##
###
On the Choice of Average Solar Zenith Angle

Timothy W. Cronin

2014
*
Journal of the Atmospheric Sciences
*

3 Simulations with idealized climate models often choose to neglect spatiotemporal variations 4 in solar radiation, but doing so comes with an important decision about how to average solar 5 radiation in space and time. Since both clear-sky and cloud albedo are increasing functions of 6 the solar zenith angle, one can choose an absorption-weighted zenith angle which reproduces 7 the spatial-or time-mean absorbed solar radiation. Here, we perform calculations for a pure 8 scattering atmosphere
## more »

... tering atmosphere and with a more detailed radiative transfer model, and find that the 9 absorption-weighted zenith angle is usually between the daytime-weighted and insolation-10 weighted zenith angles, but much closer to the insolation-weighted zenith angle in most cases, 11 especially if clouds are responsible for much of the shortwave reflection. Use of daytime-12 average zenith angle may lead to a high bias in planetary albedo of ∼3%, equivalent to a 13 deficit in shortwave absorption of ∼10 W m −2 in the global energy budget (comparable to 14 the radiative forcing of a roughly sixfold change in CO 2 concentration). Other studies that 15 have used general circulation models with spatially constant insolation have underestimated 16 the global-mean zenith angle, with a consequent low bias in planetary albedo of ∼2-6%, or 17 a surplus in shortwave absorption of ∼7-20 W m −2 in the global energy budget. 18 1 Comprehensive climate models suggest that a global increase in absorbed solar radiation 20 by 1 W m −2 would lead to an 0.6-1.1 • C increase in global-mean surface temperatures (Soden 21 and Held 2006). The amount of solar radiation absorbed or reflected by the Earth depends 22 on the solar zenith angle (ζ), or angle the sun makes with a line perpendicular to the surface. 23 When the sun is low in the sky (high ζ), much of the incident sunlight may be reflected even 24 for a clear sky; when the sun is high in the sky (low ζ), even thick clouds may not reflect 25 most of the incident sunlight. The difference in average zenith angle between the equator 26 Matching the mean insolation constrains only the product S * 0 µ * , and not either parameter 38 individually, so additional assumptions are needed. 39 The specifics of these additional assumptions are quite important to simulated climate, 40 because radiative transfer processes, most importantly cloud albedo, depend on µ (e.g., 41 Hartmann (1994) ). For instance, the most straightforward choice for a planetary-average 42 2 calculation might seem to be a simple average of µ over the whole planet, including the dark 43 half, so that S * 0S = S 0 and µ * S =1/4. However, this simple average would correspond to a sun 44 that was always near setting, only ∼15 • above the horizon; with such a low sun, the albedo 45 of clouds and the reflection by clear-sky Rayleigh scattering would be highly exaggerated. 46 A more thoughtful, and widely used choice, is to ignore the contribution of the dark half of 47 the planet to the average zenith angle. With this choice of daytime-weighted zenith angle, 48 µ * D =1/2, and S * 0D = S 0 /2. 49 A slightly more complex option is to calculate the insolation-weighted cosine of the zenith 61 of these three different choices -simple average, daytime-weighted, and insolation-weighted 62 zenith angles -is given in Figure 1 . 63 The daytime-average cosine zenith angle of 0.5 has been widely used. The early studies 64 of radiative-convective equilibrium by Manabe and Strickler (1964), Manabe and Wetherald 65 (1967), Ramanathan (1976), and the early review paper by Ramanathan and Coakley (1978), 66 all took µ * = 0.5. The daytime-average zenith angle has also been used in simulation of 67 climate on other planets (e.g., Wordsworth et al. (2010)), as well as estimation of global 68 3 radiative forcing by clouds and aerosols (Fu and Liou 1993; Zhang et al. 2013). 