• Birth, K. P., , and M. J. Downs, 1993: An updated Ellen equation for the refractive index of air. Metrologia, 30 , 155162.

  • Birth, K. P., , and M. J. Downs, 1994: Correction to the updated Ellen equation for the refractive index of air. Metrologia, 31 , 155162.

    • Search Google Scholar
    • Export Citation
  • Ciddor, P. E., 1996: Refractive index of air: New equations for the visible and near infrared. Appl. Opt., 35 , 15661574.

  • Iqbal, M., 1983: An Introduction to Solar Radiation. Academic Press, 390 pp.

  • Kasten, K., 1966: A new table and approximate formula for relative optical air mass. Arch. Meteor. Geophys. Bioklimatol. Ser. B, 14 , 206233.

    • Search Google Scholar
    • Export Citation
  • Kondratyev, K. Y., 1969: Radiation in the Atmosphere. Academic Press, 912 pp.

  • Li, J., , and H. Barker, 2005: A radiation algorithm with correlated- k distribution. Part I: Local thermal equilibrium. J. Atmos. Sci., 62 , 286309.

    • Search Google Scholar
    • Export Citation
  • Li, J., , J. S. Dobbie, , P. Räisänen, , and Q. Min, 2005: Accounting for unresolved cloud in solar radiation. Quart. J. Roy. Meteor. Soc., 131 , 16071629.

    • Search Google Scholar
    • Export Citation
  • McClatchey, R. A., , R. W. Fenn, , J. E. A. Selby, , F. E. Volz, , and J. S. Garing, 1972: Optical properties of the atmosphere. 3d ed., Tech. Rep. AFCRL-72-0497, 108 pp.

  • Owens, J. C., 1967: Optical refractive index of air: Dependence on pressure, temperature and composition. Appl. Opt., 6 , 1. 5159.

  • Robinson, N., 1966: Solar Radiation. Elsevier, 347 pp.

  • Rodgers, C. D., 1967: The radiative heat budget of the troposphere and lower stratosphere. Planetary Circulation Project, Department of Meteorology Rep. A2, Massachusetts Institute of Technology, 126 pp.

  • View in gallery

    (a) A sketch showing the geometry of the effective solar pathlength; R is the radius of the earth, H is the mean height of the atmosphere, θ0 is the solar zenith angle, l is the pathlength for plane parallel case, and l′ is the real pathlength by considering the atmospheric curvature. (b) The same as (a), but accounting for the local solar zenith angle θ(z) at height z.

  • View in gallery

    The local effective pathlength factor, 1/μλ(z), vs vertical height for three values of 1/μ0. The results are based on (3) for inclusion of refraction (λ = 0.633μm) and exclusion of refraction.

  • View in gallery

    The mean effective pathlength factors, 1/μe, vs 1/μ0 by Eq. (6) (with and without refraction) and by Kasten, Rodgers, and the new parameterizations.

  • View in gallery

    (a) A sketch showing the geometry of the effective solar pathlength; R is the radius of the earth, θ0 is the solar zenith angle, z is the height of the considered layer, Δz is the layer thickness, and Δz′ is actual pathlength through layer Δz. (b) The same as (a), but accounting for each individual effective pathlength toward two different levels within the considered region noted by the dot–dashed lines.

  • View in gallery

    (top) The results of the upward flux at TOA and the downward flux at the surface. The differences in flux by using schemes of Eq. (12) and (middle) the parameterizations of Kasten, Rodgers, and the new against PP (the results of upward flux at TOA), and (bottom) the results of downward flux at the surface. Calculations are based on standard atmospheric profile.

  • View in gallery

    (top left) The contour of the heating rate by PP; the rest of panels show the differences in heating rate by Eq. (12) and the parameterizations of Rodgers and the new against the PP. Calculations are based on the standard atmospheric profile.

  • View in gallery

    Fig. A1. A sketch showing the refraction effect for the solar beam; R is the radius of the earth, θ0 is the solar zenith angle, zi (i = 1, 2, . . .) is the height for model level i, ni (i = 1, 2, . . .) is refractive index at layer i, θ(zi) (i = 1, 2, . . .) is the local zenith angle for level i, θ′(zi) (i = 1, 2, . . .) is and refracted local zenith angle corresponding to θ(zi), and θi (i = 1, 2, . . .) is the observation angle at the surface toward the solar beam reached level i.

