## 1. Theoretical background

Radiative transfer in climate models is treated as a one-dimensional process accounting only for the vertical direction. However, within the one-dimensional limit, some realistic features of the atmosphere can still be addressed. In this note, we will discuss the effects of atmospheric curvature and refraction and their impact on the radiation calculation. If the atmosphere is plane-parallel (PP) without spherical curvature, the solar beam pathlength in the atmosphere is *H*/*μ*_{0}, where *H* is the height of the atmosphere and *μ*_{0} = cos *θ*_{0}, *θ*_{0} is the solar zenith angle.

Because of the spherical curvature of the atmosphere, when *θ*_{0} is very large the solar beam pathlength could be significantly shorter than that of the PP case. Figure 1a shows the actual solar beam pathlength for the atmosphere with spherical curvature. For solar zenith angle *θ*_{0}, if there is no curvature, the solar beam pathlength from the top of the atmosphere (TOA) to the surface is *l*, but because of the atmospheric curvature the effective pathlength is *l*′. When the solar zenith is small, *l* and *l*′ are almost the same, but for a large solar zenith angle, the difference between *l* and *l*′ could be very large, with *l* approaching infinity as *θ*_{0} nears 90°.

*μ*=

_{e}*l*′/

*H*, the following equation can be derived (Robinson 1966),where

*R*is the radius of the earth and

*H*is the considered thickness of the atmosphere as shown in Fig. 1a.

The atmosphere is vertically inhomogeneous with gaseous density exponentially decaying with height; *H* in (1) should be taken as a mean height by replacing the inhomogeneous atmosphere with an equivalent homogeneous atmosphere. In Robinson (1966), *H* is defined as *H* = *P _{g}*/

*gρ*, where

_{g}*P*and

_{g}*ρ*are the pressure and density at the surface, respectively, and

_{g}*g*is the gravitational constant. Here,

*H*is the so-called scale height of the atmosphere, typically

*H*= 7 km (global mean value).

Equation (1) is only true for mean results of the effective pathlength for the equivalent homogeneous atmosphere. In the real atmosphere, the change of the local pathlength in each model layer due to the atmospheric curvature is different. In addition, the absorbers in the atmosphere are not well mixed, which particularly is not the case for either H_{2}O or O_{3}. Therefore, the optical extinction coefficients vary with height in an inhomogeneous atmosphere. The local optical depth is the local optical extinction coefficient times the local pathlength. Thus, for radiation calculations the atmospheric curvature effect has to be considered locally in order to obtain the accurate optical depth for each model layer.

*z*be

*θ*(

*z*) (see Fig. 1b). By simple geometry,thus,where

*μ*(

*z*) = cos

*θ*(

*z*). We call 1/

*μ*(

*z*) the local pathlength factor.

*λ*is found,where

*n*(

_{λ}*z*) is the refractive index at height

*z*for wavelength

*λ*, and

*n*

_{λ}_{0}is the refractive index above TOA, which generally can be taken as the vacuum value of one. The atmospheric refraction effect at the surface is well known, which makes, for example, the sun (moon) appear to rise earlier and set later by about 3 min.

In Fig. 2 the local pathlength factor by (3) is shown for three values of 1/*μ*_{0}. Both the inclusion and exclusion of refraction is considered. It is found that the local pathlength factor decreases with height. For a small 1/*μ*_{0}, the change of 1/*μ _{λ}*(

*z*) with height is small. However for a large 1/

*μ*

_{0}, the change of 1/

*μ*(

_{λ}*z*) with height could be dramatically large. With the inclusion of refraction 1/

*μ*(

_{λ}*z*) becomes smaller. It is found that the refraction effect is important only for large values of 1/

*μ*

_{0}. Even for 1/

*μ*

_{0}= 10 (

*θ*

_{0}≈ 84°), the refraction effect is negligible. Also the refraction only has obvious impact on the lower atmosphere below 10 km. In calculations of (3) in Fig. 2, the radius of the earth

*R*= 6370 km, the spectral wavelength is 0.633

*μ*m, and the refractive index follows Ciddor (1996), which depends on pressure, temperature, water partial pressure, and CO

