### Recent Advances in Climate Change Research: Part VII - Arrhenius’ Greenhouse Rule for Carbon Dioxide

In Part II of this series (Recent Advances in Climate Change Research: Part II - Arrhenius and Blackbody Radiation) we referred to Arrhenius’ relationship between radiative forcing (heat warming) of CO_{2} and its concentration in the atmosphere, and glibly informed you that the effect is logarithmic. However, there is no simple proof as to why this is the case. In this article we investigate this, as it sheds light on understanding climate feedback and sensitivity. We go on to show how the surface temperature changes with variation in CO_{2} concentration. Our simplifications should only be regarded as the first steps toward getting a feeling of the greenhouse effect.

#### Radiative Forcing

The greenhouse effect is caused by the absorption of longwave, infrared radiation from the Earth by greenhouse gases in the atmosphere. Changes in the concentration of these gases lead to a change in the radiative energy absorbed, and thus a change in the temperature of the atmosphere, leading to a change in its radiation back to Earth. The difference in radiation received by the Earth between two defined conditions is called radiative forcing (Myhre et al., 2013). There are two common examples given in the literature. The first is the forcing believed to be caused by an increase in atmospheric CO
_{2} concentration since pre-industrial times. The second one, obtained from computer climate models, is the forcing produced as a result of doubling the concentration of atmospheric CO_{2}. The forcing can be converted to a mean global surface temperature change by multiplying it by a climate sensitivity parameter which varies between the different models. Radiative forcing is considered a direct measure of the amount by which the Earth’s energy budget is out of balance.

#### On Arrhenius’ Greenhouse Rule for CO_{2}

The history of CO_{2} and climate usually starts with Arrhenius’ 1896 paper arguing that increased levels of CO_{2} could raise global temperatures. Other scientists rejected this assertion, based on their belief that CO2 did not absorb any radiation wavelength that was not also absorbed by water vapor: the atmospheric CO_{2} level is so small compared to the level of water vapor that its effect would be insignificant. Therefore, for nearly 50 years the scientific consensus stated that CO_{2} could not affect Earth’s temperature.

However, Arrhenius believed that atmospheric CO_{2} (and other gases) had an eﬀect on surface temperatures; and he formulated a greenhouse rule for CO_{2} by stating: “If the quantity of carbonic acid increased in geometric progression, the augmentation of the temperature will increase nearly in arithmetic progression.” The augmentation of the temperature is the change in the rate of heating Earth’s surface (i.e., radiative forcing).

To interpret Arrhenius’ statement, we need to recall some basic math. Progressions (also called sequences and series) are numbers arranged such that they form a predictable order; i.e. that given some numbers, we can find the next numbers in the series. A sequence of numbers is an arithmetic progression if the difference between any two consecutive terms is always the same. When the initial term of an arithmetic progression is *a _{1}* and the common difference of successive members is

*d*, then the

*n*’th term of the sequence is given by

*a*+

_{n}= a_{1}*(n−1)d*. A sequence of numbers is called a geometric progression if the ratio of any two consecutive terms is always the same. The general form of a geometric sequence is

*a, a*, … where r is the common ratio, and a is a constant. What is of interest to us now is that geometric sequences show exponential growth (or decay), as opposed to the linear growth (or decline) of an arithmetic progression.

^{r}, ar^{2}, ar^{3}Now, according to Arrhenius’ calculations, when CO

_{2}increases in geometric progression – say, from 1 to 2 to 2

^{2}= 4 to 2

^{3}= 8 … i.e. has exponential growth – the radiative forcing increases (nearly) in arithmetic progression – i.e. shows linear growth. Since logarithmic and exponential functions are inverse functions, equation (1) below suggests itself. In particular, when

*C/C*= (1,2,4,8…) then

_{0}*ΔF*= (0,1,2,3,…)

*α*ln(2).

