## 1. Introduction

Gaseous transmission is an important process for atmospheric radiation. Over the past two decades, there has been a trend in radiative transfer schemes to replace traditional band models for gaseous transmittance with the correlated-*k*-distribution (CKD) method (Lacis and Oinas 1991; Fu and Liou 1992; Holling 1993; Kratz 1995; Edwards and Slingo 1996; Mlawer et al. 1997; Chou and Suarez 1999; Sun and Rikus 1999; Kato et al. 1999; Chou et al. 2001; Zhang et al. 2003; Li and Barker 2005; Sun 2008; Shi et al. 2009; and others). In the CKD method, sorting the absorption coefficient in ascending order transforms the problem of integrating a tortuously variable absorption coefficient over frequency space into integrating a smoothed sorted absorption coefficient using relatively few quadrature points. In principle, CKD can be applied to a single absorption line whereas band models utilize mean values for entire bands. Thus, CKD shows considerable promise in simulating atmospheric radiation more accurately and efficiently. However, there are still unresolved issues surrounding gaseous transmission in climate models.

In this work we focus on two of these unresolved problems. The first is the overlapping of solar and infrared spectra. In almost all radiation models the shortwave and longwave portions of atmospheric radiation are treated separately due to the different transfer properties within each wavelength range. Usually the solar (shortwave) portion covers the visible and ultraviolet range from 0.2 to 0.69 *μ*m and the near-infrared range from 0.69 to 4 *μ*m, while the longwave part covers the infrared range from 4 to 1000 *μ*m (or larger). However, according to Planck’s formula the solar spectral radiance should extend into the infrared range and beyond. For a blackbody source at temperature 6000 K, the spectral radiance between 4 and 1000 *μ*m is about 12 W m^{−2} or 0.88% of the total solar input of 1365 W m^{−2} at the top of the atmosphere (TOA). By the same reasoning, the infrared spectral radiance extends to wavelengths less than 4 *μ*m. For a thermal source at temperature 275 K, the spectral radiance between 0.2 and 4 *μ*m is 0.33 W m^{−2} or 0.089% of the total terrestrial spectral radiance of 368 W m^{−2}. Thus, the extension of solar energy into the longwave range is about an order of magnitude larger than the extension of infrared energy into the shortwave. Given the strong water vapor absorption in the infrared, the neglect of incoming solar energy in the infrared results in less solar absorption in the model than in the real atmosphere.

The second focus of this paper is the shortwave radiative effect of methane. The shortwave radiation algorithms in the first generation of general circulation models (GCMs) only considered water vapor and O_{3} (e.g., Lacis and Hansen 1971), with CO_{2} added later (Fouquart and Bonnel 1980). Some current shortwave radiation algorithms, including the Canadian Centre for Climate Modeling and Analysis (CCCma) fourth-generation GCM discussed in this paper, include the O_{2} Herzberg continuum and the O_{2} Schumann–Runge band using the Strobel (1978) parameterization (Li and Barker 2005). However, despite its crucial role in the longwave radiation, the shortwave radiative effect of methane has been largely overlooked. In an intercomparison of the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4) radiation schemes used in GCMs, Collins et al. (2006) showed that none include the shortwave CH_{4} effect. Line-by-line (LBL) benchmark calculations summarized in Collins et al. (2006) show that, from zero to preindustrial concentration values, the CH_{4} and N_{2}O shortwave forcing at 200 mb is about 0.35 W m^{−2} for a solar zenith angle (SZA) of 53°. Since the contribution by N_{2}O is small (likely not exceeding 0.05 W m^{−2}, based on the results of Collins et al.), most of this effect is due to CH_{4}. From the preindustrial period to present, the CH_{4} shortwave forcing at 200 mb is ∼0.13 W m^{−2}, about half of the shortwave forcing from CO_{2}. However, over the same period, the CH_{4} shortwave forcing of ∼0.53 W m^{−2} at the surface is 68% larger than that of CO_{2}. Shortwave absorption by CH_{4} thus produces a warming effect in the atmosphere that has yet to be accounted for in GCMs.

In this paper, the radiative impact of both the spectral overlap and the shortwave radiative effect of CH_{4} are quantified using both a one-dimensional radiation (CKD) model and a GCM.

