I present a discussion of the effect of increasing carbon dioxide on planetary climate, at a level suitable for insertion as a module into an upper-level Physics course. The treatment includes two key ingredients that are often missing from more elementary discussions, yet are amenable to analytic methods: First, that convection implies a dependence of surface temperature on the height of the outermost infrared-thick layer; and second, that increasing the level of CO2 closes spectral windows of absorption. These themes are applicable not only to an industrializing Earth but also to our neighboring planets.
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The author has covered this material in one week (three hours of lecture) immediately following a discussion of thermal radiation and its Planck spectrum.
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See <https://www.youtube.com/watch?v=0eI9zxZoipA> for this classroom demonstration.
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In the case of Earth, there are also reflective polar icecaps, aerosols, and so on. Although we are not yet including the effects of atmosphere, we have incorporated such reflection in the data of the table via the
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Each planet (and even some moons) also has internal energy production (geothermal energy), but this source is negligibly small for the three we have chosen to study (Ref. 4, Sec. 2.5).
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The table lists distance values computed simply as the average of the closest and farthest points on each planet's orbit.
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Perhaps surprisingly, even snow and ice have high IR absorptivity and, hence, also high emissivity.
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Note that if Earth were literally stripped of atmosphere, then its reflectivity, currently dominated by clouds, would change, affecting the prediction. The point of the estimate made here is simply the inadequacy of neglecting absorption in the atmosphere.
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The outlines of this story were already clear in the 19th century, shortly after W. Herschel's discovery of IR radiation. J. Fourier understood the role of energy balance between incoming short-wavelength and outgoing IR radiation in 1827, as well as the potential role of the atmosphere. J. Tyndall then measured the absorption spectra of many gases and correctly identified carbon dioxide and water vapor as critical for climate in 1863. At the very end of the century, S. Arrhenius constructed a simplified, but self-consistent model based in part on Tyndall's data. Modern discussions can be found in Refs. 3, 13, and 10.
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Rising warm air expands and, hence, cools, losing its buoyancy, so there is a minimal value for the gradient before convection turns on.
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A convecting region of atmosphere with the critical lapse rate is also said to follow the “dry adiabat.”
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For details, see Ref. 3. S. Manabe shared a Nobel Prize in 2021 in part for early work on radiative transport, on the role of convection, and on incorporating the effects of water condensation into a climate model.
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Section V will improve upon this simplification. We continue to assume that absorption cross section is negligibly small in the visible range.
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2023
Author(s)
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