In Table VII of his paper (see Part II), Arrhenius lists calculations of variation of radiative forcing caused by a given variation of CO

_{2}. At latitude 0, CO

_{2}ratios of 1, 1.5, 2.0, 2.5, 3.0 yield temperature increases of 0, 3.1, 4.9, 6.3, 7.2, respectively. We can easily check the validity of Arrhenius’ observation by calculating the constant

*α*for each of his calculations, which we find to be in the range 6.6–7.7 and thus in agreement with the quote we have taken from Arrhenius’ paper. The relationship between concentration and radiative forcing is nearly logarithmic.

#### Why Logarithmic?

Logarithmic equations for calculating the radiative forcing of CO_{2} are common. The functional form:*ΔF = α* ln(*C/C _{0}*) [

*W/m*] (1)

^{2}was published by Wigley (1987) using the model of Kiehl and Dickinson (1987). Here,

*C*is the CO2 concentration and

*C*is the reference concentration at the beginning of the period being studied. The form in equation (1) was used by IPCC 1990 with coefficient α derived from Hansen et al. (1988). The best estimate based on radiative transfer calculations with 3D climatological meteorological input data (Myhre et al., 1998) is

_{0}*α*= 5.35. Even though there is no theoretical basis for formula (1), it has been accepted by the scientific community as a reasonable approximation for the range from 275 ppm to 378 ppm of CO

_{2}, the levels from the beginning of the Industrial Revolution to 2005. The logarithmic relationship implies that radiative forcing will rise by roughly the same amount for each doubling of CO

_{2}concentration. Thus, increased concentrations have a progressively smaller warming effect.

Let’s calculate the radiative forcing from the beginning of the Industrial Revolution,

*C*= 275 ppm, to October 2019 when

_{0}*C*= 408.55 ppm. Equation (1) gives the warming effect

*ΔF*= 2.06 W/m

^{2}. That may not sound like much until you multiply by Earth’s total area, which gives a total warming effect of about 1,050 TW – more than 58 times the world’s average rate of energy consumption, which is currently about 18 TW.

The logarithmic dependency is intriguing. Clues of its usefulness are given also in textbooks, usually pointing to the spectroscopic features of the absorption lines. The interested reader may consult Goody and Yung (1989) or Pierrehumbert (2010). More recently, Huang and Shahabadi (2014) have proposed a simpler argument for why we would expect logarithmic dependence for monochromatic radiance, i.e. at a single wavelength.

#### Temperature Increase

A figure in Part VI (Recent Advances in Climate Change Research: Part VI - More on the Simple Greenhouse Model) depicts radiative equilibrium for the Earth system with a single-layered atmosphere. At the surface the downward radiation flux emitted by the atmosphere is *F = (β/2) σT ^{4}*, where we have substituted the atmospheric temperature with the surface temperature according to equation (2) in Part VI. Increasing, say, CO

_{2}concentration by a given amount corresponds to an increase

*Δβ*of the absorption efficiency. The radiative forcing is the radiative response to the forcing agent, taking place quickly without change in temperature; then the energy imbalance imposed on the climate system is:

*ΔF = ΔβσT*

^{4}(β)/2 (2)Now, since more energy is radiating down on Earth than is radiating back out to space, the planet gets upset; its response is to heat up. Eventually, after decades, a new equilibrium state is reached where the surface temperature has increased by

*ΔT*. Equation (1) in Part VI defines the new temperature as

*T(β + Δβ)*. For a sufficiently small perturbation, a Taylor series expansion of

*T(β + Δβ)*yields:

By use of equations (1, Part VI) and (2) above we can eliminate *Δβ* in equation (3) and obtain the linear relation:*ΔT = λΔF *(4)

where *λ* is the climate sensitivity factor:*λ = T(β)/(S(1 – a))* (5)

We proceed with a cavalier disregard for the observed limitations of the one-layer educational atmospheric model and insert Arrhenius’ rule (1) into equation (4), thereby suggesting that:*ΔT = γ *ln*(C/C _{0})* [K] ; γ =

*λα*(6)

The Earth is generally regarded as having warmed about 1°C since the beginning of the Industrial Revolution, around 1750, when the global average amount of CO

_{2}was 275 ppm. We can use that temperature T = 15°C (288.15K) as a baseline for estimating the effect of CO

_{2}doubling.

