#### Abstract

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 CO_{2} concentration is found to be 30% 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 effect. Upward radiation contributes to cooling and ensures that the absorbed energy from the sun and the terrestrial radiation can be rendered back to space and the earth’s temperature can be stabilized.

For all these processes, particularly, the interaction of radiation with infrared active molecules is of importance. These molecules strongly absorb terrestrial radiation, emitted from the earth’s surface, and they can also be excited by some heat transfer in the atmosphere. The absorbed energy is reradiated 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 modification 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.

Therefore, 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 2, we start from Einstein’s basic quantum-theoretical considerations of radiation [1] and Planck’s radiation law [2] to investigate the interaction of molecules with their own thermal background radiation under the influence of molecular collisions and at thermodynamic equilibrium [3, 4]. We show that the thermal radiation of a gas can be well understood only in terms of the spontaneous emission of the molecules. This is valid at low pressures with only few molecular collisions as well as at higher pressures and high collision rates.

In Section 3, also the influence of radiation from an external blackbody radiator and an additional excitation by a heat source is studied. The radiation and heat transfer 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. These examples right away illustrate the conversion of heat to radiation and vice versa.

In Section 4, we derive the Schwarzschild equation [5–11] as the fundamental relation for the radiation transfer in the atmosphere. This equation is deduced from pure considerations on a molecular basis, describing the thermal radiation of a gas as spontaneous emission of the molecules. This equation is investigated under conditions of only few intermolecular collisions as found in the upper mesosphere or mesopause as well as at high collision rates as observed in the troposphere. Following some modified considerations of Milne [12], 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. This equation is derived for the spectral radiance as well as for the spectral flux density (spectral intensity) as the solid angular integral of the radiance.

In Section 5, 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, reflecting the different contributions of these gases under quite realistic conditions in the atmosphere. In particular, they show the dominant influence of water vapour over the full infrared spectrum, and they explain why a further increase in the CO_{2} concentration only gives marginal corrections in the radiation budget.

It is not the objective of this paper to explain the fundamentals of the atmospheric greenhouse effect or to prove its existence within this framework. Nevertheless, the basic considerations and derived relations for the molecular interaction with radiation have some direct significance for the understanding and interpretation of this effect, 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 . 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 finally results in a population of the molecular states, given by Boltzmann’s distribution.

Such interaction first considered by Einstein [1] is replicated in the first part of this section with some smaller modifications, but following the main thoughts. In the second part of this section, also collisions between the molecules are included and some basic consequences for the description of the thermal bath are derived.

##### 2.1. Einstein’s Derivation of Thermal Radiation

The molecules are characterized by a transition between the energy states and with the transition energy where is Planck’s constant, the vacuum speed of light, the transition frequency, and is the transition wavelength (see Figure 1).

*Planckian Radiation*. The cavity radiation of a black body at temperature can be represented by its spectral energy density (units: J/m^{3}/*μ*m), which obeys Planck’s radiation law [2] with
or as a function of frequency assumes the form
This distribution is shown in Figure 2 for three different temperatures as a function of wavelength. is the refractive index of the gas.

*Boltzmann’s Relation*. Due to Boltzmann’s principle, the relative population of the states and in thermal equilibrium is [3, 4]
with and as the population densities of the upper and lower state, and the statistical weights representing the degeneracy of these states, as the Boltzmann constant, and as the temperature of the gas.

For the moment, neglecting any collisions of the molecules, between these states three different transitions can take place.

*Spontaneous Emission*. The spontaneous emission occurs independent of any external field from and is characterized by emitting statistically a photon of energy into the solid angle with the probability within the time interval is the Einstein coefficient of spontaneous emission, sometimes also called the spontaneous emission probability (units: s^{−1}).

*Induced Absorption*. With the molecules subjected to an electromagnetic field, 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 and increases its internal energy by this amount, while the radiation energy is decreasing by the same amount.

The probability for this process is found by integrating over all frequency components within the interval , contributing to an interaction with the molecules:
This process is known as induced absorption with as Einstein’s coefficient of induced absorption (units: m^{3}·Hz/J/s), as the spectral energy density of the radiation (units: J/m^{3}/Hz) and as a normalized lineshape function which describes the frequency dependent interaction of the radiation with the molecules and generally satisfies the relation:
Since is much broader than , it can be assumed to be constant over the linewidth, and with (7), the integral in (6) can be replaced by the spectral energy density on the transition frequency:
Then the probability for induced absorption processes simply becomes

*Induced Emission*. A transition from caused by the radiation is called the induced or stimulated emission. The probability for this transition is
with as the Einstein coefficient of induced emission.

