𝑚𝑛(𝜌𝐺+ 𝜌

𝐸) + 𝐶

𝑚𝑛+ 𝐴

𝑚𝑛)𝑁

𝑚,

𝑑𝑁𝑛

𝑑𝑡= −(

ℎ]𝑚𝑛

Δ]𝐵𝑛𝑚(𝜌𝐺+ 𝜌

𝐸) + 𝐶

𝑛𝑚)𝑁

𝑛

+ (ℎ]

𝑚𝑛

Δ]𝐵𝑚𝑛(𝜌𝐺+ 𝜌

𝐸) + 𝐶

𝑚𝑛+ 𝐴

𝑚𝑛)𝑁

𝑚,

𝑑𝜌𝐺

𝑑𝑡= −

ℎ]𝑚𝑛

Δ](𝐵

𝑛𝑚𝑁𝑛− 𝐵

𝑚𝑛𝑁𝑚) 𝜌

𝐺+ 𝐴

𝑚𝑛𝑁𝑚− 𝜙𝜌

𝐺,

𝑑𝜌𝐸

𝑑𝑡= +

𝑐/𝑛

2𝐿𝑃

(1 − 𝑒−(2ℎ]𝑚𝑛/(Δ]𝑐/𝑛))(𝐵𝑛𝑚𝑁𝑛−𝐵𝑚𝑛𝑁𝑚)𝐿𝑃) 𝜌

𝐸(0)

−ℎ]

𝑚𝑛

Δ](𝐵


𝑚𝑛𝑁𝑚) 𝜌

𝐸,

𝑑𝑄

𝑑𝑡= +

ℎ𝐶

𝐻𝑇

(𝑇𝐸− 𝑇

𝐺) + 𝑞SL − ℎ]𝑚𝑛 (𝐶𝑛𝑚𝑁𝑛 − 𝐶𝑚𝑛𝑁𝑚) .

(76)

To symbolize themutual coupling of these equations, here weuse a notation where the radiation induced transition ratesare represented by the Einstein coefficients and the respectivephoton densities. A change to the other representation iseasily performed applying the identities:

𝑊𝐺

𝑚𝑛=ℎ]

𝑚𝑛

Δ]𝐵𝑚𝑛𝜌𝐺= 𝐵


,

𝑊𝐸

𝑚𝑛=ℎ]

𝑚𝑛

Δ]𝐵𝑚𝑛𝜌𝐸= 𝐵

𝑚𝑛

𝑛

𝑐𝐼]𝑚𝑛

,

𝑊𝐺,𝐸

𝑛𝑚=𝑔𝑚

𝑔𝑛

𝑊𝐺,𝐸

𝑚𝑛.

(77)

It should be noticed that the rate equation for 𝜌𝐸could

also be replaced by the incident radiation, as given by (73).However, since 𝜌

𝐸approaches its equilibrium value within

a time constant 𝜏 = Δ]/(ℎ]𝑚𝑛(𝐵


𝑚𝑛𝑁𝑚)) short

compared with other processes, for a uniform representationthe differential form was preferred.

For the special case of stationary equilibrium with𝑑𝑁

𝑛/𝑑𝑡 = 𝑑𝑁

𝑚/𝑑𝑡 = 0 from the first rate equation, we get

for the population ratio:

𝑁𝑚

𝑁𝑛

=ℎ]

𝑚𝑛𝐵𝑛𝑚(𝜌𝐺+ 𝜌

𝐸) /Δ] + 𝐶

𝑛𝑚

ℎ]𝑚𝑛𝐵𝑚𝑛(𝜌𝐺+ 𝜌

𝐸) /Δ] + 𝐶

𝑚𝑛+ 𝐴

𝑚𝑛

=𝑊

𝐺

𝑛𝑚+𝑊

𝐸

𝑛𝑚+ 𝐶

𝑛𝑚

𝑊𝐺𝑚𝑛+𝑊𝐸

𝑚𝑛+ 𝐴

𝑚𝑛+ 𝐶

𝑚𝑛

(78)

which at 𝑇𝐺∼ 𝑇

𝐸= 288K and 𝜆 = 15 𝜇m due to 𝑊𝐺

𝑚𝑛=

4𝑊𝐸

𝑚𝑛= 0.037𝐴

𝑚𝑛(see (37) and (60)) in the limit of pure

spontaneous decay processes reaches its maximum value of4.4%, while in the presence of collisions with collision ratesin the troposphere of several 109 s−1, it rapidly converges to3.5%, given by the Boltzmann relation at temperature 𝑇

