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

Authors:
  • Helmut Schmidt University Hamburg

Abstract and Figures

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. The 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. The radiative forcing at doubled CO2 concentration is found to be 30% smaller than the IPCC-value.
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International Journal of Atmospheric Sciences
Volume , Article ID ,  pages
http://dx.doi.org/.//
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 earths
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 eect. Upward radiation contributes to
cooling and ensures that the absorbed energy from the sun
and the terrestrial radiation can be rendered back to space
andtheearthstemperaturecanbestabilized.
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 earths 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 modication 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 []andPlancksradi-
ation law [] to investigate the interaction of molecules with
their own thermal background radiation under the inuence
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 inuence of radiation from an
external blackbody radiator and an additional excitation by
aheatsourceisstudied.eradiationandheattransfer
processes originating from the sun and/or the earths 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 modied 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, reecting the dierent
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 eect or to
prove its existence within this framework. Nevertheless, the
basic considerations and derived relations for the molecular
interaction with radiation have some direct signicance for
the understanding and interpretation of this eect, 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
modications, but following the main thoughts. In the second
part of this section, also collisions between the molecules are
Em
En
ΔE = hmn =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=84
51
ℎ𝑐/𝑘𝑇𝐺𝜆−1,()
or as a function of frequency assumes the form
]=𝜆2
=83]3
31
]/𝑘𝑇𝐺−1.()
isdistributionisshowninFigure  for three dierent
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 dierent 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 4with 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 dierent
temperatures.
𝑚𝑛 is the Einstein coecient 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 coecient of induced absorption (units: m3Hz/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 satises 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 coecient 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 +𝑚𝑛𝑚,()
or more universally described by rate equations:
𝑚
 =+𝑛𝑚]𝑚𝑛 𝑛−𝑚𝑛]𝑚𝑛 +𝑚𝑛𝑚,
𝑛
 =−𝑛𝑚]𝑚𝑛 𝑛+𝑚𝑛]𝑚𝑛 +𝑚𝑛𝑚.()
For 𝑚/=𝑛/=0,() gets identical to (). Using
()in()gives
𝑛𝑛𝑚]𝑚𝑛 =𝑚𝑚𝑛]𝑚𝑛 +𝑚𝑛−ℎ]𝑚𝑛 /𝑘𝑇𝐺.()
Assuming that with 𝐺also ]𝑚𝑛 gets innite, 𝑚𝑛 and 𝑛𝑚
must satisfy the relation:
𝑛𝑛𝑚 =𝑚𝑚𝑛.()
en ()becomes
𝑚𝑛]𝑚𝑛 ]𝑚𝑛/𝑘𝑇𝐺=𝑚𝑛]𝑚𝑛 +𝑚𝑛,()
or resolving to ]𝑚𝑛 gives
]𝑚𝑛 =𝑚𝑛
𝑚𝑛 1
]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
83]3
𝑚𝑛 =
1
]𝑚𝑛 𝑚𝑛2
𝑚𝑛
82,()
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 coecients 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 coecient of a sample.
International Journal of Atmospheric Sciences
2.2.1. Cross Section and Absorption Coecient. 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 dierential form:
]
 =−𝑛𝑚 (])]=−𝑛𝑚 (])𝑛],()
where 𝑛𝑚(])is the absorption coecient (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
amplication of ]and is known as gain coecient.
Additionally, spontaneously emitted photons within a
considered volume element and time interval  = 