*Total Transition Rates*. Under thermodynamic equilibrium, the total number of absorbing transitions must be the same as the number of emissions. These numbers depend on the population of a state and the probabilities for a transition to the other state. According to (5)–(10), this can be expressed by
or more universally described by rate equations:
For , (12) gets identical to (11). Using (4) in (11) gives
Assuming that with also gets infinite, and must satisfy the relation:
Then (13) becomes
or resolving to gives
This expression in Einstein’s consideration is of the same type as the Planck distribution for the spectral energy density. Therefore, comparison of (3) and (16) at gives for :
showing that the induced transition probabilities are also proportional to the spontaneous emission rate , and in units of the photon energy, , are scaling with .

##### 2.2. Relationship to Other Spectroscopic Quantities

The Einstein coefficients 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 coefficient of a sample.

###### 2.2.1. Cross Section and Absorption Coefficient

Radiation propagating in -direction through an absorbing sample is attenuated due to the interaction with the molecules. The decay of the spectral energy obeys Lambert-Beer’s law, here given in its differential form:
where is the absorption coefficient (units: cm^{−1}) and is the cross section (units: cm^{2}) for induced absorption.

For a more general analysis, however, also emission processes have to be considered, which partly or completely compensate for the absorption losses. Then 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, (18) has to be expanded by two terms representing the induced and also the spontaneous emission. Quite similar to the absorption, the induced emission is given by the cross section of induced emission , the population of the upper state , and the spectral radiation density . The product now describes an amplification of and is known as gain coefficient.

Additionally, spontaneously emitted photons within a considered volume element and time interval contribute to the spectral energy density of the thermal background radiation with Then altogether this gives As we will see in Section 2.5 and later in Section 3.3 or Section 4, (19) is the source term of the thermal background radiation in a gas, and (20) already represents the theoretical basis for calculating the radiation transfer of thermal radiation in the atmosphere.

The frequency dependence of and , and thus the resonant interaction of radiation with a molecular transition can explicitly be expressed by the normalized lineshape function as
Equation (20) may be transformed into the time domain by and additionally integrated over the lineshape . When can be assumed to be broad compared to , the energy density as the integral over the lineshape of width becomes
This energy density of the thermal radiation (units: J/m^{3}) can also be expressed in terms of a photon density in the gas, multiplied with the photon energy with
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
which is identical with the balance in (12). Comparison of the first terms on the right side and applying (17) gives the identity
and therefore
Comparing the second terms in (12) and (24) results in
So, together with (21) and (27), we derive as the final expressions for and :

###### 2.2.2. Effective Cross Section and Spectral Line Intensity

Often the first two terms on the right side of (20) are unified and represented by an effective cross section . Further relating the interaction to the total number density of the molecules, it applies and becomes Integration of (30) over the linewidth gives the spectral line intensity of a transition (Figure 3): as it is used and tabulated in data bases [13, 14] to characterize the absorption strength on a transition.

###### 2.2.3. Effective Absorption Coefficient

Similar to , with (30) and (31), an effective absorption coefficient on a transition can be defined as which after replacing from (17) assumes the more common form:

##### 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. The transition rate from due to de-exciting, nonradiating collisions (superelastic collisions, of 2nd type) may be called and that for transitions from (inelastic collisions, of 1st type) as exciting collisions , respectively (see Figure 4).

###### 2.3.1. Rate Equations

Then, with (25) and the abbreviations and as radiation induced transition rates or transition probabilities (units: s^{−1})
the rate equations as generalization of (12) or (24) and additionally supplemented by the balance of the electromagnetic energy density or photon density (see (22)–(24)) assume the form:
At thermodynamic equilibrium, the left sides of (35) are getting zero. Then, also and even particularly in the presence of collisions the populations of states and will be determined by statistical thermodynamics. So, adding the first and third equation of (35), together with (4), it is found some quite universal relationship for the collision rates
showing that transitions due to inelastic collisions are directly proportional to those of superelastic collisions with a proportionality factor given by Boltzmann’s distribution. From (35), 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.

###### 2.3.2. Radiation Induced Transition Rates

When replacing in (34) by (3), the radiation induced transition rates can be expressed as
Inserting some typical numbers into (37), for example, a transition wavelength of 15 for the prominent -absorption band and a temperature of K, we calculate a ratio Assuming that , almost the same is found for the population ratio (see (4)) with . At spontaneous transition rates of the order of s^{−1} for the stronger lines in this -band then we get a radiation induced transition rate of only s^{−1}.

Under conditions as found in the troposphere with collision rates between molecules of several 10^{9} s^{−1}, any induced 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 absorption and emission processes per volume, scaling with the population density of the involved states (see (35)), can be quite significant. So, at a concentration of 400 ppm, the population in the lower state is estimated to be about m^{−3} (dependent on its energy above the ground level). Then more than 10^{19} absorption processes per m^{3} are 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

The linewidth of an optical or infrared transition is determined by different effects.