  • View in gallery

    Fig. A2. The deviation distance for the solar beam arriving at ground due to refraction; the solar wavelength is 0.633 μm.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 69 69 15
PDF Downloads 46 46 12

On the Effective Solar Pathlength

View More View Less
  • 1 Canadian Centre for Climate Modelling and Analysis, Meteorological Service of Canada, Victoria, British Columbia, Canada
  • 2 Meteorological Research Institute, Tsukuba-Shi, Japan
© Get Permissions
Full access

Abstract

The effects of atmospheric spherical curvature and refraction and their impact on radiative transfer have been studied. It is shown that formulas employed in GCMs for atmospheric curvature and refraction underestimate the effect of effective solar pathlength. A new parameterization is therefore proposed. It is emphasized that the atmospheric curvature effect on radiative transfer is a localized problem with height dependence. A method corresponding to the local effective pathlength factor is proposed. This rigorous scheme enables variations in both the pathlength and the gaseous amount along a solar direct beam to be accurately evaluated in the radiative transfer process. The results of the rigorous scheme can be used as the benchmark to the proposed parameterizations for the effective pathlength factor. It is found that the new parameterization proposed in this note has better results in flux and heating rates when compared to other parameterizations.

Corresponding author address: Dr. Jiangnan Li, Canadian Centre for Climate Modelling and Analysis, Meteorological Service of Canada, P.O. Box 1700, University of Victoria, Victoria, BC V8W 2Y2, Canada. Email: Jiangnan.Li@ec.gc.ca

Abstract

The effects of atmospheric spherical curvature and refraction and their impact on radiative transfer have been studied. It is shown that formulas employed in GCMs for atmospheric curvature and refraction underestimate the effect of effective solar pathlength. A new parameterization is therefore proposed. It is emphasized that the atmospheric curvature effect on radiative transfer is a localized problem with height dependence. A method corresponding to the local effective pathlength factor is proposed. This rigorous scheme enables variations in both the pathlength and the gaseous amount along a solar direct beam to be accurately evaluated in the radiative transfer process. The results of the rigorous scheme can be used as the benchmark to the proposed parameterizations for the effective pathlength factor. It is found that the new parameterization proposed in this note has better results in flux and heating rates when compared to other parameterizations.

Corresponding author address: Dr. Jiangnan Li, Canadian Centre for Climate Modelling and Analysis, Meteorological Service of Canada, P.O. Box 1700, University of Victoria, Victoria, BC V8W 2Y2, Canada. Email: Jiangnan.Li@ec.gc.ca

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/μ0, where H is the height of the atmosphere and μ0 = cos θ0, θ0 is the solar zenith angle.

Because of the spherical curvature of the atmosphere, when θ0 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 θ0, 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 θ0 nears 90°.

Since the problem is related to the curvature of the earth and the atmosphere, the effective pathlength should depend on the radius of the earth and considered thickness of the atmosphere. By defining 1/μe = l′/H, the following equation can be derived (Robinson 1966),
i1520-0469-63-4-1365-e1
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 = Pg/g, where Pg and ρg are the pressure and density at the surface, respectively, and 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 H2O or O3. 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.

Let the local solar zenith angle at height z be θ(z) (see Fig. 1b). By simple geometry,
i1520-0469-63-4-1365-eq1
thus,
i1520-0469-63-4-1365-e2
where μ(z) = cosθ(z). We call 1/μ(z) the local pathlength factor.
Further the refraction of the solar beam has to be considered. In the appendix, the local pathlength factor with consideration of refraction for a solar wavelength λ is found,
i1520-0469-63-4-1365-e3
where nλ(z) is the refractive index at height z for wavelength λ, and nλ0 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/μ0. Both the inclusion and exclusion of refraction is considered. It is found that the local pathlength factor decreases with height. For a small 1/μ0, the change of 1/μλ(z) with height is small. However for a large 1/μ0, the change of 1/μλ(z) with height could be dramatically large. With the inclusion of refraction 1/μλ(z) becomes smaller. It is found that the refraction effect is important only for large values of 1/μ0. Even for 1/μ0 = 10 (θ0 ≈ 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 μm, and the refractive index follows Ciddor (1996), which depends on pressure, temperature, water partial pressure, and CO2 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/μλ(z) by (3) are shown corresponding to 1/μ0 = 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/μ0 < 10 000 and z > 0, the differences in 1/μλ(z) for different atmospheric profiles are even smaller.

