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## Abstract

A one-dimensional radiative transfer algorithm that accounts for correlations between the optical thickness and the incident direct solar radiation is developed to compute the domain-averaged shortwave irradiance profile. It divides the direct irradiance into four components and treats the direct irradiance in two separate, clear and cloudy columns to account for the fact that clouds attenuate the direct irradiance more than clear sky. The horizontal inhomogeneity of clouds in the cloudy column is treated by the gamma-weighted two-stream approximation, which assumes that the optical thickness of clouds follows a gamma distribution. The algorithm inputs the cloud fraction, cumulative cloud fraction as a function of height, and a parameter expressing the shape of the probability density function of the cloud optical thickness distribution in addition to inputs required for a two-stream radiative transfer model. These cloud property inputs can be obtained using ground- and satellite-based instruments. Therefore, the algorithm can treat realistic cloud overlap features and horizontal inhomogeneity of clouds in a framework of one-dimensional radiative transfer. Heating rates computed by the algorithm using cloud fields generated by cloud resolving models agree with those computed with a Monte Carlo model.

## Abstract

A one-dimensional radiative transfer algorithm that accounts for correlations between the optical thickness and the incident direct solar radiation is developed to compute the domain-averaged shortwave irradiance profile. It divides the direct irradiance into four components and treats the direct irradiance in two separate, clear and cloudy columns to account for the fact that clouds attenuate the direct irradiance more than clear sky. The horizontal inhomogeneity of clouds in the cloudy column is treated by the gamma-weighted two-stream approximation, which assumes that the optical thickness of clouds follows a gamma distribution. The algorithm inputs the cloud fraction, cumulative cloud fraction as a function of height, and a parameter expressing the shape of the probability density function of the cloud optical thickness distribution in addition to inputs required for a two-stream radiative transfer model. These cloud property inputs can be obtained using ground- and satellite-based instruments. Therefore, the algorithm can treat realistic cloud overlap features and horizontal inhomogeneity of clouds in a framework of one-dimensional radiative transfer. Heating rates computed by the algorithm using cloud fields generated by cloud resolving models agree with those computed with a Monte Carlo model.

## Abstract

Interannual variability of the global radiation budget, regions that contribute to its variability, and what limits albedo variability are investigated using Clouds and the Earth’s Radiant Energy System (CERES) data taken from March 2000 through February 2004. Area-weighted mean top-of-atmosphere (TOA) reflected shortwave, longwave, and net irradiance standard deviations computed from monthly anomalies over a 1° × 1° region are 9.6, 7.6, and 7.6 W m^{−2}, respectively. When standard deviations are computed from global monthly anomalies, they drop to 0.5, 0.4, and 0.4 W m^{−2}, respectively. Clouds are mostly responsible for the variation. Regions with a large standard deviation of TOA shortwave and longwave irradiance at TOA are the tropical western and central Pacific, which is caused by shifting from La Niña to El Niño during this period. However, a larger standard deviation of 300–1000-hPa thickness anomalies occurs in the polar region instead of the tropics. The correlation coefficient between atmospheric net irradiance anomalies and 300–1000-hPa thickness anomalies is negative. These indicate that temperature anomalies in the atmosphere are mostly a result of anomalies in longwave and dynamical processes that transport energy poleward, instead of albedo anomalies by clouds directly affecting temperature anomalies in the atmosphere. With simple zonal-mean thermodynamic energy equations it is demonstrated that temperature anomalies decay exponentially with time by longwave emission and by dynamical processes. As a result, the mean meridional temperature gradient is maintained. Therefore, mean meridional circulations are not greatly altered by albedo anomalies on an annual time scale, which in turn provides small interannual variability of the global mean albedo.

