89. The Greenhouse Effect on Mars

In my previous three posts I have explained how the Greenhouse Effect (GHE) works on Earth, and how it is affected by changes to the carbon dioxide (CO₂) concentration in the atmosphere. The problem with studying the GHE on Earth, though, is that its operation is complicated by the presence of large amounts of water vapour in the atmosphere. As water vapour also has a broader absorption band and higher atmospheric concentration than CO₂, this means that changes in the CO₂ concentration are less important than they would be otherwise. If we want to understand and measure the GHE just due to carbon dioxide, then we need an environment with high levels of CO₂ but low levels of other greenhouse gases. In this respect one of the best places to study is Mars. 

The Atmosphere of Mars

The Martian atmosphere has some similarities with Earth but many differences. It contains many of the same gases (nitrogen, oxygen, water vapour, argon, CO₂), but the proportions are vastly different. The atmosphere of Mars is 96% CO₂ with about 2% nitrogen and 2% Argon (although different pages on Wikipedia give slightly different values such as 95% CO₂ and 3% nitrogen). There are also trace levels (< 0.1%) of other gases such as water vapour (210 ppm) and oxygen (0.15%). 

The other main difference in terms of the atmosphere is the pressure at the surface. At 610 Pa this is only 0.60% of the surface pressure on Earth, and as about 96% of this is CO₂, this means that there is a surface density of 3610 mol/m² of CO₂ on Mars compared to only 150 mol/m² on Earth. So any outgoing radiation from the surface of Mars has 24.06 times as much carbon dioxide gas to penetrate, in order to escape into outer space, compared to on Earth. What this means in practice is that the Greenhouse Effect in the Martian atmosphere should be easier to analyse because it can only have one source - CO₂.

 

The Energy Balance for Mars

Mars is also approximately 52% further from the Sun than is Earth, so one might expect that to mean that its surface is colder. This is true, but not as much as it should be based on distance alone. 

The solar radiation flux entering the Martian atmosphere is only 586 W/m² compared to the 1360 W/m² that irradiates the Earth. As this radiation is spread over a surface area (4πr²) that is four times greater than the cross-sectional area of the planet (πr²) in each case, this means that the average radiation flux at the Martian surface is 143.5 W/m². Yet this is only 22% less than the 184 W/m² that reaches the Earth's surface. This is because nearly 50% of incident radiation on Earth is either reflected by clouds or is absorbed by ozone and water vapour in the upper atmosphere. But there are no clouds, ozone or significant water vapour on Mars.

Then there is the issue of surface albedo or Bond albedo. This is the proportion of incident radiation that is reflected by the planet back out into space without being absorbed, either from clouds or the surface. For Earth this is about 31%; for Mars it is only 25%. This means that the average surface absorption on Mars is 108 W/m² compared to 161 W/m² on Earth. So while Mars only receives 43% of the solar radiation that Earth does, after absorption and reflection the Martian surface receives 67% of the radiation that the Earth's surface does. That is a relative increase of more than 50% for Mars which partially compensates for its greater distance from the Sun.

Calculating the Surface Temperature of Mars

If we now invoke the Stefan-Boltzmann law (see Eq. 13.1 in Post 13),

I = σT⁴

(89.1)

where I is the surface radiation flux, σ is the Stefan-Boltzmann constant, and T is the surface temperature in kelvins, we see that a surface radiation flux of 108 W/m² equates to a mean surface temperature on Mars of 209 K (or -64°C). Yet the true mean temperature is thought to be about 215 K (or -58°C). The difference is due to the Greenhouse Effect. 

For comparison, on Earth a solar flux at the surface of 161 W/m² would equate to a mean surface temperature of 231 K (or -42°C), yet the true mean temperature is about 289 K (or +16°C). So the GHE on Earth adds 58°C of warming while on Mars it only adds 6°C. Yet there is almost twenty-five times more carbon dioxide on Mars (3690 mol/m²) than on Earth (150 mol/m²), so you would expect the greenhouse effect due to CO₂ to be stronger. But how much stronger? The answer is: not very. In fact it is significantly less than the actual measured value on Earth.

Calculating the Strength of the GHE on Mars

In Post 87 I showed how the width of the CO₂ absorption band at 15 µm can be determined using the known concentration of CO₂ (N_o), its scattering or absorption cross-section (σ_s), the quantized frequency of rotation of the CO₂ molecules (B), and the temperature (T). From this it is possible to estimate the critical CO₂ concentration needed to absorb most of the infra-red radiation (N_th). The term N_th can be estimated to be 0.5 mol/m² based on the value of σ_s, while N_o will be about half the total CO₂ concentration, so N_o = 1845 mol/m². From this we can estimate the maximum number of excited rotational states in the absorption band, J_th, using (see Post 87)

(89.2)

where k is Boltzmann's constant, h is Planck's constant and Z is a normalization constant (see Eq. 87.2 in Post 87) equal to 185.5 in this case. The ratio term kT/hB has the value 185.3 (as hB = 0.1 meV and T = 215 K) and is always approximately equal to the Z value. 

The result we obtain is that J_th = 36.3, which, given that the spacing of the bands is 0.2 meV, means that the absorption band has a width of 14.52 meV and extends from a wavelength of 13.78 µm to 16.44 µm. This compares to a calculated range of 14.00 µm to 16.14 µm for the same band on Earth, although the measured width on Earth is actually found to be from about 13.35 µm to 17.35 µm. So even though the calculated width of the 15 µm absorption band for Mars is slightly larger than the equivalent for Earth, it is not significantly greater. But it is significantly less than the measured band width on Earth.

 

Fig. 89.1: The electromagnetic emission spectrum for the surface of Mars at a mean temperature of 215K (blue curve) together with the absorption profile due to CO₂ between 13.78 µm and 16.44 µm (red curve).

The impact of the high Martian atmospheric CO₂ concentration on the radiation feedback is demonstrated in Fig. 89.1 above. The red curve indicates the proportion of the outgoing infra-red radiation (blue curve) that is reflected by the CO₂ molecules and it amounts to f = 12.5%. This is slightly more than the 10% seen for the GHE due to backscattering from CO₂ on Earth (see Post 87) despite the absorption band being further from the peak in the emission spectrum due to the lower surface temperature on Mars.

Calculating the Temperature Rise

The radiation feedback of  f = 12.5% shown in Fig. 89.1 equates to a 14.3% increase in the surface radiation and also of T⁴ (because of Eq. 89.1), which then equates to a 3.4% increase in T. So if the initial surface temperature on Mars without the GHE was 209 K, the temperature with the GHE included will be 3.4% greater, or 216.1 K. This means that the temperature rise at the surface due to the Greenhouse Effect is expected to be 7.1 K, which is pretty close to the observed value of around 6 K (or 6°C). The reason for the small difference could be the limited availability of accurate Mars temperature data, or the uncertainty in the value of the CO₂ absorption cross-section, σ_s.

