An Analytic Model for the Clear-Sky Longwave Feedback

Climate models and observations robustly agree that Earth's clear-sky longwave feedback has a value of about -2 W m-2 K-1, suggesting that this feedback can be estimated from first principles. In this study, we derive an analytic model for Earth's clear-sky longwave feedback. Our approach uses a novel spectral decomposition that splits the feedback into four components: a surface Planck feedback and three atmospheric feedbacks from CO2, H2O, and the H2O continuum. We obtain analytic expressions for each of these terms, and the model can also be framed in terms of Simpson's law and deviations therefrom. We validate the model by comparing it against line-by-line radiative transfer calculations across a wide range of climates. Additionally, the model qualitatively matches the spatial feedback maps of a comprehensive climate model. For present-day Earth, our analysis shows that the clear-sky longwave feedback is dominated by the surface in the global mean and in the dry subtropics; meanwhile, atmospheric feedbacks from CO2 and H2O become important in the inner tropics. Together, these results show that a spectral view of Earth's clear-sky longwave feedback elucidates not only its global-mean magnitude, but also its spatial pattern and its state dependence across past and future climates.SIGNIFICANCE STATEMENT: The climate feedback determines how much our planet warms due to changes in radiative forcing. For more than 50 years scientists have been predicting this feedback using complex numerical models. Except for cloud effects the numerical models largely agree, lending confidence to global warming predictions, but nobody has yet derived the feedback from simpler considerations. We show that Earth's clear-sky longwave feedback can be estimated using only pen and paper. Our results confirm that numerical climate models get the right number for the right reasons, and allow us to explain regional and state variations of Earth's climate feedback. These variations are difficult to understand solely from numerical models but are crucial for past and future climates.