1. Introduction
It is commonly assumed that cloud longwave scattering is unimportant for estimating the atmospheric energy budget and thus is neglected in general circulation model (GCM) irradiance simulations and in radiative transfer simulations for deriving the radiation budgets from retrieved cloud optical properties. A number of studies indicate a range of overestimates of top-of-the-atmosphere (TOA) longwave irradiance resulting from neglecting longwave scattering. For example, Stephens et al. (2001) showed an overestimation of global-average outgoing longwave radiation (OLR) of approximately 8 W m−2 when cloud longwave scattering is neglected in a GCM, whereas regional overestimation could be as large as 20 W m−2. Costa and Shine (2006) showed a global overestimation of OLR of approximately 3 W m−2 by neglecting cloud longwave scattering, with low-level clouds contributing 0.9 W m−2, midlevel clouds contributing 0.7 W m−2, and high clouds contributing 1.4 W m−2. Other studies also reported OLR overestimations resulting from neglecting cloud longwave scattering with values ranging from 1.5 to 11 W m−2 (Ritter and Geleyn 1992; Joseph and Min 2003; Schmidt et al. 2006). In a recent sensitivity study, Kuo et al. (2017) showed that neglecting cloud longwave scattering could result in an overestimation of the global-average OLR of approximately 2.6 W m−2 and an overestimation of the surface net upward radiation of approximately 1.1 W m−2.
Such significant changes caused by excluding cloud longwave scattering effects suggest a pressing need for more accurate longwave radiative transfer (RT) simulations in the broadband radiance models used in GCMs. However, such a change should not substantially increase the computational burden, since the computational costs of the RT models in GCMs are already high. One approach for mitigating the computational burden is to scale the cloud absorption optical thickness, as in the scheme developed by Chou et al. (1999). The Chou et al. scheme retains the computational efficiency by neglecting the longwave scattering and improves accuracy for irradiance in the case of optically thin clouds. However, there is little improvement for optically thick clouds, as the TOA longwave irradiance is still overestimated by approximately 4 W m−2. Therefore, simply scaling the cloud optical thickness in conjunction with no-scattering simulations is insufficient for further improvement of irradiance calculations over a range of optical thicknesses.
The similarity principle for radiative transfer (van de Hulst 1974; Sobolev 1975; Joseph et al. 1976; van de Hulst 1980; Twomey and Bohren 1980; McKellar and Box 1981; Ding et al. 2017) provides an alternative approach for scaling the optical thickness. The basic concept is that radiance or irradiance simulations may have similar results with two different sets of cloud optical properties if the optical properties satisfy certain relationships. With this principle, one could consider replacing a scattering cloud layer with a nearly nonscattering cloud layer and obtain similar irradiance values. The similarity principle has been verified for TOA solar radiance calculations (with an external source and a nonemitting atmosphere) with a single-scattering albedo ω between 0.8 and 1 and an asymmetry factor g between 0.75 and 0.9 (Ding et al. 2017). However, further investigation is necessary for the case of small ω (i.e., strongly absorptive case) and for longwave irradiance from a scattering–emitting cloud layer where there is no external source term. As with the Chou et al. scheme, a simple scaling strategy also lacks the ability to provide accurate irradiance calculations for optically thick clouds. These two schemes are compared in this study (section 3) and have similar results for the irradiance simulations.
The primary goal of this study is to develop a new adjustment scheme based on the similarity principle and the Chou scaling schemes. We will show that this adjustment scheme improves the longwave irradiance calculations in a broadband radiative transfer model for both optically thin and thick ice clouds. Although the present scheme also works for low-level liquid water clouds, we will not show results for this case because the scattering effect is generally weak in the longwave regime.
The remaining part of the paper is organized as follows: Section 2 discusses the methodology. Section 3 presents the simulation results and discussions. Section 4 summarizes and concludes the study.
2. Methodology
a. Similarity principle
b. Chou’s parameterization for adjusting the cloud optical thickness
c. Adjustment scheme
In the Chou et al. (1999) scaling scheme, the major approximation is with the hemispheric isotropic radiance. The performance of the scheme highly depends on the extent to which this approximation holds true. While one can consider the ambient radiance as being approximately equal to the incident radiance in the same hemisphere, large biases may result from this approximation when the incident radiance is in the opposite hemisphere, especially for downward ambient radiance on top of a cloud layer. In this case, the downward ambient radiance is much weaker than the blackbody radiance. Hence, we reconsider the approximation for the downward ambient radiance when solving for the upward radiance.
