On CO2 and the global mean Earth’s surface temperature

Increase of global mean Earth’s surface temperature due to doubling of CO₂ estimated to be marginal.

Guest contribution by Frans van den Beemt

Abstract

The aim of this study is to get an order of magnitude value for the increase of the Earth surface temperature at doubling CO₂ concentration from 300 ppm to 600 ppm (parts per million). Here an original method is worked out that complements an earlier original study with the same aim [i]. The warming effect of higher concentrations of CO₂ appears to decrease exponentially and has largely lost its effect above 100 ppm, so it is not surprising that the atmosphere at a CO₂ doubling from 300 ppm to 600 ppm only emits 1 Wm^-2 less energy to space and that the temperature at the Earth’s surface could increase by no more than 0.2 Celsius or in theory, would not even increase at all.

Nederlandse samenvatting

Het doel van deze studie is om de invloed van verdubbeling van de CO₂ concentratie van 300 ppm naar 600 ppm (parts per million) in de aardse atmosfeer op de temperatuur van het aardoppervlak te bepalen. Dit alleen voor een fictieve globale gemiddelde aarde inclusief wolken. In een eerdere studie[i] is dat gedaan door met fysische formules een bovengrens te bepalen voor de infrarood (warmtestraling) absorptie door CO₂. Het opwarmend effect van hogere concentraties CO₂ blijkt exponentieel af te nemen en heeft zijn werking boven 100 ppm al grotendeels verloren, zodat het niet verwonderlijk is dat de atmosfeer bij een CO₂ verdubbeling van 300 ppm naar 600 ppm slechts 1 Wm^-2 energie minder uitstraalt naar de ruimte en de temperatuur aan het aardoppervlak maximaal 0,2 Celsius kan stijgen of in theorie zelfs wellicht niet zal stijgen.

INTRODUCTION

The aim of this study is to get an order of magnitude value for the increase of the Earth surface temperature at doubling the CO₂ concentration from 300 ppm to 600 ppm. Here an original method is used that complements an earlier original study with the same objective [i].

For calculating the cooling of the Earth by infra-red radiation emission to space at increasing CO₂ concentrations a commonly used computer program MODTRAN [v] is used. It is free available at the website of the University Chicago [iv]. MODTRAN calculates the infra-red emission from CO₂ gas line by line in a given Earth atmosphere model using the 1976 U.S. Standard Atmosphere.  Calculations are restricted to global mean values in combination with frequently re-measured data as found in the literature [ii]. An earlier study calculated an upper limit for the infra-red absorption by CO₂ within the troposphere without using any specialized computer model [i]. Now a different method based on emissions is used to quantify the role of CO₂ in the atmosphere.

CALCULATION AND RESULTS IN A NUTSHELL

“Greenhouse” gases like H₂O and CO₂ absorb and emit infra-red radiation in wavelengths that correspond to their molecular vibration and rotations. These specific wavelengths are known very well. Part of the CO₂ wavelengths shows strong overlap with H₂O wavelengths, except for the region of 15 μm, where the overlap is very minor. This 15 μm band contributes strongly and almost independently from H₂O to the overall infra-red emission to space. MODTRAN calculates the overall radiation (energy) emission from the Earth surface and atmosphere to space for a given composition of the atmosphere. Furthermore, 78.7% of the total by MODTRAN calculated emission, corresponds with the specific CO₂ 15 μm band.

At CO₂ concentrations above 100 ppm, CO₂ central-band emission directly to space originates in the lower stratosphere at very low temperatures of around 220K and low enough CO2 densities. At increasing CO2 concentration the direct emission to space shifts to a higher altitude with increasing temperatures caused by the Ozone layer.

Besides the central-band emission, we have the emissions of the side lobes. The central band is emitting radiation due to single vibrating and rotating molecules. But the edges (side-bands or side lobes) of that central CO₂ 15 µm band is emitting from vibrating and rotating double or triple molecules in the short moments, that they are close enough together that they can influence each other.

These emissions are known as collision broadening spectra lines. Sufficient density of these multiple molecule energy states to emit to space are found near the surface in the first few km altitudes. At increasing CO₂ concentrations direct emission to space takes place from some higher altitude in the troposphere at lower temperatures. These side-lobe-band emissions are in fact the only parts of the 15 µm CO₂ band that show a warming effect by decreasing infra-red emission to space at increasing CO₂ concentrations above 100 ppm. while clouds absorb all emissions at CO2 specific wavelengths, infra-red emission to space from CO₂ molecules from beneath these clouds is not possible. Therefore, the global mean cloudiness of 62% [ii] causes a reduction of the effectiveness of the CO₂ side-bands infra-red emissions-to-space to 38%.

