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  • View in gallery

    Schematic diagram of (left) R1,2(μ, μ0)/T1,2(μ, μ0) and (right) .

  • View in gallery

    Relative errors of the δ-2DDA, δ-2SDA, δ-4DDA, and δ-4SDA for (top two rows) direct reflection and (bottom two rows) absorption. The benchmark is δ-128S. The Henyey–Greenstein phase function is used, with the asymmetry factors g1 = 0.837 and g2 = 0.861. The absorbing media with ω1 = ω2 = 0.9 and the optical depths of the two layers are the same (τ1 = τ2).

  • View in gallery

    (top) Heating rate profiles computed from δ-128S and (bottom) the error profiles from δ-2DDA, δ-2SDA and δ-4DDA, δ-4SDA with clear skies. (left to right) Three solar zenith angles μ0 = 1, 0.5, and 0.25 are considered.

  • View in gallery

    As in Fig. 3, but for the sky containing (top two rows) low and (bottom two rows) middle clouds.

  • View in gallery

    As in Fig. 3, but for the sky containing (top two rows) high clouds and (bottom two rows) all the three cloud types.

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Doubling–Adding Method for Delta-Four-Stream Spherical Harmonic Expansion Approximation in Radiative Transfer Parameterization

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  • 1 Chinese Academy of Meteorological Sciences, University of Chinese Academy of Sciences, Beijing, and Shanghai Typhoon Institute, China Meteorological Administration, Shanghai, China
  • 2 Canadian Center for Climate Modeling and Analysis, University of Victoria, Victoria, British Columbia, Canada
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Abstract

Though the single-layer solutions have been found for the δ-four-stream spherical harmonic expansion method (SHM) in radiative transfer, there is lack of a corresponding doubling–adding method (4SDA), which enables the calculation of radiative transfer through a vertically inhomogeneous atmosphere with multilayers. The doubling–adding method is based on Chandrasekhar's invariance principle, which was originally developed for discrete ordinates approximation. It is shown that the invariance principle can also be applied to SHM and δ-four-stream spherical harmonic expansion doubling–adding method (δ-4SDA) is proposed in this paper. The δ-4SDA method has been systematically compared to the δ-Eddington doubling–adding method (δ-2SDA), the δ-two-stream discrete ordinates doubling–adding method (δ-2DDA), and δ-four-stream discrete ordinates doubling–adding method (δ-4DDA). By applying δ-4SDA to a realistic atmospheric profile with gaseous transmission considered, it is found that the accuracy of δ-4SDA is superior to δ-2SDA or δ-2DDA, especially for the cloudy/aerosol conditions. It is shown that the relative errors of δ-4SDA are generally less than 1% in both heating rate and flux, while the relative errors of both δ-2SDA and δ-2DDA can be over 6%. Though δ-4DDA is slightly more accurate than δ-4SDA in heating rates, both of them are accurate enough to obtain the cloud-top solar heating. Here δ-4SDA is superior to δ-4DDA in computational efficiency. It is found that the error of aerosol radiative forcing can be up to 3 W m−2 by using δ-2SDA at the top of the atmosphere (TOA); such error is substantially reduced by applying δ-4SDA. In view of the overall accuracy and computational efficiency, δ-4SDA is suitable for application in climate models.

Corresponding author address: Feng Zhang, Chinese Academy of Meteorological Sciences, 46 Zhong-Guan-Cun South Street, Beijing 100081, China. E-mail: feng_zhang126@126.com

Abstract

Though the single-layer solutions have been found for the δ-four-stream spherical harmonic expansion method (SHM) in radiative transfer, there is lack of a corresponding doubling–adding method (4SDA), which enables the calculation of radiative transfer through a vertically inhomogeneous atmosphere with multilayers. The doubling–adding method is based on Chandrasekhar's invariance principle, which was originally developed for discrete ordinates approximation. It is shown that the invariance principle can also be applied to SHM and δ-four-stream spherical harmonic expansion doubling–adding method (δ-4SDA) is proposed in this paper. The δ-4SDA method has been systematically compared to the δ-Eddington doubling–adding method (δ-2SDA), the δ-two-stream discrete ordinates doubling–adding method (δ-2DDA), and δ-four-stream discrete ordinates doubling–adding method (δ-4DDA). By applying δ-4SDA to a realistic atmospheric profile with gaseous transmission considered, it is found that the accuracy of δ-4SDA is superior to δ-2SDA or δ-2DDA, especially for the cloudy/aerosol conditions. It is shown that the relative errors of δ-4SDA are generally less than 1% in both heating rate and flux, while the relative errors of both δ-2SDA and δ-2DDA can be over 6%. Though δ-4DDA is slightly more accurate than δ-4SDA in heating rates, both of them are accurate enough to obtain the cloud-top solar heating. Here δ-4SDA is superior to δ-4DDA in computational efficiency. It is found that the error of aerosol radiative forcing can be up to 3 W m−2 by using δ-2SDA at the top of the atmosphere (TOA); such error is substantially reduced by applying δ-4SDA. In view of the overall accuracy and computational efficiency, δ-4SDA is suitable for application in climate models.