Alternatively, the term (1 − b) may be replaced with (1 + g)/2 in Eqs. (14) and (21) to obtain general and specific solutions based on the similarity principle. We call the scheme with this solution the similarity scaling-adjustment scheme. In Chou’s scheme, (1 − b) represents the percentage of the scattering contribution in the direction of the to-be-solved radiance from the ambient radiance in the same hemisphere, and b represents the contribution from the ambient radiance in the opposite hemisphere. In the similarity scaling-adjustment scheme, with (1 + g)/2 replacing (1 − b), the percentage contributions from the ambient radiance in the two hemispheres are (1 + g)/2 and (1 − g)/2, respectively. When g = 1, the scattering contribution to the to-be-solved radiance is purely from the ambient radiance in the same hemisphere. When g = 0, the contributions from the two hemispheres are equal, corresponding to isotropic scattering. The appendix describes the physical meaning of the similarity principle scaling and the adjustment term. In a simplified model where the atmosphere is a one-dimensional, optically thin layer, the similarity principle scaling scheme results in more accurate TOA irradiance than the neglect-scattering scheme, and the similarity scaling-adjustment scheme results in the exact solution, completely offsetting the error of the scaling scheme.
RRTM schemes and corresponding governing equations and computational times.
3. Results
We present results for the four schemes discussed in the previous section: the Chou et al. scaling approach, the similarity principle scaling approach, the Chou scaling-adjustment scheme, and the similarity scaling-adjustment scheme. For evaluation purposes, we implement all four schemes in the RRTM longwave model, a simplified version [RRTM for General Circulation Models (RRTMG)] of which has been used to simulate radiative transfer in multiple GCMs, such as the Community Earth System Model (CESM; Kay et al. 2015) and the Weather Research and Forecasting (WRF) Model (Powers et al. 2017). In the RRTM longwave model, the upward and downward radiances are calculated for a set of radiance spectral bands, without considering scattering, for a single or multiple (1–4) angles, and then integrated to provide the irradiance. Cloud scattering can also be included using the DISORT option in RRTM, but it is much slower computationally.
For the purpose of generality, we use multiple ice cloud particle models to represent the ice cloud optical properties. In addition to the built-in spherical ice particle model (Hu and Stamnes 1993; Key 1996; denoted as the spherical model) and surface-roughened single-hexagonal-column model (Fu et al. 1998; denoted as the rough column model), we implement two more ice models, the surface-roughened eight-hexagonal-column aggregate model (Yang et al. 2013) used in the Moderate Resolution Imaging Spectroradiometer (MODIS) Collection 6 satellite cloud retrieval product (Platnick et al. 2017; denoted as the aggregate model) and the two-habit model consisting of a surface-roughened single-hexagonal-column and a 20-column aggregate model as a candidate for the Clouds and the Earth’s Radiant Energy System (CERES)-MODIS Edition 5 product (Loeb et al. 2018; denoted as THM). In addition, six climatological atmospheric profiles are used: U.S. Standard Atmosphere, 1976 (COESA 1976); tropical atmosphere; midlatitude summer; midlatitude winter; high-latitude summer; and high-latitude winter. An ice cloud layer with a geometric thickness of 250 m is placed at high (11 km), middle (8 km), and lower altitudes (5 km). While clouds near the surface are generally liquid water phase, the assumption of ice phase for a cloud layer at 5 km and higher is reasonable for a sensitivity study. The cloud optical thickness varies from thin (0.1) to thick (10). The ice particle effective radius varies from 10 to 50 μm. Table 2 lists the detailed settings of atmosphere–cloud profiles used to evaluate the proposed schemes.
Atmospheric and cloud profile inputs for the 2160 RRTM simulations.