As we need to know the decrease in infra-red emissions purely caused by doubling of the CO₂ concentrations and not emissions from parts of the CO₂ spectrum that in the real atmosphere have an overlap with emissions that originate from other infra-red active gases as H₂O vapor or clouds. Hence, the total correction (15 µm band and clouds) on the emission results, as calculated by MODTRAN, becomes 78.7%*38%=29.9%.

In summary, at increasing CO2 concentrations, the central-band will emit at a higher temperature and the side-bands at lower temperatures. The net emission effect (Δemission) is calculated using MODTRAN for 300 ppm, 600 ppm, and 1000 ppm (see the results in Table 5).

The earth surface emits infra-red radiation with energy set by the formula ɛ.σ.Te⁴  and an increase of the surface temperature ΔT_s as a direct result of an increase of the CO₂ concentration from 300ppm to 600ppm is calculated with the formula:

Δ-emission=ɛ.σ.{(T_s+ΔT_s)⁴ – T_s⁴}

CALCULATION AND RESULTS IN DETAIL

The following global mean factors are used:

-A global mean surface temperature T_s of 288.2K,

-The 1976 U.S. Standard atmosphere,
-An average global cloud cover (cloudiness) of 62%  [ii],
-Emissivity, ɛ=0.9721 (derived from the difference between MODTRAN calculation and blackbody calculation and in line with water (average ɛ=0.995) and sand (average ɛ=0.955),
-Stefan-Boltzmann constant σ= 5.67 * 10^-8 W m^-2 K^-4

The infra-red (IR) active gases and clouds together shield off most of the emission of the surface to space directly. Only a small window (32% measured by satellites) of IR wavelengths emits directly from surface (288.2K) to space [ii]. Further emissions come from the atmosphere from cold cloud tops and from the atmospheric layers at different low temperatures to space.

The first 300 ppm CO₂ in the atmosphere of the Earth will shield (absorb) some emission from the surface that will not reach space. The 15µm spectral lines that originate from single CO₂ molecules contribute to IR emissions to space, originating from several different atmospheric altitudes and temperatures. For the CO₂ 15µm central lines this happens at the approximately 222K beginning at lower stratosphere and for its side-lobes, it happens between approximately 222K and 288K in the lower troposphere. These effective emitting temperatures can be read from Planck curves in satellite absorption spectra graphs but also from spectra graphs generated by MODTRAN. At  300 ppm and 600 ppm CO₂ respectively, MODTRAN shows a central line emission temperature of respectively 222K and 225K that increases further with higher CO₂ concentrations.

From these satellite absorption spectra graphs, it can be measured manually what percentage of emission energy under the Planck curve belongs to the specific 614 cm-1 – 723 cm-1 CO₂ 15µm band[i] in which the central band and the two not-yet-saturated CO₂ side-lobes are included. The average of three manually measured satellite absorption spectra (Ts around 300K) shows a CO₂ 15 µm energy fraction f=15.3% of the total spectra emission energy between 0 cm^-1 – 1800 cm^-1. See fig 1 and Table 1. At the end of this study, Table 6 provides a sensitivity test for this fraction f=0.153.

MODTRAN provides the amount of change in emission (ΔE

mission) to space for any CO₂ concentration, see Table 2. Adding CO₂ in the model atmosphere of MODTRAN results in additional absorption of the CO₂ specific spectral lines that are emitted no longer directly from the surface to space but which

will be emitted from higher altitudes at lower temperatures (in combination with lower density and pressure) and as such the integrated energy of the total emission to space will decrease with increasing CO₂ concentrations.

 

 

 

 

The specific 614 cm^-1 – 723 cm^-1 CO₂ 15µm band

 A thorough study of MODTRAN results and graphs; see Figure 3a, learns that the CO₂ ΔEmission value comes from a wide range of spectral lines, much broader than the specific 614 cm-1 – 723 cm-1 CO₂ 15µm band, which has to be taken into consideration [i]. In the actual Earth atmosphere other IR active gases overlap on both sides of this specific CO₂ band so dominantly with their emissions that CO₂ concentration differences are irrelevant. This is the reason of the focus on this specific part of the CO₂ band where CO₂ emission and absorption has the least overlap with other IR active gases.

