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On the Choice of Average Solar Zenith Angle 2994 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 71 On the Choice of Average Solar Zenith Angle TIMOTHY W. CRONIN Program in Atmospheres, Oceans, and Climate, Massachusetts Institute of Technology, Cambridge, Massachusetts (Manuscript received 6 December 2013, in final form 19 March 2014) ABSTRACT Idealized climate modeling studies often choose to neglect spatiotemporal variations in solar radiation, but doing so comes with an important decision about how to average solar radiation in space and time. Since both clear-sky and cloud albedo are increasing functions of the solar zenith angle, one can choose an absorption- weighted zenith angle that reproduces the spatial- or time-mean absorbed solar radiation. Calculations are performed for a pure scattering atmosphere and with a more detailed radiative transfer model and show that 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 in most cases, especially if clouds are re- sponsible for much of the shortwave reflection. Use of daytime-average zenith angle may lead to a high bias in planetary albedo of approximately 3%, equivalent to a deficit in shortwave absorption of approximately 22 10 W m in the global energy budget (comparable to the radiative forcing of a roughly sixfold change in CO2 concentration). Other studies that have used general circulation models with spatially constant insolation have underestimated the global-mean zenith angle, with a consequent low bias in planetary albedo of ap- 2 proximately 2%–6% or a surplus in shortwave absorption of approximately 7–20 W m 2 in the global energy budget. 1. Introduction is simply the product of the solar constant S0 and the cosine of the solar zenith angle, m [ cosz: Comprehensive climate models suggest that a global 2 increase in absorbed solar radiation by 1 W m 2 would 5 I S0 cosz, (1) lead to a 0.68–1.18C increase in global-mean surface temperatures (Soden and Held 2006). The amount of where the planetary-mean insolation is simply hIi 5 ’ 22 solar radiation absorbed or reflected by Earth depends on S0/4 342 W m (in this paper, we will denote spatial the solar zenith angle z or the angle that the sun makes averages with hxi and time averages with x). A global- with a line perpendicular to the surface. When the sun is average radiative transfer calculation requires specifying low in the sky (high z), much of the incident sunlight may both an effective cosine of solar zenith angle m*andan be reflected, even for a clear sky; when the sun is high effective solar constant S*0 such that the resulting in- in the sky (low z), even thick clouds may not reflect most solation matches the planetary-mean insolation: of the incident sunlight. The difference in average zenith h i 5 5 angle between the equator and poles is an important I S0/4 S*0 m*. (2) reason why the albedo is typically higher at high latitudes. To simulate the average climate of a planet in radiative– Matching the mean insolation constrains only the convective equilibrium, one must globally average the product S*0 m*, and not either parameter individually, so incident solar radiation and define either a solar zenith additional assumptions are needed. angle that is constant in time or that varies diurnally The details of these additional assumptions are quite (i.e., the sun rising and setting). The top-of-atmosphere important to the simulated climate, because radiative incident solar radiation per unit ground area, or insolation, transfer processes, most importantly cloud albedo, de- pend on m (e.g., Hartmann 1994). For instance, the most Corresponding author address: Timothy W. Cronin, MIT, 77 straightforward choice for a planetary-average calcula- Massachusetts Ave., Cambridge, MA 02139. tion might seem to be a simple average of m over the 5 E-mail: [email protected] whole planet, including the dark half, so that S*0S S0 DOI: 10.1175/JAS-D-13-0392.1 Ó 2014 American Meteorological Society AUGUST 2014 C R O N I N 2995 5 1 and m*S /4. However, this simple average would cor- respond to a sun that was always near setting, only about 158 above the horizon; with such a low sun, the albedo of clouds and the reflection by clear-sky Rayleigh scatter- ing would be highly exaggerated. A more thoughtful, and widely used, choice is to ignore the contribution of the dark half of the planet to the average zenith angle. With this choice of daytime-weighted zenith angle, 5 1 5 m*D /2, and S0*D S0/2. A slightly more complex option is to calculate the insolation-weighted cosine of the zenith angle: FIG. 1. Schematic example of three different choices of zenith ð angle and solar constant that give the same insolation. The solar zenith angle is shown for each of the three