Abstract
 Top of page
 Abstract
 1. Introduction
 2. Data Analyzed
 3. Conventional Analysis
 4. Nonextensive Statistical Approach
 5. Conclusion
 Acknowledgments
 References
 Supporting Information
[1] The shape and tails of the probability distribution functions of the liquid water path in stratus clouds are expressed through a model encompassing Tsallis nonextensive statistics. A model originally proposed to describe turbulent flows describes the behavior of the normalized increments of the liquid water path, at both small and large timescales, provided that the distribution of the local variability of the normalized increments can be sufficiently well fitted with a χ^{2} distribution. The transition between the smalltimescale model of a nonextensive process and the largescale Gaussian extensive homogeneous fluctuation model is found to be at around 24 h.
1. Introduction
 Top of page
 Abstract
 1. Introduction
 2. Data Analyzed
 3. Conventional Analysis
 4. Nonextensive Statistical Approach
 5. Conclusion
 Acknowledgments
 References
 Supporting Information
[2] In recent years, the formalism of nonextensive statistical mechanics, first introduced by Tsallis [1988] and later further developed by others [Abe, 2000; Abe et al., 2001; Tsallis et al., 1998], has gained considerable interest [Beck, 2000, 2001a, 2001b, 2001c, 2002a, 2002b, 2002c, 2004; Arimitsu and Arimitsu, 2000a, 2000b, 2001, 2002, 2006; Wilk and Wlodarczyk, 2000; Daniels et al., 2004]. The new theoretical approach is suitable to treat physical systems of sufficient complexity that cannot maximize the usual BoltzmannGibbsShannon (BGS) entropy, the latter leading to the usual statistical mechanics. In such a case, the system maximizes some other, more general entropy measure, such as the Tsallis entropies _{q}, which have the BGS as a limit. Various reasons may cause some physical systems not to maximize the BGS entropy, for example, longrange correlations, multifractality [Lyra and Tsallis, 1998; Campos Velho et al., 2001; Meson and Vericat, 2002; Arimitsu and Arimitsu, 2000a, 2000b, 2006], or simply the fact that the system is not in equilibrium owing to some external forcing [Tsallis and Bukman, 1996]. Recently it has been shown that nonextensive statistical mechanics is particularly useful in describing twodimensional Eulerian turbulence [Boghosian, 1996] and the stochastic properties of fully developed turbulent flows [Beck, 2000, 2001a, 2001b, 2001c, 2002a, 2002c, 2004; Beck et al., 2001; Arimitsu and Arimitsu, 2000a, 2000b, 2001; Ramos et al., 2001a; Bolzan et al., 2002], including dislocation motion in defect turbulence in inclined layer convection [Daniels et al., 2004].
[3] In this study, we apply the Beck dynamical model [Beck, 2000, 2001a, 2002a] to liquid water path fluctuations in stratus clouds in the framework of nonextensive statistical mechanics. As introduced to describe fully developed turbulence, the Beck dynamical model aims neither to solve the turbulence problem nor to reproduce fully the spatiotemporal dynamics of the NavierStokes equations, but rather to provide a simple model that captures some of the most important statistical properties of the phenomena in an analytically tractable manner [Beck, 2004].
[4] Stratus overcast conditions are associated with a neutral boundary layer. The turbulence in such a layer is generated predominately by shear production from the atmosphere toward the Earth's surface. In classical studies, the phenomenology of turbulence has been described by selfsimilar cascades, in which an identical, scaleinvariant step is repeated from large scales to small ones, as the small ones produce even smaller ones until the turbulent flow energy gets dissipated on the smallest scale [Mandelbrot, 1974]. More realistic description of turbulence is achieved by generalization of this approach to anisotropic scaling and multiplicative cascade models [Schertzer and Lovejoy, 1987]. Cascade processes generically give rise to multifractals. The resulting multifractal behavior of a random variable is scale invariant and can be determined either by the scaling of its probability distribution functions or by the scaling of its structure functions. In hydrodynamics, the velocity structure functions are expected to exhibit multiaffine scaling, for example, nonlinear scaling of the structure function exponents [Frisch, 1995].
[5] In contrast, the probability distribution functions that are obtained within the nonextensive statistical mechanics approach are not scale invariant [Tsallis, 1988]. The Beck dynamical model in the framework of Tsallis statistics describes the evolution of the timedependent probability distribution functions of a random variable for different delay times [Beck, 2000, 2001a]. The purpose of this study is to present empirical evidence that the probability distribution functions of the liquid water path fluctuations in stratus clouds are timedependent and their evolution can be sufficiently well described in the framework of Tsallis statistics.
