Dynamical model and nonextensive statistical mechanics of liquid water path fluctuations in stratus clouds

Authors


Abstract

[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 χ2 distribution. The transition between the small-timescale model of a nonextensive process and the large-scale Gaussian extensive homogeneous fluctuation model is found to be at around 24 h.

1. Introduction

[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 Boltzmann-Gibbs-Shannon (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 equation imageq, which have the BGS as a limit. Various reasons may cause some physical systems not to maximize the BGS entropy, for example, long-range 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 two-dimensional 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 Navier-Stokes 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 self-similar cascades, in which an identical, scale-invariant 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 time-dependent 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 time-dependent and their evolution can be sufficiently well described in the framework of Tsallis statistics.

2. Data Analyzed

[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 WVR-1100) 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 Δt0 = 20 s intervals. The microwave radiometer is equipped with a Gaussian-lensed microwave antenna whose small-angle 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/cm2 [Liljegren et al., 2001].

[8] We consider a 6-day stratus cloud event, 9–14 January 1998, which is an exceptionally long-lasting 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) − 〈yΔt)/σΔt, where 〈yΔt and σΔt are the mean and the standard deviation of y(t) for time lag Δt, respectively.

3. Conventional Analysis

[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.

Figure 1.

Probability distribution function pΔt(Z) of normalized increments Z(t, Δt) of liquid water path (LWP) data measured at the ARM SGP site on 9–14 January 1998, for Δt = 40 s (circles). Normalized increments are calculated as Z(t, Δt) = (y(t) − 〈yΔt)/σΔt, where 〈yΔt and σΔt are the mean and standard deviation of y(t) for time lag Δt. The solid curve represents a Gaussian distribution. The probability distribution function of normalized increments Z(t, Δt) of shuffled liquid water path signal is denoted by the crosses. Note that the fat tails disappear in the shuffled case.

[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 so-called 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∣) ∼ ∣Zκ 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.

Figure 2.

Cumulative probability distribution functions of normalized increments ∣Z(t, Δt)∣ of liquid water path signal for Δt = 40 s, 80 s, 320 s, 10 min, 1 h, 12 h, and ∼24 h (24 h 46 min 40 s). In the asymptotic regime of large values of the normalized increments ∣Z∣, the cpdf of small delay times Δt = 40 s, 80 s, 320 s, and 10 min scale as a power law. The solid line indicates a power law dependence with an exponent κ = 4.1.

[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

[14] The fat tails in the liquid water path distributions and long-range 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

equation image

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 time-dependent probability distribution p(x, t) of the stochastic variable x. The functional equation imageq is reduced to the classical extensive Boltzmann-Gibbs-Shanon 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]

equation image

for the stochastic variable x, where

equation image

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) = Cx2α 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

equation image

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 β0 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 long-range 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 χ2-distributed with degree v:

equation image

where Γ is the Gamma function and β0 = 〈β〉 with average taken over the entire data series for that specific β. The number of degrees of freedom ν can be found from

equation image

The Tsallis parameter q satisfies [Beck, 2001a]

equation image

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/σ2 is χ2-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 xZ(t, Δt), thus studying p(x) ≡ pΔt(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 χ2 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 10Δt0 to 100Δt0:

equation image

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 = 15Δt0 = 5 min. The values of the degree ν of the χ2 distribution then are obtained using equation (6). The spread [βmin, βmax] of the local variability β decreases with increasing time lag, as is expected from a χ2 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 χ2 distribution, thereby justifying the initial assumption used to obtain equation (5).

Figure 3.

Probability distribution function fΔt(β) of the local variability β [equation (8)] in terms of standard deviation of the normalized increments Z(t, Δt) of the liquid water path signal for box size n = 5 min for different time lags (symbols) (a-f): Δt = 40 s, 80 s, 320 s, 10 min, 1 h, and 89,200 s (24 h 46 min 40 s). Solid curves: χ2 distribution as given by equation (5).

[19] The impact of the α parameter on the tail behavior of the Tsallis-type 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 Kolmogorov-Smirnov distance dKS, 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 Kr (see Table 1) for the Tsallis type of distribution function, given by

equation image

where KL = 3 for a Gaussian process, is positive for all values of q < 7/5 as expected because the positiveness of Kr 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 Tsallis-type distribution function will always remain finite, as necessarily due to the type of phenomena studied here.

Figure 4.

Probability distribution function of normalized increments Z(t, Δt) of liquid water path signal (symbols) and the Tsallis-type distribution function (solid curves) for Δt = 40 s, 80 s, 320 s, 10 min, 1 h, 12 h, and ∼24 h (24 h 46 min 40 s). The last value of Δt is chosen such that at least the PDF for the positive value of the increments converges to a Gaussian distribution (dashed curve). The PDFs (symbols and curves) for each Δt are displaced by 0.1 with respect to the previous one; the top curve for Δt = 40 s is not moved. The values of the parameters for the Tsallis-type distribution function for each Δt are summarized in Table 1.

Table 1. Values of the Parameters Characterizing the Liquid Water Path Signal in the Nonextensive Thermostatistics Approacha
Δt, sqα0w = 2α/(q − 1)KrdKS
  • a

    For the definition of the Kolmogorov-Smirnov distance dKS, see the text.

401.390.960.534.9249.80.007
801.360.930.515.1713.80.005
3201.340.920.485.419.80.013
6001.310.910.465.877.10.016
36001.240.860.457.174.80.013
432001.160.780.449.753.80.016
892001.110.770.4414.003.50.040

[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 Tsallis-type distribution function for the negative values of Z fits the data whose probability distribution function still deviates from Gaussian. We consider that ΔtL = 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 small-timescale model of a nonextensive, intermittent process and the large-scale 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

equation image

Therefore the Tsallis-type 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 Tsallis-type distribution converges to a power law

equation image

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 high-precision 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].

5. Conclusion

[24] We present here a method that provides the evolution of the time-dependent probability distribution functions of the liquid water path observed in stratus clouds. We apply the Beck dynamical model that encompasses Tsallis nonextensive statistics to liquid water path data after testing that their local variability is χ2-distributed. The model describes well the shape and the tails of the probability distribution functions. We find that the transition between the small-timescale model of a nonextensive process and the large-scale Gaussian extensive homogeneous fluctuation model is found to be at around 24 h. In the limit of large values of the normalized increments, the values of the exponent of the PDF tails given by the model theoretical predictions are found to be in agreement with the slopes of the cumulative probability distribution functions. The model thus can provide a useful framework for treating liquid water path in global circulation models.

Acknowledgments

[25] KI, HNS, and EEC were supported by the Office of Biological and Environmental Research of the US Department of Energy under contracts DE-FG02-04ER63773 and DE-FG02-90ER61071. TPA was supported by the Office of Biological and Environmental Research of the US Department of Energy under contract DE-AC06-76RL01830 to the Pacific Northwest National Laboratory as part of the Atmospheric Radiation Measurement Program. The Pacific Northwest National Laboratory is operated by Battelle for the US Department of Energy.

Ancillary

Advertisement