1. Introduction

Computational efficiency has always been an issue in determining solar heating in an absorbing/scattering atmosphere in an atmospheric model, and this is dictated in part by the number of terms needed to compute the gaseous absorption. Within a spectral range that spans an entire water vapor or CO₂ absorption band in the solar spectral region, the magnitude of the absorption coefficient might vary by more than 10 orders of magnitude, and highly simplified radiative transfer schemes must be used to reduce the computational burden. One of the most widely used approaches to expediting the computation is the application of the k-distribution approximation (Ambartzumian 1936; Arking and Grossman 1972). This approximation assumes that the spectral points with a similar absorption coefficient in an atmospheric layer can be treated as a single spectral point in all other atmospheric layers. Thus, all points in a spectral band are sorted into a limited number of groups, and each group is treated as monochromatic. As a result, the computation burden is greatly reduced.

Uncertainties in radiative flux calculations due to this approximation arise from the fact that the absorption coefficient is a function of pressure and temperature, and spectral points with a similar absorption coefficient in an atmospheric layer might become dispersed in a distant layer where the pressure is greatly different. This problem can cause calculations of absorption in distant layers to be inaccurate because the absorption coefficients in a spectral group are no longer tightly packed together. A widely used approach to cope with this issue is the application of the correlated k-distribution approximation, which assumes that the distributions of the absorption coefficient at different pressures and temperatures are highly correlated (e.g., Goody et al. 1989; Lacis and Oinas 1991; Fu and Liou 1992; Mlawer et al. 1997; Nakajima et al. 2000). Accumulated k-distribution functions at a set of pressures and temperatures are computed for a spectral band. Spectral points in a given interval of the accumulated k-distribution function are assumed to be identical regardless of pressure and temperature. This approximation can be justified if the atmospheric layers and, hence, pressures are not far apart.

In the thermal infrared [longwave (LW)], the correlated k-distribution approximation has been demonstrated to be efficient in flux and cooling rate calculations for the Earth atmosphere (e.g., Fu and Liou 1992; Mlawer et al. 1997). The major reason that the correlated k-distribution approximation works well in the LW is that the cooling is primarily contributed from nearby layers, and the distant layers are of secondary importance (Wu 1980; Chou and Kouvaris 1986).

Specifically, as concerns the application of the correlated k-distribution approximation to the LW and shortwave (SW), the difference is in the source of radiation; the former comes from the emission of the atmosphere and the surface, whereas the latter is the solar radiation incident at the top of the atmosphere (TOA). The absorption of the solar radiation incident at TOA is quickly saturated in the stratosphere at those spectral points near the centers of strong absorption lines. The implication is that distant layers high in the stratosphere have a significant impact on the heating of the troposphere. However, the correlation of k distributions between layers in the stratosphere and troposphere is weak. (This view is to be elaborated later).

The k distribution without the correlation assumption has also been applied to the solar near-infrared (Chou and Lee 1996; Chou and Suarez 1999; Fomin and Correa 2005). Chou and Lee (1996) assumed a single set of the k-distribution function throughout an inhomogeneous atmosphere, which were computed at a pair of reference pressure and temperature corresponding to the height where heating is the most prominent. The absorption coefficient at other pressures and temperatures were extrapolated from the reference pressure and temperature using the “wing scaling,” assuming the dominance of the wings of absorption lines on fluxes and heating rates. This wing scaling of the k-distribution method is computationally very fast and can compute accurately the heating rate in the troposphere and lower stratosphere. However, the heating rate error increases in the upper stratosphere. Fomin and Correa (2005) developed an innovative k-distribution scheme for defining the spectral groups and the equivalent absorption coefficient for each group. The vertical profile of the equivalent absorption coefficient was derived by requiring the downward flux in a spectral group to be identical with the line-by-line (LBL) calculations at all atmospheric levels. This scheme is very fast with reasonably high accuracy in solar flux and heating rate calculations.

In Part I of a new k-distribution scheme, Chou et al. (2020) demonstrated that the new scheme without the correlated k-distribution assumption can efficiently compute LW fluxes and cooling rates. In this study, we extend this new scheme to the solar radiative transfer calculations. Unlike the pressure scaling of the absorption coefficient, as in the scheme of Chou and Lee (1996), this new k-distribution scheme can compute accurately solar heating not only in the troposphere but also in the stratosphere.

Section 2 provides information on the LBL method, the division of the SW spectrum, and the sample atmospheres for flux and heating rate calculations. Section 3 formulates the basic equations and defines some relevant parameters of the new k-distribution scheme. Section 4 details the method of the construction of the k-distribution function and the derivation of the equivalent absorption coefficient. Section 5 validates the k-distribution scheme by comparing flux and heating rate calculations with the LBL calculations. Concluding remarks are given in section 6.

