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International Journal of Atmospheric Sciences

Volume , Article ID , pages

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Research Article

Radiation 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; harde@hsu-hh.de

Received April ; Accepted July

Academic Editor: Shaocai Yu

Copyright © Hermann Harde. is 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 well

as with radiation from an external blackbody radiator. We show that the background radiation can be well understood only in

terms of the spontaneous emission of the molecules. e radiation and heat transfer processes in the atmosphere are described

by rate equations which are solved numerically for typical conditions as found in the troposphere and stratosphere, showing the

conversion of heat to radiation and vice versa. Consideration of the interaction processes on a molecular scale allows to develop

a comprehensive theoretical concept for the description of the radiation transfer in the atmosphere. A generalized form of the

radiation transfer equation is presented, which covers both limiting cases of thin and dense atmospheres and allows a continuous

transition from low to high densities, controlled by a density dependent parameter. Simulations of the up- and down-welling

radiation and its interaction with the most prominent greenhouse gases water vapour, carbon dioxide, methane, and ozone in

the atmosphere are presented. e radiative forcing at doubled CO2concentration is found to be % smaller than the IPCC-value.

1. Introduction

Radiation processes in the atmosphere play a major role in

the energy and radiation balance of the earth-atmosphere

system. Downwelling radiation causes heating of the earth’s

surface due to direct sunlight absorption and also due to

the back radiation from the atmosphere, which is the source

term of the so heavily discussed atmospheric greenhouse or

atmospheric heating eect. Upward radiation contributes to

cooling and ensures that the absorbed energy from the sun

and the terrestrial radiation can be rendered back to space

andtheearth’stemperaturecanbestabilized.

For all these processes, particularly, the interaction of

radiation with infrared active molecules is of importance.

ese molecules strongly absorb terrestrial radiation, emitted

from the earth’s surface, and they can also be excited by some

heat transfer in the atmosphere. e absorbed energy is rera-

diated uniformly into the full solid angle but to some degree

also re-absorbed in the atmosphere, so that the radiation

underlies a continuous interaction and modication process

over the propagation distance.

Although the basic relations for this interaction of

radiation with molecules are already well known since the

beginning of the previous century, up to now the correct

application of these relations, their importance, and their

consequences for the atmospheric system are discussed quite

contradictorily in the community of climate sciences.

erefore, it seems necessary and worthwhile to give a

brief review of the main physical relations and to present on

this basis a new approach for the description of the radiation

transfer in the atmosphere.

In Section , we start from Einstein’s basic quantum-

theoretical considerations of radiation []andPlanck’sradi-

ation law [] to investigate the interaction of molecules with

their own thermal background radiation under the inuence

of molecular collisions and at thermodynamic equilibrium

[,]. We show that the thermal radiation of a gas can be

well understood only in terms of the spontaneous emission

of the molecules. is is valid at low pressures with only few

molecular collisions as well as at higher pressures and high

collision rates.

International Journal of Atmospheric Sciences

In Section , also the inuence of radiation from an

external blackbody radiator and an additional excitation by

aheatsourceisstudied.eradiationandheattransfer

processes originating from the sun and/or the earth’s surface

are described by rate equations, which are solved numerically

for typical conditions as they exist in the troposphere and

stratosphere. ese examples right away illustrate the conver-

sion of heat to radiation and vice versa.

In Section ,wederivetheSchwarzschildequation[–

] as the fundamental relation for the radiation transfer

in the atmosphere. is equation is deduced from pure

considerations on a molecular basis, describing the thermal

radiation of a gas as spontaneous emission of the molecules.

is equation is investigated under conditions of only few

intermolecular collisions as found in the upper mesosphere

ormesopauseaswellasathighcollisionratesasobservedin

the troposphere. Following some modied considerations of

Milne [], a generalized form of the radiation transfer equa-

tion is presented, which covers both limiting cases of thin

and dense atmospheres and allows a continuous transition

from low to high densities, controlled by a density dependent

parameter. is equation is derived for the spectral radiance

as well as for the spectral ux density (spectral intensity) as

the solid angular integral of the radiance.

In Section , the generalized radiation transfer equation

is applied to simulate the up- and downwelling radiation

and its interaction with the most prominent greenhouse

gases water vapour, carbon dioxide, methane, and ozone in

the atmosphere. From these calculations, a detailed energy

and radiation balance can be derived, reecting the dierent

contributions of these gases under quite realistic conditions

in the atmosphere. In particular, they show the dominant

inuenceofwatervapouroverthefullinfraredspectrum,and

theyexplainwhyafurtherincreaseintheCO

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 eect or to

prove its existence within this framework. Nevertheless, the

basic considerations and derived relations for the molecular

interaction with radiation have some direct signicance for

the understanding and interpretation of this eect, and they

give the theoretical background for its general calculation.

2. Interaction of Molecules with Thermal Bath

When a gas is in thermodynamic equilibrium with its

environment it can be described by an average temperature

𝐺. Like any matter at a given temperature, which is in unison

with its surrounding, it is also a source of gray or blackbody

radiation as part of the environmental thermal bath. At the

same time, this gas is interacting with its own radiation,

causing some kind of self-excitation of the molecules which

nally results in a population of the molecular states, given by

Boltzmann’s distribution.

Such interaction rst considered by Einstein [] is repli-

cated in the rst part of this section with some smaller

modications, but following the main thoughts. In the second

part of this section, also collisions between the molecules are

Em

En

ΔE = hmn =hc/𝜆

mn

F : Two-level system with transition between states and .

included and some basic consequences for the description of

the thermal bath are derived.

2.1. Einstein’s Derivation of ermal Radiation. e molecules

are characterized by a transitionbetween the energy states 𝑚

and 𝑛with the transition energy

=𝑚−𝑛=]𝑚𝑛 =

𝑚𝑛 ,()

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

the transition frequency, and 𝑚𝑛 is the transition wavelength

(see Figure ).

Planckian Radiation. e cavity radiation of a black body

at temperature 𝐺can be represented by its spectral energy

density 𝜆(units: J/m3/m), which obeys Planck’s radiation

law []with

𝜆=84

51

ℎ𝑐/𝑘𝑇𝐺𝜆−1,()

or as a function of frequency assumes the form

]=𝜆2

=83]3

31

ℎ]/𝑘𝑇𝐺−1.()

isdistributionisshowninFigure for three dierent

temperatures as a function of wavelength. is the refractive

index of the gas.