## Abstract

Interannual variability of the global radiation budget, regions that contribute to its variability, and what limits albedo variability are investigated using Clouds and the Earth’s Radiant Energy System (CERES) data taken from March 2000 through February 2004. Area-weighted mean top-of-atmosphere (TOA) reflected shortwave, longwave, and net irradiance standard deviations computed from monthly anomalies over a 1° × 1° region are 9.6, 7.6, and 7.6 W m^{−2}, respectively. When standard deviations are computed from global monthly anomalies, they drop to 0.5, 0.4, and 0.4 W m^{−2}, respectively. Clouds are mostly responsible for the variation. Regions with a large standard deviation of TOA shortwave and longwave irradiance at TOA are the tropical western and central Pacific, which is caused by shifting from La Niña to El Niño during this period. However, a larger standard deviation of 300–1000-hPa thickness anomalies occurs in the polar region instead of the tropics. The correlation coefficient between atmospheric net irradiance anomalies and 300–1000-hPa thickness anomalies is negative. These indicate that temperature anomalies in the atmosphere are mostly a result of anomalies in longwave and dynamical processes that transport energy poleward, instead of albedo anomalies by clouds directly affecting temperature anomalies in the atmosphere. With simple zonal-mean thermodynamic energy equations it is demonstrated that temperature anomalies decay exponentially with time by longwave emission and by dynamical processes. As a result, the mean meridional temperature gradient is maintained. Therefore, mean meridional circulations are not greatly altered by albedo anomalies on an annual time scale, which in turn provides small interannual variability of the global mean albedo.

## Abstract

Vertical profiles of shortwave and longwave irradiances computed with satellite-derived cloud properties and temperature and humidity profiles from reanalysis are used to estimate entropy production. Entropy production by shortwave radiation is computed by the absorbed irradiance within layers in the atmosphere and by the surface divided by their temperatures. Similarly, entropy production by longwave radiation is computed by emitted irradiance to space from layers in the atmosphere and surface divided by their temperatures. Global annual mean entropy production by shortwave absorption and longwave emission to space are, respectively, 0.852 and 0.928 W m^{−2} K^{−1}. With a steady-state assumption, entropy production by irreversible processes within the Earth system is estimated to be 0.076 W m^{−2} K^{−1} and by nonradiative irreversible processes to be 0.049 W m^{−2} K^{−1}. Both global annual mean entropy productions by shortwave absorption and longwave emission to space increase with increasing shortwave absorption (i.e., with decreasing the planetary albedo). The increase of entropy production by shortwave absorption is, however, larger than the increase of entropy production by longwave emission to space. The result implies that global annual mean entropy production by irreversible processes decreases with increasing shortwave absorption. Input and output temperatures derived by dividing the absorbed shortwave irradiance and emitted longwave irradiance to space by respective entropy production are, respectively, 282 and 259 K, which give the Carnot efficiency of the Earth system of 8.5%.

## Abstract

Vertical profiles of shortwave and longwave irradiances computed with satellite-derived cloud properties and temperature and humidity profiles from reanalysis are used to estimate entropy production. Entropy production by shortwave radiation is computed by the absorbed irradiance within layers in the atmosphere and by the surface divided by their temperatures. Similarly, entropy production by longwave radiation is computed by emitted irradiance to space from layers in the atmosphere and surface divided by their temperatures. Global annual mean entropy production by shortwave absorption and longwave emission to space are, respectively, 0.852 and 0.928 W m^{−2} K^{−1}. With a steady-state assumption, entropy production by irreversible processes within the Earth system is estimated to be 0.076 W m^{−2} K^{−1} and by nonradiative irreversible processes to be 0.049 W m^{−2} K^{−1}. Both global annual mean entropy productions by shortwave absorption and longwave emission to space increase with increasing shortwave absorption (i.e., with decreasing the planetary albedo). The increase of entropy production by shortwave absorption is, however, larger than the increase of entropy production by longwave emission to space. The result implies that global annual mean entropy production by irreversible processes decreases with increasing shortwave absorption. Input and output temperatures derived by dividing the absorbed shortwave irradiance and emitted longwave irradiance to space by respective entropy production are, respectively, 282 and 259 K, which give the Carnot efficiency of the Earth system of 8.5%.

