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- Author or Editor: Robert F. Cahalan x

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## Abstract

We Present a simple Budyko-Sellers type climate model which is forced by a heating term whose time dependence is white noise and whose space-separated autocorrelation is independent of position and orientation on the sphere (statistical homogeneity). Such models with diffusive transport are analytically soluble by expansion into spherical harmonies. The modes are dynamically and statistically independent. Each satisfies a simple Langevin equation having a scale-dependent characteristic time. Climate anomalies in these models have an interval of predictability which can be explicitly computed. The predictability interval is independent of the wavenumber spectrum of the forcing in this class of models. We present the predictability results for all scales and discuss the implications for more realistic models.

## Abstract

We Present a simple Budyko-Sellers type climate model which is forced by a heating term whose time dependence is white noise and whose space-separated autocorrelation is independent of position and orientation on the sphere (statistical homogeneity). Such models with diffusive transport are analytically soluble by expansion into spherical harmonies. The modes are dynamically and statistically independent. Each satisfies a simple Langevin equation having a scale-dependent characteristic time. Climate anomalies in these models have an interval of predictability which can be explicitly computed. The predictability interval is independent of the wavenumber spectrum of the forcing in this class of models. We present the predictability results for all scales and discuss the implications for more realistic models.

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## Abstract

This paper treats the stability of steady-state solutions of some simple, latitude-dependent, energy-balance climate models. For north-south symmetric solutions of models with an ice-cap-type albedo feed-back, and for the sum of horizontal transport and infrared radiation given by a linear operator, it is possible to prove a “slope-stability” theorem; i.e., if the local slope of the steady-state icelinc latitude versus solar constant curve is positive (negative) the steady-state solution is stable (unstable). Certain rather weak restrictions on the albedo function and on the heat transport are required for the proof, and their physical basis is discussed in the text.

## Abstract

This paper treats the stability of steady-state solutions of some simple, latitude-dependent, energy-balance climate models. For north-south symmetric solutions of models with an ice-cap-type albedo feed-back, and for the sum of horizontal transport and infrared radiation given by a linear operator, it is possible to prove a “slope-stability” theorem; i.e., if the local slope of the steady-state icelinc latitude versus solar constant curve is positive (negative) the steady-state solution is stable (unstable). Certain rather weak restrictions on the albedo function and on the heat transport are required for the proof, and their physical basis is discussed in the text.

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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

Inhomogeneous distributions of liquid water like those observed in real clouds generally reflect less solar radiation than idealized uniform distributions assumed in plane-parallel theory. Here the authors determine cloud reflectivity and the associated plane-parallel albedo bias from distributions of liquid water path derived from 28 days of microwave radiometer measurements obtained on Porto Santo Island in the Madeiras during June 1992 as part of the Atlantic Stratocumulus Transition Experiment (ASTEX). The distributions are determined for each hour of the day, both for composites of the full act of 28 days and for a subset of 8 days having a high fraction of relatively thick cloud. Both sets are compared with results obtained from California stratocumulus during FM [First ISCCP (International Satellite Cloud Climatology Project) Regional Experiment].

In FIRE the albedo bin was dominated by variability of the cloud optical depth, as measured by a fractal parameter, 0≤ *f* 0 ≤ 1, while the ASTEX results are more complex. Mean cloud fraction above a 10 g m^{−1} threshold is about 50% in the 28-day set, compared to 76% in the 8-day subset and 82% in FIRE. Cloud fraction is sensitive to the threshold for the 28 ASTEX days, probably due to a large fraction of thin cloud below the threshold, but this is not the case for the 8-day subset or for FIRE. Clear fractions during ASTEX are generally of shorter duration than those in FIRE, as are those in the 8-day subset. The diurnal mean fractal parameter is about 0.6 in ASTEX compared to 0.5 in FIRE, while the 8-day subset has nearly the same mean but a wider range. The diurnal cycle in cloud albedo mid and albedo bias is computed from the cloud parameters for both sets, assuming zero clear-sky albedo. The total absolute albedo bias rises to values above 0.3 at sunrise and sunset, but since there is little incident energy at that time, the reflected flux is more affected by the midday bias. The total albedo bias has a 10OO LST maximum of about 0.3, largely due to a cloud fraction contribution of 0.2, absent in FIRE because in that case cloud frontier remains near 100% until after 1000 LST. The albedo bias has a second maximum of about 0.2 at noon, again mainly from cloud fraction and then drops to a minimum of about 0.1 at 1400 LST, when cloud fraction and fractal structure contribute about equally. Finally, a third maximum due to cloud fraction occurs at 1600 LST.

