## 1. Introduction

Solving the radiative transfer equation (RTE) is a key issue when dealing with radiative processes related to climate simulations. Because RTE is an integro-differential equation, it has no exact solutions. In recent decades, numerous approximation techniques have been proposed to solve RTE (e.g., Liou 1974; Liou et al. 1988; Shibata and Uchiyama 1992; Kylling et al. 1995; Li and Ramaswamy 1996; Fu et al. 1997; Li and Dobbie 1998; Thomas and Stamnes 1999; Lin et al. 2013; Zhang et al. 2010).

Since scattering is much weaker for infrared radiation than it is for shortwave radiation, an absorption approximation (AA) is used in most current climate models (Oreopoulos et al. 2012). This involves replacing the scattering phase function with a delta function, meaning that scattering is neglected in all but the forward direction. AA leads to larger errors in cloudy sky cases, where clouds play an important role in radiative transfer with different dynamics and microphysics (Lu et al. 2014, 2016). Studies have shown that AA can cause the outgoing longwave radiation to be overestimated in climate simulations, with errors of >4 W m^{−2} at the top of the atmosphere (TOA) (Toon et al. 1989; Fu et al. 1997; Li 2002).

The advantage of AA is that the upward- and downward-transfer processes are completely separated. Thus, the flux calculations are done individually layer by layer on an upward and a downward path.

Conversely, if scattering is considered, the upward and downward intensities are coupled. Under this condition, a full radiative transfer through an inhomogeneous atmosphere has to be considered by using the discrete-ordinates method (DOM) to solve the complicated layer connections. Fu (1991) derived an inverse-matrix method to deal with multiple scattering in the atmosphere by using the two-stream DOM (*δ*-2DOM) and the four-stream DOM (*δ*-4DOM). In general, more streams lead to less uncertainty, and indeed it is found that *δ*-4DOM is more accurate than *δ*-2DOM, especially for thin optical depths. Even if scattering is included, TOA errors for *δ*-2DOM can still be >2 W m^{−2} for cirrus clouds (Zhang et al. 2016). The *δ*-4DOM results tend to be very accurate, but the calculation process is complicated. The purpose of this study is to establish a more efficient infrared radiative transfer method in a scattering medium, that is, one that exhibits a comparable level of accuracy to *δ*-4DOM.

In the past two decades, the variational iteration method (VIM) has been used to deal with many nonlinear problems (He 1997, 1999, 2007). In traditional nonlinear analytical methods such as the perturbation method the perturbation parameters have to be small. VIM does not have such a limitation and it can be applied to a wide range of nonlinear problems without linearization or the introduction of small perturbations. In addition, VIM has been shown to converge faster to the exact solution than other methods for nonlinear equations do. Usually, VIM is very effective, and an accurate solution can be found for most nonlinear problems in only one or two iterations (Tatari and Dehghan 2007). Because of its flexibility, convenience, and efficiency, this method has been applied to various nonlinear equations (Tatari and Dehghan 2007; Liu et al. 2013; Jafari 2014; and many others).

In the VIM, a nonlinear problem is separated into linear and nonlinear terms, where the latter are usually difficult to deal with and are initially approximated as a first guess. Subsequently, a correction function is constructed by using a general Lagrange multiplier that can be identified optimally via variational theory. Although the infrared RTE is not a nonlinear equation, it contains an integral term for the scattering, and this is very complicated to deal with. Therefore, if VIM is applied to the infrared RTE, the integral term can be handled by the nonlinear operator. In VIM, the AA solution can be used as the zeroth-order initial approximation. The integral term is then involved in the first-order iteration process, and the final result approaching the exact solution can be derived by means of a Lagrange multiplier.

In the following section, VIM is introduced and applied to the infrared RTE, and explicit solutions for AA and DOM are derived. The calculation results and their comparison with those from the *δ*-128-stream radiative transfer scheme (*δ*-128S) (Stamnes et al. 1988) are shown in section 3. Finally, a summary is given in section 4.

## 2. Variational iteration method based on the absorption approximation

*μ*,

*τ*,

*ω*, and

*P*are the cosine of the zenith angle, the optical depth, the Planck function at optical depth

*τ*, the single-scattering albedo, and the azimuthal independent phase function, respectively. The optical depth

*τ*is equal to 0 at the top of medium and increases from the top to the bottom of the medium.

