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

MAY 2017 C O R R E S P O N D E N C E 1669

CORRESPONDENCE

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

(Manuscript received 20 June 2016, in ﬁnal form 9 January 2017)

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.

The globally averaged solar zenith angle (SZA) is an where m 5 cosu, with u being SZA, and P(m) is the important physical quantity for obtaining the averaged probability distribution. radiation variables. In climate models, most of the physical Another method is to consider the weight of solar parameterizations are applied to the global scale; thus, the insolation (Hartmann 1994; Romps 2011), which is globally averaged physical quantities are generally re- ð quired. For example, in the cloud optical property param- 1 mmS P(m) dm eterization, the cloud optical properties are first calculated 0 m*5 cosu 5 ð0 , (2) at each wavelength, then the results are weighted together I I 1 m m m S0P( ) d by the downward solar flux at a globally and temporarily 0 averaged SZA (Dobbie et al. 1999; Yang et al. 2015). The averaged SZA characterizes the spatiotemporally averaged where S0 is the total solar irradiance. For planetary aver- m 5 m 5 1 m 5 2 solar energy absorbed in the atmosphere. In simplified ages, P( ) 1(Cronin 2014), thus *D /2 and *I /3, climate models (North et al. 1981; Ballinger et al. 2015), the and the corresponding solar zenith angles are uD 5 608 and annually averaged meridional distributions of SZA need uI 5 48:198, which are called the daytime-mean SZA and to be calculated accurately and simply parameterized for insolation-weighted-mean SZA, respectively. applications. However there lacks a systematical discussion To minimize bias in solar absorption, Cronin (2014) on averaged SZA—Cronin (2014) is one of a few studies points out that SZA should be chosen to most closely addressed this issue. According to Cronin (2014),thereare match the spatial- or time-mean planetary albedo, two popular methods to calculate the averaged SZA. A and he found ‘‘the absorption-weighted zenith angle is commonlyusedmethod(Manabe and Strickler 1964; usually between the daytime-weighted and insolation- Ramanathan and Coakley 1978)is weighted zenith angles but much closer to the ð insolation-weighted zenith angle’’ (p. 2994). 1 Should the averaged SZA be determined by mini- mP(m) dm mizing the bias in planetary albedo? And is the m* 5 cosu 5 ð0 , (1) D D 1 insolation-weighted-mean SZA more accurate than the m m P( ) d daytime-mean SZA? We do not agree with either point 0 as explained in detail below. We denote the solar upward flux (or reflected flux) at Corresponding author e-mail: Dr. Jiangnan Li, jiangnan.li@ the top of the atmosphere (TOA), downward flux canada.ca (or transmitted flux) at surface and atmospheric

DOI: 10.1175/JAS-D-16-0185.1 Ó 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses). Unauthenticated | Downloaded 09/27/21 04:39 AM UTC 1670 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 74

[ Y absorption as R(m) 5 F (m, zt), T (m) 5 F (m, 0), and Y [ [ Y A(m) 5 F (m, zt) 1 F (m,0)2 [F (m, zt) 1 F (m, 0)], where F[(m, z)andFY(m, z) are the upward and downward broad band solar ﬂuxes at height z and zt 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 ð 1 R m m ð ( ) d 1 R 5 0 ð 5 R m m D 1 ( ) d . (3) dm 0 0