_{2}content. We have compared the results by using different refractive index formulae, like Owens (1967), Birth and Downs (1993, 1994). Generally the maximum relative difference is less than 1% in the results. The atmospheric profile, which provides the vertical distributions of pressure, temperature, and water vapor pressure, is the standard atmosphere (McClatchey et al. 1972). It is found that the results are insensitive to the chosen of atmospheric profiles. In Table 1 the values of 1/

*μ*(

_{λ}*z*) by (3) are shown corresponding to 1/

*μ*

_{0}= 10 000 and

*z*= 0 for four types of atmospheric profiles. The differences for different atmospheric profiles are very small. For other values of 1/

*μ*

_{0}< 10 000 and

*z*> 0, the differences in 1/

*μ*(

_{λ}*z*) for different atmospheric profiles are even smaller.

*κ*(

_{λ}*z*) is the extinction coefficient of height

*z*for wavelength

*λ*;

*λ*

_{1}and

*λ*

_{2}are the lower and upper bounds of wavelength for the considered solar spectral range,

*S*is the incoming solar flux at wavelength

_{λ}*λ*and

*δS*= ∫

^{λ2}

_{λ1}

*dS*.

_{λ}*μ*is the spectral mean and vertical mass weighted mean effective pathlength factor, from (3)where

_{e}*ρ*(

*z*) is the air density at height

*z*,

*ρ*is the air density at surface,

_{g}*H*is the effective height defined as

*ρ*= ∫

_{g}H^{∞}

_{0}

*ρ dz*.

In Fig. 3, 1/*μ _{e}* by (6) is plotted against 1/

*μ*

_{0}with and without refraction effect. The integrals of (6) are evaluated with a very high vertical resolution of 0.01 mb [converting (6) to the integral of pressure with hydrostatic approximation]. The solar spectral range is from 0.2 to 4

*μ*m. Also in Fig. 3 the results of two other parameterizations by Kasten and Rodgers are shown.

Similar to Fig. 2, the refraction plays an important role when 1/*μ*_{0} > 10. It is found that the results of Kasten’s formula and (6) without refraction are very similar. Small difference appears when 1/*μ*_{0} > 100. Compared to the result of (6) with refraction the Kasten’s formula underestimates the effect of refraction. Kasten (1966) used a formula similar to (6), but based on the ground observation angle instead of the incoming solar zenith angle (see the appendix). The atmospheric profile used in Kasten (1966) was different from what we used. Also a crude approximation for refractive index was used in Kasten (1966) and only one solar wavelength of 0.7 *μ*m was considered.

It is difficult to understand the result of Rodgers’ parameterization. In the range of 10 < 1/*μ*_{0} < 120, 1/*μ _{e}* by Rodgers’ formula is larger than that of (6). Moreover, in such range, Rodgers’ result is even lager than the result by the mean geometry of (1) for the equivalent homogeneous atmosphere without refraction. This is unreasonable in physics and indicates that Rodgers’ formula underestimates both of the atmospheric curvature effect and the refraction effect. In Rodgers (1967) it is not clear how (8) was derived and no verification is provided.

The calculations of (6) in Fig. 3 are also based on the standard atmospheric profile. In Table 1 the values of 1/*μ _{e}* at 1/

*μ*

_{0}= 10 000 are presented for four different atmospheric profiles. Again the differences for different atmospheric profiles are very small.

The three parameterizations only account for the vertical mean results of effective solar pathlength factor. As shown in (4), in order to obtain an accurate calculation the local pathlength factor, 1/*μ _{λ}*(

*z*), is required.

*z*located at height

*z*above the ground. We draw two concentric circles representing the edges of Δ

*z*. Although (2) gives the local pathlength factor at height

*z*, it actually is not the exact factor needed to calculate the local solar pathlength of layer Δ

*z*. Transmission for the direct solar beam through layer Δ

*z*is

*e*

^{−kΔz}′, where

*k*is the extinction coefficient and Δ

*z*′ is the actual pathlength along solar beam for the curved layer Δ

*z*(see Fig. 4a). In radiative transfer calculations the transmission depends on Δ

*z*as

*e*

^{−kΔz/μ(z)}. Comparing it with

*e*

^{−kΔz}′, we conclude that 1/

*μ*(

*z*) should be Δ

*z*′/Δ

*z*. With this definition, we can derive from simple geometry,where

*α*= Δ

*z*/(

*R*+

*z*). Generally the layer thickness in a climate model is less than 5 km, therefore

*α*is very small. If we assume

*α*≈ 0, then (10) reduces to (2). There is a singularity [1/

*μ*(

*z*) → ∞] in (2) at

*μ*

_{0}= 0 and

*z*= 0; while (10) is stable and with clearer physical meaning. Generally the difference between (2) and (10) is small. If the model layer Δ

*z*is less than 2 km, the differences between (2) and (10) is generally less than 2%.