### a. Absorption approximation

*δ*-2AA) and the four-stream case (

*δ*-4AA). In the following calculation, the AA solution is used as the initial approximate solution of the VIM.

### b. Variational iteration method

*L*, and

*N*are an inhomogeneous term, a linear operator, and a nonlinear operator, respectively. If

*n*th iteration, the (

*n*+ 1)th-order functional solution is

*g*is the asymmetry factor. The AA expression derived in Eq. (4) is used as the initial zeroth-order solution. By substituting Eq. (4) into Eq. (14), we obtain the first-order solution of upward intensity at

*I*

^{1}) instead of the results of AA (

*I*

^{0}) for obtaining more accurate intensity. We denote the two-stream solution of Eq. (15) as 2VIM.

In summary, the radiative transfer process in a multilayer atmosphere is based on two steps. First, AA is used to obtain the intensity solutions of

*I*

^{1}) is used as an alternative to AA result (

*I*

^{0}) in the first term of Eqs. (21). We denote the four-stream solution as 4VIM. Finally, we obtain the upward and downward fluxes:

*δ*-function adjustment is used to adjust the optical parameters following Wiscombe (1977). We refer to the VIM solutions with the

*δ*-function applied as

*δ*-2VIM and

*δ*-4VIM.

## 3. Comparison and discussion

In the following, the accuracy of *δ*-2VIM and *δ*-4VIM will be systematically investigated by comparing them with *δ*-2AA, *δ*-4AA, *δ*-2DOM, and *δ*-4DOM. In addition, *δ*-128S is used as the benchmark model.

### a. Emissivity of a single layer

*T*, the accuracy of VIM is examined in relation to the effective monochromatic emissivity. The upward and downward effective emissivities are defined as

*ω*,

*g*) are used: (0.711, 0.904) and (0.498, 0.947). The first one is for cirrostratus with an effective radius of

*r*

_{e}= 41.5

*μ*m in the spectral interval 1100–1250 cm

^{−1}(Fu 1991). The other one is for water clouds with an effective radius of

*r*

_{e}= 15

*μ*m at the infrared wavelength of 11

*μ*m (Fu et al. 1997).

For the upward and downward emissivities, the relative errors of various approximations against *δ*-128S for cirrostratus are shown in Fig. 1. The Henyey–Greenstein (HG) phase function is used. The optical depth

For the upward emissivity, the relative errors of *δ*-2AA (Fig. 1a) are 2%–5% for *δ*-2VIM (Fig. 1c) and *δ*-2DOM (Fig. 1b) have very similar accuracy for small optical depths. Both produce errors of −2% when *δ*-2DOM occur for *δ*-2VIM are <1% when *δ*-4AA (Fig. 1d) is very similar to that of *δ*-2AA when *δ*-4AA are drastically decreased for small optical depths and are 0%–2% for *δ*-4DOM (Fig. 1e) and *δ*-4VIM (Fig. 1f), as only 1% positive errors occur for both for

For the downward emissivity, *δ*-2AA (Fig. 1g) produces 0%–1% positive errors for *δ*-2AA occur for small optical depths and reach −10% for *δ*-2VIM (Fig. 1i) and *δ*-2DOM (Fig. 1h) are similar for *δ*-2VIM and *δ*-2DOM both produce positive errors when *δ*-2DOM for *δ*-2VIM are <2%. This indicates that *δ*-2VIM is more accurate than *δ*-2AA at small optical depths and more accurate than *δ*-2DOM at large optical depths. Both of *δ*-2AA and *δ*-4AA (Fig. 1j) produce negative errors in most regions, and large errors of −10% occur when *δ*-2AA and *δ*-4AA. The relative errors are also significantly reduced in *δ*-4DOM (Fig. 1k) and *δ*-4VIM (Fig. 1l), as both are bounded by 2% in most regions. Errors of 2%–5% occur at *δ*-4DOM and in the narrower *δ*-4VIM.

The above results show that *δ*-2VIM is similar to *δ*-2DOM in the small optical depth range but more accurate than *δ*-2DOM in the large optical depth range. Both *δ*-2VIM and *δ*-2DOM are more accurate than *δ*-2AA. As expected, *δ*-4DOM and *δ*-4VIM are much more accurate than *δ*-4AA.