2. Line-by-line calculations, spectral bands, and sample atmospheres

The k-distribution parameterization is based on detailed LBL calculations of fluxes and heating rates. Our LBL calculations of the absorption coefficient (Chou and Kouvaris 1986) use the Air Force Geophysical Laboratory 2012 edition of the molecular absorption parameters [High-Resolution Transmission 2012 (HITRAN2012); Rothman et al. 2013]. The molecular line shape is assumed to follow the Voigt function. Following Clough et al. (1992) and Mlawer et al. (1997), the absorption coefficient at wavenumbers > 25 cm⁻¹ from the line center is set to zero, which is equivalent to a line-cutoff of 25 cm⁻¹. The absorption coefficient is computed at spectral intervals of 0.001 cm⁻¹ throughout the atmosphere. In the middle and upper stratosphere, the Doppler broadening of the width of an absorption line is independent of pressure and increases linearly with wavenumber. Within the temperature range of Earth’s atmosphere, the Doppler line half-width is >0.001 cm⁻¹ in the solar near-infrared for both water vapor and CO₂, which is greater than the pressure-broadened line half-width. Therefore, the spectral interval of 0.001 cm⁻¹ is smaller than the Voigt line half-width. Further, we used the spectral resolution of 0.002 cm⁻¹ to test the flux and heating rate differences between the spectral resolutions of 0.001 and 0.002 cm⁻¹ in LBL calculations and found the differences are negligible. Thus, the use of δν = 0.001 cm⁻¹ is adequate for calculating flux and heating rate due to water vapor and CO₂ in the solar infrared.

Similar to Chou and Suarez (1999), the solar spectrum between 0.7 μm (14 290 cm⁻¹) and 10 μm (1000 cm⁻¹) is divided into 3 bands. Table 1 shows the spectral ranges of the three bands. The spectral distribution of the extraterrestrial solar radiation is taken from the 2000 version of the American Society for Testing and Materials (ASTM) Standard Extraterrestrial Spectrum Reference E-490-00, which has a solar constant of 1366.1 W m⁻² (https://www.nrel.gov/grid/solar-resource/spectra-astm-e490.html). More recently recommended solar spectral data are available, but it is not expected to impact the conclusion of this study. There are a number of water vapor and CO₂ molecular absorption bands across the solar near-infrared. The overall strength of both water vapor and CO₂ bands increases with decreasing wavenumber. Among the three bands, absorption, or atmospheric heating, is the strongest in band 3. The heating due to CO₂ in band 1 is negligible. Therefore, the k-distribution parameterization of the CO₂ heating only applies to bands 2 and 3.

Table 1.

Spectral bands, the number of g groups M, and the reference pressure p_r. The number M is the result of “best fits” that meets the subjectively imposed accuracy requirements in flux and heating rate parameterizations.

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For clear-sky solar heating, this study does not include contributions from O₂, O₃, and continuum absorption, as well as the H₂O absorption in the spectral region with the wavenumber > 14 290 cm⁻¹. Oxygen is a minor contributor to solar heating. The k-distribution scheme of this study can be similarly extended to cover the O₂ bands. However, the O₂ absorption is confined to a few narrow bands, and the spectral bands given in Table 1 have to be repartitioned when O₂ absorption is included and the k-distribution approximation applied. The O₃ and continuum absorption coefficients are smooth functions of the wavenumber, and each spectral interval can be treated as gray, i.e., the absorption coefficient is not a function of wavenumber. Applications of the k-distribution approximation to the O₃ and continuum absorption are unnecessary. The absorption in the visible region (25 000–14 290 cm⁻¹) by H₂O is weak and primarily near the surface where the pressure range is narrow. It can be treated as gray with the absorption coefficient a simple function of pressure.

Fluxes and heating rate computed using the k-distribution approximation for three atmospheres typical of tropical (TRP), midlatitude summer (MLS), and subarctic winter (SAW) conditions are compared with the LBL calculations. Each atmosphere is divided into 75 layers. The thickness of a layer is constant in pressure, Δp = 24 hPa, in the troposphere, and constant in Δlog₁₀p = 0.15, above the tropopause. The temperature and humidity of the three atmospheres are taken from McClatchey et al. (1972), except the specific humidity in the stratosphere is set to 3.25 × 10⁻⁶ for the tropical atmosphere, and 4 × 10⁻⁶ for the midlatitude summer and subarctic winter atmospheres. The column integrated water vapor amounts are 39.4, 27.9, and 4.3 kg m⁻², and the surface temperatures are 300, 294, and 257 K for the TRP, MLS, and SAW atmospheres, respectively. The normal, half, and doubled atmospheric CO₂ concentrations are set to 400, 200, and 800 ppmv; the 400 ppmv CO₂ concentration is the observation in 2014 at Mauna Loa Observatory, Hawaii.

The three typical atmospheres were used for developing the new k-distribution parameterization. For independent validation of the new parameterization, eight atmospheres in four diverse climatic regimes and two seasons (June and December) were sampled from the ERA5 reanalysis (Hersbach et al. 2020). Table 2 demonstrates the four sites where the sample atmospheres are located and the time of the samples. Those samples include the atmospheres in the Atmospheric Radiation Measurement (ARM) Southern Great Plains in June (SGP-J) and December (SGP-D), ARM North Slope of Alaska in June (NSA-J) and December (NSA-D), warm pool western tropical Pacific in June (WTP-J) and December (WTP-D), and cold pool eastern tropical Pacific in June (ETP-J) and December (ETP-D). These samples range from a highly cold and dry atmosphere to a warm and humid atmosphere with the skin temperature differs by 54 K and the column-integrated water vapor amount by a factor of 25.

