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## Abstract

We present a method for calculating the spectral albedo of snow which can be used at any wavelength in the solar spectrum and which accounts for diffusely or directly incident radiation at any zenith angle. For deep snow, the model contains only one adjustable parameter, an effective grain size, which is close to observed grain sizes. A second parameter, the liquid-equivalent depth, is required only for relatively thin snow.

In order for the model to make realistic predictions, it must account for the extreme anisotropy of scattering by snow particles. This is done by using the “delta-Eddington” approximation for multiple scattering, together with Mie theory for single scattering.

The spectral albedo from 0.3 to 5 μm wavelength is examined as a function of the effective grain size, the solar zenith angle, the snowpack thickness, and the ratio of diffuse to direct solar incidence. The decrease in albedo due to snow aging can be mimicked by reasonable increases in grain size (50–100 μm for new snow, growing to 1 mm for melting old snow).

The model agrees well with observations for wavelengths above 0.8 μm. In the visible and near-UV, on the other hand, the model may predict albedos up to 15% higher than those which are actually observed. Increased grain size alone cannot lower the model albedo sufficiently to match these observations. It is also argued that the two major effects which are neglected in the model, namely nonsphericity of snow grains and near-field scattering, cannot be responsible for the discrepancy. Insufficient snow depth and error in measured absorption coefficient are also ruled out as the explanation. The remaining hypothesis is that visible snow albedo is reduced by trace amounts of absorptive impurities (Warren and Wiscombe, 1980, Part II).

## Abstract

We present a method for calculating the spectral albedo of snow which can be used at any wavelength in the solar spectrum and which accounts for diffusely or directly incident radiation at any zenith angle. For deep snow, the model contains only one adjustable parameter, an effective grain size, which is close to observed grain sizes. A second parameter, the liquid-equivalent depth, is required only for relatively thin snow.

In order for the model to make realistic predictions, it must account for the extreme anisotropy of scattering by snow particles. This is done by using the “delta-Eddington” approximation for multiple scattering, together with Mie theory for single scattering.

The spectral albedo from 0.3 to 5 μm wavelength is examined as a function of the effective grain size, the solar zenith angle, the snowpack thickness, and the ratio of diffuse to direct solar incidence. The decrease in albedo due to snow aging can be mimicked by reasonable increases in grain size (50–100 μm for new snow, growing to 1 mm for melting old snow).

The model agrees well with observations for wavelengths above 0.8 μm. In the visible and near-UV, on the other hand, the model may predict albedos up to 15% higher than those which are actually observed. Increased grain size alone cannot lower the model albedo sufficiently to match these observations. It is also argued that the two major effects which are neglected in the model, namely nonsphericity of snow grains and near-field scattering, cannot be responsible for the discrepancy. Insufficient snow depth and error in measured absorption coefficient are also ruled out as the explanation. The remaining hypothesis is that visible snow albedo is reduced by trace amounts of absorptive impurities (Warren and Wiscombe, 1980, Part II).

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## Abstract

Small highly absorbing particles, present in concentrations of only 1 part per million by weight (ppmw) or less, can lower snow albedo in the visible by 5–15% from the high values (96–99%) predicted for pure snow in Part I. These particles have, however, no effect on snow albedo beyond 0.9 μm wavelength where ice itself becomes a strong absorber. Thus we have an attractive explanation for the discrepancy between theory and observation described in Part I, a discrepancy which seemingly cannot be resolved on the basis of near-field scattering and nonsphericity effects.

Desert dust and carbon soot are the most likely contaminants. But careful measurements of spectral snow albedo in the Arctic and Antarctic paint to a “grey” absorber, one whose imaginary refractive index is nearly constant across the visible spectrum. Thus carbon soot, rather than the red iron oxide normally present in desert dust, is strongly indicated at these sites. Soot particles of radius 0.1 μm, in concentrations of only 0.3 ppmw, can explain the albedo measurements of Grenfell and Maykut on Arctic Ice Island T-3. This amount is consistent with some observations of soot in Arctic air masses. 1.5 ppmw of soot is required to explain the Antarctic observations of Kuhn and Siogas, which seemed an unrealistically large amount for the earth's most unpolluted continent until we learned that burning of camp heating fuel and aircraft exhaust indeed had contaminated the measurement site with soot.