Why do 2DOM and 2VIM perform poorly when *τ*, the outgoing photons undergo only a few scattering events. Thus, the results are strongly dependent on the single-scattering direction and the radiative intensity becomes highly anisotropic. The two-stream approximation, in which only a limited angular pattern for radiative intensity is considered, cannot properly handle anisotropic radiation, therefore, a four-stream approximation can generally much improve the result for a thin optical depth case.

The results for water clouds are shown in Fig. 2. For the upward emissivity, the accuracies of *δ*-2AA (Fig. 2a), *δ*-2DOM (Fig. 2b), and *δ*-2VIM (Fig. 2c) are similar at small optical depths. Their errors are as high as −2% for *δ*-2AA, while the errors are <1% for both *δ*-2DOM and *δ*-2VIM when *δ*-4AA (Fig. 2d), *δ*-4DOM (Fig. 2e), and *δ*-4VIM (Fig. 2f), especially at thin optical depths. Positive errors of >1% occur at *δ*-4AA and at *δ*-4DOM and *δ*-4VIM.

For the downward emissivity, large negative errors also occur at small optical depths for *δ*-2AA (Fig. 2g), *δ*-2DOM (Fig. 2h), and *δ*-2VIM (Fig. 2i). Negative errors of 5% occur at *δ*-2AA and *δ*-2DOM and *δ*-2VIM. The relative errors are also significantly reduced in *δ*-4AA (Fig. 2j), *δ*-4DOM (Fig. 2k), and *δ*-4VIM (Fig. 2l), especially at thin optical depths. The errors are 1%–2% at *δ*-4AA while the errors can reach reach 2%–5% for both of *δ*-4DOM and *δ*-4VIM when *δ*-4AA is a little more accurate than *δ*-4DOM and *δ*-4VIM for a thin optical depth with a large surface emissivity. In general, *δ*-2AA, *δ*-2DOM, and *δ*-2VIM all produced large errors at small optical depths and the errors are also reduced in *δ*-4AA, *δ*-4DOM, and *δ*-4VIM. While *δ*-4DOM and *δ*-4VIM are comparable in accuracy, both are much more accurate than the two-stream results.

### b. Multilayer atmosphere

A radiation model (Li and Barker 2005) is used to study the accuracy of the VIM scheme for multiple layers within a model atmosphere. A correlation-*k* distribution scheme is used to simulate the gaseous transmission with profiles for H_{2}O, CH_{4}, CO_{2}, N_{2}O, O_{3}, and CFCs. This model is reasonably efficient because it neglects scattering for certain intervals with very large absorption coefficients and water vapor continuum at high altitudes. Nine infrared bands are adopted in this model in wavenumber ranges 0–340, 340–540, 540–800, 800–980, 980–1100, 1100–1400, 1400–1900, 1900–2200, and 2200–2500 cm^{−1}. The optical properties of ice and water clouds are calculated based on the radiative-property parameterization of Yang et al. (2015) and Lindner and Li (2000), respectively. The midlatitude winter atmospheric profiles (McClatchey et al. 1972) are used. The atmospheric profile is divided into homogeneous layers with a geometrical thickness of 0.25 km. In this model, a low cloud with an effective radius *r*_{e} = 5.89 *μ*m and liquid water content LWC = 0.22 g m^{−3} is located at 1.0–2.0 km, a middle cloud with *r*_{e} = 6.2 *μ*m and LWC = 0.28 g m^{−3} is located at 4.0–5.0 km, and a high cloud with a mean effective size *D*_{e} = 41.1 *μ*m and an ice water content IWC = 0.0048 g m^{−3} is located at 9.0–11.0 km. The surface emissivity

In the left column of Fig. 3, the benchmark heating rates calculated by *δ*-128S are given under conditions of low clouds (Fig. 3a), middle clouds (Fig. 3d), high clouds (Fig. 3g), and the all-sky case containing a combination of low, middle, and high clouds (Fig. 3j). The middle column of Fig. 3 shows the errors in the calculated heating rates of *δ*-2AA, *δ*-2DOM, and *δ*-2VIM compared to those of *δ*-128S. Furthermore, the right column of Fig. 3 shows the errors in the calculated heating rates of *δ*-4AA, *δ*-4DOM, and *δ*-4VIM compared to those of *δ*-128S.