Boltzmann’s Relation. Due to Boltzmann’s principle, the rela-

tive population of the states and in thermal equilibrium

is [,]𝑚

𝑛=𝑚

𝑛−ℎ]𝑚𝑛/𝑘𝑇𝐺,()

with 𝑚and 𝑛as the population densities of the upper and

lower state, 𝑚and 𝑛the statistical weights representing the

degeneracy of these states, as the Boltzmann constant, and

𝐺as the temperature of the gas.

For the moment, neglecting any collisions of the

molecules, between these states three dierent transitions can

take place.

Spontaneous Emission.espontaneousemissionoccurs

independent of any external eld from →and is

characterized by emitting statistically a photon of energy

]𝑚𝑛 into the solid angle 4with the probability 𝑠

𝑚𝑛 within

the time interval 𝑠

𝑚𝑛 =𝑚𝑛. ()

International Journal of Atmospheric Sciences

0246810

Wavelength 𝜆(𝜇m)

6

5

4

3

2

1

0

×10−3

Spectral energy density u𝜆(J/m3/𝜇m)

T = 2000 KT = 1500 KT = 1000 K

F : Spectral energy density of blackbody radiation at dierent

temperatures.

𝑚𝑛 is the Einstein coecient of spontaneous emission,

sometimes also called the spontaneous emission probability

(units: s−1).

Induced Absorption. With the molecules subjected to an

electromagnetic eld, the energy of the molecules can change

in that way, that due to a resonant interaction with the

radiation the molecules can be excited or de-excited. When a

molecule changes from →, it absorbs a photon of energy

]𝑚𝑛 and increases its internal energy by this amount, while

the radiation energy is decreasing by the same amount.

e probability for this process is found by integrating

over all frequency components within the interval ],con-

tributingtoaninteractionwiththemolecules:

𝑖

𝑛𝑚 =𝑛𝑚 Δ]](])]⋅. ()

isprocessisknownasinducedabsorptionwith𝑛𝑚 as Ein-

stein’s coecient of induced absorption (units: m3⋅Hz/J/s),

]as the spectral energy density of the radiation (units:

J/m3/Hz) and (])as a normalized lineshape function which

describes the frequency dependent interaction of the radia-

tion with the molecules and generally satises the relation:

∞

0(])]=1. ()

Since ]is much broader than (]), it can be assumed to

be constant over the linewidth, and with (), the integral in

() can be replaced by the spectral energy density on the

transition frequency:

Δ]](])]=]𝑚𝑛 .()

en the probability for induced absorption processes simply

becomes 𝑖

𝑛𝑚 =𝑛𝑚]𝑚𝑛 . ()

Induced Emission. A transition from →caused by the

radiation is called the induced or stimulated emission. e

probability for this transition is

𝑖

𝑚𝑛 =𝑚𝑛 Δ]](])]⋅=𝑚𝑛]𝑚𝑛 ()

with 𝑚𝑛 as the Einstein coecient of induced emission.

Total Trans i t i o n R a t e s. Under thermodynamic equilibrium,

the total number of absorbing transitions must be the same

as the number of emissions. ese numbers depend on the

population of a state and the probabilities for a transition to

the other state. According to ()–(), this can be expressed

by

𝑛𝑚]𝑚𝑛 𝑛=𝑚𝑛]𝑚𝑛 +𝑚𝑛𝑚,()

or more universally described by rate equations:

𝑚

=+𝑛𝑚]𝑚𝑛 𝑛−𝑚𝑛]𝑚𝑛 +𝑚𝑛𝑚,

𝑛

=−𝑛𝑚]𝑚𝑛 𝑛+𝑚𝑛]𝑚𝑛 +𝑚𝑛𝑚.()

For 𝑚/=𝑛/=0,() gets identical to (). Using

()in()gives

𝑛𝑛𝑚]𝑚𝑛 =𝑚𝑚𝑛]𝑚𝑛 +𝑚𝑛−ℎ]𝑚𝑛 /𝑘𝑇𝐺.()

Assuming that with 𝐺also ]𝑚𝑛 gets innite, 𝑚𝑛 and 𝑛𝑚

must satisfy the relation:

𝑛𝑛𝑚 =𝑚𝑚𝑛.()

en ()becomes

𝑚𝑛]𝑚𝑛 ℎ]𝑚𝑛/𝑘𝑇𝐺=𝑚𝑛]𝑚𝑛 +𝑚𝑛,()

or resolving to ]𝑚𝑛 gives

]𝑚𝑛 =𝑚𝑛

𝑚𝑛 1

ℎ]𝑚𝑛/𝑘𝑇𝐺−1.()

is expression in Einstein’s consideration is of the same type

as the Planck distribution for the spectral energy density.

erefore, comparison of ()and()at]𝑚𝑛 gives for 𝑚𝑛:

𝑚𝑛 =𝑛

𝑚𝑛𝑚 =𝑚𝑛3

83]3

𝑚𝑛 =

1

]𝑚𝑛 𝑚𝑛2

𝑚𝑛

82,()

showing that the induced transition probabilities are also

proportional to the spontaneous emission rate 𝑚𝑛,andin

units of the photon energy, ]𝑚𝑛,arescalingwith2.

2.2. Relationship to Other Spectroscopic Quantities. e Ein-

stein coecients for induced absorption and emission are

directly related to some other well-established quantities in

spectroscopy, the cross sections for induced transitions, and

the absorption and gain coecient of a sample.

International Journal of Atmospheric Sciences

2.2.1. Cross Section and Absorption Coecient. Radiation

propagating in -direction through an absorbing sample is

attenuated due to the interaction with the molecules. e

decay of the spectral energy obeys Lambert-Beer’s law, here

given in its dierential form:

]

=−𝑛𝑚 (])]=−𝑛𝑚 (])𝑛],()

where 𝑛𝑚(])is the absorption coecient (units: cm−1)

and 𝑛𝑚(])is the cross section (units: cm2)forinduced

absorption.

For a more general analysis, however, also emission

processes have to be considered, which partly or completely

compensate for the absorption losses. en two cases have

to be distinguished, the situation we discuss in this section,

where the molecules are part of an environmental thermal

bath, and on the other hand, the case where a directed

external radiation prevails upon a gas cloud, which will be

considered in the next section.

In the actual case, () has to be expanded by two terms

representing the induced and also the spontaneous emission.

Quitesimilartotheabsorption,theinducedemissionis

given by the cross section of induced emission 𝑚𝑛(]),the

population of the upper state 𝑚,andthespectralradiation

density ].eproduct𝑚𝑛(])⋅

𝑚now describes an

amplication of ]and is known as gain coecient.