The same is for T (m) and A(m). m m Figure 1a shows the relative errors by using *D and *I. The radiation model (Li and Barker 2005) is used, which is a correlated-k distribution scheme for gaseous trans- mission 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 R m found that the relative error of ( *D) is limited to 3%, R m but the relative error of ( *I) is about 30%. The results are similar for T (m) and A(m). Let us consider the problem from a mathematical point of view. Gaussian quadrature is an effective way to evaluate an integral like (3).Byn-node Gaussian FIG. 1. Relative errors versus surface albedo. The red solid and quadrature, the integration of moment l is evaluated by R m R m dashed lines are the relative errors of ( *D)and ( *I), re- ð 1 n spectively, the green solid and dashed lines are the relative er- l T m T m x F(x) dx ’ å b F(x ) l 5 0, 1, 2, ..., (4) rors of ( *D)and ( *I ), and the blue solid and dashed lines are i i A m A m 0 i51 the relative errors of ( *D)and ( *I). (a) The benchmark results are based on (3); (b) the benchmark results are based where F(x) is a real function of x, xi is the abscissa, and bi on (5). 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 5 1), for In Fig. 1b the benchmark results are calculated based 1 on (5). The relative error of R(m* ) is up to 25%, but the zero moment (l 5 0), x1 5 /2, and b1 5 1 and for the D 2 R(m*) is limited to 1%. This is just opposite to that of single moment (l 5 1), x 5 / 3, and b 5 0:5. These two I 1 1 m m m m Fig. 1a. Therefore the choice of *D or *I depends on values of x1 are equal to *D and *I. Gaussian quadrature 1 2 whether the averaging process is (3) or (5). For radiative tells us that the choice of x1 5 /2 or x1 5 / 3 depends on the moment of the integral function. If a physical vari- ﬂuxes, (3) should be chosen because the solar insolation able is directly averaged as (3) (zero-moment weight), has been built in the radiative transfer calculations [i.e., R m R m ’ R m weighted inside ( )] and ( *D) D. However, if *D should be used; if a physical variable is averaged by weighting of solar insolation (single-moment weight), as in a physical process, the solar insolation is weighted to R m R m ’ R ð ( )as(5), ( *I) I . 1 Cronin (2014) has proposed an effective solar con- mS R(m) dm ð 0 1 stant. According to Cronin, for m* , S* 5 1/2S , and for R 5 0 ð 5 mR m m D 0 I 1 2 ( ) d , (5) m 5 *I , S* 3/8S0,whereS0 is the solar constant. The purpose mS dm 0 0 of the effective solar constant is to make m* S* 5 S0/4. 0 D It is well known that the value of 1/4 results in the in- m *I should be used. coming solar energy being evenly distributed over Earth’s

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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): ð ð ð 1 1 1 m m m R m m m fa( ) S0 d ( ) d d 5 0 ð 5 ð0 ð 0 f a 1 1 1 m m m m m S0 d S0 d d 0 0 0 R(m* ) ’ D 5 f (m* ). (6) m a D *DS0

Therefore, though the integral is weighted by solar in- m solation, *D has to be used, because the solar insolation weight is canceled in the integral. m 5 R m m Cronin (2014) shows that f ( *I) ( *I)/ *IS0 is more m 5 R m m accurate than fa( *D) ( *D)/ *DS0. However, the ac- m curacy of f ( *I) stems from a completely wrong reason. R m Figure 1a shows that the relative error of ( *I)is m around 30%, and also the relative error of *I (refer to m *D) is 33.333%—these two positive biases largely can- m celed out in the compound variable of f ( *I). For example, in the standard U.S. proﬁle at a surface albedo of m R m 0.2, the relative error of f ( *D)isthesameas ( *D) 5 R m ( 3.862%). The relative error of ( *I)is28.606%,and m the relative error of *I is 33.333%, which makes the m 2 FIG.2.AsinFig. 1, but using the effective solar constant. relative error of f ( *I) equal to 3.546%; thus, the dif- m m ference between f ( *D) and f ( *I) becomes small. m The concept of *A is incorrect; as shown above, the surface. This is nothing to do with the averaged SZA, compound variable of planetary albedo does not pro- because SZA should not apply to the region without vide any truthful information for SZA. The averaged sunlight. SZA should not be determined by minimizing the bias in