Corresponding author address: Feng Zhang, Chinese Academy of Meteorological Sciences, 46 Zhong-Guan-Cun South Street, Beijing 100081, China. E-mail: feng_zhang126@126.com

1. Introduction

The radiative transfer equation is an integro-differential equation. The exact solution of the radiative transfer equation in a scattering and absorbing media generally does not exist; thus, the approximate methods are necessary. In the last several decades considerable attention has been paid to find the simple and effective methods for solving the radiative transfer equation (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; Lu et al. 2009; Zhang et al. 2013a,b; and others). Among these methods, the δ-two-stream approximations provide a simple answer to radiative transfer, especially the δ-Eddington approximation and δ-two-stream discrete ordinates method (DOM), are widely used in climate models.

The applications in climate modeling and remote sensing require accurate results in reflection, transmission, and absorption for a wide range of solar zenith angles and optical thicknesses. However, as shown in King and Harshvardhan (1986), even for the case of single layer, no δ-two-stream method satisfies this criterion. Recently, the accuracy of the δ-two-stream approximations has been compared under the real atmospheric profile condition with inclusion of gaseous transmission and cloud (Kay et al. 2001; Lu et al. 2009; Zhang et al. 2013b). It is found that both the δ-two-stream DOM and δ-Eddington approximation were generally accurate under the clear-sky condition, with a relative error of less than 1.5% in heating rate and flux. However, under the cloudy-sky condition, the cloud heating might have been underestimated by as much as 10%, which indicates that a four-stream or higher-order approximation is needed in order to obtain an accurate cloud solar absorption in climate models.

The single-layer analytical solutions of δ-four-stream approximations have been found (Liou et al. 1988; Shibata and Uchiyama 1992; Li and Ramaswamy 1996; Li and Dobbie 1998). Among them, Liou et al. (1988) derived the solution of δ-four-stream DOM. Li and Ramaswamy (1996) obtained the solution of the δ-four-stream spherical harmonic expansion method (SHM), which is the higher-order extension of δ-Eddington approximation. However, all these solutions are only applicable to homogeneous single layer. In current climate models, the layer connection in two-stream radiative transfer is solved generally by the doubling–adding method, which is based on the invariance principle found by Chandrasekhar (1950). The doubling–adding method allows accurate solution of the radiative transfer equation for anisotropic scattering and complicated boundaries, and works naturally with layered media and yields reflection and transmission readily. In addition, the partial cloud can be handled by doubling–adding method under a certain condition (e.g., Chou and Suarez 1999; Li et al. 2005). Because of these advantages, the doubling–adding method has been applied to δ-Eddington approximation (referred as δ-2SDA) and δ-two-stream DOM (referred as δ-2DDA). Also, the doubling–adding method has been applied to the δ-four-stream approximation (Chou 1992; Li and Ramaswamy 1996; Tian et al. 2007; Ayash et al. 2008), but the doubling–adding method used in δ-four-stream approximation is that developed for δ-Eddington approximation (Coakley et al. 1983). We refer this scheme as δ-2/4SDA. In δ-2/4SDA, the isotropic condition is imposed on the diffuse radiation; thus, the angular characteristics of radiance field are not represented properly. It is found that the accuracy of δ-2/4SDA can be even worse than δ-2SDA, especially for flux.

Based on the invariance principle, recently Zhang et al. (2013b) have derived a four-stream doubling–adding method for δ-four-stream DOM (referred as δ-4DDA). It is clearly shown that δ-4DDA can dramatically reduce the errors in flux and heating rate compared to δ-2DDA, especially in cloudy-sky cases. Since the δ-four-stream DOM and δ-four-stream SHM are two popular methods with similar accuracy (Li and Ramaswamy 1996), a further effort is needed to generate an analytical doubling–adding algorithm for δ-four-stream SHM (referred as δ-4SDA), which enables us to calculate the four-stream radiative transfer through a vertically inhomogeneous atmosphere with multilayers. In addition, it is found that δ-4SDA has a higher computational efficiency compared to δ-4DDA. Therefore, δ-4SDA is easier to implement in climate models. In the following section 2, the reflectance and transmittance matrix of the direct/diffuse radiation are presented for a single-layer solution of four-stream SHM. In section 3, the invariance principle is applied to SHM and four-stream spherical harmonic expansion doubling–adding method (4SDA) is proposed. In section 4, the accuracy of δ-4SDA is systematically investigated. Finally, a summary and conclusion is given in section 5.