###### 2.4.1. Natural Linewidth and Lorentzian Lineshape

For molecules in rest and without any collisions and also neglecting power broadening due to induced transitions, the spectral width of a line only depends on the natural linewidth which for a two-level system and is essentially determined by the spontaneous transition rate . and are the lifetimes of the lower and upper state.

The lineshape is given by a Lorentzian , which in the normalized form can be written as

###### 2.4.2. Collision Broadening

With collisions in a gas, the linewidth will considerably be broadened due to state and phase changing collisions. Then the width can be approximated by where is an additional rate for phase changing collisions. The lineshape is further represented by a Lorentzian, only with the new homogenous width (FWHM) according to (40).

###### 2.4.3. Doppler Broadening

Since the molecules are at an average temperature , they also possess an average kinetic energy with as the mass and as the velocity of the molecules squared and averaged. Due to a Doppler shift 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. This inhomogeneous broadening is determined by Maxwell’s velocity distribution and known as Doppler broadening. The normalized Doppler lineshape is given by a Gaussian function of the form: with the Doppler linewidth where is the molecular weight in atomic units and specified in .

###### 2.4.4. Voigt Profile

For the general case of collision and Doppler broadening, a convolution of and gives the universal lineshape representing a Voigt profile of the form:

##### 2.5. Spontaneous Emission as Thermal Background Radiation

From the rate equations (35), 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. This is also a consequence of (36) indicating that with an increasing rate for de-exciting transitions (without radiation), also the rate of exiting collisions is growing and just compensating for any losses, even when the branching ratio 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 , but at the same time, the number of occurring decays per time increases with .

Then the spectral power density in the gas due to spontaneous emissions is (see (19)) representing a spectral generation rate of photons of energy per volume. Photons emerging from a volume element generally spread out into neighbouring areas, but in the same way, there is a backflow 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 where is the mean free path of a photon in the gas before it is absorbed, we can write for the spectral energy density: The same result is derived when transforming (20) into the time domain and assuming a local thermodynamic equilibrium with : With from (33) and Boltzmann’s relation (4), then the spectral energy density at is found to be This is the well-known Planck formula (3) and shows that 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.

This derivation differs insofar from Einstein’s consideration leading to (16), as he concluded that a radiation field, 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 and rightfully defined by the spontaneous emission of the molecules themselves. This 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 , 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 flux 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

The 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:
with as the spectral radiance (units: W/m^{2}/Hz/sterad) and as the temperature of the emitting surface of the source (e.g., earth’s surface). The cosine term accounts for the fact that for an emission in a direction given by the azimuthal angle and the polar angle , only the projection of perpendicular to this direction is efficient as radiating surface (Lambertian radiator).

###### 3.1.2. Spectral Flux Density—Spectral Intensity

Integration over the solid angle gives the spectral flux density. Then, representing in spherical coordinates as with * as *the azimuthal angle interval and * as *the polar interval (see Figure 5), this results in:
or in wavelengths units

and , also known as spectral intensities, are specified in units of W/m^{2}/*μ*m and W/m^{2}/Hz, respectively. They represent the power flux per frequency or wavelength and per surface area into that hemisphere, which can be seen from the radiating surface element. The spectral distribution of a Planck radiator of K as a function of wavelength is shown in Figure 6.

##### 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 (18) for the spectral energy density . The same holds for the spectral radiance with where is the propagation distance of in the sample. This is valid independent of the chosen coordinate system. The letter is used when not explicitly a propagation 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 different propagation directions and brightness, some basic deviations have to be recognized. So, because of the individual propagation directions spreading over a solid angle of , 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 , is better characterized by its average expansion perpendicular to the layer surface in -direction. This means, that with respect to this direction an individual ray, propagating under an angle to the surface normal covers a distance before leaving the layer (see Figure 7). Therefore, 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 due to Lambert’s law.

This means that to first order, each individual beam direction suffers from the same absolute attenuation, and particularly the weaker rays under larger propagation angles waste relatively more of their previous spectral radiance. Thus, especially at higher absorption strengths and longer propagation lengths, the initial Lambertian distribution will be more and more modified. A criterion for an almost unaltered distribution may be that for angles the inequality is satisfied. The absorption loss for the spectral intensity as the integral of the spectral radiance (see (51)) then is given by The cosine terms under the integral sign just compensate and (54) can be written as This differential equation for the spectral intensity shows that the effective absorption coefficient 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. This last statement means that we also can assume radiation, which is absorbed at the regular absorption coefficient , but is propagating as a beam under an angle of to the surface normal . In practice, it even might give sense to deviate from an angle of , to compensate for deviations of the earth’s or oceanic surface from a Lambertian radiator, and to account for contributions due to Mie and Rayleigh scattering.

