The standard perturbation method for infrared radiative transfer in a vertically internally inhomogeneous scattering medium

Highlights: • A new scheme is proposed to solve the problem of infrared radiative transfer in a vertically internally inhomogeneous scattering medium. • The perturbation method is used in this scheme to address the vertical variation of the inherent optical properties inside a scattering medium. • It is found that the new scheme is more accurate than the standard two-stream approximation and is well suited for application in climate modeling. A new scheme based on the standard perturbation method is proposed to solve the problem of infrared radiative transfer (applicable to 4–1000 µm) in a scattering medium, in which the inherent optical properties are vertically inhomogeneous. In this scheme, we use the exponential formulas to fit the vertical variation of the optical properties first, and then use the perturbation method to resolve the nonlinear equations. In the perturbation method, the standard two-stream approximation is used as the zeroth-order solution and multiple scattering effect of the continuous changing optical properties is included in the first-order solution. By applying the new solution to an idealized medium, the new solution is found well suited for solving the infrared radiative transfer in the vertically inhomogeneous medium. The errors of the standard two-stream solution can be up to 6% for upward emissivity and 3% for downward emissivity, while the errors of the new solution is limited to 2% and 0.4%, respectively. Under the different circumstances of incident radiation from the bottom, the relative errors of downward emissivity for the new solution are also generally smaller than those for the standard two-stream solution. We also apply the new solution to two cases of water cloud in an infrared band (5–8 µm) of a radiative model and at a wavelength (11 µm) of atmospheric window. The spectrums are quite suitable for studying the optical properties of clouds. In the band of 5–8 µm, the maximum relative errors of downward emissivity can reach $-11\%$ for the standard two-stream solution and is only $-2\%$ for the new solution. At the wavelength of 11 µm, the results of upward and downward emissivity for the solution are also much more accurate than those for the standard two-stream solution. The code, coupled with the inhomogeneous infrared radiative transfer solution's code, is available from the authors upon request.