Midlatitude snowfields are likely to contain larger absolute amounts of soot and dust than their polar counterparts, but the snowfall is also much larger, so that the ppmw contamination does not differ drastically until melting begins. Nevertheless, the variations in absorbing particle concentration which *will* exist can help to explain the wide range of visible snow albedos reported in the literature.

Longwave emissivity of snow is unaltered by its soot and dust content. Thus the depression of snow albedo in the visible is a systematic effect and always results in more energy being absorbed at a snow-covered surface than would be the case for pure snow. Thus man-made carbon soot aerosol may continue to exert a significant warming effect on the earth's climate even after it is removed from the atmosphere.

## Abstract

Small highly absorbing particles, present in concentrations of only 1 part per million by weight (ppmw) or less, can lower snow albedo in the visible by 5–15% from the high values (96–99%) predicted for pure snow in Part I. These particles have, however, no effect on snow albedo beyond 0.9 μm wavelength where ice itself becomes a strong absorber. Thus we have an attractive explanation for the discrepancy between theory and observation described in Part I, a discrepancy which seemingly cannot be resolved on the basis of near-field scattering and nonsphericity effects.

Desert dust and carbon soot are the most likely contaminants. But careful measurements of spectral snow albedo in the Arctic and Antarctic paint to a “grey” absorber, one whose imaginary refractive index is nearly constant across the visible spectrum. Thus carbon soot, rather than the red iron oxide normally present in desert dust, is strongly indicated at these sites. Soot particles of radius 0.1 μm, in concentrations of only 0.3 ppmw, can explain the albedo measurements of Grenfell and Maykut on Arctic Ice Island T-3. This amount is consistent with some observations of soot in Arctic air masses. 1.5 ppmw of soot is required to explain the Antarctic observations of Kuhn and Siogas, which seemed an unrealistically large amount for the earth's most unpolluted continent until we learned that burning of camp heating fuel and aircraft exhaust indeed had contaminated the measurement site with soot.

Midlatitude snowfields are likely to contain larger absolute amounts of soot and dust than their polar counterparts, but the snowfall is also much larger, so that the ppmw contamination does not differ drastically until melting begins. Nevertheless, the variations in absorbing particle concentration which *will* exist can help to explain the wide range of visible snow albedos reported in the literature.

Longwave emissivity of snow is unaltered by its soot and dust content. Thus the depression of snow albedo in the visible is a systematic effect and always results in more energy being absorbed at a snow-covered surface than would be the case for pure snow. Thus man-made carbon soot aerosol may continue to exert a significant warming effect on the earth's climate even after it is removed from the atmosphere.

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The fundamental climatic role of radiative processes has spurred the development of increasingly sophisticated models of radiative transfer in the earth–atmosphere system. Since the basic physics of radiative transfer is rather well known, this was thought to be an exercise in refinement. Therefore, it came as a great surprise when large differences (30–70 W m^{−2}) were found among longwave infrared fluxes predicted by over 30 radiation models for identical atmospheres during the intercomparison of radiation codes used in climate models (ICRCCM) exercise in the mid-1980s. No amount of further calculation could explain these and other intermodel differences; thus, it became clear that what was needed was a set of accurate atmospheric spectral radiation data measured simultaneously with the important radiative properties of the atmosphere like temperature and humidity.

To obtain this dataset, the ICRCCM participants charged the authors to develop an experimental field program. So, the authors developed a program concept fotir the Spectral Radiance Experiment (SPECTRE), organized a team of scientists with expertise in atmospheric field spectroscopy, remote sensing, and radiative transfer, and secured funding from the Department of Energy and the National Aeronautics and Space Administration. The goal of SPECTRE was to establish a reference standard against which to compare models and also to drastically reduce the uncertainties in humidity, aerosol, etc., which radiation modelers had invoked in the past to excuse disagreements with observations. To avoid the high cost and sampling problems associated with aircraft, SPECTRE was designed to be a surface-based program.

The field portion of SPECTRE took place 13 November to 7 December 1991, in Coffeyville, Kansas, in conjunction with the FIRE Cirrus II field program, and most of the data have been calibrated to a usable form and will soon appear on a CD-ROM. This paper provides an overview of the data obtained; it also outlines the plans to use this data to further advance the ICRCCM goal of testing the verisimilitude of radiation parameterizations used in climate models.