## Abstract

Nine months of the Clouds and the Earth's Radiant Energy System (CERES)/Tropical Rainfall Measuring Mission (TRMM) broadband fluxes combined with the TRMM visible infrared scanner (VIRS) high-resolution imager measurements are used to estimate the daily average direct radiative effect of aerosols for clear-sky conditions over the tropical oceans. On average, aerosols have a cooling effect over the Tropics of 4.6 ± 1 W m^{–2}. The magnitude is ≈2 W m^{–2} smaller over the southern tropical oceans than it is over northern tropical oceans. The direct effect derived from CERES is highly correlated with coincident aerosol optical depth (*τ*) retrievals inferred from 0.63-*μ*m VIRS radiances (correlation coefficient of 0.96). The slope of the regression line is ≈−32 W m^{–2}
*τ*
^{–1} over the equatorial Pacific Ocean, but changes both regionally and seasonally, depending on the aerosol characteristics. Near sources of biomass burning and desert dust, the aerosol direct effect reaches −25 to −30 W m^{–2}. The direct effect from CERES also shows a dependence on wind speed. The reason for this dependence is unclear—it may be due to increased aerosol (e.g., sea-salt or aerosol transport) or increased surface reflection (e.g., due to whitecaps). The uncertainty in the tropical average direct effect from CERES is ≈1 W m^{–2} (≈20%) due mainly to cloud contamination, the radiance-to-flux conversion, and instrument calibration. By comparison, uncertainties in the direct effect from the Earth Radiation Budget Experiment (ERBE) and CERES “ERBE-like” products are a factor of 3–5 times larger.

## Abstract

Nine months of the Clouds and the Earth's Radiant Energy System (CERES)/Tropical Rainfall Measuring Mission (TRMM) broadband fluxes combined with the TRMM visible infrared scanner (VIRS) high-resolution imager measurements are used to estimate the daily average direct radiative effect of aerosols for clear-sky conditions over the tropical oceans. On average, aerosols have a cooling effect over the Tropics of 4.6 ± 1 W m^{–2}. The magnitude is ≈2 W m^{–2} smaller over the southern tropical oceans than it is over northern tropical oceans. The direct effect derived from CERES is highly correlated with coincident aerosol optical depth (*τ*) retrievals inferred from 0.63-*μ*m VIRS radiances (correlation coefficient of 0.96). The slope of the regression line is ≈−32 W m^{–2}
*τ*
^{–1} over the equatorial Pacific Ocean, but changes both regionally and seasonally, depending on the aerosol characteristics. Near sources of biomass burning and desert dust, the aerosol direct effect reaches −25 to −30 W m^{–2}. The direct effect from CERES also shows a dependence on wind speed. The reason for this dependence is unclear—it may be due to increased aerosol (e.g., sea-salt or aerosol transport) or increased surface reflection (e.g., due to whitecaps). The uncertainty in the tropical average direct effect from CERES is ≈1 W m^{–2} (≈20%) due mainly to cloud contamination, the radiance-to-flux conversion, and instrument calibration. By comparison, uncertainties in the direct effect from the Earth Radiation Budget Experiment (ERBE) and CERES “ERBE-like” products are a factor of 3–5 times larger.

## Abstract

The upward shortwave irradiance at the top of the atmosphere when the solar zenith angle is greater than 90° (twilight irradiance) is estimated from radiance measurements by the Clouds and the Earth's Radiant Energy System (CERES) instrument on the Tropical Rainfall Measuring Mission (TRMM) satellite. The irradiance decreases with solar zenith angle from 7.5 W m^{−2} at 90.5° to 0.6 W m^{−2} at 95.5°. The global and daily average twilight irradiance is 0.2 W m^{−2}, which is three orders of magnitude smaller than the daily and global average reflected irradiance at the top of the atmosphere. Therefore, the twilight irradiance can be neglected in global radiation budget estimate. The daily average twilight irradiance, however, can be more than 1 W m^{−2} at polar regions during seasons when the sun stays just below the horizon for a long period of time.

## Abstract

The upward shortwave irradiance at the top of the atmosphere when the solar zenith angle is greater than 90° (twilight irradiance) is estimated from radiance measurements by the Clouds and the Earth's Radiant Energy System (CERES) instrument on the Tropical Rainfall Measuring Mission (TRMM) satellite. The irradiance decreases with solar zenith angle from 7.5 W m^{−2} at 90.5° to 0.6 W m^{−2} at 95.5°. The global and daily average twilight irradiance is 0.2 W m^{−2}, which is three orders of magnitude smaller than the daily and global average reflected irradiance at the top of the atmosphere. Therefore, the twilight irradiance can be neglected in global radiation budget estimate. The daily average twilight irradiance, however, can be more than 1 W m^{−2} at polar regions during seasons when the sun stays just below the horizon for a long period of time.