𝐺and

corresponding to a local thermodynamic equilibrium.For the general case, the rate equations have to be

solved numerically, for example, by applying the finite ele-ment method. While (76) in the presented form is onlyvalid for a two-level system, a simulation under realisticconditions comparable to the atmosphere requires someextension, particularly concerning the energy transfer fromthe earth’s surface to the atmosphere. Since the main tracegases CO

2, water vapour, methane, and ozone are absorbing

the incident infrared radiation simultaneously on thousandsof transitions, as an acceptable approximation for this kindof calculation, we consider these transitions to be similar andindependent from each other, each of them contributing tothe same amount to the energy balance. In the rate equations,this can easily be included by multiplying the last term in theequation for the energy density 𝑄 with an effective number𝑀tr of transitions.

In this approximation, the molecules of an infrared activegas and even amixture of gases are represented by a “standardtransition” which reflects the dynamics and time evolution ofthemolecular populations under the influence of the incidentradiation, the background radiation, and the thermal heattransfer. 𝑀tr is derived as the ratio of the total absorbedinfrared intensity over the considered propagation length tothe contribution of a single standard transition.

An example for a numerical simulation in the tropo-sphere, more precisely for a layer from ground level upto 100m, is represented in Figure 9. The graphs show theevolution of the photon densities 𝜌

𝐺and 𝜌

𝐸, the population

densities 𝑁𝑛and 𝑁

𝑚of the states 𝑛 and 𝑚, the accumulated

heat density 𝑄 in the air, and the temperature 𝑇𝐺of the gas

(identical with the atmospheric temperature) as a function oftime and as an average over an altitude of 100m.

As initial conditions we assumed a gas and air temper-ature of 𝑇

𝐺= 40K, a temperature of the earth’s surface

of 𝑇𝐸= 288K, an initial heat flux due to convection of

𝑗𝐻𝑧

= 3 kW/m2, a latent heat source of 𝑞SL = 3mW/m3,

and an air pressure of 𝑝air = 1013 hPa. The interactionof the terrestrial radiation with the infrared active gases isdemonstrated for CO

2with a concentration in air of 380 ppm

corresponding to a molecular number density of 𝑁0

=

9.7 × 1021m−3 at 288K. The simulation was performed for

a “standard transition” as discussed before with a transitionwavelength 𝜆 = 15 𝜇m, a spontaneous transition rate 𝐴

𝑚𝑛=

1 s−1, statistical weights of 𝑔𝑚

= 35 and 𝑔𝑛

= 33, aspectral width of Δ] = 5GHz, a collisional transition rate𝐶𝑚𝑛

= 108 s−1 (we do not distinguish between rotational

and vibrational rates [10]), a penetration depth 𝐿𝑃= 100m,

and assuming a photon loss rate of 𝜙 = 7.5 × 105 s−1. Forthis calculation, a density and temperature dependence of Δ]and 𝐶

𝑚𝑛was neglected. Since under these conditions a single

transition contributes 0.069W/m2 to the total IR-absorptionof 212W/m2 over 100m,𝑀tr = 3095 transitions represent theradiative interaction with all active gases in the atmosphere.