2. Single-layer solution of four-stream spherical harmonic expansion approximation

a. Direct incident radiation

The relationship between the direct incident radiation and diffuse incident radiation are clearly demonstrated in Zhang et al. (2013a). For direct incident radiation, the azimuthally averaged solar direct incident radiative transfer equation is
e1a
e1b
e1c
where μ is the cosine of the zenith angle (we use the positive and negative μ to denote the upward and downward light beams, respectively); τ is the optical depth; ω is the single-scattering albedo; P(μ, μ′) is the azimuthally averaged scattering phase function, defining the light incidence at μ′, which is scattered in the direction μ; F0 the incoming solar flux; and μ0 is the cosine of the solar zenith angle. The quantity P(μ, μ′) can be written as
e2
where Pl is the Legendre function. The ωl can be determined from the orthogonal property of Legendre polynomials . In our notations, ω0 = 1 and ω1 = 3g, with g being the asymmetry factor.
The purpose of the spherical harmonic expansion of the intensity is to separate out the angle-dependent factor by assuming
e3
substituting (2) and (3) into (1a), and using the orthogonality relation of the Legendre function in −1 ≤ μ ≤ 1, we obtain the general relation
e4
We consider a solution with a truncation at N = 3, which corresponds to a four steam case. Equation (4) yields
e5a
e5b
e5c
e5d
where al = [(2l + 1) − ωωl] and (l = 0, 1, 2, 3). Written in matrix form, (5) becomes
e6
The 4 × 4 matrix accounts for the scattering–absorption in radiative transfer.
The solution of (6) is
e7
where f0 = 1/μ0, = [C1 D1 C2 D2]T with the superscript T indicating a matrix transpose, and the constants in matrix are determined by the boundary conditions, and the for j = 1, 2, 3, and 4 are defined as follows:
e8
The coefficients Q1,2, R1,2, , k1,2, and η0,1,2,3 are defined in the appendix.
Since P2m+1(μ) for m = 0, 1, 2 … are orthogonal in both hemispheres with −1 ≤ μ ≤ 0 and 0 ≤ μ ≤ 1, a physical quantity that is limited within a hemispheric range can be decomposed by P1(μ) and P3(μ) in a four-stream case. We therefore define
e9a
e9b
e9c
e9d
where F+ and F are the upward and downward fluxes, respectively.
Substituting (7) into (9), the solution of direct radiation can be rewritten as
e10
where , , , and . The coefficient matrix is determined by the boundary conditions. We use the so-called Marshak boundary condition (Li and Ramaswamy 1996) for the single-layer solution. The Marshak boundary condition states that at the top of the layer (τ = 0), there is no downward diffuse intensity, and thus F(τ = 0) = (τ = 0) = 0; at the bottom of the layer, there is no upward diffuse intensity, and thus F+(τ = τ0) = +(τ = τ0) = 0. Hence,
e11
where , , and (i = 1, 2). Finally, the upward fluxes at the top of the layer and downward fluxes at its bottom are given by
e12
where and .
For intensity I(τ, μ), the corresponding reflection and transmission are
e13a
e13b
The reflection and transmission can be decomposed based on P1(μ) and P3(μ):
e14a
e14b
e14c
e14d
We category the result into a 2 × 1 matrix based on the modes of P1(μ) and P3(μ) as
e15

In four-stream SHM, the parameters of , , and (i = 1, 2) are dependent on eight basic simple parameters of ai and bi (i = 0, 1, 2, 3). However, the same parameters , , and in four-stream DOM are dependent 20 basic parameters of ci,j and bi (i, j = −2, −1, 1, 2) (Liou et al. 1988; Liou 2002; Zhang et al. 2013b), each of which contains a Gaussian quadrature summation with four terms. Therefore, it is expected that four-stream SHM is superior to four-stream DOM in computational efficiency.

b. Diffuse incident radiation

In a doubling–adding process, the diffuse radiation is required, in which the contribution of the direct incident solar beam, as the term of in (1a), is set to zero. The incoming radiation is imposed through an anisotropic diffused intensity at the upper boundary of a considered layer. The radiative transfer equation for diffuse radiation is
e16a
e16b
e16c
where A(μ)/π is the incoming diffused intensity. Based on the orthogonal basis of P2m+1(μ) (m = 0, 1, 2, …) for −1 ≤ μ ≤ 0, there can be an orthogonal expansion as A(μ)/π = α1P1(μ)/π + α3P3(μ)/π + … , where α1, α3, … are constant. At the truncation of N = 3, (16a) can be written as four equations, which are the same as (5) but with bi = 0 (i = 0, 1, 2, 3). Also similar to (9), there are two types of boundary conditions corresponding to the first two terms in the orthogonal expansion of A(μ): a: [corresponding to downward flux F(τ = 0) = 1 and (τ = 0) = 0] and b: [corresponding to F(τ = 0) = 0 and (τ = 0) = 1]. Without loss of generality, the boundary conditions are normalized. For the both types of a and b, the lower boundary conditions are F+(τ = τ0) = 0 and +(τ = τ0) = 0.
Similar to (10), we obtain the solution for diffuse radiation
e17
The coefficient matrix is also determined here from the boundary condition. Under the type a boundary condition,
e18
where 3 = [1 0 0 0]T. Therefore, the upward fluxes at the top and downward fluxes at the bottom of the layer are given by
e19
where the subscript a denotes the diffuse radiation under the type a boundary condition in order to distinguish it from that in (12). Similarly, under the type b boundary condition,
e20
where 4 = [0 1 0 0]T.
Similar to (13), the reflection and transmission for diffuse intensity are
e21a
e21b
where Ia,b is the intensity corresponding to the a and b types of boundary condition, and the direct beam transmission of has been included in the diffuse transmissions . By decomposing them based on orthogonal base of P1(μ) and P3(μ),
e22a
e22b
e22c
e22d
Similar to (15), the 2 × 2 matrices for diffuse reflection and transmission are created for using in the next section
e23
Matrices in (15) are called “direct” because they are related to the direct incoming solar zenith angle μ0. Matrices in (23) are called “diffuse” because they are related to diffuse incident radiation. Actually, both the direct/diffuse reflection and transmission are used to deal with the diffuse radiance field. The result of direct solar beam is represented by , which should be added to the total transmission.

