On the direct impact of the CO₂ concentration rise to the global warming

Analysis of isotope and gas concentrations of the antarctic ice-core provides important information on the atmospheric composition and climate variations in the past four glacial-interglacial cycles [1–3]. The data show correlation between surface temperature and carbon dioxide (and methane) concentration. The role of CO₂ in these changes however remains unclear since the temperature increase seems to precede the CO₂ growth [1]. Taking into account the large amount of CO₂ dissolved in the sea and its declining solubility for higher temperatures it was suggested that the CO₂ rise is a result and amplifier of the global warming rather than its cause [4,5]. However, a recent study over 80 proxy records reports that temperature generally lags CO₂ during the last deglaciation [6].

Of particular interest is the global warming observed in the last 130 years since it could have anthropogenic reasons [7–11]. Sophisticated computer simulations were reported that estimate the impact of the growing concentrations of greenhouse gases and predict a further temperature rise of 1 to 6 K during the present century [12–18]. In addition, analyses of black carbon and tropospheric ozone reveal that other anthropogenic pollutants contribute notably to the observed temperature rise and changes of major climate zones [19,20]. It has been also suggested that periodic variations in the solar activity are the main cause for global temperature changes during the past 130 years [21]. But in spite of the close correlation observed between sunspot activity and variation of terrestrial temperature, the changes of the solar irradiation of $0.5\text{-}0.75\ \text{W/m}^2$ [22] seem to be too small compared to the measured global warming. Other authors report a close correlation between cosmic rays, ozone depletion and concentration of halocarbons in the atmosphere [23]. These observations give evidence that global warming in the last 60 years was possibly caused by the significant increase of chlorofluorocarbons. Calculations of the greenhouse effect of halocarbons on the other hand predicted a global cooling for the next five to seven decades [24]. Obviously the role of the greenhouse gases should be re-investigated on the basis of additional experimental data.

In combination with atmospheric water, CO₂ and the other greenhouse gases display a broad-band absorption of the thermal emission of the terrestrial surface, with the exception of several spectral windows. The latter fact is illustrated by the satellite measurements in fig. 1(a), taken from literature [25,26]. The far-infrared emission of the earth is plotted vs. frequency in units of wave numbers. The transmitted radiation through the atmospheric window from approximately 750 to $1000\ \text{cm}^{-1}$ is readily seen, limited on the low-frequency side by the $\nu_2$-absorption of CO₂ and by an ozone band at higher frequencies [27]. Around the $\nu_2$-band center at $667\ \text{cm}^{-1}$ [28] the surface emission is totally absorbed. However, carbon dioxide in turn gives rise to thermal radiation that is weaker because of the lower temperature of the atmosphere [29]. A second window is indicated in fig. 1(a) above $1100\ \text{cm}^{-1}$ between the ozone feature and a water band [27].

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In fig. 1(b), a schematic picture of the spectroscopic situation is presented in a larger spectral range including the $\nu_3$-band of CO₂ centered at $2349.3\ \text{cm}^{-1}$ [28]. The terrestrial surface with an assumed temperature $T_s \approx288\ \text{K}$ [7] is treated as a black radiator following Planck's law (solid curve) [29]. The spectral intensity per frequency unit is plotted vs. frequency. The nonlinear ordinate and abscissa scales of fig. 1(b) should be noted that were chosen for a better display of the details. For the assumed temperature the maximum surface emission occurs at ${\sim}565\ \text{cm}^{-1}$ close to the position of the $\nu_2$-band and drops rapidly for larger wave numbers, e.g. by a factor of ${\sim}100$ at $2000\ \text{cm}^{-1}$. The absorption regions of CO₂ and water of our model atmosphere are indicated by red and blue regions, respectively. The narrow ozone band is depicted in violet. Further contributions are buried in the strong water bands and/or have a small effect. The transmission of the atmosphere in several spectral windows, labelled (1) to (3), is readily seen. There is overlap between the low-frequency wings of the $\nu_2$ and $\nu_3$ bands of CO₂ and adjacent water bands so that the extensions of the respective red wings of the former are not relevant. The absorption bands of methane around 1300 and $3000\ \text{cm}^{-1}$ [28,30] are buried in the strong water bands and omitted in the figure. We also mention its small concentration of $\approx1.6\ \text{ppmv}$. For nitrous oxide with even smaller concentration of 0.3 ppmv the situation is similar with absorption bands around 589, 1285 and $2220\ \text{cm}^{-1}$ [30]. CH₄ and N₂O are neglected in our model atmosphere that is also considered in our computations discussed below. Thermal reemissions of the greenhouse gases are not indicated in fig. 1(b).

