On the Correlated<i>k</i>-Distribution Method for Radiative Transfer in Nonhomogeneous Atmospheres

Abstract

The correlated k-distribution method for radiative transfer in nonhomogeneous atmospheres is discussed in terms of the physical and mathematical conditions under which this method is valid. Two correlated conditions are necessary and sufficient for the exact transformation of the wavenumber integration to an integration over the cumulative probability (g), a monotonically increasing and smooth function in the absorption coefficient space. These conditions involve the use of a reference condition to define the absorption coefficient and an assumption concerning the ordering of the absorption coefficient. The correlated conditions are exact in the context of a single line, periodic lines, and the strong- and weak-line limits. In realistic atmospheres, these assumptions are best for adjacent levels but produce increasing blurring or deviations for distant levels. We investigate the blurring of the correlated assumptions on the computations of fluxes and heating rates based on “exact” line-by-line results, using a variety of atmospheric profiles and spectral intervals containing principal absorbing gases. In the thermal infrared, errors in fluxes are less than 0.2% for H2O, CO2, CH4, and N2O, and ∼2% for O3. Errors in heating rates are less than 0.01 K day−1 for these gases below ∼30 km. Larger errors of ∼0.1 K day−1 can occur at some levels above this height. For H2O lines in the solar region, errors in fluxes and heating rates are within 0.05% and 0.01 K day−1, respectively. Based on numerical experimentation, we find that the number of g values ranging from 1 (for weak bands) to ∼10 (for strong bands) are usually sufficient to achieve acceptable accuracy for flux and heating rate calculations. The correlated k-distribution method differs fundamentally from the traditional approach that employs scaling approximations and band models to separate height and wavenumber integrations for transmittance calculations. The equivalent k values for various gases computed from this approach can be directly incorporated in the multiple-scattering program involving cloud and aerosol particles.