## 2. Overlapping of solar and infrared spectra in radiative transfer

Radiative transfer processes differ for shortwave and longwave radiation. Scattering by molecules, cloud droplets, and aerosol particles is important for shortwave radiation but is weak in the infrared. Therefore, scattering is usually neglected or simply treated in longwave transmission calculations (Li 2002). In the radiation schemes commonly used in GCMs, longwave radiation arises from thermal sources inside the atmosphere that have no effect on solar radiation, while incoming solar flux at the TOA has no effect on longwave radiation. Neglecting the small shortwave spectral overlap of thermal sources (section 1), in this section we focus on modifications to the usual longwave radiative transfer method in order to accommodate the incoming solar radiance at the TOA. In the appendix, these modifications are described, both for a rigorous solution of the radiative transfer equation [Eq. (A3)] and a simple, computationally efficient solution [Eq. (A7)]. In the rigorous solution, the downward flux at a given level is directly determined by thermal emission and absorption from all layers above that level. Therefore, the computation time is quadratically proportional to the number of model layers. In the simple solution, all of the exchange contributions from outside of a layer are represented by the incoming flux at the boundary of the layer. Thus, the computation time is linearly proportional to the number of model layers. As shown in the appendix, the largest solar contribution is from the direct beam, which can be easily handled in the context of the traditional infrared radiation calculations. This simple method of accounting for the incoming solar radiance in the infrared range is expected to produce accurate results.

The left-hand panels in Fig. 1 show the downward longwave fluxes for two standard atmospheric profiles of middle latitude summer (MLS) and subarctic winter (SAW) from McClatchey et al. (1972). The calculation is based on the LBL algorithm applied to Eq. (A3) with 1999 high-resolution transmission (HITRAN) data implemented, but without the incoming solar flux term. All important radiatively active trace gases are included, with the vertical profiles of water vapor and ozone from McClatchey et al. (1972) and adopting the following uniform concentrations for the key greenhouse gases: CO_{2} = 350 ppmv, CH_{4} = 1.75 ppmv, and N_{2}O = 0.28 ppmv. The surface longwave emissivity is set to 1 and the cloud/aerosol optical depth is taken to be zero.

The difference in downward longwave flux due to inclusion of the incoming solar radiance at the TOA, based on the LBL algorithm with Eq. (A3), is also shown in Fig. 1 (middle and right panels). At solar zenith angle *θ*_{0} = 0°, the downward flux at the top model level increases to about 12 W m^{−2}, corresponding to the incoming solar flux in the 4–1000-*μ*m range. In the lower atmosphere, the difference in downward flux is larger for SAW than for MLS. This is because the atmosphere is drier in SAW, with less attenuation of solar flux by water vapor. At *θ*_{0} = 60°, the difference in downward flux due to the inclusion of solar radiance in the infrared is about 50% smaller than for *θ*_{0} = 0°. Figure 1 also shows that the difference in the downward longwave flux at the surface is small (generally less than 3 W m^{−2}), which indicates that most solar energy in the infrared range is absorbed by the atmosphere. There is no need to consider the surface reflection of the direct solar beam flux. Since the surface albedo is about 0.1, the reflected diffuse solar flux is generally less than 0.3 W m^{−2} and therefore may be neglected in comparison with the surface thermal emission flux, about a few hundred watts per square meter.

In the lower panels of Fig. 1, the same calculations are performed using a one-dimensional radiation algorithm implemented in the CCCma GCM (Li and Barker 2005). This algorithm employs the CKD method for gaseous transmission by resolving the solar spectrum in the range 0.2–4 *μ*m into 31 intervals in cumulative probability function space and the infrared spectrum in the range 4–1000 *μ*m into 46 intervals. The gaseous constituent concentrations are the same as those used in the LBL calculations. The infrared radiative transfer scheme with incoming solar radiance follows the simplified method shown in the appendix [Eq. (A7)].

The bottom left panel of Fig. 1 shows that the downward longwave flux simulated by the CKD model is very similar to that obtained in the LBL calculations based on the rigorous radiative transfer method: the error is less than 1 W m^{−2} at most levels. Also, the difference in downward flux caused by the inclusion of incoming solar radiance in the infrared range is very similar to that shown in the upper panels. This indicates that the approximate solution of the longwave radiative transfer equation given in the appendix is adequate.

Figure 2 shows the infrared heating rate and its change for the same cases as in Fig. 1. Consistent with the results of Fig. 1, the upper panels show that the additional solar flux in the infrared range enhances the heating rate by up to 10% at a solar zenith angle *θ*_{0} = 0°.