The transmission of solar direct beam from TOA to surface is
i1520-0469-63-4-1365-e4
where κλ(z) is the extinction coefficient of height z for wavelength λ; λ1 and λ2 are the lower and upper bounds of wavelength for the considered solar spectral range, Sλ is the incoming solar flux at wavelength λ and δS = ∫λ2λ1dSλ.
Generally (4) is simplified by replacing the local pathlength factor with a spectral mean and vertical mass mean effective pathlength factor,
i1520-0469-63-4-1365-e5
where 1/μe is the spectral mean and vertical mass weighted mean effective pathlength factor, from (3)
i1520-0469-63-4-1365-e6
where ρ(z) is the air density at height z, ρg is the air density at surface, H is the effective height defined as ρgH = ∫0 ρ dz.
The integral of (6) could not be evaluated easily since the refractive index varies with solar spectral wavelength. There are two popular parameterizations to (6). One is by Kasten (1966) (also see Iqbal 1983),
i1520-0469-63-4-1365-e7
and another is by Rodgers (1967),
i1520-0469-63-4-1365-e8
Today in most general circulation models (GCMs), the effective solar pathlength factor follows Rodgers’ formula.

In Fig. 3, 1/μe by (6) is plotted against 1/μ0 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 μm. 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/μ0 > 10. It is found that the results of Kasten’s formula and (6) without refraction are very similar. Small difference appears when 1/μ0 > 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 μm was considered.

It is difficult to understand the result of Rodgers’ parameterization. In the range of 10 < 1/μ0 < 120, 1/μe 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/μe at 1/μ0 = 10 000 are presented for four different atmospheric profiles. Again the differences for different atmospheric profiles are very small.

Based on the result of (6) with refraction, we propose a new parameterization
i1520-0469-63-4-1365-e9
This simple formula perfectly matches with (6) as shown in Fig. 3.

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/μλ(z), is required.

In Fig. 4a, we consider a slab layer with thickness Δz located at height z above the ground. We draw two concentric circles representing the edges of Δz. 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 Δz. Transmission for the direct solar beam through layer Δz is ekΔz′, where k is the extinction coefficient and Δz′ is the actual pathlength along solar beam for the curved layer Δz (see Fig. 4a). In radiative transfer calculations the transmission depends on Δz as ekΔz/μ(z). Comparing it with ekΔz′, we conclude that 1/μ(z) should be Δz′/Δz. With this definition, we can derive from simple geometry,
i1520-0469-63-4-1365-e10
where α = Δz/(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 μ0 = 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 Δz is less than 2 km, the differences between (2) and (10) is generally less than 2%.
With consideration of refraction, similar to (3) we have for wavelength λ,
i1520-0469-63-4-1365-e11
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.
As shown in the appendix of Li et al. (2005), the direct solar transmissions to each model level are required. This corresponds to the parallel solar beams as shown in Fig. 4b. To describe problem clearly, in the model we set level 1 as TOA and level N as the surface. Layer k is between level k and level k + 1; zi is the height from ground to level i. The same definitions are used in the appendix. At each level k, the direct transmission is eτ1,k/μ0, where τ1,k is the optical depth from TOA to level 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 be
i1520-0469-63-4-1365-eq2
where κi is the absorption coefficient for layer i, and the local pathlength factor following (11) is
i1520-0469-63-4-1365-e12
where α1 = (zizi+1)/(R + zi+1), α2 = [(R + zk)nλk]/[(R + zi+1)nλi+1], nλ i is the refractive index for layer i. In a radiation algorithm, usually the solar spectrum is split into bands. The pathlength factor for each band is therefore calculated separately. Thus λ1 and λ2 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.
In the adding process (Li et al. 2005) another variable related to the solar beam is the direct transmission through each individual layer; like eτk/μ0 for layer k, this is replaced by
i1520-0469-63-4-1365-eq3
for 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 CO2 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 (μ0 ≈ 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/μe (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 μ0 → 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/μ0 = 1, 1/μe ≠ 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 O3 and weak heating in the middle troposphere is due to CO2, O2, and H2O. 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/μe, has been evaluated accurately. Since the result of 1/μe 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.