3. Doubling–adding method for four-stream spherical harmonic expansion method

a. Two layers

1) Direct incident radiation

Let us consider a combination of two homogeneous layers as shown in Fig. 1, where τ1 and τ2 are the optical depths of the first and second layers. At the interface between the two layers, the dimensionless up and down intensities are
e24a
e24b
We can decompose the dimensionless intensities based on the orthogonal base of P1(μ) and P3(μ):
e25a
e25b
e25c
e25d
and categories the result into 2 × 1 matrices,
e26
Fig. 1.
Fig. 1.

Schematic diagram of (left) R1,2(μ, μ0)/T1,2(μ, μ0) and (right) .

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-12-0334.1

The four principles of invariance governing the reflection and transmission of light beam can be expressed as the following (Chandrasekhar 1950; Liou 2002):
e27a
e27b
e27c
e27d
The reflection and transmission functions are denoted by R1(μ, μ0) and T1(μ, μ0) for the first layer, R2(μ, μ0) and T2(μ, μ0) for the second layer, and R1,2(μ, μ0) and T1,2(μ, μ0) for the combined two layers. The similar definition is applied to the diffuse quantities. The superscript asterisk denotes that the radiation comes from below. For a homogeneous atmosphere, R(μ, μ0) = R*(μ, μ0) and T(μ, μ0) = T*(μ, μ0). The above equation is slightly different from Chandrasekhar (1950) and Liou (1992), since the direct beam transmissions of and are included in the diffuse transmissions of and , respectively.

The physical meaning of the above four principles of invariance has been stated clearly in Liou (1992) and Zhang et al. (2013b).

Chandrasekhar's invariance principle has been applied to 4DDA straightforwardly, since the integrals in (27) can be easily decomposed by two-node Gaussian quadrature (Zhang et al. 2013b). To our knowledge, the invariance principle has never been applied to four-stream spherical harmonic expansion method or higher-order spherical harmonic expansions. One of the main reasons is the difficulty in decomposing the integrals in (27). However, the orthogonality between P1(μ) and P3(μ) in the range of 0 ≤ μ ≤ 1 can help us to achieve this. We multiple 2P1(μ)dμ in both sides of (27) and do integration from 0 to 1. Also we multiple 2P3(μ)dμ in both sides of (27) and do integration from 0 to 1. Then by using (15), (23), and (26), we obtain
e28a
e28b
e28c
e28d
The detailed proof of (28) is shown in the appendix. From (28a) and (28b), we have
e29a
e29b
Substituting (29) into (28c) and (28d), we obtain the direct reflection and transmission
e30a
e30b
where is an 2 × 2 unit matrix . For a light beam incident from below, and can be obtained in a similar way as (30):
e31a
e31b

2) Diffuse incident radiation

At the interface of the two layers, the dimensionless upward and downward diffuse intensities under the a and b types of boundary condition are
e32a
e32b
where the direct beam transmission of has been included in . By decomposing the dimensionless intensities based on P1(μ) and P3(μ), we obtain
e33a
e33b
e33c
e33d
We also category the result into matrices as
e34
The application of the four principles of invariance to the diffuse radiation yields
e35a
e35b
e35c
e35d
Similar to (28), we multiple 2P1(μ)dμ in both sides of (35) and do integration from 0 to 1; also, we multiple 2P3(μ)dμ both sides of (35) and do integration from 0 to 1. Then by using (23) and (34), we obtain
e36a
e36b
e36c
e36d
From (36a) and (36b), we obtain
e37a
e37b
Substituting (37) into (36c) and (36d), we obtain the diffuse reflection and transmission
e38a
e38b
Similarly for the diffuse beam incident from below, we obtain
e39a
e39b
Equations (30) and (31) [(38) and (39)] constitute a closed set of iterative equations to compute the direct (diffuse) reflectance and transmittance for a combined layer.