The lower panels in Fig. 2 show corresponding results from the CKD radiation model. Generally, the heating rate is accurately simulated by the CKD model (left panels), with an error less than 0.1 K day^{−1} up to 1 mb. However, the differences in heating rate due to the incoming solar radiance are not simulated very accurately by the CKD model, especially in the upper atmosphere above 1 mb. In the CKD model, only about 46 intervals in cumulative probability function space lie in the infrared range due to considerations of computational efficiency. Thus, it is likely that the spectrum is not well resolved, yet the heating rate in the upper atmosphere is very sensitive to the fine structure of the gaseous absorption coefficient (Li and Barker 2005). This deficiency could likely be resolved by increasing the number of intervals in the cumulative probability function space. However, in the upper atmosphere above 1 mb, nonlocal thermodynamic equilibrium (non-LTE) effects (Fomichev et al. 1998; López-Puertas and Taylor 2001; Fomichev et al. 2004) become important, strongly influencing the heating rates. Since the upper atmosphere is not the focus of the current study, there is little motivation to introduce new parameterizations to improve the accuracy of the results in this region.

In the region below 1 mb, it is found that the difference in heating rate due to the incoming solar radiance in the infrared range is well simulated by the CKD model, with error less than 0.1 K day^{−1}. In the infrared, there is no large attenuation of solar radiance as occurs due to O_{3} in the ultraviolet range. Thus, a large portion of solar radiance in the infrared can penetrate into the troposphere, causing the heating (cooling) rate to increase (decrease).

Figure 3 shows differences in downward flux for each longwave band of the CKD scheme. The difference at the TOA represents the incoming solar energy in the infrared. The difference at the surface shows the attenuation of solar radiance by the atmosphere, which causes an increase in the downward flux at the surface. Figure 3 shows that the dominant absorption of incoming solar radiation occurs in band 7 (1400–1900 cm^{−1}), which contains most of the H_{2}O 6.3-*μ*m band. Absorption by CO_{2} at 4.3 *μ*m (bands 8 and 9, ranging from 1900 to 2500 cm^{−1}) is also quite significant.

As mentioned in the introduction, in most current radiation models the 12 W m^{−2} of solar energy that falls in the infrared range (beyond 4 *μ*m) is deposited in the shortwave spectral range (0.2–4 *μ*m). This method was used, for example, in the third-generation CCCma GCM (Scinocca et al. 2008). It is interesting to examine the effects of this incorrect treatment on the shortwave energy budget of the atmosphere. The upper left panel of Fig. 4 shows the LBL calculation of downward shortwave flux at solar zenith angle *θ*_{0} = 60° for MLS and SAW. The incoming solar flux in the range 0.2–4 *μ*m is 1365 − 11.88 = 1353.12 W m^{−2}. The surface albedo is set to 0.1 and the cloud/aerosol optical depth is taken to be zero. The same calculations were performed with a solar constant of 1365 W m^{−2}. The difference in downward flux between the two calculations is shown for two solar zenith angles in the middle and right panels of Fig. 4. For *θ*_{0} = 0°, it is found that the difference in downward flux at the upper model level is about 12 W m^{−2}, as in the longwave case summarized in Fig. 1. However, in contrast to Fig. 1, the difference in downward shortwave flux changes little from the upper to lower atmosphere. At the surface, the difference in downward flux is about 10 W m^{−2}. Hence, most of the additional incoming solar energy reaches the surface instead of being absorbed by the atmosphere as in the longwave calculations summarized above. The results for a solar zenith angle *θ*_{0} = 60° are similar to the *θ*_{0} = 0° case, but with smaller values (i.e., a difference of 5 W m^{−2} at the TOA compared with 4 W m^{−2} at the surface). In the lower panels of Fig. 4, the same calculations are performed using the CKD model, with nearly identical results.

In Fig. 5, the corresponding shortwave heating rates are shown. The left panels show the shortwave heating rate at *θ*_{0} = 60° for MLS and SAW, while the middle and right panels show differences in heating rate caused by the additional solar energy in the shortwave range, for *θ*_{0} = 0° and 60°. Additional heating occurs mostly in the upper atmosphere, mainly due to absorption by O_{3} and CO_{2}. Below about 10 mb, the heating rate increase is nearly zero, significantly smaller than that found in the longwave calculations (Fig. 2). In the lower panels of Fig. 5, it is seen that the CKD model can accurately simulate both the solar heating rate and the extra heating due to the additional solar flux at the TOA.

An alternative to depositing the solar longwave energy of 12 W m^{−2} over the entire shortwave range of 0.2–4 *μ*m is to place this energy only into the last near-infrared interval (e.g., 2.38 to 4 *μ*m in the CCCma CKD model). However, this strategy can cause unrealistic enhancement of cloud shortwave absorption since the cloud absorption is strong in this spectral range. We do not discuss the details here.

