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Understanding the Relationship between CO2 and Global Warming

Linking CO2 Emissions and Global Temperature

Roughly speaking, that link is as follows:

  • CO2 naturally has a shielding effect on the amount of radiation leaving the earth.
  • Increasing CO2 in the atmosphere increases this shielding effect. Roughly, of 240 W/m2 of radiation that would be emitted from the earth’s surface, there is a shielding of about 3 W/m2 of this – if one doubles CO2 relative to 1750 levels.
  • This causes the earth to heat up because incoming radiation is now greater than outgoing radiation.
  • As the earth increases in temperature, it emits more heat (because hotter objects emit more heat). This brings the earth to a new equilibrium temperature where incoming and outgoing radiation are balanced.
  • Further, because the earth is heating, there are changes in atmospheric water vapour, clouds and surface reflection that combine to further shield heat from leaving the atmosphere. This pushes up the equilibrium temperature at which the earth reaches an equilibrium where heat in is balanced by heat out.

The effect of Carbon Dioxide on the Earths’ Temperature

The sun’s radiation heats the earth. The earth itself then radiates heat out into space. It is the balance between these two that allows the Earth to reach an equilibrium temperature of roughly 25 degrees Celsius (or 288 K). The blue line in Fig. 1 of this blog (Fig 4 of the paper) shows how the earth would radiate heat out to space were there no atmosphere. It is a smooth curve called the “Planck Curve” and the total area underneath the curve represents the net heat emission (which is characterised by the Stefan Boltzmann law, and proportional to the temperature of the earth to the fourth power).

This model can be refined by considering that the earth has an atmosphere, and that atmosphere has the effect of absorbing radiation emitted by the earth (which happens to be mostly long wavelength radiation because of the earth’s relatively low temperature compared to the sun). The longwave radiation absorbed by the atmosphere is reemitted both out to space and back down to the earth, which has a retaining effect on heat versus if there were no atmosphere at all (in which case all long wave heat would go directly out to space). The net radiation outwards of the earth – now including this effect of the earth’s atmosphere (including today’s CO2), is represented by the black line below. The amount of radiation emitted by the earth (excluding what is reflected at the top of the atmosphere) is represented by the area beneath the black line. It is very roughly 240 W/m2.

Figure 1 of this blog: From Wijngaarden and Happer, Environmental Science, 2020

Now, finally, we can further refine this model to consider a doubling of the concentration of carbon dioxide in the atmosphere. This doubling further increases the dampening effect of the atmosphere and brings us from the black line above, to the red line. The net effect (area between the black and red lines) is the “global warming” effect of a doubling of carbon dioxide in the atmosphere, and it is approximately a reduction of 3 Watts/m2 of radiation. In the scenario of a doubling of carbon dioxide, this leads to net radiation absorbed of about 240 W/m2 and net radiation emitted of 240 – 3 = 237 W/m2 – causing an tendency for the earth to move towards a higher equilibrium temperature at which heat emission will be higher (because hotter objects emit more heat), and the balance of heat absorbed and emitted would be restored.

For a doubling of carbon dioxide, and a decrease in net heat emission of 3 W/m2, one can calculate the increase in temperature of the earth needed to restore equilibrium through the Stefan Boltzmann law, via ChatGPT, we get:

Plugging in those numbers, we get an estimated change in the earth’s temperature of less than one degree Celsius. This analysis omits effects of the earth’s emissivity (although emissivity is close to one), second order effects of the earth’s atmosphere and – perhaps most importantly – other feedback effects such as cloud cover and formation. I will build further on these equations later, but – for now – this provides the big picture as to why increasing CO2 (or other gases with an absorbing effect at low frequencies) in the earth’s atmosphere results in an increase in the earth’s temperature.

The relationship between carbon dioxide and the dampening of heat emission

As I explained above, the problem with carbon dioxide is that it re-emits a portion of heat back to the earth’s surface, causing the earth to move towards a higher equilibrium temperature (to increase heat emission and re-balance heat in with heat out). For a doubling of carbon dioxide, there is a reduction of about 3 W/m2 in heat in the outgoing direction. Now, we ask, what is the specific relationship between carbon dioxide concentration and that reduction in heat emission (called “forcing”).