First, calculate from equation (5) the climate sensitivity factor

*λ*= 0.302 km

^{2}W-1. Secondly, calculate

*γ = λα*= 1.61 and insert into equation (6) to find the temperature increase

*ΔT*= 1.61 ln(2) K = 1.12°K. Thus, the simple greenhouse model predicts a global warming of around 1.12°C for a doubling of CO

_{2}. This value is too low compared to more advanced models, which allow for positive feedback, notably from the increased water vapor due to increased temperature. A simple remedy for including this feedback process is to posit an additional increase of Δβ to approximate the effect of the increase in water vapor that would be associated with an increase in temperature.

Allowing

*Δβ*to double, the model then predicts

*ΔT*≈ 2.24°C for a doubling of CO

_{2}, roughly consistent with the IPCC understanding that climate sensitivity is somewhere between 1.5–4.5°C of warming for a doubling of pre-industrial CO

_{2}levels.

We await the IPCC’s new assessment of global warming, due in 2021. New computer models predict a warming surge, where equilibrium sensitivity looks to be 5°C. However, in assessing how fast climate may change, the next IPCC report is expected to look to other evidence as well, in particular how ancient climates and observations of recent climate change constrain sensitivity.

#### Further Reading in the 'Recent Advances in Climate Change Research' Series

**Recent Advances in Climate Change Research: Part I - Blackbody Radiation and Milankovic Cycles**

*Martin Landrø and Lasse Amundsen, NTNU / Bivrost Geo*

Geoscience will probably play an important role in mitigating carbon dioxide emissions. In part one of this series, we discuss some history and physics behind the topic of climate change including the concepts behind blackbody radiation and Millankovic Cycles.

*This article appeared in Vol. 16, No. 2 - 2019*

**Recent Advances in Climate Change Research: Part II - Arrhenius and Blackbody Radiation**

*Martin Landrø and Lasse Amundsen, NTNU / Bivrost Geo*

In Part II we look at Arrhenius’ seminal 1896 paper and see how it relates to blackbody radiation and absorption of infrared radiation by the atmosphere, taking a closer look at his model of the greenhouse effect.

*This article appeared in Vol. 16, No. 3 - 2019*

**Recent Advances in Climate Change Research: Part III - A Simple Greenhouse Model**

*Martin Landrø and Lasse Amundsen, NTNU/Bivrost Geo*

What would the temperature of Earth be without the atmosphere? By using simple physical models for solar irradiation and the Stefan-Boltzmans law for blackbody radiation, we can estimate average temperatures with and without atmosphere.

*This article appeared in Vol. 16, No. 4 - 2019*

**Recent Advances in Climate Change Research: Part IV - Challenges and Practical Issues of Carbon Capture & Storage**

*Martin Landrø, Lasse Amundsen and Philip Ringrose*

The basic idea behind CCS (Carbon Capture and Storage) is simple, but what are the main challenges and practical issues preventing a more global adoption of this method?

*This article appeared in Vol. 16, No. 5 - 2019*

**Recent Advances in Climate Change Research: Part V - Underground Storage of Carbon Dioxide**

*Eva K. Halland, Norwegian Petroleum Directorate. Series Editors: Martin Landrø and Lasse Amundsen, NTU/Bivrost Geo*

By building on knowledge from the petroleum industry and experience of over 23 years of storing CO₂ in deep geological formations, we can make a new value chain and a business model for carbon capture and storage (CCS) in the North Sea Basin.

*This article appeared in Vol. 16, No. 6 - 2019*

**Recent Advances in Climate Change Research: Part VI - More on the Simple Greenhouse Model**

*Lasse Amundsen and Martin Landrø, NTNU/Bivrost Geo*

We continue the discussion of the simple greenhouse model introduced in Part III.

*This article appeared in Vol. 17, No. 1 - 2020*