choices, which corre- mS0mP(m) dm spond to simple average, daytime-weighted, and insolation- 5 ð weighted choices of m, as in the text. m*I , (3) S0mP(m) dm that is constant in time. In this context, Hartmann (1994) where P(m) is the probability distribution function of strongly argues for the use of insolation-weighted zenith global surface area as a function of m over the illumi- angle and provides a figure with appropriate daily-mean nated hemisphere. For the purposes of a planetary av- insolation-weighted zenith angles as a function of lati- erage, P(m) simply equals 1. This can be seen by rotating tude for the solstices and the equinoxes (see Hartmann coordinates so that the North Pole is aligned with the 1994, his Fig. 2.8). Romps (2011) also uses an equatorial subsolar point, where m 5 1; then m is given by the sine of insolation-weighted zenith angle in a study of radiative– the latitude over the illuminated Northern Hemisphere, convective equilibrium with a cloud-resolving model, and since area is uniformly distributed in the sine of the though other studies of tropical radiative–convective latitude, it follows that area is uniformly distributed over equilibrium with cloud-resolving models, such as the all values of m between 0 and 1. Hereafter, when dis- work by Tompkins and Craig (1998), have used a daytime- cussing planetary averages, it should be understood that weighted zenith angle. In large-eddy simulations of integrals over m implicitly contain the probability dis- marine low clouds, Bretherton et al. (2013) advocate for tribution function P(m) 5 1. Evaluation of Eq. (3) gives the greater accuracy of the insolation-weighted zenith 5 2 5 m*I /3, and S*0I 3S0/8. Since most of the sunlight angle, noting that the use of daytime-weighted zenith 2 falling on the daytime hemisphere occurs where the sun angle gives a 20 W m 2 stronger negative shortwave is high, m*I is considerably larger than m*D . A schematic cloud radiative effect than the insolation-weighted ze- comparison of these three different choices—simple nith angle. Biases of such a magnitude would be espe- average, daytime-weighted, and insolation-weighted cially disconcerting for situations where the surface zenith angles—is given in Fig. 1. temperature is interactive, as they could lead to dra- The daytime-average cosine zenith angle of 0.5 has matic biases in mean temperatures. been widely used. The early studies of radiative–convective Whether averaging in space or time, an objective equilibrium by Manabe and Strickler (1964), Manabe decision of whether to use daytime-weighted or insolation- and Wetherald (1967), Ramanathan (1976), and the weighted zenith angle requires some known and unbiased early review paper by Ramanathan and Coakley (1978) reference point. In section 2, we develop the idea of all took m* 5 0:5. The daytime-average zenith angle has absorption-weighted zenith angle as such an unbiased also been used in simulation of climate on other planets reference point. We show that if albedo depends nearly (e.g., Wordsworth et al. 2010) as well as estimation of linearly on the zenith angle, which is true if clouds play global radiative forcing by clouds and aerosols (Fu and a dominant role in solar reflection, then the insolation- Liou 1993; Zhang et al. 2013). weighted zenith angle is likely to be less biased than the To our knowledge, no studies of global-mean climate daytime-weighted zenith angle. We then calculate the with radiative–convective equilibrium models have used planetary-average absorption-weighted zenith angle for an insolation-weighted cosine zenith angle of 2/3. The the extremely idealized case of a purely conservative above considerations regarding the spatial averaging of scattering atmosphere. In section 3, we perform calcu- insolation, however, also apply to the temporal averaging lations with a more detailed shortwave radiative transfer of insolation that is required to represent the diurnal cycle, model and show that differences in planetary albedo 5 1 5 2 or combined diurnal and annual cycles, with a zenith angle between m*D /2 and m*I /3 can be approximately 3%, 2996 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 71 equivalent to a radiative forcing difference of over where amax is the maximum albedo (for m 5 0), and aD is 2 10Wm 2.Insection 4 we show that the superiority of the drop in albedo in going from m 5 0tom 5 1. In this insolation-weighting also applies for diurnally or annu- case, we can show that the absorption-weighted zenith ally averaged insolation. Finally, in section 5, we discuss angle is exactly equal to the insolation-weighted zenith the implications of our findings for recent studies with angle, regardless of the form of P(m).
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