*λ*,However, (11) is still not proper to apply to the radiative transfer process directly. In climate models the radiative transfer is considered individually in each grid with no horizontal connection between two neighboring grids. In Fig. 4b the dot–dash lines shows a considered column (or a grid). Equation (11) only represents the variation of effective pathlength along with a special solar beam toward the surface inside the considered column. However, the effective pathlength factor at height

*z*is different for different solar direct beams passing it. In the PP case there is no need to distinguish the parallel direct solar beams. However, with atmospheric curvature, each solar beam is different in experiencing the curvature effect.

*N*as the surface. Layer

*k*is between level

*k*and level

*k*+ 1;

*z*is the height from ground to level

_{i}*i*. The same definitions are used in the appendix. At each level

*k*, the direct transmission is

*e*

^{−τ1,k/μ0}, where

*τ*

_{1,}

*is the optical depth from TOA to level*

_{k}*k*. As shown in Fig. 4b, the parallel solar beams reaching different levels in the considered column must experience different curvature variations. Therefore the transmission to level

*k*(

*k*= 2, . . . ,

*N*) should bewhere

*κ*is the absorption coefficient for layer

_{i}*i*, and the local pathlength factor following (11) iswhere

*α*

_{1}= (

*z*−

_{i}*z*

_{i}_{+1})/(

*R*+

*z*

_{i}_{+1}),

*α*

_{2}= [(

*R*+

*z*)

_{k}*n*]/[(

_{λk}*R*+

*z*

_{i}_{+1})

*n*

_{λi}_{+1}],

*n*is the refractive index for layer

_{λ i}*i*. In a radiation algorithm, usually the solar spectrum is split into bands. The pathlength factor for each band is therefore calculated separately. Thus

*λ*

_{1}and

*λ*

_{2}in (11) are the lower and upper bounds of wavelength for each band. As shown in Fig. 4b, (12) represents the local pathlength factor of layer

*i*for a beam that reaches level

*k*in the column.

*e*

^{−τk/μ0}for layer

*k*, this is replaced byfor the sake of energy conservation.

The method of (12) can be directly applied to climate models. Although the calculations become much more time consuming, but the solar heating rate at large solar zenith angles could be simulated more accurately. In addition (12) serves as the benchmark to the several proposed parameterizations.

## 2. Results in a one-dimensional radiation model

To quantitatively characterize the effective pathlength impact, we used a one-dimensional radiative transfer model to study the impact of effective pathlength on solar flux and heating rate in the atmosphere. The radiation model we used is the radiation algorithm based on Li and Barker (2005). This algorithm employs the correlated-*k* distribution method for gaseous transmission. The radiative transfer method is based on the doubling and adding scheme, in which the local solar beam pathlength is accounted for in each model layer (see the appendix in Li et al. 2005). The radiative transfer model was assessed using a standard atmospheric profile. The mixing ratio of CO_{2} is 350 ppm. A surface albedo of 0.2 is assumed.

The top panel of Fig. 5 shows the results of the downward flux at the surface and upward flux at TOA for the PP case. To show the impact of effective pathlength on the radiative transfer, the differences in the downward flux at surface and the upward flux at TOA calculated via Kasten, Rodgers and the new parameterizations against the original unmodified scheme of PP are shown (middle and lower panels). In addition the rigorous method of (12) is used to obtain the benchmark result. It is found that in general the difference in downward flux between (12) and PP is very small for a small solar zenith angle (*μ*_{0} ≈ 1). As the solar zenith angle increases, such difference increases. With consideration of the effects of atmospheric curvature and refraction, the solar pathlength to the surface is reduced. Therefore, it is easier for the solar beam to reach the ground. This yields a larger downward flux at the surface.