The absorbed spectral power density (power per volume and per frequency) with respect to the -direction then is found to be with as the initial spectral intensity at as given by (51).

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 to be small compared to the earth’s radius (consideration of an extended radiating parallel plane).

Integration of (56) over the lineshape within the spectral interval gives the absorbed power density (units: W/m^{3}), which similar to (23) may also be expressed as loss or annihilation of photons per volume ( in m^{−3}) and per time. Since the average propagation speed of in -direction is , also the differentials transform as . With from (51), this results in
or for the photon density in terms of the induced transition rates and caused by the external field:
with as an averaged absorption coefficient over the linewidth (see also (32)) and with
Similar to (37), these rates are again proportional to the spontaneous transition probability (rate) , but now they depend on the temperature of 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, different to (37), a factor of appears in this equation.

At a temperature K and wavelength *μ*m, for example, the ratio of induced to spontaneous transitions due to the external field at is
For the case of a two-level system, can also be expressed by the natural linewidth of the transition with (see (38))
Then (59) becomes
With a typical natural width of the order of only 0.1 Hz in the 15 *μ*m band of , then the induced transition rate will be less than 0.01 s^{−1}. Even on the strongest transitions around 4.2 *μ*m, the natural linewidths are only ~100 Hz, and therefore the induced transition rates are about 10 s^{−1}.

Despite of these small rates, the overall absorption of an incident beam can be quite significant. In the atmosphere, the greenhouse gases are absorbing on hundred thousands of transitions over long propagation lengths and at molecular number densities of 10^{19}–10^{23} per m^{3}. On the strongest lines in the 15 *μ*m band of , the absorption coefficient at the center of a line even goes up to . Then already within a distance of a few m, the total power will be absorbed on these frequencies. So, altogether about 85% of the total IR radiation emerging from the earth’s surface will be absorbed by these gases.

###### 3.2.3. Alternative Calculation for the Spectral Intensity

For some applications, it might be more advantageous, first to solve the differential equation (53) for the spectral radiance as a function of and also , before integrating over .

Then an integration in -direction over the length with a -dependent absorption coefficient gives and a further integration over results in First integrating over and introducing the optical depth as well as the substitution , we can write with (see (51)) and the exponential integral Since the further considerations in this paper are concentrating on the radiation transfer in the atmosphere under the influence 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 (55).

##### 3.3. Rate Equations under Atmospheric Conditions

As a further generalization of the rate equations (35), in this subsection, we additionally consider the influence of the external radiation, which together with the thermal background radiation is acting on the molecules in the presence of collisions (see Figure 8).

And different to Section 2, the infrared active molecules are considered as a trace gas in an open system, the atmosphere, which has to come into balance with its environment. Then 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. This is a consequence of energy conservation.

The radiation loss is assumed to be proportional to the actual photon density and scaling with a flux rate . This 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 (see (57)), and it can also be replaced by thermal energy. Therefore, 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 flux, 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 effective excitation over some longer volume element. Integrating (55) over the linewidth while using the definitions of (31) and (32) and then integrating over results in 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 define the penetration depth as the length , over which the initial intensity reduces to and thus the exponent in (67) gets unity with From (68), also the molecular number density is found, at which the initial intensity just drops to after an interaction length . Since characterizes 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 and also refer to the total number density of the gas.

Since at constant pressure, the number density in the gas is changing with the actual temperature due to Gay-Lussac’s law and the molecules are distributed over hundreds of states and substates, the molecular density, contributing to an interaction with the radiation on the transition , is given by [15]
where is the molecular number density at an initial temperature , is the energy of the lower level above the ground state, and is the total internal partition sum, defined as [15, 16]
In the atmosphere typical propagation, lengths are of the order of several km. So, for a stronger CO_{2} transition in the 15 *μ*m band with a spontaneous rate , , a spectral line intensity or integrated cross section of
and a spectral width at ground pressure of , the saturation density over a typical length of assumes a value of molecules/m^{3}. Since the number density of the air at 1013 hPa and 288 K is 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. This means that to first order, also the saturation density decreases with , while the gas concentration in the atmosphere, at which saturation appears, almost remains constant. The same applies for the penetration depth.

For the further considerations, it is adequate to introduce an average spectral intensity as or equivalently an average photon density which characterizes the incident radiation with respect to its average excitation over in the rate equations. In general, the incident flux onto an atmospheric layer consists of two terms, the up- and the downwelling radiation.