We know how much radiation from the Sun is arriving at Mars to high accuracy, but knowing how much is being absorbed by the planet surface is more difficult as this depends on an accurate measurement of the Bond albedo. However, conventional astronomical telescopes should be able to measure the reflected radiation to pretty good accuracy as well. That allows us to estimate the expected mean surface temperature without the GHE to fairly high precision. The problem is knowing what the actual surface temperature is with the GHE in operation. This is a difficult enough calculation to do on Earth where we have over 16,000 weather stations measuring the surface temperature on a daily basis, and numerous satellites in orbit. Sadly, none, or very little, of this exists for Mars.

So far in this blog post I have assumed a value of 215 K for the mean surface temperature of Mars, but some reports have put it as high as 225 K (or as low as 210 K). In which case Z = 194.15 and J_th = 37.0. This leads to a 15 µm band stretching from 13.76 µm to 16.47 µm, and a feedback factor of f = 13.0%. Under these circumstances the warming from the Greenhouse Effect increases slightly, but only to 7.7°C. 

 

A Comparison with Earth

For comparison, it is instructive to hypothecate the extent of warming on Earth if its atmosphere also contained 3690 mol/m² of CO₂. In that case Z = 249.3 and J_th = 41.6, which leads to a 15 µm band stretching from 13.62 µm to 16.67 µm, and a feedback factor of f = 14.1%. The resulting predicted temperature rise due to CO₂ would be 10.74 K, which is 3.26°C less than the 7.48°C rise currently predicted for Earth as was shown in Post 87). So a twenty-five fold increase in the CO₂ concentration would only result in a 3.26°C temperature increase, although as I showed in Post 87, masking by water vapour would probably reduce this by 75% to only 0.8°C. 

It is a point of note that the density of CO₂ molecules on Mars (3690 mol/m²) is more than double the combined density of all the greenhouse gases on Earth (1560 mol/m²), yet it results in a temperature rise of 5°C - 7°C that is almost ten times less than the 58°C observed for Earth. This is mainly because most of the GHE on Earth is due to water vapour as the width of the CO₂ absorption band is so much less than that for water vapour. Even increasing the concentration of CO₂ on Mars by a factor of twenty-five cannot appreciably change this.

Summary and Conclusions

What I hope I have shown in this post is that Mars is a good test bed for studying the Greenhouse Effect (GHE). Knowing only its albedo, the atmospheric concentration of CO₂, and the intensity of radiation arriving from the Sun, it is possible to accurately predict the temperature rise due to the Greenhouse Effect. This I have predicted to be about 7°C, in close agreement with the current estimate based on observational data (6°C). And this is despite the significant uncertainty over the true measured value of the mean surface temperature on Mars.

The reduced GHE on Mars relative to the Earth occurs despite its much higher (i.e. 24 times greater) atmospheric CO₂ concentration. This in turn suggests that the increasing levels of atmospheric CO₂ we are currently seeing on Earth will produce only slight temperature increases in the future. 

Yes, Mars has its own complicating factors. Heat retention on Mars is limited by the thin atmospheric blanket compared to Earth. This means that the planet does not retain heat very well, but conversely it means that the atmosphere will warm quickly when heated by the Sun. For this reason it may be better to consider Mars under direct solar illumination in daytime. Under these conditions the peak solar flux at the surface of Mars will be four times greater than stated above, or 432 W/m². This will equate to a peak surface temperature of 295 K (or +22°C). Yet the actual maximum temperature is reported to be around 303 K to 308 K (or +30°C to +35°C). So on this measure the warming from the Greenhouse Effect in daytime near the equator appears to be in the range 8°C to 13°C. Yet the predicted value based on a calculation of the width of the 15 µm absorption band is found to be 10.9°C, in other words in the mid-range of the observed values. Once again this is still much less than the total warming seen on Earth and comparable to the contribution to Earth's GHE seen just from CO₂.

87. How the Greenhouse Effect on Earth changes with increasing carbon dioxide concentration

In my previous post (Post 86) I explained how infra-red photons emitted by the Earth's surface interact with carbon dioxide (CO₂) in the atmosphere to create the Greenhouse Effect. I also showed that increasing the temperature of the planet and increasing the concentration of carbon dioxide in the atmosphere will both lead to an increase in the width of the 15 µm absorption band of CO₂. This in turn will increase the amount of radiation that is backscattered by the CO₂, and therefore increase the amount of radiation heating the surface of the planet. 

In this post I will attempt to quantify the temperature increase for different increases in the CO₂ content of the atmosphere using the results presented in Post 86 and Post 85. What I will show is that the increase in atmospheric levels of CO₂ from 280 ppm in 1750 to almost 420 ppm today can only be responsible for at most a 0.5°C increase in average temperatures. This is only about 40% of the 1.2°C claimed by the IPCC and climate scientists. In fact the actual temperature rise due to CO₂ is likely to be less than half the calculated value of 0.5°C due to the masking effects of water vapour, and could be as little as 0.1°C. To put this into context, this is less than the values I have calculated for urban heating effects from waste heat (see Post 14 and Post 29) which would persist even without the use of fossil fuels.

The maths and physics

The starting point for this analysis is the quantum structure of the absorption band. This is shown in Fig. 87.1 below and was discussed in detail in Post 86. The key issue is the height of the various absorption lines in the P and R branches. These are identified by their angular momentum quantum number, J, which is numbered for each branch from the centre of the band, Q. 

Fig. 87.1: The detailed structure of the 15 µm absorption band for CO₂ showing the absorption peaks associated with rotational transitions.

In Post 86 I also showed that the width of the 15 µm band is determined by the value of J that satisfies the following equation (see also Eq.86.9), this value being denoted as J_th.

(87.1)

In this equation T is the thermodynamic temperature in kelvins, k is the Boltzmann constant, h is Planck's constant and B is the frequency of the rotational angular momentum states. For the rotational transitions shown for CO₂ in Fig. 87.1, hB = 0.1 meV and is equal to half the energy separation of the lines in the spectrum in Fig. 87.1. The other terms will be explained below.

 

The Z term

The term Z is a normalization term equal to the total number of possible rotational states per molecule in the R (or P) branch as follows. 

(87.2)

 

In the case of Earth where the mean surface temperature T = 289 K, the term Z = 249.3. The energy term E_J = J(J+1)hB. As the degeneracy term (2J + 1) is the differential of the J component of the energy term J(J+1), it follows that for large T the summation in Eq. 87.2 reduces to an integral over all J states, in which case Z ≈ kT/hB.

 

 

The N_o term

 

The term N_o in Eq. 87.1 is equal to the total number of CO₂ molecules per unit surface area found in the R branch. This can be estimated as being equal to approximately half the molecules, with the other half being in the P branch which is assumed to be the mirror image of the R branch (but is not really as was explained in Post 86). This also neglects the significant number of CO₂ molecules (particularly at low temperatures) found in the Q peak. Nevertheless, this approach does at least set an upper limit to the width of the R branch, and thus the width of the 15 µm band as a whole. And as will be shown below, it does give results that are remarkably accurate. As the number of CO₂ molecules per unit surface area found on Earth is 150 moles per square metre, it therefore follows that N_o is equal to 75 mol/m².

Calculating N_th and J_th.