b. Multiple layers

Equations (30b) and (39a) can be applied to an atmospheric slab extending from the layer 1 to layer k, which is obtained through a downward path calculation as
e40a
e40b
where . Equations (30a) and (38a) can be applied an atmospheric slab extending from the surface (layer N) to layer k, which is obtained through a upward path calculation as
e41a
e41b
In (41), and , where , and Rs is the surface albedo. The dimensionless upward and downward internal intensities of and , at level k + 1 (lower boundary of the layer k), are determined by the results from the downward and upward calculation paths as
e42a
e42b
Finally, the upward and downward fluxes at level k + 1 are
e43a
e43b
and the upward and downward fluxes at the top of the atmosphere (TOA) are
e44a
e44b
where uk+1, dk+1, and r1,N are the first element of matrices k+1(μ0), k+1(μ0), and 1,N(μ0), respectively.

4. Comparison results and discussion

In the following, the accuracy of 4SDA will be systematically investigated. The results will be compared to another doubling–adding scheme of 4DDA. In addition, the results of comparison to 2DDA and 2SDA are included as well. As in Zhang et al. (2013b), the δ-function adjustment of scattered energy residing in the forward peak is considered in the following calculations. Therefore, the four kind of doubling–adding schemes become δ-2DDA, δ-2SDA, δ-4DDA, and δ-4SDA.

a. Double layer

First, the accuracy of δ-2DDA, δ-2SDA, δ-4DDA, and δ-4SDA is investigated in a double-layer case. The results of the double layer could be compared to those of the single layer, which has been well analyzed by Liou et al. (1988) and Li and Ramaswamy (1996). Moreover, the results from the double layer help us to understand the multilayer results shown in the next subsection. For a double layer (labeled 1 and 2 for the first and second layers), based on (43) and (44) we have reflection , total transmission , and absorption a(τ1,2, μ0) = 1 − r(τ1,2, μ0) − t(τ1,2, μ0). We consider a case of the same layer optical depth (τ1,2 = 2τ1 = 2τ2) and over wide ranges of solar zenith angles and optical depths. The asymmetry factors are g1 = 0.837 and g2 = 0.861 for the first and second layers, respectively, following Zhang et al. (2013b). The values of single-scattering albedo are set the same (ω1 = ω2 = 0.9). The rigorous standard model used in the following is the discrete ordinates numerical model (Stamnes et al. 1988). The δ-128 stream scheme (δ-128S) are used in the discrete ordinates calculations with the Henyey–Greenstein phase function.

In Fig. 2, the relative errors in reflection and absorption are shown for the four methods. For reflection, δ-2SDA yields better results for a thick optical depth, but slightly poorer results for a thin optical depth, in comparison with δ-2DDA. The errors of both are up to 20% and higher in the region of thin optical depth and small μ0. The errors are dramatically reduced in δ-4DDA and δ-4SDA. The relative errors of both are bounded by 5% for τ1,2 > 1. Although there are significant improvements, δ-4DDA and δ-4SDA cannot completely eliminate the region where the errors exceed 15%.

Fig. 2.
Fig. 2.

Relative errors of the δ-2DDA, δ-2SDA, δ-4DDA, and δ-4SDA for (top two rows) direct reflection and (bottom two rows) absorption. The benchmark is δ-128S. The Henyey–Greenstein phase function is used, with the asymmetry factors g1 = 0.837 and g2 = 0.861. The absorbing media with ω1 = ω2 = 0.9 and the optical depths of the two layers are the same (τ1 = τ2).

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-12-0334.1

For absorption, the relative errors are generally less than 15% for both δ-2DDA and δ-2SDA, except in the region of a very large solar zenith angle. Generally δ-2SDA is slightly more accurate than δ-2DDA. Again, a dramatic improvement happens for both δ-4DDA and δ-4SDA, with errors less than 5% in most regions. Generally δ-4DDA is slightly more accurate than δ-4SDA.

To show that δ-4SDA works in general, a multilayer result is presented in Table 1. Considering a case of ω = 0.9, g = 0.837, and μ0 = 0.5, we apply δ-four-stream SHM to a media as a single layer, and also apply δ-4SDA to the same media, which is divided into 2, 5, and 10 layers. Two different optical depths (τtotal = 0.1 and τtotal = 2.0) are considered. In addition, the exact values are computed from δ-128S for the single layer. It is clearly shown in Table 1 that the results of reflection, absorption, and transmission (not shown) are identical for the media containing different layers. The slight differences between the results are due to numerical roundoff errors. This indicates that the multilayer connection is properly accounted by δ-4SDA in radiative transfer process.

Table 1.

Reflection and absorption (from δ-4SDA and δ-2/4SDA) vs number of sublayers, which are divided from the medium with ω = 0.9, g = 0.837, and μ0 = 0.5; two optical depths (τtotal = 0.1, 2.0) are considered. Exact values are obtained from δ-128S for the single layer.

Table 1.