6.05.04.0

2.0

0.01.0

7.0

3.0

300270210600 30Time (h)

15090 120 180 240

Phot

on d

ensit

y𝜌 G

×10

10(m

−3)

(a)

12.010.0

8.0

4.0

0.02.0

14.0

6.0

Time (h)

Phot

on d

ensit

y𝜌E×10

7(m

−3)

300270210600 30 15090 120 180 240

(b)

10.05.00.0

15.020.025.0

Time (h)

Rela

tive p

opul

atio

nN

n/N

0(%

)

300270210600 30 15090 120 180 240

(c)

0.050.00

0.100.150.200.25

Time (h)

Rela

tive p

opul

atio

nN

m/N

0(%

)

300270210600 30 15090 120 180 240

(d)

2000

1600

1400

1200

1800

Time (h)

Ther

mal

ener

gyQ

(kJ m

−3)

300270210600 30 15090 120 180 240

(e)

300250200150

0

10050

Time (h)

Gas

tem

pera

ture

TG

(K)

300270210600 30 15090 120 180 240

(f)

Figure 9: Solution of the rate equations for the troposphere (close to the earth’s surface) representing the evolution of (a) the photon density𝜌𝐺

(b) the photon density 𝜌𝐸(c) the lower and (d) the upper population density, (e) the thermal energy density, and (f) the gas- (air)-temperature

as a function of time.

The evolution for another gas and transition can easilydisplayed by replacing the respective parameters for the othertransition and recalibrating the effective transition number.

To ensure reproducible results and to avoid any insta-bilities, the time interval for the stepwise integration ofthe coupled differential equation system has to be chosensmall enough to avoid changes comparable to the size of acomputed quantity itself, or the calculated changes have to berestrict by some upper boundaries, taking into account somesmaller repercussions on the absolute time scale. The latterprocedure was applied which accelerates the calculations byseveral orders of magnitude.

A simulation is done in such a way, that starting from theinitial conditions for the populations, the photon densities,and the temperature, in a first step, modified populationand photon densities as well as a change of the heat densityover the time interval Δ𝑡 are determined. The heat changecauses a temperature change Δ𝑇

𝐺(see (74)) and gives a

new temperature, which is used to calculate a new collisiontransfer rate 𝐶

𝑛𝑚by (36), a corrected population 𝑁

𝑛by (69)

and a new population𝑁𝑚for the upper state by (4).The total

internal partition sum for CO2is deduced from data of the

HITRAN-database [13] and approximated by a polynomial ofthe form:

𝑃 (𝑇𝐺) = 0.894 ⋅ 𝑇

𝐺+ 3 × 10

−9𝑇4

𝐺. (79)

The new populations are used as new initial conditionsfor the next time step, over which again the interaction ofthe molecules with the radiation and any heat transfer iscomputed. In this way, the time evolution of all coupledquantities shown in Figure 9 is derived. For this simulation,it was assumed that all atmospheric components still exist ingas form down to 40K.

The relatively slow evolution of the curves is due to thefact, that the absorbed IR radiation as well as the thermal fluxis mostly used to heat up the air volume which at constant airpressure is further expanding and the density decreasing.

Population changes of the CO2states, induced by the

external or the thermal background radiation, are coupledto the heat reservoir via the collisional transition rates 𝐶

𝑚𝑛

and 𝐶𝑛𝑚

(see last rate equation), which are linked to eachother by (36). As long as the calculated populations differfrom a Boltzmann distribution at the gas-temperature 𝑇

𝐺,


the two terms in the parenthesis of this rate equation do notcompensate, and their difference contributes to an additionalheating, amplified by the effective number of transitions. Rea-sons for smaller deviations from a local thermal equilibriumcan be the induced population changes due to the externalradiation, as well as temperature and density variations withaltitude and daytime, or some local effects in the atmosphere.Also vice versa, when the gas is supplied with thermal energy,the heat can be transferred to electromagnetic energy andreradiated by the molecules (radiative cooling).

The photon density 𝜌𝐺(Figure 9(a)) shows the sponta-

neous emission of the molecules and determines the thermalbackground radiation of the gas. It is mostly governed bythe population of the upper state and in equilibrium is self-adjusting on a level where the spontaneous generation rate𝐴𝑚𝑛

⋅ 𝑁𝑚is just balanced by the induced transition rates

(optical and collision induced) and the loss rate𝜙𝜌𝐺, the latter

dependent on the density and temperature variations in theatmosphere.