The final remaining parameter to calculate is N_th. Ideally, if the absorption band edge had vertical edges, it would be the threshold number of CO₂ molecules per unit area that are just sufficient to completely block the radiation and would be equal to the reciprocal of the scattering cross-section, σ_s. As σ_s for CO₂ molecules is estimated to be between 10^-24 m² and 10^-23 m² in the 15 µm band, that would imply a value for N_th of about 1 mol/m². In practice, however, the band edge is curved so the usual definition of the edge is to take the position of the half maximum. This means using a value of N_th = 0.5 mol/m² is more appropriate.

With all the parameters now set we can calculate J_th using Eq. 87.1 above. The result we get is 29.4, which when multiplied by the line spacing, 2hB, gives the width of the R branch as 47.4 cm^-1 in wavenumbers. Assuming the P branch is identical means that the 15 µm band will extend from 619.6 cm^-1 to 714.4 cm^-1, or from 14.00 µm to 16.14 µm. This is remarkably close to the 14.2 µm to 16.2 µm that is generally observed for the peak in the absorption.

Having calculated the width of the 15 µm band with an atmospheric CO₂ concentration of 420 ppm, we can also repeat the procedure for any other CO₂ concentration of our choosing. For example, an atmospheric CO₂ concentration of 280 ppm that is characteristic of global conditions in 1750 leads to a value for J_th of 27.3, which means that the width of the R branch would be 44.0 cm^-1.

The temperature rise

In Post 85 I showed how the reflection of a fraction f of outgoing infra-red radiation would reheat the Earth's surface and cause the radiation it absorbed to increase from I_o to a higher value I_T as follows

(87.3)

Then in Post 86 I showed how the width of the 15 µm absorption band could be used to determine the value of f by calculating the relative area of this band under the absorption spectrum (see Fig. 86.1). This can then be used to infer a temperature rise due to the absorption by utilizing the Stefan-Boltzmann law,

 I = σT⁴. 

(87.4)

If I_T is the intensity of radiation emitted by the Earth's surface normally (i.e. 396 W/m²), and f is the fraction of radiation reflected back by the CO₂, then the intensity of radiation emitted by the Earth's surface without the CO₂ greenhouse effect will, according to Eq. 87.3, be I_o = (1-f)I_T. We can then use Eq. 87.4 to calculate the respective surface temperatures T_f and T_o for each radiation emission intensity, I_T and I_o. The difference in the two temperatures will be the warming due to the CO₂.

I showed above that an atmospheric CO₂ concentration of 420 ppm leads to an absorption band that extends from 14.00 µm to 16.14 µm. Combining this with the black body spectrum of the Earth at T_f = 289 K (see Fig. 87.2 below) allows us to determine f to be f = 10.0%. This in turn implies that I_o = 356.1 W/m² (where I_o is the radiation intensity without CO₂ feedback), and thus T_o = 281.52 K. So the temperature rise due to CO₂ is 7.48 K.

 

Fig. 87.2: The electromagnetic emission spectrum for the Earth's surface at a mean temperature of 289K (blue curve) together with the absorption profile due to CO₂ between 14.00 µm and 16.14 µm (red curve).

Now if we reverse this calculation but use an atmospheric CO₂ concentration of 280 ppm, we find that the absorption band now extends from 14.06 µm to 16.05 µm, so f = 9.28%. The value of I_o = 356.1 W/m² will be the same as before but the addition of a different amount of CO₂ will change I_T and T_f because f is different. The new values will be I_T = 392.6 W/m² and T_f = 288.46 K. So the temperature rise from CO₂ is now only 6.94 K. This implies that the temperature rise since 1750 due to the atmospheric CO₂ concentration increasing from 280 ppm to 420 ppm is only 0.54°C (i.e. 7.48°C - 6.94°C). 

If we repeat this process for other past or future (potentially) CO₂ concentrations we can calculate a theoretical temperature rise for each. This is shown in the graph in Fig. 87.3 below with the temperature changes all measured relative to the 1750 value when the atmospheric CO₂ concentration was 280 ppm.

 

Fig. 87.3: The theoretical effect of an increasing atmospheric CO₂ concentration on the contribution of CO₂ to global warming.

What Fig. 87.3 demonstrates is that the expected temperature increase from an increase in atmospheric CO₂ has a logarithmic dependence on the CO₂ concentration. However, the trend for concentrations between 280 ppm and 420 ppm is fairly linear and leads to a 0.5°C increase. Overall it appears that doubling the CO₂ concentration leads to about 1°C (or 1 K) of warming.

The interpretation of the data

While the logarithmic trend shown in Fig. 87.3 is in general agreement with climate models, the magnitude of the temperature changes are not. Whereas Fig. 87.3 suggests that 420 ppm of CO₂ leads to about 0.52°C of warming, the IPCC and climate science are claiming the rise is much greater at about 1.2°C. The difference, I suspect, is probably down to the impact of water vapour. Unfortunately, there are many ways that water vapour can impact the temperature trend.

The conventional view from climate scientists is that water vapour is a positive amplifier; the theory being that a warmer climate causes the amount of water vapour in the troposphere to increase, thus creating even more warming. As I pointed out in Post 86, the total feedback factor for infra-red radiation, f, is about 59%, while CO₂ alone can only account for about 18%, assuming that the 15 µm CO₂ absorption band width is measured at its half-maximum points from 13.35 µm to 17.35 µm. So the assumption is that water vapour is responsible for the rest, and that as its atmospheric concentration is dependent on the temperature, its concentration will increase as the level of CO₂ increases. So this will make the feedback, f, increase about three times faster than from CO₂ alone and so massively increase the temperature change to 1.2°C. The problem with this theory is that it ignores two major snags. 

First, the 15 µm CO₂ absorption band overlaps with the H₂O absorption band, as shown in Fig. 87.4 below. At the high wavelength edge of the 15 µm CO₂ band (17 µm) the water vapour will absorb almost 100% of the outgoing radiation while at the low wavelength edge (13 µm) it will absorb about 50%. This means that about 75% of any increase in the width of the 15 µm CO₂ absorption band will be masked from the outgoing radiation by the water vapour. In which case the temperature rise will only be 25% of the predicted value, or about 0.13°C.

 

Fig. 87.4: The absorption bands of carbon dioxide and water vapour at sea level.

Secondly, the window in the H₂O absorption band extends from 8 µm to about 15 µm (using the half maxima points). This window allows only 41% of the Earth's outgoing infra-red radiation to escape. As we know that the feedback factor f = 59%, this means that water vapour could be responsible for absorbing and reflecting almost all the outgoing infra-red radiation that is absorbed and reflected. In other words, the 15 µm CO₂ band is not needed, and in fact is probably, largely redundant because it is hiding behind the water vapour. So again, a small change to the width of the CO₂ band is unlikely to cause any major temperature changes. This is why so many eminent physicists have so many serious reservations regarding the global warming predictions coming out of climate science.

 

The conclusions

1) Increasing the atmospheric CO₂ concentration will increase the width of its 15 µm absorption band.

2) In the absence of water vapour this could raise global temperatures, with an increase in CO₂ concentration from 280 ppm to 420 ppm resulting in a 0.5°C increase in global temperatures. 

3) The projected temperature increase has a logarithmic dependence on CO₂ concentration (see Fig. 87.3).

4) Water vapour masks most of the CO₂ 15 µm absorption band and so dominates the infra-red absorption. It can also account for almost all of the radiation feedback on its own.

