Saturation of the Infrared Absorption by Carbon Dioxide in the Atmosphere
Dieter Schildknecht
physics.ao-ph





















    #1.3ex
        #1-.75em1ex∼<>Dieter SchildknechtThe Saturation of the Infrared Absorption by Carbon Dioxide in the
        Atmosphere
        
        
        Fakultät für Physik, Universität Bielefeld 
         
        D-33501 Bielefeld, Germany
        
        schild@physik.uni-bielefeld.de SATURATION OF THE INFRARED ABSORPTION
        
        BY CARBON DIOXIDE IN THE ATMOSPHERE
            DIETER SCHILDKNECHT
            December 30, 2023. .../latex/6he-sdm-2/6he-sdm-2-3.tex
        ==========================================================================
        
        Day Month YearDay Month Year
        
        
        Based on new radiative transfer numerical evaluations, we reconsider an
        argument presented by Schack in 1972 that says that saturation of
        the absorption of infrared radiation by carbon dioxide in the atmosphere
        sets in as soon as the relative concentration of carbon dioxide
        exceeds a lower
        limit of approximately 300 ppm. We provide a concise brief and explicit
        representation of the greenhouse effect of the earth's atmosphere.
        We find   an equilibrium climate sensitivity (temperature increase  
        Δ T  due to doubling of atmospheric CO_2  
        concentration) of  Δ T  ≃  0.5 ^0C. We elaborate on the consistency of these results on Δ T with results observationally obtained by satellite-based measurements of short-time radiation-flux versus surface-temperature changes. 
        
        
        § INTRODUCTION
        
        
        In view of the immense literature concerning the effect of an
        (anthropogenic) increase in the relative atmospheric carbon dioxide
        concentration on the
        earth's climate, it is  appropriate to revive a simple
        argument on this subject presented by Schack<cit.> in 1972.
        The three-page
        article published in German in the journal of the German Physical Society,
        at that time called “Physikalische Blätter”, appeared under the title
        “Der Einfluß des Kohlendioxid-Gehalts der Luft auf das Klima der Welt”,
        i.e. “The influence of the carbon dioxide content of the air on the climate
        of the world”. The author, Schack, an expert on heat transfer
        in combustion devices, in 1923 had discovered the role played
        by the interaction of infrared electromagnetic radiation with carbon dioxide,
        CO_2, and water vapor, H_2O, in such devices<cit.>.
        
        The intensity of (monochromatic) radiation transversing an
        absorbing medium decreases
        exponentially with distance according to what is known as Lambert-Beer's
        law,
        
        I(z) = I (z=0) e^-κ z.
        
        Independently of the numerical value of the absorption constant κ in
        (<ref>), for sufficiently large distance, z, the limit of “complete
        absorption”, or zero intensity of the outgoing radiation, I (z →∞)
        → 0, is reached. For finite values of the exponent -κ z in (<ref>),
        the numerical value of the κ determines the distance,
        z_ϵ (κ), for “approximate saturation”,
        exp (- κ z_ϵ (κ))=ϵ≪ 1,
        or equivalently, “almost full absorption”,
        I(z=0) - I(z=z_ϵ (κ))/I(z=0) =
        1 - I(z=z_ϵ(κ))/I(z=0) = 1 - ϵ.
        
        
        In the case of CO_2 in air, the wide band absorption constant κ for
        the infrared
        electromagnetic radiation depends on the concentration, or the partial
        pressure, of the CO_2, and a natural question concerns the magnitude of
        the CO_2 concentration that leads to approximate saturation within
        the troposphere of the earth.
        
        In his 1972 article<cit.>, Schack points out that for a
        concentration of 0.03 %
        carbon dioxide in air, approximate saturation is reached within a distance
        of approximately the magnitude of the height of the troposphere. The
        absorption reaches values close to 100 % for a realistic CO_2 content
        of 0.03 %, it is concluded<cit.> that any further increase
        of (anthropogenic) CO_2 cannot lead to an appreciably stronger
        absorption of radiation, and consequently cannot affect the earth's climate.
        
        It will be useful to elaborate on the argument given by Schack in detail,
        in order to explicitly display the simplicity and generality of the
        underlying concepts that lead to a parameter-free prediction of the
        absorption of infrared radiation by CO_2.
        
        Adopting Planck's radiation law for a temperature at the surface of
        the earth, chosen as T = 293 K by Schack, and taking into account
        the well-known absorption spectrum of the CO_2 molecule,
        one finds that the radiation of
        wave lengths λ_CO_2 in the interval
        13 μ m ≤λ_CO_2≤ 17.6 μ m is relevant for the absorption
        by CO_2. The total absorption due to CO_2 in the atmosphere is
        determined by the total mass of CO_2 that is transversed by a beam of
        infrared radiation on its path from the surface at z = 0 to the upper
        end of the atmosphere, or z →∞.
        
        In the gravitational field of the earth, the pressure, p, of a gas
        decreases with increasing altitude, z, according to dp = - ρ
        g dz, where ρ denotes the density of the gas and g the
        acceleration due to gravity. For an ideal gas of temperature T, we have
        p = ρ RT/M, or ρ = pM/RT, with R being the gas constant, 
        T denoting the absolute temperature and M the molecular weight of the
        gas. The total mass per unit area transversed by a beam of infrared
        radiation on its path through the atmosphere is determined by an
        integration over the density ρ (p,T) from the surface to the upper
        end of the troposphere. The result of the integration may be represented
        in terms of an effective altitude z_0 of a fictitious atmosphere of
        homogeneous constant pressure p_0, constant temperature T and constant
        density ρ. The value of z_0 (obviously) depends on whether the
        atmosphere is treated isothermally, or rather more realistically, is described
        adiabatically.
        
        Adopting
        T =  const, the decrease of pressure with increasing altitude
        becomes equal to
        
        p(z) = p_0 exp(- Mg/RT z ),
        
        which is the well-known barometric formula. The density ρ = ρ (z)
        decreases accordingly as
        ρ (z) = M/RT p_0 exp(- z/z_0) ≡ρ_0
        exp( - z/z_0),
        
        where
        
        z_0 = RT/Mg
        denotes the scale-height of the atmosphere, the height of a fictitious
        atmosphere of homogeneous pressure p_0, temperature T and density ρ_0,
        and a mass per
        unit area,
        M_0, obtained from (<ref>),
        
        M_0 ≡∫^∞_0ρ (z) dz = ρ_0 z_0 = Mp_0/RT
        z_0.
        
