#### Radiative forcing

The greenhouse effect (or radiative forcing) is due to the low temperature of the upper layers of the troposphere, compared with the temperature of the surface of the earth. The earth-atmosphere system would not know, in fact, the greenhouse effect if it were isothermal, emitting as a whole in the infrared spectrum as would a black body[i].

The greenhouse effect increases the planet’s surface temperature by reducing energy loss to space by absorption of infrared emitted by the earth. This absorption mainly results from water vapor in the lower layers of the atmosphere, warm and humid, and from CO2 in the upper layers, cold and dry, thereby producing an insulating effect that stands between the warm surface of the earth and the cosmos whose temperature is close to absolute zero (2.7 K).

If the increase in the greenhouse effect as a result of the increase in water vapor content in the atmosphere is well understood, this causal relationship is more subtle for CO_{2} because of saturation phenomenon so that, under a first approximation, absorption does not increase with concentration. But this is not entirely true for two reasons:

- CO2 absorption line between 14 and 17 µm is not saturated in the side band: absorption thus increases weakly with CO2 concentration
- Altitude from which radiation escapes to space increases with the concentration of CO2 because of the thickening of the opaque layer in which any emission in the infrared spectrum of CO2 is reabsorbed or scattered. The temperature decreases with altitude, the emission is lower, which increases the radiative forcing of the atmosphere.

It is necessary to emphasize the special role played by water vapor on climate: the mechanism invoked to explain the warm weather on our planet, which should be -18 ° C in the absence of any greenhouse effect, is mainly attributed to it. In contrast, the thermal convection redistributes moisture and homogenizes the temperature in the layer from 0 to about 4.4 km. This influences the vertical thermal balance of the planet by reducing the average decrease in temperature with altitude that the greenhouse effect imposes to the troposphere. Without convective mechanisms, the change would be steeper and give a higher average temperature of the earth’s surface. By evaporating from the oceans water vapor draws energy, the latent heat^{[i]}, which is released during its condensation in the atmosphere. The resulting warming limits the convection from the surface of the earth, reducing heat exchange into the upper atmosphere. Added to this is the action of low clouds that have a high albedo effect and high clouds whose greenhouse effect is substantial.

**Energy balance**

The link between global warming and the increasing concentration of atmospheric carbon dioxide since the beginning of the industrial age is poorly understood by climatologists who cannot ascertain the climate impact of human activities. They presuppose a positive feedback of water vapor due to the increase in greenhouse gases, a highly speculative hypothesis.

At thermal equilibrium, the solar energy received by the earth is equal to the thermal energy re-emitted into space as infrared radiation. By writing that the *infrared flux emitted* = *Solar Flux absorbed*, we obtain the well-known relationship:

4πR^{2}σT_{e}^{4}= (1-A) πR^{2}F_{0}

where 4πR^{2} represents the surface of the earth, σT_{e}^{4} the emission of black body, (1-A) the absorption coefficient, πR^{2} the earth section, and F_{0} the solar flux outside the atmosphere; T_{e} is the radiative equilibrium temperature, A the planetary albedo, and σ the Stefan-Boltzmann constant (5.67.10^{-8}W.m^{-2} .K^{-4}).

To quantify the effect of radiative forcing on the global average temperature, assume at first a perfectly transparent atmosphere to thermal radiation re-emitted by the earth. In this case, the emitted radiative power σT_{e}^{4} would balance exactly the incident radiative power of solar shortwave radiation actually reaching the surface of the earth (1-A)F_{0}/4, i.e. 240 W/m^{2} (F_{0} =1365.8 W/m^{2}, the current albedo is close to 0.3), which gives T_{e}=-18°C (255 K).

The mean global temperature of the earth is 15 °C (288 K), which corresponds to a radiative power equal to σT_{e}^{4} = 390 W/m^{2}. The difference between the radiative power actually issued and what would be emitted if the atmosphere were perfectly transparent is 390-240= 150 W/m^{2}, corresponding to the absorption by the real atmosphere in the presence of greenhouse gases, clouds and aerosols. The effectiveness of radiative forcing, equal to (288-255)/150 = 0.22 °C/(W/m^{2}), is almost 5 times lower than that observed during the resonant forcing of solar cycles.

