The IPCC estimate
that doubling the CO2 concentration is sufficient to increase the global temperature by about 3C by
2080/2100. We can calculate the temperature increase from a
doubling of CO2 using the IPCC’s own figures and see if
the claim of a 3C increase is justifiable. First, we need to estimate the
radiative forcing from a doubling of CO2. Apparently the IPCC use this equation to calculate it: ΔRF = 5.35xLn(C1/C0). Where C1 is
the final CO2 concentration, C0 is the
reference concentration, Ln is the natural logarithm of, and
ΔRF stands for increment of radiative forcing. If we accept it for argument’s
sake we can calculate the amount of RF that the IPCC say CO2 would have.
A doubling of 390ppmv is 780ppmv. Slotting those values into C1 and C0 gives us
a net-anthropogenic RF of: ΔRF = 5.35xLn(780/390) = 3.7W/sq.m. Now that we
know the RF from a doubling of CO2 we can determine how much
radiation-enhancement this contributes to the overall greenhouse.
According to Trenberth (lead IPCC author) the greenhouse back-radiation from all sources amounts to
333W/sq.m (Trenberth 2008: Global Energy Budget).

Therefore the anthropogenic contribution from a doubling of CO2 to the entire planetary greenhouse amounts to an inconsequential 1% (i.e. 3.7/333). To estimate the resultant temperature increase we must first calculate the temperature of the Earth without a greenhouse. The effective blackbody temperature of the Earth (i.e. the assumed temperature of the planet without a greenhouse) can be calculated with the following equation: T = [(L) (1 - A) (R)^2//4σ (D)^2]^1/4. Where L is the luminosity of the Sun (6.32x10^7W/sq.m), A is the albedo of Earth (0.3), R is the radius of the Sun (6.96x10^8m), σ is the Stefan-Boltzmann constant and D is the distance from Earth to the Sun (1.496x10^11m). Slotting the values into the equation gives us an effective blackbody temperature of: T = [(6.32x10^7) (1-0.3) (6.96x10^8)^2//4σ (1.496x10^11)^2]^0.25 = 254.9K.

Because the Earth’s blackbody temperature is 255K and its average surface temperature is 288K it is suggested that the temperature difference of 33K is due to the atmospheric greenhouse. This implies that the greenhouse back-radiation of 333W/sq.m from all sources (as calculated by Trenberth) is sufficient to increase the global mean surface temperature of the Earth by 33C above its blackbody temperature of -18C. Hence this gives us a linear relationship between ΔT (at the surface) and ΔRF (by the atmospheric greenhouse) of 0.1C per 1W/sq.m (i.e. 33/333). Therefore the RF of 3.7W/sq.m produced by a doubling of CO2 by 2080/2100 is sufficient to increase the global mean surface temperature by 0.37C. This calculation assumes that the relationship between ΔT and ΔRF is linearly proportional, which it isn’t. The Stefan-Boltzmann law governs the relationship between radiation and temperature and the law deems that the absolute temperature of a body will increase according to the 4th-root of radiation that is warming it. When the Stefan-Boltzmann law is taken into account the effect is to reduce the size of the possible human component to 0.31C. However we do not need to bother with this small adjustment and can simply conclude that the total global warming on a doubling of CO2 must be no more than 0.37C.

We must now take into account the hypothesised positive feedbacks. The IPCC have a second feedback equation for this (as before, it may not be correct): ΔT = λxΔF. Where ΔT is the temperature increase, λ is the climate sensitivity parameter (a typical value is about 0.8, occasionally referred to as the ‘Hansen Factor’) and ΔF is the radiance from CO2. The IPCC’s second formula takes the radiance from CO2 and converts it into a corresponding temperature increase. The increase in surface temperature of 0.37C from CO2 corresponds to a radiance of 2W/sq.m at the mean surface temperature of 288K by the Stefan-Boltzmann law. The IPCC’s second formula tells us that the new temperature achieved after feedbacks have occurred should be as follows: ΔT = λxΔRF = 0.8x2 = 1.6C. Hence the amplification-factor implied by the IPCC’s formula is about 4. So, based on the IPCC’s own figures we should get a benign warming of just 1.6C and CO2’s direct effect can be no more than 0.37C. Clearly this figure is less than the IPCC’s model-generated figure of 3C. So to my mind, the IPCC’s claim that unchecked human CO2 emissions will cause 3C of warming by 2080/2100 is a gross exaggeration that contradicts the implications of its own “science”.