        The determination of the absorption of the atmosphere is accordingly
        reduced
        to the evaluation of the absorption by a gas pipe of length z_0
        from (<ref>)
        homogeneously filled with air, and with CO_2 of partial pressure
        p_0 ≡ p_CO_2, at a total pressure of 1 atm at temperature
        T = 293 K.
        
        For the realistic case of an adiabatically treated atmosphere relation
        (<ref>) remains valid with an appropriately chosen value of z_0
        different from (<ref>). In our approach of Section 2 the value of
        z_0 will be determined from the altitude dependence of the
        absorption of infrared radiation for given lapse rate in the atmosphere.
        
        The absorption due to (<ref>), upon introducing the
        dependence on the CO_2 concentration via the partial pressure
        p_CO_2, becomes
        
        A (p_CO_2, z_0) = 1 - exp( -a p_CO_2z_0 ),
        
        where the coefficient a must be determined from the known
        absorption spectrum of CO_2. Since the absorption cross section of
        CO_2 depends on the wave length of the radiation being absorbed,
        the absorption constant a in (<ref>), obtained by summation over a spectrum
        of a large number of spectral lines,  actually develops a
        dependence on the partial pressure p_CO_2.
        
        
        
        In his article<cit.>, addressing the question on the effect
        of an increase of the CO_2
        concentration on the climate on earth,
        Schack compares the absorption due to CO_2, 
        by a CO_2-air pipe
        as mentioned, for a partial pressure
        corresponding to a realistic value of 300 ppm = 0.03 % with the absorption
        due to an enhanced value taken as
         600 ppm = 0.06 %[ A realistic
        value for the increase of the CO_2 concentration per year is 2 ppm/year, e.g.<cit.>, 
        implying 150 years for the doubling of a CO_2 concentration
        of 300 ppm.], respectively
        corresponding to p_CO_2 = 3 × 10^-4 atm and p_CO_2 =
        6 × 10^-4 atm  at T = 293 K and
        a total pressure of 1 atm of the atmosphere. Adopting
        z_0 = 7000 m for the equivalent height of the troposphere, for the absorption,
        A [%], and the increase of absorption, Δ_2 × CO_2,
        Schack obtains the
        results that are given in Table 1.
        
        
        
        
        Saturation of infrared radiation by the realistic CO_2 content of the
        atmosphere of 0.03 % being reached within approximately 1 %
        according to Table 1, any further
        increase of the CO_2 content
        does not substantially increase the absorption of
        radiation, and accordingly does not affect<cit.> the
        climate on earth.
        
        The results on Δ_2 × CO_2 [%] in Table 1 refer to
        the radiation in the CO_2 band of 13 μ m λ_CO_2 17.6 μ m  (568 cm^-1 1/λ_CO_2 769
         cm^-1). With the assumption of a black-body radiation at
        T = 293 K, the input radiation in the CO_2 band amounts to
        87 W/m^2. The additional absorption due to CO_2 doubling is
        given by Δ_2 × CO_2× 87 W/m^2 = 0.8 ×
        10^-2× 87 W/m^2 = 0.70 W/m^2, as listed in Table 1.
        This increase of the absorption of infrared radiation in the
        atmosphere is indeed negligible in comparison with the equilibrium
        value of the total outgoing long-wave infrared radiation (LWIR) at
        the top of the atmosphere (TOA) given by S_0 = 239 W/m^2.
        [The known empirical
        value of the sun radiation of
        1365 W/m^2 with an albedo of 0.3 yields 956 W/m^2 absorption. Dividing
        by 4 to take into account the surface of the earth yields the average value
        of the radiation from the sun to be balanced by the outgoing average LWIR
        of S_0 = 239  W/m^2. <cit.> A 1 % decrease of the albedo to become ϵ = 0.297 yields S_0 = 240 W/m^2 to be used subsequently.]
        
        § ABSORPTION OF INFRARED RADIATION: CO_2-AIR PIPE VERSUS ATMOSPHERE
        
        
        Using presently available detailed data on the spectral absorption lines
        of CO_2, compare<cit.>, the absorption coefficient[Compare
        e.g. ref.<cit.> for a recent evaluation of absorption coefficients]
        a and the
        resulting absorption by a CO_2-air pipe, defined before, can be reliably
        predicted. In this Section, we compare the results on absorption by a
        CO_2-air pipe at fixed temperature and pressure, as a function of the
        length of the CO_2-air pipe, with the absorption by an adiabatically
        treated atmosphere. The results to be presented are due to a recent
        evaluation by Clark<cit.> for the ^12C ^160_2 molecule
        only, with linestrengths > 10^-23. We refer to Appendix A for further
        details.
        
        
        
        The results<cit.> for the absorption of a CO_2-air pipe for the
        spectral range of 13 μ m λ_CO_2 17.6 μ m
        (568 cm^-1 to 769 cm^-1), as a function of selected values of
        the path length (pipe length) and the relevant CO_2 concentrations
        of 0.03 % and 0.06 % are shown in Table 2. The pipe is assumed to have
        constant temperature, T = 293 K, and a total pressure of p = 1 atm. The
        input black-body infrared radiation at T = 293 K is accordingly
        given by 87 Wm^-2. The results in Table 2 explicitly demonstrate
        the expected increase of the absorption A with increasing length of
        the pipe (path length), as well as the decrease of the relative
        absorption for CO_2 doubling, Δ_2 × CO_2 [%]. The
        result of Δ_2 × CO_2 = 5 % at the pipe length of 7 km is
        considerably larger than
        the 1972 result of Δ_2 × CO_2 = 0.8 % given in Table 1.
        
        
        
        In Table 3, for illustration, we show the effect of restricting
        the spectral range of the input black-body radiation to the spectral
        range of 600 cm^-1 to 750 cm^-1 (total black-body radiation
        65 Wm^-2). Comparing the results in Table 3 with the ones
        in Table 2, we notice the more rapid approach to a (larger) asymptotic
        absorption limit with increasing path length of the infrared radiation.
        This is a consequence of the change of absorbance as a function of the
        spectral wave length taken into account.
        