The fundamental climatic role of radiative processes has spurred the development of increasingly sophisticated models of radiative transfer in the earth–atmosphere system. Since the basic physics of radiative transfer is rather well known, this was thought to be an exercise in refinement. Therefore, it came as a great surprise when large differences (30–70 W m^{−2}) were found among longwave infrared fluxes predicted by over 30 radiation models for identical atmospheres during the intercomparison of radiation codes used in climate models (ICRCCM) exercise in the mid-1980s. No amount of further calculation could explain these and other intermodel differences; thus, it became clear that what was needed was a set of accurate atmospheric spectral radiation data measured simultaneously with the important radiative properties of the atmosphere like temperature and humidity.

To obtain this dataset, the ICRCCM participants charged the authors to develop an experimental field program. So, the authors developed a program concept fotir the Spectral Radiance Experiment (SPECTRE), organized a team of scientists with expertise in atmospheric field spectroscopy, remote sensing, and radiative transfer, and secured funding from the Department of Energy and the National Aeronautics and Space Administration. The goal of SPECTRE was to establish a reference standard against which to compare models and also to drastically reduce the uncertainties in humidity, aerosol, etc., which radiation modelers had invoked in the past to excuse disagreements with observations. To avoid the high cost and sampling problems associated with aircraft, SPECTRE was designed to be a surface-based program.

The field portion of SPECTRE took place 13 November to 7 December 1991, in Coffeyville, Kansas, in conjunction with the FIRE Cirrus II field program, and most of the data have been calibrated to a usable form and will soon appear on a CD-ROM. This paper provides an overview of the data obtained; it also outlines the plans to use this data to further advance the ICRCCM goal of testing the verisimilitude of radiation parameterizations used in climate models.

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## Abstract

A new method for retrieving cloud optical depth from ground-based measurements of zenith radiance in the red (RED) and near-infrared (NIR) spectral regions is introduced. Because zenith radiance does not have a one-to-one relationship with optical depth, it is absolutely impossible to use a monochromatic retrieval. On the other side, algebraic combinations of spectral radiances, such as normalized difference cloud index (NDCI), while largely removing nonuniqueness and the radiative effects of cloud inhomogeneity, can result in poor retrievals due to its insensitivity to cloud fraction. Instead, both RED and NIR radiances as points on the “RED versus NIR” plane are proposed to be used for retrieval. The proposed retrieval method is applied to Cimel measurements at the Atmospheric Radiation Measurements (ARM) site in Oklahoma. Cimel, a multichannel sun photometer, is a part of the Aerosol Robotic Network (AERONET)—a ground-based network for monitoring aerosol optical properties. The results of retrieval are compared with the ones from microwave radiometer (MWR) and multifilter rotating shadowband radiometer (MFRSR) located next to Cimel at the ARM site. In addition, the performance of the retrieval method is assessed using a fractal model of cloud inhomogeneity and broken cloudiness. The preliminary results look very promising both theoretically and from measurements.

## Abstract

A new method for retrieving cloud optical depth from ground-based measurements of zenith radiance in the red (RED) and near-infrared (NIR) spectral regions is introduced. Because zenith radiance does not have a one-to-one relationship with optical depth, it is absolutely impossible to use a monochromatic retrieval. On the other side, algebraic combinations of spectral radiances, such as normalized difference cloud index (NDCI), while largely removing nonuniqueness and the radiative effects of cloud inhomogeneity, can result in poor retrievals due to its insensitivity to cloud fraction. Instead, both RED and NIR radiances as points on the “RED versus NIR” plane are proposed to be used for retrieval. The proposed retrieval method is applied to Cimel measurements at the Atmospheric Radiation Measurements (ARM) site in Oklahoma. Cimel, a multichannel sun photometer, is a part of the Aerosol Robotic Network (AERONET)—a ground-based network for monitoring aerosol optical properties. The results of retrieval are compared with the ones from microwave radiometer (MWR) and multifilter rotating shadowband radiometer (MFRSR) located next to Cimel at the ARM site. In addition, the performance of the retrieval method is assessed using a fractal model of cloud inhomogeneity and broken cloudiness. The preliminary results look very promising both theoretically and from measurements.