From manual measurements of real spectra’s and from MODTRAN generated spectra, it becomes clear that about 40% of the specific CO₂ band emission originates from emission temperatures at around 220K and not much, if any, from lower temperatures.  The other 60% of the specific CO₂ band emission originates from higher temperatures in the troposphere as can be read from Planck curves in the spectra graphs (see Figure 3a and caption).

With increasing CO₂ concentration, emission-to-space will originate from higher altitudes in the atmosphere at lower CO₂ densities. This has two opposite effects for the side-bands and the central part. The side-bands emit in the troposphere where the temperature drops with higher altitudes, resulting in reduced emission. The central part emits from the lower stratosphere where the temperature increases with altitude, resulting in increased emission

Furthermore, the overall emission depends not only on the temperature but also on pressure and density and the net result found with MODTRAN is a decreasing emission. The conclusion is that the amount of ΔEmission at increased CO₂ concentrations has an upper limit determined by the 60% of the CO₂ emission part or 15.3%*60% of the total emission under the Planck curve of 288.2K or 34.9 Wm^-2.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

The calculations of  MODTRAN includes emissions at wavelengths outside the specific CO₂ band of interest, omitting the overlap with other emitting gasses and hence these are too high. So a correction factor C needs to be calculated.

 Calculation of correction factor C

The maximum ΔEmission must be 60% of the specific CO₂ band emission or maximum ΔEmission=60%*f.ɛ.σ.T_e^{^4}=29.05 W m^-2 with f=15.3% and T_e=288.2K, which maximum is assumed to be reached for an arbitrary 90% level at 1000 ppm CO₂ with ΔEmission=39.90 W m^-2. Hence 90%*29.05=C*39.9 which results in C=78.7%. Obviously, MODTRAN ignores other IR active gases other than CO₂ using a CO₂ band with a factor of  1.3 times wider [i].

In Table 8 MODTRAN is used to calculate this correction factor C with a different method and the same result.

This 78.7% percentage is used to correct the MODTRAN results to the calculated emissions, corresponding with the specific 614 cm-1 – 723 cm-1 CO₂ 15µm band.  Then A further correction is necessary as clouds overrule any influence of CO₂ infra-red emissions.

 

Clouds

The former calculations assume an imaginary atmosphere limited to CO_2. N₂ and O₂ omitting other IR active gases and clouds. Clouds do absorb most if not all infrared wavelengths; consequently, the concentration of infrared active gasses below clouds is mostly if not totally irrelevant for the emission to space. MODTRAN has an option for several different cloud models without adding H₂O vapor to the atmosphere and again for an imaginary atmosphere model. In Table 7 the available cloud models in MODTRAN are listed. For example with the Standard Cirrus Model and calculated with MODTRAN, a reduction of the emission of 14% is found. In combination with the former correction factor, the necessary correction becomes 0.787*0.86=0.677 and Table 3 shows the results.

Another example; the “Alto Stratus Cloud Base 2.4 km Top 3.0 km” a 44% lower emission is found. The new correction factor becomes 0.787*0.56=0.441 and Table 4 shows the results.

These different cloud models show that clouds have a large impact but the contribution of each model to the average global cloud cover is not clear and therefore only the global mean cloudiness will be taken into account. With the global mean cloudiness of 62% [ii] the correction factor becomes 0.787*0.38=0.299 and Table 5 shows the results

Effects of greenhouse gasses
Table 8 shows MODTRAN results for different combination of greenhouse gases in a 1976 USA Standard atmosphere. In MODTRAN greenhouse gases Water Vapor, Freon and Stratospheric Ozone are standard scaled with altitude, Methane has a standard value of 1.7 ppm (parts per million) and Tropospheric Ozone has a standard value of 28 ppb (parts per billion).

The difference in emission-to-space between 600 ppm and 300 ppm: ΔEmission=1.13 Wm^-2, which would be the resulting energy imbalance at the top of the atmosphere (TOA). If this energy imbalance was to be compensated by a higher surface temperature, the temperature increase is calculated to be: ΔT_s=0.21 K.

However, in the real complex Earth atmosphere, this 1.13 Wm^-2 will not directly increase the surface temperature. There are several other effects that must be taken into consideration. At first the additionally absorbed infrared will be redistributed to all available vibration/rotation/kinetic energy levels of all air molecules, according to the Boltzmann distribution law, which is valid up to 60 km altitude where collisions between molecules are fast enough [iii].  This redistribution also causes water vapor and clouds to emit part of this additional energy to space.

