The role of surface characteristics on intermittency andzero-crossing properties of atmospheric turbulence

Daniela Cava,1 Gabriel George Katul,2,3,4 Annalisa Molini,2,3,5 and Cosimo Elefante1

Received 2 May 2011; revised 11 November 2011; accepted 14 November 2011; published 10 January 2012.

[1] Clustering and intermittency in atmospheric turbulent flows above different naturalsurfaces are investigated with reference to their dependency on surface roughness andthermal stratification. The dualism between active and quiescent phases within measuredtime series is isolated by using the telegraphic approximation (TA), which is able toeliminate the contributions to intermittency originating from amplitude variabilityassociated with the energetic states. The presence of linear correlation relating the scalingexponents of energy spectra for the original series (n) and its TA counterpart (m) within theinertial sub-range (ISR) suggests that amplitude variability acts as a de-correlationfactor for the series. Clustering exponents a estimated from velocity and scalar timeseries exhibit a weak dependence on the Taylor micro-scale Reynolds number (Rel).The average values of intermittency exponents for the original (ms) and the TA series(mTA) are linearly correlated to a for longitudinal velocity and scalars. The derivedrelationships shows that in the atmospheric surface layer (ASL), amplitude intermittencyplays a smoothing role on the clusterization of events. On the other hand, within thecanopy sublayer (CSL) above canopies, scalars are more clustered and amplitudeexcursions tend to amplify (or not alter) clusterization. Moreover, reducing surfaceroughness results in a de-correlation between a and mTA for the vertical velocity component.Additionally, the probability density functions of inter-pulse periods (Ip) were shown to bewell approximated by the law p(Ip) � Ip

�g for Ip within the ISR when g ≈ 3 � m,which also holds for sand pile models of self organized criticality in the large pile limit.

Citation: Cava, D., G. G. Katul, A. Molini, and C. Elefante (2012), The role of surface characteristics on intermittency andzero-crossing properties of atmospheric turbulence, J. Geophys. Res., 117, D01104, doi:10.1029/2011JD016167.

1. Introduction

[2] Intermittency plays a major role in atmospheric tur-bulence whether be it in describing its mixing or agglomer-ation properties [Warhaft, 2000] and is now receivingsignificant attention in a number of disciplines such as windenergy generation and turbine designs. Intermittent flowvariables exhibit quiescent states interspersed by highlyenergetic or ‘active’ events. As a consequence, two aspectscharacterize such intermittent process – one related to itstelegraphic or on-off properties (quiescent versus energeticor active states, unevenly distributed in space or time) andanother related to the amplitude variability within the

energetic states [Bershadskii et al., 2004; Kailasnath andSreenivasan, 1993; Sreenivasan and Bershadskii, 2006a;Sreenivasan et al., 1983]. A number of reviews have alreadyshown that scalar turbulence within the inertial sub-range(ISR) – the range of scales that are much smaller than thescales at which energy is produced but much larger than theviscous dissipation scales – appears more intermittent thantheir velocity counterparts [Warhaft, 2000]. These studiesoften base their conclusions on the anomalous scaling inthe higher-order structure function defined (in time) as[Sreenivasan and Antonia, 1997; Warhaft, 2000]

DpðtÞ ¼ Ds tð Þj j½ �p � tVp ; ð1Þ

where,Ds = s′(t + t) � s′(t), s′ are turbulent excursions fromthe time-averaged state �s for an arbitrary flow variable s, t istime, t is lag time, p is the moment order of the structurefunction, and Vp is a scaling function, often expressed via thelognormal model as [Chevillard et al., 2005; Katul et al.,2006b, 2009; Kolmogorov, 1962; Lashermes et al., 2008;Sreenivasan and Antonia, 1997]

&p ¼ 1

3þ 1

6ms

� �p� 1

18msp

2: ð2Þ

1Institute of Atmosphere Sciences and Climate, National ResearchCouncil, Lecce, Italy.

2Nicholas School of the Environment, Duke University, Durham, NorthCarolina, USA.

3Department of Civil and Environmental Engineering, Duke University,Durham, North Carolina, USA.

4Formerly at Dipartimento di Idraulica, Transporti ed InfrastruttureCivili, Politecnico di Torino, Turin, Italy.

5Now at Masdar Institute of Science and Technology, Abu Dhabi,United Arab Emirates.

Copyright 2012 by the American Geophysical Union.0148-0227/12/2011JD016167

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117, D01104, doi:10.1029/2011JD016167, 2012

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[3] Even at high Reynolds number, the classical intermit-tency exponent ms for passive scalars tends to exceed the oneof the velocity components [Gylfason and Warhaft, 2004;Warhaft, 2000, 2002] though some studies have reportedcomparable values with ms = 0.20 for velocity and 0.25 forair temperature [Chambers and Antonia, 1984]. Bothamplitude variations in the energetic states and the clustering(or telegraphic) properties impact ms and hence Vp inequation (1). However, it is not yet clear which of these twoaspects is actually responsible for the enhancement of ms inscalars. Are the enhancements in scalar intermittencyconnected to the nature of the surface cover or the distribu-tion and strength of sources and sinks of scalars andmomentum at the surface? To what extent atmospheric andsurface characteristics responsible for turbulent productionor destruction, such as buoyancy, friction velocity, or surfaceroughness also play a role in this enhancement? The lattertwo questions are motivated by the fact that scalar turbu-lence in the lower atmosphere often exhibits ramp-like (orinverted ramp) structures [Cava et al., 2004; Shaw et al.,1989] known to be impacted by the scalar source strengthvia a renewal process [Castellví, 2004; Castellví and Snyder,2010; Katul et al., 1996, 2006b; Paw U et al., 1992; Snyderet al., 1996] and to affect, in turn, fine-scale turbulencewithin the ISR [Katul et al., 2006b; Mahrt, 1989; Warhaft,2000]. Scalar turbulence can be more intermittent at thescales within the ISR precisely because of such interactionsbetween boundary conditions (scalar source strength), largescales (ramp-cliff shape), and fine scales (micro-front of theramp-cliff). However, a number of studies have also shownthat larger-scale eddy motion (i.e., commensurate with theintegral time scale) can impact the local kinetic energy dis-sipation rate, and hence, ms as derived from velocity timeseries [Kuznetsov et al., 1992].[4] To progress on a subset of these problems, the so-

called Telegraphic Approximation (TA) is employed. Thebinary nature (on-off) of the TA permits isolating eventclustering in time without being influenced by amplitudevariations. The TA preserves the so-called zero-crossingproperties in the original time series but eliminates ampli-tude variations associated with the energetic states[Kailasnath and Sreenivasan, 1993; Monin and Yaglom,1971; Rice, 1945; Sreenivasan et al., 1983]. This study istimely because TA properties of turbulence for a number offlow configurations have already been reported over the past7 years thereby offering a comparative framework. Exam-ples include (1) turbulence away from rigid boundaries