For the low-cloud case, the maximum errors of *δ*-2AA, *δ*-2DOM, and *δ*-2VIM are −0.8, +0.5, and +0.3 K day^{−1}, respectively, at a height corresponding to the top of the cloud. The maximum errors of *δ*-4AA, *δ*-4DOM, and *δ*-4VIM are −0.9, +0.1, and +0.03 K day^{−1}, respectively, at the same height. This shows that, for low-level water clouds, *δ*-2VIM (*δ*-4VIM) is more accurate than *δ*-2AA (*δ*-4AA) and *δ*-2DOM (*δ*-4DOM).

For the middle cloud, the maximum errors of *δ*-2AA, *δ*-2DOM, and *δ*-2VIM are approximately −0.9, +1.1, and +0.8 K day^{−1}, respectively, at a height corresponding to the top of the cloud. At the bottom of the cloud, the errors are +0.6 K day^{−1} for *δ*-2AA and just −0.05 and +0.05 K day^{−1} for *δ*-2DOM and *δ*-2VIM, respectively. At the top of the cloud, *δ*-4AA, *δ*-4DOM, and *δ*-4VIM produce biases of approximately +1.1, +0.7, and +0.5 K day^{−1}, respectively.

For the high-cloud case, the optical depth of the ice cloud is much smaller than that of the water cloud. Based on Fig. 1, the accuracies of *δ*-2DOM and *δ*-2VIM are similar and are better than that of *δ*-2AA. Furthermore, *δ*-4AA, *δ*-4DOM, and *δ*-4VIM have very similar accuracies. As seen in the third row in Fig. 3, *δ*-2AA produces an error of approximately +0.6 K day^{−1} at a height corresponding to the top of a high cloud. The accuracies of *δ*-2DOM and *δ*-2VIM are similar; the maximum errors of both are approximately +0.5 K day^{−1}. The maximum errors of *δ*-4AA, *δ*-4DOM, and *δ*-4VIM are all about +0.1 K day^{−1}. However, the difference is that the maximum errors occur at top of the high cloud for *δ*-4AA but on the bottom for *δ*-4DOM and *δ*-4VIM.

In the case of all three clouds together, *δ*-2VIM is more accurate than *δ*-2AA and *δ*-2DOM for the low and middle clouds. In general, *δ*-4DOM and *δ*-4VIM are comparable in accuracy for cloud heating rate.

The results of relative error in heating rate are shown in Table 1; we only address the altitude locations where the maximum absolute errors occur. In the low-cloud, middle-cloud, and all-cloud cases, if the maximum errors occur in clear-sky atmosphere, the relative error could be very large; if the maximum errors occur at cloud top, the relative error would be very small. For example in the low-cloud case, the maximum absolute error of *δ*-4VIM is −0.03 K day^{−1}. As it does not appear at the cloud top, the relative error reaches −43.9%. On the other hand, the maximum error of *δ*-4AA reaches −0.9 K day^{−1} but appears at the cloud top, its relative error is only −3.4%. For the high cloud, the maximum errors of all four-stream schemes are about 0.1 K day^{−1}. However, they occur either at the cloud top (*δ*-4AA) or at the cloud bottom (*δ*-4VIM and *δ*-4DOM). After considering the specific locations of cooling rate, the relative error is about 4% for *δ*-4AA and about 15% for *δ*-4VIM and *δ*-4DOM.

Comparison of *δ*-2AA, *δ*-4AA, *δ*-2DOM, *δ*-4DOM, *δ*-2VIM, and *δ*-4VIM for heating-rate relative errors (%) at which maximum errors are addressed by using the midlatitude winter atmospheric profile with

The results of upward (downward) flux at the TOA (surface) for the six schemes are presented in Table 2 for midlatitude winter profiles with the surface emissivity *δ*-2AA overestimates the upward flux at TOA by 1.8 and 2.1 W m^{−2}, respectively. The errors of the *δ*-2DOM are up to −1.0 and −1.2 W m^{−2}, while that of the *δ*-2VIM is limited to −0.7 W m^{−2}. Also, the *δ*-4VIM is generally more accurate than the *δ*-4DOM. However, the results for the high clouds are reversed, with the *δ*-2VIM errors becoming larger than that of *δ*-2DOM. For the down flux at surface, the AA schemes are generally more accurate than the corresponding DOM and VIM schemes except in the middle-cloud case.

Comparison of *δ*-2AA, *δ*-4AA, *δ*-2DOM, *δ*-4DOM, *δ*-2VIM, and *δ*-4VIM for flux (W m^{−2}) at the top and surface by using the midlatitude winter atmospheric profile with *δ*-128S are listed in parentheses.

