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- Author or Editor: William Ridgway x

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## Abstract

In this paper, the effect of cloud structure on column absorption by water vapor is investigated. Radiative fluxes above and below horizontally inhomogeneous liquid water clouds are computed using an efficient Monte Carlo technique, the independent pixel approximation, and plane-parallel theory. Cloud inhomogeneity is simulated by two related fractal models that use bounded cascades for the horizontal distribution of optical depth. The first (“clumpy”) model has constant cloud top and base, hence a constant geometrical thickness but varying extinction; the second (“bumpy”) model has constant extinction and cloud base, hence variable cloud-top and geometrical thickness. The spectral range between 0.9 and 1.0 *μ*m (with strong water vapor absorption and negligible cloud liquid water absorption) is selected for a detailed study, not only of domain-averaged quantities, but also radiation fields. Column-absorption fields are calculated as the difference between the two net fluxes above and below clouds. It is shown that 1) redistribution of cloud liquid water decreases column absorption, that is, plane-parallel absorption is larger than the independent pixel approximation one by 1%–3%; 2) 3D radiative effects enhance column absorption by about 0.6% for the clumpy model and 2% for the bumpy model, that is, Monte Carlo absorption is larger than independent pixel approximation absorption—this effect is most pronounced for the bumpy cloud model at solar zenith angle ≈45°; and 3) plane-parallel absorption is larger than 3D Monte Carlo absorption for high solar elevations and nearly equal to it for low solar elevations. Thus, for extended clouds of thickness 1–2 km or less, in an important water vapor absorption band (0.94 *μ*m), the authors do not find a significant enhancement of cloud absorption due to horizontal inhomogeneity.

## Abstract

In this paper, the effect of cloud structure on column absorption by water vapor is investigated. Radiative fluxes above and below horizontally inhomogeneous liquid water clouds are computed using an efficient Monte Carlo technique, the independent pixel approximation, and plane-parallel theory. Cloud inhomogeneity is simulated by two related fractal models that use bounded cascades for the horizontal distribution of optical depth. The first (“clumpy”) model has constant cloud top and base, hence a constant geometrical thickness but varying extinction; the second (“bumpy”) model has constant extinction and cloud base, hence variable cloud-top and geometrical thickness. The spectral range between 0.9 and 1.0 *μ*m (with strong water vapor absorption and negligible cloud liquid water absorption) is selected for a detailed study, not only of domain-averaged quantities, but also radiation fields. Column-absorption fields are calculated as the difference between the two net fluxes above and below clouds. It is shown that 1) redistribution of cloud liquid water decreases column absorption, that is, plane-parallel absorption is larger than the independent pixel approximation one by 1%–3%; 2) 3D radiative effects enhance column absorption by about 0.6% for the clumpy model and 2% for the bumpy model, that is, Monte Carlo absorption is larger than independent pixel approximation absorption—this effect is most pronounced for the bumpy cloud model at solar zenith angle ≈45°; and 3) plane-parallel absorption is larger than 3D Monte Carlo absorption for high solar elevations and nearly equal to it for low solar elevations. Thus, for extended clouds of thickness 1–2 km or less, in an important water vapor absorption band (0.94 *μ*m), the authors do not find a significant enhancement of cloud absorption due to horizontal inhomogeneity.

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## Abstract

The spectral of solar radiation in typical water clouds is determined using a radiative transfer model based on LOWTRAN transmission functions at a 20 cm^{−1} resolution and Monte Carlo simulations of photon pathlength distributions. Relative absorption by the vapor and droplets within each cloud is obtained, and both plane-parallel and horizontally finite clouds are considered.

Results indicate slightly lower absorption than found previously, with boundary layer clouds typically absorbing 9% of the extraterrestrial insolation for overhead sun. Cloud absorption depends strongly on the presence of water vapor above the cloud top and solar zenith angle, moderately on cloud aspect ratio, and (provided the cloud is neither tenuous nor broken) weakly on cloud type and thickness. The droplets, not the vapor, are shown to be the dominant absorbers within the cloud, except in the absence of water vapor above the cloud top, in which case the vapor and droplets make similar contributions to the low cloud absorption. For many of the cases modeled, the sum of the cloud and atmospheric absorption remained invariant, allowing the net solar radiation budget at the surface to be deduced from broadband satellite measurements of albedo. An explanation for this behavior is found in the analysis of the spectral absorption by the different components.