        In Fig. 1, we
        show the calculated spectrum of the absorbance for the case
        of a path length of 3 km and a CO_2 concentration of 0.03 % and
        0.06 %. The spectral ranges corresponding to Tables 2 and 3 are
        indicated in Fig. 1.
        
        
        Turning from fixed temperature T in the CO_2-air pipe to decreasing
        T and pressure p in the atmosphere, we proceed in two steps. In a
        first step, modelling a dry atmosphere,
        we consider the effect of CO_2 (at concentrations of
        0.03 % and 0.06 %), and in a second step, we add water vapor of
        fixed relative humidity (RH) chosen as RH = 85 %.
        
        
        
        
        In Table 4, we show the results for the absorption A of the atmosphere
        at several selected altitudes for a CO_2 content of 0.03 % and 0.06 %.
        The spectral range of the LWIR of 568 cm^-1 to 769 cm^-1 is
        identical to the spectral range employed for the results in Table 2. The
        evaluation of the absorption was carried out for a surface temperature
        of T = 293 K and a surface pressure p = 1 atm, and a realistic
        moist adiabatic lapse rate increasing with altitude, and an average
        value of -6.5 ^0C km^-1 was employed (compare Appendix A, Fig. A6). 
        The absorption spectrum at 0.01 cm^-1 resolution was calculated using a spatial resolution of 100 m. The absorption spectrum for each 100 m level was calculated using the pressure and temperature for that level derived from the lapse rate.
        The results
        in Table 4 show a rapid convergence to an approximately constant,
        quasi-asymptotic behavior of the absorption A that is reached above an
        altitude of approximately 5 km. By comparison of the results in Table 4
        with the ones in Table 2, we conclude that a CO_2-air pipe for a chosen
        length of 2 to 3 km represents the absorption of the
        atmosphere that reaches a quasi-asymptotic limit at altitudes of about 5 km.
        The values of the absorption
        and the increase of the absorption under CO_2 doubling of Δ_2 ×
        CO_2 = 7 % corresponding to 6 Wm^-2 in Tables 2 and 4 are in
        agreement with each other. Similar conclusions are drawn from the comparison
        of the results for the restricted spectral range in Tables 3 and 5 with
        each other.
        
        
        
        
        
        In Fig. 2, we show the absorption of the surface emission at
        293 K of 300 and 600 ppm of CO_2 for the dry atmosphere
        looking downwards from a height of 7 km. The total spectral
        range is from 550 to 800 cm^-1 and the spectral intervals are indicated. These are the same as for Fig. 1. The spectral resolution of the calculation was 0.01 cm^-1 with a spatial resolution of 100 m. The spectral resolution of the plot however, is 1 cm^-1. This spectrum includes the effects of the decrease in temperature and pressure with altitude as shown in Appendix A, Fig. A6. In this case most of the absorption occurs near the surface and the value of the total absorption tends towards an asymptotic value as the altitude increases. This is shown in Appendix A, Fig. A4.
        
        
        
        The results explicitly demonstrate that the CO_2-air pipe of path
        length of approximately 3 km represents the absorption in the atmosphere
        of a quasi-asymptotic altitude reached at 5 km.
        
        
        
        
        The important role of water vapor – semi-quantitatively discussed in
        ref.<cit.>– is illustrated by comparing the results in
        Tables 4 and 5 with the results presented in Tables 6 and 7. The
        results in Tables 6 and 7, in addition to CO_2, take into account
        water vapor with relative humidity (RH) of RH = 85 %. The presence
        of water vapor drastically reduces[Note that (obviously) the
        presence of water vapor increases the absorption of radiation. The presence
        of water vapor, however, decreases the effect of an increase of the
        CO_2 concentration. Compare ref.<cit.> for a semi-quantitative
        discussion.] the
        effect of doubling the CO_2 concentration from 6.09 Wm^-2 to
        about 2.6 W m^-2 for the infrared spectral range of 568 to
        769 cm^-1. A similar effect is seen by comparing the results in
        Tables 5 and 7 with each other.
        
        
        
        
        
        For comparison with Fig. 2, in Fig. 3,
        we show the absorption spectrum in the atmosphere for the presence of
        water vapor in addition to carbon dioxide.
        
        In summarizing this Section we conclude that a CO_2-air pipe of path
        length of 2 to 3 km represents the effect of the CO_2 increase in
        the atmosphere which has reached its quasi-asymptotic saturation limit
        at an altitude of approximately 5 km.
        Inclusion of water vapor drastically reduces the effect
        on the absorption of a rise of
        the CO_2 concentration from 0.03 % to 0.06 % from about 6 W m^-2
        to 3 Wm^-2. The value of 3 Wm^-2 additional absorption
        corresponding to 3/418 ≃ 0.7 % of the total outgoing
        radiation at the surface of the earth, we conclude that the doubling
        of the CO_2 concentration for the relevant time scale of about a
        century is a fairly negligible one. Even though the increase of
        absorption of Δ_2 × CO_2≃ 3 Wm^-2 is
        considerably larger than the 1972 value of Δ_2 × CO_2≃ 1 Wm^-2, the main conclusion from ref.<cit.>
        remains valid.
        
        
        
        
        § CO_2 DOUBLING AND SURFACE TEMPERATURE
        
        
        In Section 2, we concluded that the absorption of infrared radiation
        from the CO_2 band in the atmosphere becomes (approximately) independent
        of the altitude for altitudes larger than about 5 km; the absorption reaches
        a quasi-asymptotic saturation limit at an altitude of about 5 km. The
        magnitude of the quasi-asymptotic absorption agrees with the absorption
        of a CO_2-air pipe at constant temperature, T = 293 K, and constant
        pressure, p = 1 atm, and length of about 2 to 3 km.
        
        The evaluation<cit.>, compare Appendix A for details,
        of the absorption by the atmosphere is based on the known spectroscopic
        properties of the absorbing gases, H_2O, CO_2, and the empirically
        known average lapse rate of the atmosphere of about -6.5 K km^-1.
        The absorption reaches an approximately constant value at an
        altitude of 6 km. At and beyond that height, a constant value of infrared radiation is freely emitted to space.
        At this height, the atmosphere provides a “free window” (compare
        Appendix B) for the emittance of infrared radiation at
        temperature T_TOA = (293 - 6 × 6.5) K = 254 K ≃ 255 K.
        The outgoing intensity is given by S_space = S(T) |_T = 255 K =
        σ T^4 |_T = 255 = 240 Wm^-2.
        
