# Accurately Quantifying Clear-Sky Radiative Cooling Potentials: A Temperature Correction to the Transmittance-Based Approximation

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^{†}

## Abstract

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## 1. Introduction

^{−2}depending on the ambient temperature and total precipitable water (TPW) content [2].

^{−2}

_{,}under typical operating conditions, which is 10–23% more than the approximation itself. To address this, we apply a temperature correction that accounts for the high elevations, and thus, low temperatures, of greenhouse gases, namely, water vapor, carbon dioxide and ozone, which allows a net heat transfer to them from the earth’s surface. This reduces the underestimation of the cooling potential to 0.1–6% while retaining the useful angular resolution of the transmittance-based cosine approximation, which the MODTRAN hemispherical irradiance model does not provide. Our results suggest that recently constructed maps of radiative cooling potentials may require corrections. Moreover, they indicate that the common use of the uncorrected transmittance-based cosine approximation to verify experimental demonstrations of radiative cooling could be leading to an overestimation of performance of radiative cooling designs across the literature.

## 2. Atmospheric Irradiance and the Transmittance-Based Cosine Approximation

## 3. Issues with the Transmittance-Based Cosine Approximation

_{2}in the troposphere. In the LWIR window, however, the transparency of the atmosphere reveals the ozone layer and outer space beyond. This means that some of the radiance from the earth’s surface is absorbed by the ozone, and much of its intrinsic radiance at ~9.5 µm reaches the earth. However, because of the low temperature of the ozone layer, which can be ~70 °C lower than ${T}_{amb}$ at altitudes where ozone concentration peaks in the summer and ~40 °C lower in the winter [30], it radiates far less towards the ground than it absorbs from the latter. In other words, a net heat loss occurs from the surface to the ozone layer, and the assumption of a thermally homogeneous atmosphere no longer holds (Figure 1A). This is not captured by Equation (3), which implicitly assumes the ozone layer to be at ${T}_{amb}$, causing the downwelling irradiance from ozone to be incorrectly equal to the fraction of ${I}_{BB}$ (${T}_{amb}$) absorbed by it. Consequently, heat loss to the ozone layer is not registered, ${I}_{atm}$ is overestimated and ${P}_{cooling}$ is underestimated, as is shown in Figure 1B. This is also evident when one compares the hemispherical emittance calculated by MODTRAN with ${I}_{atm}$ calculated using the transmittance-based cosine approximation. The difference due to the ozone effect alone is about 5–18.5 Wm

^{−2}depending on the atmosphere type and temperature, which is a significant 6–21% of the net cooling potential predicted by the transmittance-based cosine approximation.

**Figure 1.**(

**A**) Conceptual framework for traditional atmospheric thermal radiation against real case atmospheric thermal radiation. The photograph in the background was taken by NASA [31], and used under the Creative Commons-CC-BY-NC-ND 2.0 License. (

**B**) The spectral hemispherical atmospheric irradiance from the transmittance-based approximation and that from MODTRAN.

^{2}m depths of the atmosphere near the earth’s surface outside the LWIR window (which is at ~${T}_{amb}$) and within 2 km of the earth’s surface (or within ~12 °C of ${T}_{amb}$) in the LWIR. The resulting temperature differences are far less than those for ozone. Consequently, the transmittance-based approximation is largely correct outside the LWIR window where the atmosphere is opaque, and shows a lower underestimation of ${P}_{cooling}$ than seen for ozone at individual LWIR wavelengths (Figure 1B). However, unlike for ozone, the underestimation occurs over a much broader bandwidth across the LWIR, and for dry atmospheres, across the 16–20 µm wavelengths, which adds to a significant total when integrated.

## 4. Temperature-Corrections of the Transmittance-Based Model

_{2}in the LWIR, which allow for heat loss from the terrestrial environment, must be taken into account. The ozone layer is kilometers-thick, and CO

_{2}and water vapor are distributed throughout the atmosphere, which means that their radiative contributions are determined by a temperature distribution along their height. However, we can simplify calculations assuming that the irradiance of the ith gaseous component arises from a specific combination of its emittance ${\epsilon}_{i}$ and effective temperature ${T}_{i}$. The irradiance ${I}_{atm}$ can then be separated into two contributions: one for ozone, which is distributed high in the atmosphere, and one for CO

_{2}+ water vapor, which is distributed throughout, as follows:

_{2}, which effectively occur below the ozone layer, is calculated using MODTRAN by setting the atmospheric ozone concentration to zero. The transmittance of ozone is calculated as follows:

_{2}where it is highly absorptive is the ambient temperature ${T}_{amb}$ and the effective temperature of the completely masked ozone layer beyond is 0 K. It thus remains to calculate the effective temperature ${T}_{Ozone}$ of the ozone layer in the LWIR and ${T}_{rest}$ of water vapor and CO