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).

Acknowledgments

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.

REFERENCES

  • Birth, K. P., , and M. J. Downs, 1993: An updated Ellen equation for the refractive index of air. Metrologia, 30 , 155162.

  • Birth, K. P., , and M. J. Downs, 1994: Correction to the updated Ellen equation for the refractive index of air. Metrologia, 31 , 155162.

    • Search Google Scholar
    • Export Citation
  • Ciddor, P. E., 1996: Refractive index of air: New equations for the visible and near infrared. Appl. Opt., 35 , 15661574.

  • Iqbal, M., 1983: An Introduction to Solar Radiation. Academic Press, 390 pp.

  • Kasten, K., 1966: A new table and approximate formula for relative optical air mass. Arch. Meteor. Geophys. Bioklimatol. Ser. B, 14 , 206233.

    • Search Google Scholar
    • Export Citation
  • Kondratyev, K. Y., 1969: Radiation in the Atmosphere. Academic Press, 912 pp.

  • Li, J., , and H. Barker, 2005: A radiation algorithm with correlated- k distribution. Part I: Local thermal equilibrium. J. Atmos. Sci., 62 , 286309.

    • Search Google Scholar
    • Export Citation
  • Li, J., , J. S. Dobbie, , P. Räisänen, , and Q. Min, 2005: Accounting for unresolved cloud in solar radiation. Quart. J. Roy. Meteor. Soc., 131 , 16071629.

    • Search Google Scholar
    • Export Citation
  • McClatchey, R. A., , R. W. Fenn, , J. E. A. Selby, , F. E. Volz, , and J. S. Garing, 1972: Optical properties of the atmosphere. 3d ed., Tech. Rep. AFCRL-72-0497, 108 pp.

  • Owens, J. C., 1967: Optical refractive index of air: Dependence on pressure, temperature and composition. Appl. Opt., 6 , 1. 5159.

  • Robinson, N., 1966: Solar Radiation. Elsevier, 347 pp.

  • Rodgers, C. D., 1967: The radiative heat budget of the troposphere and lower stratosphere. Planetary Circulation Project, Department of Meteorology Rep. A2, Massachusetts Institute of Technology, 126 pp.

APPENDIX

Refraction of Solar Direct Beam

In Fig. A1, the atmosphere is split into multiple levels. Level 1 is TOA and level N is the ground, zi is the height from ground to level i, layer i is between levels i–1 and i. The incoming solar beam with zenith angle θ0 and the local zenith angle θ(z1) at level 1 (see Fig. 1A) satisfies
i1520-0469-63-4-1365-eqa1
where R is the radius of the earth. After passing into layer 1, the refraction causes the solar beam deviating to angle of θ′(z1) and by Snell’s law,
i1520-0469-63-4-1365-eqa2
where nλ0 and nλi (i = 1, 2, . . .) are the refractive indices above TOA and of layer i for wavelength λ, respectively. Thus
i1520-0469-63-4-1365-eqa3
Now this refracted beam is supposed to reach the ground with surface observation angle θ1 (see Fig. A1), and by simple geometry
i1520-0469-63-4-1365-eqa4
then
i1520-0469-63-4-1365-eqa5
The surface locations of θ0 and θ1 are different. The distance between the two points is
i1520-0469-63-4-1365-eqa6
Generally at level i,
i1520-0469-63-4-1365-ea1
i1520-0469-63-4-1365-ea2
i1520-0469-63-4-1365-ea3
where θ(zi) is the local zenith angle for level i, θ′(zi) is the refraction angle corresponding to θ(zi), and θi is the observation angle to the solar beam at level i; θ(zi), θ′(zi), and θ(zi) are all wavelength dependent, for simplicity, the subscript of λ is not written out. The distance between θi and θi–1 is
i1520-0469-63-4-1365-eqa7
Finally for the solar beam arriving at the ground (level N) with the true observation angle θN–1, the total deviation distance to the original solar beam is
i1520-0469-63-4-1365-eqa8
Figure A2 shows the deviation distance versus 1/μ0 for a solar beam with wavelength 0.633 μm, the maximum is about 108.8 km, for μ0 → 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.
Written in continuous form, (A1) becomes
i1520-0469-63-4-1365-ea4
where nλ(z) is the refractive index of height z for wavelength λ.
From (A3)
i1520-0469-63-4-1365-eqa9
where nλg is the refractive index near the ground, and θobs is the true observation angle, that is, θobs = θN−1. Thus (A4) becomes
i1520-0469-63-4-1365-ea5
The 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/μ0 = 100 (θ0 ≈ 89.4°) the deviation distance is about 60 km, which generally is less than the length of a GCM grid cell.

