Flume experiments on intermittency and zero-crossing propertiesof canopy turbulence

Davide Poggi1,a� and Gabriel Katul2,b�

1Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili, Politecnico di Torino, Torino 10129, Italy2Nicholas School of the Environment and Department of Civil and Environmental Engineering,Duke University, Durham, North Carolina 27708-0328, USA

�Received 15 September 2008; accepted 15 April 2009; published online 4 June 2009�

How the presence of a canopy alters the clustering and the fine scale intermittency exponents andany possible connections between them remains a vexing research problem in canopy turbulence. Tobegin progress on this problem, detailed flume experiments in which the longitudinal and verticalvelocity time series were acquired using laser Doppler anemometry within and above a uniformcanopy composed of densely arrayed rods. The time series analysis made use of the telegraphicapproximation �TA� and phase-randomization �PR� methods. The TA preserved the so-calledzero-crossing properties in the original turbulent velocity time series but eliminated amplitudevariations, while the PR generated surrogate data that preserved the spectral scaling laws in thevelocity series but randomized the acceleration statistics. Based on these experiments, it was shownthat the variations in the dissipation intermittency exponents were well described by the Taylormicroscale Reynolds number �Re�� within and above the canopy. In terms of clustering, quantifiedhere using the variance in zero-crossing density across scales, two scaling regimes emerged. Forspatial scales much larger than the canopy height hc, representing the canonical scale of the vorticesdominating the flow, no significant clustering was detected. For spatial scales much smaller than hc,significant clustering was discernable and follows an extensive scaling law inside the canopy.Moreover, the canopy signatures on the clustering scaling laws were weak. When repeating theseclustering measures on the PR data, the results were indistinguishable from the original series.Hence, clustering exponents derived from variances in zero-crossing density across scales primarilydepended on the velocity correlation function and not on the distributional properties of theacceleration. In terms of the connection between dissipation intermittency and clustering exponents,there was no significant relationship. While the former varied significantly with Re�, the lattershowed only minor variations within and above the canopy sublayer. © 2009 American Institute ofPhysics. �DOI: 10.1063/1.3140032�

I. INTRODUCTION

The structure of turbulence within the canopy sublayer�CSL� is generally studied for its mixing properties. How-ever, the agglomeration of inertial particles such as aerosols,water droplets, and pollen grains near the canopy-atmosphere interface is now receiving renewed interest.Some studies suggest that agglomeration is sensitive to theinterplay between the inertia of these particles and spatialclustering of turbulent eddies at fine scales;1–5 the latter be-ing connected to fluid acceleration statistics and thus finescale intermittency.6,7

Generally, intermittency consists of two aspects: one re-lated to large amplitude variability and another related tolocal frequency oscillations.8 Within the CSL, a number ofquestions pertaining to clustering, intermittency, and theirconnection remain to be explored even for the most idealizedcases such as stationary and planar homogeneous flows inuniform canopies. Does vegetation alter the clustering prop-erties of turbulence or does it simply act to filter out turbu-

lent excursions? Are changes in clustering properties withinthe CSL primarily responding to a reduced Reynolds numberinside the vegetation or is their scaling with respect to theReynolds number altered at a more basic level? Are the clus-tering properties within the CSL anisotropic, and if so, whichof these directions exhibit higher tendencies to cluster?These questions have not been methodically tackled and arethe subject of this work.

To begin progress on these problems, the intermittencyand clustering properties of turbulence at two large bulkReynolds numbers �Reb� are explored inside a canopy com-posed of uniformly arrayed rods in a flume. Separating am-plitude variability from clustering effects is achieved by ana-lyzing the measured time series in two ways. The first usesthe so-called telegraphic approximation �TA� in which thebinary �or on-off� nature of TA allows the isolation of eventclustering in space �or time� without being influenced byamplitude variations. The TA preserves the so-called zero-crossing properties in the original time series but eliminatesamplitude variations.9,10 The second uses a surrogate datageneration scheme via the so-called phase-randomization�PR� method in which the energy spectrum of the measuredtime series is precisely reproduced but the phase angle is

a�Author to whom correspondence should be addressed. Electronic mail:davide.poggi@polito.it.

b�Electronic mail: gaby@duke.edu.

PHYSICS OF FLUIDS 21, 065103 �2009�

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randomized.11–14 By virtue of its construct, such syntheticdata generation scheme preserves the precise spectral scalinglaws of the original velocity time series but destroys anyanomalous scaling originating from velocity gradients �intime�. Even in the case of a Gaussian turbulent velocity timeseries, it was shown in a number of studies that velocitygradients often exhibit exponential rather than Gaussiantails;15–17 however, for their phase-randomized counterpart,the velocity gradients become Gaussian thereby destroyingthe anomalous scaling.

