Article Type:
Article Commentary

Comments on “On the Choice of Average Solar Zenith Angle”

Jiangnan Li Canadian Centre for Climate Modelling and Analysis, Environment and Climate Change Canada, University of Victoria, Victoria, British Columbia, Canada

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Print Publication:
01 May 2017
DOI:
https://doi.org/10.1175/JAS-D-16-0185.1
Page(s):
1669–1676
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Abstract

The daytime-mean solar zenith angle (SZA) and the solar insolation–weighted-mean SZA are discussed from a global scale and from a latitude-dependent local-scale perspective. It is found that the choosing of daytime-mean SZA or insolation-weighted-mean SZA depends on whether the averaging process is zero-moment or single-moment weighted. It is a misleading to state that the solar insolation–weighted-mean SZA is more accurate than the daytime-mean SZA when averaging a radiation variable, as claimed by Cronin.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Dr. Jiangnan Li, jiangnan.li@canada.ca

The original article that was the subject of this comment/reply can be found at http://journals.ametsoc.org/doi/abs/10.1175/JAS-D-13-0392.1.

Abstract

The daytime-mean solar zenith angle (SZA) and the solar insolation–weighted-mean SZA are discussed from a global scale and from a latitude-dependent local-scale perspective. It is found that the choosing of daytime-mean SZA or insolation-weighted-mean SZA depends on whether the averaging process is zero-moment or single-moment weighted. It is a misleading to state that the solar insolation–weighted-mean SZA is more accurate than the daytime-mean SZA when averaging a radiation variable, as claimed by Cronin.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author e-mail: Dr. Jiangnan Li, jiangnan.li@canada.ca

The original article that was the subject of this comment/reply can be found at http://journals.ametsoc.org/doi/abs/10.1175/JAS-D-13-0392.1.

Keywords: Radiative fluxes; Climate models; Parameterization
The globally averaged solar zenith angle (SZA) is an important physical quantity for obtaining the averaged radiation variables. In climate models, most of the physical parameterizations are applied to the global scale; thus, the globally averaged physical quantities are generally required. For example, in the cloud optical property parameterization, the cloud optical properties are first calculated at each wavelength, then the results are weighted together by the downward solar flux at a globally and temporarily averaged SZA (Dobbie et al. 1999; Yang et al. 2015). The averaged SZA characterizes the spatiotemporally averaged solar energy absorbed in the atmosphere. In simplified climate models (North et al. 1981; Ballinger et al. 2015), the annually averaged meridional distributions of SZA need to be calculated accurately and simply parameterized for applications. However there lacks a systematical discussion on averaged SZA—Cronin (2014) is one of a few studies addressed this issue. According to Cronin (2014), there are two popular methods to calculate the averaged SZA. A commonly used method (Manabe and Strickler 1964; Ramanathan and Coakley 1978) is
e1
where , with θ being SZA, and is the probability distribution.
Another method is to consider the weight of solar insolation (Hartmann 1994; Romps 2011), which is
e2
where is the total solar irradiance. For planetary averages, (Cronin 2014), thus and ⅔, and the corresponding solar zenith angles are and , which are called the daytime-mean SZA and insolation-weighted-mean SZA, respectively.

To minimize bias in solar absorption, Cronin (2014) points out that SZA should be chosen to most closely match the spatial- or time-mean planetary albedo, and he found “the absorption-weighted zenith angle is usually between the daytime-weighted and insolation-weighted zenith angles but much closer to the insolation-weighted zenith angle” (p. 2994).

Should the averaged SZA be determined by minimizing the bias in planetary albedo? And is the insolation-weighted-mean SZA more accurate than the daytime-mean SZA? We do not agree with either point as explained in detail below.

We denote the solar upward flux (or reflected flux) at the top of the atmosphere (TOA), downward flux (or transmitted flux) at surface and atmospheric absorption as , , and , where and are the upward and downward broad band solar fluxes at height z and is the height of TOA.

In the solar radiative transfer equation (Li and Ramaswamy 1996; Liou 2002; Zhang et al. 2014), the solar insolation has been included in the source term of the equation. There is no physical reason to weight the solar insolation again in averaging a radiation variable. Thus, the daytime-averaged solar upward flux is
e3
The same is for and .