As shown in Fig. 3, the new parameterization of (9) generally produces smaller 1/*μ _{e}* (i.e., shorter pathlength) in comparison with the parameterizations of Kasten and Rodgers. Therefore it a larger downward flux by (9) is expected in comparison with the results by the other two parameterizations. In Fig. 5 it is found that the difference in the downward flux at surface between (9) and Rodgers’ parameterizations could be more then 1 W m

^{−2}, and the result of Kasten’s is usually in between them.

The result of the rigorous method (12) is very close to the result of (9). This indicates that the new parameterization catches the physics of the effective solar pathlength more accurately than the previous parameterizations.

As *μ*_{0} → 1 there should be neither the atmospheric curvature effect nor the refractive effect. The results by any parameterization should be the same as the result of PP. However, this is not true for Kasten’s parameterization as shown in Fig. 5. By (7) when 1/*μ*_{0} = 1, 1/*μ _{e}* ≠ 1. This is a shortcoming of Kasten’s parameterization.

The influence of the effective solar pathlength is much smaller for the upward flux at TOA, since the upward flux is mostly determined by multiple scattering. The reduction of pathlength makes it easier for the scattered photons to escape to outer space, but it also enhances the downward flux. These two factors partly cancel each other and result in the less sensitivity in upward flux to the schemes used. Again the bottom panel of Fig. 5 shows that the result of (9) is mostly close to the result of (12).

Figure 6 shows the corresponding results of the heating rate. The upper left panel shows the heating rate by PP. The strong heating rate around the stratopause is mostly due to O_{3} and weak heating in the middle troposphere is due to CO_{2}, O_{2,} and H_{2}O. The rest panels show the difference in heating rate by other schemes against PP. It is found that the difference in the heating rate between (12) and PP is generally small with a maximum absolute difference less than 0.5 K day^{−1}. Also, most of the differences occur above 100 mb. As discussed above, the reduction of solar effective pathlength causes less solar energy to be absorbed in each model layer. Therefore the solar heating rate decreases.

In Fig. 6 the results of Rodgers parameterization and (9) are also shown. The result of Kasten is not presented since it is usually in between the results of the other two parameterizations. Above 1 mb, the result of Rodgers parameterization is slightly closer to that of (12) compared to the result of (9). However, below 1 mb, the result of (9) becomes much closer to that of (12) compared to Rodgers parameterization. Therefore (9) is more accurate in heating rate. This is consistent with the results in flux.

Generally the results of the heating rate by (9) do not match with the rigorous calculations as well as the results of flux shown in Fig. 5. In some extent the flux at surface or TOA is a vertical integrated result. At the point the vertical mass mean effective pathlength factor [see discussion of (4) and (5)] could lead to the accurate results. As for the heating rate distribution in the atmosphere the local pathlength has to be properly accounted for at every layer along a solar direct beam. Thus the mean effective pathlength factor scheme could not represent the physics very well, which results in relatively poorer results compared to the case of flux. Here we only present the results for the standard atmosphere, since it is found that the results for other atmospheric profiles are similar.

## 3. Conclusions

In this note, we have systematically investigated the effective solar pathlength problem and its impact on radiative transfer. The effective solar pathlength is determined by the atmospheric curvature effect and refraction effect. Generally the curvature effect is more important in the upper atmosphere and the refraction effect is more important in the lower atmosphere. In this note the vertical mass mean effective pathlength factor, 1/*μ _{e}*, has been evaluated accurately. Since the result of 1/

*μ*is not sensitive to the chosen of the atmospheric profile, it is relatively easy to propose a new general parameterization of effective pathlength factor for radiative transfer in climate models.

_{e}It is found that the parameterizations proposed about 40 yr ago by Kasten and Rodgers underestimate the effective solar pathlength effect (overestimate the effective pathlength factor). In the real inhomogeneous atmosphere the curvature effect is localized with height dependence as in (3). A more mathematically rigorous formula for the local effective zenith angle has been derived as (11). However (11) could not be applied to climate models directly, since the solar radiation attenuation along the beam shown in Fig. 4a could not represent the direct solar beams reaching different levels in the considered grid (see Fig. 4b).