/contribute to the spectral energy density of the thermal
background radiation with
]=]𝑚𝑛𝑚𝑛𝑚(])=]𝑚𝑛
/ 𝑚𝑛𝑚(]).()
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
82]𝑚𝑛 𝑛
=
1
]𝑚𝑛 0
𝑛𝑚𝑛]𝑚𝑛 ,()
and therefore
0
𝑛𝑚 =𝑚
𝑛𝑚𝑛2
𝑚𝑛
82.()
Comparing the second terms in ()and() results in
0
𝑚𝑛 =𝑚𝑛2
𝑚𝑛
82=𝑛
𝑚0
𝑛𝑚.()
So, together with ()and(), we derive as the nal
expressions for 𝑛𝑚(])and 𝑚𝑛(]):
𝑚𝑛 (])=𝑛
𝑚𝑛𝑚 (])=𝑚𝑛2
𝑚𝑛
82(]),
𝑚𝑛 (])=]𝑚𝑛
/ 𝑚𝑛(]),
𝑛𝑚 (])=]𝑚𝑛
/ 𝑛𝑚(]).
()
2.2.2. Eective Cross Section and Spectral Line Intensity.
Oen the rst two terms on the right side of ()areunied
and represented by an eective cross section 𝑛𝑚(]).Further
International Journal of Atmospheric Sciences
Cross section (cm2)
Snm
Frequency
Δmn
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 (])𝑛1𝑛
𝑚𝑚
𝑛
=]𝑚𝑛
/ 𝑛𝑚(])𝑛1𝑛
𝑚𝑚
𝑛,
()
and 𝑛𝑚(])becomes
𝑛𝑚 (])=]𝑚𝑛
/ 𝑛𝑚 𝑛
1𝑛
𝑚𝑚
𝑛(]).()
Integration of () over the linewidth gives the spectral line
intensity 𝑛𝑚 of a transition (Figure ):
𝑛𝑚 =Δ]𝑛𝑚 (])]=]𝑚𝑛
/ 𝑛𝑚 𝑛
1𝑛
𝑚𝑚
𝑛, ()
as it is used and tabulated in data bases [,] to characterize
the absorption strength on a transition.
2.2.3. Eective Absorption Coecient. Similar to 𝑛𝑚(]),with
()and(), an eective absorption coecient on a transi-
tioncanbedenedas
𝑛𝑚 (])=𝑛𝑚 (])
=]𝑚𝑛
/ 𝑛𝑚𝑛1𝑛
𝑚𝑚
𝑛(])
=𝑛𝑚(])=0
𝑛𝑚(]),
()
which aer replacing 𝑛𝑚 from () assumes the more
common form:
𝑛𝑚 (])=𝑚𝑛2
82]2𝑛𝑚
𝑛𝑚
𝑛(])
=𝑚𝑛2
82𝑛𝑚
𝑛𝑚
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 +𝑛𝑚𝑛−𝐺
𝑚𝑛 +𝑚𝑛 +𝑚𝑛𝑚,
𝑛
 =−𝐺
𝑛𝑚 +𝑛𝑚𝑛+𝐺
𝑚𝑛 +𝑚𝑛 +𝑚𝑛𝑚,
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
]𝑚𝑛/𝑘𝑇𝐺−1,
𝐺
𝑛𝑚 =𝑚
𝑛𝑚𝑛
]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 signicant. 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 dierent
eects.
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
2𝑚+1
2𝑛𝑚𝑛
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/22,
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
2𝑚𝑛 +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
2𝐺=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/22ln 2 ()
with the Doppler linewidth
]𝐷=2]𝑚𝑛2𝐺
𝐺2ln 21/2 =7.1610−7]𝑚𝑛𝐺
,()
where is the molecular weight in atomic units and 𝐺
specied in K.
2.4.4. Voigt Prole. For the general case of collision and
Doppler broadening, a convolution of 𝐷(]󸀠)and 𝐿(]
]󸀠)gives the universal lineshape 𝑉(])representing a Voigt
prole of the form:
𝑉(])=
0𝐷]󸀠𝐿]]󸀠]󸀠
=ln 2
]𝑚𝑛
]𝐷
×
0exp −]󸀠]𝑚𝑛2
]𝐷/22ln 2
]󸀠
]]󸀠2+]𝑚𝑛/22.
()
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 backow 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 ()andBoltzmannsrelation(), then the
spectral energy density at ]𝑚𝑛 is found to be
]𝑚𝑛 =83]3
𝑚𝑛
31
𝑚𝑛/𝑛𝑚−1
=83]3
𝑚𝑛
31
]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 diers 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:
]𝐸cos ]
=2]32
21
]/𝑘𝑇𝐸−1cos ], ()
with ](𝐸)as the spectral radiance (units: W/m2/
Hz/sterad) and 𝐸as the temperature of the emitting surface
ofthesource(e.g.,earthssurface).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 ecient 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]𝐸cos 
=2𝜋
𝜑=0 𝜋/2
𝛽=0 ]𝐸cos sin 
=2]32
21
]/𝑘𝑇𝐸−1
()
or in wavelengths units
𝜆=
2]=223
51
ℎ𝑐/𝑘𝑇𝐸𝜆−1.()
𝜆and ], also known as spectral intensities, are specied
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
]
 =−𝑛𝑚 (])],()
where is the propagation distance of ]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 dierent 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 ]cos duetoLambertslaw.
is means that to rst order, each individual beam
direction suers 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 satised. e absorption
International Journal of Atmospheric Sciences
loss for the spectral intensity as the integral of the spectral
radiance ](see ()) then is given by
∫]()cos 