2. Data Analyzed
 Top of page
 Abstract
 1. Introduction
 2. Data Analyzed
 3. Conventional Analysis
 4. Nonextensive Statistical Approach
 5. Conclusion
 Acknowledgments
 References
 Supporting Information
[6] The data used in this study are the liquid water path (LWP) time series measured with the microwave radiometer at the Southern Great Plains (SGP) site of the US Department of Energy Atmospheric Radiation Measurements (ARM) program. The microwave radiometer (Radiometrics, Model WVR1100) measures the radiances, expressed as brightness temperatures, at the frequencies of 23.8 and 31.4 GHz from which the vertical column amounts of cloud liquid water and water vapor are retrieved [Westwater, 1993].
[7] The microwave radiometer measures the downwelling atmospheric radiance and records the data as brightness temperatures at Δt_{0} = 20 s intervals. The microwave radiometer is equipped with a Gaussianlensed microwave antenna whose smallangle receiving cone is steered with a rotating flat mirror (http://www.arm.gov). The field of view of the microwave radiometer is 5.7° at 23.8 GHz and 4.6° at 31.4 GHz. The brightness temperature is measured with a radiance error of ±0.5 K. The atmosphere is not optically thick at these two microwave radiometer frequencies during cloudy conditions. Hence these two frequencies can be used to retrieve the total column amounts of water vapor and cloud liquid water. The error for the liquid water path retrieval is estimated to be less than about 0.005 g/cm^{2} [Liljegren et al., 2001].
[8] We consider a 6day stratus cloud event, 9–14 January 1998, which is an exceptionally longlasting cloudy period for the Southern Great Plains site, that allows us to analyze a high resolution and unusually long time series of 25,772 data points.
[9] In this study, we are concerned with the distribution of the normalized increments of the liquid water path signal y(t) for various values of the time lag Δt. Normalized increments are calculated as Z(t, Δt) = (y(t) − 〈y〉_{Δt})/σ_{Δt}, where 〈y〉_{Δt} and σ_{Δt} are the mean and the standard deviation of y(t) for time lag Δt, respectively.
3. Conventional Analysis
 Top of page
 Abstract
 1. Introduction
 2. Data Analyzed
 3. Conventional Analysis
 4. Nonextensive Statistical Approach
 5. Conclusion
 Acknowledgments
 References
 Supporting Information
[10] The distribution of the normalized increments Z(t, Δt) of the liquid water path signal for the period 9–14 January 1998 for Δt = 40 s is plotted in Figure 1. A fit is attempted first with a Gaussian distribution (solid curve in Figure 1) for small values of the increments, i.e., the central part of the distribution. Even for small values of the increments, however, the probability distribution function (PDF) is not fit well with a Gaussian curve.
[11] To illustrate that the correlations of the fluctuations of the liquid water path signal are expressed in the tails of the probability distribution function, we shuffle the LWP signal by randomly changing the order of the measurements in the time series. The PDF of the normalized increments of the shuffled signal is plotted with crosses in Figure 1. The socalled fat tails, for example, tails of the PDF showing probabilities higher than the Gaussian distributions would predict, almost vanish.
[12] To test the tails of the PDF further, we calculate the probability that the increment is larger than ∣Z∣; that is, we calculate the cumulative probability distribution (cpdf) of the increments P_{>}(∣Z∣) for the different values of the time lag Δt = 40 s, 80 s, 320 s, 10 min, 1 h, 12 h, and 89,200 s (24 h 46 min 40 s). These cumulative distributions are plotted in Figure 2. In the asymptotic regime for large values of the increments ∣Z∣ > 4, the cumulative distributions for small values of the time lag Δt = 40 s, 80 s, 320 s, and 10 min decay as a power law P_{>}(∣Z∣) ∼ ∣Z∣^{κ} with κ ≈ 4, as shown by the solid line in Figure 2. The cpdf of the liquid water path data for larger values of the time lag approach Gaussian behavior.
[13] Previous empirical studies of the tropical atmospheric boundary layer report values of the exponent κ ≈ 5 for wind velocity in the vertical [Lazarev et al., 1994], κ ≈ 7 for wind velocity in the horizontal [Chigirinskaya et al., 1994], and κ ≈ 7 for the time series of wind velocity at the finest available resolution [Schmitt et al., 1994]. In contrast, the cumulative probability distribution of hourly rainfall data in Valentia, Ireland, at the resolution of the data set is found to scale with an exponent κ = 4 [Kiely and Ivanova, 1999], and the value of the scaling exponent is equal to κ = 3.5 for daily rainfall accumulations in France [Ladoy et al., 1991]. The value of the exponent κ obtained in this study is closer to its value for rainfall probably because the liquid water path in stratus clouds is quantitatively closely related to precipitation.