Typically, the empirical relationship used (e.g. Myhre et al. Geophysical Research Letters, 1998) between forcing in W/m2 and carbon dioxide concentration is a logarithmic one, with:

As such, the action of increasing carbon dioxide in the atmosphere – according to this equation leads to a reaction that is logarithmic, i.e. one that is dampened. 

To recap so far:

  • Carbon dioxide absorbs and re-emits heat in the atmosphere – specifically the long wave heat that is emitted by the earth’s surface. This reduces the outward radiation of the planet, requiring the planet to increase in temperature and emit more radiation such that a balance is re-gained with incoming radiation. 
  • The extent of radiative forcing (dampening of emission), if carbon dioxide is to double, is about 3 W/m2, compared to outward radiation of about 240 W/m2, i.e. a 1.25% effect. And, this relationship is logarithmic.

The next questions to ask are:

  1. Is this forcing significant relative to other natural and human caused changes in radiation and temperature?
  2. Owing to increased radiative forcing, are there second order effects (feedbacks), such as changes in oceans and cloud cover, that have secondary shielding effects on heat emission?

Is the radiative forcing caused by carbon dioxide important?

Our confidence that carbon dioxide is causing radiative forcing is grounded in our understanding of the physics of gases and their absorption and emission spectrums. Our confidence largely does not come from direct measurements of radiative forcing because – as calculated above – the net effect on outgoing radiation is small (~1%) and, accordingly, the measurement error is high.

The radiative forcing of carbon dioxide (and other greenhouse gases) is real, and my next question is whether it is important? One vantage point is to compare the forcing caused by CO2 to the forcing of other natural and human caused phenomena. Human caused forcings are shown in Fig. 2 below, indicating that – of all causes – the effect of carbon dioxide emissions and of methane are the largest, followed and counteracted somewhat by the negative forcing (i.e. cooling) of sulphur emissions (which significantly counter the forcings of CO2 and methane).

Figure 2, Page 92 of the IPCC AR6 Physical Science Basis Report.

IPCC reports also address certain natural forcings, such as the the effects of changes in solar activity and the effect of volcanos. Quoting from page 192 of the sixth annual report:

“The net radiative forcing from changes in solar activity and volcanic activity in 1850–1900, compared to the period around 1750, is estimated to be smaller than ±0.1 W m–2, but note there were several large volcanic eruptions between 1750 and 1850 (Cross-Chapter Box 1.2, Figure 1).”

Going further, as to whether forcings today of ~2-3 W/m2 by carbon dioxide and methane are significant relative to periods prior to 1750, this question is difficult, again quoting IPCC, now on Page 197:

“Uncertainties also exist regarding past emissions and radiative forcings. These are especially important for simulations of paleoclimate time periods, such as the Pliocene, Last Glacial Maximum or the last millennium, but are also relevant for the CMIP historical simulations of the instrumental period since 1850. In particular, historical radiative forcings due to anthropogenic and natural aerosols are less well constrained by observations than the GHG radiative forcings. There is also uncertainty in the size of large volcanic eruptions (and in the location for some that occurred before around 1850), and the amplitude of changes in solar activity, before satellite observations. ”

Where this leaves us – and my treatment of the topic is very short relative to what is in IPCC reports and literature – is that the magnitude of radiative forcing caused by human activities is indeed significant relative to natural variations over the past few centuries – with the exception of short term volcanic episodes. Said differently, human emissions very likely exert a net effect on radiative forcing (we haven’t yet gotten to effect on global temperature!) that is substantially larger than natural variations seen over the period after 1750.

What are the problems and risks of increased radiative forcing?