## Abstract

The spectral of solar radiation in typical water clouds is determined using a radiative transfer model based on LOWTRAN transmission functions at a 20 cm^{−1} resolution and Monte Carlo simulations of photon pathlength distributions. Relative absorption by the vapor and droplets within each cloud is obtained, and both plane-parallel and horizontally finite clouds are considered.

Results indicate slightly lower absorption than found previously, with boundary layer clouds typically absorbing 9% of the extraterrestrial insolation for overhead sun. Cloud absorption depends strongly on the presence of water vapor above the cloud top and solar zenith angle, moderately on cloud aspect ratio, and (provided the cloud is neither tenuous nor broken) weakly on cloud type and thickness. The droplets, not the vapor, are shown to be the dominant absorbers within the cloud, except in the absence of water vapor above the cloud top, in which case the vapor and droplets make similar contributions to the low cloud absorption. For many of the cases modeled, the sum of the cloud and atmospheric absorption remained invariant, allowing the net solar radiation budget at the surface to be deduced from broadband satellite measurements of albedo. An explanation for this behavior is found in the analysis of the spectral absorption by the different components.

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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

Water vapor contributes a maximum of 1°C/day to the middle atmospheric thermal infrared (IR) cooling. This magnitude is small but not negligible. Because of the small amount of mass involved and the extremely narrow molecular absorption lines at pressures less than 1 mb, only a few existing parameterizations can compute accurately the water vapor cooling in this region. The accuracy and efficiency of two IR parameterizations are examined in this study. One is the correlated-*k* distribution method, and the other is the table look-up using precomputed transmission functions. Both methods can accurately compute the cooling rate from the earth's surface to 0.01 mb with an error of only a few percent. The contribution to the cooling rate at pressures <1 mb comes from a very small fraction (<0.005) of the spectrum near the centers of the absorption bands, where the absorption coefficient varies by four orders of magnitude. It requires at least 100 terms of the *k*-distribution function to accurately compute the cooling profile. The method of table look-up is, therefore, much faster than the correlated-*k* distribution method for computing the water vapor cooling profile involving both the middle and lower atmospheres.

## Abstract

Water vapor contributes a maximum of 1°C/day to the middle atmospheric thermal infrared (IR) cooling. This magnitude is small but not negligible. Because of the small amount of mass involved and the extremely narrow molecular absorption lines at pressures less than 1 mb, only a few existing parameterizations can compute accurately the water vapor cooling in this region. The accuracy and efficiency of two IR parameterizations are examined in this study. One is the correlated-*k* distribution method, and the other is the table look-up using precomputed transmission functions. Both methods can accurately compute the cooling rate from the earth's surface to 0.01 mb with an error of only a few percent. The contribution to the cooling rate at pressures <1 mb comes from a very small fraction (<0.005) of the spectrum near the centers of the absorption bands, where the absorption coefficient varies by four orders of magnitude. It requires at least 100 terms of the *k*-distribution function to accurately compute the cooling profile. The method of table look-up is, therefore, much faster than the correlated-*k* distribution method for computing the water vapor cooling profile involving both the middle and lower atmospheres.

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## Abstract

A medium-sized band model for water vapor and CO_{2} absorption is developed using the one-parameter scaling approximation. The infrared spectrum is divided into 10 bands. The Planck-weighted diffuse transmittance is reduced to a function dependent only upon the scaled absorber amount and fit by an exponential sum. By selecting specific sets of absorption coefficients for exponential-sum fitting, computations of fluxes and cooling rate are made very fast. Compared to a broadband model, the accuracy, speed, and versatility are all enhanced. With absorption due to water vapor line, continuum, CO_{2} as well as O_{3} included, the parameterization introduces an error of < 1.5 W m^{−2} in fluxes and <0.15°C day^{−1} in the tropospheric and lower stratospheric cooling rates.