## Abstract

This reply addresses a comment on the study by Kato and Rose (herein referred to as KR2020). The comment raises four points of criticism. These are 1) on notations used, 2) on a steady-state assumption made, 3) on the result of entropy production change with Earth’s albedo, and 4) disputing the statement that a simple energy balance model cannot produce absorption temperature change with Earth’s albedo. We concur on points 2 and 3 raised by the comment and recognize the significance of entropy storage due to ocean heating in the analysis of how entropy production changes with the shortwave absorptivity of Earth. Once entropy storage is considered, the results of KR2020 indicate that the increase of entropy production rate by irreversible processes, including by radiative processes, is smaller than the increase of entropy storage when absorptivity is increased. This is a manifestation of the primary contribution of positive top-of-atmosphere net irradiances (i.e., energy input to Earth) to heating the ocean and is consistent with an energy budget perspective. Once entropy storage is separated, the entropy production by irreversible processes increases with the shortwave absorptivity.

## Abstract

This reply addresses a comment on the study by Kato and Rose (herein referred to as KR2020). The comment raises four points of criticism. These are 1) on notations used, 2) on a steady-state assumption made, 3) on the result of entropy production change with Earth’s albedo, and 4) disputing the statement that a simple energy balance model cannot produce absorption temperature change with Earth’s albedo. We concur on points 2 and 3 raised by the comment and recognize the significance of entropy storage due to ocean heating in the analysis of how entropy production changes with the shortwave absorptivity of Earth. Once entropy storage is considered, the results of KR2020 indicate that the increase of entropy production rate by irreversible processes, including by radiative processes, is smaller than the increase of entropy storage when absorptivity is increased. This is a manifestation of the primary contribution of positive top-of-atmosphere net irradiances (i.e., energy input to Earth) to heating the ocean and is consistent with an energy budget perspective. Once entropy storage is separated, the entropy production by irreversible processes increases with the shortwave absorptivity.

## Abstract

An algorithm is developed for the gamma-weighted discrete ordinate two-stream approximation that computes profiles of domain-averaged shortwave irradiances for horizontally inhomogeneous cloudy atmospheres. The algorithm assumes that frequency distributions of cloud optical depth at unresolved scales can be represented by a gamma distribution though it neglects net horizontal transport of radiation. This algorithm is an alternative to the one used in earlier studies that adopted the adding method. At present, only overcast cloudy layers are permitted.

## Abstract

An algorithm is developed for the gamma-weighted discrete ordinate two-stream approximation that computes profiles of domain-averaged shortwave irradiances for horizontally inhomogeneous cloudy atmospheres. The algorithm assumes that frequency distributions of cloud optical depth at unresolved scales can be represented by a gamma distribution though it neglects net horizontal transport of radiation. This algorithm is an alternative to the one used in earlier studies that adopted the adding method. At present, only overcast cloudy layers are permitted.

## Abstract

The Clouds and Earth’s Radiant Energy System (CERES) provides coincident global cloud and aerosol properties together with reflected solar, emitted terrestrial longwave, and infrared window radiative fluxes. These data are needed to improve the understanding and modeling of the interaction between clouds, aerosols, and radiation at the top of the atmosphere, surface, and within the atmosphere. This paper describes the approach used to estimate top-of-atmosphere (TOA) radiative fluxes from instantaneous CERES radiance measurements on the *Terra* satellite. A key component involves the development of empirical angular distribution models (ADMs) that account for the angular dependence of the earth’s radiation field at the TOA. The CERES *Terra* ADMs are developed using 24 months of CERES radiances, coincident cloud and aerosol retrievals from the Moderate Resolution Imaging Spectroradiometer (MODIS), and meteorological parameters from the Global Modeling and Assimilation Office (GMAO)’s Goddard Earth Observing System (GEOS) Data Assimilation System (DAS) V4.0.3 product. Scene information for the ADMs is from MODIS retrievals and GEOS DAS V4.0.3 properties over the ocean, land, desert, and snow for both clear and cloudy conditions. Because the CERES *Terra* ADMs are global, and far more CERES data are available on *Terra* than were available from CERES on the *Tropical Rainfall Measuring Mission* (*TRMM*), the methodology used to define CERES *Terra* ADMs is different in many respects from that used to develop CERES *TRMM* ADMs, particularly over snow/sea ice, under cloudy conditions, and for clear scenes over land and desert.