Figure 1a shows the relative errors by using and . The radiation model (Li and Barker 2005) is used, which is a correlated-k distribution scheme for gaseous transmission with O3, H2O, O2, CH4, and CO2 included in the solar spectrum range. The surface albedo varies from 0.1 to 0.6, covering most of the surface albedos of Earth. It is found that the relative error of is limited to 3%, but the relative error of is about 30%. The results are similar for and .

Fig. 1.
Fig. 1.

Relative errors versus surface albedo. The red solid and dashed lines are the relative errors of and , respectively, the green solid and dashed lines are the relative errors of and , and the blue solid and dashed lines are the relative errors of and . (a) The benchmark results are based on (3); (b) the benchmark results are based on (5).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

Let us consider the problem from a mathematical point of view. Gaussian quadrature is an effective way to evaluate an integral like (3). By n-node Gaussian quadrature, the integration of moment l is evaluated by
e4
where is a real function of x, is the abscissa, and is the weight. The values of abscissa and weight for different nodes and moments are listed in Abramowitz and Stegun (1965). Considering the single node (n = 1), for the zero moment (l = 0), , and and for the single moment (l = 1), ⅔, and . These two values of are equal to and . Gaussian quadrature tells us that the choice of or ⅔ depends on the moment of the integral function. If a physical variable is directly averaged as (3) (zero-moment weight), should be used; if a physical variable is averaged by weighting of solar insolation (single-moment weight), as
e5
should be used.

In Fig. 1b the benchmark results are calculated based on (5). The relative error of is up to 25%, but is limited to 1%. This is just opposite to that of Fig. 1a. Therefore the choice of or depends on whether the averaging process is (3) or (5). For radiative fluxes, (3) should be chosen because the solar insolation has been built in the radiative transfer calculations [i.e., weighted inside ] and . However, if in a physical process, the solar insolation is weighted to as (5), .

Cronin (2014) has proposed an effective solar constant. According to Cronin, for , , and for , , where is the solar constant. The purpose of the effective solar constant is to make . It is well known that the value of ¼ results in the incoming solar energy being evenly distributed over Earth’s surface. This is nothing to do with the averaged SZA, because SZA should not apply to the region without sunlight.

Figure 2 is the same as Fig. 1, but we replace with . It is found that the relative errors are much enhanced compared to Fig. 1. In Fig. 2, is not applied to the benchmark calculations of (3) or (5). If is applied to the benchmark calculations, Fig. 2 becomes the same as Fig. 1, because the effect of the effective solar constant will be canceled out. Dr. Cronin pointed to me (T. W. Cronin 2016, personal communication) that should be used in the benchmark calculations for both of and . Then it is difficult to understand why the solar constant should be different in the benchmark and approximation calculations related to . The concept of an effective solar constant does not help solve the problem but only causes confusion.

Fig. 2.
Fig. 2.

As in Fig. 1, but using the effective solar constant.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

Cronin (2014) has proposed a concept of absorption-weighted SZA, , which is from the evaluation of planetary albedo. All studies in Cronin (2014) are based on by comparing it with and . The standard definition of planetary albedo is , and the planetary albedo shown in (12) of Cronin (2014) is based on it.

To address the output solar energy at TOA, the planetary albedo should be averaged by weighting of solar insolation (Cronin 2014):
e6
Therefore, though the integral is weighted by solar insolation, has to be used, because the solar insolation weight is canceled in the integral.

Cronin (2014) shows that is more accurate than . However, the accuracy of stems from a completely wrong reason. Figure 1a shows that the relative error of is around 30%, and also the relative error of (refer to ) is 33.333%—these two positive biases largely canceled out in the compound variable of . For example, in the standard U.S. profile at a surface albedo of 0.2, the relative error of is the same as (=3.862%). The relative error of is 28.606%, and the relative error of is 33.333%, which makes the relative error of equal to −3.546%; thus, the difference between and becomes small.

The concept of is incorrect; as shown above, the compound variable of planetary albedo does not provide any truthful information for SZA. The averaged SZA should not be determined by minimizing the bias in planetary albedo.