To summarize the results of this section, the results of Figs. 1 –5 indicate that the usual practice of depositing the solar longwave energy of 12 W m^{−2} into the shortwave range instead of the infrared underestimates heating in the atmosphere and overestimates the absorption of heat at the surface. This may have important implications for the energy budget and temperatures in the atmosphere and at the surface.

## 3. The shortwave radiative influence of CH_{4}

Methane has two main bands of atmospheric absorption, one in the infrared from 1100 to 1900 cm^{−1} (5.26–9.09 *μ*m) and the other in the near-infrared from 2500 to 6000 cm^{−1} (1.66–4 *μ*m). The CH_{4} absorption coefficient at the surface over the entire range is shown in Fig. 6. The magnitude of the absorption in the infrared (Fig. 6, top) varies significantly with wavelength and spans some 10 orders of magnitude. The strong infrared absorption makes CH_{4} an important contributor to the atmospheric radiative forcing change since the preindustrial era. In most radiation algorithms, this effect is well simulated. The CH_{4} absorption coefficient in the near-infrared is shown in the lower panel of Fig. 6. In this range the CH_{4} absorption lines are clustered in three isolated groups, each with a similar range of absorptivity and wavenumber. The magnitude of the absorption coefficient in this higher wavenumber range is of the same order as that in the infrared. Despite this, according to Collins et al. (2006) and Forster et al. (2007), none of the radiation algorithms in the IPCC AR4 GCMs includes the shortwave effect of CH_{4} in this range.

LBL models reveal that the shortwave radiative forcing by CH_{4} since the preindustrial era is about 0.13 W m^{−2} at 200 mb (near the tropopause) for a MLS profile and *θ*_{0} = 53°, smaller than the corresponding longwave radiative forcing of about 0.76 W m^{−2}. However, the surface shortwave radiative forcing by CH_{4} since the preindustrial era, 0.53 W m^{−2}, exceeds both the CH_{4} longwave forcing of 0.46 W m^{−2} and the CO_{2} shortwave forcing at the surface (Collins et al. 2006). Shortwave absorption by CH_{4} is therefore a component of the radiative balance of the atmosphere that needs to be incorporated in climate models.

Following the strategy of Li and Barker (2005) for gaseous absorption, CH_{4} absorption in the shortwave spectral range was added to the CKD model. In any CKD model, spectral overlap with other gaseous constituents is a key issue. Since absorption by CH_{4} is relatively weak compared to that of water vapor in the near-infrared, the so-called “alternate mapping” method is used to handle the overlap of the spectral lines among water vapor, CO_{2}, and CH_{4} in that range. This method sorts each individual gas absorption coefficient alternately in different intervals of the cumulative probability function space. In this application, only two intervals in the cumulative probability function space are sorted based on the CH_{4} absorption coefficient. Details of the alternate mapping method can be found in Li and Barker (2005).

Figure 7 shows the difference in net radiative flux and the difference in heating rate due to the inclusion of shortwave absorption by CH_{4}. The solar energy in the infrared is included both with and without the shortwave effect of CH_{4}, and the mixing ratios of greenhouse gases are the same as those adopted in section 2. The upper panels in Fig. 7 show the results of the LBL calculations, while the lower panels show the corresponding CKD model results. To include the overlap between solar and infrared spectra as considered in section 2, the calculations are performed from 0.2 to 1000 *μ*m, with the thermal Planck function set to zero in the 4–1000-*μ*m range. Thus, the solar energy absorption by CH_{4} occurs over the entire 0.2–1000-*μ*m range, but longwave emission by CH_{4} is not allowed to occur. The left and middle panels are the results of the difference in net radiative flux (downward minus upward) over 0.2–1000 *μ*m with solar zenith angle *θ* = 0° and 60°, respectively.

Mainly as a result of shortwave absorption by CH_{4}, both upward and downward shortwave flux decrease throughout the atmosphere. The difference in net flux at the surface reaches −2.5 W m^{−2} for overhead sun conditions. The difference in net flux at the TOA is considerably smaller, approximately +0.5 W m^{−2} for *θ*_{0} = 0. At the surface the sign is negative, since the downward flux is reduced by the CH_{4} absorption, but the upward flux is less affected because of the small value of surface albedo. At the TOA the sign becomes positive, since the reflected flux is reduced owing to the CH_{4} absorption, but the downward flux is unchanged. As shown in Fig. 6 the CH_{4} absorption lines are mostly in the near-infrared and infrared. In this range, Rayleigh scattering is very weak, so the change of upward flux is mainly caused by the reduction of reflected upward flux due to the absorption of CH_{4}. For a small surface albedo, such as 0.1 as assumed in Fig. 7, the reflected upward flux from the surface is about one order of magnitude smaller than the downward flux. Therefore, the net flux difference at the TOA is expected to be one order of magnitude smaller than that at the surface. The CH_{4} net flux forcing depends on the solar zenith angle, and the middle panel shows the result for *θ*_{0} = 60°. The lower left and middle panels show the corresponding CH_{4} forcing results obtained by the CKD model. The model simulates the net shortwave flux difference due to CH_{4} quite accurately.

