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Lazaros Oreopoulos and Robert F. Cahalan

Abstract

Two full months (July 2003 and January 2004) of Moderate Resolution Imaging Spectroradiometer (MODIS) Atmosphere Level-3 data from the Terra and Aqua satellites are analyzed in order to characterize the horizontal variability of vertically integrated cloud optical thickness (“cloud inhomogeneity”) at global scales. The monthly climatology of cloud inhomogeneity is expressed in terms of standard parameters, initially calculated for each day of the month at spatial scales of 1° × 1° and subsequently averaged at monthly, zonal, and global scales. Geographical, diurnal, and seasonal changes of inhomogeneity parameters are examined separately for liquid and ice phases and separately over land and ocean. It is found that cloud inhomogeneity is overall weaker in summer than in winter. For liquid clouds, it is also consistently weaker for local morning than local afternoon and over land than ocean. Cloud inhomogeneity is comparable for liquid and ice clouds on a global scale, but with stronger spatial and temporal variations for the ice phase, and exhibits an average tendency to be weaker for near-overcast or overcast grid points of both phases. Depending on cloud phase, hemisphere, surface type, season, and time of day, hemispheric means of the inhomogeneity parameter ν (roughly the square of the ratio of optical thickness mean to standard deviation) have a wide range of ∼1.7 to 4, while for the inhomogeneity parameter χ (the ratio of the logarithmic to linear mean) the range is from ∼0.65 to 0.8. The results demonstrate that the MODIS Level-3 dataset is suitable for studying various aspects of cloud inhomogeneity and may prove invaluable for validating future cloud schemes in large-scale models capable of predicting subgrid variability.

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Gerald R. North and Robert F. Cahalan

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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David A. Short and Robert F. Cahalan

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The interannual variability (IAV) in monthly averaged outgoing infrared radiation (IR, from the NOAA polar orbiting satellites) is observed to be larger during summer than during winter over the north Pacific Ocean. A statistical analysis of the daily observations shows the daily variance to be similar during both seasons while the autocorrelation function is quite different. This leads to a seasonal difference in estimates of the climatic noise level, i.e., the variances expected in summer and winter monthly averages due to the number of effectively independent samples in each average. Because of a less vigorous tropospheric circulation, monthly means of IR during summer are affected by the passage of fewer synoptic-scale disturbances and their attendant cloudiness. Fewer independent samples imply a larger variance in the time averages. While the observed IAV is less in winter, the ratio of the observed IAV to the climatic noise level is larger, suggesting that signals of climatic variability in outgoing IR may be more readily diagnosed during winter in this region. The climatic noise level in monthly averaged IR and cloudiness is also estimated for two other climatic regimes—the quiescent subtropical north Pacific and the ITCZ in the western Pacific.

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Joachim H. Joseph and Robert F. Cahalan

Abstract

Histograms of nearest neighbor spacings of fair weather cumulus at 15 locations Over the world's oceans are presented based on the analysis of high resolution LANDSAT 3 Multispectral Scanner images for amounts of cloud cover ranging from 0.6% to 37.6%. These histograms are found to be essentially the same at all locations analysed, similarly to our previous findings on the size distributions and the fractal dimensions of the perimeters for this cloud type.

The nearest neighbor spacings are linearly dependent on the effective cloud radii, with a proportionality factor ranging from five to twenty. The histograms peak at about 0.5 km. Nearest-neighbor spacings smaller than about a kilometer, associated with cumulus clouds with an effective radius less than a few hundred meters, have a distribution of cloud centers that is almost independently distributed in the horizontal plane and show a tendency for the formation of clumps. Larger spacings of up to thirty kilometers occur and are associated with the larger clouds. These latter spacings are not independent.

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Robert F. Cahalan and Joachim H. Joseph

Abstract

Landsat Multispectral Scanner (MSS) and Thematic Mapper (TM) data, with 80 and 30 m spatial resolution, respectively, have been employed to study the spatial structure of boundary-layer and intertropical convergence zone (ITCZ) clouds. The probability distributions of cloud area and cloud perimeters are found to approximately follow a power-law, with a different power (i.e., fractal dimension) for each cloud type. They are better approximated by a double power-law behavior, indicating a change in the fractal dimension at a characteristic size which depends upon cloud type. The fractal dimension also changes with threshold. The more intense cloud areas are found to have a higher perimeter fractal dimension, perhaps indicative of the increased turbulence at cloud top. A detailed picture of the inhomogeneous spatial structure of various cloud types will contribute to a better understanding of basic cloud processes, and also has implications for the remote sensing of clouds, for their effects on remote sensing of other parameters, and for the parameterization of clouds in general circulation models, all of which rely upon plane-parallel radiative transfer algorithms.