For each atmosphere–cloud profile, we use four newly implemented schemes along with the original neglect-scattering scheme to calculate irradiances. Broadband radiances in one to four angles for both upward or downward directions are solved and integrated to obtain irradiances with scattering neglected. For a reference, the band radiances in 16 streams (eight angles for both upward and downward radiation) are calculated using DISORT with the cloud scattering fully considered using a 16-term Legendre expansion.
a. Ice cloud optical property parameterizations
Ice cloud scattering properties depend on the particle shape and size distribution, which vary temporally and spatially (Baran 2005, 2012). For our simulations, ice particle shapes are fixed and the size distribution is parameterized with a single parameter, that being the effective radius (Re). It is proportional to the ratio of the total volume to the total projected area for the particles in the habit/size distribution (Hansen and Travis 1974; Foot 1988), whereas the size distribution width (effective variance) is fixed. We implement two ice cloud optical property parameterizations in addition to the spherical and rough column parameterizations built into RRTM as mentioned previously. For each ice model, the optical properties are calculated for a number of wavelengths within each band before being averaged; the band-averaged optical properties are used for the calculations. A Planck function with a temperature of 250 K (about the temperature in the middle of the troposphere) is used as the weighting function for averaging. The size distribution is a gamma function with an effective variance fixed to be 0.1, consistent with that used in the MODIS Collection 6 and CERES-MODIS retrieval products.
Figure 1 shows the scattering property parameterizations of four ice particle models for two effective radii, 10 and 30 μm, as functions of the RRTM bands. Note that smaller particles have larger ω and smaller g, both of which lead to stronger scattering effects. For the atmospheric window bands (about 700–1200 cm−1), ω is about 0.4–0.7 and g is about 0.9. This means ice clouds scatter nearly as much thermal radiation as they absorb. Because the gas absorption in the window bands is weak, ice cloud scattering plays a more important role in the radiative transfer than in other bands. The difference between the four ice models is roughly 0.1 for ω and 0.03 for g. In particular, the rough column model tends to have the largest ω and smallest g and thus has the strongest scattering effect. The aggregate model has moderate ω and the largest g. Natural ice cloud particles have even more complicated geometries than the four ice models presented (Baran 2009). However, we use the four ice particle models to represent the natural variability of particle geometries and to test the performance of the newly developed schemes without losing too much generality.
Scattering property parameterizations of four ice particle models for RRTM, which divides the longwave spectral region (10–3250 cm−1) into 16 bands for radiation calculations (vertical dashed lines). Columns are for Re = (left) 10 and (right) 30 μm. Rows are (top) single-scattering albedo and (bottom) asymmetry factor.
Citation: Journal of the Atmospheric Sciences 75, 7; 10.1175/JAS-D-18-0014.1
b. RRTM simulations
We implement the four schemes discussed in section 2 into the standard RRTM release (version 3.3): the Chou scaling approach, the similarity principle scaling approach, the Chou scaling-adjustment scheme, and the similarity scaling-adjustment scheme. In addition, there are two schemes available in RRTM: one neglects scattering and the other uses DISORT to more accurately solve the radiative transfer equation.
For each of the 2160 atmosphere–cloud profiles, we performed irradiance calculations with these six schemes. The DISORT calculations are performed with 16 streams and a 16-order Legendre expansion, the results of which serve as a reference for the error estimation of the other five schemes. The DISORT calculation was compared to the rigorous line-by-line DISORT (Turner et al. 2003; Turner 2005) benchmark and was within 2 W m−2 in terms of TOA irradiance. (We compared RRTM 16-stream simulations with the line-by-line DISORT counterparts with a 0.01-cm−1 spectral resolution for 540 of the atmosphere–cloud profiles in Table 2 with the ice particle chosen to be the aggregate model.) The surface emissivity is assumed to be unity (no reflection), which is reasonable for calculations over ocean and snow, since their emissivity is very close to 1 (about 0.97–0.99) at around 11 μm (Masuda et al. 1988; Li et al. 2013), the central wavelength of the thermal infrared regime, though all of the schemes are able to deal with a nonunity surface albedo by adding the reflection of the surface downward irradiance and the surface emission to obtain the surface upward irradiance. Figure 2 shows the error of five schemes in terms of TOA upward and surface downward longwave irradiances. The considerable errors in the TOA upward irradiance for the original neglect-scattering scheme are noted, with an overestimation up to 14 W m−2. The two optical thickness scaling schemes reduce the maximum overestimation to about 8 W m−2. The two adjustment schemes further reduce the overestimation to less than 2 W m−2, which is in the range of the RRTM error when compared to the more rigorous line-by-line DISORT method (not shown).