Why does VIM not perform better for ice clouds whether for the relative error of emissivity in a single layer or absolute error of heating rate and flux in multilayers? In VIM, a nonlinear problem is separated into linear and nonlinear terms and the nonlinear term is initially approximated with a first guess. Subsequently, a correction function is constructed by a general Lagrange multiplier. The solution of AA is used as a first guess for the infrared radiative transfer equation. Figures 1 and 2 indicate that AA is generally poor in accuracy for a thin optical depth. This affects the performance of VIM on ice clouds, since the optical depth of an ice cloud is always small.

For climate modeling, the efficiency of radiative transfer parameterization is also very important. Table 3 lists the computing times required for *δ*-2AA, *δ*-4AA, *δ*-2DOM, *δ*-4DOM, *δ*-2VIM, and *δ*-4VIM, which were computed by HP EliteDesk 880 G1 TWR with 8 Intel(R) Core(TM) i7-4790 CPUs, 32-bit operating system, and 8GB memory. The results are normalized to that of *δ*-2AA. The computational efficiency of *δ*-2VIM is slightly better than that of *δ*-2DOM, which took more than double the time of *δ*-2AA. However, *δ*-4VIM is more than twice as fast as *δ*-4DOM for the radiation algorithm alone and the radiation model.

Computing times of *δ*-2AA, *δ*-4AA, *δ*-2DOM, *δ*-4DOM, *δ*-2VIM, and *δ*-4VIM (normalized by the computing time of *δ*-2AA).

## 4. Summary and conclusions

VIM differs from other analytical methods for solving nonlinear differential equations in that it requires neither linearization nor small perturbations. The optimal result is constructed through variation by a Lagrange multiplier. The aim of this study was to use VIM to establish a more efficient infrared radiative transfer method for a scattering medium, whose accuracy would be comparable to that of *δ*-4DOM. This was the first time that VIM had been applied to solve the infrared RTE. It was found that the scattering term in the infrared RTE could be dealt with as a nonlinear operator in VIM. By taking the AA solution as the zeroth-order solution, the scattering effect was properly included in the first-order iterative solution.

The six schemes of *δ*-2AA, *δ*-4AA, *δ*-2DOM, *δ*-4DOM, *δ*-2VIM, and *δ*-4VIM were compared systematically against the benchmark results provided by *δ*-128S for two single-layer cases and a multilayer case. For the cirrostratus case, the results of *δ*-2VIM were similar to those of *δ*-2DOM, which is more accurate than *δ*-2AA for small optical depths, and is more accurate than *δ*-2DOM for large optical depths. A dramatic improvement occurred for both *δ*-4DOM and *δ*-4VIM, with relative errors becoming <2%. For the water cloud case, *δ*-2AA, *δ*-2DOM, and *δ*-2VIM all produced large errors at small optical depths. The errors were also reduced in *δ*-4AA, *δ*-4DOM, and *δ*-4VIM. For a multilayer atmosphere, VIM gave more accurate results than those of DOM and AA for the low and middle clouds in both the two- and four-stream cases. However, the errors from VIM for high clouds were similar to those from AA.

Computationally, *δ*-2VIM was slightly faster than *δ*-2DOM, which took more than double the time of *δ*-2AA. However, *δ*-4VIM was more than twice as fast as *δ*-4DOM for both the pure radiation algorithm and the radiation model in a layered, cloudy atmosphere. In general, the main benefit of *δ*-2VIM was an improved accuracy with a computational time similar to that of *δ*-2DOM. The main benefit of *δ*-4VIM was improved computational efficiency with accuracy similar to that of *δ*-4DOM.

In view of their overall high accuracy and computational efficiency, the conclusion is that *δ*-2VIM and *δ*-4VIM are well suited for parameterizing the infrared radiation in climate models.

## Acknowledgments

The authors thank the four anonymous reviewers for their constructive comments and Professor P. Yang for his editorial efforts. The work is supported by the State Key Development Program for Basic Research of China (2015CB452805), National Natural Science Foundation of China (41305004, 91537213, and 41675003), Grant-in-Aid for Scientific Research (B) (15H03729) of the Japan Society for the Promotion of Science, Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD), and Startup Foundation for Introducing Talent of NUIST.

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