Figure 2 is the same as Fig. 1, but we replace S0 with planetary albedo. S*. It is found that the relative errors are much enhanced In Cronin (2014) the local latitude-dependent SZA compared to Fig. 1.InFig. 2, S* is not applied to the is discussed as well. The local SZA is estimated using benchmark calculations of (3) or (5).IfS*isappliedto the results from spherical trigonometry (Jacobson the benchmark calculations, Fig. 2 becomes the same 2005): as Fig. 1, because the effect of the effective solar constant will be canceled out. Dr. Cronin pointed to cosu 5 sinu sind 1 cosu cosd cosh, (7) me (T. W. Cronin 2016, personal communication) that S* 5 1/2S0 should be used in the benchmark calculations where d is the declination of the sun, u is the local lati- m m for both of *D and *I. Then it is difﬁcult to understand tude, and h is the local solar time. The local solar time is why the solar constant should be different in the bench- from 0 to 24 h or converted to hour angle as 24 h 5 2p. m mark and approximation calculations related to *I.The The declination is accurately estimated by using the concept of an effective solar constant does not help solve parameters of Earth’s orbit: the problem but only causes confusion. Cronin (2014) has proposed a concept of absorption- 3608 weighted SZA, m* , which is from the evaluation of d 5 arcsin sin(2j ) cos (N 1 10) A o 365:24 planetary albedo. All studies in Cronin (2014) are based 8 8 on m* by comparing it with m* and m*. The standard 1 360 : 360 2 A D I p 0 0167 sin : (N 2) , (8) deﬁnition of planetary albedo is fa(m) 5 R(m)/mS0, and 365 24

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j 5 : 8 where o 23 44 (Earth’s axial tilt angle) and N is the For a given day at a given latitude, the SZA is the day of the year beginning with N 5 0 at midnight co- function of local solar time h; thus, the diurnal-mean ordinated universal time. cosine of SZA is

ð 8 h 0 > sinu sind 1 (cosu cosd sinh )/h ; 21 # 2tanu tand # 1 m dh <> 0 0 m 5 ð0 5 sinu sind; 2tanu tand ,21 : (9) d h > 0 :> dh 0; 2tanu tand . 1 0

In the region j2tanu tandj # 1, we can define cosh0 5 the summer profiles being used from April to September 2tanu tand,whereh0 is the hour angle for sunset (2h0 and the winter profiles being used for the rest of months, for sunrise), as cosu 5 0in(7).Hourangleh 5 0corre- and the subarctic profiles being extended to 908N. In sponds to the local solar noon time with the lowest SZA Fig. 3, the contour lines are not smooth in some regions, (maximum m, which is equal to sinu sind 1 cosu cosd). In which is due to the change of different atmospheric the region of 2tanu tand ,21, h0 5 2p or 24 h, there is profiles. m R m no sunset. By using d of (9),theresultsof ( d) are very 2 The daytime-mean solar upward flux at TOA is cal- accurate, with error about 2 W m 2 in almost all the T m 22 culated as domains. The errors of ( d) can be over 10 W m , ð especially in the summer season. However, in this h0 R m region the downward flux at the surface is generally ( ) dh 22 0 over 500 W m , the relative errors are only about 2%. R 5 ð . (10) 2 d h A m 2 0 For ( d), the errors are larger than 4 W m in the dh tropics and during the summer season in high-latitude 0 regions. m Using the diurnal-mean d from (9), the approximate The small error in solar flux and atmospheric solar R m result by a single time calculation is ( d), with error of absorption indicates that the daytime-mean SZA can R m 2 R ( d) d. The same calculations are used for the solar characterize the solar insolation distribution in the at- T R m ’ R T m ’ T A m ’ A flux at the surface d and atmospheric solar absorption mosphere, as ( d) d, ( d) d, and ( d) d. A R m d. The left column of Fig. 3 shows the errors of ( d), The analytic form of (9) makes it very easy to apply the T m A m 8 8 ( d), and ( d), for latitudes from 0 to 90 N and day daytime-mean SZA in climate models. numbers from 0 to 365. The five atmospheric profiles Similar to (9) if the solar insolation is weighted, the (McClatchey et al. 1972) apply to the calculations, with diurnal insolation-weighted cosine of SZA is