5) The impact of water vapour means that an atmospheric CO₂ concentration rising from 280 ppm to 420 ppm could result in as little as a 0.13°C increase in global temperatures. This is ten times less than is currently claimed by climate science.

The caveats and discrepancies

In this analysis there are major uncertainties over the value of the CO₂ scattering cross-section, σ_s, and the widths of the CO₂ and H₂O absorption bands. This is also due to the difficulty in estimating the parameters N_o, N_th and J_th. However, the overall level of agreement between this analysis and real data and existing theory is encouraging.

The major discrepancy is between the measured width of the CO₂ 15 µm absorption band at its half maximum, where it extends from 13.35 µm to 17.35 µm, with the predicted width based on J_th where it extends from 14.00 µm to 16.14 µm. This difference could be due to line broadening from pressure broadening and temperature.

86. How photons interact with carbon dioxide molecules

In my previous post (Post 85) I countered a number of myths surrounding the greenhouse effect, and outlined how it really works. Many people falsely believe that it is caused by a heating of the atmosphere by outgoing infra-red radiation, but this is not the case. The key concept at the heart of the greenhouse effect is photon scattering, or more accurately, the absorption and re-emission of infra-red photons by molecules of the greenhouse gases. There are three main gases that perform this role: carbon dioxide (CO₂), water vapour (H₂O), and methane (CH₄). Each of these compounds absorbs electromagnetic radiation at its own set of unique wavelengths, but it is the absorption in the wavelength range between 6 µm and 90 µm that is key as this is where 96% of the thermal emission from the Earth's surface takes place. As CO₂ is seen as the most important player in terms of anthropogenic greenhouse gas emissions I will consider its role in greatest detail.

The emission and absorption spectra

In Fig. 85.1 of Post 85 I showed the main absorption bands of carbon dioxide in the infra-red part of the electromagnetic spectrum. In total there are four main bands at 2 µm, 2.7 µm, 4 µm and 15 µm, but only the 15 µm band is of any importance as the other three have energies that are way beyond the peak of the infra-red emission spectrum for Earth's outgoing radiation. This is illustrated in Fig. 86.1 below, where the blue curve represents the Earth's emission spectrum at 289 K and the red curve shows which frequencies are absorbed by CO₂. The area under the blue curve thus represents the total power of the radiation emitted at the Earth's surface, while the area under the red curve is the amount of radiated heat that can be absorbed and then reflected by CO₂.

 

Fig. 86.1: The electromagnetic emission spectrum for the Earth's surface at a mean temperature of 289K (blue curve) together with the absorption profile due to CO₂ between 14.2 µm and 16.2 µm (red curve).

The graph in Fig. 86.1 shows that only the 15 µm band is important as it is the only one close to the radiation peak. It is centred at 667 wavenumbers. Note: wavenumbers are the number of wavelengths of a radiation type that can fit in a distance of one centimetre. They are used because they are a measure of frequency, they result in much smaller numbers compared to other units of frequency like hertz, and are proportional to the photon energy of the radiation. The absorption bands at 2 µm, 2.7 µm and 4 µm correspond to wavenumbers of 5000, 3704 and 2500 cm^-1 respectively. Consequently, they can only interact with the tail of the frequency spectrum where there is very little energy being emitted (i.e. less than 1% of the total).

What the curves in Fig. 86.1 show is that only about 9% of the Earth's outgoing radiation can be absorbed and then reflected back by the carbon dioxide molecules in the atmosphere. In other words, the area under the red curve is only 9% of the area under the blue curve. Yet in Post 85 I showed that in reality 59% of the radiation is reflected back. The difference in these two numbers is largely due to water vapour which not only accounts for most of that difference (i.e. 50%), but also overlaps the 15 µm absorption peak of CO₂ (see Fig. 85.1 in Post 85), and so may render much of its function redundant. This is one reason why the claim that increasing the amount of CO₂ in the atmosphere will necessarily increase the strength of the greenhouse effect is disputed by many.

 

Fig. 86.2: The absorption bands of carbon dioxide and water vapour at sea level.

 

The other complicating factor is the actual width of the 15 µm band. The peak of maximum absorption generally stretches from about 14.2 µm to 16.2 µm, or from about 700 to 620 wavenumbers (or a width of 10.8 meV), and results in a feedback factor f = 9.2% and a temperature rise of 6.88°C. 

The band width, though, is dependent on both temperature and pressure. The full band at its tails can stretch from 12.5 µm to 18.5 µm (800 to 540 wavenumbers or a width of 32.2 meV), but absorption in the outer parts of the band decreases dramatically (see Fig. 86.2 above). If we were to use the full width of the band to determine the backscatter it would result in 26.8% of the outgoing radiation being reflected (see Fig. 86.3 below) and a temperature rise of 21.7°C; still much less than the 59% backscatter and the 58°C temperature rise that is actually seen.

Fig. 86.3: The electromagnetic emission spectrum for the Earth's surface at a mean temperature of 289K (blue curve) together with the potential absorption profile due to CO₂ between 12.5 µm and 18.5 µm (red curve).

A more realist estimate would be achieved by considering the width at the half maximum points of the band. This will be from about 13.35 µm to 17.35 µm (749 to 576 wavenumbers or a width of 21.4 meV) and would result in an absorption and reflection of 18.1%. This equates to a temperature rise at the surface of the Earth of 14.1°C.

Backscattering by Rayleigh scattering

The photons of infra-red radiation emitted by the Earth's surface interact with the carbon dioxide in two ways. One is by elastic scattering and the other via absorption and re-emission.

In elastic scattering the photons basically bounce off the CO₂ molecules like the collision of two billiard balls. This is the basis of what is known as Rayleigh scattering, named after the nineteenth-century British physicist Lord Rayleigh (John William Strutt), and it is a result of the electromagnetic interaction of the photon with the charged particles (electrons and protons) in the molecule. After collision the photons will end up travelling in a different direction. Some will be reflected back while others will carry on in a forward direction. It is the ones that are reflected that contribute to the greenhouse effect. The intensity of photons at a distance R from the molecule after being scattered through an angle θ is given by the equation

(86.1)

It can be seen that the overall scattering probability depends on the wavelength of the radiation (λ) and the strength of the polarizability of the molecule doing the scattering (α). The strong inverse dependence on wavelength explains why the sky is blue in daylight. This happens because blue light photons with wavelengths of about 400 nm scatter sixteen times more often as they travel through the Earth's atmosphere than red light photons of wavelength 800 nm do. It also means that Rayleigh scattering from carbon dioxide at wavelengths of 15 µm is about a billion times less likely to occur than the scattering of visible light. This is because the wavelengths of the photons being scattered are at least twenty times greater, so the scattering rate of 15 µm photons by CO₂ molecules is at least 160,000 times less than for visible light. But in addition, the proportion of CO₂ in the atmosphere is only 0.042% of all the molecules. This equates to about 150 moles of CO₂ per square metre of the Earth's surface.

Another way of describing the strength of the scattering is via a scattering cross-section, σ_s. This is a measure of the effective cross-sectional area of each molecule that a photon will see, or alternatively how effective that molecule is at blocking the radiation. The scattering cross-section is usually (but not always) much less than the physical dimension of the molecule. 

