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 down- ward 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 re- sults 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 dif- ferent 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 Unauthenticated | Downloaded 09/27/21 04:39 AM UTC MAY 2017 C O R R E S P O N D E N C E 1671 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).