## 1. Introduction

Planetary albedo *α* exerts a fundamental control on climate. For certain atmospheric profiles, especially those with optically thin clouds, the planetary albedo decreases near linearly with the cosine of the solar zenith angle *μ*. That is to say, when the sun lies higher in the sky, a smaller fraction of sunlight is reflected than when the sun lies lower in the sky. Because of this covariance of albedo and solar zenith angle, choosing a simple time-mean solar zenith angle may not give the correct time-mean solar flux, even if the insolation *α* is a linear function of *μ* and was shown by Cronin (2014) to outperform simple daytime weighting for cloudy skies in both the RRTMG model and a simpler pure-scattering atmosphere. In essence, the proposed approach considers the zenith angle corresponding to the average photon incident on a point or region, rather than the zenith angle corresponding to the average daylight hour.

The comment of Li (2017) claims that the insolation-weighted zenith angle leads to large errors when compared to the daytime-weighted zenith angle. The criticism of Li (2017), however, is based on a serious misinterpretation of the method proposed by Cronin (2014). Specifically, Li (2017) repeatedly fails to modify the solar constant to hold the insolation constant when comparing different choices of average zenith angle. The result is predictably large but spurious “errors” in fluxes calculated with the insolation-weighted zenith angle, because the sun is higher in the sky than it is in the time mean, but has not been turned down in strength to compensate and hold insolation constant.

To summarize this reply: Eq. (3) of Li (2017) is indeed the correct benchmark, and his calculations performed with the daytime-weighted zenith angle also appear to be correct, but his Eq. (5) is not the implementation of insolation-weighting advocated for in Cronin (2014). As a consequence, the rest of Li (2017) is an invalid critique, as it repeatedly compares the correct approach to daytime weighting with an incorrect approach to insolation weighting.

## 2. Hold the insolation constant

See also Fig. 1 of Cronin (2014) for a depiction of how the solar constant must be reduced when the sun is put higher in the sky.A global-average radiative transfer calculation requires specifying both an effective cosine of solar zenith angle

and an effective solar constant such that the resulting insolation matches the planetary-mean insolation…. Matching the mean insolation constrains only the product , and not either parameter individually, so additional assumptions are needed.

The insolation-weighted “benchmark” Eq. (5) of Li (2017) does not hold insolation fixed when compared to the correct benchmark [his Eq. (3)]. Suppose albedo is constant and does not depend on zenith angle, so that

Figures 1 and 3 of Li (2017) also fail to hold insolation constant and obtain large “errors” as a result for the insolation-weighted zenith angle. These “errors” can easily be predicted by comparing to a correct time-mean calculation with time-varying zenith angle and fluxes. If

Li (2017) attempts to address this point in his Fig. 2a, using effective solar constants from Cronin (2014), yet still fails to perform comparisons at fixed insolation. Li’s “benchmark calculation” in his Eq. (3) is performed only for the daylit half of the planet, whereas effective solar constants *U.S. Standard Atmosphere, 1976*—clear.” A central result of Cronin (2014) was that insolation-weighted calculations are superior to daytime-weighted calculations primarily in cloudy skies.

Assuming that Li’s confusion resulted from lack of clarity in Cronin (2014), it is perhaps worth stating once more, decisively: When the zenith angle is modified from its time-mean value, the solar constant must also be adjusted to keep the insolation fixed.

Li (2017) also makes a peculiar false assertion toward the end of the paper, which deserves comment: “There is no physical meaning to study the averaged SZA over cloudy sky since clouds change from time to time.” Regions with near-permanent clouds, such as stratocumulus-covered subtropical eastern ocean basins, are one obvious counterexample to this point—and a situation in which the daytime-weighted zenith angle has been shown to lead to overestimation of diurnal-average reflection by ~20 W m^{−2} (Bretherton et al. 2013). More importantly, from a standpoint of climate, the temporal variation in cloud cover at a fixed location is immaterial (so long as it does not covary with zenith angle). Over long time scales, different cloud scenes at a point are sampled at many different times of day, and all-sky radiative transfer under suitably averaged illumination is far more relevant than the clear-sky radiative transfer in determining long-term mean radiative fluxes.

## 3. Implementation

Averages over the full annual and diurnal cycles of insolation cannot be computed analytically, but we can easily tabulate and plot them (Table 1 and Fig. 1). As in Cronin (2014), Fig. 1 shows that the insolation-weighted cosine zenith angle is larger than the daytime-weighted cosine zenith angle by about 20% across latitudes. Figure 1 also includes the appropriate effective solar constant that should be used in the case of insolation weighting. As recognized by Li (2017), fits to the annual-mean profiles are simpler for daytime weighting—

Annual-average normalized insolation, insolation-weighted and daytime-weighted zenith angles, and effective solar constants, as a function of latitude, for a planet with a circular orbit and obliquity 23.5°.

I hope this comment clarifies and makes more accessible the approach of using the insolation-weighted zenith angle. Although not superior in all cases to the daytime-weighted zenith angle, it is likely a far better approach when clouds lead to a substantial fraction of shortwave reflection—as is the case for most skies on Earth.

## REFERENCES

Bretherton, C. S., P. N. Blossey, and C. R. Jones, 2013: Mechanisms of marine low cloud sensitivity to idealized climate perturbations: A single-LES exploration extending the CGILS cases.

,*J. Adv. Model. Earth Syst.***5**, 316–337, doi:10.1002/jame.20019.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.Li, J., 2017: Comments on “On the choice of average solar zenith angle.”

,*J. Atmos. Sci.***74**, 1669–1676, doi:10.1175/JAS-D-16-0185.1.