A new scheme based on perturbation method is presented to solve the problem of solar/infrared radiative transfer (SRT/IRT) in a scattering medium, in which the inherent optical properties (IOPs) are vertically inhomogeneous. The Eddington approximation for SRT and the two-stream approximation for IRT are used as the zeroth-order solution, and multiple-scattering effect of inhomogeneous IOPs is included in the first-order solution. Observations show that the stratocumulus clouds are vertically inhomogeneous, and the accuracy of SRT/IRT for stratocumulus clouds by different solutions is evaluated. In the spectral band of 0.25–0.69 μm, the relative error in absorption with inhomogeneous SRT solution is 1.4% at most, but with the homogeneous SRT solution, it can be up to 7.4%. In the spectral band of 5–8 μm, the maximum relative error of downward emissivity can reach −11% for the homogeneous IRT solution but only −2% for the inhomogeneous IRT solution.
- perturbation method
- radiative transfer
- vertical inhomogeneity
Solving the radiative transfer equation (RTE) is a key issue in radiation scheme for climate model and remote sensing. In most numerical radiative transfer algorithms, the atmosphere is divided into many homogeneous layers. The inherent optical properties (IOPs) are then fixed within each layer and the variations of IOPs inside each layer are ignored, effectively regarding each layer as internally homogeneous. The standard solar/infrared radiative transfer (SRT/IRT) solutions are based on this assumption of internal homogeneity [1, 2, 3, 4], which cannot resolve within-layer vertical inhomogeneity.
It has been well established by observation that cumulus and stratocumulus clouds (hereinafter, collectively referred to as cumulus clouds) are inhomogeneous, both horizontally and vertically [5, 6, 7, 8, 9]. Inside a cumulus cloud, the liquid water content (LWC) and the cloud droplet size distribution vary with height, and so the IOPs of cloud droplets depend on vertical height.
How to deal with vertical internal inhomogeneity in SRT/IRT models is an interesting topic for researchers. Li developed a Monte Carlo cloud model that can be used to investigate photon transport in inhomogeneous clouds by considering an internal variation of the optical properties . Their model showed that when overcast clouds become broken clouds, the difference in reflectance at large solar zenith angles between vertically inhomogeneous clouds and their plane-parallel counterparts can be as much as 10%.
However, the Monte Carlo method is very expensive in computing and not applicable to climate models or remote sensing . The albedo of inhomogeneous mixed-phase clouds at visible wavelengths could be obtained by using a Monte Carlo method to compare such clouds with plane-parallel homogeneous clouds .
In principle, the vertical inhomogeneity problem of the SRT/IRT process can be solved by increasing the number of layers of the climate model. However, it is time-consuming to increase the vertical resolution of a climate model. Typically, there are only 30–100 layers in a climate model , which is not high enough to resolve the cloud vertical inhomogeneity. To completely address the problem of vertical inhomogeneity by using a limited number of layers in a climate model, the standard SRT method must be extended to deal with the vertical inhomogeneity inside each model layer. The primary purpose of this study is to introduce a new inhomogeneous SRT/IRT solution presented by Zhang and Shi. This solution follows a perturbation method: the zeroth-order solution is the standard Eddington approximation for SRT and two-stream approximation for IRT, with a first-order perturbation to account for the inhomogeneity effect. In Section 2, the basic theory of SRT/IRT is introduced, and the new inhomogeneous SRT/IRT solution is presented. In Section 3, the inhomogeneous SRT/IRT solution is applied to cloud as realistic examples to demonstrate the practicality of this new method. A summary is given in Section 4.
2. SRT/IRT solution for an inhomogeneous layer
2.1. SRT solution
The azimuthally averaged solar radiative transfer equation [1, 2, 3, 4, 10, 11, 12] is
where μ is the cosine of the zenith angle (μ > 0 and μ < 0 refer to upward and downward radiation, respectively), is the scattering phase function, is the optical depth ( and refer to the top and bottom of the medium, respectively), is the single-scattering albedo, and is the incoming solar flux. For the Eddington approximation, (−1 < μ <1) and are the asymmetry factors. For the scattering atmosphere, the irradiance fluxes in the upward and downward directions can be written as
To simulate a realistic medium such as cloud or snow, we consider and to vary with , and we use exponential expressions here to simplify the process. The single-scattering albedo and asymmetry factor are written as
where is the optical depth of the layer, is the single-scattering albedo at , and is the asymmetry factor at the same place. Both and are small parameters that are far less than and , respectively, in a realistic medium.