        The approximate stability of the climate on earth, a necessary condition
        for life on earth, requires equilibrium of the outgoing infrared radiation,
        predicted by S_space = S(T) |_T = 255 K, with the effective
        radiation provided by the sun. The effective radiation by the sun amounts
        [Compare footnote 3] to S_0 = 240 Wm^-2. The agreement of
        the above parameter-free prediction of the radiative transfer calculation,
        with the effective radiation from the sun,
        
        S_space = S(T) |_T = 255 K = S_0 ,
        
        provides a non-trivial
        support of the radiative-transfer calculation based on the empirical
        lapse rate of 6.5 Km^-2. Indeed, the radiative-transfer analysis
        explains the approximate stability of the climate on earth.
        
        For a concise representation of the radiative “greenhouse effect”
        in terms of the radiation from the sun, S_0, and the earth's LWIR within
        as well as outside the “atmospheric window”, as quantified by S_W,
        and the fractional transmittance, f^'≪ 1, respectively,
        compare Appendix B.
        
        
        We turn to the effect of CO_2 doubling on the surface temperature. We
        assume the existence of radiative equilibrium of the emitted black-body LWIR,
        given by S(T), for any given CO_2 content (within the presently
        discussed limits). Absorption under doubling of the CO_2 concentration,
        Δ_2 × CO_2, for compensation requires increase of radiation
        S (T) → S (T) + Δ S, i.e.
        
        S(T) + Δ S = S(T + Δ T) = S(T) + dS/dT|_T Δ T,
        
        where Δ S ≡Δ_2 × CO_2 and T denotes the surface
        temperature, chosen as T = 293 K in our treatment. Note that the
        detailed mechanism occurring in the atmosphere leading to the required
        increase of Δ S = Δ_2 × CO_2 need not be specified. 
        According to (<ref>), the temperature has to rise by
        Δ T = Δ_2 × CO_2/( dS/dT)_T
        = T Δ_2 × CO_2/4 S(T),
        
        where S(T) ∼ T^4 for a Planck radiator is used in the last step.
        Relation (<ref>) specifies the rise of the surface temperature connected
        with restoration of equilibrium at the end of a CO_2-air pipe, or
        equivalently, at the altitude of the atmosphere where the quasi-asymptotic
        limit of the absorption has been reached. Inserting T = 293 K, and
        S(T=293K) = 418 Wm^-2, as well as
        Δ_2 × CO_2 = 6.09 W m^-2 from
        Tables 2 and 4 into (<ref>), we obtain an increase of surface
        temperature of Δ T = 1.07 K, compare Table 8. Insertion of
        Δ_2 × CO_2 = 2.6 W m^-2 from Table 6 into (<ref>)
        shows the strong effect of water vapor of RH = 85 %. The strong
        absorption due to H_2O diminishes the surface-temperature change to
        Δ T = 0.46 K for the assumed increase of the
        CO_2 concentration from 0.03 % to 0.06 %.
        
        
        
        § RADIATIVE ABSORBANCE UNDER CHANGE OF ATMOSPHERIC CONDITIONS.
        
        
        A very elaborate radiative transfer calculation, recently presented by Harde<cit.>, 
        within a two-layer climate model,
        explores[Compare refs.<cit.> for alternative
        dynamical climate models.]
        the infrared absorbance by CO_2 under a variety of atmospheric conditions,
        such as surface temperature, humidity, clouds etc. Results<cit.>
        on the radiative absorbance for CO_2 contents of 380 ppm and 760 ppm, under
        variation of the atmospheric conditions, are shown in Table 9.
        
        
        
        We note that the results
        for the absorbance in Table 9 give the absorption due to CO_2 doubling
        as a fraction of the total LWIR of S(T) = 396 Wm^-2 at T = 289 K employed
        by Harde, compared to Tables 2, 3 through 6, 7 that
        give the results for the increase of absorbance from CO_2 doubling
        as a fraction of the radiation in the CO_2 band. For a detailed
        discussion, we refer to ref.<cit.>.
        
        In the context of the present discussion, the essential and important
        point of the results in Table 9 is the approximate independence of the
        absorption due to CO_2 doubling, Δ_2 × CO_2≅ 1 %,
        from the
        various atmospheric conditions evaluated. Compare the results for
        Δ_2 × CO_2 in column 3 of Table 9. The spread of the
        results for Δ T in column 5 of Table 9 between Δ T =
        0.36 K and Δ T = 0.92 K, and the average value of Δ T
        = 0.63 K, are consistent with the result from our straight-forward
        evaluation of Δ T = 0.46 K in Table 8.
        
        The consistency or approximate agreement of our result of
        Δ T = 0.46 ≃ 0.5 K in Table 8, and the result of
        Δ T≅ 0.6 K in Table 9 requires additional
        comments. The agreement strengthens the validity of our result
        based on what may be called a single-layer (surface) model,
        particularly evident from the equivalence of the atmosphere and
        the CO_2-air pipe running at surface temperature and surface
        pressure. On the other hand, we conclude that much of the
        involved details of various flux terms of a two-layer model
        can apparently be summarized by our simple radiative equilibrium
        assumption for Δ T in (<ref>). For a supplementary discussion on Δ T we refer to the last part of Appendix B.
        
        
        
        § ON OBSERVATIONALLY TESTING THE RADIATIVE TRANSFER RESULTS
        
        
        The experimental tests of predictions on the climate sensitivity in general require decadal or even centennial time scales. In pioneering papers by Lindzen and Choi <cit.> it was pointed out that data on fluctuations of the earth's (sea-)surface temperature at short time scales, once combined with satellite-based radiation-flux measurements <cit.>, can be employed to gain important empirical information on the earth's climate sensitivity. In response to immediate reactions in the literature <cit.> on their results <cit.>, an additional more expanded and refined investigation was presented by Lindzen and Choi 
        <cit.>. Compare also ref. <cit.> for a more recent related investigation.
        
        In what follows, we shall point out that the results from combining the data of sea-surface temperature fluctuations with data on radiation-flux intensities from the ERBE <cit.> and CERES <cit.> experiments empirically support our results on Δ T from the radiative-transfer analysis given in Sections 2 to 4. We restrict ourselves to elaborating on the basic underlying assumptions and the final conclusion from them. Entering a discussion on the technical details involved in the analysis of the experimental data is far beyond our expertise.
        