Any heat flux, supplied to the gas volume, contributes to a volume expansion, and via collisions and excitation of molecules, this can also be transferred to radiation energy. Simultaneously absorbed radiation can be released as heat in the gas. Therefore, the rate equations are additionally supplemented by a balance for the heat energy density of the air [J/m^{3}], which under isobaric conditions and making use of the ideal gas equation in the form can be written as
or after integration
Here represents the specific heat capacity of the air at constant pressure with , where are the degrees of freedom of a molecule (for N_{2} and O_{2}: ) and is the universal gas constant (at room temperature: ). is the specific weight of the air (at room temperature and ground pressure: kg/m^{3}), which can be expressed as with as the mass of an air molecule, the mol weight, Avogadro’s number, the number density, and the pressure of the air.

The thermal energy can be supplied to a volume element by different means. So, thermal convection and conductivity in the gas contribute to a heat flux [J/(s·m^{2})], causing a temporal change of proportional to , or for one direction proportional to . is a vector and points from hot to cold. Since this flux is strongly dominated by convection, it can be well approximated by with as the heat transfer coefficient, typically of the order of –15 W/m^{2}/K. At some more or less uniform distribution of the incident heat flux over the troposphere (altitude km), the temporal change of due to convection may be expressed as with mW/m^{3}/K.

Another contribution to the thermal energy originates from the absorbed sunlight, which in the presence of collisions 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 [J/(s·m^{3})].

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 .

Altogether, this results in a set of coupled differential 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 and/or a neighbouring layer, this all in the presence of collisions and under the influence of heat transfer processes: To symbolize the mutual coupling of these equations, here we use a notation where the radiation induced transition rates are represented by the Einstein coefficients and the respective photon densities. A change to the other representation is easily performed applying the identities: It should be noticed that the rate equation for could also be replaced by the incident radiation, as given by (73). However, since approaches its equilibrium value within a time constant short compared with other processes, for a uniform representation the differential form was preferred.

For the special case of stationary equilibrium with from the first rate equation, we get for the population ratio:
which at and due to (see (37) and (60)) in the limit of pure spontaneous decay processes reaches its maximum value of 4.4%, while in the presence of collisions with collision rates in the troposphere of several 10^{9} s^{−1}, it rapidly converges to 3.5%, given by the Boltzmann relation at temperature and corresponding to a local thermodynamic equilibrium.

For the general case, the rate equations have to be solved numerically, for example, by applying the finite element method. While (76) 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 the earth’s surface to the atmosphere. Since the main trace gases CO_{2}, 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 effective number 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 reflects the dynamics and time evolution of the molecular populations under the influence of the incident radiation, the background radiation, and the thermal heat transfer. 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 troposphere, more precisely for a layer from ground level up to 100 m, is represented in Figure 9. The graphs show the evolution of the photon densities and , the population densities and of the states and , the accumulated heat density in the air, and the temperature of the gas (identical with the atmospheric temperature) as a function of time and as an average over an altitude of 100 m.

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**(e)**

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As initial conditions we assumed a gas and air temperature of , a temperature of the earth’s surface of , an initial heat flux due to convection of , a latent heat source of , and an air pressure of . The interaction of the terrestrial radiation with the infrared active gases is demonstrated for CO_{2} with a concentration in air of 380 ppm corresponding to a molecular number density of at 288 K*.* The simulation was performed for a “standard transition” as discussed before with a transition wavelength , a spontaneous transition rate s^{−1}, statistical weights of and , a spectral width of , a collisional transition rate s^{−1} (we do not distinguish between rotational and vibrational rates [10]), a penetration depth , and assuming a photon loss rate of s^{−1}. For this calculation, a density and temperature dependence of and was neglected. Since under these conditions a single transition contributes 0.069 W/m^{2} to the total IR-absorption of 212 W/m^{2} over 100 m, transitions represent the radiative interaction with all active gases in the atmosphere.

The evolution for another gas and transition can easily displayed by replacing the respective parameters for the other transition and recalibrating the effective transition number.

To ensure reproducible results and to avoid any instabilities, the time interval for the stepwise integration of the coupled differential equation system has to be chosen small enough to avoid changes comparable to the size of a computed quantity itself, or the calculated changes have to be restrict by some upper boundaries, taking into account some smaller repercussions on the absolute time scale. The latter procedure was applied which accelerates the calculations by several orders of magnitude.

A simulation is done in such a way, that starting from the initial conditions for the populations, the photon densities, and the temperature, in a first step, modified population and photon densities as well as a change of the heat density over the time interval are determined. The heat change causes a temperature change (see (74)) and gives a new temperature, which is used to calculate a new collision transfer rate by (36), a corrected population by (69) and a new population for the upper state by (4). The total internal partition sum for CO_{2} is deduced from data of the HITRAN-database [13] and approximated by a polynomial of the form:
The new populations are used as new initial conditions for the next time step, over which again the interaction of the molecules with the radiation and any heat transfer is computed. In this way, the time evolution of all coupled quantities shown in Figure 9 is derived. For this simulation, it was assumed that all atmospheric components still exist in gas form down to 40 K.