Figure 9(b) represents the effective photon density 𝜌𝐸

available to excite the gas over the layer depth𝐿𝑃by terrestrial

radiation and back radiation from the overlaying atmosphere.The downwelling part was assumed to originate from a layerof slightly lower temperature than the actual gas temperature𝑇𝐺with a lapse rate also varying with 𝑇

𝐺. The curve first

declines, since with slightly increasing temperature, the lowerstate, which is not the ground state of CO

2, is stronger

populated (see also Figure 9(c)), and therefore the absorptionon this transition increases, while according to saturationeffects, the effective excitation over 𝐿

𝑃decreases. With

further growing temperature 𝜌𝐸again increases, caused by

the decreasing gas density and also reduced lower population,accompanied by a smaller absorption.

Figures 9(c) and 9(d) show the partial increase as wellas the depletion of the lower state and the simultaneousrefilling of the upper state due to the radiation interactionand thermal heating.The populations are displayed in relativeunits, normalized to the CO

2number density𝑁

0at 288K.

The heat density of the air and the gas (air) temperature,are plotted in Figures 9(e) and 9(f). Up to about 80K (first70 h), heating is determined by convection, while at highertemperatures radiative heatingmore andmore dominates. Aninitial heat flux of 3 kW/m2 seems relatively large but quicklyreduces with increasing temperature and at a difference of 3 Kbetween the surface and lower atmosphere, similar to latentheat, does not contribute tomore than 40W/m2, representingonly 10% of the terrestrial radiation.

In principle, it is expected that a gas heated by an externalsource cannot get warmer than the source itself. This isvalid for a heat flux caused by convection or conduction inagreement with the second law of thermodynamics, but itleads to some contradictionwith respect to a radiation source.

In a closed system as discussed in Section 2, thermalradiation interacting with the molecules is in unison withthe population of the molecules, and both the radiationand the gas can be characterized by a unique temperature𝑇𝐺. In an open system with additional radiation from an

external source of temperature 𝑇𝐸, the prevailing radiation,

interacting with the molecules, consists of two contributions,which in general are characterized by different temperaturesand different loss processes. So, 𝑇

𝐸determines the spectral

intensity and distribution of the incident radiation and inthis sense defines the power flux which can be transferredto the molecules, when this radiation is absorbed. But itdoes not describe any direct heat transfer, which alwaysrequires a medium for the transport and by nomeans definesthe temperature of the absorbing molecules. At thermalequilibrium and in the presence of collisions, the moleculesare better described by a temperature, deduced from thepopulation ratio𝑁

𝑚/𝑁

𝑛given by the Boltzmann relation (4).

This temperature also determines the spontaneous emissionand together with the losses defines the thermal backgroundradiation.

Since an absorption on a transition takes place as longas the population difference 𝑁

𝑛− 𝑁

𝑚is positive (equal

populations signify infinite temperature), molecules can bewell excited by a thermal radiation source, which has a tem-perature lower than that of the absorbing gas. Independent ofthis statement, it is clear that at equilibrium the gas can onlyreradiate that amount of energy that was previously absorbedfrom the incident radiation and/or sucked up as thermalenergy.

In any way, for the atmospheric calculations, it seemsreasonable, to restrict the maximum gas temperature tovalues comparable to the earth’s surface temperature. Anadequate condition for this is a radiation loss rate

𝜙 ≥𝑐/𝑛

4𝐿𝑃

(1 − 𝑒−(2ℎ]𝑚𝑛/(Δ]𝑐/𝑛))(𝐵𝑛𝑚𝑁𝑛−𝐵𝑚𝑛𝑁𝑚)𝐿𝑃) , (80)

as derived from the rate equations at stationary equilibriumand with up- and downwelling radiation of the same magni-tude.