In Cronin (2014) the local latitude-dependent SZA is discussed as well. The local SZA is estimated using the results from spherical trigonometry (Jacobson 2005):
e7
where δ is the declination of the sun, φ is the local latitude, and h is the local solar time. The local solar time is from 0 to 24 h or converted to hour angle as 24 h = 2π. The declination is accurately estimated by using the parameters of Earth’s orbit:
e8
where (Earth’s axial tilt angle) and N is the day of the year beginning with at midnight coordinated universal time.
For a given day at a given latitude, the SZA is the function of local solar time h; thus, the diurnal-mean cosine of SZA is
e9
In the region , we can define , where is the hour angle for sunset ( for sunrise), as in (7). Hour angle corresponds to the local solar noon time with the lowest SZA (maximum μ, which is equal to ). In the region of , or 24 h, there is no sunset.
The daytime-mean solar upward flux at TOA is calculated as
e10
Using the diurnal-mean from (9), the approximate result by a single time calculation is , with error of . The same calculations are used for the solar flux at the surface and atmospheric solar absorption . The left column of Fig. 3 shows the errors of , , and , for latitudes from 0° to 90°N and day numbers from 0 to 365. The five atmospheric profiles (McClatchey et al. 1972) apply to the calculations, with the summer profiles being used from April to September and the winter profiles being used for the rest of months, and the subarctic profiles being extended to 90°N. In Fig. 3, the contour lines are not smooth in some regions, which is due to the change of different atmospheric profiles.
Fig. 3.
Fig. 3.

The errors of the (top) upward solar flux at TOA, (middle) downward solar flux at the surface, and (bottom) atmospheric solar absorption. The benchmark results are calculated following (10). Results based on (left) diurnal-mean SZA of (9) and (right) diurnal insolation-weighted SZA of (11).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

By using of (9), the results of are very accurate, with error about 2 W m−2 in almost all the domains. The errors of can be over 10 W m−2, especially in the summer season. However, in this region the downward flux at the surface is generally over 500 W m−2, the relative errors are only about 2%. For , the errors are larger than 4 W m−2 in the tropics and during the summer season in high-latitude regions.

The small error in solar flux and atmospheric solar absorption indicates that the daytime-mean SZA can characterize the solar insolation distribution in the atmosphere, as , , and . The analytic form of (9) makes it very easy to apply the daytime-mean SZA in climate models.

Similar to (9) if the solar insolation is weighted, the diurnal insolation-weighted cosine of SZA is
e11
where and .

By using the insolation-weighted mean, large errors occur as shown in the right column of Fig. 3. For and , errors are larger than 30 W m−2 in most of the regions. For , the errors can be over 180 W m−2!

As in Fig. 1, if the solar upward flux at TOA is an insolation-weighted mean
e12
then the error of will be much smaller than that of .
Equation (9) can be extended to average over days as a temporal mean:
e13
where and are two day numbers, and the weighting factor presents the fraction of a day that is illuminated at latitude φ on day N,
eq1
where is the sunset angle shown in (9). A local illuminated hour angle is , and hour angle for a whole day is .
Note that is a function of latitude φ. In Fig. 4, the annual-mean meridional distributions of are shown. To include the leap year, the average is performed for four consecutive years. There were several parameterizations of annual-mean before. The most popular parameterization was proposed by North et al. (1981):
e14
where is the second-order Legendre function; the choice of an even-order Legendre function is due to the symmetry of the Northern and Southern Hemispheres.
Fig. 4.
Fig. 4.

(a) The annual- and seasonal-averaged latitudinal distributions of diurnal-mean SZA from (13); the solid line is the result of North et al. (1981). (b) As in (a), but for diurnal insolation-weighted SZA. Seasons are June–August (JJA) and December–February (DJF).

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

Figure 4a shows the comparison of the annual-mean SZA values. The parameterization proposed by North et al. (1981) is very close to the benchmark result, with relative errors of about 4% in the midlatitude region. In Ballinger et al. (2015), the annual-mean meridional distribution of SZA is parameterized as well. That parameterization strongly overestimates SZA in the higher latitudes.

Equation (13) is a general result based on an analytical formula of (9). By (13) the monthly or seasonal-mean SZA can be easily obtained, and this will be very useful for simplified climate models (Ballinger et al. 2015).

