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Anthony B. Davis and Alexander Marshak

Abstract

Here, previous work using photon diffusion theory to describe radiative transfer through dense plane-parallel clouds at nonabsorbing wavelengths is extended. The focus is on the scaling of space- and time-domain moments for transmitted light with respect to cloud thickness H and optical depth τ; and the new results are as follows: accurate prefactors for asymptotic scaling, preasymptotic correction terms in closed form, 3D effects for internal variability in τ, and the rms transit time or pathlength. Mean pathlength is ∝H for dimensional reasons and, from random-walk theory, we already know that it is also ∝(1 – g)τ for large enough τ (g being the asymmetry factor). Here, it is shown that the prefactor is precisely 1/2 and that corrections are significant for (1 – g)τ < 10, which includes most actual boundary layer clouds. It is also shown that rms pathlength is not much larger than the mean for transmittance (its prefactor is ≈ 0.59); this proves that, in sharp contrast with reflection, pathlength distributions are quite narrow in transmission. If the light originates from a steady point source on a cloud boundary, a fuzzy spot is observed on the opposite boundary. This problem is formally mapped to the pulsed source problem, and it is shown that the rms radius of this spot slowly approaches H as τ increases; it is also shown that the transmitted spot shape has a flat top and an exponential tail. Because all preasymptotic corrections are computed here, the diffusion results are accurate when compared to Monte Carlo counterparts for τ ≥ 5, whereas the classic scaling relations apply only for τ ≥ 70, assuming g = 0.85. The temporal quantities shed light on observed absorption properties and optical lightning waveforms. The spatial quantity controls the three-dimensional radiative smoothing process in transmission, which was recently observed in spectral analyses of time series of zenith radiance at 725 nm. Opportunities in ground-based cloud remote sensing using the new developments are described and illustrated with simulations of 3D solar radiative transfer in realistic models of stratocumulus. Finally, since this analytical diffusion study applies only to weakly variable stratus layers, extensions to more complex cloud systems using anomalous diffusion theory are discussed.

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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

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

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 μm (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 μm), the authors do not find a significant enhancement of cloud absorption due to horizontal inhomogeneity.

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Mikael K. Witte, Hugh Morrison, Anthony B. Davis, and Joao Teixeira

Abstract

Coarse-gridded atmospheric models often account for subgrid-scale variability by specifying probability distribution functions (PDFs) of process rate inputs such as cloud and rainwater mixing ratios (qc and qr, respectively). PDF parameters can be obtained from numerous sources: in situ observations, ground- or space-based remote sensing, or fine-scale modeling such as large-eddy simulation (LES). LES is appealing to constrain PDFs because it generates large sample sizes, can simulate a variety of cloud regimes/case studies, and is not subject to the ambiguities of observations. However, despite the appeal of using model output for parameterization development, it has not been demonstrated that LES satisfactorily reproduces the observed spatial structure of microphysical fields. In this study, the structure of observed and modeled microphysical fields are compared by applying bifractal analysis, an approach that quantifies variability across spatial scales, to simulations of a drizzling stratocumulus field that span a range of domain sizes, drop concentrations (a proxy for mesoscale organization), and microphysics schemes (bulk and bin). Simulated qc closely matches observed estimates of bifractal parameters that measure smoothness and intermittency. There are major discrepancies between observed and simulated qr properties, though, with bulk simulated qr consistently displaying the bifractal properties of observed clouds (smooth, minimally intermittent) rather than rain while bin simulations produce qr that is appropriately intermittent but too smooth. These results suggest fundamental limitations of bulk and bin schemes to realistically represent higher-order statistics of the observed rain structure.

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Igor N. Polonsky, Steven P. Love, and Anthony B. Davis

Abstract

The Wide-Angle Imaging Lidar (WAIL), a new instrument that measures cloud optical and geometrical properties by means of off-beam lidar returns, was deployed as part of a multi-instrument campaign to probe a cloud field at the Atmospheric Radiation Measurement (ARM) Southern Great Plain (SGP) site on 25 March 2002. WAIL is designed to determine physical and geometrical characteristics using the off-beam component of the lidar return that can be adequately modeled within the diffusion approximation. Using WAIL data, the extinction coefficient and geometrical thickness of a dense cloud layer is estimated, from which optical thickness is inferred. Results from the new methodology agree well with counterparts obtained from other instruments located permanently at the SGP ARM site and from the WAIL-like airborne instrument that flew over the site during our observation period.

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Christopher Davis, Chris Snyder, and Anthony C. Didlake Jr.

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Tropical cyclone formation over the eastern Pacific during 2005 and 2006 was examined using primarily global operational analyses from the National Centers for Environmental Prediction. This paper represents a “vortex view” of genesis, adding to previous work on tropical cyclone formation associated with tropical waves. Between 1 July and 30 September during 2005 and 2006, vortices at 900 hPa were tracked and vortex-following diagnostic quantities were computed. Vortices were more abundant during periods of an enhanced “Hadley” circulation with monsoon westerlies around 10°N in the lower troposphere. This zonally confined Hadley circulation was significantly stronger during the genesis of developing vortices. Developing vortices were stronger at the outset, with a deeper potential vorticity maximum, compared to nondeveloping vortices. This implies that developing disturbances were selected early on by favorable synoptic-scale features.

