In previous work, we discussed the mathematical nature of the climate response to “forcing.” We determined that there were several difficulties in determining the “climate sensitivity”-one needs to know the forcing at work, one needs to know the response time of the system, the forcing needs to be spatially uniform, and so on. I have figured out a possible way around the response time problem. On a timescale of billions of years, the difference between the actual temperature and the equilibrium value is negligible, so we may neglect it and use a zero dimensional model for radiation balance. While I do tend to believe that we are mostly concerned with the climate response over centuries a most, the long term evolution of the Earth system might still offer some constraints on the kind of response we might expect. Of course, over thousands of years, feedback processes emerge in the carbon cycle and ice sheets that would alter the value over hundreds of years: to the extent that those are positive feedbacks, the long term sensitivity is higher than the sensitivity we care about. To the extent that there is an even longer term carbon cycle feedback involving silicate weathering (Walker et al 1981, Volk 1987) which is probably negative, we may underestimate the sensitivity. However, I have made s0me improvements over the model in the earlier post: I am treating radiation balance more explicitly (using the Stefan-Boltzmann law and estimates of the current solar “constant” from Kopp and Lean 2011, the Albedo from Goode et al 2001 and a tuned value of the emissivity that would yield the current climate of ~288 K mean temperature as “baseline” values.) and allowing for the possibility of nonlinear feedback (by treating feedback not as a linear dependence of radiation fluxes on temperature, but as whatever dependence of the ratio of 1-albedo/emissivity on temperature will balance the system). The climate change we are assessing, is the “Faint Young Sun Paradox” first identified by Sagan and Mullen (1972). Essentially, the problem is that our understanding of the evolution of the sun (based upon physics and assessment of sun like stars) calls for the sun to have been substantially dimmer than it presently is billions of years ago, gradually brightening over time (Gough 1981): a constant reflectance of the Earth and a constant tendency of the atmosphere to impede radiative emission would imply that about 2.7 billion years ago, the Earth’s mean surface temperature should have just risen above the freezing point of water (I have produced this result from my own model). But liquid water and a stable ocean was present on Earth by 3.9 billion years ago (Pinti 2005), and possibly as early as 4.4 billion years ago (Wilde et al 2001). To be sure, the presence of liquid water does not require the mean temperature be above zero, since the water may be present in the Tropics, as long as they remain significantly above the mean temperature. Nevertheless, there is an absence of evidence of widespread glaciations through much of the period when the mean temperature should have been below freezing (Young 1991) although a mid-latitude glaciation probably occurred 2.9 billion years ago (Young et al 1998). Some might take the position that an absence of evidence for glaciation requires a mean climate warmer than the present, but this is frankly implausible on physical grounds-simply put, the greenhouse effect from higher CO2 at the time would not have produced sufficient forcing to lead to warmer temperatures (there is an extensive overview of this discrepancy here starting at page 99), and cooling temperatures over time would lead to, by the aforementioned weathering feedback, long term increases, not decreases, in CO2 levels-which, combined with a brightening sun, surely should have lead to warming temperatures, even though we just stipulated that they were cooling. If there really was an absence of ice, it might mean that there was a change in the meridional heat transport, which would possibly render assessment of the climate sensitivity more difficult. I will instead assume here that the warming over time was slow, such that the Earth was on average above freezing for 4.5 billion years, and we might plausibly claim that the temperatures for much of Earth’s history were sufficiently similar to the present so as to have little ice present (the glacial cycles in recent climate history were associated with mean temperatures about 5 degrees colder than the present, which happens, with our gradual warming, about 1.5 billion years ago) and keep in mind, slower warming would demand a lower climate sensitivity. (Note that if the Earth was still generating a significant amount of it’s own heat for several hundred million years after it’s formation, the very early Earth should have been very warm and I’d have to chop out a significant time period of Earth’s history from my model as invalid. However, the result is not particularly dependent on the climate of the Hadean, it is mostly dependent on the Earth having warmed at a rate since the Archean to the present such that if extrapolated back to the formation of the Earth, would imply freezing temperatures at the Earth’s formation. Although that period is technically included in the calculations of feedback, the feedback is explicitly not linear and is different in the assumed much colder climate of that era that is perhaps incorrect.) Anyway, I’ve kept you all in suspense as to the results of this work: I get a sensitivity to doubling CO2 from roughly present conditions of about .65 K. This is very low compared to “official” estimates from models, although it is compatible with careful assessment of the radiation flux data from CERES (but not careless analysis). It is possible this low sensitivity is due to a long term negative CO2 feedback, which might mean that the sensitivity we’d get over centuries rather than billions of years would be higher to some extent. Still, a long term sensitivity which is much higher than that throws into question how the Earth could possibly have stayed unfrozen through so much of it’s history. I note that I am not the first person to suggest a low sensitivity could resolve the Faint Young Sun Paradox, and for much of the inspiration for this I am indebted to Rondanelli and Lindzen (2010). A key difference between their work and mine is that they identify a specific mechanism which might lead to low sensitivity, I have shown that resolution of the paradox can be achieve by low climate sensitivity regardless of the origin of the negative feedback-other than that, their study is more comprehensive in it’s treatment of climate physics. While this does not constitute absolute “proof” it is does make, in my opinion, a good argument for sensitivity being pretty low. Anyway, you can download a spreadsheet that goes through the determination of this value for sensitivity here.
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