        In the radiative-transfer calculations in Sections 2 and 3, we proceeded in two distinct steps. In a first step, we treated a dry atmosphere (zero humidity), in the second step the realistic case of non-zero humidity was investigated. It will be useful to introduce a notation referring to these two cases explicitly. Specializing (<ref>) to the cases of zero and non-zero humidity, we define
        Δ T^(0) = Δ^(0)_2 × CO_2/(
        dS/dT)_T≡ G (T) Δ^(0)_2 × CO_2, 
        and
        Δ T^(H) = Δ^(H)_2 × CO_2/(
        dS/dT)_T≡ G (T) Δ^(H)_2 × CO_2, 
        related to zero and non-zero humidity, respectively. We recall the numerical values of G = G (T = 293 K) = 293/4 × 418 = 0.175 KW^-1 m^2, as well as Δ^(0)_2 × C0_2 = 6.09 Wm^-2 and Δ^(H)_2 × C0_2 = 2.6 Wm^-2 given in Section 3.
        
        Identically rewriting Δ T^(H) as
        Δ T^(H) = Δ T^(0) + Δ T^(H) - Δ T^(0)≡Δ T^(0) + f Δ T^(H),
        
        we find that the commonly employed so-called feedback factor f,
        with (<ref>) and (<ref>), is given by
        
        f = 1 - Δ T^(0)/Δ T^(H) = 1 -
        Δ^(0)_2 × CO_2/Δ^(H)_2 × CO_2,
        
        or equivalently,
        Δ T^(H) = Δ T^(0)/1-f.
        
        Numerically, from (<ref>) and Table 8, we find the value of 
        f from the radiative-transfer calculation, henceforth called the theoretical value of f = f_th, to be given by
        
        f_th = 1 - 6.09/2.6 = - 1.34.
        
        The negative feedback factor, upon insertion into (<ref>), thus identically reproduces 
        Δ T^(H) = Δ T^(0)/2.34 = 1.07/2.34 = 0.46 K < 1.07 K
        
        from Table 8.
        
        Replacing Δ^(H)_2 × CO_2 in (<ref>) by
        Δ T^(H), by employing (<ref>), the factor f may be rewritten as
        
        f = 1 - G (T) Δ^(0)_2 × CO_2/Δ T^(H).
        
        According to (<ref>), the factor f is determined by dividing the radiation increase Δ^(0)_2 × CO_2 resulting from CO_2 doubling at zero humidity by the surface temperature increase Δ T^(H) necessary for equilibrium of the earth's atmosphere for CO_2 doubling in the case of non-zero humidity.
        
        Identifying Δ^(0)_2 × CO_2 / Δ T^(H) in
        (<ref>) with the experimentally measured slope (Δ  Flux/Δ T)_exp of the outgoing radiation at the TOA measured as a function of the sea-surface temperature,
        Δ^(0)_2 × CO_2/Δ T^(H) =
        ( Δ  Flux/Δ T)_exp,
        
        we find that the factor f can be represented by 
        
        f ≡ f_exp = 1 - G(T) ( Δ Flux/Δ T)_exp.
        
        Relation (<ref>) is due to Lindzen and Choi <cit.>[Compare equations (5) and (6) in ref. <cit.>. Equation (6) contains an additional factor c via f → cf introduced for sharing of tropical feedbacks over the globe. The value of c depends on how the slope Δ Flux / Δ T is extracted from measurements. A value of c = 2 is used in ref. <cit.>. For a global value of Δ Flux / Δ T one has c ≅ 1.]
        It allows one to deduce the climate sensitivity for non-zero humidity, Δ T^(H), see (<ref>), from the experimentally determined radiation flux by inserting f = f_exp according to (<ref>), and the value of Δ T = Δ
        T^(0)≃ 1.1 K for CO_2 doubling at zero humidity.
        
        Using satellite measurements of the ERBE <cit.> and CERES <cit.> collaborations, the detailed analysis in ref. <cit.> led to a value of
        ( Δ Flux/Δ T)_exp = 6.9 ±
        1.8 W m^-2 K^-1
        (see Table 2 in ref. <cit.>). Inserting this value into (<ref>), we find
        
        f_exp≅ 1 - 0.175 × 6.9 ≅ - 0.21,
        
        with 1 - f_exp≅ 1.21, and Δ T^(0)≅ 1.07 K, from Table 8, from (<ref>), we obtain
        Δ T_exp = 1.07/1.21≃ 0.9 K.
        
        Introducing the sharing factor c=2 from ref. <cit.>, compare footnote g, we have
        f_exp≅ - 0.1 and Δ T_exp becomes
        Δ T_exp = 1.07/1.1≃ 1.0 K.
        
        
        Lindzen and Choi, <cit.>, employing c = 2,
        find f_exp = -0.5 and consequently Δ T_exp≅ 0.7 K (confidence interval 0.5 K to 1.3 K at 99 % confidence level).[The difference between our result of f_exp = - 0.1 for c = 2 and Δ T_exp≃ 1 K, and the result from ref. <cit.>, namely f_exp≃ - 0.5 for c = 2 and Δ T_exp≃ 0.7 K, is due to the
        difference between our value of G = G (T = 293 K) ≅ 0.175 KW^-1 m^2 and G (T = 255 K) ≅ 0.3 used in ref.
        <cit.>, or, equivalently C^(0)_2 × CO_2 = 6.09 Wm^-2 versus C^(0)_2 × CO_2 = Δ Q = 3.7 Wm^-2 implying T^(0)≃ 1.1 K in both cases.]
        
        In view of the complexity of the subject matter, we conclude that there is satisfactory agreement between the theoretical result of
        Δ T_th = 0.46 K ≃ 0.5 K
        
        from Table 8,[Compare also the variation of the theoretical results in Table 9 under various atmospheric conditions.] and T_exp≃ 0.9 K to 1.0 K empirically obtained upon identifying the ratio of Δ^(0)_2 × CO_2/Δ T^(H) with the satellite-based radiation flux measurements, Δ  Flux / Δ T. In fact, we conclude
        that the results from the satellite-based radiation-flux measurement support the results on the climate sensitivity from the detailed and reliable radiation-transfer calculations.
        