[Sreenivasan and Bershadskii, 2006a], hereafter referredto as SB06, (2) convection at high Rayleigh number[Bershadskii et al., 2004], hereafter referred to as B04,(3) velocity components within and immediately above adense canopy composed of densely arrayed rods situated in aflume [Poggi and Katul, 2009, 2010], hereafter referred to asPK09, and (4) velocity and air temperature measurementscollected inside a dense hardwood canopy [Cava and Katul,2009], hereafter referred to as CK09. Comparisons withthese experiments can offer clues on how the boundaryconditions, including roughness effects and thermal stratifi-cation, modify one key aspect of intermittency in the ISRconnected to event clustering. At fine scales, understandingagglomeration of inertial particles such aerosols, water dro-plets, and pollen grains near the canopy-atmosphere inter-face can benefit from quantifying aspects of turbulentclustering. In fact, agglomeration is known to be sensitiveto the interplay between the inertia of these particles andthe spatial clustering of turbulent eddies at fine scales[Thoroddsen and Van Atta, 1992; Sreenivasan and Antonia,1997]. At larger scales, turbulent structures whose cluster-ing properties and inter-pulse durations appear to be closeto white-noise (versus structured) as discussed in SB06 canalso be delineated by the approaches here.[5] To address how intermittency and clustering in the ISR

are impacted by atmospheric stratification and surfaceroughness, turbulent time series collected above differentsurface cover types ranging from ice sheets to tall forests(described in section 2) are analyzed. The methodology andvariants on it employed in SB06 is reviewed in section 3 andits application to the data sets is presented in section 4.Finally, conclusions and comparison with previous studiesare presented in section 5.

2. Data Sets

[6] Six data sets collected above different surface covertypes and summarized in Table 1 were analyzed. These datasets were divided in two broad groups: The first group refersto measurements collected above three tall-forested ecosys-tems, where the measurement height (zm) is situated withinthe so-called canopy sublayer (CSL). The second grouprefers to measurements collected well above the mean sur-face obstacle height in the atmospheric surface layer (ASL).Atmospheric stability conditions were broadly classified asunstable when zst ≤ �0.5, near neutral when �0.5 < zst <0.5, and stable when zst ≥ 0.5, where zst = (zm � d)/Lmo,

Table 1. Summary of the Site Characteristics for Each Data Seta

Site Designation zm (m) hc (m) or [zo]LAI

(m2 m�2)SamplingFrequency

VariablesAnalyzed

Group 1: Canopy Sublayer (CSL)Alpine Hardwood Lav_H 33 28 9.6 20 Hz s = u, w, T, q, COak-Hickory Hardwood Duke_H 39 25 7 10 HzLoblolly Pine Duke_P 22 16 7 10 Hz

Group 2: Atmospheric Surface Layer (ASL)Grass Duke_G 5.2 1.0 3 56 Hz s = u, w, TBare Soil BSoil 2–3 [0.02] - 10 HzIce Sheet Ice 2, 4.5, 10 [0.001–0.0001] - 20.8 Hz

aHere hc is the canopy height, zo is the momentum roughness length, and LAI is the all sided leaf area index. The variables q and C refer to water vaporand CO2 concentration fluctuations.

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d being the zero-plane displacement assumed to be 2/3 ofthe canopy height, and Lmo being the Obukhov length. Thischoice was based on a trade-off between ensuring sufficientdata points in each stability class and minimizing the effectsof thermal stratification for near-neutral conditions acrosssites. Details about the sites, data sets, and their processingfor the ice sheet [Cava et al., 2001; Giostra et al., 2002], thebare soil surface [Katul, 1994; Katul et al., 1995], the grass-covered surface [Katul et al., 1997b, 1998b], the Loblollypine forest [Cava et al., 2004; Katul and Albertson, 1998;Siqueira and Katul, 2002], the Oak-Hickory hardwood for-est [Katul et al., 1997a, 1998a], and the Alpine hardwoodcanopy [Cava and Katul, 2008; Cava et al., 2006, 2008;Cava and Katul, 2009] are discussed elsewhere. The numberof runs employed in each stability class at each site is sum-marized in Table 2. For the ASL studies, the variables

analyzed are s = u, w, T where u and w are the longitudinaland vertical velocities, and T is the virtual air temperature.For the CSL studies, two additional scalar concentrationfluctuations were also analyzed (s = q, C, where q is thewater vapor concentration and C is the carbon dioxideconcentration).

3. Method of Analysis

[7] Because B04, SB06, PK09, and CK09 already pre-sented a broad range of results on intermittency and cluster-ing properties of turbulence time series, comparisonsbetween these studies and the atmospheric turbulence datasummarized in Table 1 were conducted first. The conven-tions and definitions used in those studies are employedthroughout. For completeness, the nomenclature and thedetermination of various scaling exponents are brieflyreviewed in the following section.

3.1. The Telegraphic Approximation (TA)

[8] B04 and SB06 defined the telegraphic approximation(TA) as

TA sð Þ ¼ 1

2

s′ tð Þjs′ tð Þj þ 1

� �: ð3Þ

where s′(t) = s(t) � �s and over-bar indicates time averaging.For consistency with these previous studies, this definition isadopted throughout though thresholds other than the zero-mean can be used. Figure 1 shows an example of tempera-ture series (Figure 1, top) and the corresponding TA

Table 2. Number of Runs in Each Stability Class at EachExperimental Site

Site Unstable Near Neutral Stable

Group 1: Canopy Sublayer (CSL)Alpine Hardwood 100 100 100Oak-Hickory Hardwood 387 160 122Loblolly Pine 109 32 54

Group 2: Atmospheric Surface Layer (ASL)Grass 82 9 0Bare Soil 8 1 5Ice Sheet 53 143 128

Figure 1. (top) Example of a measured temperature fluctuation (T′) time series and (bottom) its tele-graphic approximation (TA). Because TA(T′) can only take on binary values of 1 or 0 (i.e., no amplitudevariations), only clustering becomes relevant when analyzing the TA series.

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(Figure 1, bottom). Because TA(s) can either take on valueof 1 (when s′(t) > 0) or 0 (when s′(t) < 0), there are no‘amplitude variations’ in the telegraphic approximation’.