Additionally, spontaneously emitted photons within a

considered volume element and time interval = ⋅

/contribute to the spectral energy density of the thermal

background radiation with

]=]𝑚𝑛𝑚𝑛𝑚(])=]𝑚𝑛

/ 𝑚𝑛𝑚(]).()

en altogether this gives

]

=−𝑛𝑚 (])𝑛]+𝑚𝑛 (])𝑚]+]𝑚𝑛

/ 𝑚𝑛𝑚(]).

()

As we will see in Section . and later in Section . or

Section ,() is the source term of the thermal background

radiation in a gas, and () already represents the theoretical

basis for calculating the radiation transfer of thermal radia-

tion in the atmosphere.

e frequency dependence of 𝑛𝑚(])and 𝑚𝑛(]),and

thus the resonant interaction of radiation with a molecular

transition can explicitly be expressed by the normalized

lineshape function (])as

𝑚𝑛 (])=0

𝑚𝑛(]),

𝑛𝑚 (])=0

𝑛𝑚(]).()

Equation () may be transformed into the time domain by

= /⋅and additionally integrated over the lineshape

(]).When]canbeassumedtobebroadcomparedto(]),

the energy density astheintegraloverthelineshapeofwidth

]becomes

Δ]]

]=]𝑚𝑛

]=

=−

0

𝑛𝑚𝑛−0

𝑚𝑛𝑚]𝑚𝑛 +]𝑚𝑛𝑚𝑛𝑚.

()

is energy density of the thermal radiation (units: J/m3)

can also be expressed in terms of a photon density 𝐺[m−3]

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

=]𝑚𝑛 ]=𝐺⋅]𝑚𝑛.()

Since each absorption of a photon reduces the population of

state and increases by the same amount—for an emission

it is just opposite—this yields

𝐺

=𝑛

=−𝑚

=−/

]𝑚𝑛 0

𝑛𝑚𝑛−0

𝑚𝑛𝑚]𝑚𝑛 +𝑚𝑛𝑚

()

whichisidenticalwiththebalancein(). Comparison of

the rst terms on the right side and applying ()givesthe

identity

𝑛𝑚]𝑚𝑛 𝑛=𝑚

𝑛

1

]𝑚𝑛 𝑚𝑛2

𝑚𝑛

82]𝑚𝑛 𝑛

=

1

]𝑚𝑛 0

𝑛𝑚𝑛]𝑚𝑛 ,()

and therefore

0

𝑛𝑚 =𝑚

𝑛𝑚𝑛2

𝑚𝑛

82.()

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

0

𝑚𝑛 =𝑚𝑛2

𝑚𝑛

82=𝑛

𝑚0

𝑛𝑚.()

So, together with ()and(), we derive as the nal

expressions for 𝑛𝑚(])and 𝑚𝑛(]):

𝑚𝑛 (])=𝑛

𝑚𝑛𝑚 (])=𝑚𝑛2

𝑚𝑛

82(]),

𝑚𝑛 (])=]𝑚𝑛

/ 𝑚𝑛(]),

𝑛𝑚 (])=]𝑚𝑛

/ 𝑛𝑚(]).

()

2.2.2. Eective Cross Section and Spectral Line Intensity.

Oen the rst two terms on the right side of ()areunied

and represented by an eective cross section 𝑛𝑚(]).Further

International Journal of Atmospheric Sciences

Cross section (cm2)

Snm

Frequency

Δmn

mn

F : For explanation of the spectral line intensity.

relating the interaction to the total number density of the

molecules, it applies

𝑛𝑚 (])=𝑛𝑚 (])𝑛−𝑚𝑛 (])𝑚

=𝑛𝑚 (])𝑛1− 𝑛

𝑚𝑚

𝑛

=]𝑚𝑛

/ 𝑛𝑚(])𝑛1− 𝑛

𝑚𝑚

𝑛,

()

and 𝑛𝑚(])becomes

𝑛𝑚 (])=]𝑚𝑛

/ 𝑛𝑚 𝑛

1− 𝑛

𝑚𝑚

𝑛(]).()

Integration of () over the linewidth gives the spectral line

intensity 𝑛𝑚 of a transition (Figure ):

𝑛𝑚 =Δ]𝑛𝑚 (])]=]𝑚𝑛

/ 𝑛𝑚 𝑛

1− 𝑛

𝑚𝑚

𝑛, ()

as it is used and tabulated in data bases [,] to characterize

the absorption strength on a transition.

2.2.3. Eective Absorption Coecient. Similar to 𝑛𝑚(]),with

()and(), an eective absorption coecient on a transi-

tioncanbedenedas

𝑛𝑚 (])=𝑛𝑚 (])

=]𝑚𝑛

/ 𝑛𝑚𝑛1− 𝑛

𝑚𝑚

𝑛(])

=𝑛𝑚(])=0

𝑛𝑚(]),

()

which aer replacing 𝑛𝑚 from () assumes the more

common form:

𝑛𝑚 (])=𝑚𝑛2

82]2𝑛𝑚

𝑛−𝑚

𝑛(])

=𝑚𝑛2

82𝑛𝑚

𝑛−𝑚

𝑛(]).()

Nm

Nn

WG

mn Amn

Cmn

WG

nm Cnm

F : Two-level system with transition rates due to stimulated,

spontaneous, and collisional processes.

2.3. Collisions. Generally the molecules of a gas underlie

collisions, which may perturb the phase of a radiating

molecule, and additionally cause transitions between the

molecular states. e transition rate from →due to

de-exciting, nonradiating collisions (superelastic collisions,

of nd type) may be called 𝑚𝑛 and that for transitions from

→(inelastic collisions, of st type) as exciting collisions

𝑛𝑚,respectively(seeFigure ).