The present work is restricted to a discussion of the global warming as revealed by reliable empirical data. Only the role of CO₂ will be discussed relating new spectroscopic data with 1-dimensional global models and using some recent empirical results. We confine ourselves to a simplified physical picture of the greenhouse effect so that only semi-quantitative results can be expected.

We have obtained precise data on the transmission of optically dense CO₂ samples in the wave number range $550\text{-}4000\ \text{cm}^{-1}$. Our experimental system was a Tensor 27 Fourier-transform infrared spectrometer (FTIR, Bruker Optics) with a spectral resolution of $0.4\ \text{cm}^{-1}$. The sample cell was 10 cm long with KBr windows. To remove impurities, the sample was filled and evacuated several times with high purity CO₂ (99.995%). The pressure was kept at 2 bar for measurements in the temperature range 220–295 K. The temperature was controlled by a cryostat and directly measured in the sample. An overview spectrum is presented in fig. 2(a). The wings of the $\nu_2$-band at $667\ \text{cm}^{-1}$ are extended by various hot bands [28] included in our computations. The situation is similar for the $\nu_3$-band with the low-frequency wing extended by the corresponding mode of ¹³C¹⁶O₂ at $2284\ \text{cm}^{-1}$ [28]. For the concentration range of interest ${>}250\ \text{ppmv}$ and the thickness of the atmosphere we estimate the optical density $(OD)$ in the band center of $\nu_2$ to be $OD >800\ \text{mol}^{-1}$ (for $\nu_3$: $OD>1100\ \text{mol}^{-1}$). In other words there is total absorption around the band centers not depending on concentration changes. Thus only the wings of the $\nu_2$ and $\nu_3$ absorptions are relevant for concentration changes of CO₂, indicated in the figure by red circles. The relatively weak absorption by hot bands at 791, 961 and $1064\ \text{cm}^{-1}$ are also important because they modify the spectral windows of the greenhouse effect and were carefully measured.

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Some experimental results on the temperature dependence of the CO₂ absorption bands are presented in figs. 2(b) and (c). The measured $OD$ of the high-frequency wings is plotted on a logarithmic scale vs. wave number. For atmospheric conditions, the experimental OD-values of the ordinate scales have to be multiplied by a factor of approximately 15.6. The measured OD of the sample was converted to the situation in the atmosphere using Beer's law and the barometric formula for an effective thickness of the atmosphere of 7990 m. The concentration dependences of the wings are readily deduced from our data. A concentration rise of 33% (290 to 385 ppmv) [31] in the atmosphere with mean temperature 255 K increases the effective $\nu_2$-width by $2.0\ \text{cm}^{-1}$ (relative broadening 2.3% of the band). The effective width is defined here as the halfwidth at 50% band transmission (HWHM). The corresponding number for the $\nu_3$-band is $4.7\ \text{cm}^{-1}$ (relative broadening 5.5%).

For computations of the greenhouse effect we introduce the integrated absorptivity η of the atmosphere for the infrared emission of the surface. Due to the spectral windows only a fraction $\eta<1$ is absorbed by the atmosphere while the remaining part $(1- \eta)$ of the surface emission is transmitted into the universe. The quantity depends on concentration (molar fraction x, in units of ppmv) and the mean temperature T_at of the absorbing CO₂ molecules, $\eta = \eta(x, T_{at},T_s)$, and is evaluated from our spectroscopic data. The other greenhouse gases are assumed to be constant. The surface temperature T_s comes in since the spectral intensity distribution of the surface emission is temperature-dependent (compare with Wien's law) [29]. For the approximate global values $T_{at}=255\ \text{K}$ and $T_s=288\ \text{K}$ [7] and using the effective bandwidth changes mentioned above we estimate an increase of the efficiency of $\Delta \eta=2.0\times10^{-3}$ for the concentration change from 290 to 385 ppmv.

