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

“Order and simplification are the first steps toward the mastery of a subject.”Thomas Mann (1875–1955), 1929 Nobel Prize in Literature laureate

The model discussed in Recent Advances in Climate Change Research: Part III, which is widely used to explain the greenhouse effect in a conceptual way, succeeds in its main purpose: to demonstrate that an atmosphere which absorbs and re-emits some of the radiation from the Earth’s surface results in a surface that is warmer than if there were no atmosphere. The Earth’s surface temperature was determined as a function of the atmosphere’s absorptivity of the terrestrial longwave infrared (IR) radiation by energy balancing the incoming and outgoing shortwave and longwave radiation. Here, using the same model, we derive the temperature of the atmosphere from Kirchhoff’s law of thermal radiation.

The energy emitted from the Earth and its atmosphere is commonly referred to as thermal infrared radiation or terrestrial radiation. Trapping of this radiation by atmospheric gases is called the atmospheric effect – commonly known as the greenhouse effect because it is similar to the way the glass that covers a greenhouse transmits shortwave solar radiation and absorbs longwave thermal infrared radiation.

In Part III we introduced an educational model of the greenhouse effect, a short summary of which is: the Earth’s system is driven and maintained by the radiation exchange between the system and space. The Earth absorbs shortwave solar radiation energy – in the form of visible light and ultraviolet radiation – after reflecting about a third of the incident solar radiation back to space. The shortwave radiation is converted to heat at the Earth’s surface, and then re-radiated as terrestrial (i.e. longwave or infrared) radiation back to space. However, some of the terrestrial radiation is trapped by greenhouse gases and re-radiated to the Earth, resulting in the warming of its surface – known as the greenhouse effect. Under a steady state, the absorbed shortwave radiation energy is balanced with the emitted longwave radiation energy.

A greenhouse gas is any gas that has the property of absorbing infrared radiation (in the wavelength range 4–50 micrometres) emitted from the Earth’s surface and re-radiating it back to the surface. The important greenhouse gases are those present at concentrations sufficiently high to absorb a significant fraction of the Earth’s IR radiation. They include H_{2}O, CO_{2}, CH_{4}, N_{2}O, O_{3} and chlorofluorocarbons (CFCs). By far the most important greenhouse gas is water vapour because of its abundance and its extensive IR absorption capacity. Although greenhouse gases make up only a fraction of all atmospheric gases they have a profound effect on the energy budget of the Earth’s system.

#### Energetic Equilibrium

In Part III, we viewed the atmosphere as an isothermal layer (see figure above). The layer is transparent to shortwave solar radiation and absorbs a fraction β of the terrestrial IR radiation. The temperature of the Earth’s surface is *T* and the temperature of the atmospheric layer is *T _{g}*. We obtained the surface temperature:

where *S* = 1,365 W/m^{2} is the average annual solar radiation arriving at the top of the Earth’s atmosphere, *σ* = 5.67 · 10^{-8} Wm^{-2}K^{-4} is Stefan Boltzmann’s constant, and *a* = 0.3 is Earth’s average albedo. The clouds, aerosols, snow and ice reflect a lot of solar radiation; even water reflects a small percentage. On average, about 30% of the solar radiation is reflected to outer space.*T _{0}* = 255K (-18°C) is the frigid temperature of a non-absorptive atmosphere, as was the case in the early history of the Earth, 4 billion years ago. In fact, it was even lower during the solar evolution, when solar radiation was only around 1,000 W/m

^{2}; the temperature at the time was 236K (-37°C).

To reproduce today’s observed global mean surface temperature of 16°C, the atmospheric layer must absorb 80% of terrestrial radiation (

*β*= 0.8); then T(

*β*= 0.8) = 289K (16°C).

*β*= 0.8 implies that 20% of the terrestrial radiation escapes directly to outer space.

As a result of the greenhouse process, the Earth’s surface is 34°C warmer than it would be without it; for this we must be grateful. However, increasing concentrations of greenhouse gases increases the absorption efficiency

*β*of the atmosphere. Equation (1) reveals that an increase in

*β*results in an increase in the surface temperature

*T*.

Equation (1) follows from the radiative balance of the Earth. The Earth must be in energetic equilibrium between the radiation it receives from the Sun and the radiation it emits out to space. If Earth were not in radiative balance, the climate on the planet would not be stable. The simple model in the figure succeeds in its main purpose: to demonstrate how an atmosphere that absorbs and re-emits some of the radiation from the Earth’s surface results in a surface that is warmer than if there were no atmosphere.