        As pointed out in refs. <cit.> and <cit.>, there is definite disagreement between the results from the radiation-flux measurements of f_exp < 0, and the predictions of a large number of atmospheric models that arrive at a positive feedback factor f > 0 and an enhanced value of the climate sensitivity of Δ T ( atm. models) ≫ 1 K.
        
        We add a reference to a recent paper <cit.> that responds to the claim <cit.> of empirical support for a significant climate sensitivity, Δ T ≫ 1K, due to CO_2 doubling, based on the analysis of the observed changes of
        Δ T in the 25-year period from 1983 to 2008. It is pointed out <cit.> that the climate models employed <cit.> for this analysis fail to correctly incorporate the important influence of the low-cloud-cover
        contribution on Δ T in the period under investigation. Correctly incorporating the cloud cover effect yields consistency with Δ T < 1 K<cit.> in disagreement with the results of Δ T ≫ 1 K<cit.>.
        
        
        
        § CONCLUSIONS
        
        
        It has been the aim of this paper to estimate the increase in
        temperature Δ T (“climate sensitivity”) of the surface
        of the earth due to a doubling of the CO_2 concentration in the
        atmosphere.
        The estimate is obtained in a concise and transparent
        manner without oversimplification. All necessary steps are
        explicitly elaborated upon.
        
        The basic assumption of associating a uniform constant temperature T
        with the surface of the earth, and a black-body long-wave infrared
        radiation S(T), is by no means trivial, implicitly or explicitly, however,
        common to 
        main-stream investigations on this matter. Our results are based on a new
        radiative-transfer evaluation, the details being presented in Appendix A.
        The absorption of the
        atmosphere in the CO_2 spectral range can be, and is reliably determined,
        and leads to an approximately constant value beyond an altitude of about
        5 km, or a length of the horizontal CO_2-air pipe of about 3 km at surface
        temperature and pressure.
        
        Assuming restoration of equilibrium upon
        doubling of the CO_2 concentration by an associated increase of the
        temperature then implies a definite estimate of the increase of the surface
        temperature Δ T, given by Δ T ≅ 0.5 ^0C 
        (compare Sections 3 and 4).
        
        In terms of the widely employed feedback parameter f, the result of Δ T ≅ 0.5^0 C corresponds to a negative feedback of f < 0. This result is empirically supported by satellite-based measurements of short-time fluctuations of the outgoing radiation flux at the TOA as a function of (sea-)surface temperature. A consistent picture emerges by combining theoretical radiation-transfer results with radiation-flux measurements (compare Section 5). This picture disagrees with an abundant number of predictions from climate models that imply positive feedbacks, f > 0.
        
        The quantitative result of Δ T ≃ 0.5 to 0.6  ^0C valid for
        the drastic increase of doubling of the CO_2 content in air from
        380 ppm to 760 ppm to be related to one century, confirms that the effect of
        an anthropogenic CO_2 increase
        on the climate on earth is fairly negligible. This conclusion is in strong
        contrast to the values of Δ T ∼ 1.5 - 4.5  ^0C quoted in the
        2013 IPCC report<cit.>.
        The published results on Δ T fill an even larger interval between
        Δ T ≃ 0.4 ^0C to Δ T ≃ 8 ^0C.
        There is a systematic
        tendency of the results on Δ T published between the years 2000 to 2018
        to decrease<cit.> with increasing publication date,
        the results coming
        closer to our result of Δ T ≃ 0.5  ^0C.[Compare e.g. Fig. 1 in ref. <cit.> and the associated list of tens of references.]
        
        We found a concise and brief derivation of the earth's greenhouse effect
        (compare Appendix B, in particular Fig. B1). The greenhouse effect is
        explicitly derived. It is understood as a shift of the radiative
        equilibrium from the earth's surface to an atmospheric level above
        approximately five kilometers.
        
        
        
        § ACKNOWLEDGEMENTS
        
        
        The author thanks Konrad Kleinknecht for discussions that initiated
        the present investigation. The author thanks Peng-Sheng Wei for a
        useful correspondence. Particular thanks go to Roy Clark for providing
        the radiative transfer results in Appendix A, for much help in
        tracing the relevant literature on the subject matter, as well as a critical
        reading of the present paper.
        
        𝐂O_2 Absorbance 568 to 770 cm^-1,
        7 km path, 296 K, 300, 600 ppm
        
        The configurations used for the pipe and atmospheric absorption calculations
        are shown in 
        Figures A1a and A1b.  
        
        
        
        For the `pipe' configuration, a variable horizontal
        path length L, from 1 to 
        7 km long was used.  The CO_2 concentration was varied from 280 to 1000 ppm.
        A total pressure 
        of 1 atmosphere dry air at 293 K was used.  
        
        For the atmospheric absorption
        calculation, a vertical 
        path, h, from 1 to 11 km was used.  The surface temperature was set
        to 293 K at 1 atmosphere 
        pressure.  Two sets of calculations were performed.  The first used dry
        air with a lapse rate of - 6.5 K km^-1.  The second used a
        surface relative humidity of 85 % with a lapse rate of - 6.5 K
        km^-1 up to the saturation level.  Above the saturation level,
        the meteorological formula for the 
        saturated lapse rate is used<cit.>.  The lapse rates used are plotted
        below in Fig. A6.  
        The vertical atmospheric profile was calculated using a spatial resolution
        of 100 m.
        
        The high resolution CO2 absorbance spectra were calculated using
        spectral data from the 
        HITRAN database<cit.>, ^12C^16O_2 lines only with linestrengths > 1e-23.
        The spectral resolution was 0.01 cm^-1.  The temperature was
        set to 293 K.  For the `pipe' calculations, the large spectral data 
        files were divided into seven spectral ranges with the absorption
        coefficient calculated for each 
        spectral range.  The calculations were performed using Excel workbooks
        with each spectral 
        range in a separate workbook.  The weighted averages over the spectral
        ranges 568 to 770 cm^-1 
        and 600 to 750 cm^-1 were then calculated. The `pipe' calculation
        is illustrated in Table A1.
        Path lengths of 1, 2, 3, 5
        and 7 km were used.  The 
        results are given in Tables A2 and A3.
        Plots of the data are shown in Figures A2 and A3.
        