A calculation for conditions as found in the upper strato-sphere at about 60 km altitude is shown in Figure 10. The airpressure is 0.73 hPa, the regular temperature at this altitude is242.7 K, and the number density of air is𝑁air = 2.2×10

22m−3.The density of CO

2reduces to 𝑁 = 8.3 × 1018m−3 at a

concentration of 380 ppm.At this lower pressure, the collisioninduced linewidth is three orders of magnitude smaller anddeclines to about Δ] = 5MHz. Under these conditions,the remaining width is governed by Doppler broadeningwith 34MHz. At this altitude, any terrestrial radiation foran excitation on the transition is no longer available. Theonly radiation interacting with the molecules originates fromneighbouring molecules and the own background radiation.

As initial conditions we used a local gas and air temper-ature 𝑇

𝐺= 250K, slightly higher than the environment with

𝑇𝐸= 242.7K, the latter defining the radiation (up and down)

from neighbouring molecules. The heat flux 𝑗𝐻𝑧

from or toadjacent layers was assumed to be negligible, while a heatwell due to sun light absorption by ozone with ∼15W/m2over ∼50 km in this case is the only heat source with 𝑞SL =0.3mW/m3. The calculation was performed for a collisionaltransition rate 𝐶

𝑚𝑛= 10

5 s−1, a photon loss rate of 𝜙 =7.5 × 10

4 s−1, a layer depth of 𝐿𝑃= 1 km, and an effective

number of transitions 𝑀tr = 1430 (absorption on standard


3.1

3.0

2.9

3.2

2.82550

Time (min)10 15 20

Phot

on d

ensit

y𝜌 G

×10

8(m

−3)

(a)

4.00

4.10

3.95

4.05

3.90

Phot

on d

ensit

y𝜌E×10

5(m

−3)

2550Time (min)

10 15 20

(b)

8.908.80

9.009.109.20

Rela

tive p

opul

atio

nN

n/N

0(%

)

2550Time (min)

10 15 20

(c)

0.190

0.186

0.194

0.198

0.202

Rela

tive p

opul

atio

nN

m/N

0(%

)

2550Time (s)

10 15 20

(d)

1.420

1.4141.4121.410

1.4161.418

Ther

mal

ener

gyQ

(kJ m

−3)

2550Time (min)

10 15 20

(e)

250

248

246

242

244

Gas

tem

pera

ture

TG

(K)

2550Time (min)

10 15 20

(f)

Figure 10: Solution of the rate equations for the stratosphere (60 km altitude) representing the evolution of (a) the photon density 𝜌𝐺, (b)

the photon density 𝜌𝐸, (c) the lower and (d) the upper population density, (e) the thermal energy density, and (f) the gas (atmospheric)

temperature as a function of time.

transition: 2.8 × 10−4W/m2; total IR-absorption over 1 km:0.4W/m2).

Similar to Figure 9, the graphs show the evolution ofthe photon densities 𝜌

𝐺and 𝜌

𝐸, the population densities

𝑁𝑛and 𝑁

𝑚of the states 𝑛 and 𝑚, the heat density 𝑄 in

air, and the temperature 𝑇𝐺

of the gas as a function oftime. But different to the conditions in the troposphere,where heating of the atmosphere due to terrestrial heat andradiation transfer was the dominating effect, the simulationsin Figure 10 demonstrate the effect of radiative cooling in thestratosphere, where incident radiation from adjacent layersand locally released heat is transferred to radiation andradiated to space. An increased reradiation will be observed,as represented by the upper graph till the higher temperaturediminishes and the radiated energy is in equilibriumwith thesupplied energy flux.

Due to the lower density and heat capacity of the air atthis reduced pressure, the system comes much faster to astationary equilibrium.

3.4. Thermodynamic and Radiation Equilibrium. Thermody-namic equilibrium in a gas sample means that the population

of themolecular states is given by the Boltzmann distribution(4) and the gas is characterized by an average temperature𝑇

𝐺.