## Abstract

The Clouds and Earth’s Radiant Energy System (CERES) provides coincident global cloud and aerosol properties together with reflected solar, emitted terrestrial longwave, and infrared window radiative fluxes. These data are needed to improve the understanding and modeling of the interaction between clouds, aerosols, and radiation at the top of the atmosphere, surface, and within the atmosphere. This paper describes the approach used to estimate top-of-atmosphere (TOA) radiative fluxes from instantaneous CERES radiance measurements on the *Terra* satellite. A key component involves the development of empirical angular distribution models (ADMs) that account for the angular dependence of the earth’s radiation field at the TOA. The CERES *Terra* ADMs are developed using 24 months of CERES radiances, coincident cloud and aerosol retrievals from the Moderate Resolution Imaging Spectroradiometer (MODIS), and meteorological parameters from the Global Modeling and Assimilation Office (GMAO)’s Goddard Earth Observing System (GEOS) Data Assimilation System (DAS) V4.0.3 product. Scene information for the ADMs is from MODIS retrievals and GEOS DAS V4.0.3 properties over the ocean, land, desert, and snow for both clear and cloudy conditions. Because the CERES *Terra* ADMs are global, and far more CERES data are available on *Terra* than were available from CERES on the *Tropical Rainfall Measuring Mission* (*TRMM*), the methodology used to define CERES *Terra* ADMs is different in many respects from that used to develop CERES *TRMM* ADMs, particularly over snow/sea ice, under cloudy conditions, and for clear scenes over land and desert.

## Abstract

The ocean surface albedo is responsible for the distribution of solar (shortwave) radiant energy between the atmosphere and ocean and therefore is a key parameter in Earth’s surface energy budget. In situ ocean observations typically do not measure upward reflected solar radiation, which is necessary to compute net solar radiation into the ocean. Instead, the upward component is computed from the measured downward component using an albedo estimate. At two NOAA Ocean Climate Station buoy sites in the North Pacific, the International Satellite Cloud Climatology Project (ISCCP) monthly climatological albedo has been used, while for the NOAA Global Tropical Buoy Array a constant albedo is used. This constant albedo is also used in the Coupled Ocean–Atmosphere Response Experiment (COARE) bulk flux algorithm. This study considers the impacts of using the more recently available NASA Cloud and the Earth’s Radiant Energy System (CERES) albedo product for these ocean surface heat flux products. Differences between albedo estimates in global satellite products like these imply uncertainty in the net surface solar radiation heat flux estimates that locally exceed the target uncertainty of 1.0 W m^{−2} for the global mean, set by the Global Climate Observing System (GCOS) of the World Meteorological Organization (WMO). Albedo has large spatiotemporal variability on hourly, monthly, and interannual time scales. Biases in high-resolution SW_{net} (the difference between surface downwelling and upwelling shortwave radiation) can arise if the albedo diurnal cycle is unresolved. As a result, for periods when satellite albedo data are not available it is recommended that an hourly climatology be used when computing high-resolution net surface shortwave radiation.

## Abstract

The ocean surface albedo is responsible for the distribution of solar (shortwave) radiant energy between the atmosphere and ocean and therefore is a key parameter in Earth’s surface energy budget. In situ ocean observations typically do not measure upward reflected solar radiation, which is necessary to compute net solar radiation into the ocean. Instead, the upward component is computed from the measured downward component using an albedo estimate. At two NOAA Ocean Climate Station buoy sites in the North Pacific, the International Satellite Cloud Climatology Project (ISCCP) monthly climatological albedo has been used, while for the NOAA Global Tropical Buoy Array a constant albedo is used. This constant albedo is also used in the Coupled Ocean–Atmosphere Response Experiment (COARE) bulk flux algorithm. This study considers the impacts of using the more recently available NASA Cloud and the Earth’s Radiant Energy System (CERES) albedo product for these ocean surface heat flux products. Differences between albedo estimates in global satellite products like these imply uncertainty in the net surface solar radiation heat flux estimates that locally exceed the target uncertainty of 1.0 W m^{−2} for the global mean, set by the Global Climate Observing System (GCOS) of the World Meteorological Organization (WMO). Albedo has large spatiotemporal variability on hourly, monthly, and interannual time scales. Biases in high-resolution SW_{net} (the difference between surface downwelling and upwelling shortwave radiation) can arise if the albedo diurnal cycle is unresolved. As a result, for periods when satellite albedo data are not available it is recommended that an hourly climatology be used when computing high-resolution net surface shortwave radiation.