Using the definition of scattering cross-section Eq. 86.1 becomes

(86.2) 

 

where the scattering cross-section, σ_s(λ,α) is a function of both the wavelength of the radiation (λ) and the polarizabilty (α) as follows

(86.3)

All molecules have their own unique polarizabilities; a list of some of the more common ones can be found here. Most of the common atmospheric gases have similar values for α, although the value for CO₂ is about 50% higher than most. So scattering from CO₂ is about twice that from oxygen and nitrogen. Nevertheless, this still means that with only about one in a billion 15µm photons being scattered by the 150 moles per square metre of CO₂, the Rayleigh scattering cross-section of CO₂ at this wavelength equates to only about 10^-35 m². This is a million billion times smaller than the actual size of the molecule.

The key point, therefore, is that Rayleigh scattering is of negligible importance when discussing the greenhouse effect because it is a billion times weaker than the Rayleigh scattering we see of visible light. Absorption and re-emission of photons by the 15 µm band are the more important processes, but as I will explain next, their combined behaviour in terms of scattering is very similar to Rayleigh scattering, but much stronger.

Backscattering by absorption and re-emission

Almost all backscattering by carbon dioxide occurs via a process of absorption and re-emission of the infra-red photons with wavelengths in the region of the 15 µm band (i.e. 12.5-18.5 µm). The closer these photons are in wavelength to the centre of the band the more strongly they are absorbed.

Once excited, the molecule may then collide with other molecules and exchange its extra energy with them, thereby heating the gas as a whole. But once all the gas is heated equally, thermal equilibrium will be established where some of the excited CO₂ molecules will re-emit photons at the same rate that other CO₂ molecules are absorbing them. This situation physicists refer to as the steady state. At this point, for every photon being absorbed, another is being re-emitted by another molecule in a random direction.

Ultimately, every molecule wants to get to its lowest energy state, so while bombarding it with radiation will force its energy state to increase, it also increases the rate at which it tries to lose energy. So eventually an equilibrium is achieved where the amount of radiation being absorbed by the gas balances the amount that is lost through re-emission. The key point to note, though, is that the re-emission is a random process that can result in the emitted photons being ejected in any direction. In contrast, most of the absorbed photons come from a single direction: the Earth's surface. This means that fewer of the re-emitted photons end up travelling away from the surface as some are reflected back. Those photons that are reflected back then provide additional heat to the surface, thereby raising its temperature even further. This is the origin of the Greenhouse Effect. 

 

Structure of the absorption band

The infra-red photons emitted from the Earth's surface are absorbed by carbon dioxide via a process of molecular excitation where the molecule is promoted to a higher energy state. Usually this process is in the form of electrons within the molecule moving to higher energy levels, but those transitions are usually of very high energy (several electron volts or eV) and so they only result from the absorption of visible or ultraviolet light. 

The 15 µm transition in carbon dioxide is in the infra-red part of the spectrum and so it is of a much lower energy (about 83 meV). Instead it results from a bond stretching excitation where the molecule bends, as shown in mode v₂ in Fig. 86.4 below, under the action of the incoming electromagnetic wave of the photon. This bond stretching occurs because of an uneven distribution of electric charge within the molecule which then interacts with the electric field of the incoming electromagnetic wave or photon.

Fig. 86.4: The three vibrational modes of carbon dioxide. The symmetric stretch mode (v₁) occurs at 7.5 µm but does not absorb because of symmetry violation. The bend mode (v₂) corresponds to absorption at 15 µm, while the asymmetric stretch mode (v₃) corresponds to absorption at 4.3 µm.

In addition to the bond stretching, a much smaller amount of energy from the incoming photon may go towards increasing the angular momentum or rotational spin of the molecule. This leads to "wings" or branches to the central band as shown in Fig. 86.5 below, with the lower energy branch being denoted as the P branch, and the higher energy one the R branch. It is these branches that cause the band to broaden from a single sharp peak Q at 15 µm to a broader band stretching from 12.5 µm to 18.5 µm.

 

Fig. 86.5: The detailed structure of the 15 µm absorption band for CO₂ showing the absorption peaks associated with rotational transitions.

 

The branches P and R are each comprised of a set of evenly spaced discrete transitions, each one matched to a different change in rotational energy of the molecule due to a change in its angular momentum state J by an amount ∆J = ±1. This is because the incoming photon has an angular momentum quantum of J = +1 and angular momentum must be conserved in the absorption process. The R branch arises from absorption processes where the photon causes the molecule to spin faster, while the P branch arises from absorption processes where the photon causes the molecule to spin more slowly (i.e. the angular momentum of the photon is in the opposite direction to that of the molecule). The two branches are, in theory, mirror images of each other, but in practice centrifugal distortion and rotational-vibrational coupling lead to the line spacing increasing at lower energies.

As the energy of each rotational state J of the molecule is (ideally) equal to J(J+1)hB where h is Planck's constant and B is given by

(86.4)

with I being the moment of inertia of the molecule about its spin axis, the energy change associated with a transition from state J to J+1 when a photon is absorbed will be ±2hB(J+1) (see here). So the absorption peaks due to changes in rotation have an ideal energy separation of 2hB, or about 1.54 wavenumbers (cm^-1) or 0.2 meV for lines in the 15 µm band of carbon dioxide. 

These individual line transitions are also broadened via doppler broadening due to the different speeds and directions of motion of the various molecules, and pressure broadening due to collisions with other molecules in the atmosphere. Both of these broadening mechanisms are temperature dependent and increase as the temperature increases. The result is that the width and overall scattering cross-section of each line will increase with temperature thereby narrowing or closing completely the gaps between the individual rotational lines in the spectrum. For lines in the 15 µm absorption band of CO2 the typical scattering cross-section is about 10^-23 m², which is about one trillion (i.e. 10¹²) times greater than the scattering cross-section of each molecule due to Rayleigh scattering at the same wavelength.

In addition, an increase in temperature will shift the position of the peak or maximum in both the P and R branches, J_max (again see here). This is because the height of each line in the 15 µm band is dependent on both its J number and the temperature. Each line has a degeneracy of 2J+1, while the probability of exciting that rotational J mode depends exponentially on both energy and temperature. This means that the relative occupancy of each mode, N_J, is given by 

(86.5)

where the energy E_J of the mode J is given by J(J+1)hB, N_o is the total number of excited modes, and Z is a normalization term that is equal to the sum of the probabilities.

 (86.6)

This also explains why the P and R branches in Fig. 86.5 have peaks in their distributions. The occupancy of each rotational state N_J increases linearly with J via the degeneracy term 2J+1, but for large J decreases more rapidly due to the energy term E_J in the exponential. This leads to a maximum in the distribution, the position of which relative to the band centre at Q can be determined by differentiation with respect to J. The result is given by 

(86.7)

What this shows is that as the temperature of the gas increases, the position of the maxima will move further from the band centre at Q, so the overall width of the band will increase with temperature as well. This is because higher temperatures will lead to a higher proportion of CO₂ molecules in higher energy rotational states with larger J values. Generally, though, it takes a large temperature change to significantly widen the band. As the position of the R branch maximum relative to Q increases with temperature T as √T, it is reasonable to expect the width of the band to do the same.