        
        
        The atmospheric radiative transfer calculations were performed using
        the same CO_2 lines that were used for the `pipe' calculations with
        ^1H_2^16O lines with linestrengths > 1e-23 added for the 
        85 % RH cases.  The calculations were performed using Excel VBA macros
        at a spectral resolution of 0.01 cm^-1.  However, the data were
        binned into 1 cm^-1 intervals for plotting.  The 
        conditions of temperature and CO_2 concentration and the spectral ranges
        were the same as for the 
        `pipe' data.  The H_2O concentration was set to a relative humidity
        of 85 % at 293 K for the first 100 m layer.  The vertical 
        path length was varied from 1 to 11 km in 1 km steps.
        The results are given in Tables A4 and 
        A5.  Plots of the data are shown in Figures A4 and A5.
        
        A Concise Representation of the Radiative “Greenhouse Effect” in
        the Atmosphere
        The surface of the earth receives a net radiation from the sun, S_0,of
        approximately S_0 ≃ 240 Wm^-2, see footnote b
        in the main text. The infrared radiation from the surface of the earth,
        the LWIR,
        contains a component, S_W, restricted in its spectral range to the
        so-called “atmospheric window”, and a second component, S_up.
        The second component consists of a small part, f^' S_up 
        (f^'≪ 1), emitted to space, and a larger fraction, (1-f^') 
        S_up that is absorbed by the so-called
        “greenhouse gases”, essentially H_2O and CO_2, in the atmosphere,
        and subsequently re-emitted, partly to space, A_up, and partly down
        to the surface, A_down, compare Fig. B1.
        
        The “transmittance parameter”f^', with 0 ≤ f^'≤ 1,
        realistic
        values being restricted to f^'≪ 1, quantifies the transmittance
        of the earth's surface radiation in the spectral range
        outside the atmospheric window. Compare the incomplete saturation of
        the absorption in Fig. 4a at altitudes larger than (approximately) 9 km.
        
        
        
        
        Global radiative equilibrium of the earth is due to an equality between
        the intensity of the short-wave radiation received from the sun, S_0,
        and the infrared radiation sent out to space, i.e.
        
        S_0 = S_W + f^' S_up + A_up.
        
        
        
        Equilibrium at the upper atmosphere (the top of the atmosphere, TOA) requires
        
        (1 - f^') S_up = A_up + A_down = 2 A_up,
        
        
        where equality of the upward and the downward emission from the greenhouse
        gases has been employed in the second step. From (<ref>) and
        (<ref>), we deduce the radiation from the surface, S_up, given by
        
        S_up = 2/1 + f^' (S_0 - S_W),
        
        
        as well as the radiation from the TOA,
        
        A_up = A_down = 1 - f^'/1+f^' (S_0 - S_W).
        
        
        
        Both, S_up and A_up are expressed in terms of S_0 - S_W, that
        is the radiation from the sun upon reduction by the radiation, S_W,
        that is associated with the transmission of radiation through the
        atmospheric window.
        
        For given S_0, from (<ref>), we find that the total emission
        by the surface, S_tot, surf (S_W, f^'), is given by
        
        S_tot,surf (S_W, f^') = S_W + S_up = S_0 ( 2/1+f^'
        - S_W/S_01 - f^'/1+f^') ≡
        S_0 c ( S_W/S_0, f^').
        
        
        
        Employing (<ref>) and (<ref>), one may explicitly check
        the validity of equilibrium at the surface, 
        
        S_tot,surf = S_W + S_up = S_0 + A_down = S_0 + A_up.
        
        
        
        The radiation to space at the TOA,
        
        S_tot, TOA (S_W, f^') = S_W + f^' S_up + A_up,
        
        
        upon substitution of the results for S_up and for A_up from
        (<ref>) and (<ref>) becomes
        
        S_tot, TOA (S_W, f^') = S_0.
        
        
        Global radiative equilibrium (<ref>) is realized by the equality
        of the radiation intensities emitted from the TOA, S_tot, TOA,
        and received from the sun, S_0.
        
        It will be illuminating to numerically evaluate our main result, the
        expression for the surface radiation (<ref>), for several
        relevant values of S_W and f^' for fixed S_0 = 240 Wm^-2.
        
        Upon assuming a Planck black-body radiation spectrum for the surface
        radiation (<ref>),
        
        S_tot, surf (S_W, f^') = σ( T_surf (S_W, f^')
        )^4,
        
        
        where σ≅ 5.670 × 10^-8 Wm^-2 K^-4, we obtain the
        associated surface temperature,
        
        T_surf(S_W, f^')     = ( S_0/σ)^1/4( 2/1 + f^' - S_W/S_01-f^'/1+f^')^1/4
            = 255.069 ( 2/1+f^' - S_W/S_01-f^'/1+f^')^1/4 K 
            = 255.069 c (S_W/S_0, f^')^1/4 K.
        
        
        
        Relations (<ref>) and (<ref>) give the equilibrium surface
        radiation and surface temperature for given values of S_0, S_W and
        f^'.
        
        The numerical results for S_tot,surf from (<ref>) and
        T_surf from (<ref>) for a few representative values of
        S_W are collected in Table B1.
        
        
        
        Turning to a discussion of the results in Table B1, we start with
        the results for the empirical input of S_W = 43 ± 15 Wm^-2, or
        58 ≥ S_W ≥ 28 Wm^-2, deduced<cit.> from meteorological
        measurements of the long-wave infrared radiation (LWIR). The value of
        f^' = 0.1 follows from Table 6 in the main text. At the saturation
        limit (for C0_2 content of 300 ppm) the absorption reaches a value of
        A = 0.90. This corresponds to (1 - f^') S_up = 0.90 S_up,
        or f^' = 0.1.
        The results
        for the surface temperature, T_surf, from (<ref>) given
        by 288 ≤ T_surf≤ 292 K, or 15^0 C to 19^0 C, are consistent with the very
        elaborate theoretical analysis in refs.<cit.> and<cit.>.
        
        The case of S_W = S_0 = 240 Wm^-2 in the first line of Table B1, according
        to (<ref>) and (<ref>), corresponds to S_up = A_up = 0.
        This is the hypothetical case of an atmosphere allowing for direct
        total emission from the surface to space, or zero absorption, no
        greenhouse gas present. Comparing S_tot surf (S_W = S_0, f^')
        with (<ref>), we find the identity
        
        S_tot TOA (S_W, f^') = S_tot surf (S_W = S_0, f^').
        
