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

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

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

An Earth Without Atmosphere

View of the Sun on October 14, 2011 from NASA’s Solar Dynamics Observatory, taken at 171 Angstrom, showing the corona and upper transition region of the Sun. The irradiance of solar radiation peaks in the visible wavelengths. About half of the energy is in visible wavelengths below 0.7 μm. The solar spectrum can be approximated by a blackbody spectrum. Source: NASA. In his paper, Arrhenius stated, without any reference, that without its atmosphere the mean temperature of the Earth would be very different, and significantly less. Let’s use a simple Earth energy model to investigate this further. 

The energy that keeps any planet warm, including the Earth, comes from the sun. But the Earth needs to send this energy back; otherwise it would become warmer and warmer. The equilibrium temperature of the earth can be estimated by realizing that the incoming energy flux must equal the outgoing energy flux. The incoming energy is the solar irradiance – the total amount of energy per unit time and unit area measured at the Earth. The sun’s output energy is not constant. It fluctuates (< 0.15 %) over a course of 11 years, from a relatively quiet state to a peak in intensity; however, the average annual solar radiation arriving at the top of the Earth's atmosphere is roughly S = 1,365 W/m2. But this energy does not continuously reach each square meter of the Earth's spherical surface. As shown in the figure on the opposite page, the area of the Earth that collects the sunlight is effectively its cross-section. The incoming energy on the cross-section of the Earth is

Ei= πR2 • S [1]

where R is the radius of the Earth.

The outgoing energy consists of two parts. First, part of the incoming energy is reflected from the Earth’s surface. The fraction of the sun’s incident radiation that is reflected is called albedo (α), so that the reflected energy is Er = αEi. Albedo is close to 1 for white surfaces (like fresh snow), and close to 0 for black surfaces (like dark and wet soil). Earthorbiting satellites measure the Earth’s albedo to be α = 0.33. Secondly, the Earth’s blackbody radiation is also outgoing energy. The entire surface of the Earth emits this energy, which according to the Stefan-Boltzmann law is proportional to the fourth power of its temperature. Then, the energy related to the blackbody radiation is:

Eb = 4πR2 • σT4 [2]

Energy balance, Ei = Er + Eb, now gives the temperature equation:

σT4 = (1 – α) S/4 [3]

For the Earth we now can calculate the effective temperature that would exist at its surface if the planet had no atmosphere. We find T = 252K, or -21°C, in fair agreement with Arrhenius’s observations.

The Moon, which has a very tenuous atmosphere and no clouds, has an albedo of 0.12. Venus, covered by dense clouds, has an albedo of 0.76. Since the Moon has a similar distance as Earth to the sun, the solar radiation is the same. Inserting Moon’s albedo into equation 3 gives the blackbody temperature T = 270K (-3°C). Average temperatures of the Moon's surface have been reported as low as -20°C, but temperatures in different areas vary greatly depending upon whether they are in sunlight or shadow. Daytime maximum temperatures are 387–397K (114–124°C) at the equator, dropping to 95K (-178°C) just before sunrise.

A Simple Greenhouse Effect Model

The area collecting sunlight is the cross-sectional area of the Earth, while the area emitting radiation is the entire surface of the Earth. Observe that we have calculated a temperature of the Earth that is (fortunately) much lower than the observed temperature. The reason, of course, is that we have neglected the fact that the Earth's atmosphere contains gases that absorb the longwave radiation emitted from the Earth's surface. Atmospheric absorption gives rise to the greenhouse effect, and thus warmer conditions. The warmest temperature ever recorded on Earth was 70.7°C, in the Lut Desert of Iran in 2005, while the coldest was -89.2°C, at the Soviet Vostok Station on the Antarctic Plateau in 1983. The average surface temperature on Earth is approximately 16°C.

The blackbody curves discussed in Part I and Part II show that the solar radiance is almost completely separated spectrally from the infrared (IR) thermal emission from Earth. This fact motivates a simple but illustrative model showing how the atmosphere warms the surface of the Earth. Let us assume that the atmospheric layer absorbs β% of the Earth’s IR radiation. As the atmosphere absorbs this longwave radiation, the layer also warms up. By treating the atmosphere as a blackbody with temperature Tg the figure left shows that the atmosphere must emit radiation through both its upper and lower surface in equal amounts. The model thus assumes:

  1. The atmosphere is transparent to solar radiation at all wavelengths <2μm.
  2. The area of the Earth that collects sunlight is its cross-section. The Earth is a blackbody where emission occurs over its entire spherical surface area, so the average radiation at the surface is S/4.
  3. The atmosphere, which absorbs β% of terrestrial radiation, is a blackbody at terrestrial wavelengths <2μm.