4. Nonextensive Statistical Approach
 Top of page
 Abstract
 1. Introduction
 2. Data Analyzed
 3. Conventional Analysis
 4. Nonextensive Statistical Approach
 5. Conclusion
 Acknowledgments
 References
 Supporting Information
[14] The fat tails in the liquid water path distributions and longrange temporal correlations [Davis et al., 1994; Marshak et al., 1997; Ivanova et al., 2000, 2002] indicate that nonextensive statistical mechanics may offer an appropriate framework to quantify the corresponding statistics. At present, the most consistent one seems to be the one based on the generalized entropies given by
as postulated by Tsallis [1988]. In equation (1), q is a parameter and k is a normalization constant. The main ingredient in equation (1) is the timedependent probability distribution p(x, t) of the stochastic variable x. The functional _{q} is reduced to the classical extensive BoltzmannGibbsShanon form in the limit of q 1. The Tsallis parameter q characterizes the nonextensivity of the entropy. Subject to certain constraints, the functional in equation (1) yields a probability distribution function of the form [Beck, 2001a, 2002a; Ramos et al., 2001a, 2001b]
for the stochastic variable x, where
in which Γ is the gamma function, C is a constant, and 0 < α ≤ 1 is the power law exponent of the potential U(x) = C∣x∣^{2α} that provides the “restoring force” F(x) in the Beck model of turbulence [Beck, 2001a, 2001b; Beck et al., 2001]. The latter is described by a Langevin equation
where γ and σ are parameters and R(t) is a Gaussian white noise. A nonzero value of γ corresponds to providing energy to (or draining from) the system [Sattin, 2003]. The parameter β_{0} in equations (2) and (3) is the mean of the fluctuating variability β, i.e., the local standard deviation of x over a certain window of size n [Ramos et al., 2001a, 2001b].
[15] Although the nonextensive formalism originally was suggested to have physical applications to equilibrium systems with longrange interactions, it has been suggested recently that the nonextensive formalism is of particular physical relevance for nonequilibrium systems with fluctuating temperature [Wilk and Wlodarczyk, 2000] or fluctuating energy dissipation [Beck, 2001a, 2001c, 2002a, 2002b]. Beck [2002a] showed that the nonextensive behavior is a consequence of integrating over all possible values of the fluctuating variable β, provided that β is χ^{2}distributed with degree v:
where Γ is the Gamma function and β_{0} = 〈β〉 with average taken over the entire data series for that specific β. The number of degrees of freedom ν can be found from
The Tsallis parameter q satisfies [Beck, 2001a]
Significantly, equation (1) is obtained exactly for systems that are governed by the nonlinear Langevin equation (4) for which parameters γ and σ fluctuate in such a way that β = γ/σ^{2} is χ^{2}distributed with degree ν (equation (5) [Beck, 2002a]).
[16] We use the Beck model assuming that the normalized increments Z(t, Δt) represent the stochastic variable x, as in equation (1). We search whether equation (2) is obeyed for x ≡ Z(t, Δt), thus studying p(x) ≡ p_{Δt}(Z) for various time lags Δt.
[17] In order to test if the distribution of the “local” variability of the normalized increments Z(t, Δt) is of the form of a χ^{2} distribution, we have checked the distribution of the normalized increments of the liquid water path signal for the different values of the time lag Δt = 40 s, 80 s, 320 s, 10 min, 1 h, 12 h, and 89,200 s (24 h 46 min 40 s). We have calculated the standard deviation of the normalized increments within various nonoverlapping windows of size n, ranging from 10Δt_{0} to 100Δt_{0}:
where N is the length of the time series.
[18] In doing so, we have considered different various numbers of nonoverlapping windows for various time lags Δt, and we have searched for the most efficient size of the window in order not to lose data points and therefore information. The resulting empirically obtained distributions of the “local” variability [equation (8)] of normalized increments for the different time lags of interest are plotted in Figure 3 for an intermediate case n = 15Δt_{0} = 5 min. The values of the degree ν of the χ^{2} distribution then are obtained using equation (6). The spread [β_{min}, β_{max}] of the local variability β decreases with increasing time lag, as is expected from a χ^{2} distribution function because of the exponential function in equation (5) for large values of the degrees of freedom ν. On the basis of these results, for example, Figure 3, we accept that the β distributions of the increments Z(t, Δt) for each of the Z(t, Δt) obtained for the different values of the time lag can be sufficiently well fitted for our purpose with a χ^{2} distribution, thereby justifying the initial assumption used to obtain equation (5).