In Li and Ramaswamy (1996), the two-stream doubling–adding method was used to connect the multilayer medium with δ-four-stream spherical harmonic expansion solution (as δ-2/4SDA). The corresponding results of δ-2/4SDA are also listed in Table 1. It is shown that the results of reflection, absorption, and transmission (not shown) vary with the media containing different layers. In general, the greater the number of sublayers, the worse is the result. Therefore, δ-2/4SDA is inappropriate to calculate the radiative transfer through a multilayer atmosphere.

b. Multilayer atmosphere

Radiative transfer in the atmosphere is a complicated process. It depends not only on the single-layer direct reflection and transmission, but also on the diffuse results and the gaseous transmission, cloud–aerosol scattering and absorption, etc. It is important to evaluate errors in radiation under a variety of atmospheric conditions by using a radiation algorithm. The Fu–Liou radiation model (Fu and Liou 1992) is used in this study. This model adopts the correlated-k distribution method for gaseous transmission, including five major greenhouse gases H2O, CO2, O3, N2O, and CH4.

In benchmark calculations, the discrete ordinates model (Stamnes et al. 1988) with a δ-128-stream scheme is incorporated with the gaseous transmission scheme of Fu–Liou radiation model by replacing the existing radiative transfer algorithm in the model. Also the δ-2DDA, δ-2SDA, δ-4DDA, and δ-4SDA schemes are incorporated with the same gaseous transmission scheme. The atmosphere was vertically divided into 280 layers, each of thickness 0.25 km. The midlatitude winter of atmospheric profile (McClatchey et al. 1972) is used with surface albedo of 0.2 for each band. The mixing ratios of CO2, N2O, and CH4 are assumed to be uniform throughout the atmosphere with concentrations of 330, 0.28, and 1.6 ppmv, respectively. For ice clouds, the optical properties are parameterized in terms of ice water content (IWC) and mean effective size (De) (Fu and Liou 1993). For water cloud, the optical properties are parameterized in terms of liquid water content (LWC) and effective radius (re) (Fu 1991). Five calculations are performed: 1) clear sky; 2) a low cloud (LWC = 0.22 g m−3, re = 5.89 μm) positioned from 1.0 to 2.0 km; 3) a middle cloud (LWC = 0.28 g m−3, re = 6.2 μm) positioned from 4.0 to 5.0 km; 4) a high cloud (IWC = 0.0048 g m−3, De = 41.1 μm) positioned from 10.0 to 12.0 km; and 5) low, middle, and high clouds together. The visible optical depths for low, middle, and high clouds are ~60, ~72, and ~0.8, respectively. Three solar zenith angles of μ0 = 1, 0.5, and 0.25 are generally considered.

In the first row of Fig. 3, the benchmark results of clear-sky heating rate are shown for three solar zenith angles. The absolute errors of δ-2DDA, δ-2SDA, δ-4DDA, and δ-4SDA against the benchmark results are shown in the second row. Generally, both δ-2DDA and δ-2SDA are relatively accurate under the clear-sky condition. However, δ-2DDA and δ-2SDA can produce relatively large error in heating rate in the region near the surface, especially for δ-2DDA. The relative error of δ-2DDA is about 2.0% near the surface. The δ-2SDA is generally better than δ-2DDA in clear-sky heating rate. It is shown in Fig. 3 that both δ-4DDA and δ-4SDA yield substantially more accurate results in heating rate.

Fig. 3.
Fig. 3.

(top) Heating rate profiles computed from δ-128S and (bottom) the error profiles from δ-2DDA, δ-2SDA and δ-4DDA, δ-4SDA with clear skies. (left to right) Three solar zenith angles μ0 = 1, 0.5, and 0.25 are considered.

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-12-0334.1

The corresponding errors in upward flux at TOA and the downward flux at the surface are shown in Tables 24 for μ0 = 1, 0.5, and 0.25, respectively. For a small solar zenith angle (e.g., μ0 = 1.0), δ-2SDA generally produces larger relative errors compared to δ-2DDA. The results become opposite for a large solar zenith angle (e.g., μ0 = 0.25). Similar to the result of heating rate, the accuracy in flux is considerably improved by δ-4DDA and δ-4SDA.

Table 2.

Comparison of upward flux at TOA and downward flux at the surface (W m−2) for the four doubling–adding schemes. The solar zenith μ0 = 1. The numbers in parentheses give differences (W m−2) between the two schemes and δ-128S.

Table 2.
Table 3.

As in Table 2, but for the solar zenith μ0 = 0.5.

Table 3.
Table 4.

As in Table 2, but for the solar zenith μ0 = 0.25.

Table 4.

For the low and middle clouds, Fig. 4 shows that the absolute errors in heating rate for δ-2DDA and δ-2SDA become much larger, compared to the result of clear sky. When μ0 = 1, the absolute errors of δ-2DDA are up to about 1.5 and 2.3 K day−1 (relative errors about 6%) for the low and middle clouds. The error of δ-2SDA is smaller than δ-2DDA. When μ0 = 0.25, the result becomes opposite as the error of δ-2SDA becomes larger than δ-2DDA. By using δ-4DDA and δ-4SDA, the relative errors are much suppressed. In general the relative error becomes less than 1% for the three solar zenith angles. This indicates both δ-4DDA and δ-4SDA are accurate enough to obtain the cloud-top solar heating.