Error of TOA upward and surface downward irradiances vs reference TOA upward irradiance scatterplot of five schemes: neglect scattering (black), (left) similarity scaling (cyan) and similarity adjustment (red), and (right) Chou scaling (cyan), and Chou adjustment (red). An RRTM calculation with scattering fully considered using 16-stream DISORT serves as the reference. Scattering is not explicitly considered in the five schemes; therefore, they consume similar computational time. For each scheme, radiances are calculated at three angles independently and integrated to obtain irradiance. Irradiances calculated using two to four angles (two and four angle solutions are not shown) are very similar, except when using one angle, which has large errors even for clear sky. A total of 2160 atmosphere–cloud profiles, including various atmospheric profiles, ice cloud particle shape models, cloud heights, optical thicknesses, and effective radii, are used.
Citation: Journal of the Atmospheric Sciences 75, 7; 10.1175/JAS-D-18-0014.1
The original neglect-scattering scheme underestimates the surface downward irradiance by less than 2 W m−2 because the cloud reflection caused by scattering of the upward radiation is ignored. The scaling schemes increase the cloud optical thickness and therefore increase the downward blackbody radiation and reduce the underestimation. With the adjustment schemes, the ignored cloud reflection is compensated for in the downward irradiance and therefore the underestimation of surface downward irradiance is further reduced to less than 0.5 W m−2. Both of the adjustment schemes have similar performance in reducing the surface downward and TOA upward irradiance errors. The irradiances shown in Fig. 2 are obtained by integrating radiances calculated for three angles in the upward and downward directions. We observe that the results obtained using two, three, and four angles are nearly the same. However, use of only a single angle results in significant errors as shown in Fig. 3 even though the scaling and adjustment schemes have reduced the error. This significant error could be due to the quadrature selection. RRTM uses the Gaussian quadrature for the radiance calculation and integration [for a single angle the zenith angle is cos−1(1/31/2)], which has been suggested to be less accurate than an empirical choice of cos−1(1/1.66) (Fu et al. 1997). This issue is solved in the RRTMG by optimizing the angle for a few different bands for a range of total water column values.
As in Fig. 2, but with the radiance calculated for one zenith angle for both upward and downward radiation.
Citation: Journal of the Atmospheric Sciences 75, 7; 10.1175/JAS-D-18-0014.1
To demonstrate the performance of the new adjustment schemes, Fig. 4 shows the comparison of errors between the Chou scaling-adjustment scheme and a four-stream approximation (listed in Table 2), which explicitly considers the cloud scattering. The two schemes have similar accuracies with errors smaller than about 1 W m−2 in terms of TOA upward irradiance, but the four-stream approximation is demonstrably worse for optically thin clouds. The four-stream approximation consumes a computational time of about 20 times that of the adjustment scheme. Therefore, the adjustment schemes have both better accuracy and computational efficiency than the four-stream approximation.
Comparison of TOA upward irradiance error between the adjustment scheme based on Chou’s scaling scheme (cyan) and a four-stream approximation (red).
Citation: Journal of the Atmospheric Sciences 75, 7; 10.1175/JAS-D-18-0014.1
Figure 5 shows the TOA upward irradiance error as a function of visible optical thickness for different values of the cloud height and cloud effective radius. The original neglect-scattering scheme always overestimates the TOA upward irradiance and is at a maximum for intermediate optical thickness (~1), which is consistent with Kuo et al. (2017). Qualitatively, the scattering is less effective for optically thin clouds, while for optically thick clouds the effect of scattering is partly offset by the absorption. The overestimation decreases with Re because the impact of scattering decreases with Re as a result of ω increasing and g decreasing with Re in the atmospheric window spectral region. The overestimation increases with cloud height because the upward irradiance resulting from scattering in a higher cloud layer is less affected by the atmosphere above the cloud. The two scaling schemes reduce the overestimation significantly for small and intermediate optical thicknesses but approach the neglect-scattering scheme for large optical thicknesses. In fact, the no-scattering simulations always approach a constant value for large optical thickness. The two adjustment schemes further reduce the overestimation to less than 2 W m−2 over the entire range of optical thicknesses.
TOA longwave irradiance errors of five schemes as functions of visible optical thickness: neglect scattering (solid black), similarity scaling (dashed red), Chou scaling (dashed blue), similarity adjustment (solid red), and Chou adjustment (solid blue). Columns are for different effective radii of (left) 10, (middle) 30, and (right) 50 μm. Rows are for different cloud heights at (top) 11, (middle) 8, and (bottom) 5 km. Radiances at three angles are calculated and integrated to obtain irradiances. The rough column ice model and the U.S. Standard Atmosphere, 1976, profile are used.