Therefore the anthropogenic contribution from a doubling of CO2 to the entire planetary greenhouse amounts to an inconsequential 1% (i.e. 3.7/333). To estimate the resultant temperature increase we must first calculate the temperature of the Earth without a greenhouse. The effective blackbody temperature of the Earth (i.e. the assumed temperature of the planet without a greenhouse) can be calculated with the following equation: T = [(L) (1 - A) (R)^2//4σ (D)^2]^1/4. Where L is the luminosity of the Sun (6.32x10^7W/sq.m), A is the albedo of Earth (0.3), R is the radius of the Sun (6.96x10^8m), σ is the Stefan-Boltzmann constant and D is the distance from Earth to the Sun (1.496x10^11m). Slotting the values into the equation gives us an effective blackbody temperature of: T = [(6.32x10^7) (1-0.3) (6.96x10^8)^2//4σ (1.496x10^11)^2]^0.25 = 254.9K.

Because the Earth’s blackbody temperature is 255K and its average surface temperature is 288K it is suggested that the temperature difference of 33K is due to the atmospheric greenhouse. This implies that the greenhouse back-radiation of 333W/sq.m from all sources (as calculated by Trenberth) is sufficient to increase the global mean surface temperature of the Earth by 33C above its blackbody temperature of -18C. Hence this gives us a linear relationship between ΔT (at the surface) and ΔRF (by the atmospheric greenhouse) of 0.1C per 1W/sq.m (i.e. 33/333). Therefore the RF of 3.7W/sq.m produced by a doubling of CO2 by 2080/2100 is sufficient to increase the global mean surface temperature by 0.37C. This calculation assumes that the relationship between ΔT and ΔRF is linearly proportional, which it isn’t. The Stefan-Boltzmann law governs the relationship between radiation and temperature and the law deems that the absolute temperature of a body will increase according to the 4th-root of radiation that is warming it. When the Stefan-Boltzmann law is taken into account the effect is to reduce the size of the possible human component to 0.31C. However we do not need to bother with this small adjustment and can simply conclude that the total global warming on a doubling of CO2 must be no more than 0.37C.

We must now take into account the hypothesised positive feedbacks. The IPCC have a second feedback equation for this (as before, it may not be correct): ΔT = λxΔF. Where ΔT is the temperature increase, λ is the climate sensitivity parameter (a typical value is about 0.8, occasionally referred to as the ‘Hansen Factor’) and ΔF is the radiance from CO2. The IPCC’s second formula takes the radiance from CO2 and converts it into a corresponding temperature increase. The increase in surface temperature of 0.37C from CO2 corresponds to a radiance of 2W/sq.m at the mean surface temperature of 288K by the Stefan-Boltzmann law. The IPCC’s second formula tells us that the new temperature achieved after feedbacks have occurred should be as follows: ΔT = λxΔRF = 0.8x2 = 1.6C. Hence the amplification-factor implied by the IPCC’s formula is about 4. So, based on the IPCC’s own figures we should get a benign warming of just 1.6C and CO2’s direct effect can be no more than 0.37C. Clearly this figure is less than the IPCC’s model-generated figure of 3C. So to my mind, the IPCC’s claim that unchecked human CO2 emissions will cause 3C of warming by 2080/2100 is a gross exaggeration that contradicts the implications of its own “science”.