        
        The infrared radiation at the TOA, S_tot TOA (S_W, f^') = S_0,
        is equal to the surface radiation of a non-absorbing
        hypothetical atmosphere[Equivalently, the non-absorbing atmosphere
        may be described by inserting f^' = 1 in (<ref>), implying
        S_tot surf (S_W, f^' = 1) = S_0 and A_up = A_down = 0 according
        to (<ref>).]. The associated temperature is given by
        T_TOA = T_surf = 255 K. The “greenhouse effect” of the
        atmosphere containing “greenhouse gases” consists of shifting the
        origin of the outgoing infrared radiation (responsible for equilibrium
        with the radiation from the sun) from the surface to the TOA.
        
        The hypothetical limit of S_W = 0 and f^' = 0 in
        the last line of Table B1
        describes the case of a hypothetical “closed atmospheric window”,
        total absorption of the emitted surface radiation of S_tot, surf
        (S_W = 0, f^' = 0) = 2 S_0, implying the maximally possible
        equilibrium surface temperature of T_surf = 303 K = 30 ^0C.
        
        A comment on the consistency of our results with the 
        radiation and energy budget analysis in ref.<cit.> is
        appropriate. Trenberth et al.<cit.>, using S_0^' =
        239 Wm^-2 as input solar radiation, find a total surface radiation,
        S^'_tot surf = 396 Wm^-2 corresponding to T^' = 289 K =
        16 ^0C, consistent with our result from Table B1, S_tot surf≃
        401 Wm^-2 and T = 17 ^0C. The consistency is not accidental, but can
        be understood as follows.
        
        According to ref.<cit.>, in our notation, the downward radiation
        is given by A^'_down = 333 Wm^-2. Including the amount of
        78 Wm^-2<cit.> of solar radiation absorbed by the atmosphere
        into the solar radiation absorbed by the surface, one effectively has
        A^''_down = 333 - 78 = 255 Wm^-2.
        Thermals and evaporation amount to 97 Wm^-2<cit.>. From
        our point of view they should be considered as being independent from the
        radiation budget. The amount of 97  Wm^-2 quantifies the
        energy conversion of an adiabatic heat engine working between the
        temperatures of T = 289 K on the surface, and T = 255 K at the TOA.
        The heat engine, working in the gravitational field of the earth,
        solely continuously converts internal energy of the air at T = 289 K
        back and forth into potential energy at T = 255 K. This process does not
        affect the radiative balance, and accordingly, A^'_down = 333 Wm^-2
        is reduced to A^'''_down = 333 - 78 - 97 = 158 Wm^-2,
        a value consistent with A_down = 161 Wm^-2 from (<ref>).
        
        The radiation balance obtained by Trenberth et al.<cit.>
        accordingly is consistently reduced to the simple representation
        shown in Fig. C1 that contains the essential features of the
        greenhouse effect.
        
        We turn to the change of the surface radiation under doubling
        of the atmospheric CO_2 content, and the associated change of the
        surface temperature Δ T discussed in section 3 of the main
        text.
        
        For a small change of the parameter f^'→ f^' + Δ f^',
        with S_W = const, from (<ref>) and (<ref>) we find
        dS_up (S_W, f^')/df^' = dS_tot, surf (S_W,f^')/df^' = - 2(S_0 - S_W)/(1+f^')^2,
        
        or
        Δ f^' = - (1+f^')^2/2 (S_0 - S_W)Δ S_up =
        - (1+f^')^2/2(S_0 - S_W)Δ_2 × CO_2,
        
        where Δ S_up = Δ S_tot, surf = Δ_2 × CO_2,
        due to CO_2 doubling, was inserted in the second step. The negative
        value of Δ f^' < 0 for Δ_2 × CO_2 > 0 implies
        less direct radiation to space and, accordingly, a rise of the surface
        temperature, Δ T > 0.
        
        For the change of the surface temperature T_surf, from (<ref>), we
        obtain
        Δ T = dT/df^' df^' = - T_surf/42 ( 1 - S_W/S_0)/2 (1+f^') -
        S_W/S_0 (1 - f^' 2)Δ f^' =
        T_surf/4Δ_2 × CO_2/S_tot, surf (S_W, f^'),
        
        where Δ f^' from (<ref>) was inserted in the last step.
        The expression resulting from the insertion was simplified to yield
        the expression for S_tot, surf (S_W, f^') in (<ref>)
        in the denominator of (<ref>).
        
        The agreement of Δ T in (<ref>) with Δ T in
        (<ref>) of the main text comes without surprise. We note however, that
        the present derivation of (<ref>), relying on (<ref>) and
        (<ref>), assures equilibrium of radiation at the earth's surface and
        at the TOA for any given values of S_0, S_W, f^', and for
        f^'→ f^' + Δ f^' in particular. In slight
        distinction from the present derivation of (<ref>), in (<ref>),
        we relied on using restoration of equilibrium upon CO_2 doubling.
        It is noteworthy, moreover, that T_surf in (<ref>) and
        Δ T in (<ref>) is predicted from the empirically known
        values of S_0, S_W and f^', while T_surf in (<ref>) was
        used as an input parameter.
        
        
        
        The numerical results in Table B2 are based on the input parameters, S_0
        = 240 Wm^-2 and S_W = 43 Wm^-2, as well as f^' = 0.1 and
        f^' = 0.1 + Δ f^' = 0.092015, as obtained by evaluation
        of (<ref>) for Δ_2 × CO_2 = 2.6 Wm^-2 taken from
        Table 6.
        
        In a first step, upon evaluating (<ref>) and (<ref>), we obtained
        the surface radiation and surface temperature shown in the second and
        third lines of Table B2. The subtraction of the two surface temperatures from
        each other then yields Δ T = 0.47 K shown in the last line of
        Table B2. In a second step, we confirm Δ T = 0.47 K [The
        discrepancy between Δ T = 0.47 K in Table B2 and Δ T =
        0.46 K in Table 8 of the main text is due to the different input of
        T = 293 K in the main text.] by evaluating (<ref>) by either
        inserting Δ f^' from (<ref>) or by directly inserting
        Δ_2 × CO_2 = 2.6 Wm^-2.
        
        
        
        
        
        
        
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