8 2 1 1 : 1 : 2 ð >a h0 2ab sinh0 [0 5h0 0 25 sin(2h0)]b h > ; 21 # 2tanu tand # 1 0 > 1 mm <> ah0 b sinh0 S0 dh m 5 ð0 5 i h , (11) 0 >a 1 0:5b2/a; 2tanu tand ,21 mS dh > 0 > 0 : 0; 2tanu tand . 1

ð where a 5 sinu sind and b 5 cosu cosd. h0 R m m By using the insolation-weighted mean, large errors ( ) S0 dh R m R 5 0 ð , (12) occur as shown in the right column of Fig. 3.For ( i) i h0 A m 22 m and ( i), errors are larger than 30 W m in most of S0 dh T m 0 the regions. For ( i), the errors can be over 2 180 W m 2! R m 2 R As in Fig. 1, if the solar upward ﬂux at TOA is an then the error of ( i) i will be much smaller than R m 2 R insolation-weighted mean that of ( d) i.

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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).

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Equation (9) can be extended to average over days as a temporal mean:

N2 å m G u d ( , N) N5N mT 5 1 d , (13) N2 å G(u, N) 5 N N1 where N1 and N2 are two day numbers, and the weighting factor G(u, N) presents the fraction of a day that is illuminated at latitude u on day N, 8 >2h0 > ; 21 # 2tanu tand # 1 <2p G u 5 ( , N) > 2 u d ,2 , >1; tan tan 1 :> 0; 2tanu tand . 1 where h0 is the sunset angle shown in (9). A local illuminated hour angle is h0 2 (2h0) 5 2h0, and hour angle for a whole day is 2p. mT u Note that d is a function of latitude .InFig. 4, the mT annual-mean meridional distributions of d are shown. To include the leap year, the average is performed for four consecutive years. There were several parameteri- mT zations of annual-mean d before. The most popular parameterization was proposed by North et al. (1981):

mT 5 : 2 : u d 0 5[1 0 482P2(sin )], (14) where P2 is the second-order Legendre function; the FIG. 4. (a) The annual- and seasonal-averaged latitudinal dis- choice of an even-order Legendre function is due to the tributions of diurnal-mean SZA from (13); the solid line is the symmetry of the Northern and Southern Hemispheres. result of North et al. (1981). (b) As in (a), but for diurnal insolation- Figure 4a shows the comparison of the annual-mean weighted SZA. Seasons are June–August (JJA) and December– February (DJF). SZA values. The parameterization proposed by North et al. (1981) is very close to the benchmark result, with mT relative errors of about 4% in the midlatitude region. In of i shows a turning point at a high latitude, which is not Ballinger et al. (2015), the annual-mean meridional distri- found in the result of Cronin (2014). At high latitudes over bution of SZA is parameterized as well. That parameteri- 668N [the annual minimum value of u 5 arctan(2cotd)], zation strongly overestimates SZA in the higher latitudes. there could be no sunrise in the winter season and no Equation (13) is a general result based on an analytical sunset in the summer season. This causes a turning point at 8 mT mT formula of (9).By(13) the monthly or seasonal-mean SZA 66 N in the distributions of d and i . can be easily obtained, and this will be very useful for Equation (9) can also be extended to an average over simpliﬁed climate models (Ballinger et al. 2015). latitude as spatial mean, mT mT Similar to d , i can also be obtained based on the ðu m 2 insolation-weighted i in (11).InFig. 4b, the meridi- m G u u u d ( , N) cos d u onal distributions of insolation-weighted annual- and S 1 m 5 ðu , (15) seasonal-mean SZA are shown. The annual-mean result d 2 G(u, N) cosu du is obviously different from that shown in Cronin (2014). u 1 mT mT From (9) and (11), the results of d and i should be the 8 u u 8 same at the latitude of 90 N, but they are different in where 1 and 2 are two latitudes ( ). For a global av- Cronin (2014). In addition, the annual-mean distribution erage, the integral interval becomes [2p/2, u*] for the

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In summary, the averaged SZA cannot be determined by minimizing the bias in planetary albedo. The accu- racy 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.