Isolating amplitude variability from clustering was re-cently undertaken in studies dealing with turbulent convec-tion at high Rayleigh number and with turbulence far fromany boundary,8,18 hereafter referred to as B04 and SB06, re-spectively. The SB06 and B04 results are used as referencewhen exploring how the presence of a dense canopy altersclustering and intermittency exponents.

II. EXPERIMENTAL SETUP

The details of the recirculating flume setup, the rodcanopy configuration, velocity measurements, and data pro-cessing are described elsewhere.19–23 But, briefly, the flumeexperiments were carried out at the hydraulics Laboratory,DITIC Politecnico di Torino, in a recirculating rectangularchannel that is 18 m long, 0.90 m wide, and 1 m high. Ver-tical steel cylinders, 120 mm tall and 4 mm in diameter�=dr� were arranged in a regular pattern along the 9 m longtest section to represent the canopy. The rod density was1072 rods m−2 and resulted in a drag coefficient comparableto drag coefficient estimates reported for crops and denselyforested canopies.24–26

Two experiments were carried out at different flow ratesresulting in a friction velocity at the canopy top of u�

=0.053 and 0.098 m s−1. The concomitant canopy Reynoldsnumbers �Re�=u�hc /�� for these two experiments wereabout 6000 and 12000, respectively. The longitudinal �u� andvertical �w� velocity time series were sampled at 2500–3000Hz for 300 s using a two-component laser Doppler anemom-etry employed in a forward scattering mode. This samplingfrequency was sufficiently large to resolve the entire inertialsubrange and about some half a decade of viscous dissipationrange �at z /hc=1.9�. The sampling location in a plane withincanopy was selected so that the mean velocity at the selectedlocation was identical to the temporally and spatially aver-aged velocity between the rods. The details of how theselocations were selected are discussed elsewhere.19 The 600mm uniform water depth �hw� was sampled vertically every10 mm, although the focus here is on the region extending up2.6hc as it is often linked with the thickness of the CSL.27–31

Using the vertically averaged velocity �Ub� across the entirehw, the Reb�=Ubhw /�� for the two experiments were about115000 and 170000, differing by about a factor of 1.5, where� is the kinematic viscosity of water.

Figure 1 compares across the two experiments the pro-

files of mean longitudinal velocity �U�, the velocity standarddeviations �u= �u�2�1/2 and �w= �w�2�1/2, the mean turbulentkinetic energy dissipation rate �, and the Taylor microscaleReynolds number Re�=��u /�, where �= �15��u

2 /��1/2 is theTaylor microscale, overbar is time averaging, and primedquantities are turbulent excursions. The � was estimated us-ing the isotropic relationship32

� = 15�� �u�

�x�2

, �1�

where x is the longitudinal distance. Using Taylor’s frozenturbulence hypothesis,33–35 the spatial gradients in Eq. �1�

0 0.5 10

0.5

1

1.5

2

2.5

3z/

h c

U (m/s)

Re*=12000

Re*=6000

0 2 4 60

0.5

1

1.5

2

2.5

3

z/h c

ε hc/u

*3

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

z/h c

σu/u

*,σ

w/u

*

0 500 10000

0.5

1

1.5

2

2.5

3z/

h c

Rλ

Re*=6000

Re*=12000

FIG. 1. �Color online� Profiles of the normalized mean

velocity �U�, longitudinal and vertical velocity standarddeviations ��u=�u�2, �w=�w�2�, and the mean turbu-lent kinetic energy dissipation rate computed from �=15���u� /�x�2, and the Taylor microscale Reynoldsnumber �Re�=��u /�� for the two Reb. All velocity andlength scales are normalized by the friction velocity�u�� and canopy height �hc�. Note the strong inflection

point in �U� at the canopy top, the dampening of �u andthe enhanced � inside the canopy, and the similarity inRe�=��u /� despite large differences in bulk Reynoldsnumber.

065103-2 D. Poggi and G. Katul Phys. Fluids 21, 065103 �2009�

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can be determined from temporal gradients. This estimate of� agreed well with an estimate derived as a residual of aone-dimensional turbulent kinetic energy budget equationwhose terms �mechanical, wake, and transport components�were independently measured.

Figure 1 illustrates few canonical features about the

measured bulk properties of CSL turbulence. The U insidethe canopy is generally small but finite with a strong inflec-tion point near the canopy-top resembling mixing layers.