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Robert F. Cahalan and Gerald R. North

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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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Lazaros Oreopoulos, Robert F. Cahalan, and Steven Platnick

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The authors present the global plane-parallel shortwave albedo bias of liquid clouds for two months, July 2003 and January 2004. The cloud optical properties necessary to perform the bias calculations come from the operational Moderate Resolution Imaging Spectroradiometer (MODIS) Terra and MODIS Aqua level-3 datasets. These data, along with ancillary surface albedo and atmospheric information consistent with the MODIS retrievals, are inserted into a broadband shortwave radiative transfer model to calculate the fluxes at the atmospheric column boundaries. The plane-parallel homogeneous (PPH) calculations are based on the mean cloud properties, while independent column approximation (ICA) calculations are based either on 1D histograms of optical thickness or joint 2D histograms of optical thickness and effective radius. The (positive) PPH albedo bias is simply the difference between PPH and ICA albedo calculations. Two types of biases are therefore examined: 1) the bias due to the horizontal inhomogeneity of optical thickness alone (the effective radius is set to the grid mean value) and 2) the bias due to simultaneous variations of optical thickness and effective radius as derived from their joint histograms. The authors find that the global bias of albedo (liquid cloud portion of the grid boxes only) is ∼+0.03, which corresponds to roughly 8% of the global liquid cloud albedo and is only modestly sensitive to the inclusion of horizontal effective radius variability and time of day, but depends strongly on season and latitude. This albedo bias translates to ∼3–3.5 W m−2 of bias (stronger negative values) in the diurnally averaged global shortwave cloud radiative forcing, assuming homogeneous conditions for the fraction of the grid box not covered by liquid clouds; zonal values can be as high as 8 W m−2. Finally, the (positive) broadband atmospheric absorptance bias is about an order of magnitude smaller than the albedo bias. The substantial magnitude of the PPH bias underlines the importance of predicting subgrid variability in GCMs and accounting for its effects on cloud–radiation interactions.

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Anthony Davis, Alexander Marshak, Warren Wiscombe, and Robert Cahalan

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This study investigates the internal structure of marine stratocumulus (Sc) using the spatial fluctuations of liquid water content (LWC) measured along horizontal flights off the coast of southern California during the First ISCCP Regional Experiment (FIRE) in summer of 1987. The results of FIRE 87 data analyses are compared to similar ones for marine Sc probed during the Atlantic Stratocumulus Transition Experiment (ASTEX) in summer 1992 near the Azores. In this first of two parts, the authors use spectral analysis to determine the main scale-invariant regimes, defined by the ranges of scales where wavenumber spectra follow power laws; from there, they discuss stationary issues. Although crucial for obtaining meaningful spatial statistics (e.g., in climate diagnostics), the importance of establishing stationarity—statistical invariance under translation—is often overlooked. The sequel uses multifractal analysis techniques and addresses intermittency issues. By improving our understanding of both nonstationarity and intermittency in atmospheric data, we are in a better position to formulate successful sampling strategies.

Comparing the spectral responses of different instruments to natural LWC variability, the authors find scale breaks (characteristic scales separating two distinct power law regimes) that are spurious, being traceable to well-documented idiosyncrasies of the Johnson–Williams probe and forward scattering spectrometer probes. In data from the King probe, the authors find no such artifacts; all spectra are of the scale-invariant form k −β with exponents β in the range 1.1–1.7, depending on the flight. Using the whole FIRE 87 King LWC database, the authors find power-law behavior with β = 1.56 ± 0.06 from 20 m to 20 km. From a spectral vantage point, the ASTEX cloud system behaves statistically like a scaled-up version of FIRE 87: a similar exponent β = 1.43 ± 0.08 is obtained, but the scaling range is shifted to [60 m, 60 km], possibly due to the 2–3 times greater boundary layer thickness.

Finally, the authors reassess the usefulness of spectral analysis:

  1. • Its main shortcoming is ambiguity: very different looking stochastic processes can yield similar, even identical, spectra. This problem impedes accurate modeling of the LWC data and, ultimately, is why multifractal methods are required.

  2. • Its main asset is applicability in stationary and nonstationary situations alike and, in conjunction with scaling, it can be used to detect nonstationary behavior in data.

Having β > 1, LWC fields in marine Sc are nonstationary within the scaling range and stationary only at larger scales. Nonstationarity implies long-range correlations, and we demonstrate the damage these cause when tying to estimate means and standard deviations with limited amounts of LWC data.

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Anthony Davis, Alexander Marshak, Robert Cahalan, and Warren Wiscombe

Abstract

Several studies have uncovered a break in the scaling properties of Landsat cloud scenes at nonabsorbing wavelengths. For scales greater than 200–400 m, the wavenumber spectrum is approximately power law in k −5/3, but from there down to the smallest observable scales (50–100 m) follows another k β law with β > 3. This implies very smooth radiance fields. The authors reexamine the empirical evidence for this scale break and explain it using fractal cloud models, Monte Carlo simulations, and a Green function approach to multiple scattering theory. In particular, the authors define the “radiative smoothing scale” and relate it to the characteristic scale of horizontal photon transport. The scale break was originally thought to occur at a scale commensurate with either the geometrical thickness Δz of the cloud, or with the “transport” mean free path l t = [(1 − g)σ]−1, which incorporates the effect of forward scattering (σ is extinction and g the asymmetry factor of the phase function). The smoothing scale is found to be approximatelyltΔz at cloud top; this is the prediction of diffusion theory which applies when (1 − g)τ = Δz /l t ≳ 1 (τ is optical thickness). Since the scale break is a tangible effect of net horizontal radiative fluxes excited by the fluctuations of τ, the smoothing scale sets an absolute lower bound on the range where one can neglect these fluxes and use plane-parallel theory locally, even for stratiform clouds. In particular, this constrains the retrieval of cloud properties from remotely sensed data. Finally, the characterization of horizontal photon transport suggests a new lidar technique for joint measurements of optical and geometrical thicknesses at about 0.5-km resolution.

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Steven M. Gollmer, Harshvardhan, Robert F. Cahalan, and Jack B. Snider

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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