The right-hand panels in Fig. 7 show the difference in heating rate due to inclusion of the CH_{4} absorption for MLS and SAW with a solar zenith angle *θ*_{0} = 60°. The change in heating rate due to CH_{4} is generally small, less than 0.1 K day^{−1} in the region below 1 mb and less than 0.2 K day^{−1} above 1 mb. The lower right panel shows the corresponding result from the CKD model calculation. Here the heating rate of CH_{4} is not simulated accurately, especially in the upper atmosphere above 10 mb. The reason for this deficiency is the same as that found for the spectral overlap calculations in section 2; that is, the spectrum is not well resolved due to the use of only a small number of intervals in cumulative probability function space.

In the previous section, a treatment of the overlap between solar and infrared spectra was introduced. Therefore, it is natural to ask how much of the CH_{4}-induced change in net radiative flux lies in the infrared (i.e., the 4–1000-*μ*m range). In the leftmost panel of Fig. 8, the net flux difference due to CH_{4} in the infrared range is shown based on the LBL calculations. This is the same calculation as displayed in Fig. 7, but only within the infrared range. The contribution by CH_{4} to the radiative flux change in the infrared is small, less than 0.1 W m^{−2} at the surface, as most of the CH_{4} absorption lies beyond 7 *μ*m where the solar energy input is very small. The corresponding contributions for O_{3}, CO_{2}, and H_{2}O are also shown in Fig. 8. The contribution to the net flux difference by O_{3} is slightly larger than for CH_{4}, while that for CO_{2} is about 2 W m^{−2} at the surface for overhead sun conditions. The largest contribution is from water vapor, up to 5 W m^{−2} at the surface for overhead sun conditions. This is not surprising since the water vapor spectrum spans nearly the entire infrared range. Also, water vapor has the largest concentration of any gaseous constituent in the lower atmosphere and very strong absorption in the vibration–rotation band. As in section 2, the CKD model accurately simulates the net flux anomaly for each individual gas (not shown).

In the calculations presented here, the mixing ratio of CH_{4} is assumed to be spatially uniform. In reality, the CH_{4} concentration decreases approximately exponentially above the tropopause (e.g., Randel et al. 1998). Therefore, the increase of heating rate due to CH_{4} in this region shown in Fig. 7 is likely an overestimate.

## 4. GCM simulations

In this section, we study the impacts of the radiation code modifications described in the previous two sections on present-day, model-simulated climate. The climate simulations are performed with the CCCma fourth-generation atmospheric general circulation model (AGCM4), which employs a spectral resolution of T47 with 35 vertical levels from the surface up to 1 mb. AGCM4 is based on the third-generation CCCma AGCM (McFarlane et al. 2005; Scinocca et al. 2008) with the addition of the CKD radiation scheme described above along with numerous other improvements. The model is forced by a repeated annual cycle of sea surface temperature and sea ice extent averaged over the second Atmospheric Model Intercomparison Project (AMIP II) period (Gleckler 1996). Ten-year climate integrations were performed to determine the impact of the overlap between solar and infrared spectra and shortwave radiative effect of CH_{4}, with the last 10 years averaged to reduce the influence of natural variability.

### a. Overlap of solar and infrared spectra

To investigate the climate impact of the solar spectral overlap in the infrared, two GCM experiments were performed: one based on the usual method of depositing solar energy from the infrared into the visible and near-infrared range (hereafter CTL) and the other accounting for the longwave solar energy in the infrared radiative transfer equation as described in section 2 (hereafter OVL). Differences between the two simulations (i.e., OVL minus CTL) are described in this subsection.