2.3.1. Rate Equations. en, with ()andtheabbreviations

𝐺

𝑚𝑛 and 𝐺

𝑛𝑚 as radiation induced transition rates or transi-

tion probabilities (units: s−1)

𝐺

𝑚𝑛 =𝑚𝑛]𝑚𝑛 =

1

]𝑚𝑛 0

𝑚𝑛]𝑚𝑛 ,

𝐺

𝑛𝑚 =𝑛𝑚]𝑚𝑛 =

1

]𝑚𝑛 0

𝑛𝑚]𝑚𝑛 =𝑚

𝑛𝐺

𝑚𝑛

()

therateequationsasgeneralizationof()or() and addi-

tionally supplemented by the balance of the electromagnetic

energy density or photon density (see ()–()) assume the

form:

𝑚

=+𝐺

𝑛𝑚 +𝑛𝑚𝑛−𝐺

𝑚𝑛 +𝑚𝑛 +𝑚𝑛𝑚,

𝑛

=−𝐺

𝑛𝑚 +𝑛𝑚𝑛+𝐺

𝑚𝑛 +𝑚𝑛 +𝑚𝑛𝑚,

𝐺

=−𝐺

𝑛𝑚𝑛+𝐺

𝑚𝑛𝑚+𝑚𝑛𝑚.()

At thermodynamic equilibrium, the le sides of ()are

getting zero. en, also and even particularly in the presence

of collisions the populations of states and will be

determined by statistical thermodynamics. So, adding the

rst and third equation of (), together with (), it is found

some quite universal relationship for the collision rates

𝑛𝑚 =𝑚

𝑛−ℎ]𝑚𝑛/𝑘𝑇𝐺𝑚𝑛,()

showing that transitions due to inelastic collisions are directly

proportional to those of superelastic collisions with a pro-

portionality factor given by Boltzmann’s distribution. From

(), it also results that states, which are not connected by an

allowed optical transition, nevertheless will assume the same

populations as those states with an allowed transition.

International Journal of Atmospheric Sciences

2.3.2. Radiation Induced Transition Rates. When replacing

]𝑚𝑛 in ()by(), the radiation induced transition rates can

be expressed as

𝐺

𝑚𝑛 =𝑚𝑛

ℎ]𝑚𝑛/𝑘𝑇𝐺−1,

𝐺

𝑛𝑚 =𝑚

𝑛𝑚𝑛

ℎ]𝑚𝑛/𝑘𝑇𝐺−1.()

Inserting some typical numbers into (), for example, a

transition wavelength of mfortheprominentCO

2-

absorption band and a temperature of 𝐺= 288K, we

calculate a ratio 𝐺

𝑚𝑛/𝑚𝑛 =0.037.Assuming that 𝑚=𝑛,

almost the same is found for the population ratio (see ())

with 𝑚/𝑛=0.036. At spontaneous transition rates of the

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

band then we get a radiation induced transition rate of only

𝐺

𝑚𝑛 =𝐺

𝑛𝑚 =0.03-0.04s−1.

Under conditions as found in the troposphere with colli-

sion rates between molecules of several 9s−1,anyinduced

transition rate due to the thermal background radiation is

orders of magnitude smaller, and even up to the stratosphere

and mesosphere, most of the transitions are caused by

collisions, so that above all they determine the population of

the states and in any case ensure a fast adjustment of a local

thermodynamic equilibrium in the gas.

Nevertheless, the absolute numbers of induced absorp-

tion and emission processes per volume, scaling with the

population density of the involved states (see ()), can be

quite signicant. So, at a CO2concentration of ppm,

thepopulationinthelowerstateis estimated to be about

𝑛∼7×10

20 m−3 (dependent on its energy above the

ground level). en more than 19 absorption processes per

m3are expected, and since such excitations can take place

simultaneously on many independent transitions, this results

in a strong overall interaction of the molecules with the

thermal bath, which according to Einstein’s considerations

even in the absence of collisions leads to the thermodynamic

equilibrium.

2.4. Linewidth and Lineshape of a Transition. e linewidth

of an optical or infrared transition is determined by dierent

eects.

2.4.1. Natural Linewidth and Lorentzian Lineshape. For

molecules in rest and without any collisions and also neglect-

ing power broadening due to induced transitions, the spectral

width of a line only depends on the natural linewidth

]𝑚𝑛 =1

2𝑚+1

2𝑛≈𝑚𝑛

2 ()

which for a two-level system and 𝑛

𝑚is essentially

determined by the spontaneous transition rate 𝑚𝑛 ≈1/

𝑚.

𝑛and 𝑚are the lifetimes of the lower and upper state.

e lineshape is given by a Lorentzian 𝐿(]),whichinthe

normalized form can be written as

𝐿(])=]𝑚𝑛/2

]−]𝑚𝑛2+]𝑚𝑛/22,

∞

0𝐿(])]=1.

()

2.4.2. Collision Broadening. With collisions in a gas, the

linewidth will considerably be broadened due to state and

phase changing collisions. en the width can be approxi-

mated by

]𝑚𝑛 ≈1

2𝑚𝑛 +𝑚𝑛 +𝑛𝑚 +phase, ()

where phase is an additional rate for phase changing colli-

sions. e lineshape is further represented by a Lorentzian,

only with the new homogenous width ]𝑚𝑛 (FWHM)

according to ().

2.4.3. Doppler Broadening. Since the molecules are at an

average temperature 𝐺, they also possess an average kinetic

energy

kin =3

2𝐺=𝐺

2V2,()

with 𝐺as the mass and V2as the velocity of the molecules

squared and averaged. Due to a Doppler shi of the moving

particles, the molecular transition frequency is additionally

broadened and the number of molecules interacting within

their homogeneous linewidth with the radiation at frequency

]is limited. is inhomogeneous broadening is determined

by Maxwell’s velocity distribution and known as Doppler

broadening. e normalized Doppler lineshape is given by a

Gaussian function of the form:

𝐷(])=2ln 2

]𝐷exp −]−]𝑚𝑛2

]𝐷/22ln 2 ()

with the Doppler linewidth

]𝐷=2]𝑚𝑛2𝐺

𝐺2ln 21/2 =7.16⋅10−7]𝑚𝑛𝐺

,()

where is the molecular weight in atomic units and 𝐺

specied in K.

2.4.4. Voigt Prole. For the general case of collision and

Doppler broadening, a convolution of 𝐷(])and 𝐿(]−

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

prole of the form:

𝑉(])=∞

0𝐷]𝐿]−]]

=ln 2

]𝑚𝑛

]𝐷

×∞

0exp −]−]𝑚𝑛2

]𝐷/22ln 2

⋅]

]−]2+]𝑚𝑛/22.

()

International Journal of Atmospheric Sciences

2.5. Spontaneous Emission as ermal Background Radiation.

Fromtherateequations(), it is clear that also in the

presence of collisions, the absolute number of spontaneously

emitted photons per time should be the same as that without

collisions. is is also a consequence of () indicating

that with an increasing rate 𝑚𝑛 for de-exciting transitions

(without radiation), also the rate 𝑛𝑚 of exiting collisions is

growing and just compensating for any losses, even when the

branching ratio =𝑚𝑛/𝑚𝑛 of radiating to nonradiating

transitions decreases. In other words, when the molecule

is in state , the probability for an individual spontaneous

emission act is reduced by the ratio ,butatthesametime,

thenumberofoccurringdecayspertimeincreaseswith𝑚𝑛.

enthespectralpowerdensityinthegasdueto

spontaneous emissions is (see ())

]

=]𝑚𝑛𝑚𝑛𝑚(]),()

representing a spectral generation rate of photons of energy

]𝑚𝑛 per volume. Photons emerging from a volume element

generally spread out into neighbouring areas, but in the same

way, there is a backow from the neighbourhood, which

in a homogeneous medium just compensates these losses.