Finally, if the width of the absorption band can change with temperature, then it can also change with CO₂ concentration. In this case the width will be set by the highest value of J for which N_J exceeds some threshold value N_th, this being the number of modes required to almost completely absorb the outgoing radiation at that wavelength. According to Eq. 86.5 this will be the value of J such that

(86.8)

Rearranging Eq. 86.8 gives the following result for the threshold J value, J_th, that defines the edge of the band

(86.9)

So Eq. 86.9 not only confirms that the band width varies approximately as √T, but also that it varies approximately as √ln(N_o). 

Conclusions

The dominant mechanism by which CO₂ molecules scatter infra-red radiation in the 15 µm band is by absorption and re-emission.

The absorption and re-emission process has a similar effect to that of Rayleigh scattering, but it is almost a trillion (i.e. 10¹²) times stronger. 

The width of the 15 µm band increases with temperature as √T, but increases much more slowly with CO₂ concentration (see Eq. 86.9).

29. Lateral thought #1 - suburban heating

Question

How much does the average home heat up its environment?

This is really a question that ties in with what I wrote on Post 14. Surface Heating, but I think it illustrates the point at a level that most people can relate to.
 

Answer

Well, we know from Trenberth et al. that the average power of solar radiation incident on the Earth's surface is approximately 161 W/m² (see Fig. 14.1). We also know that this leads to a mean surface temperature for the Earth of about 288 K. We also know from Post 12 (black body radiation and Planck's law) that the emitted surface radiation density scales as T ⁴, where T is the absolute temperature of the surface measured in kelvins, and the emitted radiation must balance the incoming radiation. In other words, both incoming and outgoing surface radiation densities will be proportional to T ⁴. For the outgoing radiation the constant of proportionality will be the Stefan-Boltzmann constant, while for the incoming radiation it will be the Stefan-Boltzmann constant divided by the feedback amplification factor. It therefore follows that if the mean surface temperature were to increase by 1 K to 289 K, then the quantity T ⁴ must increase by 1.40 %. And there are two main ways that this could be achieved. 

The first is to increase the feedback or radiative forcing through an increase in the strength of the Greenhouse Effect. This is what most climate scientists concentrate on, and what they believe is responsible for any temperature changes. But the second possibility is to increase the radiation power absorbed by the surface before the feedback amplification occurs. This could happen if the strength of the Sun's output changes, but more realistically it will happen whenever extra heat is liberated at the surface of the Earth. The amount of heat required to do this will be 1.40 % of 161 W/m², or 2.25 W/m². So an increase of 2.25 W/m² in the incident surface energy density will result in a 1 °C temperature rise (see Post 13 - Case 2).

As I pointed out in Post 14, a major source for such additional heat liberation at the Earth's surface is energy generation and consumption by humans, often for industrial needs. This leads to direct anthropogenic surface heating (DASH) that can raise the temperature of whole countries by as much as 1 °C. But it is not just industry that can significantly heat the local environment.

Consider a typical home. The average household in the UK uses at least 10,000 kWh of energy per year. That equates to an average rate of usage of energy of 1.14 kW throughout the year.

The average land area of homes in the UK is at most 500 m². Most modern housing developments have more than 30 new homes per hectare (see PPG3 guidance paragraphs 57-58); older suburban developments are generally a lot less dense than this; inner city flats and terraced houses are clearly a lot more.

All of this means that the power density for heat escaping from homes will be at least 2.28 W/m² (i.e 1140 ÷ 500). In other words, the energy used by a typical household is more than sufficient to increase the local surface temperature by more than 1 °C. And remember, all this heating has got nothing to do with CO₂ emissions. Nor does this calculation include the energy consumption of commercial buildings, industry or transport.


Conclusion

The energy used by the average household in the UK each day raises the temperature of their local environment by at least 1 °C compared to pre-industrial levels. That will be true irrespective of the source of the energy. Renewables will not help. Nor will cutting your level of CO₂ emissions. This is all down to heat, entropy and thermodynamics.

13. The Earth's energy budget

In order to understand how the Earth is heating up, you need to understand why it is warm in the first place. That means you need to know where the energy is coming from and where it is going. That is the basis of the Earth's energy budget or energy balance.

The purpose of this post is to analyse that energy balance, and to determine which parts of it can change, and what the effects of those changes are likely to be. Specifically, this post will try to relate various possible changes in the energy balance to any consequential changes in global temperatures. In so doing, it will also be necessary to critically ascertain the degree of confidence that there is surrounding the various estimates, and measurements, regarding the energy flows in the different parts of the atmosphere.

As I pointed out in the last post, virtually all the energy that is present on Earth originated in the Sun. The amount of energy per second arriving from the Sun at the top of the Earth’s atmosphere is 1361 watts per square metre (W/m²), and as I also pointed out, because the area this energy is ultimately required to heat up (4πr² where r is the Earth's radius) is four times the cross-sectional area that actually captures the energy (πr²), that means that the mean power density (remember: power is rate of flow of energy) that the Earth receives is only a quarter of the incoming 1361 W/m², or about 341 W/m². However as I also showed in Fig. 12.1, not all this energy reaches the Earth's surface. In fact only about 161 W/m² does. The rest is either absorbed by the atmosphere (78 W/m²), reflected by the atmosphere and clouds (79 W/m²), or is reflected by the Earth's surface (23 W/m²). This is shown diagrammatically in Fig. 13.1 below.

  

Fig. 13.1: The Earth's energy budget as postulated by Trenberth et al. (2009).

The image in Fig. 13.1 is taken from a 2009 paper by Kevin Trenberth, John Fasullo and Jeffrey Kiehl (Bull. Amer. Meteor. Soc. 90 (3): 311–324). It is not necessarily the most definitive representation of the energy flows (as we shall see there are other models and significant disparaties and uncertainties in the numbers), but it is probably the most cited. The data it quotes specifically relates to the energy budget for the period March 2000 - May 2004.

Fig. 13.2: The Earth's energy budget as postulated by Kiehl and Trenberth (1997).

The 2009 Trenberth paper is not the first or last paper he has produced on the subject. The energy budget it describes is actually a revision of an earlier attempt from 1997 (J. T. Kiehl and K. E. Trenberth, Bull. Amer. Meteor. Soc., 78, 197–208) shown in Fig 13.2 above, and has since been revised again in 2012 (K. E. Trenberth and J. T.  Fasullo, Surv. Geophys. 33, 413–426) as shown in Fig. 13.3 below.

Fig. 13.3: The Earth's energy budget as postulated by Trenberth and Fasullo (2012).

The only real difference between the energy budget in Fig. 13.3 and that from 2009 in Fig. 13.1 is the magnitude of the atmospheric window for long wave infra-red radiation (revised down from 40 W/m² to 22 W/m²), but I still think this highlights the level of uncertainty that there is regarding these numbers. This is further emphasised by a contemporary paper from Stephens et al. (Nature Geoscience 5, 691–696 (2012) ) shown below in Fig. 13.4.

Fig. 13.4: The Earth's energy budget as postulated by Graeme L. Stephens et al. (2012).