Roughly, the measured U /u��3.3 is consistent with a broadrange of field experiments.28,36 The velocity variances aredamped inside the canopy, and there is a region of enhanceddissipation rate in the upper layers of the canopy. The high �and decreased �u

2 simultaneously act to reduce Re� inside thecanopy. Figure 1 shows that the Re� profiles appear compa-rable despite the large differences in Reb across these twoexperiments. It should be emphasized that within an experi-ment, variations in Re� with z /hc are large ��10�.

III. METHOD OF ANALYSIS

As already mentioned, SB06 already reported results onintermittency and clustering properties of turbulencesampled far from boundaries as a function of Re�. For com-pleteness, a brief review of these results and on how theclustering and the intermittency exponent parameters weredetermined from time series along with other analyses con-ducted to address the study objectives is presented next.

A. TA and phase randomization

For an arbitrary zero-mean flow variable s�t�, B04 andSB06 defined the TA of s�t� as

TA�s� =1

2� s��t�

s��t�+ 1� , �2�

where s��t�=s�t�− s. For consistency with these two studies,this definition is adopted throughout. SB06 also showed that

this TA definition is not sensitive to the zero-mean threshold.Because TA�s� can either take on values of 1 or 0, there areno “amplitude variations” in the TA series.

For the phase randomization �PR�, a synthetic time se-ries possessing the same power spectrum as the original se-ries was generated but the phase angle randomized.14 Theapproach adopted was as follows: The Fourier transform ofthe velocity component time series is first computed, thesquared Fourier amplitudes are then determined at each fre-quency, a phase angle is randomly selected between 0 and2� and assigned to each frequency, real and imaginary com-ponents of the Fourier coefficients are then regenerated, andan inverse Fourier transform is carried out. Because the PRdoes not guarantee a real-valued series, only the real compo-nents �in the time domain� are considered. The resulting syn-thetic time series has an identical power spectrum as theoriginal series, but the probability distribution of the gradi-ents �or acceleration� is Gaussian. Hereafter, we refer to thephase-randomized series of s�t� as PR�s�.

As an illustration, Fig. 2 shows a u time series, its TA�u�and PR�u� counterparts, and the histogram of the velocitytemporal gradients. Notice from Fig. 2 that TA�u� preservesthe zero-crossing properties in the u series and that the PR�u�resembles the u series in terms of duration of apparent co-herent events but the distributional properties of its temporalgradients behave as Gaussian rather than exponential.

Because agglomeration of inertial particles occurs withinzones of zero acceleration �or acceleration stagnation re-gimes�, the PR transformation preserves the scaling laws andthe global correlation function in the measured velocity timeseries but randomizes its acceleration statistics. Hence, thecontrast between the PR and the original velocity series isintended to highlight the role of acceleration on the cluster-ing statistics within the CSL when all the scaling laws in thevelocity spectra are preserved.

0 10 20 30−0.4

−0.2

0

0.2

0.4

u

t (s)

−200 −100 0 100 200

10−6

10−4

10−2

pdf

du/dt, duPR

/dt (m/s2)

0 10 20 30−0.4

−0.2

0

0.2

0.4

u

t (s)

0 10 20 300

0.5

1

1.5

u TA

t (s)

FIG. 2. �Color online� Time series of u at z /hc=2.66�top left�, along with its TA�u� �bottom left� and PR�u��top right� counterparts, and the histogram of du /dt�open circles� and dPR�u� /dt �stars�. Temporal gradi-ents were estimated from forward differencing.

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B. Spectra

The relationships between the spectral scaling exponentof s�t�, which is impacted by both amplitude variability andclustering, and TA�s�, which is impacted by clustering only,are reviewed here. For a wide range of stochastic processes,including turbulence far from rigid boundaries, it was em-pirically demonstrated by SB06 that when the spectrum ofs�t� scales as f−n, the spectrum of TA�s� scales as f−m with

m =n + 1

2, �3�

where f is the frequency. Moreover, SB06 noted that thespectral scaling regime for TA�s� is more extensive than fors�t�. When n�1 �typically, n=5 /3 for turbulence away fromboundaries at high Reynolds number�, the expression bySB06 predicts an m�n. Stated differently, when n�1, thereis more “memory” in TA�s� due to the sequence of on-off oroff-on switching in time when compared to s�t�. Hence, ac-cording to Eq. �1�, when n�1, amplitude variability may bedecorrelating the series.

A pertinent question for the objectives here is whether alinear relationship between m and n exists in the CSL. If so,is this relationship invariant when the analyses are repeatedon the PR data?