Nevertheless, photons have an average lifetime, before they

are annihilated due to an absorption in the gas. With an

average photon lifetime

ph =ph

=

𝑛𝑚,()

where ph =1/𝑛𝑚 isthemeanfreepathofaphotoninthe

gas before it is absorbed, we can write for the spectral energy

density:

]=]𝑚𝑛𝑚𝑛ph𝑚(])=

]𝑚𝑛 𝑚𝑛

𝑛𝑚 𝑚(]).()

e same result is derived when transforming ()into

the time domain and assuming a local thermodynamic

equilibrium with ]/=0:

𝑛𝑚 (])𝑛−𝑚𝑛 (])𝑚]=𝑛𝑚]

=

]𝑚𝑛𝑚𝑛𝑚(]).()

With 𝑛𝑚(])from ()andBoltzmann’srelation(), then the

spectral energy density at ]𝑚𝑛 is found to be

]𝑚𝑛 =83]3

𝑚𝑛

31

𝑚𝑛/𝑛𝑚−1

=83]3

𝑚𝑛

31

ℎ]𝑚𝑛/𝑘𝑇𝐺−1.()

isisthewell-knownPlanckformula()andshowsthat

without an additional external excitation at thermodynamic

equilibrium, the spontaneous emission of molecules can be

understood as nothing else as the thermal radiation of a gas

on the transition frequency.

is derivation diers insofar from Einstein’s consid-

eration leading to (), as he concluded that a radiation

eld, interacting with the molecules at thermal and radiation

equilibrium, just had to be of the type of a Planckian

radiator, while here we consider the origin of the thermal

radiation in a gas sample, which exclusively is determined

andrightfullydenedbythespontaneousemissionofthe

molecules themselves. is is also valid in the presence of

molecular collisions. Because of this origin, the thermal

background radiation only exists on discrete frequencies,

determined by the transition frequencies and the linewidths

of the molecules, as long as no external radiation is present.

But on these frequencies, the radiation strength is the same

as that of a blackbody radiator.

Since this spontaneous radiation is isotropically emitting

photons into the full solid angle 4, in average half of the

radiation is directed upward and half downward.

3. Interaction of Molecules with Thermal

Radiation from an External Source

In this section, we consider the interaction of molecules with

an additional blackbody radiation emitted by an external

source like the earth’s surface or adjacent atmospheric layers.

We also investigate the transfer of absorbed radiation to heat

in the presence of molecular collisions, causing a rise of

the atmospheric temperature. And vice versa, we study the

transfer of a heat ux to radiation resulting in a cooling of the

gas. An appropriate means to describe the mutual interaction

of these processes is to express this by coupled rate equations

which are solved numerically.

3.1. Basic Quantities

3.1.1. Spectral Radiance. e power radiated by a surface

element on the frequency ]in the frequency interval ]

and into the solid angle element is also determined by

Planck’s radiation law:

],Ω 𝐸cos ]

=2]32

21

ℎ]/𝑘𝑇𝐸−1cos ], ()

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

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

ofthesource(e.g.,earth’ssurface).ecosinetermaccounts

for the fact that for an emission in a direction given by the

azimuthal angle and the polar angle ,onlytheprojection

of perpendicular to this direction is ecient as radiating

surface (Lambertian radiator).

3.1.2. Spectral Flux Density—Spectral Intensity. Integration

over the solid angle gives the spectral ux density. en,

representing in spherical coordinates as =sin

International Journal of Atmospheric Sciences

𝜑

𝛽

dA

d𝜑

d𝛽

F : Radiation from a surface element into the solid angle

=sin .

with as the azimuthal angle interval and as the polar

interval (see Figure ), this results in:

]=2𝜋

Ω],Ω 𝐸cos

=2𝜋

𝜑=0 𝜋/2

𝛽=0 ],Ω 𝐸cos sin

=2]32

21

ℎ]/𝑘𝑇𝐸−1

()

or in wavelengths units

𝜆=

2]=223

51

ℎ𝑐/𝑘𝑇𝐸𝜆−1.()

𝜆and ], also known as spectral intensities, are specied

in units of W/m2/mandW/m

2/Hz, respectively. ey

represent the power ux per frequency or wavelength and per

surface area into that hemisphere, which can be seen from

theradiatingsurfaceelement.espectraldistributionofa

Planck radiator of =299K as a function of wavelength is

shown in Figure .

3.2. Ray Propagation in a Lossy Medium

3.2.1. Spectral Radiance. Radiation passing an absorbing

sample generally obeys Lambert-Beer’s law, as already applied

in ()forthespectralenergydensity].esameholdsfor

thespectralradiancewith

],Ω

=−𝑛𝑚 (])],Ω,()

where is the propagation distance of ],Ω in the sample.

is is valid independent of the chosen coordinate system.

e letter is used when not explicitly a propagation

35

30

25

20

15

10

5

00 102030405060

Wavelength (𝜇m)

Spectral intensity (Wm −2𝜇m−1)

Earth

299 K

F : Spectrum of a Planck radiator at K temperature.

z

𝛽

Δz

I,Ω cos 𝛽dΩ

Δz/cos 𝛽

F : Radiation passing a gas layer.

perpendicular to the earth’s surface or a layer (-direction)

is meant.

3.2.2. Spectral Intensity. For the spectral intensity, which due

to the properties of a Lambertian radiator consists of a bunch

of rays with dierent propagation directions and brightness,

somebasicdeviationshavetoberecognized.So,becauseof

the individual propagation directions spreading over a solid

angle of 2, only for a homogeneously absorbing sphere and

in a spherical coordinate system with the radiation source in

the center of this sphere the distances to pass the sample, and

thus the individual contributions to the overall absorption,

would be the same.