3.2. Spectra

[9] The relationship between the spectral scaling exponentof s(t) (impacted by both amplitude variability and cluster-ing) and its TA(s) (impacted by only clustering) are pre-sented here to investigate the correlation structure of the’switches’ from 1 to 0 or 0 to 1 in the TA series (that neednot to be entirely random if associated with coherent inter-mittent events). Consider the spectra of the series Es( f ) andof the TA ETA( f ) to scale as f �n and f �m respectively.[10] For turbulence far away from boundaries, SB06

heuristically demonstrated that the spectral exponents ofthe full series and its TA approximation are linearly relatedin the form

m ¼ anþ b; ð4Þ

with a = b = 1/2. Hence, it follows that when n = 5/3 (typicalfor ISR scaling), the expression by SB06 predicts m = 4/3.Such a relationship is obtained by assuming that the velocityis mono-fractal, with Hurst exponent H, so that the fractaldimension of the zero-crossing set Z over the line is D(Z) =1� H, and the second order structure functions of s(t) and itsTA scale with co-dimensions 2H and H respectively. For thespectral exponents, it follows that n = 1 + 2H and m = 1 + H,and when combined together result in equation (4) with a =b = 1/2 as shown in B04. Using similar scaling arguments,B04 also obtained this result for a multifractal series m =1 + V1′, where V1′ is the first order scaling exponent of the TA.However, they argued that for turbulence away from a wall,spectral exponents obtained based on the mono-fractal andmultifractal hypothesis are only slightly different. Theybased this argument on the assumption that the origin ofintermittency buildup across scales is entirely endogenousto the turbulent kinetic energy dissipation rate with no otherexternal factors (i.e., those responsible for turbulence gen-eration) being relevant. PK09 found that to assume a = b =1/2 was still reasonable above their rod canopy. Inside therod canopy, a linear relationship between m and n wasreported but with exponents different from what wasreported in SB06 for u (a ≈ 0.7 and b ≈ 0.1). However, forw, the exponents reported in SB06 agreed reasonably wellwith PK09. For the Lavarone site, CK09 actually showedthat a ≈ 0.66 and b ≈ 0.09 is in better agreement than a =b = 1/2 when combining all flow variables (includingscalars) and measurement levels inside the canopy. For a ≈0.66, b ≈ 0.09 and when n = 5/3 results in m ≈ 1.19 ratherthan 1.33. While the precise values of a and b differedamong the various studies, the common finding fromSB06, PK09 and CK09 is as follows.[11] 1. The linearity in equation (4) seems to hold across a

wide range of boundaries and flow conditions and for bothvelocity and scalar time series. This result confirms SB06’sconclusion that TA spectra contain significant informationabout the scaling laws of the measured turbulent velocityand concentration time series;[12] 2. The exponent m < n and a ∈ [0.5, 0.7] < 1 and

implies that there is more ‘memory’ in TA(s) due to the

sequence of on-off or off-on switching in time when com-pared to s(t). Stated differently, amplitude variability maybe de-correlating the series.

3.3. Non-Gaussianity

[13] Another question to explore in TA series is the pres-ervation of some of the non-Gaussian properties encoded inthe intermittency of the original time series (including bothclustering and amplitude variability). The non-Gaussianityof turbulent velocity gradients results from the nonlineardynamics of the Navier–Stokes equations. Intermittencyeffects result in non-Gaussian statistics, whereas K41appears consistent with the concept of an inertial cascade ifthe velocity statistics within the inertial subrange do notdiffer significantly from Gaussian for homogeneous andisotropic turbulence [Katul et al., 1994; Giostra et al., 2002].In fact, one of the characteristics about inhomogeneous andintermittent turbulence is the skewness of the velocity dis-tribution and its derivatives required to produce, respec-tively, energy flux and energy spectral transfer [Lumley andPanofsky, 1964].[14] Strong linkages do exist between the ejection-sweep

cycle and the skewness of a series [Cava et al., 2006; Feret al., 2004; Katul and Albertson, 1998; Katul et al., 1997a,1997c, 2006a; Maitani and Ohtaki, 1987; Poggi and Katul,2009; Poggi et al., 2004; Raupach, 1981; Shaw et al., 1983].In fact, one of the defining characteristics of flows in acanopy sublayer is the positively skewed u and the nega-tively skewed w related to cycles of ejections and sweepsand to their contribution to energy, momentum and masstransfer inside the canopy. Atmospheric stability is alsoknown to impact the skewness of velocity and scalars [Chuet al., 1996; Jiménez and Cuxart, 2006; Kaimal et al.,1976; Tillman, 1972; Wyngaard and Weil, 1991].[15] The TA partially preserves these properties via the

fraction of time TA = 1 (= G+). To illustrate this point, recallthat the skewness of an arbitrary flow variable is defined as

Sks ¼ sð Þ3 ; s ¼ s′

ss: ð5Þ

To link Sks with G+ analytically, a third-order cumulantexpansion (CEM) of the probability density function p(s) isemployed and results in [Katul et al., 1997a, 2006a;Nakagawa and Nezu, 1977; Poggi et al., 2004; Raupach,1981]

p sð Þ ≈ 1ffiffiffiffiffiffi2p

p exp � s2

2

� �� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

Gaussian

1þ 1

6Sks s3 � 3s� � �

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}First�order correctionto Gaussian

: ð6Þ

The fraction of time s > 0 (= G+) can be computed and isgiven by

Gþ ¼Z∞0

p xð Þdx ¼ 1

2� 1

12Sks

ffiffiffi2

p

r: ð7Þ

This relationship can be employed to assess how well the TAseries preserves non-Gaussianity in the original series.

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3.4. Clustering and Intermittency Exponents

[16] Within the TA framework, clustering is linked withthe correlations in space or time of the on-off or off-onbinary switches, and the probability distribution of the inter-pulse durations. For certain classes of stochastic processessuch as fractional Brownian motion (or fBm), formal rela-tionships between the inter-pulse duration and the correla-tion functions have been derived using both the Rice’sformula (e.g., see review in PK09) and simple scalingsymmetries [Ding and Yang, 1995]. A number of measuresused to characterize clustering and its (empirical) connectionto intermittency in the actual series are presented next.[17] The clustering exponent (a) was determined by SB06

from scaling relationships applied to the density nt(t) ofzero-crossings. A time series of nt(t) was computed from thenumber of zero-crossings for the portion s′(t) to s′(t + t)normalized by the total number of points in the time intervalt. The clustering exponent was then determined from

dn2t1=2 � t�a ð8Þ

where dnt(t) = nt(t) � ntðtÞ, and ntðtÞ is the long-term meanzero-crossing density. For white-noise, where no significantclustering is present, a = 1/2. SB06 demonstrated that forwell-developed turbulence extrapolated to infinite Taylormicro-scale Reynolds number (Rel), a = 0.1. Here, Rel =lsu/n, where l is the Taylor micro-scale or the smallest

scale within the ISR, su =ffiffiffiffiffiffiu′2

qis the standard deviation of

the longitudinal velocity, and n is the kinematic viscosity.For the finite Rel ∈ [200, 20000] in SB06, a ranged from0.25 to 0.40 within the ISR but approached the white-noisevalue of 1/2 at larger scales. This is not surprising as anumber of studies already suggested that (for near-neutralconditions) near-Gaussian statistics describe well amplitudeof the velocity differences at large scales (decorrelated intime) and scale-dependent non-Gaussian statistics existwithin the ISR [Katul, 1994; Katul et al., 1994, 2009;Meneveau, 1991].[18] Because variance (or energy) dissipation rate is pro-

portional to the squared spatial gradients, which can beconverted to temporal gradients via Taylor’s frozen turbu-lence hypothesis, the quantities

x tð Þ ¼ dsðtÞdt

2 ð9Þ

xt ¼1

t

Ztþt

t

x tð Þdt ð10Þ

can be simultaneously used to infer the intermittency prop-erties of a turbulent series from the scaling of the moments(e.g., B04):

xtðtÞð ÞqxtðtÞð Þq

� t�mq : ð11Þ

[19] The scaling exponents mq were used by a number ofauthors in turbulence research [Katul et al., 2001, 2009;