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

Population changes of the CO_{2} states, induced by the external or the thermal background radiation, are coupled to the heat reservoir via the collisional transition rates and (see last rate equation), which are linked to each other by (36). As long as the calculated populations differ from a Boltzmann distribution at the gas-temperature , the two terms in the parenthesis of this rate equation do not compensate, and their difference contributes to an additional heating, amplified by the effective number of transitions. Reasons for smaller deviations from a local thermal equilibrium can be the induced population changes due to the external radiation, as well as temperature and density variations with altitude and daytime, or some local effects in the atmosphere. Also vice versa, when the gas is supplied with thermal energy, the heat can be transferred to electromagnetic energy and reradiated by the molecules (radiative cooling).

The photon density (Figure 9(a)) shows the spontaneous emission of the molecules and determines the thermal background radiation of the gas. It is mostly governed by the population of the upper state and in equilibrium is self-adjusting on a level where the spontaneous generation rate is just balanced by the induced transition rates (optical and collision induced) and the loss rate , the latter dependent on the density and temperature variations in the atmosphere.

Figure 9(b) represents the effective photon density available to excite the gas over the layer depth by terrestrial radiation and back radiation from the overlaying atmosphere. The downwelling part was assumed to originate from a layer of slightly lower temperature than the actual gas temperature with a lapse rate also varying with . The curve first declines, since with slightly increasing temperature, the lower state, which is not the ground state of CO_{2}, is stronger populated (see also Figure 9(c)), and therefore the absorption on this transition increases, while according to saturation effects, the effective excitation over decreases. With further growing temperature again increases, caused by the decreasing gas density and also reduced lower population, accompanied by a smaller absorption.

Figures 9(c) and 9(d) show the partial increase as well as the depletion of the lower state and the simultaneous refilling of the upper state due to the radiation interaction and thermal heating. The populations are displayed in relative units, normalized to the CO_{2} number density at 288 K.

The heat density of the air and the gas (air) temperature, are plotted in Figures 9(e) and 9(f). Up to about 80 K (first 70 h), heating is determined by convection, while at higher temperatures radiative heating more and more dominates. An initial heat flux of 3 kW/m^{2} seems relatively large but quickly reduces with increasing temperature and at a difference of 3 K between the surface and lower atmosphere, similar to latent heat, does not contribute to more than 40 W/m^{2}, representing only 10% of the terrestrial radiation.

In principle, it is expected that a gas heated by an external source cannot get warmer than the source itself. This is valid for a heat flux caused by convection or conduction in agreement with the second law of thermodynamics, but it leads to some contradiction with respect to a radiation source.

In a closed system as discussed in Section 2, thermal radiation interacting with the molecules is in unison with the population of the molecules, and both the radiation and the gas can be characterized by a unique temperature . In an open system with additional radiation from an external source of temperature , the prevailing radiation, interacting with the molecules, consists of two contributions, which in general are characterized by different temperatures and different loss processes. So, determines the spectral intensity and distribution of the incident radiation and in this sense defines the power flux which can be transferred to the molecules, when this radiation is absorbed. But it does not describe any direct heat transfer, which always requires a medium for the transport and by no means defines the temperature of the absorbing molecules. At thermal equilibrium and in the presence of collisions, the molecules are better described by a temperature, deduced from the population ratio given by the Boltzmann relation (4). This temperature also determines the spontaneous emission and together with the losses defines the thermal background radiation.

Since an absorption on a transition takes place as long as the population difference is positive (equal populations signify infinite temperature), molecules can be well excited by a thermal radiation source, which has a temperature lower than that of the absorbing gas. Independent of this statement, it is clear that at equilibrium the gas can only reradiate that amount of energy that was previously absorbed from the incident radiation and/or sucked up as thermal energy.

In any way, for the atmospheric calculations, it seems reasonable, to restrict the maximum gas temperature to values comparable to the earth’s surface temperature. An adequate condition for this is a radiation loss rate as derived from the rate equations at stationary equilibrium and with up- and downwelling radiation of the same magnitude.

A calculation for conditions as found in the upper stratosphere at about 60 km altitude is shown in Figure 10. The air pressure is 0.73 hPa, the regular temperature at this altitude is 242.7 K, and the number density of air is m^{−3}. The density of CO_{2} reduces to m^{−3} at a concentration of 380 ppm. At this lower pressure, the collision induced linewidth is three orders of magnitude smaller and declines to about *.* Under these conditions, the remaining width is governed by Doppler broadening with 34 MHz. At this altitude, any terrestrial radiation for an excitation on the transition is no longer available. The only radiation interacting with the molecules originates from neighbouring molecules and the own background radiation.