Similar to , can also be obtained based on the insolation-weighted in (11). In Fig. 4b, the meridional distributions of insolation-weighted annual- and seasonal-mean SZA are shown. The annual-mean result is obviously different from that shown in Cronin (2014). From (9) and (11), the results of and should be the same at the latitude of 90°N, but they are different in Cronin (2014). In addition, the annual-mean distribution of shows a turning point at a high latitude, which is not found in the result of Cronin (2014). At high latitudes over 66°N [the annual minimum value of ], there could be no sunrise in the winter season and no sunset in the summer season. This causes a turning point at 66°N in the distributions of and .

Equation (9) can also be extended to an average over latitude as spatial mean,
e15
where and are two latitudes (°). For a global average, the integral interval becomes for the winter season and for the summer season, where . In (15), represents area weight, and now the weighting factor represents the area fraction of a latitude band that is illuminated on day N.

Figure 5 shows the global- and regional-averaged daytime-mean SZA. The results generally depend on the chosen latitude region and day number. However the global averaged , which is independent of day number. Though this globally averaged value is the same as (1), the physical meaning is different. When Earth’s hemisphere is facing the sun, occurs only at its central point. Along a radius from the central point to an edge point of the hemisphere, μ changes from 1 to 0. This variation range of μ is the same as (1); thus, the integral in (1) represents an averaged SZA along any radius. Therefore, (1) is an averaged SZA on global scale, whereas (15) is an averaged SZA from a global integral over local daytime-mean SZA. It is interesting to find that the two different approaches lead to the same result. The globally averaged SZA is a constant regardless of the day number and Earth’s axial tilt angle.

Fig. 5.
Fig. 5.

The global- and regional-averaged diurnal-mean SZA. The tropics cover 0°–15°N, the midlatitudes cover 15°–45°N, and the subarctic covers 45°–60°N.

Citation: Journal of the Atmospheric Sciences 74, 5; 10.1175/JAS-D-16-0185.1

Our calculations are based on the clear-sky condition. There is no physical meaning to study the averaged SZA over cloudy sky since clouds change from time to time. This is why the averaged SZA is usually applied to a clear sky, especially in the stratosphere, which is cloud free (Hogan and Hirahara 2016). However, if we assume that clouds remain the same in shape and location over an integral time period as done in Cronin (2014), the result of cloudy sky is very similar to that of clear sky as shown above.

In summary, the averaged SZA cannot be determined by minimizing the bias in planetary albedo. The accuracy of the insolation-weighted-mean SZA in planetary albedo is caused by the cancellation of two positive errors. On both a global scale and a latitude-dependent local scale, it is misleading to say that the insolation-weighted-mean SZA is more accurate than the daytime-mean SZA. The choice of daytime-mean SZA or insolation-weighted-mean SZA depends on the averaging process, as (3) or (5) for the global scale, and (10) or (12) for the latitude-dependent local scale. For radiation variables, the weighting process should follow (3) or (10) because the solar insolation has been built into the radiative transfer calculations, and the daytime-mean SZA should be used. However, if the solar insolation is weighted to a physical variable as in (5) or (12), the insolation-weighted-mean SZA should be used.

Acknowledgments

The author thanks Dr. H. Barker and anonymous reviewers for their help comments and Professor P. Yang for his editorial efforts.

REFERENCES

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  • Dobbie, J. S., J. Li, and P. Chýlek, 1999: Two- and four-stream optical properties for water clouds and solar wavelengths. J. Geophys. Res., 104, 2067–2079, doi:10.1029/1998JD200039.

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  • Hartmann, D. L., 1994: Global Physical Climatology. International Geophysics Series, Vol 56, Academic Press, 409 pp.

    • Crossref
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  • Hogan, R. J., and S. Hirahara, 2016: Effect of solar zenith angle specification in models on mean shortwave fluxes and stratospheric temperatures. Geophys. Res. Lett., 43, 482–488, doi:10.1002/2015GL066868.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jacobson, M. Z., 2005: Fundamentals of Atmospheric Modeling. Cambridge University Press, 317 pp.