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## Abstract

By analyzing aircraft measurements of individual drop sizes in clouds, it has been shown in a companion paper that the probability of finding a drop of radius *r* at a linear scale *l* decreases as *l ^{D(r)},* where 0 ≤

*D*(

*r*) ≤ 1. This paper shows striking examples of the spatial distribution of large cloud drops using models that simulate the observed power laws. In contrast to currently used models that assume homogeneity and a Poisson distribution of cloud drops, these models illustrate strong drop clustering, especially with larger drops. The degree of clustering is determined by the observed exponents

*D*(

*r*). The strong clustering of large drops arises naturally from the observed power-law statistics. This clustering has vital consequences for rain physics, including how fast rain can form. For radiative transfer theory, clustering of large drops enhances their impact on the cloud optical path. The clustering phenomenon also helps explain why remotely sensed cloud drop size is generally larger than that measured in situ.

## Abstract

By analyzing aircraft measurements of individual drop sizes in clouds, it has been shown in a companion paper that the probability of finding a drop of radius *r* at a linear scale *l* decreases as *l ^{D(r)},* where 0 ≤

*D*(

*r*) ≤ 1. This paper shows striking examples of the spatial distribution of large cloud drops using models that simulate the observed power laws. In contrast to currently used models that assume homogeneity and a Poisson distribution of cloud drops, these models illustrate strong drop clustering, especially with larger drops. The degree of clustering is determined by the observed exponents

*D*(

*r*). The strong clustering of large drops arises naturally from the observed power-law statistics. This clustering has vital consequences for rain physics, including how fast rain can form. For radiative transfer theory, clustering of large drops enhances their impact on the cloud optical path. The clustering phenomenon also helps explain why remotely sensed cloud drop size is generally larger than that measured in situ.

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## Abstract

Monte Carlo radiative transfer methods are employed here to estimate the plane-parallel albedo bias for marine stratocumulus clouds. This is the bias in estimates of the mesoscale-average albedo, which arises from the assumption that cloud liquid water is uniformly distributed. The authors compare such estimates with those based on a more realistic distribution generated from a fractal model of marine stratocumulus clouds belonging to the class of “bounded cascade” models. In this model the cloud top and base are fixed, so that all variations in cloud shape are ignored. The model generates random variations in liquid water along a single horizontal direction, forming fractal cloud streets while conserving the total liquid water in the cloud field. The model reproduces the mean, variance, and skewness of the vertically integrated cloud liquid water, as well as its observed wavenumber spectrum, which is approximately a power law. The Monte Carlo method keeps track of the three-dimensional paths solar photons take through the cloud field, using a vectorized implementation of a direct technique. The simplifications in the cloud field studied here allow the computations to be accelerated. The Monte Carlo results are compared to those of the independent pixel approximation, which neglects net horizontal photon transport. Differences between the Monte Carlo and independent pixel estimates of the mesoscale-average albedo are on the order of 1% for conservative scattering, while the plane-parallel bias itself is an order of magnitude larger. As cloud absorption increases, the independent pixel approximation agrees even more closely with the Monte Carlo estimates. This result holds for a wide range of sun angles and aspect ratios. Thus, horizontal photon transport can be safely neglected in estimates of the area-average flux for such cloud models. This result relies on the rapid falloff of the wavenumber spectrum of stratocumulus, which ensures that the smaller-scale variability, where the radiative transfer is more three-dimensional, contributes less to the plane-parallel albedo bias than the larger scales, which are more variable. The lack of significant three-dimensional effects also relies on the assumption of a relatively simple geometry. Even with these assumptions, the independent pixel approximation is accurate only for fluxes averaged over large horizontal areas, many photon mean free paths in diameter, and not for local radiance values, which depend strongly on the interaction between neighboring cloud elements.