**(a)**

**(b)**

**(c)**

**(d)**

**(e)**

**(f)**

As initial conditions we used a local gas and air temperature K, slightly higher than the environment with K, the latter defining the radiation (up and down) from neighbouring molecules. The heat flux from or to adjacent layers was assumed to be negligible, while a heat well due to sun light absorption by ozone with ~15 W/m^{2} over ~50 km in this case is the only heat source with mW/m^{3}. The calculation was performed for a collisional transition rate , a photon loss rate of s^{−1}, a layer depth of km, and an effective number of transitions (absorption on standard transition: W/m^{2 }; total IR-absorption over 1 km: 0.4 W/m^{2}).

Similar to Figure 9, the graphs show the evolution of the photon densities and , the population densities and of the states and , the heat density in air, and the temperature of the gas as a function of time. But different to the conditions in the troposphere, where heating of the atmosphere due to terrestrial heat and radiation transfer was the dominating effect, the simulations in Figure 10 demonstrate the effect of radiative cooling in the stratosphere, where incident radiation from adjacent layers and locally released heat is transferred to radiation and radiated to space. An increased reradiation will be observed, as represented by the upper graph till the higher temperature diminishes and the radiated energy is in equilibrium with the supplied energy flux.

Due to the lower density and heat capacity of the air at this reduced pressure, the system comes much faster to a stationary equilibrium.

##### 3.4. Thermodynamic and Radiation Equilibrium

Thermodynamic equilibrium in a gas sample means that the population of the molecular states is given by the Boltzmann distribution (4) and the gas is characterized by an average temperature . On the other hand, spectral radiation equilibrium requires that the number of absorptions in a considered spectral interval is equal to the number of emissions. While without an external radiation generally, any thermodynamic equilibrium in a sample will be identical with the spectral radiation equilibrium, in the presence of an additional field, these cases have to be distinguished, since the incident radiation also induces transitions and modifies the populations of the molecular states, deviating from the Boltzmann distribution.

Therefore, it is appropriate to investigate the population ratio more closely and to discover how it looks like under some special conditions. This is an adequate means to determine what kind of equilibrium is found in the gas sample.

For this, we consider the total balance of absorption and emission processes as given by the rate equations (76). Under stationary equilibrium conditions (see (78)), it is With (17), (34) and (59) we can write At local thermodynamic equilibrium, and are linked to each other by (36). But even more generally, we can conclude that (36) is valid, as long as the molecules can be described by a Maxwellian velocity distribution and can be characterized by an average temperature (see also [12]).

Thus, applying (36) and introducing some abbreviations: after elementary rearrangement, (82) becomes: It is easy to be seen that for large and thus high collision rates, (84) approaches to a Boltzmann distribution. Also for (no collisions) and (no external radiation), this gives the Boltzmann relation, while for and , (84) changes to This equation shows that under conditions as found in the upper atmosphere (small collision rates), the populations are approximated by a modified Boltzmann distribution (right hand side: for ), which with increasing spectral intensity is more and more determined by the temperature of the external radiation source. Under such conditions, the normal thermodynamic equilibrium according to (4) is violated and has to be replaced by the spectral radiation equilibrium.

#### 4. Radiation Balance and Radiation Transfer in the Atmosphere

A more extensive analysis of the energy and radiation balance of an absorbing sample not only accounts for the net absorbed power from the incident radiation, as considered in (53)–(66), but it also includes any radiation originating from the sample itself as well as any re-radiation due to the external excitation. This was already investigated in some detail in Section 3.3, however, with the view to a local balance and its evolution over time. In this section, we are particularly concentrating on the propagation of thermal radiation in an infrared active gas and how this radiation is modified due to the absorption and re-emission of the gas.

Two perspectives are possible, an energy balance for the sample or a radiation balance for the in- and outgoing radiation. Here we discuss the latter case. Scattering processes are not included.

We consider the incident radiation from an external source with the spectral radiance as defined in (50). Propagation losses in a thin gas layer obey Lambert-Beer’s law. Simultaneously, this radiation is superimposed by the thermal radiation emitted by the gas sample itself, and as already discussed in Section 2.5, this thermal contribution has its origin in the spontaneous emission of the molecules. For small propagation distances in the gas, both contributions can be summed up yielding The last term is a consequence of (19) or (45) when transforming the spectral energy density of the gas to a spectral intensity and considering only that portion emitted into the solid angle interval . Equation (86) represents a quite general form of the radiation balance in a thin layer, and it is appropriate to derive the basic relations for the radiation transfer in the atmosphere under different conditions. The differential indicates that the absolute propagation length in the gas is meant.