As extension of the well-known 2-layer model for the greenhouse effect [32,33] we have developed models where the atmosphere is arbitrarily divided into two and more parts with respective temperatures T_i. In this way the temperature variation of the atmosphere is included. The spectral efficiencies $\eta_i$ of the layers have to be distinguished. Transport phenomena between the layers are included by additional parameters but are found to have little quantitative effect on the global surface warming by CO₂. The efficiencies $\eta_i$ are computed from our spectroscopic data, the temperatures T_i being self-consistently determined from thermal equilibrium. Radiative equilibrium is considered between the solar input, and the thermal emissions of the surface and the various layers of the atmosphere. In spite of the simplicity of our models the results are believed to be significant, since the global properties of the earth have to be consistent with energy conservation arguments explicitly treated by the models. The use of a 1-dimensional model is fully justified if the CO₂ concentration dependences of the geographic and seasonal variations of the surface temperature need to be expanded to first order, only. The problem is thus linearized and a global average over geographic and seasonal changes can be carried out first before evaluating the mean concentration-induced temperature rise, i.e. the 1-dimensional treatment is sufficient. The first order approach is believed to be an acceptable approximation because of the moderate concentration increase of $\approx30\%$ of CO₂ and the small rise of the mean surface temperature.

The level scheme in the case of a 4-level model is shown in fig. 3. Assuming equal absorption of the three atmospheric parts the thickness of the layers is estimated from the barometric formula to be 3.2 and 5.6 km (layer 1 and 2, respectively), while the top layer extends above 8.8 km into the universe. The spectral efficiencies $\eta_{ij}$ are calculated from the measured absorption properties of CO₂ without a fitting parameter. The surface (0) with temperature T_s is treated as a black radiator with intensity $\sigma T_s^4$ (spectral efficiency of 1, Stefan-Boltzmann constant σ). The atmospheric layers are treated as selective emitters with temperatures T_i, $(i=1 - 3)$. Quasi-equilibrium between the intensity input and output of the individual layers leads to the following equations:

$$\begin{equation} \eta_{10} T_1^4 + \eta_{20}T_2^4 + \eta_{30}T_3^4 = T_s^4 - (S_{net}-B_1)/\sigma,\end{equation} \tag{ 1 }$$

$$\begin{equation} \eta_{01} T_s^4\!+\! \eta_{21}T_2^4\!+\! \eta_{31}T_3^4\!=\!2\eta_{11}T_{1}^4\!-\!(A_1\!+\!B_1\!-\!B_2)/\sigma,\end{equation} \tag{ 2 }$$

$$\begin{equation} \eta_{02} T_s^4\!+\!\eta_{12}T_1^4\!+\! \eta_{32}T_3^4\! =\! 2\eta_{22}T_2^4\! -\! (A_2\!+\!B_2\!-\!B_3)/\sigma,\end{equation} \tag{ 3 }$$

$$\begin{equation} \eta_{03} T_s^4 + \eta_{13}T_1^4 + \eta_{23}T_2^4 = 2\eta_{33}T_3^4 - (A_3+B_3)/\sigma.\end{equation} \tag{ 4 }$$

Here $\eta_{ij}(T_i,T_j,x)$ denotes the spectral efficiency by absorption of layer j, while subscript i labels the origin of the thermal radiation. The parameters are evaluated from the measured OD-values by integration over the spectrum without a fitting parameter. Details will be described elsewhere. A numerical solution of the above equations delivers the temperatures of the 4-layer model of, respectively, 231 K, 261 K and 277 K for the upper, middle and lower part of the troposphere, while the surface temperature is taken to be $T_s=288.0\ \text{K}$ (for $x=290.4\ \text{ppmv}$ in 1880). The specific parameter values of A_i and B_i determine the temperatures of layers (1) to (3) but have little influence on the CO₂ concentration effect of the surface temperature as shown by the data of tables 1 and 2.