In the, 8-day subset the 1000 LST maximum becomes dominated by the frontal structure, since the cloud fraction remains near 100% until 1000 LST, as in FIRE. The noon maximum receives roughly equal contributions, while the 1400 LST minimum bias is mainly due to fractal structure. Finally, the 1600 LST maximum and the evening limb bias are similar to those of the full 28-day set. These results show lids cloud fractal and radiative properties can vary considerably from one site and time to another mid at different times within the same site, as meterological conditions change.

## Abstract

Inhomogeneous distributions of liquid water like those observed in real clouds generally reflect less solar radiation than idealized uniform distributions assumed in plane-parallel theory. Here the authors determine cloud reflectivity and the associated plane-parallel albedo bias from distributions of liquid water path derived from 28 days of microwave radiometer measurements obtained on Porto Santo Island in the Madeiras during June 1992 as part of the Atlantic Stratocumulus Transition Experiment (ASTEX). The distributions are determined for each hour of the day, both for composites of the full act of 28 days and for a subset of 8 days having a high fraction of relatively thick cloud. Both sets are compared with results obtained from California stratocumulus during FM [First ISCCP (International Satellite Cloud Climatology Project) Regional Experiment].

In FIRE the albedo bin was dominated by variability of the cloud optical depth, as measured by a fractal parameter, 0≤ *f* 0 ≤ 1, while the ASTEX results are more complex. Mean cloud fraction above a 10 g m^{−1} threshold is about 50% in the 28-day set, compared to 76% in the 8-day subset and 82% in FIRE. Cloud fraction is sensitive to the threshold for the 28 ASTEX days, probably due to a large fraction of thin cloud below the threshold, but this is not the case for the 8-day subset or for FIRE. Clear fractions during ASTEX are generally of shorter duration than those in FIRE, as are those in the 8-day subset. The diurnal mean fractal parameter is about 0.6 in ASTEX compared to 0.5 in FIRE, while the 8-day subset has nearly the same mean but a wider range. The diurnal cycle in cloud albedo mid and albedo bias is computed from the cloud parameters for both sets, assuming zero clear-sky albedo. The total absolute albedo bias rises to values above 0.3 at sunrise and sunset, but since there is little incident energy at that time, the reflected flux is more affected by the midday bias. The total albedo bias has a 10OO LST maximum of about 0.3, largely due to a cloud fraction contribution of 0.2, absent in FIRE because in that case cloud frontier remains near 100% until after 1000 LST. The albedo bias has a second maximum of about 0.2 at noon, again mainly from cloud fraction and then drops to a minimum of about 0.1 at 1400 LST, when cloud fraction and fractal structure contribute about equally. Finally, a third maximum due to cloud fraction occurs at 1600 LST.

In the, 8-day subset the 1000 LST maximum becomes dominated by the frontal structure, since the cloud fraction remains near 100% until 1000 LST, as in FIRE. The noon maximum receives roughly equal contributions, while the 1400 LST minimum bias is mainly due to fractal structure. Finally, the 1600 LST maximum and the evening limb bias are similar to those of the full 28-day set. These results show lids cloud fractal and radiative properties can vary considerably from one site and time to another mid at different times within the same site, as meterological conditions change.