## Abstract

Monte Carlo radiative transfer methods are employed here to estimate the plane-parallel albedo bias for marine stratocumulus clouds. This is the bias in estimates of the mesoscale-average albedo, which arises from the assumption that cloud liquid water is uniformly distributed. The authors compare such estimates with those based on a more realistic distribution generated from a fractal model of marine stratocumulus clouds belonging to the class of “bounded cascade” models. In this model the cloud top and base are fixed, so that all variations in cloud shape are ignored. The model generates random variations in liquid water along a single horizontal direction, forming fractal cloud streets while conserving the total liquid water in the cloud field. The model reproduces the mean, variance, and skewness of the vertically integrated cloud liquid water, as well as its observed wavenumber spectrum, which is approximately a power law. The Monte Carlo method keeps track of the three-dimensional paths solar photons take through the cloud field, using a vectorized implementation of a direct technique. The simplifications in the cloud field studied here allow the computations to be accelerated. The Monte Carlo results are compared to those of the independent pixel approximation, which neglects net horizontal photon transport. Differences between the Monte Carlo and independent pixel estimates of the mesoscale-average albedo are on the order of 1% for conservative scattering, while the plane-parallel bias itself is an order of magnitude larger. As cloud absorption increases, the independent pixel approximation agrees even more closely with the Monte Carlo estimates. This result holds for a wide range of sun angles and aspect ratios. Thus, horizontal photon transport can be safely neglected in estimates of the area-average flux for such cloud models. This result relies on the rapid falloff of the wavenumber spectrum of stratocumulus, which ensures that the smaller-scale variability, where the radiative transfer is more three-dimensional, contributes less to the plane-parallel albedo bias than the larger scales, which are more variable. The lack of significant three-dimensional effects also relies on the assumption of a relatively simple geometry. Even with these assumptions, the independent pixel approximation is accurate only for fluxes averaged over large horizontal areas, many photon mean free paths in diameter, and not for local radiance values, which depend strongly on the interaction between neighboring cloud elements.

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## Abstract

Studies of paleoclimate and modern observations indicate that evaporative effects limit thermal response in equatorial regions. We develop a latitude-resolved, steady-state energy balance model which incorporates the effect of an evaporative constraint on the variation of equatorial temperature with solar luminosity. For a diffusive model of surface heat transport the constraint requires the diffusion coefficient to vary with insolation. We find that the movement of the iceline with insolation is four times larger than in standard energy balance models with a constant thermal diffusion coefficient. This is a consequence of the global energy balance which forces temperature changes to occur at high latitudes when they are evaporatively buffered at the equator. Nonlinear temperature-ice albedo feedback at high latitudes then amplifies the response leading to greater sensitivity in the vicinity of current climate.

## Abstract

Studies of paleoclimate and modern observations indicate that evaporative effects limit thermal response in equatorial regions. We develop a latitude-resolved, steady-state energy balance model which incorporates the effect of an evaporative constraint on the variation of equatorial temperature with solar luminosity. For a diffusive model of surface heat transport the constraint requires the diffusion coefficient to vary with insolation. We find that the movement of the iceline with insolation is four times larger than in standard energy balance models with a constant thermal diffusion coefficient. This is a consequence of the global energy balance which forces temperature changes to occur at high latitudes when they are evaporatively buffered at the equator. Nonlinear temperature-ice albedo feedback at high latitudes then amplifies the response leading to greater sensitivity in the vicinity of current climate.

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## Abstract

Due to the spatially inhomogeneous nature of clouds there are large uncertainties in validating remote sensing retrievals of cloud properties with traditional in situ cloud probes, which have sampling volumes measured in liters. This paper introduces a new technique called in situ cloud lidar, which can measure extinction in liquid clouds with sampling volumes of millions of cubic meters. In this technique a laser sends out pulses of light horizontally from an aircraft inside an optically thick cloud, and wide-field-of-view detectors viewing upward and downward measure the time series of the number of photons returned. Diffusion theory calculations indicate that the expected in situ lidar time series depends on the extinction and has a functional form of a power law times an exponential, with the exponential scale depending on the distance to the cloud boundary. Simulations of 532-nm wavelength in situ lidar time series are made with a Monte Carlo radiative transfer model in stochastically generated inhomogeneous stratocumulus clouds. Retrieval simulations are performed using a neural network trained on three parameters fit to the time series of each detector to predict 1) the extinction at four volume-averaging scales, 2) the cloud geometric thickness, and 3) the optical depth at four averaging scales. Even with an assumed 20% lidar calibration error the rms extinction and optical depth retrieval accuracy is only 12%. Simulations with a dual wavelength lidar (532 and 1550 nm) give accurate retrievals of liquid water content and effective radius. The results of a mountain-top demonstration of the in situ lidar technique show the expected power-law time series behavior.