**Warming and anthropogenic CO2**

*Thermal radiation in the *CO_{2}* absorption band*

*Thermal radiation in the*CO

_{2}

*absorption band*

Knowing the effectiveness of radiative forcing, let’s estimate the impact on the global temperature of a change in the concentration of atmospheric CO2 by involving thermal radiation in the CO2 absorption band between 630 and 700 cm^{-1}. When the concentration increases the altitude from which the thermal radiation is emitted toward the cosmos also increases. As shown in the table representing the partial CO2 pressure (product of the pressure by the concentration) as a function of altitude for concentrations of 300, 400 and 500 ppmv, the temperature of the emitted radiation can be determined using the » normal temperature and pressure « (NTP) which allow to overcome the variations of these two parameters depending on location and time considered. The NTP pressure is expressed by the equation P=1013.25 x (1-0.0065 x z/288.15)^{5.255} where z is the altitude (m). At the sea level the pressure is 1013.25 hPa, the temperature 15 °C and the lapse rate -6.5 °C/km. To 11 km altitude the pressure drops to 226.38 hPa and the temperature to -56 °C. Between 11 km and 20 km (stratosphere) the temperature remains constant. In order to overcome the singularity of temperature at 11000 m resulting from the equation change, an adjustment by a second order polynomial is performed.

Considering thermal radiation corresponding to 400 ppm of CO2, the emitted radiative power derived from the satellite observed infrared spectrum, reduced to the terrestrial sphere, is 38.55/4=9.64 W/m^{2} to a temperature equal to 220 K, which corresponds to an altitude of 12350 m and a CO2 partial pressure of 0.0729 hPa.

Now we can deduce what was the radiative power emitted at the beginning of the industrial age when the CO2 concentration was 300 ppmv, which corresponds to a 10550 m altitude emission and a temperature of 224.93 K. Application of Stefan-Boltzmann’s law gives 9.64 x (224.93/220)^{4} = 10.53 W/m^{2}, i.e. a power gain of 10.53-9.64 = 0.89 W/m^{2}. This corresponds to a decrease in temperature of 0.19 °C from the effectiveness of radiative forcing.

In terms of the climatic impact in the case of doubling the concentration of anthropogenic CO2, a concentration of 500 ppmv corresponds to a 13680 m altitude emission and a temperature of 217.63 K. The re-emitted radiative power derived from the Stefan-Boltzmann’s law is 9.64 x (217.63/220)^{4} = 9.23 W/m^{2}, i.e. a power loss of 9.64-9.23 = -0.41 W/m^{2}, which corresponds to a temperature increase of 0.09 ° C.

*Effect of broadening of the CO2 absorption band*

*Effect of broadening of the CO2 absorption band*

To anthropogenic warming resulting from increasing altitude from which the thermal radiation escapes to the cosmos adds the effect of the broadening of the CO_{2} absorption line when the concentration of carbon dioxide increases. If an atmosphere of average humidity is considered, the increase, since the beginning of the industrial era, of radiative forcing ∆e (W/m^{2}) resulting from radiation re-emitted to the earth in the band 14 – 17 µm is expressed, depending on the relative content of CO_{2} compared to what it was in 1850, i.e. (CO_{2})/(CO_{2})_{1850 }, by the relationship:

∆e=2.94*Log_{2} [(CO_{2})/(CO_{2})_{1850}]

*Effects associated with the increase in water vapor*

*Effects associated with the increase in water vapor*

The increase ∆*Tmg* = 1.1 °C of the Earth’s average temperature *Tmg* leads to an increase in the content of water vapor in the atmosphere because of increased evaporation of the oceans. The increased flux of latent heat *q _{l}* that follows can be written:

*q _{l}* = d

*Q*/d

_{l}*Tmg*× ∆

*Tmg*= 2.303 ×

*Q*×

_{l}*f*’× ∆

*Tmg*

where *Q _{l}* is the latent heat flux to the considered location and

*f'(Tmg)*is the derivative with respect to the temperature

*Tmg*of

*f*(

*Tmg*) (Gill, 1982):

*f*(*Tmg*)=(0.7859+0.03477×*Tmg*)/(1+0.00412×*Tmg*)

Considering at mid-latitude *Tmg* =15 °C, *Q _{l }*= -50 W/m

^{2}(European Reanalysis ERA40 « Kallberg et al, 2005) is obtained

*q*=-3.5 W/m

_{l}^{2}.

Global energy transfer by latent heat is -80 W/m^{2} (Trenberth et al., 2008). Combining the albedo effect due to low clouds and the greenhouse effect due to high clouds, radiative forcing is about -20 W/m^{2} (ERBE program, 1996). Therefore, reduced to the increased flux of latent heat *q _{l}*, the net radiative energy balance resulting in cloudiness is about -20 × 3.2/80 = -0.80 W/m

^{2}.

Considering an average humidity of the air, radiative forcing due to increased water vapor, reduced to the flux of latent heat *q _{l}*, is 1.3 W/m

^{2}(Zhou et al., 2007), so that the resultant radiative effects induced by water vapor and clouds is about 0.5 W/m

^{2}: this includes both shortwave and longwave radiation fluxes. This radiative forcing corresponds to an increase in the temperature

*Tmg*of 0.1 °C by considering an efficiency of 0.22 °C/(W/m

^{2}). It is therefore apparent that, despite simplifying assumptions, it can be stated that

**warming does not involve significant feedback due to increased water vapor in the atmosphere, which results from the combined effects of the increase in albedo and the greenhouse effect**.