    • Crossref
    • Export Citation

  • Li, J., and V. Ramaswamy, 1996: Four-stream spherical harmonic expansion approximation for solar radiative transfer. J. Atmos. Sci., 53, 1174–1186, doi:10.1175/1520-0469(1996)053<1174:FSSHEA>2.0.CO;2.

    • Crossref
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  • Li, J., and H. W. Barker, 2005: A radiation algorithm with correlated-k distribution. Part I: Local thermal equilibrium. J. Atmos. Sci., 62, 286–309, doi:10.1175/JAS-3396.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Liou, K., 2002: An Introduction to Atmospheric Radiation. 2nd ed. International Geophysics Series, Vol. 84, Academic Press, 583 pp.

  • Manabe, S., and R. F. Strickler, 1964: Thermal equilibrium of the atmosphere with a convective adjustment. J. Atmos. Sci., 21, 361–385, doi:10.1175/1520-0469(1964)021<0361:TEOTAW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • McClatchey, R., R. Fenn, J. A. Selby, F. Volz, and J. Garing, 1972: Optical properties of the atmosphere. 3rd ed. Air Force Cambridge Research Laboratories Rep. AFCRL-72-0497, NTIS N7318412, 108 pp. [Available online at http://www.dtic.mil/dtic/tr/fulltext/u2/753075.pdf.]

  • North, G. R., R. F. Cahalan, and J. A. Coakley, 1981: Energy balance climate models. J. Atmos. Sci., 19, 91–121.

  • Ramanathan, V., and J. A. Coakley, 1978: Climate modeling through radiative-convective models. Rev. Geophys., 16, 465–489, doi:10.1029/RG016i004p00465.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Romps, D. M., 2011: Response of tropical precipitation to global warming. J. Atmos. Sci., 68, 123–138, doi:10.1175/2010JAS3542.1.

  • Yang, P., K. Liou, L. Bi, C. Liu, B. Yi, and B. Baum, 2015: On the radiative properties of ice clouds: Light scattering, remote sensing, and radiation parameterization. Adv. Atmos. Sci., 32, 32–63, doi:10.1007/s00376-014-0011-z.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhang, F., X. Zhou, H. Zhang, X. Peng, and Z. Wang, 2014: On the relationship between direct and diffuse radiation. Infrared Phys. Technol., 65, 5–8, doi:10.1016/j.infrared.2014.02.002.

    • Crossref
    • Search Google Scholar
    • Export Citation
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Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences

  • Abramowitz, M., and I. A. Stegun, Eds., 1965: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, 1046 pp.

  • Ballinger, A. P., T. M. Merlis, I. M. Held, and M. Zhao, 2015: The sensitivity of tropical cyclone activity to off-equatorial thermal forcing in aquaplanet simulations. J. Atmos. Sci., 72, 2286–2302, doi:10.1175/JAS-D-14-0284.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Cronin, T. W., 2014: On the choice of average solar zenith angle. J. Atmos. Sci., 71, 2994–3003, doi:10.1175/JAS-D-13-0392.1.

  • Dobbie, J. S., J. Li, and P. Chýlek, 1999: Two- and four-stream optical properties for water clouds and solar wavelengths. J. Geophys. Res., 104, 2067–2079, doi:10.1029/1998JD200039.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Hartmann, D. L., 1994: Global Physical Climatology. International Geophysics Series, Vol 56, Academic Press, 409 pp.

    • Crossref
    • Export Citation

  • Hogan, R. J., and S. Hirahara, 2016: Effect of solar zenith angle specification in models on mean shortwave fluxes and stratospheric temperatures. Geophys. Res. Lett., 43, 482–488, doi:10.1002/2015GL066868.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Jacobson, M. Z., 2005: Fundamentals of Atmospheric Modeling. Cambridge University Press, 317 pp.

    • Crossref
    • Export Citation

  • Li, J., and V. Ramaswamy, 1996: Four-stream spherical harmonic expansion approximation for solar radiative transfer. J. Atmos. Sci., 53, 1174–1186, doi:10.1175/1520-0469(1996)053<1174:FSSHEA>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Li, J., and H. W. Barker, 2005: A radiation algorithm with correlated-k distribution. Part I: Local thermal equilibrium. J. Atmos. Sci., 62, 286–309, doi:10.1175/JAS-3396.1.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Liou, K., 2002: An Introduction to Atmospheric Radiation. 2nd ed. International Geophysics Series, Vol. 84, Academic Press, 583 pp.

