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Abstract The cooling-to-space (CTS) approximation says that the radiative cooling of an atmospheric layer is dominated by that layer’s emission to space, while radiative exchange with layers above and below largely cancel. Though the CTS approximation has been demonstrated empirically and is thus fairly well accepted, a theoretical justification is lacking. Furthermore, the intuition behind the CTS approximation cannot be universally valid, as the CTS approximation fails in the case of pure radiative equilibrium. Motivated by this, we investigate the CTS approximation in detail. We frame the CTS approximation in terms of a novel decomposition of radiative flux divergence, which better captures the cancellation of exchange terms. We also derive validity criteria for the CTS approximation, using simple analytical theory. We apply these criteria in the context of both gray gas pure radiative equilibrium (PRE) and radiative–convective equilibrium (RCE) to understand how the CTS approximation arises and why it fails in PRE. When applied to realistic gases in RCE, these criteria predict that the CTS approximation should hold well for H2O but less so for CO2, a conclusion we verify with line-by-line radiative transfer calculations. Along the way we also discuss the well-known “τ = 1 law,” and its dependencemore »on the choice of vertical coordinate.« less

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 day-1, 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 day-1 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 ~ 220more »K. A further, ancillary result is that gray models fail to reproduce basic features of atmospheric radiative cooling.« less

This paper addresses issues of statistical misrepresentation of the a priori parameters (henceforth called ancillary parameters) used in geophysical data estimation. Parameterizations using ancillary data are frequently needed to derive geophysical data of interest from remote measurements. Empirical fits to the ancillary data that do not preserve the distribution of such data may induce substantial bias. A semianalytical averaging approach based on Taylor expansion is presented to improve estimated cirrus ice water content and sedimentation flux for a range of volume extinction coefficients retrieved from spaceborne lidar observations by CALIOP combined with the estimated distribution of ancillary data from in situ aircraft measurements of ice particle microphysical parameters and temperature. It is shown that, given an idealized distribution of input parameters, the approach performs well against Monte Carlo benchmark predictions. Using examples with idealized distributions at the mean temperature for the tropics at 15 km, it is estimated that the commonly neglected variance observed in in situ measurements of effective diameters may produce a worst-case estimation bias spanning up to a factor of 2. For ice sedimentation flux, a similar variance in particle size distributions and extinctions produces a worst-case estimation bias of a factor of 9. The value ofmore »the bias is found to be mostly set by the correlation coefficient between extinction and ice effective diameter, which in this test ranged between all possible values. Systematic reporting of variances and covariances in the ancillary data and between data and observed quantities would allow for more accurate observational estimates.

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