C. Clustering exponent

The clustering exponent ��� can be derived from scalingrelationships applied to the zero-crossing density �n�t��,whose statistics are briefly reviewed. For a time series havinga length Ns�i=1,2 , . . . ,Ns� and sampled at times ti at a ratedt= ti+1− ti, the times at which zero crossings occur are de-fined by the indicator function I�ti� given as

I�ti� = 1 if s��ti�s��ti+1� � 0,

0 otherwise.�

The overall density of zero crossings is given by

10−1

100

101

102

10−2

100

102

z/hc=0.05

10−1

100

101

102

z/hc=0.3

10−1

100

101

102

10−2

100

102

z/hc=0.5

10−2

100

102

E(u

)

z/hc=0.83 z/h

c=1.05

10−2

100

102

z/hc=1.24

10−1

100

101

102

10−2

100

102

z/hc=1.51

10−1

100

101

102

(f hc)/U

z/hc=1.79

10−1

100

101

102

10−2

100

102

z/hc=2.66

FIG. 3. �Color online� The spectra of u and TA�u� at various levels within and above the canopy as a function of scale. The scaling ranges �vertical solid lines�used to compute m and n, and the 4/3 and 5/3 power laws are shown for reference. The two length scales, canopy height hc and rod diameter dr, are alsoshown �vertical dashed lines�.

065103-4 D. Poggi and G. Katul Phys. Fluids 21, 065103 �2009�

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nd =�i=1

Ns−1I�ti�Ns − 1

.

Time series of zero-crossing densities at time scales can beconstructed from the density of zero crossings for the portionof the time series bounded between s��t� and s��t+� givenas

n�ti� =�i=1

i+jI�ti� j

,

where j = jdt, with j=2,3 , . . . .. The so-called clustering ex-ponent ���, related to the standard deviation of the zero-crossing density at different scales, can be determined fromscaling relationships by varying so that �SB06�

�n21/2 −�, �4�

where �n�t�=n�t�−n�t� and n�t� is the average zero-crossing density for a given j. For white noise, taken here toimply no significant clustering, �=1 /2. SB06 demonstratedthat for well-developed turbulence far from any rigid bound-ary, the clustering exponent varied only with Re� as

� � 0.1 +3/2

ln�Re��. �5�

Hence, as Re�→�, ��0.1 and clustering remains finiteeven at infinite Taylor microscale Reynolds number.37 Forfinite Re�� �200,20000�, data in SB06 suggest that � rangedfrom 0.25 to 0.40 for inertial subrange �ISR� scales but ap-proached white-noise values � 1 /2� at much larger scales�or �.

D. Intermittency exponent

The so-called dissipation intermittency exponent can bedetermined from the time series of

��t� = 15�� 1

U

du��t�dt �2

by defining a new series as

�t� =1

�

t

t+

��t�dt , �6�

and using the scaling of the moments �e.g., SB06�,

10−1

100

101

102

10−2

100

102

z/hc=0.21

10−1

100

101

102

z/hc=0.34

10−1

100

101

102

10−2

100

102

z/hc=0.6

10−2

100

102

E(w

)

z/hc=0.87 z/h

c=1.08

10−2

100

102

z/hc=1.24

10−1

100

101

102

10−2

100

102

z/hc=1.47

10−1

100

101

102

(f hc)/U

z/hc=1.74

10−1

100

101

102

10−2

100

102

z/hc=2.29

FIG. 4. �Color online� Same as Fig. 3 but for the spectra of w and TA�w�.

065103-5 Intermittency and zero-crossing properties Phys. Fluids 21, 065103 �2009�

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� �t��q

� �t��q −�q. �7�

This intermittency measure has been used by a number ofauthors in turbulence research.7,38 For comparisons withSB06, only q=2 is considered. The �2 computed for s isreferred to as �s for notational simplicity.

SB06 found that �s�Re��=3 /2 / ln�Re��, suggesting aone-to-one correlation between clustering and intermittencyexponents as Re� varies �i.e., �=�s+0.1�. Moreover, whenRe�→�, �s�0.

IV. RESULTS

How the canopy modifies the spectral, intermittency, andclustering exponents is evaluated using the velocity seriescollected at the two high Reb. Whether these canopy-inducedexponent modifications are originating from amplitude vari-ability or shifts in the zero-crossing statistics are explored bycontrasting between the velocity series and their TA or PRsurrogates.

A. Spectra

The spectra of u and w,and TA�u� and TA�w� are shownin Figs. 3 and 4, respectively, for z /hc ranging from 0.05 to2.66. Frequency was converted to length scale using U viaTaylor’s frozen turbulence hypothesis. For reference, twolength scales are also shown in these figures: the canopyheight and the rod diameter. The latter is presented becausewake production and the generation of Von Karman streetsinside the canopy occur at scales comparable to dr.