At a first level, there is the matter of quantifying the human and economic impact of increased radiative forcing, through changes in climate and weather events. This can be further sub-divided into questions of:

  1. What changes have been observed – irrespective of understanding their root cause – and what is the human and economic impact of those changes? I address some of these topics in an earlier post about reading the data of climate change.
  2. How does one quantify the relationship between increased carbon dioxide, increased radiative forcing, and the causation of changes in climate and weather events? I have described the relationship between carbon dioxide and increased forcing above, and I will next describe the relationship between radiative forcing and global average temperature change. The final step in closing the loop of causation would be to characterise the relationship between radiative forcing and specific weather events. Doing so involves building a model using coupled non-linear heat and mass transfer equations, with a granularity in space and time that captures the multi-scale nature of our weather systems all the way from small cloud formations up to regional winds. Such a model requires not just detail in characterising the weather at every scale, but is highly sensitive to how one sets the initial conditions, for clouds, for temperatures, for rain, across the entire planet. In light of these strong non-linearities AND the insufficient long-term granular data to characterise the climate, I am skeptical of being able to characterise this relationship accurately (specifically, between radiative forcing and specific weather events). IPCC reports do provide some coverage of this modelling. For now though, I will focus establishing a causal relationship between radiative forcing and the notion of “average global temperature”.

The relationship between radiative forcing and global temperature increase

It is useful to consider the global average temperature because it provides a measure that can be related – via physics – to radiative forcing at an aggregate scale.

Indeed, I have already expressed a simple form of this relationship earlier when calculating the rise in the earth’s temperature that would be required for a new equilibrium under the conditions of a doubling of carbon dioxide concentrations. As carbon dioxide concentrations increase in the atmosphere, radiative forcing increases (albeit logarithmically). This encourages the earth to the heat up, and, as it does so, it radiates out heat more strongly (according to Stefan-Boltzmann) until it reaches a new equilibrium. This increased heat emission (or loss) as the earth warms is called the “Planck Feedback”, and it serves to self-limit the increase in temperature of the earth.

Now, I will build on this model to include second order feedback effects resulting from changes to the average global temperature. This adds an extra layer to the model where one can identify physics that might cause instabilities and accelerations in the reaction of the atmosphere and earth.

Figure 3 below illustrates the Planck Feedback (the effect of the earth emitting/losing more heat when it gets warmer), and the three primary opposing feedbacks that arise because of the earth’s change in temperature:

  1. Water Vapour (and Lapse Rate) – as global temperatures increase, water vapour content increases in the atmosphere. This causes an additional shielding effect on outgoing radiation.
  2. Surface Albedo – as the earth’s average temperature increases the reflection of the earth’s surface changes (e.g. due to changes in ice coverage). This tends to reduce the amount of radiation emitted by the earth through the atmosphere.
  3. Cloud Cover – increases in global average temperature cause changes in cloud coverage and patterns that tend to create a shielding effect on outgoing radiation.

To summarise a little differently, because the earth emits more heat as it warms, this moderates the heating of the planet (Planck feedback). However, increasing the temperature of the planet affects the water vapour content of the atmosphere, which moderates the moderation of the Planck effect. To a lesser degree, changes in the reflectance of the planet as it warms also moderate the moderation of the Planck effect. Lastly – while uncertainty is high – changes to clouds also moderate the moderation of the Planck effect.

All in all, the increase in carbon dioxide and methane causes radiative forcing, but that radiative forcing is moderated somewhat by other pollutants, notably sulphur. Then, that net radiative forcing tends to cause the planet to heat, but that heating is moderated by the Planck effect (hotter objects lose/emit more heat). However, that Planck effect is itself moderated because changes to water vapour content, surface albedo (reflection) and cloud cover (changes which are all due to increasing temperature) tend to reduce the emission of radiation from the planet’s surface.

Figure 3, Page 979 of Annual Report 6 from the IPCC.

Taking the net radiative forcing of 2.72 W/m2 from Fig. 2 above, and combining it with net climate feedback of 1.2 W/m2/K in Fig. 3, leads to a temperature increase of the earth of about 2.3 degrees C for the period of 1750 to 2019.

In conclusion: The link between carbon dioxide and global temperatures

Carbon dioxide affects global average temperature because it causes radiative forcing, namely, it reduces the amount of heat passing from the earth’s surface to outer space. The effect on radiative forcing depends on the logarithm of the concentration of carbon dioxide in the atmosphere. So, increasing carbon dioxide increases radiative forcing, but more slowly than the rate at which carbon dioxide is increasing. Roughly speaking, the effect of doubling carbon dioxide from 1750 levels is a decrease of 1% in net outward radiation.