Citation: Journal of the Atmospheric Sciences 75, 7; 10.1175/JAS-D-18-0014.1
Figure 6 is similar to Fig. 5, but it shows the error of surface downward irradiance as a function of visible optical thickness. The original neglect-scattering scheme underestimates the surface downward irradiance and similarly has a maximum value for intermediate optical thickness. The error also decreases with Re but decreases with height, because the cloud reflection of upward radiation is less affected by the atmosphere beneath the clouds. The two scaling schemes again reduce the error significantly for thin and intermediate optical thicknesses but approach the results from the neglect-scattering scheme for high optical thicknesses. Both of the adjustment schemes reduce the errors for all optical thickness values to less than 0.5 W m−2.
As in Fig. 5, but for the surface downward irradiance error.
Citation: Journal of the Atmospheric Sciences 75, 7; 10.1175/JAS-D-18-0014.1
Figure 7 shows the vertical distribution of the heating rate error for each of the five schemes for an intermediate (~1) cloud optical thickness. The original neglect-scattering scheme basically underestimates the heating rate beneath cloud by about 0.1 K day−1, indicating a cooling effect on the atmosphere, consistent with the overestimation of the TOA upward irradiance. The cloud scattering plays a role in reflecting upward thermal radiance and warming the atmosphere below the cloud. The two scaling schemes reduce this heating rate error to less than 0.01 K day−1, whereas the two adjustment schemes further reduce it to almost zero. The vertical average heating rate error is reduced by the two scaling schemes and further reduced by the two adjustment schemes. An accurate calculation of the thermal heating rate is especially important at high latitudes, where the thermal radiation is comparable to, or may even dominate, the contribution from solar radiation.
As in Fig. 5, but for heating rate error of the five schemes as a function of height. The cloud optical thickness is 1.
Citation: Journal of the Atmospheric Sciences 75, 7; 10.1175/JAS-D-18-0014.1
The ice cloud parameterization also leads to great uncertainties in the radiative transfer simulations. It is uncertain whether the longwave scattering effect is negligible compared to the effect of ice cloud parameterization. To gain further insight, we investigate the effect of ice cloud parameterization on the thermal radiative transfer. Figure 8 shows the TOA longwave radiation as a function of visible optical thickness for the four different ice cloud parameterizations. For the same Re and visible optical thickness, different ice cloud parameterizations have the same ice water path. The rough column model always results in the smallest TOA thermal loss, leading to a warming effect in contrast with the other three models. The aggregate model and THM result in the largest values of the TOA thermal radiation and are nearly the same with a difference of less than 2 W m−2. The differences between four ice models are larger for intermediate optical thickness values, at most 20 W m−2. The differences are larger for high clouds and larger cloud effective radii because the optical properties of larger particles are more sensitive to particle shape. The sensitivity of TOA irradiance to Re is relatively weak compared with the parameterizations, especially for the aggregate model and THM. Thus, in GCMs, it seems reasonable to assume a constant value of Re for longwave calculations. However, for the rough column model, the TOA irradiance for an Re of 10 μm is still larger than that for an Re of 30 μm by up to about 15 W m−2, indicating a potential cooling effect when the cloud particle size is reduced, for example, because of aerosol pollution.
As in Fig. 5, but for TOA longwave irradiance as a function of visible cloud optical thickness for four different ice cloud models: spherical (dark blue), rough column (green), aggregate (red), and THM (light blue). RRTM with 16-stream DISORT is used in the simulations.
Citation: Journal of the Atmospheric Sciences 75, 7; 10.1175/JAS-D-18-0014.1
4. Discussion and conclusions
This study demonstrates that neglecting cloud longwave scattering can result in a significant overestimation of the top-of-the-atmosphere (TOA) thermal radiation, which in turn leads to a spurious cooling effect below the clouds. The overestimation can be as large as 14 W m−2 depending on the cloud thickness, altitude, and optical properties. In general, the overestimation reaches its highest values for a higher cloud of an intermediate optical thickness with relatively small cloud particles. Additionally, neglecting cloud longwave scattering leads to an underestimate in the surface downward irradiance—or in other words, it overestimates the surface net upward irradiance by up to 2 W m−2. Neglecting multiple scattering in lower clouds results in larger errors in the surface downward irradiance.