But radiation, emerging from a plane parallel surface

and passing an absorbing layer of thickness ,isbetter

characterized by its average expansion perpendicular to the

layer surface in -direction. is means, that with respect to

this direction an individual ray, propagating under an angle

to the surface normal covers a distance =/cos before

leaving the layer (see Figure ). erefore, such a ray on one

side contributes to a larger relative absorption, and on the

other side, it donates this to the spectral intensity only with

weight ],Ωcos duetoLambert’slaw.

is means that to rst order, each individual beam

direction suers from the same absolute attenuation, and

particularly the weaker rays under larger propagation angles

waste relatively more of their previous spectral radiance.

us, especially at higher absorption strengths and longer

propagation lengths, the initial Lambertian distribution will

bemoreandmoremodied.Acriterionforanalmost

unaltered distribution may be that for angles ≤

max the

inequality 𝑛𝑚 ⋅/cos 1is satised. e absorption

International Journal of Atmospheric Sciences

loss for the spectral intensity as the integral of the spectral

radiance ],Ω (see ()) then is given by

∫],Ω ()cos

=−𝑛𝑚 (])2𝜋

0𝜋/2

0],Ω cos

cos sin . ()

e cosine terms under the integral sign just compensate and

()canbewrittenas

],𝐸

=−2𝑛𝑚 (])],𝐸. ()

is dierential equation for the spectral intensity shows that

the eective absorption coecient is twice that of the spectral

radiance, or in other words, the average propagation length of

the radiation to pass the layer is twice the layer thickness. is

last statement means that we also can assume radiation, which

is absorbed at the regular absorption coecient 𝑛𝑚,butis

propagating as a beam under an angle of 60∘to the surface

normal (1/cos 60∘=2). In practice, it even might give sense

todeviatefromanangleof60∘,tocompensatefordeviations

of the earth’s or oceanic surface from a Lambertian radiator,

and to account for contributions due to Mie and Rayleigh

scattering.

e absorbed spectral power density ](power per

volume and per frequency) with respect to the -direction

then is found to be

]()=]()

=−2𝑛𝑚 (])](0)−2𝛼𝑛𝑚(])𝑧,()

with ](0)as the initial spectral intensity at =0as given by

().

An additional decrease of ]with due to a lateral

expansion of the radiation over the hemisphere can be

neglected, since any propagation of the radiation is assumed

tobesmallcomparedtotheearth’sradius(considerationof

an extended radiating parallel plane).

Integration of ()overthelineshape(])within the

spectral interval ]gives the absorbed power density (units:

W/m3), which similar to () may also be expressed as loss

or annihilation of photons per volume (𝐸in m−3)andper

time. Since the average propagation speed of ]in -direction

is /2, also the dierentials transform as =/2⋅.With

]from (), this results in

()=2

Δ]]

]=2

]𝑚𝑛

]=2𝐸

]𝑚𝑛

=−𝑚𝑛

2𝑛𝑚

𝑛−𝑚

𝑛]𝑚𝑛

ℎ]𝑚𝑛/𝑘𝑇𝐸−1−2𝛼𝑛𝑚𝑧,()

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

tion rates 𝐸

𝑛𝑚 and 𝐸

𝑚𝑛 caused by the external eld:

𝐸

=−𝐸

𝑛𝑚𝑛+𝐸

𝑚𝑛𝑚

=−𝑚𝑛

4𝑚

𝑛𝑛−𝑚1

ℎ]𝑚𝑛/𝑘𝑇𝐸−1−2𝛼𝑛𝑚𝑧,()

with 𝑛𝑚 = 0

𝑛𝑚/]=

𝑛𝑚/]as an averaged absorption

coecient over the linewidth (see also ()) and with

𝐸

𝑚𝑛 ()=𝑚𝑛2

𝑚𝑛

82]𝑚𝑛 ]𝑚𝑛 ()=0

𝑚𝑛

]𝑚𝑛 ]𝑚𝑛 ()

=𝑚𝑛

4−2𝑆𝑛𝑚𝑁𝑧/Δ]

ℎ]𝑚𝑛/𝑘𝑇𝐸−1,

𝐸

𝑛𝑚 =𝑚

𝑛𝐸

𝑚𝑛.

()

Similar to (), these rates are again proportional to the

spontaneous transition probability (rate) 𝑚𝑛,butnowthey

depend on the temperature 𝐸of the external source and the

propagation depth .

Due to the fact that radiation from a surface with a

Lambertian distribution and only from one hemisphere is

acting on the molecules, dierent to (), a factor of 1/4

appears in this equation.

At a temperature 𝐸=288K and wavelength =15m,

for example, the ratio of induced to spontaneous transitions

duetotheexternaleldat=0is

𝐸

𝑚𝑛

𝑚𝑛 =0.0093≈1%.()

For the case of a two-level system, 𝑚𝑛 can also be expressed

by the natural linewidth of the transition with (see ())

]𝑁

𝑚𝑛 =𝑚𝑛

2 .()

en ()becomes

𝐸

𝑚𝑛 ()=

2]𝑁

𝑚𝑛 −2𝑆𝑛𝑚𝑁𝑧/Δ]

ℎ]𝑚𝑛/𝑘𝑇𝐸−1.()

With a typical natural width of the order of only . Hz in the

mbandofCO

2, then the induced transition rate will be

less than . s−1. Even on the strongest transitions around

. m,thenaturallinewidthsareonly∼ Hz, and therefore

the induced transition rates are about s−1.

Despite of these small rates, the overall absorption of an

incident beam can be quite signicant. In the atmosphere,

the greenhouse gases are absorbing on hundred thousands of

transitions over long propagation lengths and at molecular

number densities of 19–23 per m3. On the strongest lines

in the mbandofCO

2, the absorption coecient at the

center of a line even goes up to 1m−1.enalreadywithina

distanceofafewm,thetotalpowerwillbeabsorbedonthese

frequencies. So, altogether about % of the total IR radiation

emergingfromtheearth’ssurfacewillbeabsorbedbythese

gases.

3.2.3. Alternative Calculation for the Spectral Intensity. For

some applications, it might be more advantageous, rst to

solve the dierential equation () for the spectral radiance

as a function of and also , before integrating over .

International Journal of Atmospheric Sciences

Nm

Nn

WG

mn

Amn

Cmn

WG

nm

WE

mn

WE

nm

Cnm

F : Two-level system with transition rates including an

external excitation.

en an integration in -direction over the length /cos

with a -dependent absorption coecient gives

],Ω ()=],Ω (0)−(1/cos 𝛽) ∫𝑧

0𝛼𝑛𝑚(𝑧)𝑑𝑧,()

and a further integration over results in

]()=2𝜋

0𝜋/2

0],Ω ()cos sin

=],Ω (0)2𝜋

0𝜋/2

0−(1/cos 𝛽)∫𝑧

0𝛼𝑛𝑚(𝑧)𝑑𝑧

×cos sin .