##### 4.1. Schwarzschild Equation in Dense Gases

In the troposphere, the population of molecular states is almost exclusively determined by collisions between the molecules even in the presence of stronger radiation. Then, due to the previous discussion, a well-established local thermodynamic equilibrium can be expected, and from (47) or (48), we find with as the spectral radiance of the gas, also known as Kirchhoff-Planck-function and given by (50), but here at temperature . To distinguish between the external radiation and the radiating gas, we use the letter (from background radiation), which may not be mixed with the Einstein coefficients or . Then we get as the final result: This equation is known as the Schwarzschild equation [5–7], which describes the propagation of radiation in an absorbing gas and which additionally takes into account the thermal background radiation of the gas. While deriving this equation, no special restrictions or conditions concerning the density of the gas or collisions between the molecules have to be made, only that (36), is valid and the thermodynamic equilibrium is given. In general, this is the case in the troposphere up to the stratosphere.

*Another Derivation. *Generally, the Schwarzschild equation is derived from pure thermodynamic considerations. The gas in a small volume is called to be in local thermodynamic equilibrium at temperature , when the emitted radiation from this volume is the same as that of a blackbody radiator at this temperature. Then Kirchhoff’s law is strictly valid, which means that the emission is identical to the absorption of a sample.

The total emission in the frequency interval during the time may be . The emissivity may not be mixed with in (83). Then with the spectral radiance as given by (50) and with the absorption coefficient , it follows from Kirchhoff’s law (see, e.g., [8–11]) For an incident beam with the spectral radiance and propagating in -direction we then get: which is identical with (88).

##### 4.2. Radiation Transfer Equation for the Spectral Intensity

Normally the radiation and by this the energy, emitted into the full hemisphere, is of interest, so that (88) has to be integrated over the solid angle . As already discussed in Section 3.2, for this integration, we have to have in mind that a beam, propagating under an angle to the layer normal (see Figure 11), only contributes an amount to the spectral intensity (spectral flux density) due to Lambert’s law. The same is assumed to be true for the thermal radiation emitted by a gas layer under this angle.

On the other hand, the propagation length through a thin layer of depth is increasing with , so that the -dependence for both terms and disappears.

Therefore, analogous to (54) and (55) integration of (88) over gives for the spectral intensity, propagating in -direction:
or with the definition for and analogous :
Equation (92) is a 1st-order differential equation with the general solution:
or after some transformation:
Usually the density of the gas, the total pressure and the temperature are changing over the propagation path of the radiation. Therefore, (94) has to be solved stepwise for thin layers of thickness , over which, and can be assumed to be constant (see Figure 12). With the running index for different layers, then (94) can be simplified as follows:
The intensity in the th layer is calculated from the previous intensity of the layer with the values and of the *i*th layer. In this way, the propagation over the full atmosphere is stepwise calculated. The first term in (95) describes the transmission of the incident spectral intensity over the layer thickness, while the second term represents the self-absorption of the thermal background radiation into forward direction and is identical with the spontaneous emission of the layer into one hemisphere.

##### 4.3. Schwarzschild Equation in a Thin Gas

Equation (88) was derived under the assumption of a local thermodynamic equilibrium in the gas, which is generally the case at higher gas pressures and therefore higher collision rates of the molecules. At low pressures, however, the population of the molecular states can also significantly be determined by induced transitions caused by the incident radiation (see Section 3.4), and then it gives more sense to consider the monochromatic or spectral radiation equilibrium, which means that the gas re-emits all monochromatic radiation it has absorbed.

Since this absorption is caused by the total spectral radiation incident within the solid angle has to be integrated over . Then the absorbed spectral intensity over the interval is found from (51) together with (53). This must be equal to the spectral power density emitted by the gas as spontaneous emission into the full solid angle . Thus, we obtain Only of (96) contributes to the spectral radiance, and (86) changes to With , (97) becomes This differential equation for the spectral radiance in thin gases was derived, only assuming the absorption and subsequent emission due to the external radiation and in this respect orientated at the standard considerations as found in the literature (see, e.g., [12]). A more rigorous derivation of this equation, however, also has to take into account the emission originating from the thermal background radiation of the molecules, as long as their temperature and thus their kinetic energy are not zero. This is considered in the next subsection.

*Extended Schwarzschild Equation in a Thin Gas.* We start from (86), which already describes quite generally the propagation of radiation under conditions when a sample not only absorbs but also emits on the same frequencies as the incident radiation. Again requesting radiation equilibrium, the total balance of absorption and emission processes is given by the rate equations (76), which under stationary conditions results in (see (81))
With (34) and (59) we find:
Some rearrangement of (100) and multiplication with gives
At local thermodynamic equilibrium, we know that due to (36), the term will be zero, and trivially it is also zero in the absence of any collisions . Thus, we can expect that in a thin gas with only few collisions even under radiation equilibrium, this term is quite small and therefore can be neglected. The general case, including this term, will be discussed in the next paragraph. So, with (33) and dividing (101) by as well as using the relations and , we get