## Abstract

Due to the spatially inhomogeneous nature of clouds there are large uncertainties in validating remote sensing retrievals of cloud properties with traditional in situ cloud probes, which have sampling volumes measured in liters. This paper introduces a new technique called in situ cloud lidar, which can measure extinction in liquid clouds with sampling volumes of millions of cubic meters. In this technique a laser sends out pulses of light horizontally from an aircraft inside an optically thick cloud, and wide-field-of-view detectors viewing upward and downward measure the time series of the number of photons returned. Diffusion theory calculations indicate that the expected in situ lidar time series depends on the extinction and has a functional form of a power law times an exponential, with the exponential scale depending on the distance to the cloud boundary. Simulations of 532-nm wavelength in situ lidar time series are made with a Monte Carlo radiative transfer model in stochastically generated inhomogeneous stratocumulus clouds. Retrieval simulations are performed using a neural network trained on three parameters fit to the time series of each detector to predict 1) the extinction at four volume-averaging scales, 2) the cloud geometric thickness, and 3) the optical depth at four averaging scales. Even with an assumed 20% lidar calibration error the rms extinction and optical depth retrieval accuracy is only 12%. Simulations with a dual wavelength lidar (532 and 1550 nm) give accurate retrievals of liquid water content and effective radius. The results of a mountain-top demonstration of the in situ lidar technique show the expected power-law time series behavior.

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## Abstract

An increase in the planetary albedo of the earth-atmosphere system by only 10% can decrease the equilibrium surface temperature to that of the last ice age. Nevertheless, albedo biases of 10% or greater would be introduced into large regions of current climate models if clouds were given their observed liquid water amounts, because of the treatment of clouds as plane parallel. Past work has addressed the effect of cloud shape on albedo; here the focus is on the within-cloud variability of the vertically integrated liquid water. The main result is an estimate of the “plane-parallel albedo bias” using the “independent pixel approximation,” which ignores net horizontal photon transport, from a simple fractal model of marine stratocumulus clouds that ignores the cloud shape. The use of the independent pixel approximation in this context will be justified in a separate Monte Carlo study.

The focus on marine stratocumulus clouds is due to their important role in cloud radiative forcing and also that, of the wide variety of earth's cloud types, they are most nearly plane parallel, so that they have the least albedo bias. The fractal model employed here reproduces both the probability distribution and the wavenumber spectrum of the stratocumulus liquid water path, as observed during the First ISCCP Regional Experiment (FIRE). The model distributes the liquid water by a cascade process, related to the upscale cascade of energy transferred from the cloud thickness scale to the mesoscale by approximately 2D motions. For simplicity, the cloud microphysical parameters are assumed homogeneous, as is the geometrical cloud thickness; and the mesoscale-averaged vertical optical thickness is kept fixed at each step of the cascade. A single new fractal parameter, 0 ≤ *f* ≤ 1, is introduced and determined empirically by the variance of the logarithm of the vertically integrated liquid water. In the case of conservative scattering, the authors are able to estimate the albedo bias analytically as a function of the fractal parameter *f*, mean vertical optical thickness *T _{ν}*, and sun angle

*θ*. Typical observed values are

*f*= 0.5,

*T*= 15, and

_{ν}*θ*= 60°, which give an absolute bias of 0.09, or a relative bias equal to 15% of the plane-parallel albedo of 0.60. The reduced reflectivity of fractal stratocumulus clouds is approximately given by the plane-parallel reflectivity evaluated at a reduced “effective optical thickness,” which when

*f*= 0.5 is

*T*

_{eff}≈ 10.

Study of the diurnal cycle of stratocumulus liquid water during FIRE leads to a key unexpected result: the plane-parallel albedo bias is largest when the cloud fraction reaches 100%, that is, when any bias associated with the cloud fraction vanishes. This is primarily due to the variability increase with cloud fraction. Thus, the within-cloud fractal structure of stratocumulus has a more significant impact on estimates of its mesoscale-average albedo than does the cloud fraction.

## Abstract

An increase in the planetary albedo of the earth-atmosphere system by only 10% can decrease the equilibrium surface temperature to that of the last ice age. Nevertheless, albedo biases of 10% or greater would be introduced into large regions of current climate models if clouds were given their observed liquid water amounts, because of the treatment of clouds as plane parallel. Past work has addressed the effect of cloud shape on albedo; here the focus is on the within-cloud variability of the vertically integrated liquid water. The main result is an estimate of the “plane-parallel albedo bias” using the “independent pixel approximation,” which ignores net horizontal photon transport, from a simple fractal model of marine stratocumulus clouds that ignores the cloud shape. The use of the independent pixel approximation in this context will be justified in a separate Monte Carlo study.