()

First integrating over andintroducingtheopticaldepth

∫𝑧

0𝑛𝑚()=as well as the substitution =/cos ,

we can write

]()=2],Ω (0)2∞

𝜏−𝑢

3

=](0)21()+−𝜏 1

2−1

()

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

1()=∞

𝜏−𝑢

=−0.5772−ln +∞

𝑘=1 (−1)𝑘+1𝑘

⋅! .()

Since the further considerations in this paper are concen-

trating on the radiation transfer in the atmosphere under

the inuence of thermal background radiation, it is more

appropriate to describe the interaction of radiation with the

molecules by a stepwise propagation through thin layers of

depth as given by ().

3.3. Rate Equations under Atmospheric Conditions. As a

further generalization of the rate equations (), in this

subsection, we additionally consider the inuence of the

external radiation, which together with the thermal back-

ground radiation is acting on the molecules in the presence

of collisions (see Figure ).

And dierent to Section , the infrared active molecules

are considered as a trace gas in an open system, the

atmosphere, which has to come into balance with its environ-

ment. en molecules radiating due to their temperature and

thus loosing part of their energy have to get this energy back

from the surrounding by IR radiation, by sensitive or latent

heat or also by absorption of sunlight. is is a consequence

of energy conservation.

e radiation loss is assumed to be proportional to the

actual photon density 𝐺and scaling with a ux rate .

is loss can be compensated by the absorbed power of

the incident radiation, for example, the terrestrial radiation,

which is further expressed in terms of the photon density

𝐸(see ()), and it can also be replaced by thermal energy.

erefore, the initial rate equations have to be supplemented

by additional relations for these two processes.

It is evident that both, the incident radiation and the

heat ux, will be limited by some genuine interactions. So,

the radiation can only contribute to a further excitation, as

long as it is not completely absorbed. At higher molecular

densities, however, the penetration depth of the radiation

in the gas is decreasing and therewith also the eective

excitation over some longer volume element. Integrating ()

over the linewidth while using the denitions of ()and()

andthenintegratingoverresults in

]𝑚𝑛 ()=]𝑚𝑛 (0)−2𝑆𝑛𝑚𝑁𝑧/Δ]()

which due to Lambert-Beer’s law describes the averaged

spectral intensity over the linewidth as a function of the

propagation in -direction. From this, it is quite obvious to

dene the penetration depth as the length 𝑃,overwhichthe

initial intensity reduces to 1/and thus the exponent in ()

gets unity with

𝑃=]

2𝑛𝑚𝑆.()

From (), also the molecular number density 𝑆is found, at

which the initial intensity just drops to 1/aer an interaction

length 𝑃.Since𝑆characterizes the density at which an

excitation of the gas gradually comes to an end and in this

sense saturation takes place, it is known as the saturation

density, where 𝑆and also 𝑛𝑚 refer to the total number

density of the gas.

Since at constant pressure, the number density in the

gasischangingwiththeactualtemperatureduetoGay-

Lussac’slawandthemoleculesaredistributedoverhundreds

of states and substates, the molecular density, contributing to

an interaction with the radiation on the transition →,is

given by []

𝑛=00

𝐺𝑛−𝐸𝑛/𝑘𝑇𝐺

𝐺,()

where 0is the molecular number density at an initial

temperature 0,𝑛is the energy of the lower level above

International Journal of Atmospheric Sciences

thegroundstate,and(𝐺)is the total internal partition sum,

dened as [,]

𝐺=

𝑖𝑖−𝐸𝑖/𝑘𝑇𝐺.()

In the atmosphere typical propagation, lengths are of the

order of several km. So, for a stronger CO2transition in the

mbandwithaspontaneousrate=1s−1,𝑛/ =

𝑛/0=0.07,aspectrallineintensity𝑛𝑚 or integrated cross

section of

𝑛𝑚 =0

𝑛𝑚 =𝑚𝑛2

82𝑛

1− 𝑛

𝑚𝑚

𝑛

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

()

and a spectral width at ground pressure of ]=5GHz,

the saturation density over a typical length of 𝑃=1km

assumes a value of 𝑆≈4×10

18 molecules/m3.Sincethe

number density of the air at hPa and K is air =

2.55 × 1025 m−3, saturation on the considered transition

already occurs at a concentration of less than one ppm. It

should also be noticed that at higher altitudes and thus lower

densities, also the linewidth ]due to pressure broadening

reduces. is means that to rst order, also the saturation

density decreases with ], while the gas concentration in

the atmosphere, at which saturation appears, almost remains

constant. e same applies for the penetration depth.

For the further considerations, it is adequate to introduce

an average spectral intensity ]𝑚𝑛 as

]𝑚𝑛 =1

𝑃𝐿𝑃

0]𝑚𝑛 ()

=]

2𝑛𝑚𝑃]𝑚𝑛 (0)⋅1−−2𝑆𝑛𝑚𝑁𝐿𝑃/Δ]()

or equivalently an average photon density

𝐸=

]

]𝑚𝑛 ]𝑚𝑛 =]

2𝑛𝑚𝑃𝐸(0)⋅1−−2𝑆𝑛𝑚𝑁𝐿𝑃/Δ]

()

which characterizes the incident radiation with respect to its

average excitation over 𝑃intherateequations.Ingeneral,

the incident ux 𝐸(0)onto an atmospheric layer consists of

two terms, the up- and the downwelling radiation.

Anyheatux,suppliedtothegasvolume,contributes

to a volume expansion, and via collisions and excitation of

molecules, this can also be transferred to radiation energy.

Simultaneously absorbed radiation can be released as heat

in the gas. erefore, the rate equations are additionally

supplemented by a balance for the heat energy density of the

air [J/m3], which under isobaric conditions and making use

of the ideal gas equation in the form air =air ⋅⋅𝐺can be

written as

=𝑝air 𝐺=

2+1𝐺

air air

𝐴air

𝐺𝐺

=7

2air

𝐺𝐺,()

or aer integration

𝐺=7

2air 𝑇𝐺

𝑇01

𝐺

𝐺=7

2air ln 𝐺

0.()

Here 𝑝represents the specic heat capacity of the air at

constant pressure with 𝑝=(/2+1)⋅𝐺/air ,whereare

the degrees of freedom of a molecule (for N2and O2:=5)

and 𝐺=⋅

𝐴is the universal gas constant (at room

temperature: 𝑝= 1.01kJ/kg/K). air is the specic weight

of the air (at room temperature and ground pressure: air =

1.29kg/m3), which can be expressed as air =

air ⋅air =

air/𝐴⋅air =air /𝐴⋅air /(𝐺)with air as the mass of

an air molecule, air the mol weight, 𝐴Avogadro’s number,

air the number density, and air the pressure of the air.

e thermal energy can be supplied to a volume element

by dierent means. So, thermal convection and conductivity

in the gas contribute to a heat ux j𝐻[J/(s⋅m2)], causing

atemporalchangeofproportional to div j𝐻, or for one

direction proportional to 𝐻𝑧 /.j𝐻is a vector and points

from hot to cold. Since this ux is strongly dominated by

convection, it can be well approximated by 𝐻𝑧 =𝐶⋅(𝐸−

𝐺)with 𝐶as the heat transfer coecient, typically of the

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

distribution of the incident heat ux over the troposphere

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

convection may be expressed as 𝐻𝑧/=𝐶/𝑇⋅(𝐸−𝐺)

with 𝐶/𝑇≈1mW/m3/K.