As the 2009 Trenberth paper appears to be the most cited it is probably best to use this as the basis for the following discussion, but to bear in mind the amount of uncertainty regarding the actual numbers.

In Fig. 13.1 the three most significant numbers are those for the direct surface absorption from the Sun (161 W/m²), the upward surface radiation (396 W/m²), and the long-wave infra-red back radiation due to the Greenhouse Effect (333 W/m²). Of these it is the upward surface radiation (396 W/m²) that determines the temperature but its value is set by the other two.

As I explained in the last post the emission of electromagnetic radiation from a hot object is governed by the Stefan-Boltzmann law as shown below

  

(13.1)

where I(T) is the power density (per unit area) of the emitted radiation, σ = 5.67 x 10^-8 Wm^-2K^-4 is the Stefan-Boltzmann constant, and the term ε is the relative emissivity of the object. The emissivity defines the proportion of the emission from that object at that wavelength compared to a black body at the same temperature, and it varies with wavelength. It is also different for different materials. In the case of planet Earth, it is generally assumed to be very close to unity all over the surface for all emission wavelengths, but this is not always the case.

It is Eq. 13.1 that allows us to determine the surface temperature (T = 289 K) from the upward surface radiation (396 W/m²) or visa versa. It also allows us to calculate the change in upward surface radiation that would result from a given increase in the surface temperature. It turns out that an increase in surface temperature of 1 °C would necessitate the upward surface radiation increasing from 396 W/m² to 401 W/m², in other words a 1.39% increase. A 2 °C increase would require a 2.80% increase in the upward surface radiation.

I also explained in the last post how the total upward surface radiation (I_T) was related to the direct surface absorption from the Sun (I_o) via a feedback factor f which represented the fraction of upward surface radiation that was reflected back via the Greenhouse Effect.

(13.2)

This model assumed that all the energy absorbed by the greenhouse gases came from one source, though, namely surface upward radiation, and was driven by a single input, the surface absorption of solar radiation, I_o. As Fig. 13.1 indicates, this is not the case. This means that Eq. 13.2 will need to be modified.

The aim here is to determine what changes to the energy flows in Fig. 13.1 would result in a particular temperature rise, specifically a rise of 1 °C in the surface temperature. Realistically, there are only three things that could bring about any significant change. The first is a change in the amount of energy coming from the Sun. The second is is a change in the direct absorption of radiation at the surface, I_o. The third is a change in the strength of the Greenhouse Effect, f.


Case 1: Changes to the incoming solar radiation.

This is probably the easiest of the three propositions to analyse. If the incoming solar radiation at the top of the atmosphere were to change by 1.39%, then we would expect virtually all the projected heat flows in Fig. 13.1 to change by the same amount, including the upward surface radiation (from 396 W/m² to 401 W/m²). This is because almost all the scattering mechanisms and absorption processes in Fig. 13.1 are linear and proportional. The two exceptions are likely to be the thermals (17 W/m²) and the evapo-transpiration (80 W/m²), the former of which will be governed more by temperature differences, and the latter by the non-linear Clausius-Clapeyron equation. While changes to these two components are likely to be linear for small changes, they are unlikely to be proportional. However, as the changes to these two components are likely to be fairly small and comparable to other errors, we can probably ignore these deficiencies. So, if the incoming solar radiation (1361 W/m²) were to increase by 1.39% we could see a global temperature rise of 1 °C.

The problem is that there is no evidence to suggest the Sun's solar output has changed by anything like 1.39% over the last 100 years, and no obvious theoretical mechanism to suggest that it could. The only evidence of change is from satellite measurements over the last 40 years or so that suggest an oscillation in solar output with an eleven year period and an amplitude of about 0.05% (see Fig. 13.5 below). This would give a maximum temperature change of about 0.1 °C.

Fig. 13.5: Changes in the Sun's output since 1979 (from NOAA).

The only other known mechanism is the Milankovitch cycle. This can produce temperature oscillations of over 10 °C in magnitude (peak to trough) but is only seen over 120,000 year cycles (see red curve in Fig. 13.6 below). 

Fig. 13.6: Changes to temperature in the southern oceans (red curve) derived from isotope analysis of the Vostok ice core in Antarctica.

These temperature oscillations are mainly due to changes in the Earth's orbit around the Sun (changes to a more elliptical ortbit), or changes in the Earth's angle of inclination or tilt, or an increased precession that then exposes the polar regions to higher levels of solar radiation. Such effects may be responsible for the cycle of ice ages, but cannot be responsible for changes thought to have happened over the last 100 years. As the data in Fig. 13.6 indicates, even the periods of fastest climate change amounted to only a 10 °C increase over 10,000 years, or 0.1 °C per century, and we do not appear to be in one of those warming periods. If anything, the planet should be slowly cooling by about 0.01 °C per century.

The conclusion, therefore, is that global temperatures may fluctuate by 0.1 °C across the decade due to changes in solar output, but there is no evidence or credible mechanism that would support a long-term warming trend.


Case 2: Changes to the direct absorption of radiation at the surface.

The second possible driver of global warming comes from changes at the surface, specifically to the thermal energy absorbed there, I_o. This will then impact on the total upward surface radiation I_T and thereby also on the back radiation. According to Eq. 13.2 the changes to I_o and I_T should be proportional. As Eq. 13.1 indicates that a 1 °C change to the surface temperature, T_o, should result in a 1.39% change to I_T, it follows that a 1.39% change to I_o should result in a 1 °C change to T_o. Unfortunately there are three additional complications that we need to consider: the thermals (I_th = 17 W/m²), the evapo-transpiration (I_E = 80 W/m²), and the incoming solar radiation absorbed by the atmosphere (I_A = 78 W/m²).

The thermals (17 W/m²) and evapo-transpiration (80 W/m²) in Fig. 13.1 transfer heat from the surface into the upper atmosphere (top of the tropopause) by mass transfer (convection) rather than radiation. This may potentially provide a route for heat to escape from the Earth via a by-passing of the greenhouse mechanism. However, I would expect this energy to eventually get dumped in the atmosphere somewhere before the top of the tropopause (at a height of 20 km). When this happens it will merely add to the long-wave infra-red radiation being emitted from the surface, and so should still be reflected by the greenhouse gases. So while these heat sources will not contribute to the surface temperature as defined in Eq. 13.1, they should be included in the feedback factor f in Eq. 13.2.

So too will some of the power absorbed by the atmosphere directly from the incoming solar radiation (78 W/m²). Here again things are complicated because if the energy is absorbed before the bottom of the stratosphere (at 20 km altitude), the Greenhouse Effect will actually reflect some of that heat back into space. To account for this we can include an additional parameter μ as a variable that specifies the proportion of the incoming solar absorbed by the atmosphere that is absorbed in the lower atmosphere where it can be reflected backwards the surface. The fraction (1-μ) absorbed in the upper atmosphere will escape and therefore will not contribute to the back radiation.

In all there are seven energy terms that we need to consider.

1. Initial surface absorption (I_o = 161 W/m²).
2. Thermals (I_th = 17 W/m²).
3. Evapo-transpiration (I_E = 80 W/m²).
4. Upward surface long-wavelength radiation (I_up = 396 W/m²).
5. Long-wavelength back radiation (I_RF = 333 W/m²).
6. Incoming solar absorbed by the atmosphere (I_A = 78 W/m²).
7. Net radiation permanently absorbed by the Earth's surface (I_net = 0.9 W/m²).