Kuznetsov et al., 1991, 1992; Obukhov, 1962; Shi et al.,2005; Sreenivasan and Antonia, 1997] as classical inter-mittency measures, and in particular, the second orderexponent m2 is often referred to as the intermittency expo-nent (as already shown in equation (2)). For comparisonswith SB06, CK09, and PK09, the intermittency exponents msand mTA for s and TA(s) are here considered. SB06 demon-strated that when ms was inferred from s = u, ms linearlydecreased with decreasing 1/log(Rel), approaching 0 atinfinite Rel. The rationale for the 1/log(Rel) representationis discussed elsewhere [Sreenivasan and Bershadskii,2006b] and is not repeated here. SB06 also found thatms(Rel) = a(Rel) � 0.1, suggesting a linear correlationbetween clustering and intermittency exponents. This findingin SB06 has significant implications – it suggests that clus-tering at fine scales explains all the variations in ms (impactedby both clustering and amplitude variability) as Rel varies.Whether such relationships also hold across a wide range offlow variables, atmospheric stability states, and surfaceconditions remain an open question to be explored.

3.5. Inter-pulse Period and Linkages Between VariousExponents

[20] A relationship between durations of inter-pulse peri-ods and temporal correlations of ‘on-off’ or ‘off-on’switches is expected for TA series. For a weighted super-imposition of Poisson processes, it was shown in B04 andelsewhere that [Jensen, 1998]

m ¼ 3� g; ð12Þ

where g is the exponent of p(Ip) � Ip�g, p(.) is the probability

density function (pdf) of the distribution of inter-pulse per-iods Ip(i) = t i+1 � t i, and t i is the time for which TA(s(ti))switches. Interestingly, the same relationship also holds forclassical “sand pile” models of self organized criticality(SOC) in the “large pile” limit [Jensen, 1998], which hasbeen used as one analogy to a number of turbulent flows [DeMenech and Stella, 2002; Sreenivasan et al., 2004]. Thestudy in B04 then proposed an ‘intermittency’ correction toequation (9) yielding m = (3� m2/2)� g. SB06 later showedthat turbulence maybe classified as ‘active’ or ‘passive’depending on the scaling of the p(Ip). SB06 noted that for‘active’ scalars, p(Ip) � Ip

�g while for ‘passive’ scalars, theyfound that

pðIpÞ � exp aq2 þ bqþ c�

; ð13Þ

where q = log(Ip), a is only related to the variance of q, and band c are related to the mean (= b) and variance (= s2) of q,given by

a ¼ � 1

2s2; b ¼ �1þ 2ba; c ¼ � log

ffiffiffiffiffiffi2p

ps

� �� b2a: ð14Þ

[21] This finding was rather surprising because d-corre-lated processes (i.e., white noise) exhibit a lognormal dis-tribution of inter-pulses, while according to SB06, p(Ip) �Ip�g results in an energy spectrum more consistent with thetraditional Lorentzian form of processes with exponentiallydistributed inter-arrival times. The term ‘active’ as used inSB06 refers to flow variables where the source strength

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determines the scalar dynamics such as for temperature inconvective turbulence, where the amount of heat injecteddetermines the degree of convection. CK09 found thatwithin the CSL of the Lavarone site and for all series ana-lyzed, the above formulation describes well the p(Ip) exceptfor temperature in the crown region of the canopy, where adouble regime was observed (power law followed by alognormal distribution).

4. Results

4.1. Properties of the Telegraphic Approximation forthe Flow Variables

[22] Turbulence in the ASL and CSL possess twoimportant properties: (1) spectral scaling laws in the ISRand (2) non-Gaussian statistics. How these two propertiesare preserved in the telegraphic approximation is explored.With regards to the first property, Figures 2 and 3 show theensemble-averaged measured spectral densities for s = u,w, T along with their TA counterparts for all sites andstability classes, respectively, as a function of the normal-ized frequency. It is evident from Figures 2 and 3 that

scaling regimes do exist for both the actual series and theirTA counterparts against the normalized frequency f (zm � d)/U > 1, where U is the mean longitudinal velocity. Moreover,consistent with PK09, the scaling regimes do not appear tobe appreciably more ‘extensive’ in the TA spectra, at leastfor the range of frequencies resolved here.[23] Figure 4 compares the scaling exponents of energy

spectra for the original series (n) and its TA counterpart (m)for all individual runs across all sites, variables, and stabilityclasses. Spectral exponents have been computed for nor-malized frequency f (zm � d)/U > 5, but removing dataassociated with the highest frequencies when these fre-quencies show clear distortion effects (i.e., aliasing, sonicpath averaging). Linear regression analysis on the combineddata sets in Figure 4 resulted in a = 0.58 and b = 0.26 (seeequation (4)).[24] Figure 4 shows a large range of variability of n values

and significant deviations from n = 5/3 predicted by K41theory [Kolmogorov, 1941] for locally homogeneous andisotropic turbulence at very high Reynolds number. Unlikedata sets away from boundaries analyzed in SB06, aneffective local isotropy is not always or completely realized

Figure 2. Ensemble-averaged normalized energy spectra of the longitudinal velocity (u′), verticalvelocity (w′) and air temperature (T′) as a function of the normalized frequency for the three stabilityclasses: (a, b, c) unstable, (d, e, f) near-neutral and (g, h, i) stable. Different symbols refer to differentmeasurement sites: gray hexagrams (Lav_H), gray diamonds (Duke_H), gray circles (Duke_P), emptypentagrams (Duke_G), empty triangles (BSoil), empty circles (Ice). Spectra are artificially shifted togroup data within CSL (gray symbols) and ASL (empty symbols). The ISR scaling (n = 5/3) is shownfor reference (black line). Vertical dotted line refers to a unity reduced frequency.

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in the analyzed range of frequencies for the data sets here,which are influenced by roughness and characteristics of theunderlying surfaces, even at ‘small scales’ (smaller thanproduction scales, but not purely inertial scales). It is note-worthy that, even above a wall turbulent flow, an open

problem is related to how large-scale (or very large scale)motion influences smaller scale characteristics and at whichscale the effective local isotropy is realized [Van Atta, 1991;Sreenivasan and Antonia, 1997; Biferale and Vergassola,2001; Giostra et al., 2002]. The potential sources of small-

Figure 3. Same as Figure 2 but for the ensemble-averaged normalized energy spectra of the TelegraphicApproximation (TA).

Figure 4. Scatterplot of the spectral exponents (n) for all individual runs and their telegraphic approxi-mations (m) for all flow available variables and for the six measurement sites. Black dashed line refersto the linear regression fit. As a reference the relationship proposed by SB06 (gray continuous line) is alsoshown.