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Table 1:.  CO₂ concentration effect on the layer temperatures from 1880 to 2010 keeping the surface temperature in 1880 constant at 288.0 K. The direct solar input into the atmospheric layers $A_i=\epsilon_i A_{in}\ (i=1\text{-}3)$ is varied. The resulting surface temperature change $\Delta T_s$ is approximately constant.

Ain (W/m2)	$\epsilon_1$ / $\epsilon_2$ %	$\Delta T_s$ (mK)	$\Delta T_1$ / $\Delta T_2$ / $\Delta T_3$ (mK)
53.51	10 / 20	254	165 / 101 /   44
63.43	20 / 40	262	146 /   72 /   34
67.42	30 / 33	261	137 /   70 /   34
84.87	45 / 45	268	101 /   29 /   20
99.91	70 / 20	268	69 /   22 /   13

Table 2:.  The same as table 1 varying the input parameters $B_1\text{-}B_3$ describing the heat transport from the surface to the atmospheric layers. The small effect on $\Delta T_s$ is noteworthy.

Ain (W/m2)	B1 / B2 / B3 (W/m2)	$\Delta T_s$ (mK)	$\Delta T_1$ / $\Delta T_2$ / $\Delta T_3$ (mK)
100.47	10 /   8 /   4	268	111 /   37 /   23
84.52	25 / 20 / 10	267	113 /   38 /   24
76.33	30 / 30 / 12	267	120 /   38 /   25
73.36	30 / 30 / 18	265	125 /   49 /   28
29.98	70 / 70 / 28	267	134 /   39 /   27

The quantities $S=343.3\ \text{W/m}^2$ (total solar input), albedo${} = 0.30$ and the surface temperature $T_s=288.0\ \text{K}$ are kept constant as empirical facts. The direct solar input A_in into the atmosphere is adjusted correspondingly to maintain this temperature. With respect to previous results a range of 50 to $100\ \text{W/m}^2$ is considered (see column 1 in the tables). Column 2 of table 1 with $\epsilon_i$ values indicates how the input A_in is distributed over the atmospheric parts $(\epsilon_1 + \epsilon_2 + \epsilon_3=100\%)$. The numbers 30 and 33% (so that $\epsilon_3=37\%$) refer to the situation where the solar energy deposition into the three layers is approximately equal. The corresponding temperature increase $\Delta T_s$ of the surface for a CO₂ concentration rise from 290.4 (in 1880) to 385 ppmv in 2010 (see below) is listed in column 3. It is important to note that the numbers are approximately constant within a few mK. The column 4 states the computed CO₂-induced temperature rise of the atmosphere with small variations of $\Delta T_i\ (i=1\text{-}3)$ in spite of the considerable change of the solar input parameter.

The effect of parameters B_i describing the heat transport is addressed by table 2 (see column 2). Quite obviously varying the energy loss of the surface (parameter B₁) requires a compensation by a similar change of A_in in column 1. The third column again shows that the computed concentration effect for 1880 to 2010 on the surface temperature is remarkably insensitive to the differing input parameters with $\Delta T_s=260\pm1\ \text{mK}$. Column 4 again indicates the concentration effect on the layers with variations of $\Delta T_i$ below 100 mK. The corresponding temperature values T₁ to T₃ of the atmosphere in tables 1 and 2 for $x=290.4\ \text{ppmv}$ cover ranges of 273–284 K (1), 254–269 K (2) and 230.5–230.9 K (3), respectively.

We now discuss the causal relationship between CO₂ concentration and global surface temperature. We recall that an empirical correlation does not answer the question what factor determines the other one. The situation is illustrated by the reported data of fig. 4. The measured CO₂ concentration x(t) of the atmosphere in the years 1880 to 2010 is plotted in fig. 4(a) [31]. Note the monotonic increase from 290 to 385 ppmv. The measured global rise of the surface temperature [34] is depicted in fig. 4(b). A value of $\Delta T_{\textit{meas}}=1.2\pm0.2\ \text{K}$ is indicated for the period 1880 to 2010. It is interesting to see the non-monotonic temperature behaviour as suggested by the fitted blue line with a certain decrease from 1940 to 1970 that does not follow the simultaneous CO₂ growth in the same period. The feature is at variance with a simple correlation between concentration x and temperature T_s; i.e. other factor(s) notably effect the surface temperature.