We must then consider energy conservation at the surface and in the atmosphere. At the surface the law of conservation of energy (1st law of thermodynamics) requires that

 (13.3)

while in the atmosphere similar considerations mean that the total energy entering the atmosphere must equal the total that is emitted. As f is the proportion that is reflected back it follows that

(13.4)

The parameter μ is a variable that specifies the proportion of the incoming solar absorbed by the atmosphere (I_A) that is absorbed in the lower atmosphere where it can be reflected back towards the surface. The fraction (1-μ) absorbed in the upper atmosphere will escape and therefore will not contribute to I_RF. It therefore follows that

(13.5)

Using Eq. 13.5 we can work out a value for f, but only if we know μ, which we don't. However, using Eq. 13.4 and the knowlege that μ must lie in the range 0 <  μ < 1, we can say that f will be in the range 0.583 to 0.675 and that when μ = 0.5, f = 0.626. This allows us to estimate the change required in I_o to generate a 1 °C change in T_o, but to do that we will need to make some assumptions given the number of variables that there are.

First we can probably assume that f, μ and I_A remain unchanged even when I_o changes. We know that a 1 °C increase in T_o will result in a 1.39% increase in I_up to 401.5 W/m² and a 2 °C increase in T_o will result in a 2.80% increase in I_up to 407.1 W/m². The question is what happens to the thermals (I_th), the evapo-transpiration (I_E) and the net surface absorption (I_net)? They will probably increase as well, but by how much? A good starting point is to assume that they will increase by the same percentage as the upward surface long-wavelength radiation (I_up). A benchmark control is to assume that they stay constant. This gives us the following two scenarios.

If I_th, I_E and I_net scale with I_up and the scaling factor due to the increase in temperature T_o is g, then Eq. 13.5 can be rearranged to give

(13.6)

whereas if I_th, I_E and I_net are constant then

(13.7)

We know that g = 1.0139 for a 1 °C rise in T_o and g = 1.0280 for a 2 °C rise in T_o. So combining the two options in Eq. 13.56 and Eq. 13.7 implies that I_o is in the range 162.8-163.9 W/m². That implies an excess direct heating at the surface of ∆I_o = 2.33 ± 0.54 W/m², with the error range being set by the range of possible values for f, μ, I_E, I_net and I_th. A 2 °C increase in surface temperature would require a change in direct heating at the surface of ∆I_o = 4.69 ± 1.09 W/m².

The conclusion, therefore, is that a 1 °C increase in global temperatures would require an increase in the initial surface absorption of ∆I_o = 2.3 ± 0.5 W/m². How this might be achieved will be explored further in the next post.


Case 3: Changes to the feedback factor.

The most obvious and heavily reported mechanism by which global temperatures could increase is via changes to the Greenhouse Effect due to increased carbon dioxide concentrations in the atmosphere. The specific change that will ensue will be in the value of the feedback term, f, and hence the value of the back radiation, I_RF. As in the previous case, some of the heat flow parameters in Fig. 13.1 would change and some would stay the same. For example, we can confidently assume that I_A and I_o will remain unchanged, but if f changes, so might μ. But as before, the main question is what happens to the thermals (I_th) and the evapo-transpiration (I_E)?

Rearranging Eq. 13.5 once more gives

 (13.8)

while for the case that I_th, I_E and I_net are constant we get

 (13.9)

It turns out there is very little difference in the results using the two methods. The biggest factor affecting f is the value of μ. When there is no warming (g = 1.0) f = 0.629. A warming of 1 °C (g = 1.0139) requires f to increase to 0.634, and a warming of 2 °C (g = 1.0280) requires f to increase to 0.638. These values all correspond to values for μ of 0.5, but the possible spread of values for μ leads to an error in f of ±0.047 in all cases.

What this shows is that the increase in feedback factor needed for a 1 °C rise in global temperatures will be about 0.005. This is a small change, but at the end of the last post (Post 12) I calculated that the fraction of the long-wave infra-red radiation that could be absorbed and reflected by the carbon dioxide in its main absorption band (the frequency range 620-720 wavenumbers or the wavelength range 13.89 - 16.13 μm). The result was at best 10.5%. This implies that only about 15% of the Greenhouse Effect is due to CO₂, and the rest is due to other agents, mainly water vapour.

The conclusion, therefore, is that a 1 °C increase in global temperatures would require an increase in the width or strength of the carbon dioxide absorption band by at least 5% relative to its current size in order achieve this temperature rise.


The final point to note is the size of the potential measurement errors in the various energy flows, and the effect of rounding errors. A particular egregious anomaly occurs at the top of the atmosphere in Fig. 13.1 (and remains uncorrected in Fig. 13.3) where the rounded value of the incoming solar (341 W/m²) radiation balances the rounded outgoing values (239 W/m² and 102 W/m²). This is inconsistent with the rest of the diagram as there should be a 0.9 W/m² difference to account for the net absorption at the surface. In the more exact values quoted (341.3 W/m², 238.5 W/m² and 101.9 W/m²) this difference is specified correctly. So the problem is a rounding issue initially, but it then has a knock-on effect for the values quoted within the atmosphere.

For consistency it would therefore be better in this instance to round the 238.5 W/m² value down (to 238 W/m²) rather than up (to 239 W/m²). That would ensure that there was a net inflow of about 1 W/m² that balanced the net absorbed value at the surface (0.9 W/m²). It would also eliminate the false imbalance within the atmosphere itself. Here the net inflow should balance the net outflow (currently there is a 1 W/m² deficit). There can be no 0.9 W/m² energy gain in the atmosphere otherwise the atmosphere would heat up, and heat up by more than 2.7 °C per annum. What should remain invariant at various points from the surface to the top of the atmosphere is the following energy balance

(13.10)

where I_TOA = 238.5 W/m² is the outgoing long-wave radiation at the top of the atmosphere. A correction for this error requires the stated value for the power emitted upwards by the atmosphere (169 W/m²) in Fig. 13.1 to be reduced to 168 W/m².

It is also important to note that some of the errors in the energy flows in Fig. 13.1-Fig.13.4 are considerable, either in magnitude, or as a percentage. A comparison of the data in Fig. 13.1 and Fig. 13.4 illustrates how variable the results can be. The back radiation values, for example, do not agree within the noted error range, and the net surface absorption is 50% higher in Trenberth's papers than it is in the Stephens paper (Fig. 13.4). I shall look at the net surface absorption in more detail later as it has important implications for sea level rise, but the fact that this value is so small, not just relative to the other energy flows, but also in comparison to their errors, is a cause for concern with respect to its own accuracy. It should also be noted that the net surface absorption should also be measurable directly at the top of the atmosphere using satellite technology to measure both the solar energy going in and the Earth's thermal energy flowing out. Yet the discrepancies seen there between incoming and outgoing energy flows currently far exceed 0.9 W/m². The result is that most of the energy flows shown in Fig. 13.1-Fig.13.4 are at best estimates, and are often based more on climate models than on actual data.