Extension of the weak-line approximation and application to correlated-k methods

Extension of the weak-line approximation and application to correlated-k methods A.J. Conley a, * , W.D. Collins b,c a National Center for Atmospheric Research, 1850 Table Mesa Dr., Boulder, CO 80305, United States b Department of Earth and Planetary Science, University of California, Berkeley, United States c Earth Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, U.S. abstract Global climate models require accurate and rapid computation of the radiative transfer through the atmosphere. Correlated-k methods are often used. One of the approximations used in correlated-k models is the weak- line approximation. We introduce an approximation Tg which reduces to the weak-line limit when optical depths are small, and captures the deviation from the weak-line limit as the extinction deviates from the weak-line limit. This approximation is constructed to match the ﬁrst two moments of the We gamma distribution to the k-distribution of the transmission. compare the errors of the weak-line approximation with Tg in the context of a water vapor spectrum. The extension Tg is more accurate and converges more rapidly than the weak-line approximation. 1. Introduction Accurate treatments of the interactions of incident sunlight and emitted thermal radiation with Earth system components are essential for models of the climate system. While computations that resolve the line structure of the gases using line-by- line methods can provide a reference computation of radiative transfer for the known spectral characteristics of the known atmospheric com- position, they are computationally too expensive—by several orders of magnitude—for routine use in global climate models. Band transmission models have been used in earlier generations of global climate models to estimate the radiative ﬂuxes through the earth’s atmosphere [1]. According to recent surveys of the climate modeling community (Q. Fu, private communication), correlated-k methods are currently the most widely used method of approximating the solution while maintaining acceptable accuracy and speed [2,3]. In these methods, the integrals over frequency required for broadband ﬂuxes and heating rates are replaced via Laplace transforms with mathematically identical integrals over speciﬁc extinction. A signiﬁcant source of error in correlated-k methods is the quadrature error associated with the parameterization of the integration over the distribution of extinctions. The line strengths of radiatively active gases can vary over several orders of magnitude through a band; there are often many lines in a band, making accurate and fast integration through the corresponding range of extinctions difﬁcult. Instead, the integration through the extinctions (or frequencies) is often parameterized by a weighted sum of exponentials [4–7]. This weighted sum is called the weighted sum of gray gases, or the exponential sum ﬁtting of transmission. A constructive method for approximating the transmission with a rigorous and predeﬁned error constraint at all path lengths is found in [8]. This method appears to overestimate the terms required to construct an approximation satisfying the