FIG. 5. The global- and regional-averaged diurnal-mean SZA. Acknowledgments. The author thanks Dr. H. Barker The tropics cover 08–158N, the midlatitudes cover 158–458N, and and anonymous reviewers for their help comments and 8 8 the subarctic covers 45 –60 N. Professor P. Yang for his editorial efforts. winter season and [u*, p/2] for the summer season, where u* 5 arctan(2cotd). In (15),cosu represents area weight, REFERENCES G u and now the weighting factor ( , N) represents the area Abramowitz, M., and I. A. Stegun, Eds., 1965: Handbook of fraction of a latitude band that is illuminated on day N. Mathematical Functions with Formulas, Graphs, and Mathe- Figure 5 shows the global- and regional-averaged matical Tables. Dover Publications, 1046 pp. daytime-mean SZA. The results generally depend on Ballinger, A. P., T. M. Merlis, I. M. Held, and M. Zhao, 2015: The the chosen latitude region and day number. However sensitivity of tropical cyclone activity to off-equatorial thermal forcing in aquaplanet simulations. J. Atmos. Sci., 72, 2286– mS 5 1 the global averaged d /2, which is independent of day 2302, doi:10.1175/JAS-D-14-0284.1. number. Though this globally averaged value is the same Cronin, T. W., 2014: On the choice of average solar zenith angle. as (1), the physical meaning is different. When Earth’s J. Atmos. Sci., 71, 2994–3003, doi:10.1175/JAS-D-13-0392.1. hemisphere is facing the sun, m 5 1 occurs only at its Dobbie, J. S., J. Li, and P. Chýlek, 1999: Two- and four-stream central point. Along a radius from the central point to an optical properties for water clouds and solar wavelengths. m J. Geophys. Res., 104, 2067–2079, doi:10.1029/1998JD200039. edge point of the hemisphere, changes from 1 to 0. Hartmann, D. L., 1994: Global Physical Climatology. International This variation range of m is the same as (1); thus, the Geophysics Series, Vol 56, Academic Press, 409 pp. integral in (1) represents an averaged SZA along any Hogan, R. J., and S. Hirahara, 2016: Effect of solar zenith angle radius. Therefore, (1) is an averaged SZA on global specification in models on mean shortwave fluxes and strato- scale, whereas (15) is an averaged SZA from a global spheric temperatures. Geophys. Res. Lett., 43, 482–488, doi:10.1002/2015GL066868. integral over local daytime-mean SZA. It is interesting Jacobson, M. Z., 2005: Fundamentals of Atmospheric Modeling. to find that the two different approaches lead to the Cambridge University Press, 317 pp. same result. The globally averaged SZA is a constant Li, J., and V. Ramaswamy, 1996: Four-stream spherical harmonic regardless of the day number and Earth’s axial tilt angle. expansion approximation for solar radiative transfer. J. Atmos. , Our calculations are based on the clear-sky condition. Sci., 53, 1174–1186, doi:10.1175/1520-0469(1996)053 1174: FSSHEA.2.0.CO;2. There is no physical meaning to study the averaged SZA ——, and H. W. Barker, 2005: A radiation algorithm with correlated-k over cloudy sky since clouds change from time to time. This distribution. Part I: Local thermal equilibrium. J. Atmos. Sci., 62, is why the averaged SZA is usually applied to a clear sky, 286–309, doi:10.1175/JAS-3396.1. especially in the stratosphere, which is cloud free (Hogan Liou, K., 2002: An Introduction to Atmospheric Radiation. 2nd ed. and Hirahara 2016). However, if we assume that clouds International Geophysics Series, Vol. 84, Academic Press, 583 pp. Manabe, S., and R. F. Strickler, 1964: Thermal equilibrium of remainthesameinshapeandlocationoveranintegraltime the atmosphere with a convective adjustment. J. Atmos. period as done in Cronin (2014), the result of cloudy sky is Sci., 21, 361–385, doi:10.1175/1520-0469(1964)021,0361: very similar to that of clear sky as shown above. TEOTAW.2.0.CO;2.

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