Another contribution to the thermal energy originates

from the absorbed sunlight, which in the presence of col-

lisions is released as kinetic or rotational energy of the

molecules. Similarly, latent heat can be set free in the air. In

the rate equations, both contributions are represented by a

source term SL [J/(s⋅m3)].

Finally, the heat balance is determined by exciting and de-

exciting collisions, changing the population of the states and

, and reducing or increasing the energy by ⋅]𝑚𝑛.

Altogether, this results in a set of coupled dierential

equations, which describe the simultaneous interaction of

molecules with their self-generated thermal background

radiation as well as with radiation from the earth’s surface

International Journal of Atmospheric Sciences

and/or a neighbouring layer, this all in the presence of

collisions and under the inuence of heat transfer processes:

𝑚

=+]𝑚𝑛

]𝑛𝑚 𝐺+𝐸+𝑛𝑚𝑛

−]𝑚𝑛

]𝑚𝑛 𝐺+𝐸+𝑚𝑛 +𝑚𝑛𝑚,

𝑛

=−]𝑚𝑛

]𝑛𝑚 𝐺+𝐸+𝑛𝑚𝑛

+]𝑚𝑛

]𝑚𝑛 𝐺+𝐸+𝑚𝑛 +𝑚𝑛𝑚,

𝐺

=−]𝑚𝑛

]𝑛𝑚𝑛−𝑚𝑛𝑚𝐺+𝑚𝑛𝑚−𝐺,

𝐸

=+/

2𝑃1−−(2ℎ]𝑚𝑛/(Δ]𝑐/𝑛))(𝐵𝑛𝑚𝑁𝑛−𝐵𝑚𝑛𝑁𝑚)𝐿𝑃𝐸(0)

−]𝑚𝑛

]𝑛𝑚𝑛−𝑚𝑛𝑚𝐸,

=+𝐶

𝑇𝐸−𝐺+SL −]𝑚𝑛 𝑛𝑚𝑛−𝑚𝑛𝑚.

()

To symbolize the mutual coupling of these equations, here we

use a notation where the radiation induced transition rates

are represented by the Einstein coecients and the respective

photon densities. A change to the other representation is

easily performed applying the identities:

𝐺

𝑚𝑛 =]𝑚𝑛

]𝑚𝑛𝐺=𝑚𝑛]𝑚𝑛 ,

𝐸

𝑚𝑛 =]𝑚𝑛

]𝑚𝑛𝐸=𝑚𝑛

]𝑚𝑛,

𝐺,𝐸

𝑛𝑚 =𝑚

𝑛𝐺,𝐸

𝑚𝑛 .

()

It should be noticed that the rate equation for 𝐸could

also be replaced by the incident radiation, as given by ().

However, since 𝐸approaches its equilibrium value within

atimeconstant=]/(]𝑚𝑛(𝑛𝑚𝑛−

𝑚𝑛𝑚))short

compared with other processes, for a uniform representation

the dierential form was preferred.

Forthespecialcaseofstationaryequilibriumwith

𝑛/=𝑚/ =0from the rst rate equation, we get

for the population ratio:

𝑚

𝑛=]𝑚𝑛𝑛𝑚 𝐺+𝐸/]+𝑛𝑚

]𝑚𝑛𝑚𝑛 𝐺+𝐸/]+𝑚𝑛 +𝑚𝑛

=𝐺

𝑛𝑚 +𝐸

𝑛𝑚 +𝑛𝑚

𝐺

𝑚𝑛 +𝐸

𝑚𝑛 +𝑚𝑛 +𝑚𝑛

()

which at 𝐺∼

𝐸= 288Kand=15mdueto𝐺

𝑚𝑛 =

4𝐸

𝑚𝑛 = 0.037𝑚𝑛 (see ()and()) in the limit of pure

spontaneous decay processes reaches its maximum value of

.%, while in the presence of collisions with collision rates

in the troposphere of several 9s−1,itrapidlyconvergesto

.%, given by the Boltzmann relation at temperature 𝐺and

corresponding to a local thermodynamic equilibrium.

For the general case, the rate equations have to be

solved numerically, for example, by applying the nite ele-

ment method. While () in the presented form is only

valid for a two-level system, a simulation under realistic

conditions comparable to the atmosphere requires some

extension, particularly concerning the energy transfer from

theearth’ssurfacetotheatmosphere.Sincethemaintrace

gases CO2, water vapour, methane, and ozone are absorbing

the incident infrared radiation simultaneously on thousands

of transitions, as an acceptable approximation for this kind

of calculation, we consider these transitions to be similar and

independent from each other, each of them contributing to

the same amount to the energy balance. In the rate equations,

this can easily be included by multiplying the last term in the

equation for the energy density with an eective number

tr of transitions.

In this approximation, the molecules of an infrared active

gas and even a mixture of gases are represented by a “standard

transition” which reects the dynamics and time evolution of

the molecular populations under the inuence of the incident

radiation, the background radiation, and the thermal heat

transfer. tr is derived as the ratio of the total absorbed

infrared intensity over the considered propagation length to

the contribution of a single standard transition.

An example for a numerical simulation in the tropo-

sphere, more precisely for a layer from ground level up

to m, is represented in Figure . e graphs show the

evolution of the photon densities 𝐺and 𝐸,thepopulation

densities 𝑛and 𝑚of the states and , the accumulated

heat density in the air, and the temperature 𝐺of the gas

(identical with the atmospheric temperature) as a function of

time and as an average over an altitude of m.

As initial conditions we assumed a gas and air temper-

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

of 𝐸= 288K,aninitialheatuxduetoconvectionof

𝐻𝑧 =3kW/m2,alatentheatsourceofSL =3mW/