According to the Eddington approximation, the radiative intensity can be written as
Using Eqs. (1), (2), and (4), we obtain
where , , and ; is the optical depth of the single layer; and ( ) is the diffuse (resp., direct) reflection from the layer below or the diffuse (direct) surface albedo. Substituting , , and into Eq. (3) and ignoring the small second-order parameters , , and , we get
where , , , , , , , and .
By perturbation theory , the corresponding flux can also be expanded by using the perturbation coefficients and :
Substituting Eqs. (6) and (7) into Eq. (5) yields
where . And, Eq. (8) can be rewritten as separate equations for , , and . We obtain the following equations for the scattered flux :
Eq. (9) is the standard SRT equation for a homogeneous layer  and has the following solution:
where , , , , , and . And, the equations for the perturbation terms (i = 1, 2) are
where . Letting and , Eq. (11a) and (11b) yields
where , , , , , and .
From Eq. (12), we obtain
where , , , , , and .
The solutions of Eq. (13) are
where , , and . Finally, we can obtain and as
where , , , ,
, , , , , , and . and are determined by the boundary conditions as
All detailed calculation about solar radiation can be found at .
2.2. IRT solution
The azimuthally averaged infrared radiative transfer equation for intensity is [1, 2, 3, 4, 10, 11, 12]
where , , , and are same as in Eq. (1). is the Planck function at temperature , which represents the internal infrared emission of the medium.
The Planck function is approximated lineally as a function of optical depth  as
where and are the total optical depth of the medium. The Planck functions and are evaluated by using the temperature of the top ( ) and the bottom ( ) of the medium.
According to the two-stream approximation, the intensities can be written as and , respectively, where is a diffuse factor that converts radiative intensity to flux . can be written as
where is the asymmetry factor.
Using Eqs. (17) and (19), we can obtain
where , , and .
For IRT, we also use Eq. (3) to represent an inhomogeneous medium such as cloud or snow, in which and vary with . By substituting Eq. (3) into , , and and by ignoring the second order of the small parameters of , , and , we can obtain
In the above formula, , , and (i = 1, 2, 3) are the known factors of and . These known factors are introduced for simplifying original expressions, in which , , , , , , , and .
Same as in Eq. (7), the upward and downward intensity can be written as
By substituting Eqs. (21)–(22) into Eq. (20), we obtain
By removing the second-order and higher-order perturbation terms, we can also separate Eq. (23) into three equations of (i = 0, 1, 2). The equations of can be written as
where and are surface temperature and surface emissivity, respectively. Eq. (24) is the standard homogeneous two-stream infrared radiative transfer equation [3, 15] with solutions
where , , , , , , and .
The equations for (i = 1, 2) are
Let and . Eq. (26a) and (26b) yields
where , , , , , , , , , and .
From Eq. (27), we can obtain
where , , , , , , , , , , and . Thus, the solutions are
where , , , , , , , and .
The expressions of are
where , , , , . (j = 1, 2, 3, 4, 5, 6, 8), , and and are determined by boundary conditions. By substituting Eq. (30) into the boundary conditions of Eq. (26c), we can obtain
Finally, the upward and downward fluxes are obtained by
All detailed calculation about solar radiation can be found at .
3. Results and discussion
We apply the two schemes to idealized medium to investigate its accuracy, and the result has been shown on  and .