Figure 9a shows the difference of upward reflected shortwave flux at the TOA under clear-sky conditions, for which (top) all cloud effects are ignored, and (bottom) under all-sky conditions. In the clear-sky case, the reflected shortwave flux is generally lower in the OVL experiment. This is consistent with the results found in the offline calculations, discussed in section 2, since more solar energy is absorbed in the infrared range. The reduction is less than 1 W m^{−2} in most regions, except over the continents at Northern Hemisphere mid and high latitudes, where it can be as large as a few watts per square meter. In the GCM simulations, climate feedbacks can play an important role. As shown in section 2, the additional solar energy in the infrared causes a warming effect. The temperature change inside the atmosphere can affect the distribution of water vapor, which in turn influences the resulting reflected solar flux. Also, surface heating due to both solar infrared and water vapor feedbacks could have an impact on snow and ice cover, which can affect the surface albedo. Therefore, in some continental land regions the reduction of the upward solar flux is larger than over the surrounding ocean. Over the ocean surface albedo change is strongly limited by the repeated annual cycle of sea surface temperature used in both experiments.

In the all-sky case (Fig. 9b), the pattern of upward flux change is much noisier than in the clear-sky case. Some regions with negative differences in Fig. 9a become positive in Fig. 9b. This is due to differences in cloud location and thickness between the two simulations (see Fig. 11 for changes in the cloud zonal distribution). Clouds, especially low clouds, can dramatically affect the reflected solar flux. Although the anomaly pattern of the upward flux at the TOA is noisy under cloudy sky conditions, there are still more areas showing a reduction in reflected shortwave flux, as can be confirmed by a global mean calculation.

Figure 10 shows the zonal mean changes of longwave and shortwave heating rate and temperature due to the two different treatments of incoming solar energy. The shaded regions in the figures indicate points at which the difference is significant at the 95% confident level, according to a difference of means *t* test using sample variances computed from the annual mean values for individual years. We comment only on statistically significant changes in the following.

Figure 10a shows a small but widespread increase in the longwave heating rate due to the extra solar incoming energy in the infrared. This increase is less than 0.05 K day^{−1} in the troposphere, while in the stratosphere the heating rate difference increases with height, reaching 0.04 K day^{−1} at 10 mb in the tropics. In the stratosphere, the GCM-simulated heating rate differences are smaller than those shown in Fig. 2, which range from 0.08–0.12 K day^{−1} from 50 up to 10 mb for *θ*_{0} = 60° (close to the daily averaged value). However, since there is no solar energy input at night, the heating rate difference is likely about half this value. Therefore, the increase of less than 0.05 K day^{−1} seen in the GCM simulations appears reasonable.

Since ∼12 W m^{−2} of additional solar energy is deposited in the range beyond 4 *μ*m, there is less solar energy in the shortwave range in OVL compared to CTL, which agrees with the negative differences seen in Fig. 10b. Combining the results shown in Figs. 10a and 10b, an increase of the net heating rate in the free atmosphere is evident. This would be expected to cause an increase of atmospheric temperature, which is indeed seen in Fig. 10c. It is remarkable that the temperature difference in the tropical stratosphere can exceed 1 K. In the tropical tropopause layer (TTL), defined as the region between approximately 150 and 70 hPa, the temperature increase is about 0.6 K. The results of Fig. 10 suggest that the neglect of solar energy deposition in the infrared can lead to a cold bias in the TTL. Gettelman et al. (2010, chapter 7) show that a lot of GCMs have colder bias in the TTL to some extent. The TTL is a complicated problem that can be affected by cirrus cloud near the tropopause. However, our result shows that the mistreatment of the overlap of solar and infrared spectra could be one cause for colder bias in the TTL.

In Fig. 11, the zonal mean difference in cloudiness (OVL minus CTL, in percent) is shown. Although few of the changes are statistically significant, the inclusion of solar energy in the infrared range does lead to differing cloud distributions in the upper tropical troposphere. In the TTL in particular, there is a reduction of high cloud. The upper portions of high clouds generally produce a local cooling effect (Li 2002). This cooling is therefore reduced in experiment OVL and this helps explain the warming in the TTL shown in Fig. 10.

Finally, the effect of the incoming solar longwave flux on the outgoing longwave radiation (OLR) was also examined. OLR increases in both clear and cloudy-sky conditions (not shown) due to the overall warming of the atmosphere. However, because OLR is more sensitive to surface temperature, which is held fixed in these simulations, the differences in OLR are much smaller than those in outgoing shortwave radiation. The influence of these radiation scheme changes on OLR should be investigated using a coupled ocean–atmosphere GCM.

### b. Shortwave radiative effect of CH_{4}

To investigate the climate impact of CH_{4} shortwave absorption, two 10-yr GCM integrations were performed, with and without the CH_{4} effect included (as in the offline calculations, the solar overlap in infrared range was included in both GCM simulations). Differences between 10-yr averages of the two simulations (i.e., with the CH_{4} shortwave effect minus without) are described below.