  • Manabe, S., and R. F. Strickler, 1964: Thermal equilibrium of the atmosphere with a convective adjustment. J. Atmos. Sci., 21, 361–385, doi:10.1175/1520-0469(1964)021<0361:TEOTAW>2.0.CO;2.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • McClatchey, R., R. Fenn, J. A. Selby, F. Volz, and J. Garing, 1972: Optical properties of the atmosphere. 3rd ed. Air Force Cambridge Research Laboratories Rep. AFCRL-72-0497, NTIS N7318412, 108 pp. [Available online at http://www.dtic.mil/dtic/tr/fulltext/u2/753075.pdf.]

  • North, G. R., R. F. Cahalan, and J. A. Coakley, 1981: Energy balance climate models. J. Atmos. Sci., 19, 91–121.

  • Ramanathan, V., and J. A. Coakley, 1978: Climate modeling through radiative-convective models. Rev. Geophys., 16, 465–489, doi:10.1029/RG016i004p00465.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Romps, D. M., 2011: Response of tropical precipitation to global warming. J. Atmos. Sci., 68, 123–138, doi:10.1175/2010JAS3542.1.

  • Yang, P., K. Liou, L. Bi, C. Liu, B. Yi, and B. Baum, 2015: On the radiative properties of ice clouds: Light scattering, remote sensing, and radiation parameterization. Adv. Atmos. Sci., 32, 32–63, doi:10.1007/s00376-014-0011-z.

    • Crossref
    • Search Google Scholar
    • Export Citation

  • Zhang, F., X. Zhou, H. Zhang, X. Peng, and Z. Wang, 2014: On the relationship between direct and diffuse radiation. Infrared Phys. Technol., 65, 5–8, doi:10.1016/j.infrared.2014.02.002.

    • Crossref
    • Search Google Scholar
    • Export Citation
  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Relative errors versus surface albedo. The red solid and dashed lines are the relative errors of and , respectively, the green solid and dashed lines are the relative errors of and , and the blue solid and dashed lines are the relative errors of and . (a) The benchmark results are based on (3); (b) the benchmark results are based on (5).

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    As in Fig. 1, but using the effective solar constant.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    The errors of the (top) upward solar flux at TOA, (middle) downward solar flux at the surface, and (bottom) atmospheric solar absorption. The benchmark results are calculated following (10). Results based on (left) diurnal-mean SZA of (9) and (right) diurnal insolation-weighted SZA of (11).

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    (a) The annual- and seasonal-averaged latitudinal distributions of diurnal-mean SZA from (13); the solid line is the result of North et al. (1981). (b) As in (a), but for diurnal insolation-weighted SZA. Seasons are June–August (JJA) and December–February (DJF).

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    The global- and regional-averaged diurnal-mean SZA. The tropics cover 0°–15°N, the midlatitudes cover 15°–45°N, and the subarctic covers 45°–60°N.

Fig. 1.

Relative errors versus surface albedo. The red solid and dashed lines are the relative errors of and , respectively, the green solid and dashed lines are the relative errors of and , and the blue solid and dashed lines are the relative errors of and . (a) The benchmark results are based on (3); (b) the benchmark results are based on (5).
Fig. 2.

As in Fig. 1, but using the effective solar constant.
Fig. 3.

The errors of the (top) upward solar flux at TOA, (middle) downward solar flux at the surface, and (bottom) atmospheric solar absorption. The benchmark results are calculated following (10). Results based on (left) diurnal-mean SZA of (9) and (right) diurnal insolation-weighted SZA of (11).
Fig. 4.

(a) The annual- and seasonal-averaged latitudinal distributions of diurnal-mean SZA from (13); the solid line is the result of North et al. (1981). (b) As in (a), but for diurnal insolation-weighted SZA. Seasons are June–August (JJA) and December–February (DJF).
Fig. 5.

The global- and regional-averaged diurnal-mean SZA. The tropics cover 0°–15°N, the midlatitudes cover 15°–45°N, and the subarctic covers 45°–60°N.
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