The characteristic time-mean reversal of the meridional gradient of absolute vorticity in the lower troposphere was found to nearly vanish when the aggregate contribution of strong vortices was removed from the time-mean vorticity. This finding implies that it is difficult to unambiguously attribute development to a preexisting enhancement of vorticity on the synoptic scale. The time-mean enhancement of cyclonic vorticity primarily results from the accumulated effect of vortices. It is suggested that horizontal deformation in the background state helps distinguish developing vortices from nondevelopers, and also biases the latitude of development poleward of the climatological ITCZ axis.

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Linda Forster, Anthony B. Davis, David J. Diner, and Bernhard Mayer

Abstract

For passive satellite imagers, current retrievals of cloud optical thickness and effective particle size fail for convective clouds with 3D morphology. Indeed, being based on 1D radiative transfer (RT) theory, they work well only for horizontally homogeneous clouds. A promising approach for treating clouds as fully 3D objects is cloud tomography, which has been demonstrated for airborne observations. However, more efficient forward 3D RT solvers are required for cloud tomography from space. Here, we present a path forward by acknowledging that optically thick clouds have “veiled cores” (VCs). Sunlight scattered into and out of this deep region does not contribute significant information about the inner structure of the cloud to the spatially detailed imagery. We investigate the VC location for the MISR and MODIS imagers. While MISR provides multiangle imagery in the visible and near-infrared (IR), MODIS includes channels in the shortwave IR, albeit at a single view angle. This combination will enable future 3D retrievals to disentangle the cloud’s effective particle size and extinction fields. We find that, in practice, the VC is located at an optical distance of ~5, starting from the cloud boundary along the line of sight. For MODIS’s absorbing wavelengths the VC covers a larger volume, starting at smaller optical distances. This concept will not only lead to a reduction in the number of unknowns for the tomographic reconstruction but also significantly increase the speed and efficiency of the 3D RT solver at the heart of the algorithm by applying, say, the photon diffusion approximation inside the VC.

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

Abstract

This is the second of two papers analyzing the internal liquid water content (LWC) structure of marine stratocumulus (Sc) based on observations taken during the First ICCP (International Commission on Cloud Physics) Regional Experiment (FIRE) 1987 and Atlantic Stratocumulus Transition Experiment (ASTEX) 1992 field programs. Part I examined wavenumber spectra and the three-decade scale range (tens of meters to tens of kilometers) over which scale invariance holds; the inability of spectral analysis to distinguish between different random processes was also underscored. This indetermination is removed in this part by applying multifractal analysis techniques to the LWC fields, leading to a characterization of the role of intermittency in marine Sc.

Two multiscaling statistics are computed and associated nonincreasing hierarchies of exponents are obtained: structure functions and H(q), singular measures and D(q). The real variable q is the order of a statistical moment (e.g., q = 1.0 yields a mean); D(q) quantifies intermittency, H(q) nonstationarity. Being derived from the slopes of lines on log(statistic) versus log(scale) plots, these exponents are only defined when those lines are reasonably straight and where this happens defines the scale-invariant range. Being nonconstant, the derived H(q) and D(q) indicate multifractality rather than monofractality of LWC fields.

Two exponents can serve as first-order measures of nonstationarity and intermittency: H 1 = H(1) and C 1 = 1 − D(1). For the ensemble average of all FIRE and all ASTEX data, the authors find the two corresponding points in the (H 1, C 1) plane to be close: (0.28, 0.10) for FIRE and (0.29, 0.08) for ASTEX. This indicates that the dynamics determining the internal structure of marine Sc depend little on the local climatology. In contrast, the scatter of spatial averages for the individual flight around the ensemble average illustrates ergodicity violation. Finally, neither multiplicative cascades (with H 1 = 0) nor additive Gaussian models such as fractional Brownian motions (with C 1 = 0) adequately reproduce the LWC fluctuations in marine Sc.

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Nicolas Ferlay, François Thieuleux, Céline Cornet, Anthony B. Davis, Philippe Dubuisson, Fabrice Ducos, Frédéric Parol, Jérôme Riédi, and Claudine Vanbauce

Abstract

New evidence from collocated measurements, with support from theory and numerical simulations, that multidirectional measurements in the oxygen A band from the third Polarization and Directionality of the Earth’s Reflectances (POLDER-3) instrument on the Polarization and Anisotropy of Reflectances for Atmospheric Sciences coupled with Observations from a Lidar (PARASOL) satellite platform within the “A-Train” can help to characterize the vertical structure of clouds is presented. In the case of monolayered clouds, the standard POLDER cloud oxygen pressure product P O2 is shown to be sensitive to the cloud geometrical thickness H in two complementary ways: 1) P O2 is, on average, close to the pressure at the geometrical middle of the cloud layer (MCP) and methods are proposed for reducing the pressure difference P O2 − MCP and 2) the angular standard deviation of P O2 and the cloud geometrical thickness H are tightly correlated for liquid clouds. Accounting for cloud phase, there is thus the potential to obtain a statistically reasonable estimate of H. Such derivation from passive measurements, as compared with or supplementing other observations, is expected to be of interest in a broad range of applications for which it is important to define better the macrophysical cloud parameters in a practical way.

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