Fig. 4.
Fig. 4.

As in Fig. 3, but for the sky containing (top two rows) low and (bottom two rows) middle clouds.

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-12-0334.1

The corresponding errors in flux are also shown in Tables 24. Here δ-2DDA overestimates the upward flux at TOA by ~11, ~6, and ~3 W m−2 for μ0 = 1, 0.5, and 0.25, respectively. Generally the error in flux for δ-2SDA is smaller than that of δ-2DDA. The only exception is in the case of μ0 = 0.25, where the errors of downward flux at the surface by δ-2SDA is larger than δ-2DDA. When δ-4DDA and δ-4SDA are implemented, the errors are substantially reduced. Generally the relative errors are less than 0.8%.

The results of high cloud are shown in Fig. 5. The δ-4DDA is better than δ-4SDA in heating rate, especially for the large solar zenith angle of μ0 = 0.25. The optical depth of the high cloud is very thin and, consequently, the solar heating rate is very low. Therefore, the choice of radiative transfer method does not appear to make a big difference. The corresponding results in flux are also shown in Tables 24; both δ-4DDA and δ-4SDA are very accurate with relative error less than 0.8%. Different from the result of heating rate, δ-4SDA is not inferior to δ-4DDA in flux.

Fig. 5.
Fig. 5.

As in Fig. 3, but for the sky containing (top two rows) high clouds and (bottom two rows) all the three cloud types.

Citation: Journal of the Atmospheric Sciences 70, 10; 10.1175/JAS-D-12-0334.1

In Fig. 5, the results of heating rate for all the low, middle, and high clouds are presented as well. The corresponding results in flux are shown in Table 24. Again, both δ-4SDA and δ-4DDA are superior to δ-2DDA and δ-2SDA in accuracy.

In Fig. 2, it is shown that the relative errors of δ-2DDA and δ-2SDA are considerably large in the region of small optical depth. Generally the aerosol optical depth in the atmosphere is very small. The global mean dust optical depth is less than 0.1 (Peng et al. 2012). To investigate the error in aerosol radiative forcing, we consider a dust aerosol profile with a column optical thickness of τa = 0.1, which is uniformly distributed from the surface to 2 km. The asymmetry factors and single-scattering albedo are set as ga = 0.679 and ωa = 0.988, which corresponds to optical property of dust aerosol with effective radii of 0.5 μm at a wavelength of 0.55 μm (Fu et al. 2009).

The results in flux are listed in Table 5. As expected, both δ-2DDA and δ-2SDA produce relatively large errors in the upward flux at TOA and downward flux at the surface. For example, the error of upward flux at TOA by δ-2SDA is up to 5.4 W m−2 at μ0 = 1. The corresponding error in the clear sky without aerosol is 3.9 W m−2 (see Table 2). Thus, the aerosol radiative forcing is overestimated by 1.5 W m−2. When μ0 = 0.25, the errors of upward flux at TOA are −0.59 W m−2 and −3.58 W m−2, respectively, for the clear sky without and with aerosol (Table 4). Thus the aerosol radiative forcing is underestimated by 2.99 W m−2. Since δ-2SDA is widely used in current climate models, the model estimated aerosol forcing is not reliable. The results of δ-2DDA are slightly better than δ-2SDA for a small solar zenith angle but become worse for a large solar zenith angle. It is shown that the errors in flux are dramatically reduced by applying δ-4DDA or δ-4SDA.

Table 5.

Comparison of upward flux at TOA and downward flux at the surface (W m−2) for the four doubling–adding schemes. The aerosol optical depth τa = 0.1. The value in the μ0 column is the cosine of the solar zenith angle. The values in parentheses are the differences between the four schemes and δ-128S.

Table 5.

The computational efficiency is a key issue for climate modeling. Table 6 lists the comparison results of computing time for δ-2SDA and δ-4SDA. The computational efficiency of δ-2SDA is not shown since it is close to that of δ-2SDA. In Fu–Liou model, an inverse matrix formulation is used to solve the connection of all the layers in δ-four-stream DOM radiative transfer (denoted as δ-4SMODEL). As proposed by Zhang et al. (2013b), δ-4DDA has the very similar efficiency as δ-4SMODEL. For the pure radiative transfer calculations without considering the gaseous transmission, it is found that the computational time of δ-4SDA is about 70% of δ-4SMODEL. As discussed above, the single-layer solution of δ-four-stream SHM is much simpler than that of δ-four-stream DOM. For the whole model process including the calculations of gaseous absorption and cloud optical properties, the computational time of δ-4SDA is about 80% of δ-4SMODEL. In view of the overall high accuracy and computational efficiency, it appears that δ-4SDA is well suited for radiation calculation in weather and climate models.