Figure 12 shows the results of the difference in reflected solar flux at the TOA between the two experiments under clear-sky (Fig. 12a) and all-sky conditions (Fig. 12b). In the clear-sky case, inclusion of the CH_{4} shortwave absorption causes a decrease of the reflected shortwave flux at the TOA in most regions. This decrease is generally less than 0.5 W m^{−2}, consistent with the results of the offline calculations (Fig. 7). However, similar to Fig. 9, the reflected shortwave flux at TOA increases in some regions.

As in the spectral overlap experiment (Fig. 9), the pattern of reflected shortwave flux at the TOA in the all-sky case is much noisier than in the clear-sky case. The extra CH_{4} solar absorption warms the atmosphere, which enhances the water vapor amount and changes the cloud amount and distribution, thereby strongly influencing the reflected shortwave flux.

Figure 13 shows the zonal mean changes of longwave and shortwave heating rate and temperature due to inclusion of the shortwave CH_{4} effect. The presence of shortwave-absorbing CH_{4} causes an increase in the shortwave heating rate, mostly in the stratosphere. The heating rate difference increases with height, consistent with the offline calculations (Fig. 7), but is of smaller magnitude because of the inclusion of nighttime conditions in the GCM. An associated and widespread increase of temperature is also clearly seen in Fig. 13c. The maximum of the temperature difference is located in the lower stratosphere (upper TTL) at ∼70 mb and is generally smaller than that caused by the solar spectral overlap in the infrared, shown in Fig. 10. In the vicinity of the TTL, the increase of temperature due to the shortwave CH_{4} effect is about 0.4 K, which would further augment the TTL temperature bias due to inclusion of the solar longwave effect (section 4a). Based on Figs. 10 and 13, the combined effect of solar–infrared overlap and CH_{4} shortwave absorption is expected to result in an increase in zonal temperature of over 1 K throughout the tropical and midlatitude stratosphere. However, this assumes that the two effects add linearly, which should be confirmed by further simulations.

In the two simulations discussed here, no changes were made to the longwave radiation algorithm. Despite this, Fig. 13a shows a decrease in the longwave heating rate over most of the stratosphere due to inclusion of the shortwave CH_{4} effect. This heating rate decrease, which corresponds to increased longwave emission, is likely a result of Stefan–Boltzmann feedback due to the local temperature increase from CH_{4} shortwave heating. Note that this interdependence of shortwave and longwave radiative changes was not seen in the solar–infrared overlap experiments of section 4a since the longwave heating increase seen there does not directly affect the shortwave heating rate.

## 5. Summary and conclusions

This work focused on two issues: treatment of the infrared spectral contribution of incoming solar radiation and the shortwave radiative effect of CH_{4}. These two phenomena are related insofar as the CH_{4} longwave transmission is affected by the solar infrared flux, while the overall longwave emission responds to changes in shortwave heating.

A straightforward method was proposed that accounts for the contribution of the solar direct beam flux to longwave radiative transfer. It is shown that, in most cases, the offline CKD model gives results in good agreement with the benchmark results based on LBL calculations. Compared to the usual method of depositing solar energy from the infrared into the visible and near-infrared range, the current approach results in more shortwave absorption in the atmosphere and reduced solar flux at the surface. In the second part of the paper, CH_{4} shortwave absorption was included in a CKD model, with the resulting changes in flux and heating rate being accurately simulated compared to LBL calculations.

The impact of these radiative changes on model-simulated climate was examined by performing a series of decadal GCM integrations. These experiments show that the deposition of solar energy in the infrared range produces a significant amount of warming throughout the atmosphere, with peak values in the midstratosphere where the temperature increase can exceed 1 K. The GCM simulations also show that including CH_{4} shortwave absorption causes widespread warming of the upper atmosphere and a reduction of the upward shortwave flux at the TOA. In particular, the sum of these changes has a significant warming effect on the TTL region, the locus of many chemical and dynamical processes of interest (Fueglistaler et al. 2009). Finally, inclusion of the CH_{4} shortwave effect would be expected to lead to enhanced atmospheric warming in simulations of future climate under scenarios of CH_{4} increase. The consequences of this atmospheric warming for the surface energy budget could be investigated using a coupled ocean–atmosphere GCM.

The authors thank Professor M.-D. Chou and two anonymous reviewers for constructive comments.