=−𝑛𝑚 (])2𝜋
0𝜋/2
0]cos
cos sin . ()
e cosine terms under the integral sign just compensate and
()canbewrittenas
],𝐸
 =−2𝑛𝑚 (])],𝐸. ()
is dierential equation for the spectral intensity shows that
the eective absorption coecient 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 coecient 𝑛𝑚,butis
propagating as a beam under an angle of 60to the surface
normal (1/cos 60=2). In practice, it even might give sense
todeviatefromanangleof60,tocompensatefordeviations
of the earths 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
tobesmallcomparedtotheearthsradius(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 dierentials transform as =/2.With
]from (), this results in
()=2
Δ]]
]=2
]𝑚𝑛
 ]=2𝐸
 ]𝑚𝑛
=−𝑚𝑛
2𝑛𝑚
𝑛𝑚
𝑛]𝑚𝑛
]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
]𝑚𝑛/𝑘𝑇𝐸−1−2𝛼𝑛𝑚𝑧,()
with 𝑛𝑚 = 0
𝑛𝑚/]=
𝑛𝑚/]as an averaged absorption
coecient over the linewidth (see also ()) and with
𝐸
𝑚𝑛 ()=𝑚𝑛2
𝑚𝑛
82]𝑚𝑛 ]𝑚𝑛 ()=0
𝑚𝑛
]𝑚𝑛 ]𝑚𝑛 ()
=𝑚𝑛
4−2𝑆𝑛𝑚𝑁𝑧]
]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, dierent to (), a factor of 1/4
appears in this equation.
At a temperature 𝐸=288K and wavelength =15m,
for example, the ratio of induced to spontaneous transitions
duetotheexternaleldat=0is
𝐸
𝑚𝑛
𝑚𝑛 =0.00931%.()
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]
]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 signicant. 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 coecient at the
center of a line even goes up to 1m−1.enalreadywithina
distanceofafewm,thetotalpowerwillbeabsorbedonthese
frequencies. So, altogether about % of the total IR radiation
emergingfromtheearthssurfacewillbeabsorbedbythese
gases.
3.2.3. Alternative Calculation for the Spectral Intensity. For
some applications, it might be more advantageous, rst to
solve the dierential 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 coecient gives
]()=](0)−(1/cos 𝛽) 𝑧
0𝛼𝑛𝑚(𝑧󸀠)𝑑𝑧󸀠,()
and a further integration over results in
]()=2𝜋
0𝜋/2
0]()cos sin 
=](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](0)2
𝜏−𝑢
3
=](0)21()+−𝜏 1
21
 ()
with ](0)=](0)(see ()) and the exponential integral
1()=
𝜏−𝑢
=−0.5772ln +
𝑘=1 (−1)𝑘+1𝑘
⋅! .()
Since the further considerations in this paper are concen-
trating on the radiation transfer in the atmosphere under
the inuence 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 inuence 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 dierent 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 eective
excitation over some longer volume element. Integrating ()
over the linewidth while using the denitions 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
dene the penetration depth as the length 𝑃,overwhichthe
initial intensity reduces to 1/and thus the exponent in ()
gets unity with
𝑃=]
2𝑛𝑚𝑆.()
From (), also the molecular number density 𝑆is found, at
which the initial intensity just drops to 1/aer 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-
Lussacslawandthemoleculesaredistributedoverhundreds
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,
dened 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
82𝑛
1𝑛
𝑚𝑚
𝑛
=6×10−13 m2Hz =2×10−19 cm−1/moleculescm−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 ()
=]
2𝑛𝑚𝑃]𝑚𝑛 (0)⋅1−−2𝑆𝑛𝑚𝑁𝐿𝑃]()
or equivalently an average photon density
𝐸=
]
]𝑚𝑛 ]𝑚𝑛 =]
2𝑛𝑚𝑃𝐸(0)⋅1−−2𝑆𝑛𝑚𝑁𝐿𝑃]
()
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.
Anyheatux,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
𝐺𝐺
=7
2air
𝐺𝐺,()
or aer integration
𝐺=7
2air 𝑇𝐺
𝑇01
󸀠
𝐺󸀠
𝐺=7
2air ln 𝐺
0.()
Here 𝑝represents the specic 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 specic 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 dierent means. So, thermal convection and conductivity
in the gas contribute to a heat ux j𝐻[J/(sm2)], 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 coecient, 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/(sm3)].
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 dierential
equations, which describe the simultaneous interaction of
molecules with their self-generated thermal background
radiation as well as with radiation from the earths surface
 International Journal of Atmospheric Sciences
and/or a neighbouring layer, this all in the presence of
collisions and under the inuence of heat transfer processes:
𝑚
 =+]𝑚𝑛
]𝑛𝑚 𝐺+𝐸+𝑛𝑚𝑛
−]𝑚𝑛
]𝑚𝑛 𝐺+𝐸+𝑚𝑛 +𝑚𝑛𝑚,
𝑛
 =−]𝑚𝑛
]𝑛𝑚 𝐺+𝐸+𝑛𝑚𝑛
+]𝑚𝑛
]𝑚𝑛 𝐺+𝐸+𝑚𝑛 +𝑚𝑛𝑚,
𝐺
 =−]𝑚𝑛
]𝑛𝑚𝑛−𝑚𝑛𝑚𝐺+𝑚𝑛𝑚−𝐺,
𝐸
 =+/
2𝑃1−(2ℎ]𝑚𝑛/(Δ]𝑐/𝑛))(𝐵𝑛𝑚𝑁𝑛−𝐵𝑚𝑛𝑁𝑚)𝐿𝑃𝐸(0)
]𝑚𝑛
]𝑛𝑚𝑛−𝑚𝑛𝑚𝐸,

 =+𝐶
𝑇𝐸−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 coecients 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 dierential 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 =
4𝐸
𝑚𝑛 = 0.037𝑚𝑛 (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
theearthssurfacetotheatmosphere.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 eective 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 reects the dynamics and time evolution of
the molecular populations under the inuence 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 earths surface
of 𝐸= 288K,aninitialheatuxduetoconvectionof
𝐻𝑧 =3kW/m2,alatentheatsourceofSL =3mW/