Table 2.

Locations, time, skin temperature, and column-integrated water vapor amount of the eight sample atmospheres used for independent validation of the k-distribution approximation. The letters J and D indicate the months of June and December, respectively.

View Table

3. The k-distribution approximation

It is noted that all of the above equations apply to the k-distribution approximation both with and without the assumption of the correlation of the k-distribution function between atmospheric layers. The difference between them is the definition of the spectral points contained in a given g group, i.e., R_g in Eqs. (3.4) and (3.6). In this study, the spectral points sorted into a given g group, R_g, stays in the same group, i.e., independent of pressure and temperature. Therefore, the equivalent absorption coefficient k_g is derived from the same spectral points throughout the atmosphere. In the correlated k-distribution approximation, on the other hand, a spectral point could be sorted into different g groups at different pressures, and the spectral points contained in a given g group are not identical at different atmospheric layers. It follows that k_g along an atmospheric path are derived from different spectral points in a correlated k-distribution scheme. It is also noted that the weight H_g is computed in this study using Eqs. (3.6) and (3.10), whereas it is specified in the correlated k-distribution approximation (e.g., Fu and Liou 1992; Mlawer et al. 1997).

4. Construction of the k-distribution scheme

b. Grouping of spectral points and derivation of the k-distribution function

For each spectral band, grouping of spectral points is based on the absorption coefficient at the reference pressure and temperature. With the reference pressures selected and the reference temperature fixed at 250 K for a spectral band Δν, the g groups are defined by first specifying the number of the groups M. Those M groups are divided into subgroups corresponding to each of the reference pressures. Following Chou et al. (2020), steps for grouping spectral points are given below:

• Starting from the smallest reference pressure p_r1, high spectral resolution absorption coefficients k_ν(p_r1, θ_r) are computed using the LBL method.

• Initially, use a large number of M, say 20, and divide the range of log₁₀[k_ν(p_r1, θ_r)] into M intervals with a constant width

$\mathrm{\Delta }{x}_{\nu }\left(\hspace{0.17em}{p}_{r1},{\theta }_r\right)={\displaystyle \frac{x^{\max }\left(\hspace{0.17em}{p}_{r1},{\theta }_r\right)-x^{\min }(\hspace{0.17em}{p}_{r1},{\theta }_r)}{M}},$(4.1)

where x_ν(p_r1, θ_r) = log₁₀[k_ν(p_r1, θ_r)], and x^max and x^min are the maximum and minimum of x_ν in the band Δν. By dividing the range of the absorption coefficient into intervals with a constant size of Δlog₁₀[k_ν(p_r1, θ_r)], the g groups have more comparable ranges of transmittance values than if a constant size of Δk_ν(p_r, θ_r) is used.

• A g group is defined by the upper and lower boundaries of the absorption coefficient $x^{U_j}\left(\hspace{0.17em}{p}_{r1},{\theta }_r\right)$ and $x^{L_j}\left(\hspace{0.17em}{p}_{r1},{\theta }_r\right)$, j = 1, 2, …, M. It is noted that the heating in the stratosphere associated with the first reference pressure is attributable to the spectral points with the largest absorption coefficient. Therefore, the grouping of spectral points is in descending order of k_ν(p_r1, θ_r),

$x^{L_j}\left(\hspace{0.17em}{p}_{r1},{\theta }_r\right)=x^{U_j}(\hspace{0.17em}{p}_{r1},{\theta }_r)-\mathrm{\Delta }{x}_{\nu }(\hspace{0.17em}{p}_{r1},{\theta }_r)$

and

$x^{U_1}\left(\hspace{0.17em}{p}_{r1},{\theta }_r\right)=x^{\max }\left(\hspace{0.17em}{p}_{r1},{\theta }_r\right),$

$x^{L_M}\left(\hspace{0.17em}{p}_{r1},{\theta }_r\right)=x^{\min }\left(\hspace{0.17em}{p}_{r1},{\theta }_r\right).$

• Based on k_ν(p_r1, θ_r), each spectral point is identified with a g group corresponding to the first reference pressure g₁(ν), where g₁ is one of the M groups ranging from 1 to M.

• For each one of the M groups, compute the heating rate profiles of all the spectral points ν contained in the group using the LBL method.

• Compute the total heating rate profiles of each of the M groups by summing over all the heating rate profiles of the spectral points contained in the group.

• Identify judiciously the first m₁ groups that have a substantial contribution to the heating at p_r1. Thus, grouping of the first m₁ groups is complete with g₁(ν) identified, where g₁ = 1, 2, 3, …, m₁.