19–22,39

This mechanism signifies a new injection of energy at scalescommensurate with the rod diameter, thereby disrupting theusual ISR, and fine scale analysis must be conducted atscales even smaller than dr. Such a mechanism is entirelyabsent in turbulent flows away from boundaries studied bySB06.

The analyses in these figures are suggestive that the TApower-law scaling ranges are no more extensive than theoriginal series �also shown in the two figures�, unlike SB06.The spectral shapes of the flow variables and their TA arequalitatively similar in the sense that wake production wasdiscernable in both and at a scale comparable to dr.

Figure 5 shows the spectral scaling exponents, deter-

0

1

2

3

z/h c

Su

−2 −10

1

2

3

z/h c

Su

STAu

−2.5 −2 −1.5 −1

−2

−1.5

−1

STA

uS

u

Re*=12000

Re*=6000

0

1

2

3

z/h c

Sw

−2 −10

1

2

3

z/h c

Sw

STAw

−2.5 −2 −1.5 −1

−2

−1.5

−1

STA

w

Sw

Re*=12000

Re*=6000

FIG. 5. �Color online� Relationshipbetween the spectral scaling exponentsm and n determined for all z /h levelsfor u �left� and w �right� along with theregression lines and the relationship inSB06 �top panels�. The vertical pro-files of n �middle panels� and m �bot-tom panels� are shown for both canopyReynolds numbers �Re*� along withthe expected values from classical ISRscaling �n=5 /3� and proposed SB06scaling �m=4 /3�, presented here asvertical lines for reference.

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mined from regression analysis across the range of scaleshighlighted in Figs. 3 and 4, as a function of z /hc for u andw components and their TA counterparts. As expected fromthe empirical findings of SB06, above the canopy, the spec-tral exponents are near 5/3 for the turbulent flow variablesand 4/3 for their TA counterpart. These “above-canopy” re-sults appear to be insensitive to the variations in Reb here.Inside the canopy, the variations in the exponents are depen-dent on both-z and Reb. However, what must be emphasizedis that vertical variations in the spectral exponents and theirTA counterpart remain consistent, suggesting that the spectraof the TA series account for both the height variations andReb dependence. Figure 5 also compares the spectral expo-nents for the flow variables and their TA, along with theempirical relationship in SB06. The agreement between thedata and this relationship is reasonable for slopes that do notsignificantly diverge from 5/3. However, the agreement de-grades for slopes far from 5/3 �i.e., inside the canopy� al-though the relationship between m and n maintains its linear-ity, again, suggestive that the TA spectra contain significantinformation about the scaling laws of the velocity series.

The analysis in Fig. 5 was repeated for the phase-randomized series using the higher Reb runs �for illustration�and the outcome is presented in Fig. 6. Again, the scaling

exponents are identical by virtue of their construct. For theTA exponents of the PR data, they do not appreciably divergefrom 4/3 above the canopy. Inside the canopy, the resultingexpression between m and n remains linear but differentfrom what was derived in Fig. 5. When contrasting the basicrelationship between m and n in Figs. 5 and 6, the PR pri-marily alters the intercept.

B. Dissipation intermittency

Using the computed ��t� series, the moment scalingfunction � �t��2 / � �t��2 and the inference of �s for u and ware presented in Fig. 7 along with their PR counterparts �gen-erated from u�. It is clear from Fig. 7 that for the velocityseries, a more extensive scaling regime inside the canopyexists when compared to their counterpart above the canopy.Also, the analysis here suggests that there is no significantdifference between the dissipation scaling functions of u andw. The PR series exhibited no significant intermittency scal-ing, except at small . Using the range of scales identified inFigs. 3 and 4 �and repeated in Fig. 7 for reference�, the �s fors=u and w were computed and presented as a function ofz /hc in Fig. 8 along with SB06 predictions based on�s�Re��=3 /2 / ln�Re��. Here, the values of Re� were taken

0

1

2

3

z/h c

Su

−2 −10

1

2

3

z/h c

Su

STAu

−2.5 −2 −1.5 −1

−2

−1.5

−1

STA

uS

u

Original

PR

0

1

2

3

z/h c

Sw

−2 −10

1

2

3

z/h c

Sw

STAw

−2.5 −2 −1.5 −1

−2

−1.5

−1

STA

w

Sw

Original

PR

FIG. 6. �Color online� Same as Fig. 5but using u and w and PR�u� andPR�w� for the higher Reb.