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scale anisotropy could be the direct interaction between largeand small scales or a possible influence of temperaturefluctuations on the inertial subrange motions.[25] Moreover, in the case of data collected just above a

vegetated canopy (or in general within a CSL) the failure ofK41 theory (and of the classical surface layer theory) can beexplained by the importance of turbulent transport terms andtheir imprint on disrupting the balance between turbulentproduction and dissipation rates of turbulent kinetic energy[Kaimal and Finnigan, 1994].[26] Despite all these deviations from K41 scaling, the

relationship between m and n remained linear with a slope (=0.58) not too distinct from SB06 (= 0.5). It appears thatmuch of the departure from K41 impacted the intercept (=0.27) when comparing to SB06 (= 0.5).[27] The correlation between the measured G+ (determined

from the TA series) and finite Sks (determined from theoriginal time series) can be used to assess how well the TAseries preserves non-Gaussian properties of the originalseries. Figure 5 shows that measured and predicted G+ fromequation (7) using measured Sks agree thereby confirmingthat the non-Gaussian properties of the original series arepreserved in the TA series (via G+). It should also be notedhere that the measured G+ includes the integrated effects ofall higher-order cumulants (beyond order 3) on p(s). How-ever, given the agreement in Figure 5, it is safe to state thatthe dominant cumulant appears to be Sks for all the flowvariables and stability classes as shown earlier in PK09 andother studies [Katul et al., 1997a].

4.2. Clustering and Intermittency

[28] Figure 6 shows the ensemble-averaged variations indn2t

1/2 (equation (8)) as a function of the normalized fre-quency (zm � d)/(tU). At large scales (i.e., (zm � d)/(tU) <1), dn2t

1/2 resembles a white-noise process (a = 0.5: dotted-pointed lines). However, for higher frequencies (i.e., (zm �d)/(tU) > 1), the clustering exponent a differs among flowvariables and stability conditions, though all the exponents

remain larger than a = 0.1 (continuous lines) estimated bySB06 in the limit as 1/log(Rel) → 0. In the following anal-ysis, a refers to clustering exponents for the high frequencyrange.[29] Variations in atmospheric stability conditions and

surface properties can significantly modify the Taylor micro-scale Reynolds number Rel. According to SB06, variationsin clustering exponent a and global intermittency exponentms can be entirely explained by variations in Rel. Theanalysis by PK09 for u and w also suggested that within theCSL, variations in a and ms are not entirely (but partially)explained by Rel. Hence, to what extent variations in a(shown in Figure 6) are explained by variations in Rel forthe CSL and ASL should be explored. Unfortunately, sonicanemometry is not ideally suited for the direct estimation ofthe Taylor micro-scale l needed in the determination of Relbecause the averaging path length of the sonic anemometerand l are generally comparable [Katul et al., 1997b]. Hence,the use of indirect methods is necessary to estimate l. Sincems is sensitive to departures from the 5/3 scaling in theenergy spectrum of velocities, determining l using dissipa-tion rates inferred from energy spectra or structure functionscan introduce artificial correlations between the computed land ms. To avoid this potential artificial correlation, l wasindependently inferred by assuming that the production anddissipation of turbulent kinetic energy are in balance result-ing in [Katul et al., 1997b]

l≈su

u*

kvðzm � dÞu*

15nfɛðVstÞ

� �1=2

; ð15Þ

where kv is the Von Karman constant, fɛ(st) is the turbulentkinetic energy dissipation rate stability correction functiongiven as [Hsieh and Katul, 1997; Kader, 1992]

fɛ Vstð Þ ¼ 0:410þ 7:5ð�VstÞ þ 6:25ð�VstÞ2

4þ 2:5ð�VstÞ

!; Vst < 0

1þ 4ð�VstÞ; Vst > 0

:

8><>:

ð16Þ

Figure 5. Scatterplot of the skewness of original time series (Sks) for all variables and for the six mea-surement sites against the fraction of time TA(s) = 1 (G+). Dashed line refers to the linear regression fit(G+ = 0.5 � 0.081 Sks, R

2 = 0.68); continuous line refers to the relationship derived using a third ordercumulant expansion (equation (14)).

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[30] This assumption may be biased by as much as a factorof 2 in the CSL [Cava et al., 2006; Katul and Albertson,1998] and a factor of 1.5 in the ASL [Hsieh and Katul,1997] for the mean dissipation rate calculations (or alterna-tively lumped into fɛ(&st)). However, it is not the intent hereto evaluate the mean turbulent kinetic energy dissipation rateper se but to estimate a quantity that depends on its log-transformed value (i.e., [log(Rel)]

�1), which is more robustto such errors given that l � (fɛ)

�1/2. Figure 7a shows thevariations in a (in the high frequency range) as a function ofthe computed [log(Rel)]

�1 for all the runs along with thelinear relationship reported in SB06. The a range ([0.1, 0.4])in Figure 7 appears consistent with the range reported bySB06 and CK09. Moreover, despite the large scatter, theaveraged values of clustering exponent a follows surprisingwell the SB06 relationship reported for the ISR. The sameanalysis was repeated for the intermittency exponents for theoriginal (ms) and the TA (mTA) series and is shown inFigures 7b and 7c as well. The values of ms are generallylarger than those reported in SB06 and show no significantdependency on the inferred [log(Rel)]

�1. These ms valuesare also larger than the ones reported for ASL turbulence byothers [Chambers and Antonia, 1984; Sreenivasan andAntonia, 1997] that put the estimate of ms ≈ 0.25 � 0.05(at least for velocity). Some studies did report ms ≈ 0.35–0.40

for s = T [Meneveau et al., 1990] though these values may bebiased by the method of inference [Warhaft, 2000].[31] In a separate analysis, variations in ms for s = u and s =

w appear comparable, roughly varying between 0.28 and0.38, and linearly related across sites and stability classes.The values of ms for s = u were consistently smaller thanthose for s = T (varying between 0.33 and 0.48) at the samesites and stability conditions. Moreover, there was no sig-nificant correlation between the values of ms for s = u and s =T. The analysis in Figure 7 also shows that mTA does not varywith the estimated [log(Rel)]

�1. Moreover, the relationshipderived in SB06 appears to set a ‘lower-bound’ on theobserved values suggesting ‘extra’ variability not entirelyexplained by the Taylor micro-scale Reynolds number.Interestingly, the mean value of ms (≈0.30 � 0.06) appearlower that the corresponding mean value of mTA (≈0.36 �0.06) along all the estimated range of [log(Rel)]

�1 vari-ability; that is, on average, amplitude intermittency seems toplay a smoothing role on the clusterization effect (B04).[32] In more detail, for the ASL and CSL experiments, it

was found that, on average, mTA > ms only for s = u sug-gesting that amplitude variations mitigate intermittency onlyin the longitudinal velocity (Figure 8). For the CSL sites andfor all stability classes, mTA ≈ ms for s = T, q, C suggesting

Figure 6. Ensemble-averaged standard deviations for the running zero-crossing density fluctuations ofthe longitudinal velocity (u′), the vertical velocity (w′) and air temperature (T′) for the three atmosphericstability classes: (a, b, c) unstable, (d, e, f) near-neutral and (g, h, i) stable. Different symbols refer todifferent measurement sites: gray hexagrams (Lav_H), gray diamonds (Duke_H), gray circles (Duke_P),empty pentagrams (Duke_G), empty triangles (BSoil), empty circles (Ice). As a reference, the values forwhite noise (a = 1/2: point dashed line) and for Re → ∞ (a = 0.1: continuous line) derived from thelaboratory measurements by SB06 are also shown. Vertical dotted line refers to a unity reducedfrequency.

