Development and Application of the Delta-DM Method for Computing Shortwave Radiative Fluxes in a Vertically Inhomogeneous Atmosphere

Abstract

<p>This study describes the development and application of the Delta-DM method for computing shortwave radiative fluxes planetary albedo and heating rates in vertically inhomogeneous atmospheres. Delta-DM was designed to minimize computer time requirements, furnish accurate flux estimates and permit a wide range of shortwave radiative transfer problems to be studied. Phase function approximation is variable and there is no restriction on either the number of layers into which the atmosphere may be divided or on composition of individual layers.</p> <p>The Delta-DM algorithm is based on the discrete ordinate method for solving the radiative transfer equation and includes several major refinements: (i) replacement of Gauss-Legendre quadrature for the source function with shifted-Legendre quadrature; (ii) replacement of quasi-analytical solutions for eigenanalyses with fast, efficient subroutines based on the QR algorithm; and (iii) inclusion of Wiscombe's δ-M method for phase function truncation. Comparison of results with the doubling method indicate low order approximations provide two to three significant digit accuracy for a large combination of atmospheric optical parameters.</p> <p>A block-tridiagonal algorithm is applied to solve the sparse system of equations defining monochromatic fluxes in a multi-layer atmosphere. This approach is found to be both efficient and accurate and makes Delta-DM particularly well suited for problems where fluxes or derived quantities are desired for the same atmosphere but different boundary conditions.</p> <p>Absorption of radiation by water vapour and ozone is based on distribution of absorption coefficients determined from line parameter data (Chou and Arking, 1981) for water vapour and low resolution transmittance model LOWTRAN4 (McClatchey et aI., 1974) for ozone. Comparison of modelled heating rates with line-by-line calculations reveals maximum errors in Delta-DM heating rates of ~ 0.02°C day¹. Comparison of atmospheric albedo and absorptivity, computed from Delta-DM, with benchmark radiative transfer calculations of Braslau and Dave [1972] demonstrates excellent agreement: maximum differences are less than three percent.</p> <p>Delta-DM was applied to compute vertical profiles of downward, upward, and net shortwave fluxes for a variety of atmospheric conditions over the tropical North Atlantic Ocean. Cloud droplet and aerosol (Saharan dust) size distributions were not measured on these occasions so that these parameters had to be prescribed. Comparison of estimated fluxes with measured fluxes from aircraft traverses revealed root mean square errors of estimated fluxes of less than 10 Wmˉ² for cloudless, dust-free atmosphere; ~ 20 Wmˉ² for cloudless but hazy conditions; and ~ 75-100 Wmˉ² for combined cloud-haze conditions. Measured and estimated cloud absorption were in virtual agreement. The larger root mean square errors for the cloud cases also reflect the difficulties of accurately measuring cloud thickness and cloud liquid water contents. Comparison with a similar study (Slingo et aI., 1982) using measured cloud droplet distributions suggests that errors in estimating cloud thickness, liquid water contents, and employing model clouds to determine cloud optical parameters, amount to approximately 60 Wmˉ²</p>