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from Fig. 1. From Fig. 8, it is evident that variations in Re�

explain much of the observed variations in �s for both ve-locity components. Moreover, for z /hc�2, the range of �s

reported here is in good agreement with SB06 �0.15–0.30�.The �s for w was generally higher than �s for u, althoughthis difference may be due to the limited scaling regime usedin the determination of �s above the canopy. Deeper in thecanopy, �s increased for both velocity components, but thisincrease can be primarily explained by the decrease in Re�

noted in Fig. 1. Hence, in a first order analysis, the effects ofthe canopy on �s are primarily linked with the reduced Re�.Whether these variations in �s are connected to clustering, aswas the case in SB06, is explored later.

C. Non-Gaussianity, TA, and zero-crossing statistics

Up to this point, it was demonstrated that TA�s� pre-serves much of the spectral scaling laws in the velocity se-ries. Hence, a follow-up question is whether other attributesconnected with the distributional properties of s are also pre-

served in TA�s�. As we show here, the TA series does pre-serve important attributes about the non-Gaussian propertiesof the original series. However, key informations about theprobability density of the velocity gradients are not well pre-served by variability in zero-crossing properties. To illustratethese two points, the variations with z /hc of the fraction oftime the TA�s�=1 �Fig. 9� and variations in the standarddeviation of the zero-crossing density as captured by the

scaling relationship between �n21/2 and for various z /hc

�Fig. 10� are discussed.One of the defining features of CSL flows is the posi-

tively skewed u and the negatively skewed w, both con-nected with the ejection sweep cycle and its contribution tomomentum transfer inside the canopy.19,28,29,40–42 It is dem-onstrated that the TA series preserves such features via thefraction of time TA�s�=1, which is equivalent to the fractionof time, the turbulent excursions in the original series arepositive �hereafter referred to as �+�.

The skewness of an arbitrary flow variable is defined as

10−3

100

103

1

3

10 z/hc=0.21

10−3

100

103

z/hc=0.34

10−3

100

103

1

3

10z/hc=0.6

1

3

10

<ετ2 >/

<ετ>2

z/hc=0.87 z/h

c=1.08

1

3

10z/hc=1.24

10−3

100

103

1

3

10 z/hc=1.47

10−3

100

103

(τ U)/hc

z/hc=1.74

10−3

100

103

1

3

10z/hc=2.29

FIG. 7. �Color online� The inference of the dissipation intermittency exponent �s from � �t��2 / � �t��2 as a function of scale �=U /hc� for u, PR�u�, and wat various heights above the ground. The range of scales used to infer m and n in Figs. 3 and 4 are used to infer �s and are repeated here for reference. Starsrefer to the PR series.

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Sks = �s�3, s =s�

�s,

and is shown for s=u ,w in Fig. 9 for z /hc ranging from 0.05to 2.66. Also shown in Fig. 9 are sample u and w, along withtheir TA�u� and TA�w� time series inside and above thecanopy �for contrast�. It is clear from these time series illus-trations that a connection between the time fraction the TA isunity and the skewness in the series must exist. In fact, uponplotting �+ against Sks for these CSL experiments, a consis-tent near-linear relationship across the two flow variables �uand w� and the two Reb emerged. Hence, Fig. 9 empiricallysupports the notion that much of the skewness in u and w iscaptured by �+. To further explore the genesis of this near-linear expression analytically, a cumulant expansion of theprobability density function p�s� is considered and, for scal-ing purposes, all cumulants beyond order 3 are discarded,resulting in20,30,42–44

p�s� � � 1�2�

exp�−s2

2���1 +

1

6Sks�s3 − 3s�� . �8�

The fraction of time s�0 is given by

�+ = �0

�

p� �d =1

2−

1

12Sks� 2

�. �9�

Figure 9 demonstrates the good agreement between mea-sured and modeled �+ based on measured Sks. In fact, thisagreement suggests that measured �+ �preserved in the TAseries� includes the integrated effects of all higher order cu-mulants of p�s� although the dominant cumulant appears tobe Sks here. Having shown that non-Gaussianity in the origi-nal series is preserved in the TA series through �+, we con-sider next the mechanisms that govern variations in zero-crossing densities at various time scales and z /hc andwhether they capture the distributional properties of the ve-locity gradients �or acceleration statistics�. Recall that zero-crossing densities are not connected to �+ but are connectedwith the one-time step differences in TA�s�.