On the other hand, spectral radiation equilibrium requiresthat the number of absorptions in a considered spectralinterval is equal to the number of emissions. While withoutan external radiation generally, any thermodynamic equilib-rium in a sample will be identical with the spectral radiationequilibrium, in the presence of an additional field, thesecases have to be distinguished, since the incident radiationalso induces transitions and modifies the populations of themolecular states, deviating from the Boltzmann distribution.

Therefore, it is appropriate to investigate the populationratio more closely and to discover how it looks like undersome special conditions. This is an adequate means todetermine what kind of equilibrium is found in the gassample.

For this, we consider the total balance of absorption andemission processes as given by the rate equations (76). Understationary equilibrium conditions (see (78)), it is

(𝑊𝐺

𝑛𝑚+𝑊

𝐸

𝑛𝑚+ 𝐶

𝑛𝑚)𝑁

𝑛= (𝑊

𝐺

𝑚𝑛+𝑊

𝐸

𝑚𝑛+ 𝐴

𝑚𝑛+ 𝐶

𝑚𝑛)𝑁

𝑚.

(81)


With (17), (34) and (59) we can write

(𝐵𝑛𝑚(𝑢] +

𝑛

𝑐⋅ 𝐼]) + 𝐶𝑛𝑚)𝑁𝑛

= (𝐵𝑚𝑛(𝑢] +

𝑛

𝑐⋅ 𝐼]) + 𝐴𝑚𝑛 + 𝐶𝑚𝑛)𝑁𝑚.

(82)

At local thermodynamic equilibrium,𝐶𝑚𝑛

and𝐶𝑛𝑚

are linkedto each other by (36). But even more generally, we canconclude that (36) is valid, as long as the molecules can bedescribed by a Maxwellian velocity distribution and can becharacterized by an average temperature (see also [12]).

Thus, applying (36) and introducing some abbreviations:

𝑢] +𝑛

𝑐⋅ 𝐼] = 𝑈],

𝐶𝑛𝑚

𝐵𝑛𝑚

= 𝜀,

𝐴𝑚𝑛

𝐵𝑛𝑚

=𝑔𝑛

𝑔𝑚

8𝜋𝑛2

𝜆2𝑚𝑛

ℎ]𝑚𝑛

𝑐/𝑛=𝑔𝑛

𝑔𝑚

𝜅

(83)

after elementary rearrangement, (82) becomes:

𝑁𝑚

𝑁𝑛

=𝑔𝑚

𝑔𝑛

𝑈] + 𝜀

𝑈] + 𝜅 + 𝜀 ⋅ 𝑒ℎ]𝑚𝑛/(𝑘𝑇𝐺)

. (84)

It is easy to be seen that for large 𝜀 and thus high collisionrates, (84) approaches to a Boltzmann distribution. Also for𝜀 = 0 (no collisions) and 𝐼] = 0 (no external radiation), thisgives the Boltzmann relation, while for 𝜀 = 0 and 𝐼] ≫ (𝑐/𝑛) ⋅𝑢], (84) changes to

𝑁𝑚

𝑁𝑛

≈𝑔𝑚

𝑔𝑛

𝐼]

𝐼] + (𝑐/𝑛) 𝜅=𝑔𝑚

𝑔𝑛

1

1 + 4 (𝑒ℎ]𝑚𝑛/(𝑘𝑇𝐸) − 1) 𝑒2𝑆𝑛𝑚𝑁𝑧/Δ]

≈1

4

𝑔𝑚

𝑔𝑛

𝑒−ℎ]𝑚𝑛/𝑘𝑇𝐸𝑒

−2𝑆𝑛𝑚𝑁𝑧/Δ].

(85)

This equation shows that under conditions as found in theupper atmosphere (small collision rates), the populations areapproximated by a modified Boltzmann distribution (righthand side: for ℎ]

𝑚𝑛> 𝑘𝑇

𝐸), which with increasing spectral

intensity is more and more determined by the temperature𝑇𝐸of the external radiation source. Under such conditions,

the normal thermodynamic equilibrium according to (4) isviolated and has to be replaced by the spectral radiationequilibrium.