_{2}in the LWIR within the 16–20 µm range. To do so, we first obtain, from MODTRAN, the effective hemispherical irradiance of CO

_{2}and water vapor ${I}_{rest}$(i.e., without ozone) and that of the ozone layer ${I}_{Ozone}$ by subtracting ${I}_{rest}$ from the hemispherical irradiance of the whole atmosphere (${I}_{atm}$). In parallel, we also calculate the hemispherical emittance of water vapor + CO

_{2}, and the ozone layer from the directional values ${\epsilon}_{rest}\left(\lambda ,\theta \right)$ and ${\epsilon}_{Ozone}\left(\lambda ,\theta \right)$ calculated earlier. We then solve for ${T}_{i}\left(\lambda \right)$ using the equation:

_{2}, which the traditional transmittance-based model does not capture.

^{−2}or 12 to 29 % depending on the temperature and atmosphere type. The underestimations are large, particularly for high values of ${I}_{atm}$. Our corrected model, by comparison, is within 1–8 Wm

^{−2}or 0.1–7% of the MODTRAN model, irrespective of the temperature, and the different total precipitable water and other greenhouse gas levels represented by the different atmospheres. The corrected transmission-based model thus provides an accurate irradiance relative to MODTRAN, while also providing angle-resolved irradiance values that the traditional transmission-based model provides.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**$\mathsf{\Delta}{T}_{ozone}$ and $\Delta {T}_{rest}$ for the six MODTRAN standard atmospheres: Tropical, Midlatitude summer, Midlatitude winter, US 1976 Standard Atmosphere, Subarctic summer, Subarctic winter.

**Figure 3.**Spectral hemispherical atmospheric irradiance ${I}_{atm}$ for the transmittance-based model and our corrected model for the standard MODTRAN atmospheres.

**Figure 4.**Maximum cooling potential (${P}_{cooling}={I}_{BB}\left({T}_{amb}\right)-{I}_{atm}$) calculated using the traditional transmittance-based model, the corrected model and MODTRAN for the six standard atmospheric profiles, but with ${T}_{amb}$ varied from 262 to 332 K. Dots indicate the calculated data, and the lines indicate fits with R

^{2}> 0.99. Equations for the fits are provided. The data at low temperatures for the Tropical, Midlatitude Summer and Subarctic summer atmospheric profiles should be used with caution because at such temperatures, the atmosphere does not typically hold the high TPW levels that characterize the standard cases.

**Table 1.**Analytical expressions of the ${P}_{cooling}$ corrections between the MODTRAN, transmittance-based and corrected models.

$\Delta {\mathit{P}}_{\mathit{c}\mathit{o}\mathit{o}\mathit{l}\mathit{i}\mathit{n}\mathit{g}}$ | Tropical | Midlatitude Summer | Midlatitude Winter | US 1976 Standard | Subarctic Summer | Subarctic Winter |
---|---|---|---|---|---|---|

P_{MODTRAN}-P_{Corrected} | 0.0011T^{2} − 0.57T + 72.7 | 0.0012T^{2} − 0.6T + 72.6 | −0.002T^{2} + 0.15T − 32.7 | −0.0005T^{2} + 0.32T − 53.32 | 0.0008T^{2} − 0.385T + 45 | −0.0007T^{2} + 0.49T − 82.2 |

P_{MODTRAN}-P_{Traditional} | 0.001T^{2} − 0.4T + 41.9 | 0.0009T^{2} − 0.31T + 26.1 | −0.001T^{2} + 0.767T − 127.2 | −0.0001T^{2} + 0.29T − 59.8 | 0.0005T^{2} − 0.071T − 7.9 | −0.002T^{2} + 1.39T − 214.8 |

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**MDPI and ACS Style**

Mandal, J.; Huang, X.; Raman, A.P.
Accurately Quantifying Clear-Sky Radiative Cooling Potentials: A Temperature Correction to the Transmittance-Based Approximation. *Atmosphere* **2021**, *12*, 1195.
https://doi.org/10.3390/atmos12091195

**AMA Style**

Mandal J, Huang X, Raman AP.
Accurately Quantifying Clear-Sky Radiative Cooling Potentials: A Temperature Correction to the Transmittance-Based Approximation. *Atmosphere*. 2021; 12(9):1195.
https://doi.org/10.3390/atmos12091195

**Chicago/Turabian Style**

Mandal, Jyotirmoy, Xin Huang, and Aaswath P. Raman.
2021. "Accurately Quantifying Clear-Sky Radiative Cooling Potentials: A Temperature Correction to the Transmittance-Based Approximation" *Atmosphere* 12, no. 9: 1195.
https://doi.org/10.3390/atmos12091195