Then the additionally absorbed and reradiated ΔEmission in the lower troposphere causes additional water evaporation at ground level which takes up latent energy, furthermore, it steepens the temperature gradient which increases convection and condensation and cloud formation in the troposphere, releasing the latent energy again. Consequently, these processes transport the energy from the surface and lower levels aloft, compensating for the additional back radiation and thus counteracting additional warming of the surface.

Furthermore, the increased cloudiness reflects more shortwave sunlight radiation to space and emits more infrared upwards as a near black body into space. It can be concluded that these factors inhibit an increase of T_s due to a doubling of CO₂ and the resulting ΔEmission of 1.13 Wm²-.

This one-dimensional model is based on global mean values and has its limitations, therefore these findings only must be regarded as a first order estimation for our complex Earth climate system.

In the former study [i] the additional absorbed energy (complementary to ΔEmission) in the atmosphere was considered to be  half to be lost to space and half be absorbed at the surface, causing a temperature increase.

From these calculations based on MODTRAN it can be concluded that under global mean cloudiness 62% condition the ultimate possible contribution to an increase (ΔTs) of the global mean surface temperature Ts caused by an increase of the CO₂ concentration even until 1000 ppm will not exceed 0.4 K or ΔTs<0.4 K and that there are strong indications for a ΔTs=0.

Sensitivity test

A sensible test for the factor f “CO₂ band share value of 15.3” is provided for +/- 10% different values for f, see next Table 6. The values as found for the surface temperature increase does also change within 10%.

Then Table 7 shows all the cloud models as they are available in MODTRAN. Table 8 shows MODTRAN results for different combinations of greenhouse gases in a 1976 USA Standard atmosphere. In MODTRAN greenhouse gases Water Vapor, Freon and Stratospheric Ozone are standard scaled with altitude, Methane has a standard value of 1.7 ppm (parts per million) and Tropospheric Ozone has a standard value of 28 ppb parts per billion.

Conclusions
Based on emission to space calculations a doubling of CO₂ concentration until 600 ppm results in a surface temperature increase of ΔTs<0.21 K at most, if any. This result is consistent with to the earlier calculation of ΔTs<0.16K based on a different upper limit troposphere absorption calculation [i]. Consequently, for a first order effect –considering the complexity of the Earth system- any CO₂ related surface temperature influence on today’s climate can be disregarded.

 

About Frans van den Beemt
Frans obtained his MSc physics in June 1973 at the University of Technology, Eindhoven, The Netherlands and his PhD in Science Studies, June 2000, University of Leiden, The Netherlands
He held positions as Teaching Assistant, University of Technology, Eindhoven, The Netherlands, then as Research Fellow, Department of Experimental Nuclear Physics, University of Technology, Eindhoven, The Netherlands. Next he was teacher in Physics and Mathematics, Eckhart College, Eindhoven, The Netherlands, System Engineer, Machines & System Group Holec, Ridderkerk, The Netherlands and Program Director, Technology Foundation STW, Utrecht, The Netherlands He is founder of consulting firm: VdBeemt 2G Advies and founder of European academia consulting firm: HandsonGrants. He authored numurous publications and performed many specialized functions.

Notes and references
[i] The physics of doubling CO₂ (full version) Frans van den Beemt www.sciencetalks.nl/the-physics-of-doubling-co2-full-version/ ; Computing HITRAN data, basic global temperature increase for doubling CO₂ is limited to 0.16K-0.20K
[ii] Martin Wild, Doris Folini, Christoph Schär, Norman Loeb, Ellsworth G. Dutton, and Gert König-Langlo (2013). “The global energy balance from a surface perspective.” Climate Dynamics 40 (11-12), pp. 3107–3134. DOI 10.1007/s00382-012-1569-8
[iii] R. M. Goody and Y. L. Yung, Radiation: Theoretical Basis, Oxford University Press, Oxford, UK, 1989.
[iv] [Uchicago] MODTRAN Infrared Light in the Atmosphere http://climatemodels.uchicago.edu/modtran/
[v]. [MODTRAN] http://modtran.spectral.com/ The MODTRAN® (MODerate resolution atmospheric TRANsmission) computer code is used worldwide by research scientists in government agencies, commercial organizations, and educational institutions for the prediction and analysis of optical measurements through the atmosphere.