Figure 10 presents variations in �n21/2 with for the u

and PR�u� within and above the canopy. As in SB06, tworegimes emerged with �n

21/2 decaying approximately as −1/2

for large and approximately as −1/3 for small . In fact,

when converting to an equivalent length scale via U, the−1/2 scaling �i.e., no significant clustering� only occurs for

−0.4 −0.2 0

0

1

2

3

4

z/h c

Longitudinal

Re*=12000

Re*=6000

0

1

2

3

4

z/h c

PR

Orig

−0.4 −0.2 00.1

0.15

0.2

0.25

0.3

1/lo

g(R

e λ)

µs

−0.4 −0.2 0

0

1

2

3

4

Vertical

Re*=12000

Re*=6000

0

1

2

3

4PR

Orig

−0.4 −0.2 00.1

0.15

0.2

0.25

0.3

µs

FIG. 8. �Color online� The variations the intermittencyexponents �s as a function of distance from the groundand Re�. Top panels: The variations of �s with z /hc foru �left� and w �right� for both the Re�. Predictions fromSB06 are also shown �solid lines�. Middle panels: Sameas above but for u and PR�u� �left� and w and PR�w��right� but for the high Re�. Bottom panels: The varia-tions of �s, for u and PR�u� �left� and w and PR�w�,with Re� along with predictions from SB06.

065103-9 Intermittency and zero-crossing properties Phys. Fluids 21, 065103 �2009�

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spatial scales much larger than the canonical scale of thevortices dominating the flow �e.g., hc within the CSL�.However, the scaling regimes of �n

21/2 with −1/3 �significantclustering� are fairly extensive inside the canopy, and unlikethe power spectra in Figs. 3 and 4, there are no signatures ofwake production even for scales commensurate with dr. Infact, for all practical purposes, the canopy signatures on�n

21/2 versus remain weak.The PR�u� series also exhibited identical �n

21/2 with scaling as the u series within and above the canopy. Thisagreement in scaling can only be suggestive that � primarilydepends on the correlation function of the velocity series andnot on the distributional properties of the acceleration. Thisstatement is consistent with Rice’s formula45,46 in which theexpected density of zero crossings for a correlated Gaussianseries is entirely described by its autocorrelation function��� via

nd �1

����−

d2���d2 ��

=0.

The velocity time series is not Gaussian, especially withinthe CSL as evidenced by Fig. 9, but such non-Gaussianity isalready captured by �+ within the TA series. The fact that �nis based on one-time-step differences within the TA seriesmakes them robust to the Gaussianity assumption but muchmore linked with the behavior of ��� for small . The ro-bustness of Rice’s formula to the Gaussian assumption is notnew and has been alluded to in other turbulence studies. Forexample, noting that

2

�2 = �− �d2���d2 ��

=0,

it was already demonstrated that Rice’s formula accuratelyreproduced the Taylor microscale from measured nd evenwhen the turbulent velocity time series wasnon-Gaussian.9,10,47 In summary, Fig. 10 demonstrates thatthe �n

21/2- scaling appears to be dependent only on thespectral �or correlation function� properties of the velocity�through Rice’s formula� and not on the precise statistics ofthe acceleration. Stated differently, while the TA series cap-tures the key attributes of the velocity time series probabilitydensity function �through �+�, the TA does not capture thekey attributes of the gradient probability density function.This finding is somewhat disappointing because clustering ofinertial particles may be linked with the acceleration statis-tics, and � appears not to be an appropriate scalar index thatcan be used to discern eddy clustering in CSL flows.

Figure 11 shows the profile of � for s=u ,w computedfor the fine scales already shown in Fig. 10. It is clear herethat the canopy does not significantly alter �, although thereare persistent patterns in the shape of these profiles across thetwo Reynolds numbers and velocity components that cannotbe ignored. The clustering exponent appears to be highestnear the canopy top although the overall variation in � re-mained small.

In terms of connection between �s and �, there appearsto be no clear relationship—in disagreement with SB06 whofound a strong relationship between �s and � for turbulence

0 10 20

−0.5

0

0.5

1

1.5Lev. 1

t (s)

w/(

6σw

)

0 10 20

−0.5

0

0.5

1

1.5Lev. 1

t (s)

u/(6

σ u)

0 5 10

Lev. 2

t (s)

0 5 10

Lev. 2

t (s)

−2 −1 0 1

0

0.5

1

1.5

2

2.5

3

Lev. 1

Lev. 2

z/h c

sku,sk

w

−2 −1 0 10.35

0.4

0.45

0.5

0.55

0.6

0.65

Γ +

sku,sk

w

FIG. 9. �Color online� Same time series of positively skewed u �top left� and negatively skewed w �bottom left� along with their TA visually illustrating theconnection between zero-crossing density and skewness �Sk�. The variation of Sku and Skw with z /hc is also shown �top left� for the two Reb. The linearrelationship between the “global” zero-crossing density n and Sk is presented �bottom right�. The regression line �solid� and prediction from the third-ordercumulant expansion �n�1 /2−1 /12Sks

�2 /�� model �dashed� are also shown.