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that amplitude variations do not affect or slightly amplifyintermittency for scalars (Figure 8).[33] Figure 9 directly shows the relationship between

clustering exponent a and global intermittency exponent ms.In spite of the scatter, the average values of ms appear line-arly correlated to a for s = u, T, q, C (the case of s = w willbe discussed later). The regression slopes and intercepts(mS = a′a + b′) do slightly vary across sites (a′ ≈ 0.66–0.99,b′ ≈ 0.10–0.19) even if there is no clear dependence onsurface roughness alone. The mean regression (mS = 0.85a +0.15), computed considering all the six experimental sites,shows that the significant difference from the SB06 rela-tionship mS = a � 0.1 (continuous line in Figure 9) is, again,in the intercept: in fact when a → 0.1 (for Rel → ∞, asshown in Figure 7a), mS ≈ 0.23, while mS → 0 in the SB06relationship.[34] The correlations between a and TA intermittency

exponent of TA series mTA were significant for s = u, T, q, Cacross all sites and stability classes as shown in Figure 10.The regression slopes (mTA = a″a + b″) do vary across siteswith the bare soil and the ice sheet showing a reduced a″ ≈1.0–1.1, which is smaller than the a″ derived for the

vegetated sites (a″ ≈ 1.41–1.65) and is closer to �a′ = 0.85 andto the value found in the SB06 relationship (a′ = 1).[35] The comparison between linear regressions mS =

a′a + b′ and mTA = a″a + b″ confirm results deduced inFigure 8: for ASL mTA > ms ∀ a, variable and atmosphericstability condition. On the other hand, for CSL the tworegression lines intercept; in particular above forested sites(Figures 10a, 10b, and 10c) i) mTA > ms if a > 0.2 and ii)mTA ≤ ms if a < 0.2. Because Figure 8 shows that the secondcase correspond to s = T, q, C, this result demonstrates thatin the CSL scalars are more clustered (a ∈ [0.1, 0.2]) andamplitude excursions tend to amplify (or to not alter)clusterization.[36] Figures 9 and 10 also show that for progressively

smoother surfaces (i.e., forests to ice sheet), a surprising ‘de-correlation’ of ms and mTA with a occurs for s = w, the flowvariable most constrained by the presence of the surface (anda variable not considered in SB06). The analysis for the icesheet was repeated across three heights and the de-correla-tions between mTA and a reported in Figures 9 and 10 (as wellas the correlations for s = u, T) remain virtually un-alteredwith height from the surface. It is noteworthy that thelargest decorrelation corresponds to high value of a, thatis to less clusterized vertical velocity time series. Hence,the relationship between intermittency exponents and a fors = w appears to be the most sensitive to surface roughnesseffects among all the flow variables considered.

4.3. Inter-pulse Periods

[37] The probability density functions p(Ip) of the inter-pulse period Ip are shown in Figure 11 for all sites, flowvariables, and stability classes. The inter-pulse periods inFigure 11 are normalized by their respective integral timescale (ITS), a measure of the time scale associated with large-scale events. As noted in section 3.4, two types of modelshave been proposed for p(Ip) by SB06: A power law and alognormal distribution. The SB06 study demonstrated theonset of an apparent lognormal distribution for p(Ip) at scalesthat span the inertial to viscous dissipation range and not theintegral (or longer) time scales, a major departure from theanalysis here.[38] The results that emerge from the comparisons in

Figure 11 to the canonical shapes for p(Ip) proposed bySB06 are as follows.[39] 1. Over forested sites, the canonical shape of p(Ip)

resembles an apparent lognormal instead of a power law.Here, the apparent lognormal shape describes p(Ip) for allthree flow variables and all three stability classes.[40] 2. Over the bare soil and ice sheet sites, the canonical

shape of p(Ip) resembles a lognormal distribution only fors = w but a power law better describes p(Ip) for s = u, T andfor a wide range of Ip. These canonical shapes generallypersisted for all three stability classes as well. The grass siteclosely follows these shapes.[41] Further analysis of the p(Ip) in Figure 11 suggests that

when the lognormal model is fitted via a maximum likeli-hood approach across the entire range of Ip (also shown inFigure 11), the fit tends to underestimate p(Ip) at small Ip.Moreover, it is evident from Figure 11 that at small Ip values(i.e., smaller than the integral time scale), the power law

Figure 7. Variations of (a) the clustering exponents (a),(b) intermittency exponents (mS), and (c) intermittency expo-nents for TA of signals (mTA) against [log(Rel)]

�1 for bothvelocity and scalars relative to the measured 30 min timeseries at the different experimental sites and stability condi-tions. Grey points and vertical bars refer, respectively tomean values and standard deviations of variables. For refer-ence, the relationships proposed by SB06 (Figure 7a: a ≃0.1 + 3=2

log Reð Þ ; dotted line) and (Figures 7b and 7c: mS ≃3=2

log Reð Þ; dotted line) are also shown.

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description of p(Ip) is optimum. For large Ip (i.e., com-mensurate or larger than the integral time scale), a quasi-exponential cutoff censoring the extrapolation of the powerlaw beyond the integral time scale emerges thereby invitinga model of the form

p Ip� � I�g

p exp � IpIref

� �; ð17Þ

where g is the power law exponent, and Iref is a characteristictime scale at which the randomization effects (i.e., expo-nential cutoff) of the surface become significant. In thismodel, if Ip ≪ Iref, then p(Ip) � Ip

�g, and if Ip ≫ Iref, thepower law in p(Ip) is censored by an exponential decay. Theonset of the exponential cutoff may be attributed to the factthat coherent structures are randomly deformed by the sur-face and split into several sub-structures, thereby loweringthe probability of large structures and long inter-arrivaltimes, and reducing the process behavior to a simple Poissonwith independent arrival times and exponential p(Ip). Hence,a logical choice of Iref is perhaps the integral time scale ofthe series (ITS). The effect of the surface roughness on theequality between Iref and (ITS) is discussed in the context ofFigure 11.