• In defining the next m₂ groups associated with the second reference pressure, p_r2, the spectral points already identified with the groups of the first reference pressure, i.e., $x_{\nu }\left(\hspace{0.17em}{p}_{r1},{\theta }_r\right)>x^{L_{{}_{m_1}}}\left(\hspace{0.17em}{p}_{r1},{\theta }_r\right)$, are excluded and there are only (M − m₁) groups left to be defined.

• Compute the absorption coefficient of the remaining spectral points at the second reference pressure k_ν(p_r2, θ_r) using the LBL method. Equation (4.1) becomes

$\mathrm{\Delta }{x}_{\nu }\left(\hspace{0.17em}{p}_{r2},{\theta }_r\right)={\displaystyle \frac{x^{\max }\left(\hspace{0.17em}{p}_{r2},{\theta }_r\right)-x^{\min }\left(\hspace{0.17em}{p}_{r2},{\theta }_r\right)}{M-m_1}}$(4.2)

for the second reference pressure. Here x^max and x^min are the maximum and minimum of x_ν in the band Δν with those spectral points identified with the first m₁ groups excluded. Therefore, each of the (M − m₁) g groups associated with the second reference pressure p_r2 are defined by the upper and lower boundaries of the absorption coefficient

$x^{L_j}\left(\hspace{0.17em}{p}_{r2},{\theta }_r\right)=x^{U_j}(\hspace{0.17em}{p}_{r2},{\theta }_r)-\mathrm{\Delta }{x}_{\nu }\left(\hspace{0.17em}{p}_{r2},{\theta }_r\right),$

where j = 1, 2, …, M − m₁. It is noted that

$x^{U_1}\left(\hspace{0.17em}{p}_{r2},{\theta }_r\right)=x^{\max }\left(\hspace{0.17em}{p}_{r2},{\theta }_r\right),$

$x^{L_{M-m1}}\left(\hspace{0.17em}{p}_{r2},{\theta }_r\right)=x^{\min }\left(\hspace{0.17em}{p}_{r2},{\theta }_r\right).$

It is also noted that $x^{U_1}\left(\hspace{0.17em}{p}_{r2},{\theta }_r\right)\ne x^{L_{m_1}}\left(\hspace{0.17em}{p}_{r1},{\theta }_r\right)$ as the reference pressures are different.

• Identify each spectral point with a g group corresponding to the second reference pressure g₂(ν). Here g₂ is one of the (M − m₁) groups ranging from 1 to (M − m₁).

• For each g₂ group, compute the heating rate profiles of all the spectral ν contained in the group g₂(ν), using the LBL method.

• The total heating rate profile of the group is then derived by summing over all the heating rate profiles of those spectral points in the group; there are a number of (M − m₁) total heating rate profiles.

• Identify judiciously the first m₂ groups that have a substantial contribution to the heating at p_r2.

• The same procedures are repeated for defining the last m₃ g groups if three reference pressures are used. For this case, m₃ = M − m₁ − m₂, and

$\mathrm{\Delta }{x}_{\nu }\left(\hspace{0.17em}{p}_{r3},{\theta }_r\right)={\displaystyle \frac{x^{\max }\left(\hspace{0.17em}{p}_{r3},{\theta }_r\right)-x^{\min }\left(\hspace{0.17em}{p}_{r3},{\theta }_r\right)}{M-m_1-m_2}}.$(4.3)

• Compose the grouping of spectral points for the spectral band Δν by combining the first m₁ group of the g₁(ν), m₂ group of the g₂(ν), and m₃ group of the g₃(ν) for the case of three reference pressures. The composed g(ν) is R_g in Eq. (3.4).

• The large number of M is reduced and the above steps repeated until errors of fluxes and the heating rate calculated using the new k-distribution scheme are greater than the prespecified accuracies.

It is found that two reference pressures are adequate for accurate computations of fluxes and heating rate in the H₂O and CO₂ bands. It is noted that the spectral heating rate profiles were computed using the MLS atmosphere only for identifying the reference pressures and the number of g groups associated with the reference pressures on which spectral points are grouped, i.e., g(ν). Other typical atmospheres can also be used for the same purposes. The peak of the heating rate profile shifts up and down with the solar zenith angle only slightly. It does not have much impact on the judiciously and empirically selected reference pressures. Figure 1 is the flowchart demonstrating the grouping of spectral points. It is for the case of three reference pressures, but can also be applied to the case of two reference pressures of this study.

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Grouping of spectral points in each of the bands shown in Table 1. The diagram is for the case of three reference pressures: p_r1, p_r2, and p_r3.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0278.1

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Finally, the flux-weighted k-distribution function H_g is computed from Eqs. (3.6) and (3.10), where the set of spectral points R_g is derived from g(ν) by composing g₁(ν) and g₂(ν) for the case of two reference pressures.

5. Validations with line-by-line calculations

By setting the surface reflectivity to 0, i.e., r_s = 0, the downward flux and heating rate of some sample atmospheres are computed using the k-distribution scheme of Eqs. (3.8) and (3.18), and results are compared with LBL calculations. The equivalent absorption coefficient k_g is interpolated from tables as a function of pressure and temperature precomputed using Eq. (4.6), and the flux-weighted k-distribution function H_g given by Eq. (3.10) is precomputed and specified for each band, gas, and g group. The sample atmospheres include the three typical atmospheres of MLS, TRP, and SAW, two artificial atmospheres that mix the temperature and humidity of the TRP and SAW atmospheres, as well as eight atmospheres sampled from four diverse climatic regimes and two seasons.

