Simple spectral models for atmospheric radiative cooling

Atmospheric radiative cooling is a fundamental aspect of Earth’s greenhouse effect, and is intrinsically connected to atmospheric motions. At the same time, basic aspects of longwave radiative cooling, such as its characteristic value of 2 K day²¹, its sharp decline (or ‘‘kink’’) in the upper troposphere, and the large values of CO₂ cooling in the stratosphere, are difficult to understand intuitively or estimate with pencil and paper. Here we pursue such understanding by building simple spectral (rather than gray) models for clear-sky radiative cooling. We construct these models by combining the cooling-to-space approximation with simplified greenhouse gas spectroscopy and analytical expressions for optical depth, and we validate these simple models with line-by-line calculations. We find that cooling rates can be expressed as a product of the Planck function, a vertical emissivity gradient, and a characteristic spectral width derived from our simplified spectroscopy. This expression allows for a pencil-and-paper estimate of the 2 K day²¹ tropospheric cooling rate, as well as an explanation of enhanced CO₂ cooling rates in the stratosphere. We also link the upper-tropospheric kink in radiative cooling to the distribution of H₂O absorption coefficients, and from this derive an analytical expression for the kink temperature T_kink ’ 220 K. A further, ancillary result is that gray models fail to reproduce basic features of atmospheric radiative cooling.