[42] Over forests, the exponential cutoff term censoringthe power law in p(Ip) appears to be significant in all flowvariables for Ip/ITS ≥ 1. However, p(Ip) appears to follow apower law distribution even for Ip/ITS > 1 in the ASL exceptfor w. In the case of w, the exponential cutoff becomes sig-nificant even for Ip/ITS < 1. Hence, with progressivelysmoother surfaces (forest to ice sheet), Iref becomes pro-gressively larger than the integral time scale for s = u, T andprogressively smaller for s = w.[43] Not withstanding the precise onset of the exponential

cutoff at Iref, it is safe to state that for the range of Ip/Iref ≪ 1,p(Ip) � Ip

�g (shown in Figure 11) for all the flow variableswhether they are collected in the CSL or ASL. Table 3presents all the regression statistics and estimates of g. Asmentioned in equation (12), for a weighted superimpositionof Poisson processes and for some classical models of self-organized criticality (SOC), the scaling exponents of TAspectra m = 3 � g. Figure 12 shows the measured andmodeled m using equation (12) with g inferred fromFigure 11 for Ip /Iref ≪ 1. It is clear that the m = 3 � grelationship describes well the variations in m from thevariations in g. Additional intermittency corrections (i.e.,m = (3 � m/2) � g) produce some worsening in suchcomparisons (Figure 12), presumably due to the fact thatsuch intermittency here appears to be ‘contaminated’ by

Figure 8. Mean and standard deviation values of the ratio of intermittency exponents (mTA/mS) at finescales computed for all variables for the six data sets and for the three stability classes: (a) unstable,(b) near-neutral, and (c) stable. As a reference, the value for mTA = mS is also shown (dotted line).

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external factors not endogenous to the energy cascade (atleast when compared to the findings in SB06).

5. Conclusions

[44] Velocity, air temperature, and scalar concentrationfluctuations measurements collected above natural surfacesranging from ice sheets to tall forests were analyzed via thetelegraphic approximation (TA). The overall goal was toexplore how intermittency and clustering are impacted bysimultaneous changes in thermal stratification and surfaceroughness. The findings here in relation to previous con-clusions from studies that employed the TA can be sum-marized as follows:[45] 1. A number of experiments have already shown that

the spectral exponents of the series (n) and its telegraphicapproximation (m) are related via m = an + b. However, theprecise values of a and b differed among these studies. SB06heuristically obtained a = b = 1/2 via scaling arguments andreported good agreement with measurements for longitudi-nal velocity far from boundaries. When all the data sets arecombined here, the linearity between m and n is still wellpreserved but with a = 0.58 and b = 0.26. Hence, for n = 5/3,m = 1.23 rather than 4/3 earlier noted for turbulence far fromboundaries. However, the linear relationship found betweenthe two exponents confirmed there is more ‘memory’ in TA(s) when compared to s(t), suggesting that amplitude

variability may be decorrelating the series. Moreover, itappears that the intercept of this linear relationship is farmore impacted than the slope when comparing the findingshere with SB06.[46] 2. The TA approximation was shown to preserve

some of the non-Gaussian properties (e.g., skewness) of theoriginal series via the fraction of time describing the ‘on’phases. Previous investigations did demonstrate that non-Gaussianity is related to amplitude variability (an aspect ofintermittency) [e.g., Katul et al., 1994; Giostra et al., 2002].This study demonstrated that also clusterization in theintermittent structures is related to the non-Gaussianity ofthe turbulent flow.[47] 3. A previous study demonstrated that for well-

developed turbulence far from boundaries (SB06) and withextrapolations to infinite Taylor micro-scale Reynoldsnumber (Rel), the clustering exponent is constant (a = 0.1).SB06 also found that ms(Rel) = a(Rel) � 0.1 and suggesteda linear correlation between clustering and intermittencyexponents. Like SB06, the analysis here revealed moreclustering (i.e., lower a) at finer time scales when comparedto larger scales. Moreover, ensemble-averaged values of a atfiner scale for the combined sites and flow variables fol-lowed quite well the SB06 relationship, even if individualruns did not. However, the data sets here did not reveal anysignificant ms (global intermittency exponent including bothamplitude and clusterization effect) or mTA (intermittency

Figure 9. Scatterplot of the mean global intermittency exponent (mS) versus the clustering exponent (a)computed for all three stability classes and for the variables (s = u, T, q, C) in the different experimentalsites: (a) Lav_H, (b) Duke_H, (c) Duke_P, (d) Duke_G, (e) BSoil, and (f) Ice. Vertical bars refer to standarddeviations and dashed lines represent the corresponding linear fits (Figure 9a: mS = 0.81a + 0.19; Figure 9b:mS = 0.88a + 0.14; Figure 9c: mS = 0.94a + 0.14; Figure 9d: mS = 0.66a + 0.17; Figure 9e: mS = 0.99a +0.10; Figure 9f: mS = 0.76a + 0.15). Grey points refer to the same quantities obtained for 30 min time seriesof vertical velocity component (s = w). For reference, the relationship proposed by SB06 (a: mS = a � 0.1;continuous line) is also shown.

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exponent of TA series including only clusterization effect)dependency on estimated [log(Rel)]

�1 when all the sites andflow variables were combined. The ranges of variability ofmTA were comparable to ms though the mean value of ms(≈0.30� 0.06) appear lower than that of mTA (≈0.36� 0.06);in other words, on average, the magnitude intermittencyseems to play a smoothing role on the clusterization effect.[48] 4. The average values of ms were linearly correlated to

a for longitudinal velocity and scalars. The mean regression(mS = 0.85a + 0.15), when all experimental sites werecombined, differ from SB06 relationship (mS = a � 0.1),mainly in the intercept value. The correlations between aand mTA were significant for longitudinal velocity and scalarsacross all sites and stability classes, with regression slopes

(a″) increasing with the roughness of surface (a″ ≈ 1.0–1.1for the bare soil and the ice sheet; a″ ≈ 1.41–1.65 for vege-tated sites). The comparison between the regression linesfound for the two intermittency exponents (mS = a′a + b′ andmTA = a″a + b’’) displayed the following: (1) for ASL mTA >ms ∀ a, for each variable and atmospheric stability condition:that is, intermittency in the magnitude of energetic eventsplays a smoothing role on the clusterization effect, consistentwith results obtained by SB06 and B04 for turbulence awayfrom boundaries; (2) for CSL mTA > ms for longitudinalvelocity and it is associated to a > 0.2 On the other hand,CSL scalars are more clusterized (a ∈ [0.1, 0.2]) andamplitude excursions tend to amplify (or to not alter) clus-terization (mTA ≤ ms). These findings are consistent with the

Figure 10. Scatterplot of the mean TA intermittency exponent (mTA) versus the clustering exponent (a)computed for all three stability classes and for the variables (s = u, T, q, C) in the different experimentalsites: (a) Lav_H, (b) Duke_H, (c) Duke_P, (d) Duke_G, (e) BSoil, and (f) Ice. Vertical bars refer to stan-dard deviations and dotted lines represent the corresponding linear fits (Figure 10a: mTA = 1.52a + 0.04;Figure 10b: mTA = 1.57a � 0.002; Figure 10c: mTA = 1.41a + 0.05; Figure 10d: mTA = 1.65a + 0.03;Figure 10e: mTA = 1.02a + 0.18; Figure 10f: mTA = 1.10a + 0.14). Grey points refer to the same quantitiesobtained for 30 min time series of vertical velocity component (s = w). For comparison, the regressionlines found for mS in Figure 9 (dashed lines) are also shown.