Figure 2 demonstrates the total heating rate of water vapor in bands 1, 2, and 3 (1000–14 290 cm⁻¹), and the difference between the KD and LBL calculations for three cases: the MLS atmosphere with a solar zenith angle φ₀ = 60° (Figs. 2a,b); the SAW atmosphere with φ₀ = 0° (Figs. 2c,d); the TRP atmosphere with φ₀ = 75° (Figs. 2e,f). The choice of φ₀ for the SAW and TRP cases is to enhance the contrast in the water vapor path; the path-integrated water vapor amount of TRP is 35 times larger than that of SAW. There is a heating peak in the lower troposphere which is due to a large amount of water vapor. Among these three cases, the SAW has the largest heating in the lower troposphere because of the largest solar radiation incident at TOA. The maximum difference in the heating rate between KD and LBL is <0.05 K day⁻¹ in the troposphere and <0.02 K day⁻¹ in the stratosphere.

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The total-band water vapor heating rate and heating rate difference between the KD and LBL calculations for (top) a solar zenith angle of 60° in the midlatitude summer atmosphere, (middle) a solar zenith angle of 0° in the subarctic winter atmosphere, and (bottom) a solar zenith angle of 75° in the tropical atmosphere. Here, the total band refers to the sum over bands 1, 2, and 3.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0278.1

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In addition to the three typical atmospheres, two artificial atmospheres that mix the TRP atmosphere with the SAW atmosphere are used for independent validation of the k-distribution scheme. One has the TRP temperature and SAW water vapor, TRP-T: SAW-W, and the other has the SAW temperature and TRP water vapor, SAW-T: TRP-W. These two artificial atmospheres differ by a factor of ~10 in the column-integrated water vapor (0.34 vs 4.0 g cm⁻²) and by 43K in the surface temperature (257 vs 300 K). Figures 3a and 3b demonstrate the heating rate and the heating rate difference between the KD and LBL for the case TRP-T: SAW-W with φ₀ = 0°, respectively, whereas Figs. 3c and 3d are the case SAW-T: TRP-W with φ₀ = 75°. The choice of the two solar zenith angles, φ₀ = 0° and 75°, is to enhance the difference in the path-integrated water vapor amount between these two artificial atmospheres by a factor of 40. The heating rate difference between the KD and LBL is <0.08 K day⁻¹ for both cases. Table 3 demonstrates the downward surface radiation F_s and the difference in the downward surface radiation between KD and LBL, F_s(KD) − F_s(LBL), for the three typical and two extreme atmospheres, as well as two solar zenith angles, 0° and 75°. The difference in the downward surface radiation is <1.2 W m⁻².

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As in Fig. 2, but for (top) a solar zenith angle of 0° in the TRP-T: SAW-W atmosphere and (bottom) a solar zenith angle of 75° in the SAW-T: TRP-W atmosphere.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0278.1

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Table 3.

The total-band downward surface flux F_s and the difference between the k-distribution (KD) and LBL values, F_s(KD) − F_s(LBL), due to the absorption by water vapor in the three typical and two extreme atmospheres (as described in the text). φ₀ is the solar zenith angle, and F₀ is the incident solar radiation at the top of the atmosphere. The total band refers to the sum over bands 1, 2, and 3. Units are W m⁻².

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Figure 4 demonstrates the total heating rate in bands 2 and 3 (1000–8200 cm⁻¹), of a normal CO₂ concentration of 400 ppmv (1 × CO₂) and the difference between the KD and LBL for the same three cases of the typical atmospheres as Fig. 2: the MLS atmosphere with φ₀ = 60°, the SAW atmosphere with φ₀= 0°, and the TRP atmosphere with φ₀ = 75°. As revealed in the figure, the CO₂ solar heating in the stratosphere is very large, greater than 7 K day⁻¹ at 0.01 hPa for φ₀ = 0°, but insignificant in the troposphere. Band 3 contributes to most of the heating in the stratosphere. The heating rate difference between the KD and LBL is generally <0.2 K day⁻¹ in the stratosphere, except in the upper stratosphere where the maximum is 0.45 K day⁻¹. In the troposphere, the difference is <0.03 K day⁻¹.