The focus on marine stratocumulus clouds is due to their important role in cloud radiative forcing and also that, of the wide variety of earth's cloud types, they are most nearly plane parallel, so that they have the least albedo bias. The fractal model employed here reproduces both the probability distribution and the wavenumber spectrum of the stratocumulus liquid water path, as observed during the First ISCCP Regional Experiment (FIRE). The model distributes the liquid water by a cascade process, related to the upscale cascade of energy transferred from the cloud thickness scale to the mesoscale by approximately 2D motions. For simplicity, the cloud microphysical parameters are assumed homogeneous, as is the geometrical cloud thickness; and the mesoscale-averaged vertical optical thickness is kept fixed at each step of the cascade. A single new fractal parameter, 0 ≤ *f* ≤ 1, is introduced and determined empirically by the variance of the logarithm of the vertically integrated liquid water. In the case of conservative scattering, the authors are able to estimate the albedo bias analytically as a function of the fractal parameter *f*, mean vertical optical thickness *T _{ν}*, and sun angle

*θ*. Typical observed values are

*f*= 0.5,

*T*= 15, and

_{ν}*θ*= 60°, which give an absolute bias of 0.09, or a relative bias equal to 15% of the plane-parallel albedo of 0.60. The reduced reflectivity of fractal stratocumulus clouds is approximately given by the plane-parallel reflectivity evaluated at a reduced “effective optical thickness,” which when

*f*= 0.5 is

*T*

_{eff}≈ 10.

Study of the diurnal cycle of stratocumulus liquid water during FIRE leads to a key unexpected result: the plane-parallel albedo bias is largest when the cloud fraction reaches 100%, that is, when any bias associated with the cloud fraction vanishes. This is primarily due to the variability increase with cloud fraction. Thus, the within-cloud fractal structure of stratocumulus has a more significant impact on estimates of its mesoscale-average albedo than does the cloud fraction.

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An international program of intercomparison of radiation models has been initiated because of the central role of radiative processes in many proposed climate change mechanisms. Models ranging from the most detailed (line-by-line) to the most-highly parameterized have been compared with each other and with selected aircraft observations. Although line-by-line-model fluxes tend to agree with each other to within one percent (if the water-vapor–continuum absorption is ignored), the less-detailed models show a spread of 10–20 percent. The spread is even larger (30–40 percent) for the sensitivities of the models to changes in important radiation variables, such as carbon dioxide amounts and water-vapor amounts. These spreads are disturbingly large.

Lacking highly accurate flux observations from within the atmosphere, it has been customary to regard line-by-line–model results as “the truth.” However, uncertainties in the physics of line wings and in the proper treatment of the water-vapor continuum make it impossible for the line-by-line models to provide an absolute reference for evaluating less-detailed models. Therefore, a dedicated surface-based field measurement program is recommended in order to properly evaluate model performance; the goal would be to use sophisticated spectrometers to measure accurately spectral radiances rather than integrated fluxes.

An international program of intercomparison of radiation models has been initiated because of the central role of radiative processes in many proposed climate change mechanisms. Models ranging from the most detailed (line-by-line) to the most-highly parameterized have been compared with each other and with selected aircraft observations. Although line-by-line-model fluxes tend to agree with each other to within one percent (if the water-vapor–continuum absorption is ignored), the less-detailed models show a spread of 10–20 percent. The spread is even larger (30–40 percent) for the sensitivities of the models to changes in important radiation variables, such as carbon dioxide amounts and water-vapor amounts. These spreads are disturbingly large.

Lacking highly accurate flux observations from within the atmosphere, it has been customary to regard line-by-line–model results as “the truth.” However, uncertainties in the physics of line wings and in the proper treatment of the water-vapor continuum make it impossible for the line-by-line models to provide an absolute reference for evaluating less-detailed models. Therefore, a dedicated surface-based field measurement program is recommended in order to properly evaluate model performance; the goal would be to use sophisticated spectrometers to measure accurately spectral radiances rather than integrated fluxes.