Figure 11. Ensemble-averaged probability density function (pdf) of the inter-pulse period normalized by the integral timescale for longitudinal velocity (a), vertical velocity (b) and temperature (c) for the three atmospheric stability conditions andfor data collected above the Lavarone Hardwood forest. The pdfs for the different stability conditions are shifted to permitcomparisons. Continuous lines represent the lognormal fits (for comparison with SB06) and dashed lines represent the powerlaw fits for ISR scales. Note that the lognormal model does not retain the apparent power law evident in the measurementsfor small Ip/ITS. (d, e, f) Same as Figures 11a, 11b, and 11c but for data collected above the Hardwood forest at Duke. (g, h, i)Same as Figures 11a, 11b, and 11c but for data collected above the Pine forest at Duke. (j, k, l) Same as Figures 11a, 11b, and11c but for data collected above the grass at Duke. (m, n, o) Same as Figures 11a, 11b, and 11c but for data collected abovethe Bare Soil. Continuous and dashed lines represent the lognormal and power law fit, respectively. (p, q, r) Same asFigures 11a, 11b, and 11c but for data collected above the Ice Sheet in Antarctica. Continuous and dashed lines representthe lognormal and power law fit, respectively.

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Figure

11

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fact that large-scale structures such as ramps, that are lessdistorted (maintain their clusterization) in the CSL whencompared to their ASL counterparts, do impact ISR scalingfor scalars.

[49] 5. The relationship between the clustering exponent(a) and the intermittency parameters (mS, mTA) appearedmost sensitive to surface roughness for the vertical velocitycomponent (w). With progressively reduced surface rough-ness, a de-correlation between a and mS (and mTA) wasobserved for w but not for the other flow variables. Thisde-correlation between a and intermittency parameters forw with progressively increasing smoothness of the surfaceis perhaps the strongest evidence that roughness effectsmay partially impact the structure of fine scale turbulencewithin the ISR. Also, this de-correlation was tested acrossmultiple heights above the ice sheet surface and was notfound to change with variations in measurement level.[50] 6. The probability density functions p(Ip) of the

inter-pulse period (Ip) was shown to be well approximatedby a power law distribution with exponential cut-off abovethe forested ecosystems and by a simple power law abovethe smoother surfaces for all flow variables except w for theentire range of Ip. However, when restricting Ip to eventsshorter than the integral time scale (i.e., to time scalescomparable to those within the ISR), a power law distri-bution (p(Ip) � Ip

�g) emerged for all flow variables andstability classes (even for w). The power law exponent gappears to be well related to m via the expression m = 3 � g,an expression also applicable for systems exhibiting self-organized criticality (SOC).[51] More broadly, whether the TA properties of atmo-

spheric turbulence actually share attributes with SOC israther intriguing, and may be traced back to Per Bak’soriginal work [Bak et al., 1988]. Analogies between plasmaturbulence, convection turbulence, and SOC have receivedsome attention in the physics literature [De Menech andStella, 2002; Sreenivasan et al., 2004]. In plasma turbu-lence, it was shown that instabilities governed by threshold-like behavior may lead to SOC by producing transport eventsat all scales (e.g., analogous to avalanches in the Per Bakmodel). These avalanches arise due to accumulation of localenergy, leading to an increased gradient, which whenexceeding a specific threshold, result in a burst of activitythereby expelling this accumulated energy [Sreenivasan

Table 3. Exponent (g) of Power Law Approximation (p(Ip) � Ip�g)

of the Probability Density Function of the Distribution of theInter-pulse Periods (Ip) When Ip Is Smaller Than the Integral TimeScale for Longitudinal Velocity Component (u), Vertical VelocityComponent (w), Temperature (T), Water Vapor (q) and CO2 Con-centration (C)

Site Variable Unstable Near Neutral Stable

Group 1: Canopy Sublayer (CSL)Alpine Hardwood u 1.58 1.65 1.56

w 1.60 1.77 1.70T 1.60 1.78 1.46q 1.65 1.87 2.02C 1.85 2.19 1.69

Oak-Hickory Hardwood u 1.87 1.86 2.15w 1.84 1.76 2.00T 1.87 1.81 1.88q 1.58 1.79 1.91C 1.71 1.85 1.78

Loblolly Pine u 1.73 1.65 1.71w 1.82 1.65 1.67T 1.73 1.57 1.56q 1.54 1.64 1.69C 1.59 1.74 1.82

Group 2: Atmospheric Surface Layer (ASL)Grass u 1.77 1.76 -

w 1.71 1.49 -T 1.83 1.70 -

Bare Soil u 1.85 1.60 1.87w 1.58 1.42 1.36T 1.78 1.78 2.03

Ice Sheet u 2.00 2.18 1.97w 1.92 1.30 1.50T 1.97 2.21 1.97

Figure 12. Scatterplot of modeled (from g) versus ensemble-averaged measured spectral exponents ofTelegraphic Approximation (TA) for all variables, atmospheric stability conditions, and sites.

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et al., 2004]. It is conceivable that atmospheric flows, withtheir unique co-existence of mechanical and buoyancy pro-duced turbulent kinetic energy, and the fact that this pro-duction is embedded in a background of well-developedturbulence excited at many scales might exhibit analogousbehavior. Kelvin-Helmholtz vortices characterizing CSLflows, convection plumes, attached eddies characterizingboundary layers all have some form of a threshold-like‘origin of hydrodynamic instability’ governing their forma-tion. Once these instabilities occur and develop, the ensuingeddies do have a local production source (ground heating,local mean velocity gradient) and are characterized (in time)by some form of a burst or excited state. However, thesimilarity between SOC and ASL/CSL turbulence, if any,cannot be viewed except as a statistical analogy within theconfines of telegraphic properties pertaining to p(Ip).

[52] Acknowledgments. Cava and Katul acknowledge support from“Cooperazione Italia-USA su Scienza e Tecnologia dei Cambiamenti Cli-matici, Anno 2006–2008.” G. G. Katul acknowledges support from theFulbright-Italy Distinguished Scholars program, the National ScienceFoundation (AGS-1102227, EAR-1013339, IOS-0920355, and CBET-1033467), and the United States Department of Agriculture (USDA-FSAGRMN’T SRS 09-CA11330140-059 and 2011-67003-30222). A. Moliniacknowledges support of National Science Foundation (NSF EAR1063717).

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D. Cava and C. Elefante, Institute of Atmosphere Sciences and Climate,National Research Council, I-73100 Lecce, Italy. (d.cava@le.isac.cnr.it;c.elefante@le.isac.cnr.it)G. G. Katul, Nicholas School of the Environment, Duke University,

Box 90328, Durham, NC 27708-0328, USA. (gaby@duke.edu)A. Molini, Masdar Institute of Science and Technology, PO Box 54224,

Abu Dhabi, United Arab Emirates. (amolini@masdar.ac.ae)

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