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As in Fig. 2, but for the total-band CO₂ heating rate and heating rate difference. The atmospheric CO₂ concentration is set to 400 ppmv. Here, the total band refers to the sum over bands 2 and 3.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0278.1

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Figure 5 demonstrates the total-band heating rate and heating rate difference due to the overlapping absorption of water vapor and the normal CO₂ concentration of 400 ppmv for the same three cases as in Fig. 4. Since the k-distribution parameterization for CO₂ does not apply to band 1, the results demonstrated in the figure cover only bands 2 and 3. For the overlapping absorption, the transmission function is computed from Eq. (3.18). The maximum heating occurs in the upper stratosphere in all three cases. It reaches 8 K day⁻¹ for the case of φ₀ = 0°. By comparison with Fig. 4, the large heating in the stratosphere is primarily due to CO₂. There is also a heating peak in the middle troposphere (300–600 hPa) ranging from 0.6 to 1.6 K day⁻¹. The tropospheric heating is mostly due to water vapor. The heating rate difference between the KD and LBL is <0.05 K day⁻¹ in the troposphere and <0.1 K day⁻¹ in the stratosphere, except in the region above the 0.1-hPa level.

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As in Fig. 4, but for the overlapping absorption due to H₂O and a CO₂ concentration of 400 ppmv.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0278.1

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In addition to the normal CO₂ concentration of 400 ppmv, the new k-distribution scheme is also validated with a reduced CO₂ concentration of 200 ppmv (0.5 × CO₂) and a doubled CO₂ concentration of 800 ppmv (2 × CO₂). Figure 6 is the same as Fig. 5 for the total-band (bands 2 and 3) heating rate and heating rate difference due to the overlapping absorption of water vapor and CO₂, except for a doubled CO₂ concentration of 800 ppmv. Compared with Fig. 5, the heating increases in the stratosphere due to a doubled CO₂ but remains nearly unchanged in the troposphere because of the weak CO₂ heating. The difference in the heating between the KD and LBL also increases in the stratosphere due to a doubled CO₂ but remains nearly unchanged in the troposphere. For the case of the reduced CO₂ concentration of 200 ppmv, the results are smaller than those shown in Figs. 5 and 6 (not demonstrated in the figures).

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As in Fig. 5, but for the overlapping absorption due to H₂O and a doubled CO₂ concentration of 800 ppmv.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0278.1

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Table 4 demonstrates the downward surface flux F_s and the difference between the KD and LBL values ΔF_s due to the absorption by CO₂ in bands 2 and 3 for the solar zenith angles of 0° and 75°. The difference in the downward surface radiation is only ~0.5 W m⁻², except for the cases of a doubled CO₂ and a solar zenith angle of 75° which have a difference of ~1.0 W m⁻². Table 5 is the same as Table 4, except for the overlapping absorption of water vapor and CO₂. The atmospheres with a doubled CO₂ and φ₀ = 0° have the largest difference of 2.4 W m⁻² in the downward surface radiation. This difference is substantially larger than that given in Tables 3 and 4 for the absorption due to, respectively, water vapor and CO₂. It is caused by applying the multiplication rule of Eq. (3.18) for the transmission function of overlapping absorption in wide spectral bands. The difference can be reduced by dividing the solar infrared into more than the three bands given in Table 1.

Table 4.

The total-band downward surface flux F_s and the difference between the KD and LBL values ΔF_s [=F_s(KD) − F_s(LBL)], due to the absorption by CO₂. The cases (0.5 × CO₂), (1 × CO₂), and (2 × CO₂) refer to the CO₂ concentrations of 200, 400, and 800 ppmv, respectively. φ₀ is the solar zenith angle. The total band refers to the sum over bands 2 and 3. Units are W m⁻². The downward solar flux at the top of the atmosphere is 286.45 W m⁻² for φ₀ = 0°.

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Table 5.

As in Table 4, but for the overlapping absorption of water vapor and CO₂.

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The error in the impact of a doubling of CO₂ induced from using the new k-distribution scheme can be readily derived from the data in Table 5. It is the difference in ΔF_s between 2 × CO₂ and 1 × CO₂. For the case of a solar zenith angle of 75°, the percentage error is large, ~50%, but the absolute error is small, <0.3 W m⁻². For the case of a solar zenith angle of 0°, both percentage and absolute errors are insignificant.

The important parameter that affects the climate due to changing CO₂ is the TOA flux but not the surface flux. The TOA flux is important because it is the heating of the Earth-atmosphere system. Surface heating has an impact on the stability of the lower atmosphere near the surface. The extra heat, either positive or negative, is redistributed to the atmosphere through convection. Therefore, the absolute change of the surface heating is important, whereas the relative change is of secondary importance. The error of 0.3 W m⁻² in the impact of a doubled CO₂ on the surface flux can be considered small.

For independent validation of the new k-distribution approximation scheme, we sampled eight atmospheres in four diverse climatic regimes and two seasons (June and December). The locations and the time of the eight sample atmospheres are demonstrated in Table 2. Clear-sky temperatures and humidities of these eight atmospheres were sampled from the ERA5 global reanalysis (Hersbach, et al. 2020). The ERA5 temperature and humidity available to us are defined at 37 pressure levels ranging from 1 to 1000 hPa. In flux and heating rate calculations, we artificially added a layer above 1 hPa and set the temperature and gas concentration to the values at 1 hPa.