Owing to radiative forcing, the earth has a tendency to heat up – since incoming radiation is now higher than outgoing radiation to outer space. As the earth heats up, it emits more heat (Planck feedback), which stabilises its temperature at a new, higher, level. The theoretical temperature at which the earth stabilises (given sufficient time) depends also on second order effects due to increasing global temperatures causing shielding effects on outward radiation due to:

  • Increased atmospheric water vapour content
  • Changing cloud patterns/types
  • Changing surface reflectance

Combining the elements of this simple model:

  • net radiative forcing of 2.72 W/m2 from emissions (Fig. 2 above), and
  • net climate feedback of ~1.2 W/m2/K from the combination of increasing heat loss owing to a earth’s increasing temperature and the three effects above,

leads to a temperature increase of the earth of about 2.3 degrees C for the period of 1750 to 2019, noting that – as carbon dioxide and methane concentrations increase – this estimate then increases logarithmically with concentration.

One can go further with this. For example, by considering the error estimates on the input values to these calculations, one can calculate an estimated error on this eventual warming estimate (although be warned in interpreting errors, because climate processes are not necessarily Gaussian).

One can then ask questions such as:

  • How does this estimate compare to historical temperature measurements and estimates of the global average temperature in 1750 and 2019?
  • Are there direct ways of measuring and validating the relationship between carbon dioxide concentrations and the earth’s average temperature, for example via satellite measurements or other approaches?

The IPCC’s annual reports (the latest Physic Science report being AR6) are highly detailed and a trove of information for those who wish to dig further.

For those shorter on time, my hope is that – armed with the simple model above – one can start to appreciate the main levers and sensitivities associated with increasing greenhouse gases in the earth’s atmosphere.

—APPENDIX—

This is the text I posted on LinkedIn and X as a means of introducing this piece.

🌍 Seeking Understanding over Alarmism 🌍

My view is that the effects of carbon dioxide emissions should be taken seriously, but are not cause for alarmism. 

➡️ I sense the average person is well calibrated on this

The average person cares about the impact of emissions, along with many other important matters: the cost of groceries, the cost of energy, employment, safe streets and neighbourhoods, education for their family, health, and more.

I also sense that those conducting scientific studies underpinning the Intergovernmental Panel on Climate Change’s reports are well calibrated, as are those who operate our electrical grids (e.g., the Irish grid).

➡️ BUT, at the political level, there is a tendency for alarmism to beat out understanding.

This is exemplified by the German energy policy leading up to 2022, where reliance on Russian gas put Europe’s energy, economic and military strength at risk, i.e., nuclear alarmism (see https://ronanmcgovern.com/november-newsletter-putin-uber-and-compensation/). It is separately exemplified by political pressures pushing the Irish energy grid towards externalising the costs and burden of grid stability to the grid operator and natural gas, i.e., a lack of political understanding around the second order effects of subsidising intermittent power instead of reliable power mixes (see https://ronanmcgovern.com/firmed-energy-tenders-a-resilient-approach-to-energy-subsidies/).

➡️ Three winds are buffering the sails of alarmism over understanding:

1. Professionalisation/specialisation of communications departments in political parties and corporations

This results in a divorce between messaging and understanding.

2. Social pressure to delegate understanding to experts.

This is not to downplay the value of experts. This is a call to set our aspirations higher for what we (and leaders!) seek to understand ourselves. At the margin, our expectations for individual understanding are not general enough!

3. A growing misconception that science is about consensus.

Rather, science is about the creation and criticism of explanations. Neither voting, social status nor consensus can make an explanation more true. However, criticism can make ideas false and leave the best explanations standing. The misconception of science being about consensus is double trouble because a) it is a flawed way of identifying the best explanations (Galileo had a better explanation, not consensus!), and b) it discourages us from seeking and criticising explanations ourselves. 

➡️ My personal take

I am an optimist and believer that we can shift the tide towards understanding.

With that, I’ve linked below my explanation of the relationship between the carbon dioxide we pump into the atmosphere and its effect on global average temperature. This piece does not the issue of ocean acidification, which is also worthy of serious consideration. I welcome your criticism!

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