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A rough estimate of the global warming by CO₂ can be derived from the spectral forcing $\Delta S$ that results from the concentration rise via the increase of the spectral efficiency. In the well-known 2-layer model a value of $\Delta S\approx1\ \text{W/m}^2$ is estimated (the more detailed 4-layer model predicts $1.4\ \text{W/m}^2$). Treating the surface as a black radiator the thermal re-emission amounts to $S = \sigma T_s^4$ [29]. By differentiation we readily see that the temperature rise $\Delta T$ required to maintain thermal equilibrium is given by $\Delta T \approx \Delta S /4\sigma T_s^4 \approx0.2\ \text{K}$ for $T_s\approx288\ \text{K}$.

We have computed different global models for the greenhouse effect where radiative equilibrium is considered between the solar input, and the thermal emissions of the surface and the various layers of the atmosphere. Transport phenomena between the layers are included by additional parameters. Our 5-layer model (surface plus 4 atmospheric layers developed in analogy with the scheme shown in fig. 3) predicts reasonable temperatures of approximately 286, 277, 263 and 232 K for the atmospheric parts. The surface temperature of $T_s=288\ \text{K}$ for $x=290\ \text{ppmv}$ is calculated for a proper choice of the solar warming of the atmosphere by $75\ \text{W/m}^2$, while the direct solar input of the surface is taken to be $163.91\ \text{W/m}^2$ (total input $341.3\ \text{W/m}^2$, global albedo of 0.300). The minor influence of the CO₂ increase on the solar input by absorption in the near infrared and leading to the opposite effect of surface cooling is omitted.

Some results for the 5-layer model are presented in fig. 4(b) by the red curves. A direct effect of $\Delta T_s=0.26\pm0.01\ \text{K}$ is shown for the period 1880–2010 (dashed red line). The simultaneous global warming of the atmosphere, e.g. by 0.15 K for the lowest layer, leads to a growing water content of the atmosphere. The latter effect is omitted in our model computations. As reported by Wang et al. [30] the rising H₂O concentration enhances the greenhouse effect by a factor of approximately 1.5. Including this feedback factor leads to the solid red line in the fig. 4(b). Comparing our estimated results with the measured rise of the surface temperature (black data points, fitted blue line) only a modest contribution to the terrestrial warming of less than 33% is derived from the FIR properties of CO₂.

The question remains what other factor(s) may be more important for the global temperature increase. A possible source could be the albedo factor (taken to be 0.30 in our calculations). It is obvious that the surface temperature depends on the net solar input reduced by backscattering and reflexion of solar radiation as described by the albedo factor. A decrease by a few percent would be required for a surface warming of 1 K. Changes of the factor of opposite sign $(\pm0.02)$ were reported in the years 1985–1997 and 1997–2004 [35,36], without a noticeable effect on the temperature data in fig. 4(b). Other authors provided evidence that the global albedo value remained fairly constant during the past decades [37]. It has been recognized recently that black carbon (BC) in the atmosphere is highly relevant, much more important than the concentration changes of ozone, methane and nitrous oxide [20,38]. As best estimate of industrial-era climate forcing of black carbon a value of ${\sim}1.1\ \text{W/m}^2$ was obtained and considered to be the second most important human emission after carbon dioxide [39]. The present authors feel that the absorption of environmental pollution in the far infrared should be given more attention. BC absorbs in this spectral region [40,41] and is thus likely to modify the greenhouse effect. Our model calculations suggest that a transmission decrease of 2.3% of the atmosphere in its spectral windows would be sufficient to raise the surface temperature by 1.0 K.

In conclusion we wish to say that we have performed a detailed study of the infrared properties of carbon dioxide to estimate its contribution to the global warming. Our results suggest that CO₂ is not predominant for the terrestrial temperature increase. Only a moderate effect of less than 33% is found. Scenarios aiming to reduce further warming should take into account that carbon dioxide is not the dominant cause.