4. Radiation Balance and Radiation Transferin the Atmosphere

Amore extensive analysis of the energy and radiation balanceof an absorbing sample not only accounts for the net absorbedpower from the incident radiation, as considered in (53)–(66), but it also includes any radiation originating from thesample itself as well as any re-radiation due to the externalexcitation. This was already investigated in some detail inSection 3.3, however, with the view to a local balance andits evolution over time. In this section, we are particularlyconcentrating on the propagation of thermal radiation in an

infrared active gas and how this radiation is modified due tothe absorption and re-emission of the gas.

Two perspectives are possible, an energy balance forthe sample or a radiation balance for the in- and outgoingradiation.Herewe discuss the latter case. Scattering processesare not included.

We consider the incident radiation from an externalsource with the spectral radiance 𝐼],Ω(𝑇𝐸) as defined in (50).Propagation losses in a thin gas layer obey Lambert-Beer’slaw. Simultaneously, this radiation is superimposed by thethermal radiation emitted by the gas sample itself, and asalready discussed in Section 2.5, this thermal contributionhas its origin in the spontaneous emission of the molecules.For small propagation distances 𝑑𝑟 in the gas, both contribu-tions can be summed up yielding

𝑑𝐼],Ω (𝑟)

𝑑𝑟= −𝛼

𝑛𝑚(]) 𝐼],Ω (𝑟, 𝑇𝐸) +

ℎ]𝑚𝑛

4𝜋𝐴𝑚𝑛𝑁𝑚𝑔 (]) . (86)

The last term is a consequence of (19) or (45) when trans-forming the spectral energy density 𝑢] of the gas to a spectralintensity (𝑐/𝑛) ⋅𝑢] and considering only that portion (𝑐/4𝜋𝑛) ⋅𝑢] emitted into the solid angle interval 𝑑Ω. Equation (86)represents a quite general form of the radiation balance in athin layer, and it is appropriate to derive the basic relationsfor the radiation transfer in the atmosphere under differentconditions. The differential 𝑑𝑟 indicates that the absolutepropagation length in the gas is meant.

4.1. Schwarzschild Equation in Dense Gases. In the tropo-sphere, the population of molecular states is almost exclu-sively determined by collisions between the molecules evenin the presence of stronger radiation. Then, due to theprevious discussion, a well-established local thermodynamicequilibrium can be expected, and from (47) or (48), we find

ℎ]𝑚𝑛

4𝜋𝐴𝑚𝑛𝑁𝑚𝑔 (]) =

𝑐

4𝜋𝑛𝛼𝑛𝑚𝑢] = 𝛼𝑛𝑚𝐵],Ω (𝑇𝐺) (87)

with 𝐵],Ω(𝑇𝐺) = (𝑐/4𝜋𝑛) ⋅ 𝑢] as the spectral radiance of thegas, also known as Kirchhoff-Planck-function and given by(50), but here at temperature 𝑇

𝐺. To distinguish between the

external radiation and the radiating gas, we use the letter 𝐵(from background radiation), which may not be mixed withthe Einstein coefficients 𝐵

𝑛𝑚or 𝐵

𝑚𝑛. Then we get as the final

result:

𝑑𝐼],Ω (𝑟)

𝑑𝑟= −𝛼

𝑛𝑚(]) 𝐼],Ω (𝑟, 𝑇𝐸) + 𝛼𝑛𝑚 (]) 𝐵],Ω (𝑇𝐺 (𝑟)) .

(88)

This equation is known as the Schwarzschild equation [5–7], which describes the propagation of radiation in anabsorbing gas and which additionally takes into account thethermal background radiation of the gas. While deriving thisequation, no special restrictions or conditions concerning thedensity of the gas or collisions between the molecules haveto be made, only that (36), is valid and the thermodynamic


z

𝛽

dz

I�,Ω cos 𝛽dΩ

B�,Ω cos 𝛽dΩ

dz/cos 𝛽

Figure 11: Radiation passing a gas layer.

equilibrium is given. In general, this is the case in thetroposphere up to the stratosphere.