Again, we exploit the fact that the incoming energy flux must equal the outgoing energy flux. In the following, we drop the area of the bodies that enter the energy calculations. Outside the layer, energy balance yields:

At the surface of the Earth, the energy balance gives:

By inserting equation (4) into equation (5), we obtain:

simple model of the greenhouse effect. The temperature at the surface of the Earth can be calculated by exploiting blackbody radiation and the physical principle that the incoming energy equals the outgoing energy. The atmosphere is represented as one layer that absorbs β of all Earth’s IR radiation; when β = 1 all terrestrial radiation is absorbed. The atmosphere heats and re-radiates half of the energy to space and half back to the surface. In this way, the atmosphere acts like a blanket, causing some of the heat from Earth to be retained. T(β = 1) = 300K (27°C) is the temperature for a fully IR absorptive atmosphere. T(β = 0) = 252K (-21°C) is the temperature for a non-absorptive atmosphere. However, part of Earth’s IR radiation escapes directly into space, which allows it to maintain a cool surface temperature. To reproduce today’s average temperature of 16°C, we need to select β = 0.844: T(β = 0.844) = 289K (16°C).

The model is a gross simplification of the real climate system, but the simple calculations which contain the basic ingredients of the climate system demonstrate that the temperature increases due to greenhouse gases. Furthermore, the temperature is very sensitive to how much terrestrial radiation is absorbed in the atmosphere (β – value).

The blackbody temperature of Venus is found from equation (6) by using Venus’s albedo and its solar irradiance S = 2,601 (explained below). We arrive at the chilly temperature of 229K (-44°C). However, the very strong greenhouse effect of Venus’ atmosphere changes this temperature to an average of 737K (464°C). The model in the figure on the previous page does not apply to Venus, partly because it has such a dense atmosphere that the assumption that the short wavelength solar radiation reaches the surface of the planet does not hold. The atmosphere, composed of more than 96% CO2 and 3.5% nitrogen, heats, trapping much of the infrared radiation in the dense atmosphere and thick cloud layers. This trapped radiation heats the lower atmosphere, raising the surface temperature by hundreds of degrees.

Total Solar Irradiance

Temperature at the surface of the Earth T(β). When β = 1 all terrestrial radiation is absorbed and when β = 0 no terrestrial radiation is absorbed. The total solar irradiance (TSI) is the amount of solar radiation received at the top of Earth's atmosphere. It has been measured since 1978 by a series of satellite experiments to be 1,365 kW/m². Variations of this number have been discovered on many timescales, linked to several physical processes known to occur in the Sun's interior, including the 11-year sunspot (Schwabe) cycle, the 88-year Gleisberg cycle, the 208-year DeVries cycle and the 1,000-year Eddy cycle. TSI provides the energy that drives Earth's climate, so continuation of the TSI time series database is critical to understanding the role of solar variability in climate change.

Painting by Thomas Wyke of the Thames Frost Fair in 1682. In 1894 the English astronomer Edward Walter Maunder pointed out that very few sunspots had been observed between 1645 and 1715. This ‘Maunder minimum’, which featured exceptionally low numbers of sunspots, is known to coincide with the coldest part of the ‘Little Ice Age’ (ca. 1500–1850) in Europe, North America and China. In these cold years, the Thames River in London froze over during winter, and Norwegian farmers demanded that the Danish king recompensed them for lands occupied by advancing glaciers. On a much longer time scale, over its 4.55 billion year lifespan, it is also known that the sun will increase its luminosity significantly.

We can calculate the sun’s surface temperature Ts by using the principle of energy conservation, where the energy on spherical shells centered on the sun is constant. Considering the sun to be a blackbody, the total radiated energy at the surface is 4πRs2 • σTs4, where Rs = 695,508 km is the radius of the sun. The total solar irradiance is measured at the top of Earth's atmosphere, at a distance Rse = 149,600,000 km from the center of the sun. The energy in a spherical shell at this distance is 4πRse2 • S. The energies must be equal; therefore, the temperature can be solved from the equation:

4πRs2 • σTs4 = 4πRse2 • S

This yields Ts = 5,778K.

Venus is Rsv = 108,200,000 km from the sun. The total solar irradiance measured at the top of Venus’s atmosphere is

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 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 - 2020

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

Part VI - More on the Simple Greenhouse Model
Lasse Amundsen and Martin Landrø, NTU/Bivrost Geo
We continue the discussion of the simple greenhouse model introduced in Part III.
This article appeared in Vol. 17, No. 1 - 2020

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