Fig. 1.
Fig. 1.

(a) A sketch showing the geometry of the effective solar pathlength; R is the radius of the earth, H is the mean height of the atmosphere, θ0 is the solar zenith angle, l is the pathlength for plane parallel case, and l′ is the real pathlength by considering the atmospheric curvature. (b) The same as (a), but accounting for the local solar zenith angle θ(z) at height z.

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3682.1

Fig. 2.
Fig. 2.

The local effective pathlength factor, 1/μλ(z), vs vertical height for three values of 1/μ0. The results are based on (3) for inclusion of refraction (λ = 0.633μm) and exclusion of refraction.

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3682.1

Fig. 3.
Fig. 3.

The mean effective pathlength factors, 1/μe, vs 1/μ0 by Eq. (6) (with and without refraction) and by Kasten, Rodgers, and the new parameterizations.

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3682.1

Fig. 4.
Fig. 4.

(a) A sketch showing the geometry of the effective solar pathlength; R is the radius of the earth, θ0 is the solar zenith angle, z is the height of the considered layer, Δz is the layer thickness, and Δz′ is actual pathlength through layer Δz. (b) The same as (a), but accounting for each individual effective pathlength toward two different levels within the considered region noted by the dot–dashed lines.

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3682.1

Fig. 5.
Fig. 5.

(top) The results of the upward flux at TOA and the downward flux at the surface. The differences in flux by using schemes of Eq. (12) and (middle) the parameterizations of Kasten, Rodgers, and the new against PP (the results of upward flux at TOA), and (bottom) the results of downward flux at the surface. Calculations are based on standard atmospheric profile.

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3682.1

Fig. 6.
Fig. 6.

(top left) The contour of the heating rate by PP; the rest of panels show the differences in heating rate by Eq. (12) and the parameterizations of Rodgers and the new against the PP. Calculations are based on the standard atmospheric profile.

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3682.1

i1520-0469-63-4-1365-fa01

Fig. A1. A sketch showing the refraction effect for the solar beam; R is the radius of the earth, θ0 is the solar zenith angle, zi (i = 1, 2, . . .) is the height for model level i, ni (i = 1, 2, . . .) is refractive index at layer i, θ(zi) (i = 1, 2, . . .) is the local zenith angle for level i, θ′(zi) (i = 1, 2, . . .) is and refracted local zenith angle corresponding to θ(zi), and θi (i = 1, 2, . . .) is the observation angle at the surface toward the solar beam reached level i.

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3682.1

i1520-0469-63-4-1365-fa02

Fig. A2. The deviation distance for the solar beam arriving at ground due to refraction; the solar wavelength is 0.633 μm.

Citation: Journal of the Atmospheric Sciences 63, 4; 10.1175/JAS3682.1

Table 1.

The values of local pathlength factor 1/μλ(z) (z = 0) at λ = 0.633 μm and the effective pathlength factor 1/μe for different atmospheric profiles at 1/μ0 = 10 000.

Table 1.
Save