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# APPENDIX

## Solution of the Infrared Radiative Transfer Equation with Incoming Solar Flux

In most current radiation models the shortwave and longwave radiation are treated separately, with the respective algorithms demarcated by a cutoff wavelength of about 4 *μ*m. However, some modelers have extended the shortwave algorithm beyond 4 *μ*m by adding a separate longwave band for the solar contribution only. For example, Mlawer et al. (1997) added a solar band to the Rapid Radiative Transfer Model (RRTM) from 820 to 2600 cm^{−1}, and Chou and Suarez (1999) did the same in their radiation model for 1000–2500 cm^{−1}. Treating the solar longwave contribution in this manner is valid since the solar portion can be represented as a linear term in the infrared radiative transfer equation, independent of the terrestrial contribution [see Eq. (A3) below]. However, the method described here is conceptually simpler, insofar as both solar and terrestrial longwave radiation utilize the same parameterization of gaseous transmission; no additional calculations are required. Also, the calculation can be easily extended to the whole infrared spectral range.

*I*(

*τ*,

*μ*) is the diffuse infrared intensity at wavelength

*λ*,

*μ*= cos

*θ*,

*θ*(

*θ*

_{0}) is the local (solar) zenith angle,

*ω̃*is the single scattering albedo,

*B*[

*T*(

*τ*)] is the Planck function for temperature

*T*,

*πS*

_{0}is the solar flux, and

*P*(

*μ*,

*μ*′) is the azimuthally averaged phase function. The last term on the rhs of Eq. (A1) accounts for incoming solar radiation

*S*

_{0}in the infrared range. For simplicity, we neglect the subscript

*λ*in what follows.

*P*(

*μ*,

*μ*′) = 2

*δ*(

*μ*−

*μ*′) (Li 2002). Thus, the transfer equation (A1) becomesThe general solution for the downward radiance corresponding to the general equationis

*I*(

*τ*, −

*μ*) =

*I*(0, −

*μ*)

*e*

^{−κ/μ}, where

*κ*= (1 −

*ω̃*)

*τ*is the absorption depth. Using the general solution, we can obtain the particular solution for the second term on the rhs of Eq. (A2). The corresponding downward flux is (Liou 2002)whereis the flux transmittance defined in terms of

*κ*(

*p*,

*p*′) for a slab of atmosphere between

*p*and

*p*′. Under the two-stream approximation, Ψ[

*κ*(

*p*,

*p*′)] =

*πe*

^{−κ(p,p′)/μ1}; 1/

*μ*

_{1}=

*e*

^{1/2}= 1.648 721 3 is the diffusivity factor (Li 2000).

*T*′ is the temperature at pressure

*p*′ and

*κ*(

*p*, 0) is the absorptance depth from the TOA to

*p*. The solution for the upward flux is the same as in Liou (2002). In Eq. (A3), the solar absorption by cloud/aerosol is included in the direct solar beam term as the absorptance depth is used.

The solar flux term on the rhs of Eq. (A3) can be included in an existing radiation scheme by adding additional solar bands in the infrared, as was done, for example, by Mlawer et al. (1997) and Chou and Suarez (1999). However, this is not necessary since the attenuation and transmission of solar energy can be calculated simultaneously with the thermal emission and attenuation. In a CKD model such as that used in this work, the same parameterization used for gaseous transmission in the infrared is used for transmission of the solar beam.

In Eq. (A3), the downward flux at a given pressure level is determined by thermal emission and absorption from all layers above that level; the contributions exchanged between all layers are considered explicitly. Li (2002) derived a more efficient solution in which all exchange contributions from outside a layer are represented by the incoming flux at the boundary of the layer. We now extend this solution to include the solar source term.

*i*is between levels

*i*and (

*i*+ 1), and the Planck function within the

*i*th layer iswhere

*α*=

_{i}*B*

_{i+1}−

*B*, with

_{i}*B*and

_{i}*B*

_{i+1}being the Planck functions at levels

*i*and

*i*+ 1, respectively, and

*τ*the optical depth of layer

_{i}*i*. By Eq. (A2) we obtain the subsequent solution of the downward radiance at level

*i*+ 1, from the TOA (

*i*= 1),where

*i*. In this case, the particular solution to the solar source can be solved as the last term in Eq. (A5), which, however, is negligible since

*ω̃*≈ 0. Therefore, the diffuse downward flux at level

*i*+ 1 based on the two-stream approximation iswhere

*ζ*=

_{i}*πα*

_{i}μ_{1}/

*κ*,

_{i}*B̃*=

_{i}*πB*, and

_{i}*S̃*

_{0}=

*πS*

_{0}. At the TOA (

*i*= 1), the downward diffuse flux

*F*

_{1}

^{d−}= 0.