Another Derivation. Generally, the Schwarzschild equation isderived from pure thermodynamic considerations. The gasin a small volume is called to be in local thermodynamicequilibrium at temperature 𝑇

𝐺, when the emitted radiation

from this volume is the same as that of a blackbody radiator atthis temperature.Then Kirchhoff ’s law is strictly valid, whichmeans that the emission is identical to the absorption of asample.

The total emission in the frequency interval (], ] + 𝑑])during the time 𝑑𝑡may be 4𝜋𝜀(]).The emissivity 𝜀(])may notbe mixed with 𝜀 = 𝐶

𝑛𝑚/𝐵

𝑛𝑚in (83). Then with the spectral

radiance 𝐵],Ω(𝑇𝐺) as given by (50) and with the absorptioncoefficient 𝛼

𝑛𝑚(]), it follows from Kirchhoff ’s law (see, e.g.,

[8–11])

𝜀 (]) = 𝛼𝑛𝑚(]) ⋅ 𝐵],Ω (𝑇𝐺) . (89)

For an incident beam with the spectral radiance 𝐼],Ω andpropagating in 𝑟-direction we then get:

𝑑𝐼],Ω

𝑑𝑟= −𝛼

𝑛𝑚(]) 𝐼],Ω (𝑟, 𝑇𝐸) + 𝜀 (])

= −𝛼𝑛𝑚(]) 𝐼],Ω (𝑟, 𝑇𝐸) + 𝛼𝑛𝑚 (]) 𝐵],Ω (𝑇𝐺) ,

(90)

which is identical with (88).

4.2. Radiation Transfer Equation for the Spectral Intensity.Normally the radiation and by this the energy, emitted intothe full hemisphere, is of interest, so that (88) has to beintegrated over the solid angle Ω = 2𝜋. As already discussedin Section 3.2, for this integration, we have to have in mindthat a beam, propagating under an angle𝛽 to the layer normal(see Figure 11), only contributes an amount 𝐼],Ω cos𝛽𝑑Ω tothe spectral intensity (spectral flux density) due to Lambert’slaw.The same is assumed to be true for the thermal radiationemitted by a gas layer under this angle.

On the other hand, the propagation length through a thinlayer of depth 𝑑𝑧 is increasing with 𝑑𝑙 = 𝑑𝑧/cos𝛽, so that the𝛽-dependence for both terms 𝐼],Ω and 𝐵],Ω disappears.

Therefore, analogous to (54) and (55) integration of(88) over Ω gives for the spectral intensity, propagating in𝑧-direction:

∫𝑑𝐼],Ω (𝑧) cos𝛽𝑑Ω𝑑𝑧

= 𝛼𝑛𝑚(]) ∫

2𝜋

0

∫

𝜋/2

0

(−𝐼],Ω (𝑧, 𝑇𝐸) + 𝐵],Ω (𝑇𝐺 (𝑧)))

× sin𝛽𝑑𝛽𝑑𝜑,

(91)

or with the definition for 𝐼] = 𝜋 ⋅ 𝐼],Ω and analogous 𝐵] =𝜋 ⋅ 𝐵],Ω:

𝑑𝐼] (𝑧)

𝑑𝑧= 2𝛼

𝑛𝑚(]) (−𝐼] (𝑧, 𝑇𝐸) + 𝐵] (𝑇𝐺 (𝑧))) . (92)

Equation (92) is a 1st-order differential equation with thegeneral solution:

𝐼] (𝑧) = 𝑒−2∫𝑧

0𝛼𝑛𝑚(],𝑧

)𝑑𝑧

× [2∫

𝑧

0

𝛼𝑛𝑚(], 𝑧) 𝐵] (𝑇𝐺 (𝑧

))

×𝑒2 ∫𝑧


)𝑑𝑧

𝑑𝑧+ 𝐼] (0) ]

(93)

or after some transformation:

𝐼] (𝑧) = 𝐼] (0) 𝑒−2∫𝑧


)𝑑𝑧

+ 2∫

𝑧

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