Table 6.

The timing for the radiative transfer calculation (normalized to the timing ratio of Fu–Liou model) by the algorithm only and by the radiation model.

Table 6.

5. Summary and conclusions

Even though several single-layer solutions of δ-four-stream approximation have been found in the last several decades, less attention has been paid to the derivation of an analytical result for a corresponding doubling–adding calculation, which would enable us to calculate the radiative transfer through a vertically inhomogeneous atmosphere with multilayers. The recent work of δ-4DDA (Zhang et al. 2013b) was the first effort in this aspect. Among the various δ-two-stream approximations, the δ-Eddington approximation and δ-two-stream DOM are widely used in radiation algorithms. The higher-order extensions of the two methods are the δ-four-stream SHM and δ-four-stream DOM. Therefore, it is natural to seek a doubling–adding scheme for δ-four-stream SHM. In this work, first the simplified single-layer solution of four-stream SHM is shown; second, the invariance principle is applied to four-stream SHM and an analytical method of δ-4SDA is obtained.

In a double-layer case, the accuracy of δ-4SDA is systematically compared with a wide range of solar zenith angles and optical depths. Similar to the comparison for a single layer, δ-4SDA provides much more accurate result compared to δ-2DDA and δ-2SDA. The idealized medium test shows that the δ-4SDA scheme can exactly solve the multilayer connection in a radiative transfer process.

The δ-4SDA, along with δ-2DDA, δ-2SDA, and δ-4DDA, has been applied to a realistic atmospheric profile with inclusion of gaseous transmission. It is found the accuracy of δ-4SDA is comparable to δ-4DDA; both of them are superior to δ-2DDA and δ-2SDA in most cases, especially under the cloudy-sky condition. For δ-2DDA and δ-2SDA, the relative error in heating rate at cloud-top layer can be as large as 6%, and the errors in flux are up to 12 W m−2 and 9 W m−2, respectively. These errors are substantially reduced by using δ-4SDA or δ-4DDA. The errors of both are less than 1.5 W m−2.

It is found that the aerosol radiative forcing can be significantly misestimated, owing to the large errors from δ-2DDA and δ-2SDA under the thin optical depth condition. For a typical case of dust, the error of aerosol radiative forcing at TOA can be over 3 W m−2. The only way to eliminate the large error in aerosol radiative forcing is to use δ-4DDA or δ-4SDA. The errors of both are less than 0.72 W m−2.

Both δ-4DDA and δ-4SDA are analytical method so that the computing time involved is minimal. It is shown that δ-4SDA is more efficient than δ-4DDA. Therefore, δ-4SDA could be more easily implemented in climate models, where both the accuracy and efficiency are required.

Acknowledgments

The authors thank the anonymous reviewers for their constructive comments. The work is supported by the State Key Development Program for Basic Research of China (Grants 2009CB421506 and 2011CB403405), the key program of CAMS (Grant 2012Z001), the National Natural Science Foundation of China (Grants 41221064, 41075056, 41205116, and 41075015), and the Program of China Meteorological Administration (Grant GYHY200906002).

APPENDIX

The Coefficients of (2) and the Proof of (28)

This appendix contains two parts: 1) the coefficients of (7) and 2) the proof of (28).

1)
eq1
where and ;
eq2
where , , , , and , with .
2) Proving (28a) is equivalent to finding out the relation between the direct and diffuse
ea1
the subscripts in R(μ, μ′) and are neglected.
Because of the orthogonal relation of P1(μ) and P3(μ) in the hemisphere, The following subparts of (25)
eA2a
eA2b
indicate a hemispherical harmonic expansion of D(μ′, μ0)μ′ in the range of 0 ≤ μ ≤ 1,
ea3
where the terms beyond truncation are neglected.
From (13a), R(μ, μ0) ~ I(μ)/μ0, where I(μ) is the reflected direct intensity. In (A1), μ0 is replaced by the local μ′ for all direction of 0 ≤ μ′ ≤ 1. From (21), Ra,b ~ Ia,b, where Ia,b is the reflected diffused intensity. It is shown in (26a) of Zhang et al. (2013a) that the angular averaged R(μ, μ′)μ′ (as integral to μ′) corresponds to the diffused reflection. In four-stream SHM, R(μ, μ′)μ′ should be further decomposed to P1(μ′) and P3(μ′) according to the a and b types of boundary condition as shown in (19)(21). Therefore, following (22),
eA4a
eA4b
eA4c
eA4d
Because of the orthogonal relation of P1(μ) and P3(μ), by (A4a) and (A4c) we have
ea5
and by (A4b) and (A4d) we have
ea6
From (A5) and (A6), based on the orthogonal relation of P1(μ) and P3(μ), we obtain
ea7
In (A5)(A7), the terms beyond truncation are neglected. Finally, it is proven that
ea8
ea9
Equations (28b) and (28d) can also be proven in a similar way as was (28a).

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