In Part I of this series, we introduced the concept of the blackbody – a perfect absorber of energy – and stated that this idea is useful in the study of radiation phenomena. In this respect it is worth observing that the Earth is not a blackbody at visible wavelengths since the absorption efficiency of solar radiation by the Earth is only 1 –

*a*= 0.7. However, Earth radiates almost exclusively in the infrared, where the Earth’s absorption efficiency is in fact near unity. For example, clouds and snow reflect visible radiation but absorb IR radiation. Therefore, in the infrared range the Earth can be considered a blackbody – a result of Kirchhoﬀ’s law of thermal radiation – and we approximated the emission heat flux from the Earth as that of a blackbody of temperature

*T*.

#### A Correction

Order is a first step towards the mastery of a subject. In Part III, the energy balance equations (4)–(5) had a misprint, where the factor ½ should be replaced by the factor *β*. The same misprint appeared in the figure describing the simple model of the greenhouse effect, so that figure is redrawn above in its correct form. These misprints did not influence the surface temperature equation (1). In this article we go on to derive the atmospheric temperature directly by invoking Kirchhoﬀ’s law of thermal radiation. In 1860, Gustav Robert Kirchhoff (1824–1887) stated that “at thermal equilibrium, the power radiated by an object must be equal to the power absorbed.” This leads to the observation that if an object absorbs 100% of the radiation incident upon it, it must re-radiate 100%. Kirchhoff termed a body that absorbed all incident heat radiation a blackbody. Thus, the Earth is a blackbody in the IR. For the atmosphere, the absorptivity is *β* since 1- *β* of the terrestrial radiation, called transmittance, goes to outer space. Following Kirchhoff’s law, as absorptivity is *β* then the emissivity of the atmosphere must also be *β*. The atmosphere is often termed a ‘greybody’ since part of the incident terrestrial radiation is being transmitted through the body. When articles describe the atmosphere as ‘grey’, it means that its absorptivity, and equivalently its emissivity, are constant as a function of radiation frequency.

#### Temperature of the Atmospheric Layer

Stefan-Boltzmann’s law (1879, 1884), discussed in Part II, states that the total radiant heat power emitted from a surface is *B* = *σT4* where T is the surface temperature of the body. This total emissive power is the sum of the radiation emitted over all wavelengths. The law applies only to blackbodies, perfect absorbers with unit absorptivity. From Kirchhoff’s law, a blackbody in thermal equilibrium also has unit emissivity.

Assume that the atmospheric layer of temperature *Tg* is a greybody with emissivity equal to absorptivity equal to *β*. Since the layer obviously has both upward- and downward-facing surfaces, each emitting a radiation flux *β σTg ^{4}*, its total radiant heat power is

*E*

_{out}= 2 β*σT*. The energy balance principle applied to the atmosphere – energy in (

_{g}^{4}*E*) equals energy out – yields the ratio of the two temperatures,

_{in}= β σT^{4}This equation tells us that the atmosphere is always cooler than the ground. Specifically, *T _{g}*(

*β*= 0.8) = 243K (–30°C), which is roughly the observed temperature at the height 9 km of the atmosphere (see above).

This simple greenhouse model assumes a constant atmospheric temperature. In a later part of this series, we will extend the model by viewing the atmosphere as vertically layered. The energy balance equation can then be applied to the elemental slabs of atmosphere and expressions for the temperature of each layer can be derived analytically. The result is a decrease of temperature with altitude. When the number of layers gets large, the absorption

*β*in each layer becomes small. Equation (2) is valid for the last layer; the temperature at the top of atmosphere (TOA) will approach:

a temperature which is fairly consistent with typical tropopause

observations. This modelled temperature is the coldest temperature achievable in the atmosphere in the absence of absorption of solar radiation by gas molecules. The figure above shows that by the time you reach the top of the troposphere the temperature has fallen to a chilly 216K (-57°C).

The temperature of the first layer of a layered atmosphere is

In the real atmosphere, there is no discontinuity in the temperature between the ground and the lowest part of the atmosphere; heat will be transferred by convection.

#### Further Reading on Climate Change Research

More articles from the "Recent Advances in Climate Change Research" Series:**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*

**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*

**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*

**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*

**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*