065103-10 D. Poggi and G. Katul Phys. Fluids 21, 065103 �2009�

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far from boundaries. While �s varied significantly �mainlydue to variations in Re��, � showed only minor variationswithin the CSL.

V. CONCLUSIONS

How the presence of vegetation alters clustering, inter-mittency, and possible relationships between their exponentswas explored using flume experiments for a canopy com-posed of densely arrayed rods. Based on these experiments,we found the following.

�1� The spectral shapes of the velocity components and theirTAs were qualitatively similar. Above the canopy, therelationship between the spectral exponents of the seriesand its TA followed closely predictions by SB06. Insidethe canopy, wake production was detected in the spectraof both the velocity and the TAs at scales comparable tothe rod diameter.

�2� The TA “encodes” the non-Gaussian properties of thevelocity time series via the fraction of time the telegraphseries takes on nonzero values ��+�. It was demonstratedthat the near-linear relationship between the velocityskewness and �+ observed in the data for both velocitycomponents can be reproduced using a third-order cu-

mulant expansion. Hence, when this finding is taken to-gether with the previous one, the TA does preserve keyattributes of the spectra and the non-Gaussian propertiesof the velocity time series.

�3� The vertical variation in the dissipation intermittency ex-ponent �s was explained by the variations in1 / log�Re��. Inside the canopy, large reductions in Re�

are expected due to enhanced turbulent kinetic energydissipation rate and reduced velocity variances. Hence,in a first order analysis, the effects of the canopy on �s

appear to be captured by their effects in reducing Re�.�4� The clustering analysis for the velocity series inside and

above the canopy revealed two scaling regimes. For spa-tial scales much larger than the canonical scale of thevortices dominating the flow, no significant clusteringwas detected. However, for all other scales, significantclustering was discernable and followed an extensive1/3 scaling law inside the canopy. Moreover, thecanopy signatures on these scaling laws were weak �un-like the spectra�. When the analysis was repeated on thephase-randomized surrogate data, identical clusteringpatterns emerged. This agreement suggests that the“clustering measures” used in SB06 are primarily sensi-tive to the correlation function and not to the distribu-

10−3

100

103

10−3

10−2

10−1

z/hc=0.05z/h

c=0.05

10−3

100

103

z/hc=0.3

10−3

100

103

10−3

10−2

10−1

z/hc=0.5

10−3

10−2

10−1

z/hc=0.83

<δn

τ2 >1/2

z/hc=1.05

10−3

10−2

10−1

z/hc=1.24

10−3

100

103

10−3

10−2

10−1

z/hc=1.51

10−3

100

103

z/hc=1.79

(τ U)/hc

10−3

100

103

10−3

10−2

10−1

z/hc=2.66

FIG. 10. �Color online� The variations in �n21/2 with for the u �circles� and PR�u� �stars� within and above the canopy. The two regimes −1/2 for large

and as −1/3 for small are also shown. The clustering exponent � was determined for the same range of scales used to infer the spectral exponents in Figs.3 and 4.

065103-11 Intermittency and zero-crossing properties Phys. Fluids 21, 065103 �2009�

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tional properties of the acceleration. This finding is con-sistent with Rice’s formula. In terms of connectionbetween dissipation intermittency and clustering expo-nent, we found no significant relationship. While theformer varied significantly with Re�, the latter showedonly minor variations within and above the CSL. Forturbulent flows far from boundaries, it was demonstratedthat much of the dissipation intermittency variations lin-early scale with the clustering exponent. Hence, it isconceivable that the presence of a canopy decorrelatesthe dissipation intermittency exponent from the cluster-ing exponent.

ACKNOWLEDGMENTS

G. Katul acknowledges support from the NationalScience Foundation �Grant Nos. NSF-EAR 06-35787,NSF-EAR-06-28432, and NSF-ATM-0724088� and the Bi-national Agricultural Research and Development �Grant No.IS3861-06�.

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−0.5 −0.4 −0.3

0

1

2

3

4z/

h c

Longitudinal

Re*=12000

Re*=6000

0

1

2

3

4

z/h c

PR

Orig

−0.5 −0.4 −0.30.1

0.15

0.2

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1/lo

g(R

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α

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4Re

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Vertical

0

1

2

3

4PR

Orig

−0.5 −0.4 −0.30.1

0.15

0.2

0.25

0.3

α

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065103-12 D. Poggi and G. Katul Phys. Fluids 21, 065103 �2009�

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065103-13 Intermittency and zero-crossing properties Phys. Fluids 21, 065103 �2009�

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