69 To our knowledge, no studies of global-mean climate with single-column models have 70 used an insolation-weighted cosine zenith angle of 2/3. However, the above considerations 71 regarding spatial averaging over variations in insolation also apply to the temporal averaging 72 of insolation that is required to represent the diurnal cycle, or combined diurnal and annual 73 cycles of insolation, with a zenith angle that is constant in time. In this context, Hartmann 74 (1994) strongly argues for the use of insolation-weighted zenith angle, and provides a figure 75 with appropriate daily-mean insolation-weighted zenith angles as a function of latitude for 76 the solstices and the equinoxes (see Hartmann (1994), Figure 2.8). Romps (2011) also 77 uses an equatorial insolation-weighted zenith in a study of radiative-convective equilibrium 78 with a cloud-resolving model; though other studies that have focused on tropical radiative-79 convective equilibrium, such as the work by Tompkins and Craig (1998), have used a daytime-80 weighted zenith angle. In large-eddy simulations of marine low clouds, Bretherton et al. 81 (2013) advocate for the greater accuracy of the insolation-weighted zenith angle, noting that 82 the use of daytime-weighted zenith angle gives a 20 W m −2 stronger negative shortwave 83 cloud radiative effect than the insolation-weighted zenith angle. Biases of such a magnitude 84 would be especially disconcerting for situations where the surface temperature is interactive, 85 as they could lead to dramatic biases in mean temperatures. 86 Whether averaging in space or time, an objective decision of whether to use daytime-87 weighted or insolation-weighted zenith angle requires some known and unbiased reference 88 point. In section 2, we develop the idea of absorption-weighted zenith angle as such an un-89 biased reference point. We show that if albedo depends nearly linearly on the zenith angle, 90 which is true if clouds play a dominant role in solar reflection, then the insolation-weighted 91 zenith angle is likely to be less biased than the daytime-weighted zenith angle. We then 92 calculate the planetary-average absorption-weighted zenith angle for the extremely idealized 93 case of a purely conservative scattering atmosphere. In section 3, we perform calculations 94 with a more detailed shortwave radiative transfer model, and show that differences in plan-95 157 µ * I ; the first three rows of Table 1 summarize our findings for a pure scattering atmosphere. 158 For a clear sky, the daytime-weighted zenith angle is a slightly more accurate choice than 159 the insolation-weighted zenith angle. On the other hand, for a cloudy sky with moderate 160 optical thickness, the insolation-weighted zenith angle is essentially exact, and a daytime-161 weighted zenith angle may overestimate the planetary albedo by over 7%. For Earthlike 162 conditions, with a mixed sky that has low optical thickness in clear regions, and moderate 163 optical thickness in cloudy regions, a cosine-zenith angle close to but slightly less than the 164 planetary insolation-weighted mean value of 2/3 is likely the best choice. The common 165 choice of µ * =1/2 will overestimate the negative shortwave radiative effect of clouds, while 166 choices of µ * that are larger than 2/3 will underestimate the negative shortwave radiative 167 effect of clouds. Our calculations here, however, are quite simplistic, and do not account for 168 atmospheric absorption or wavelength-dependent optical properties. In the following section, 169 we will use a more detailed model to support the assertion that the insolation-weighted zenith 170 angle leads to smaller albedo biases than the daytime-weighted zenith angle. 171 3. Calculations with a Full Radiative Transfer Model 172 The above calculations provide a sense for the magnitude of planetary albedo bias that 173 may result from different choices of average solar zenith angle. In this section, we calculate 174 albedos using version 3.8 of the shortwave portion of the Rapid Radiative Transfer Model, 175 for application to GCMs (RRTMG SW, v3.8; Iacono et al. (2008); Clough et al. (2005)); 176 hereafter we refer to this model as simply "RRTM" for brevity. Calculations with RRTM 177 allow for estimation of biases associated with different choices of µ when the atmosphere has 178 more realistic scattering and absorption properties than we assumed in the pure scattering 179 355

doi:10.1175/jas-d-13-0392.1
fatcat:xw6g4zdp5rc2zlpltz4w2aoaum