[19] The impact of the α parameter on the tail behavior of the Tsallistype distribution function for fixed q and the impact of q at fixed α are investigated elsewhere [Ausloos and Ivanova, 2003] and find as expected that the tails of the distribution functions approach a Gaussian type when q approaches 1.
[20] Next, we calculate the probability distributions of the normalized increments of the liquid water path signal for the different values of the time lag Δt = 40 s, 80 s, 320 s, 10 min, 1 h, 12 h, and 89,200 s (24 h 46 min 40 s). They are shown in Figure 4 together with the curves representing the best fit to the Tsallis type of distribution function. In Table 1, the statistical parameters related to the Tsallis type of distribution function are summarized, including a criterion for the goodness of the fit, i.e., the KolmogorovSmirnov distance d_{KS}, which is defined as the maximum Euclidean distance between the cumulative probability distributions of the data and the fitting curves. Note that the flatness coefficient K_{r} (see Table 1) for the Tsallis type of distribution function, given by
where K_{L} = 3 for a Gaussian process, is positive for all values of q < 7/5 as expected because the positiveness of K_{r} is related directly to the occurrence of intermittency [Sreenivasan and Antonia, 1997; Ramos et al., 2001a, 2001b]. Moreover, the limit q < 7/5 also implies that the second moment of the Tsallistype distribution function will always remain finite, as necessarily due to the type of phenomena studied here.
Table 1. Values of the Parameters Characterizing the Liquid Water Path Signal in the Nonextensive Thermostatistics Approach^{a}Δt, s  q  α  Cβ_{0}  w = 2α/(q − 1)  K_{r}  d_{KS} 


40  1.39  0.96  0.53  4.92  49.8  0.007 
80  1.36  0.93  0.51  5.17  13.8  0.005 
320  1.34  0.92  0.48  5.41  9.8  0.013 
600  1.31  0.91  0.46  5.87  7.1  0.016 
3600  1.24  0.86  0.45  7.17  4.8  0.013 
43200  1.16  0.78  0.44  9.75  3.8  0.016 
89200  1.11  0.77  0.44  14.00  3.5  0.040 
[21] In order to obtain an estimate for the value of the scale at which the distribution converges to Gaussian, we observe that equation (2) fits well the normalized increments for Δt = 24 h 46 min 40 s and q = 1.03 (Table 1 and Figure 4). The α parameter (α = 0.7) in this case plays an important role in controlling the tails such that the Tsallistype distribution function for the negative values of Z fits the data whose probability distribution function still deviates from Gaussian. We consider that Δt_{L} = 24 h 46 min 40 s is a good estimate for the time lag at which the PDF converges to Gaussian (dashed curve in Figure 4), based on the limited duration of the stratus cloud event. This result indicates where the transition occurs between the smalltimescale model of a nonextensive, intermittent process and the largescale Gaussian extensive homogeneous fluctuation model [Ramos et al., 2001a, 2001b; Tsallis, 1988].
[22] One can explore the theoretically predicted Tsallis type of the probability distribution function equation (2) in two limits. For small values of the normalized increments Z, the probability distribution function converges to the form
Therefore the Tsallistype distribution function converges to a Gaussian, i.e., α 1, for small values of the normalized increments, for any Δt investigated here (see Figure 4).
[23] In the limit of large values of normalized increments Z, i.e., the tails of the probability distribution function, the Tsallistype distribution converges to a power law
with an exponent w = 2α/(q − 1) (see Table 1). The value of w at small scales is ∼5 that is in good agreement with the values of the slope of the cumulative probability distributions (Figure 2) κ = w − 1 ∼ 4. At larger scales, the tail exponent w increases to ∼10. A similar trend though different values of the tail exponent is found experimentally in the case of fully developed turbulence. In the highprecision measurements during experimentally observed fully developed turbulence [Beck et al., 2001], the tail exponent was estimated to be ∼9 for very small spatial scales, increasing to ∼15 at medium scales. Different values of the tail exponent w for the liquid water path data versus those for fully developed turbulence are understood to be owing to the differences that occur when analyzing vector versus passive scalar quantities related to atmospheric turbulence [Sreenivasan and Antonia, 1997].