Theoretically, clouds scatter longwave upward radiation to the side and backward directions and therefore reduce the thermal radiation reaching the TOA while also increasing the thermal radiation reflected back to surface. Thus, neglecting scattering overestimates TOA thermal radiation and underestimates surface downward radiation. By increasing the cloud absorptive optical thickness while still neglecting scattering, less thermal radiation reaches the TOA and more blackbody radiation emanating from clouds reaches the surface, and thus both TOA and surface radiation errors are reduced. However, for optically thick clouds, increasing the cloud absorptive optical thickness offers little improvement because the irradiance changes little with increasing optical thickness. We applied the similarity principle and Chou et al.’s (1999) scaling schemes to RRTM and verified the conclusion stated above. For optically thick clouds, the TOA irradiance overestimation can still reach 8 W m−2.
We developed adjustment schemes for both the similarity principle and Chou et al. (1999) scaling schemes, in which the downward radiances are calculated and then used to adjust the upward radiances. In an iterative process, the upward radiances are again used to adjust the downward radiances. A negative adjustment term for the upward radiance is derived that increases with cloud optical thickness and approaches a constant for very thick clouds. The adjustment term for the downward radiance is similar but positive. In such a way, the TOA and surface irradiance errors are further reduced. We applied these two schemes to RRTM and found that the TOA upward and surface downward irradiance errors are reduced to less than 2 and 0.5 W m−2, respectively, over a wide range of optical thickness values. The comparison with the four-stream approximation shows that the new adjustment schemes are more accurate than the four-stream approximation and about 20 times faster computationally.
The simulations are performed for various atmospheric profiles, cloud heights, optical thicknesses, ice particle sizes, and cloud optical property parameterizations. As a reference to evaluate the error, the cloud scattering is considered with a 16-stream DISORT computation in RRTM, which has been shown to agree with rigorous line-by-line DISORT benchmarks within 2 W m−2 for all sky conditions.
In addition, considering that the similarity principle failed to improve the two-stream approximation for multiple layers combined with the adding method, we further verified the new schemes for multiple cloud layers and obtained very similar errors as for single layers (not shown). We conclude that the adjustment schemes are effective in improving the accuracy of longwave radiative transfer for virtually all sky conditions.
In the RRTM simulations, we found that the radiance computed at a single Gaussian quadrature angle is insufficient for the irradiance integral, similar to the study by Fu et al. (1997). More than one Gaussian quadrature angle results in relatively accurate irradiance integrals. Thus, we suggest using at least two Gaussian quadrature angles to solve the longwave radiative transfer equation in a broadband radiative transfer model such as RRTM.
Cloud longwave scattering has a warming effect beneath clouds of up to 0.1 K day−1, but this is neglected in most GCMs. The scaling and adjustment schemes help to account for such a warming effect.
We also found that the broadband irradiances are sensitive to the ice cloud optical property parameterization. The differences between different parameterizations are of similar magnitude to the scattering effect itself. As studies involving the remote sensing of clouds are leading to more accurate cloud microphysics parameterizations, the new schemes, in conjunction with more realistic cloud microphysical illustrations, will help improve the energy budget estimation.
Finally, we note that RRTMG, as a GCM version of RRTM, uses one angle following Fu et al. (1997) with optimizations in a few bands as a function of the total column water. Further improvement of the RRTMG by using more than one angle and the adjustment schemes will be explored in future work.
Acknowledgments
This study was supported by the National Science Foundation (AGS-1632209) and partly by the endowment funds related to the David Bullock Harris Chair in Geosciences at the College of Geosciences, Texas A&M University. The effort of coauthor Huang is supported by the U.S. Department of Energy, Office of Science, Office of Biological and Environmental Research, Climate and Environmental Science Division under Award DE-SC0012969 to the University of Michigan. The simulations are conducted through the Texas A&M supercomputing facility.
APPENDIX
Physical Meaning of the Similarity Scaling and Adjustment
The error of the similarity principle scaling solution τcω[(1 − g)/2]πBc is basically smaller than that of the neglect-scattering solution τcω[(1 − g)/2]πBS because the cloud temperature is lower than the surface temperature, especially when the cloud layer is high. The error of the similarity scaling scheme comes from the increase of the cloud thermal radiation resulting from larger absorptive optical thickness.
Error of the schemes in terms of TOA upward irradiance for the simplified single-atmospheric-layer model.
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