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## Abstract

To improve radiative transfer calculations for inhomogeneous clouds, a consistent means of modeling inhomogeneity is needed. One current method of modeling cloud inhomogeneity is through the use of fractal parameters. This method is based on the supposition that cloud inhomogeneity over a large ranges of scales is related. An analysis technique named wavelet analysis provides a means of studying the multiscale nature of cloud inhomogeneity. In this paper, the authors discuss the analysis and modeling of cloud inhomogeneity through the use of wavelet analysis.

Wavelet analysis as well as other windowed analysis techniques are used to study liquid water path (LWP) measurements obtained during the marine stratocumulus phase of the First ISCCP (International Satellite Cloud Climatology Project) Regional Experiment. Statistics obtained using analysis windows, which are translated to span the LWP dataset, are used to study the local (small scale) properties of the cloud field as well as their time dependence. The LWP data are transformed onto an orthogonal wavelet basis that represents the data as a number of times series. Each of these time series lies within a frequency band and has a mean frequency that is half the frequency of the previous band. Wavelet analysis combined with translated analysis windows reveals that the local standard deviation of each frequency band is correlated with the local standard deviation of the other frequency bands. The ratio between the standard deviation of adjacent frequency bands is 0.9 and remains constant with respect to time. This ratio defined as the variance coupling parameter is applicable to all of the frequency bands studied and appears to be related to the slope of the data's power spectrum.

Similar analyses are performed on two cloud inhomogeneity models, which use fractal-based concepts to introduce inhomogeneity into a uniform cloud field. The bounded cascade model does this by iteratively redistributing LWP at each scale using the value of the local mean. This model is reformulated into a wavelet multiresolution framework, thereby presenting a number of variants of the bounded cascade model. One variant introduced in this paper is the “variance coupled model”, which redistributes LWP using the local standard deviation and the variance coupling parameter. While the bounded cascade model provides an elegant two parameter model for generating cloud inhomogeneity, the multiresolution framework provides more flexibility at the expense of model complexity. Comparisons are made with the results from the LWP data analysis to demonstrate both the strengths and weaknesses of these models.

## Abstract

To improve radiative transfer calculations for inhomogeneous clouds, a consistent means of modeling inhomogeneity is needed. One current method of modeling cloud inhomogeneity is through the use of fractal parameters. This method is based on the supposition that cloud inhomogeneity over a large ranges of scales is related. An analysis technique named wavelet analysis provides a means of studying the multiscale nature of cloud inhomogeneity. In this paper, the authors discuss the analysis and modeling of cloud inhomogeneity through the use of wavelet analysis.

Wavelet analysis as well as other windowed analysis techniques are used to study liquid water path (LWP) measurements obtained during the marine stratocumulus phase of the First ISCCP (International Satellite Cloud Climatology Project) Regional Experiment. Statistics obtained using analysis windows, which are translated to span the LWP dataset, are used to study the local (small scale) properties of the cloud field as well as their time dependence. The LWP data are transformed onto an orthogonal wavelet basis that represents the data as a number of times series. Each of these time series lies within a frequency band and has a mean frequency that is half the frequency of the previous band. Wavelet analysis combined with translated analysis windows reveals that the local standard deviation of each frequency band is correlated with the local standard deviation of the other frequency bands. The ratio between the standard deviation of adjacent frequency bands is 0.9 and remains constant with respect to time. This ratio defined as the variance coupling parameter is applicable to all of the frequency bands studied and appears to be related to the slope of the data's power spectrum.

Similar analyses are performed on two cloud inhomogeneity models, which use fractal-based concepts to introduce inhomogeneity into a uniform cloud field. The bounded cascade model does this by iteratively redistributing LWP at each scale using the value of the local mean. This model is reformulated into a wavelet multiresolution framework, thereby presenting a number of variants of the bounded cascade model. One variant introduced in this paper is the “variance coupled model”, which redistributes LWP using the local standard deviation and the variance coupling parameter. While the bounded cascade model provides an elegant two parameter model for generating cloud inhomogeneity, the multiresolution framework provides more flexibility at the expense of model complexity. Comparisons are made with the results from the LWP data analysis to demonstrate both the strengths and weaknesses of these models.

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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.