## Abstract

The respective errors caused by the gamma-weighted two-stream approximation and the effective thickness approximation for computing the domain-averaged broadband shortwave irradiance are evaluated using cloud optical thicknesses derived from 1 h of radiance measurements by the Moderate Resolution Imaging Spectrometer (MODIS) over footprints of Clouds and the Earth’s Radiant Energy System (CERES) instruments. Domains are CERES footprints of which dimension varies approximately from 20 to 70 km, depending on the viewing zenith angle of the instruments. The average error in the top-of-atmosphere irradiance at a 30° solar zenith angle caused by the gamma-weighted two-stream approximation is 6.1 W m^{−2} (0.005 albedo bias) with a one-layer overcast cloud where a positive value indicates an overestimate by the approximation compared with the irradiance computed using the independent column approximation. Approximately one-half of the error is due to deviations of optical thickness distributions from a gamma distribution and the other half of the error is due to other approximations in the model. The error increases to 14.7 W m^{−2} (0.012 albedo bias) when the computational layer dividing the cloud layer is increased to four. The increase is because of difficulties in treating the correlation of cloud properties in the vertical direction. Because the optical thickness under partly cloudy conditions, which contribute two-thirds of cloudy footprints, is smaller, the error is smaller than under overcast conditions; the average error for partly cloudy condition is −2.4 W m^{−2} (−0.002 albedo bias) at a 30° solar zenith angle. The corresponding average error caused by the effective thickness approximation is 0.5 W m^{−2} for overcast conditions and −21.5 W m^{−2} (−0.018 albedo bias) for partly cloudy conditions. Although the error caused by the effective thickness approximation depends strongly on the optical thickness, its average error under overcast conditions is smaller than the error caused by the gamma-weighted two-stream approximation because the errors at small and large optical thicknesses cancel each other. Based on these error analyses, the daily average error caused by the gamma-weighted two-stream and effective thickness approximations is less than 2 W m^{−2}.

## Abstract

The respective errors caused by the gamma-weighted two-stream approximation and the effective thickness approximation for computing the domain-averaged broadband shortwave irradiance are evaluated using cloud optical thicknesses derived from 1 h of radiance measurements by the Moderate Resolution Imaging Spectrometer (MODIS) over footprints of Clouds and the Earth’s Radiant Energy System (CERES) instruments. Domains are CERES footprints of which dimension varies approximately from 20 to 70 km, depending on the viewing zenith angle of the instruments. The average error in the top-of-atmosphere irradiance at a 30° solar zenith angle caused by the gamma-weighted two-stream approximation is 6.1 W m^{−2} (0.005 albedo bias) with a one-layer overcast cloud where a positive value indicates an overestimate by the approximation compared with the irradiance computed using the independent column approximation. Approximately one-half of the error is due to deviations of optical thickness distributions from a gamma distribution and the other half of the error is due to other approximations in the model. The error increases to 14.7 W m^{−2} (0.012 albedo bias) when the computational layer dividing the cloud layer is increased to four. The increase is because of difficulties in treating the correlation of cloud properties in the vertical direction. Because the optical thickness under partly cloudy conditions, which contribute two-thirds of cloudy footprints, is smaller, the error is smaller than under overcast conditions; the average error for partly cloudy condition is −2.4 W m^{−2} (−0.002 albedo bias) at a 30° solar zenith angle. The corresponding average error caused by the effective thickness approximation is 0.5 W m^{−2} for overcast conditions and −21.5 W m^{−2} (−0.018 albedo bias) for partly cloudy conditions. Although the error caused by the effective thickness approximation depends strongly on the optical thickness, its average error under overcast conditions is smaller than the error caused by the gamma-weighted two-stream approximation because the errors at small and large optical thicknesses cancel each other. Based on these error analyses, the daily average error caused by the gamma-weighted two-stream and effective thickness approximations is less than 2 W m^{−2}.