Idealized greenhouse model
The temperatures of a planet's surface and atmosphere are governed by a delicate balancing of their energy flows. The idealized greenhouse model is based on the fact that certain gases in the
Essential features of this model where first published by
Simplified energy flows
The model will find the values of Ts and Ta that will allow the outgoing radiative power, escaping the top of the atmosphere, to be equal to the absorbed radiative power of sunlight. When applied to a planet like Earth, the outgoing radiation will be longwave and the sunlight will be shortwave. These two streams of radiation will have distinct emission and absorption characteristics. In the idealized model, we assume the atmosphere is completely transparent to sunlight. The planetary albedo αP is the fraction of the incoming solar flux that is reflected back to space (since the atmosphere is assumed totally transparent to solar radiation, it does not matter whether this albedo is imagined to be caused by reflection at the surface of the planet or at the top of the atmosphere or a mixture). The flux density of the incoming solar radiation is specified by the solar constant S0. For application to planet Earth, appropriate values are S0=1366 W m−2 and αP=0.30. Accounting for the fact that the surface area of a sphere is 4 times the area of its intercept (its shadow), the average incoming radiation is S0/4.
For longwave radiation, the surface of the Earth is assumed to have an emissivity of 1 (i.e. it is a black body in the infrared, which is realistic). The surface emits a radiative flux density F according to the Stefan–Boltzmann law:
where σ is the
The infrared flux density out of the top of the atmosphere is computed as:
In the last term, ε represents the fraction of upward longwave radiation from the surface that is absorbed, the absorptivity of the atmosphere. The remaining fraction (1-ε) is transmitted to space through an
The energy balance solution
Zero net radiation leaving the top of the atmosphere requires:
Zero net radiation entering the surface requires:
Energy equilibrium of the atmosphere can be either derived from the two above equilibrium conditions, or independently deduced:
Note the important factor of 2, resulting from the fact that the atmosphere radiates both upward and downward. Thus the ratio of Ta to Ts is independent of ε:
Thus Ta can be expressed in terms of Ts, and a solution is obtained for Ts in terms of the model input parameters:
or
The solution can also be expressed in terms of the effective emission temperature Te, which is the temperature that characterizes the outgoing infrared flux density F, as if the radiator were a perfect radiator obeying F=σTe4. This is easy to conceptualize in the context of the model. Te is also the solution for Ts, for the case of ε=0, or no atmosphere:
With the definition of Te:
For a perfect greenhouse, with no radiation escaping from the surface, or ε=1:
Application to Earth
Using the parameters defined above to be appropriate for Earth,
For ε=1:
For ε=0.78,
- .
This value of Ts happens to be close to the published 287.2 K of the average global "surface temperature" based on measurements.[7] ε=0.78 implies 22% of the surface radiation escapes directly to space, consistent with the statement of 15% to 30% escaping in the greenhouse effect.
The radiative forcing for doubling carbon dioxide is 3.71 W m−2, in a simple parameterization. This is also the value endorsed by the IPCC. From the equation for ,
Using the values of Ts and Ta for ε=0.78 allows for = -3.71 W m−2 with Δε=.019. Thus a change of ε from 0.78 to 0.80 is consistent with the radiative forcing from a doubling of carbon dioxide. For ε=0.80,
Thus this model predicts a global warming of ΔTs = 1.2 K for a doubling of carbon dioxide. A typical prediction from a
Tabular summary with K, C, and F units
ε | Ts (K) | Ts (C) | Ts (F) |
---|---|---|---|
0 | 254.8 | -18.3 | -1 |
0.78 | 288.3 | 15.2 | 59 |
0.80 | 289.5 | 16.4 | 61 |
0.82 | 290.7 | 17.6 | 64 |
1 | 303.0 | 29.9 | 86 |
Extensions
The one-level atmospheric model can be readily extended to a multiple-layer atmosphere.[8][9] In this case the equations for the temperatures become a series of coupled equations. These simple energy-balance models always predict a decreasing temperature away from the surface, and all levels increase in temperature as "greenhouse gases are added". Neither of these effects are fully realistic: in the real atmosphere temperatures increase above the tropopause, and temperatures in that layer are predicted (and observed) to decrease as GHG's are added.[10] This is directly related to the non-greyness of the real atmosphere.
An interactive version of a model with 2 atmospheric layers, and which accounts for convection, is available online.[11]
See also
References
- ^ "What is the Greenhouse Effect?" (PDF). Intergovernmental Panel on Climate Change. 2007.
- .
- ^ Marshall J., Plumb R.A., Atmosphere, Ocean and Climate Dynamics, AP 2007, Chapter 2, The global energy balance
- ^ "ACS Climate Science Toolkit - How Atmospheric Warming Works". American Chemical Society. Retrieved 2 October 2022.
- Penn StateCollege of Mineral and Earth Sciences - Department of Meteorology and Atmospheric Sciences. Retrieved 2 October 2022.
- ^ Kleeman, Richard. "Zero Dimensional Energy Balance Model". math.nyu.edu.
- .
- ^ "ACS Climate Science Toolkit - Atmospheric Warming - A Multi-Layer Atmosphere Model". American Chemical Society. Retrieved 2 October 2022.
- ^ "Energy Recirculation and a Layered Radiative Atmosphere Model". Climate Puzzles. 17 April 2021. Retrieved 1 June 2023.
- .
- ^ "2-Layer Atmosphere with Solar, Longwave, & Convection". Denning Research Group, Colorado State University. Retrieved 1 June 2023.
Additional bibliography
- Bohren, Craig F.; Clothiaux, Eugene E. (2006). "1.6 Emissivity and Global Warming". Fundamentals of Atmospheric Radiation. Chichester: ISBN 978-3-527-40503-9.
- Petty, Grant W. (2006). "6.4.3 Simple Radiative Models of the Atmosphere". A First Course in Atmospheric Radiation (2nd ed.). Madison, Wisconsin: Sundog Pub. pp. 139–143. ISBN 978-0-9729033-1-8.
External links
- Media related to Idealized greenhouse model at Wikimedia Commons
- Computing wikipedia's idealized greenhouse model