We proposed a rigorous method of (12), which accounts for the pathlength variation for each individual direct solar beams. By (12), the pathlength along a solar beam is properly evaluated, and the variation of gaseous amount along such beam can also be accurately assessed. This is particularly important for ozone with a highly inhomogeneous distribution. This effect has been fully discussed in this note and to our knowledge no attention has been paid to it in any one-dimensional radiative transfer processes in climate modeling. In addition (12) can be used as the benchmark to the proposed parameterizations.

It is found that effective solar pathlength has little impact on flux and heating rate, since the thickness of the atmosphere is very small in comparison with the radius of the earth. Among the three proposed parameterizations, the results of the new one are generally closest to the rigorous results of (12).

The authors thank Drs. C. Curry, V. Fomichev, K. v. Salzen, L. Solheim, and two anonymous reviewers for their helpful comments. The work is partly supported by Aeolian Dust Experiment on Climate Impact ADEC project sponsored by the Ministry of Education, Culture, Sports, Science, and Technology of Japan.

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# APPENDIX

## Refraction of Solar Direct Beam

*N*is the ground,

*z*is the height from ground to level

_{i}*i*, layer

*i*is between levels

*i*–1 and

*i*. The incoming solar beam with zenith angle

*θ*

_{0}and the local zenith angle

*θ*(

*z*

_{1}) at level 1 (see Fig. 1A) satisfieswhere

*R*is the radius of the earth. After passing into layer 1, the refraction causes the solar beam deviating to angle of

*θ*′(

*z*

_{1}) and by Snell’s law,where

*n*

_{λ}_{0}and

*n*(

_{λi}*i*= 1, 2, . . .) are the refractive indices above TOA and of layer

*i*for wavelength

*λ*, respectively. ThusNow this refracted beam is supposed to reach the ground with surface observation angle

*θ*

_{1}(see Fig. A1), and by simple geometrythenThe surface locations of

*θ*

_{0}and

*θ*

_{1}are different. The distance between the two points isGenerally at level

*i*,where

*θ*(

*z*) is the local zenith angle for level

_{i}*i*,

*θ*′(

*z*) is the refraction angle corresponding to

_{i}*θ*(

*z*), and

_{i}*θ*is the observation angle to the solar beam at level

_{i}*i*;

*θ*(

*z*),

_{i}*θ*′(

*z*), and

_{i}*θ*(

*z*) are all wavelength dependent, for simplicity, the subscript of

_{i}*λ*is not written out. The distance between

*θ*and

_{i}*θ*

_{i}_{–1}isFinally for the solar beam arriving at the ground (level

*N*) with the true observation angle

*θ*

_{N}_{–1}, the total deviation distance to the original solar beam isFigure A2 shows the deviation distance versus 1/

*μ*

_{0}for a solar beam with wavelength 0.633

*μ*m, the maximum is about 108.8 km, for

*μ*

_{0}→ 0. It takes about 3.8 min to pass through such a distance by the rotation of the earth. The wavelength dependence of

*D*is very small.

*n*(

_{λ}*z*) is the refractive index of height

*z*for wavelength

*λ*.

*n*is the refractive index near the ground, and

_{λg}*θ*

_{obs}is the true observation angle, that is,

*θ*

_{obs}=

*θ*

_{N}_{−1}. Thus (A4) becomesThe derivation of (A5) could also be found in Kondratyev (1969) with a different approach. The study of Kasten (1966) on effective solar pathlength is based on (A5), which emphasizes the location of the ground observation angle.

For parallel incoming solar beams as shown in Fig. 4b, the locations of the observation angle inside a grid cell of a climate model are different for different beams. They are unknown for the radiative transfer process. In addition, though there is the deviation for solar beam reaching the ground, the deviation distance is usually small since the earth’s radius is much larger than the atmospheric thickness. From Fig. A2, even at 1/*μ*_{0} = 100 (*θ*_{0} ≈ 89.4°) the deviation distance is about 60 km, which generally is less than the length of a GCM grid cell.

The values of local pathlength factor 1/*μ _{λ}*(

*z*) (

*z*= 0) at

*λ*= 0.633

*μ*m and the effective pathlength factor 1/

*μ*for different atmospheric profiles at 1/

_{e}*μ*

_{0}= 10 000.