For true cloud medium, because ice clouds' optical properties strongly depend on the complex particle habits [19, 20, 21]. Therefore, we limit our discussion here to water cloud only. According to the observation, the internal LWC (g m−3) and droplet radius of the cloud tend to increase with height . To take this feature into account, LWC and droplet cross-sectional area (DCA; cm−2, m−3) should increase linearly from the cloud base to the position near the top of the cloud:
where 0 < z < z0. The terms z and z0 denote the height from the cloud base and the height of the cloud top, respectively. From Eq. (33a) to (33b), the cloud effective radius ( ; μm) and liquid water path (LWP; g m−2) can be obtained:
where ρ (g m−3) is the liquid water density. In this case, LWC varies from 0.22 to 0.30 g m−3, and varies from 2.06 to 16.50 μm, in which both ranges are consistent with observation . According to , we choose LWP = 260 (g m−2) to represent low cloud. In the benchmark calculations, z0 is divided into 100 internal homogeneous sub-layers, although other numbers can be chosen (e.g., 200). In principle, more internal sub-layers should result in more accurate results. We use 100 internal sub-layers throughout this study because having any more makes little difference to the calculated results. Using 100 sub-layers are sufficiently accurate to resolve the vertical internal inhomogeneity of the medium. We use the optical properties of a water cloud in the solar spectral band of 0.25–0.69 μm and at 0.94 μm and in the infrared spectral band of 5–8 μm and 11 μm.
In Figure 1a and b, the benchmark values of the inhomogeneous IOPs and the parameterized results for the spectral band of 0.25–0.69 μm are shown. The parameterized inhomogeneous IOPs are
where . The corresponding results for reflection and absorption are shown in Figure 1c–f. For reflection, the relative error with the homogeneous solution increases from 0.25 to 0.71% as μ0 increases from 0.01 to 1, whereas the relative error with the inhomogeneous solution increases from 0.05 to 0.14%. For absorption, the relative error is not sensitive to μ0; it is around 7.4% with the homogeneous solution but around only 1.4% with the inhomogeneous solution.
In Figure 2a and b, the benchmark values of the inhomogeneous IOPs and the parameterized results for the wavelength 0.94 μm are shown. The parameterized inhomogeneous IOPs are
where . Figure 2c–f shows the corresponding results for reflection and absorption. For reflection, the relative error with the homogeneous solution increases from 1.1 to 3.0% as μ0 increases from 0.01 to 1, whereas the relative error with the inhomogeneous solution increases from 0.7 to 2.0%. For absorption, the relative error is not sensitive to μ0; it is around 10% with the homogeneous solution but around only 5.7% with the inhomogeneous solution.
The benchmark values of IOPs and parameterized results for the band of 5–8 μm are shown in Figure 3a and b. Here, we assume
where . For upward emissivity (Figure 3c and d), the relative errors of both solutions are not sensitive to ; the errors are around −3% for homogeneous solution and around 1% for inhomogeneous solution. For downward emissivity (Figure 3e and f), the relative error of homogeneous solution is 4% when , while the error of inhomogeneous solution is only 1%. With increasing from 0 to , the error of homogeneous solution decreases to 0 firstly but then negatively increases to around −10%. The error of inhomogeneous solution shows a similar decreasing-increasing pattern, but the negative increase only reaches about −2%.
The benchmark values of IOPs and parameterized results for the band of 11 μm are shown in Figure 4a and b. In this case, we assume
where . For upward emissivity (Figure 4c and d), the relative error of homogeneous solution is −1.2%, while the error of inhomogeneous solution is less than 0.5%. For downward emissivity (Figure 4e and f), with increasing from 0 to , the error of homogeneous (inhomogeneous) solution varies from 3 to −11% (from 0 to −1%).
4. Summary and conclusions
In the above, we have considered the vertically inhomogeneous structures of only cloud and snow, whereas all physical quantities in the atmosphere are vertically inhomogeneous (e.g., the concentrations of all types of gases and aerosols). In current climate models, the vertical layer resolution is far from that required to resolve such vertical inhomogeneity. In this study, we have proposed a new inhomogeneous SRT/IRT solution to address the vertical inhomogeneity 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 to the standard solution when the medium is homogeneous.
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 the band of 5–8 μm, the homogeneous IRT solution is not sensitive to , and its relative error may reach −3.2% for upward emissivity, whereas the error of inhomogeneous IRT solution is only 1%. With increasing from 0 to , the error of downward emissivity 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 around −1% when .
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 stratocumulus clouds are not very different [5, 7, 8].
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 inhomogeneous SRT/IRT solution is available from the authors upon request.
The work is supported by National Natural Science Foundation of China (41675003) and the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD).
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