Author details

by introducing an internal variation of IOPs inside each model layer. This scheme is based on standard perturbation theory and allows us to use the standard solar Eddington solution and standard infrared two-stream solution for homogeneous layers to identify a zeroth-order equation and a first-order equation that includes the inhomogeneous effect. The new SRT/IRT solution can accurately express the inhomogeneous effect in each model layer, and it reduces

The new inhomogeneous SRT/IRT solution is a good way to resolve cloud vertical inhomogeneity. In the spectral band of 0.25–0.69 μm, the relative error in the inhomogeneous SRT solution is no more than 1.4%, whereas the error with the homogeneous SRT solution can be up to 7.4%. At the specific wavelength of 0.94 μm, the relative error with the inhomogeneous solution is not more than 5.7% but can be up to 10% with the homogeneous SRT solution. In

error may reach �3.2% for upward emissivity, whereas the error of inhomogeneous IRT

ity for homogeneous solution varies from 4 to �10%, while the error ranges from 1 to �2% for inhomogeneous IRT solution. In the band of 11 μm, the relative error of homogeneous IRT solution is around �1.2% for upward emissivity, and the error of inhomogeneous IRT solution is only less than 0.5%. For downward emissivity, the maximum error of homogeneous IRT solution can be up to �11%, and the maximum error of inhomogeneous IRT solution is only

In specific spectral bands or at particular wavelengths, the vertical variations in IOPs can typically be fitted easily into Eq. (3) to obtain the required parameters. A simple fitting program can be easily incorporated into a climate model to produce the inhomogeneous IOPs of stratocumulus clouds. If no such cloud inhomogeneity information is available in the current climate models, the vertical variation rates of cloud LWC and DCA can be derived empirically from observations, which show that the vertical variation rates of LWC and DCA

In this study, we presented only a single-layer inhomogeneous SRT/IRT solution. To implement the new solution in a climate model, the adding process for layer-to-layer connections has to be solved. Under the homogeneous condition, the single-layer result in reflection and transmission is the same for an upward path and a downward path, but this is not true for an inhomogeneous layer. Therefore, the adding process has to be modified. We will present an algorithm for this multilayer adding process in our next study, in which the climatic impact of inhomogeneous clouds and inhomogeneous snows will be explored. The code base for the

The work is supported by National Natural Science Foundation of China (41675003) and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).

inhomogeneous SRT/IRT solution is available from the authors upon request.

<sup>I</sup> ð Þ τ<sup>0</sup> increasing from 0 to 5πB Tð Þ, the error of downward emissiv-

<sup>I</sup> ð Þ τ<sup>0</sup> , and its relative

to the standard solution when the medium is homogeneous.

156 Perturbation Methods with Applications in Science and Engineering

<sup>I</sup> ð Þ¼ τ<sup>0</sup> 5πB Tð Þ.

in stratocumulus clouds are not very different [5, 7, 8].

solution is only 1%. With F<sup>þ</sup>

around �1% when F<sup>þ</sup>

Acknowledgements

the band of 5–8 μm, the homogeneous IRT solution is not sensitive to F<sup>þ</sup>

Yi-Ning Shi1 , Feng Zhang<sup>1</sup> \*, Jia-Ren Yan<sup>1</sup> , Qiu-Run Yu1 and Jiangnan Li<sup>2</sup>

\*Address all correspondence to: fengzhang@nuist.edu.cn

1 Key Laboratory of Meteorological Disaster, Ministry of Education, Nanjing University of Information Science and Technology, Nanjing, China

2 Canadian Centre For Climate Modelling and Analysis, Environment and Climate Change Canada, University of Victoria, Victoria, British Columbia, Canada