**This is indirectly confirmed by the variability of the effectiveness of radiative forcing inferred from climate archives.**

*Cumulative effects of gyral waves and human activity*

*Cumulative effects of gyral waves and human activity*

Considering the atmospheric concentration of CO_{2} observed and predicted assuming the current increase would continue at the rate of 2 ppmv/year, the anthropogenic effect adds to natural climate variability resulting from resonant forcing of gyral waves.

**Conclusion**

**Despite the simplifying assumptions (in particular the NTP conditions are used, a more complete model should involve the variable altitude of the tropopause), anthropogenic warming, which is 0.30 °C in 2015 and will be around 0.35 °C in 2045, explains only a part, of the order of a third, of the warming observed over the second half of the 20 ^{th} century, most of which is attributable to natural climate variability.** The anthropogenic contribution, added to the natural variability, allows replicate closely the

*Tgm*until today. Forecasts show that natural variability will be decisive in the coming decades, the two models that take into account anthropogenic warming indicating a decline in global temperature

*Tmg*beyond 2020. The different assumptions about the increase of CO

_{2}over the next few decades has little influence on results; this is why only the most pessimistic scenario corresponding to an increase of 2 ppmv/year is considered.

Anthropogenic warming has increased significantly in the early 2000. Its slow growth over the next decades results 1) from the altitude of emission of thermal radiation in the CO_{2} absorption band which, in first approximation, is now located in the stratosphere where the temperature depends little on the altitude 2) according to the Beer-Lambert law, from the logarithmic growth in radiative forcing to the wings of the absorption band when the CO_{2} content increases.

**References**

Clive Best, Doubling CO2 and basic physics, http://clivebest.com/blog/?p=1169

Hanel R.A., B. Schlachman, D. Rogers, D. Vanous, The Numbus 4 Michelson interferometer, Appl Opt. 1971 Jun 1;10(6):1376-82. doi: 10.1364/AO.10.001376.

IPCC Fifth Assessment Report – Climate Change 2013 – www.ipcc.ch/report/ar5/wg1/

Dufresne J-L and Treiner J. (2011) L’effet de serre atmosphérique : plus subtil qu’on ne le croit !, http://www.udppc.asso.fr/bupdoc/consultation/article-bup.php?ID_fiche=21046 (in French)

Anthony Watts, The Logarithmic Effect of Carbon Dioxide, http://wattsupwiththat.com/2010/03/08/the-logarithmic-effect-of-carbon-dioxide/

ERBE (The Earth Radiation Budget Experiment) NASA program, 1996, FS-1996-05-03-LaRC, http://www.nasa.gov/centers/langley/news/factsheets/ERBE.html

Gill AE (1982) Atmosphere–Ocean Dynamics, International Geophysics Series, 30, Academic Press, 662 pp.

Kallberg, P., Berrisford, P., Hoskins, B., Simmons, A., Uppala, S. and Lamy-Thepaut, S (2005) ‘Atlas of the atmospheric general circulation’. ECMWF ERA-40 Project Report Series, No. 19. European Centre for Medium-Range Weather Forecasts, Shinﬁeld, Reading, UK (available from **www.ecmwf.int/publications**)

Trenberth K. E., J. T. Fasullo, and J. Kiehl (2008) Earth’s Global Energy Budget, American Meteorological Society, DOI:10.1175/2008BAMS2634.1

Zhou, Y., D. P. Kratz, A. C. Wilber, S. K. Gupta, and R. D. Cess (2007), An improved algorithm for retrieving surface downwelling longwave radiation from satellite measurements, J. Geophys. Res., 112, D15102, doi:10.1029/2006JD008159.

**Glossary**

[i] **The latent heat **is heat-exchanged in the change of state of the seawater during the process of vaporization.

[i] A **black body** in thermal equilibrium (that is, at a constant temperature) emits electromagnetic radiation called black-body radiation. The radiation is emitted according to Planck’s law, meaning that it has a spectrum that is determined by the temperature alone, not by the body’s shape or composition. For example the Sun can be considered as emitting black-body radiation that exhibits a distribution in energy characteristic of the temperature T=5780 K: the photosphere contains photons of light nearly in thermal equilibrium, and some escape into space while producing negligible effect upon the equilibrium of the radiation inside the photosphere.

Escaping radiation from the earth also approximates black-body emission that exhibits a distribution in energy characteristic of the temperature T=288K. Photons whose energy corresponds to the saturated absorption bands of the atmosphere (water vapor, CO2) are also in thermal equilibrium if they escape from the opaque layer: the temperature gradually decreases with altitude in the troposphere (-6.5°C/km).