## Abstract

A medium-sized band model for water vapor and CO_{2} absorption is developed using the one-parameter scaling approximation. The infrared spectrum is divided into 10 bands. The Planck-weighted diffuse transmittance is reduced to a function dependent only upon the scaled absorber amount and fit by an exponential sum. By selecting specific sets of absorption coefficients for exponential-sum fitting, computations of fluxes and cooling rate are made very fast. Compared to a broadband model, the accuracy, speed, and versatility are all enhanced. With absorption due to water vapor line, continuum, CO_{2} as well as O_{3} included, the parameterization introduces an error of < 1.5 W m^{−2} in fluxes and <0.15°C day^{−1} in the tropospheric and lower stratospheric cooling rates.

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## Abstract

Parameterizations for infrared radiation (IR) in clear atmosphere can be made fast and accurate by grouping spectral regions with similar radiative properties, and by separating the low pressure region of the atmosphere from the high pressure region. Various approaches are presented in this study to parameterizing the broadband transmission functions for use in numerical climate models. For water vapor and carbon dioxide (CO_{2}) bands, the transmission functions are parameterized separately for the middle atmosphere (0.01–30 mb) and for the region below. In the middle atmosphere where the dependence of absorption on pressure and temperature is not strong, the diffuse transmission functions are derived from that at a reference pressure and temperature. In the lower stratosphere and the troposphere, the spectra are grouped into band-center regions and band-wing regions. One-parameter scaling is applied to approximate a nonhomogeneous path with an equivalent homogeneous path, and the diffuse transmittances are either fit by analytical functions or interpolated from precomputed tables.

As opposed to the one-parameter scaling, which applies only to a relatively narrow pressure range, the two-parameter scaling (commonly called the Curtis-Godson approximation) is applied to parameterizing the carbon dioxide (CO_{2}) and ozone (O_{3}) transmission functions in both the middle and the lower atmosphere. The diffuse transmission functions are simply interpolated from three small precomputed tables. The accuracies of cooling rates in the 15-μm band computed using the approximation for both the middle and the lower atmospheres are comparable with that using the parameterizations separately for the middle and the lower atmospheres. The radiative effect of nitrous oxide (N_{2}O) and methane (CH_{4}) is also examined. Parameterizations are presented for the N_{2}O and CH_{4} diffuse transmission functions.

## Abstract

Parameterizations for infrared radiation (IR) in clear atmosphere can be made fast and accurate by grouping spectral regions with similar radiative properties, and by separating the low pressure region of the atmosphere from the high pressure region. Various approaches are presented in this study to parameterizing the broadband transmission functions for use in numerical climate models. For water vapor and carbon dioxide (CO_{2}) bands, the transmission functions are parameterized separately for the middle atmosphere (0.01–30 mb) and for the region below. In the middle atmosphere where the dependence of absorption on pressure and temperature is not strong, the diffuse transmission functions are derived from that at a reference pressure and temperature. In the lower stratosphere and the troposphere, the spectra are grouped into band-center regions and band-wing regions. One-parameter scaling is applied to approximate a nonhomogeneous path with an equivalent homogeneous path, and the diffuse transmittances are either fit by analytical functions or interpolated from precomputed tables.

As opposed to the one-parameter scaling, which applies only to a relatively narrow pressure range, the two-parameter scaling (commonly called the Curtis-Godson approximation) is applied to parameterizing the carbon dioxide (CO_{2}) and ozone (O_{3}) transmission functions in both the middle and the lower atmosphere. The diffuse transmission functions are simply interpolated from three small precomputed tables. The accuracies of cooling rates in the 15-μm band computed using the approximation for both the middle and the lower atmospheres are comparable with that using the parameterizations separately for the middle and the lower atmospheres. The radiative effect of nitrous oxide (N_{2}O) and methane (CH_{4}) is also examined. Parameterizations are presented for the N_{2}O and CH_{4} diffuse transmission functions.

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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.