Using the KD and LBL schemes, we computed the heating rate profiles and the difference between the KD and LBL values for each of the eight sample atmospheres and three solar zenith angles, φ₀ = 0°, 60°, and 75°. For a given absorbing gas, we selected from the eight sample atmospheres and the three solar zenith angles the one having the least satisfactory KD-calculated heating rate profile when compared with the LBL calculations. For water vapor, this case is identified as WTP-D with φ₀ = 0°, and the results are demonstrated in Fig. 7. The heating rate fluctuates greatly with height in the troposphere, yet the KD can reproduce the fine heating fluctuation. The maximum heating rate of 5 K day⁻¹ is located near 750 hPa, and the maximum difference between KD and LBL is 0.09 K day⁻¹ at 950 hPa. Table 6 demonstrates the maximum difference in the heating rate profiles between the KD and LBL in the stratosphere (Δh_stra) and troposphere (Δh_trop), and the difference in the surface flux (ΔF_s) due to the absorption of solar radiation by water vapor in the eight sample ERA5 atmospheres with two solar zenith angles, φ₀ = 0° and 75°. The flux and heating rate are the sums over bands 1, 2, and 3. For all of these atmospheres and solar zenith angles, ΔF_s is <1.4 W m⁻², Δh_trop is <0.09 K day⁻¹, and Δh_stra is negligible.

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As in Fig. 2, but for a solar zenith angle of 0° in the western tropical Pacific in December (WTP-D) atmosphere.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0278.1

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Table 6.

The maximum difference in the water vapor heating rate profiles between the KD and LBL in the stratosphere (Δh_stra) and troposphere (Δh_trop), and the difference in the surface flux (ΔF_s) in bands 1–3. φ₀ is the solar zenith angle, and F₀ is the incident solar radiation at the top of the atmosphere. The difference Δ is KD minus LBL. Units are K day⁻¹ for the heating rate and W m⁻² for the flux.

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For CO₂, the least satisfactory KD-calculated heating rate profile is identified as NSA-D with φ₀ = 0°, and the results for the normal, doubled, and reduced CO₂ cases are demonstrated in Fig. 8. In those cases, the maximum difference between KD and LBL is ~0.06 K day⁻¹ in the stratosphere and ~0.03 K day⁻¹ in the troposphere. Table 7 is similar to Table 6, except for the absorption of solar radiation by CO₂. Tables 7a, 7b, and 7c are cases with atmospheric CO₂ concentrations of 200, 400, and 800 ppmv, respectively. Differences in the heating rate and the surface flux are small; Δh_stra is <0.06 K day⁻¹, Δh_trop is <0.03 K day⁻¹, and ΔF_s is <1.1 W m⁻². Table 8 is the same as Table 7, except for the overlapping absorption of water vapor and CO₂. Differences in the heating rate and the surface flux are Δh_stra is <0.08 K day⁻¹, Δh_trop is <0.09 K day⁻¹, and ΔF_s is <2.6 W m⁻².

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The total-band heating rates and heating rate differences for a solar zenith angle of 0° in the North Slope of Alaska in December (NSA-D) atmosphere with CO₂ concentrations of (top) 400 and (bottom) 800 ppmv. Here, the total band refers to the sum over bands 2 and 3.

Citation: Journal of the Atmospheric Sciences 78, 9; 10.1175/JAS-D-20-0278.1

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Table 7a.

As in Table 6, but for the absorption in bands 2 and 3 due to a CO₂ concentration of 200 ppmv.

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Table 7b.

As in Table 7a, but for the absorption due to a CO₂ concentration of 400 ppmv.

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Table 7c.

As in Table 7a, but for the absorption due to a CO₂ concentration of 800 ppmv.

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Table 8a.

As in Table 7a, but for the overlapping absorption of water vapor and a CO₂ concentration of 200 ppmv.

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Table 8b.

As in Table 7b, but for the overlapping absorption of water vapor and a CO₂ concentration of 400 ppmv.

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Table 8c.

As in Table 7c, but for the overlapping absorption of water vapor and a CO₂ concentration of 800 ppmv.

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The results presented above were computed using wide ranges of the absorber amount and solar zenith angle. The path-integrated water vapor amount differs by a factor of 40, and the CO₂ concentration ranges from half, normal, and double of the current value. In addition to the standard atmospheres, the sample atmospheres were randomly selected from diverse climate regimes and seasons. Those results demonstrate that the new k-distribution approximation is an accurate scheme for calculations of both stratospheric and tropospheric heating due to water vapor, CO₂, and overlapping absorption of these two gases, all in a computationally efficient way.

6. Concluding remarks

A new k-distribution scheme for radiative transfer due to water vapor and CO₂ absorption in the solar spectral region is developed. Two major features of this scheme are the grouping of spectral points and derivation of the equivalent absorption coefficient:

1. In grouping the spectral points of a spectral band, a couple of pressure levels, or reference pressures, where solar heating is substantial are identified. By fixing the temperature at 250 K, the range of the absorption coefficient at a reference pressure k_r is divided into a small number of intervals with a constant Δ log₁₀(k_r). A spectral point is identified with a log₁₀(k_r) interval, or g group, according to its log₁₀(k_r) value. Different from the correlated k-distribution approximation, a spectral point can only be identified with the same g group in all atmospheric layers.

2. Transmission is a highly nonlinear function of the absorption coefficient, and the equivalent absorption coefficient of a g group should be adjusted to a value smaller than the linearly averaged absorption coefficient of all the spectral points in the g group. Further, using an equivalent absorption coefficient to represent a group of absorption coefficients is equivalent to approximate nongray radiation by gray radiation, which overestimates absorption when atmospheric layers are combined in flux calculations. For these two reasons, the equivalent absorption coefficient is adjusted empirically to obtain a balanced accuracy between the atmospheric heating rate and the surface radiation.

The incident extraterrestrial solar radiation at spectral points with a large absorption coefficient is quickly absorbed to heat the stratosphere and, hence, does not contribute substantially to the heating of the troposphere. Those spectral points are identified from the absorption coefficient at a low reference pressure chosen in the stratosphere. By excluding those spectral points from a spectral band, the range of the absorption coefficient of the remaining spectral points is narrowed, and the k-distribution approximation can be applied to accurately compute fluxes and heating rate in both the stratosphere and the troposphere. For the absorption due to water vapor and CO₂, random overlapping of the absorption is assumed, which is only for clear skies.

The new k-distribution scheme is applied to compute fluxes and heating rates in three typical atmospheres, two artificial extreme atmospheres, and eight atmospheres sampled from four diverse climatic regimes and two seasons. With the solar infrared divided into three wide bands as demonstrated in Table 1 and the use of a total of 52 g groups in the water vapor and CO₂ bands, the heating rate difference between the k-distribution and line-by-line calculations in the spectral region 1000–14 290 cm⁻¹ is <0.09 K day⁻¹ for water vapor, and <0.2 K day⁻¹ for CO₂ except in regions near 0.01 hPa. The difference in the downward surface radiation is <1.4 W m⁻² for water vapor and <0.6 W m⁻² for CO₂. With the overlapping absorption of water vapor and CO₂, the difference in heating rate is nearly the same as water vapor in the troposphere and CO₂ in the stratosphere. However, the difference in the downward surface flux increases to 2.6 W m⁻². The results are sensitive to the number of g groups, which is equivalent to the spectral resolution of the k-distribution scheme. Results can be improved by dividing the solar infrared into more than three bands and increasing the number of g groups. In summary, this new k-distribution scheme demonstrates 1) the stability of the parameterization in light of the wide range of water vapor column amounts and CO₂ amounts, 2) the accurate parameterization of the effect of the overlap of water vapor and CO₂, and 3) the accurate parameterization of both stratospheric and tropospheric heating, all in a computationally efficient way.

The current wide spectral limits used have their limitations. For example, variations in the scattering properties of clouds with wavenumbers are not small in the three wide spectral bands of Table 1, and treatments of the overlapping of cloud scattering with gaseous absorption in cloudy skies can be more accurate by increasing the number of spectral bands. In addition to the issue of the scattering properties of clouds, there is also the issue of overlapping of absorption due to multiple gases. In this study, we used the multiplication rule to compute the mean transmission function of a band [Eqs. (3.18) and (3.19)] in clear skies by assuming the spectral overlapping of the gaseous absorption coefficients is random. For a wide spectral band that encloses at least an entire H₂O or CO₂ absorption band, the assumption of random overlapping of the absorption coefficients would degrade the flux and heating rate calculations. Therefore, a partition of narrower spectral bands than those given in Table 1 is necessary for improving solar heating calculations even in clear skies.

However, there is an ensuing issue on the computational economy when the spectrum is divided into more narrower bands. To reach a similar level of accuracy, the total number of g groups, i.e., M, of the narrow bands would be larger than that of the broad band, and flux calculations would be more expensive. This is because of the large spectral variation of the absorption coefficient, which is dominated by the absorption coefficient of single molecular lines. If a broad band having an absorption coefficient ranges by an order of 10 is divided into two narrow bands, the sum of the ranges of the absorption coefficient in the two narrow bands should be larger than 10.

The scope of this study is limited to the computation of solar heating in nongray spectral regions where the variation of the absorption coefficient with wavenumber is not smooth and the k-distribution (or the correlated k-distribution) approximation applies. It does not include the heating due to O₃, the minor heating due to O₂, and the continuum absorption. The O₃ and continuum absorptions are smooth with respect to wavenumber and do not need the application of the new k-distribution (or the correlated k-distribution) approximation. The new k-distribution can be extended to compute the heating in the minor O₂ bands. The scattering due to clouds and aerosols and the Rayleigh scattering are also beyond the scope of this study.

In terms of applications, the correlated k-distribution method and the new k-distribution method are identical, except the flux-weighted k-distribution function [the term H_g in Eq. (3.10)] and the equivalent absorption coefficient [the term k_g in Eq. (3.7)] are precomputed (or specified) differently. Thus, this new k-distribution approximation can be used as an option to modify the existing full-scale radiative transfer models, such as the RRTMGP (Pincus et al. 2019) and the model of Li and Barker (2005), which use the correlated k-distribution approximation in computing the nongray gaseous absorption.