roughness effects on ﬁne-scale anisotropy and anomalous … · 2020-03-26 · roughness effects...

Roughness effects on fine-scale anisotropy and anomalous scalingin atmospheric flows

G. G. Katul,1,2,a� A. Porporato,2,1,b� and D. Poggi3,c�

1Nicholas School of the Environment and Earth Sciences, Duke University, Durham,North Carolina 27708, USA2Department of Civil and Environmental Engineering, Pratt School of Engineering, Duke University,Durham, North Carolina 27708, USA3Dipartimento di Idraulica, Trasporti ed Infrastrutture Civili, Politecnico di Torino, Torino, Italy

�Received 11 April 2008; accepted 24 January 2009; published online 26 March 2009�

The effects of surface roughness on various measures of fine-scale intermittency within the inertialsubrange were analyzed using two data sets that span the roughness “extremes” encountered inatmospheric flows, an ice sheet and a tall rough forest, and supplemented by a large number ofexisting literature data. Three inter-related problems pertaining to surface roughness effects on �i�anomalous scaling in higher-order structure functions, �ii� generalized dimensions and singularityspectra of the componentwise turbulent kinetic energy, and �iii� scalewise measures such localflatness factors and stretching exponents were addressed. It was demonstrated that surface roughnesseffects do not impact the fine-scale intermittency in u �the longitudinal velocity component�,consistent with previous laboratory experiments. However, fine-scale intermittency in w �the verticalvelocity component� increased with decreasing roughness. The consequence of this externalintermittency �i.e., surface roughness induced� is that the singularity spectra of the scaling exponentsare much broader for w when compared u in the context of the multifractal formalism for the localkinetic energy �instead of the usual conservative cascade studied for the dissipation rate�. Thescalewise evolution of the flatness factors and stretching exponents collapse when normalized usinga global Reynolds number Rt=�LI /�, where � is the velocity standard deviation, LI is the integrallength scale, and � is the fluid viscosity. © 2009 American Institute of Physics.�DOI: 10.1063/1.3097005�

I. INTRODUCTION

The structure of turbulence in the inertial subrange hasreceived considerable research attention in the past six de-cades, based on the fact that universal or quasiuniversal theo-ries for turbulence, if they exist, may reveal themselves forthis range of scales.1,2 The inertial subrange encompasseseddies much larger than the viscous dissipation scales �� yetmuch smaller than the integral length scale �LI� of the flow.The basic premise for the emergence of universal scalinghere is that large-scale anisotropic characteristics �i.e., intro-duced by boundary effects or turbulent production mecha-nism� are lost during the energy cascade process, assumed tobe dissipationless and involving only neighboring scales,thereby achieving local isotropy and universality.1–4

However, several experiments and simulations over thepast two decades suggested persistent anisotropy at these so-called inertial scales, even for very high Reynolds numbersand after many cascading steps.2,5,6 The departure from theso-called Kolmogorov7 view �K41� of universal scaling andsubsequent refinements such as K62 �Ref. 8� is now sup-ported by several observations, simulations, and theoreticalarguments regarding the anomalous scaling in measuredhigher-order velocity structure functions. One of the impor-

tant manifestations of anomalous scaling is that the exponentof each higher-order structure function cannot be determinedindividually in a trivial manner or guessed from dimensionalarguments.2,6 The anomalous scaling is commonly attributedto some short-circuiting of the energy cascade process due tothe existence of organized large-scale features �e.g., vortexstructures with large enstrophy�, which themselves are influ-enced by boundary conditions and directly influence small-scale turbulence statistics.2,6,9–13

To remove any anisotropic effects originating from theboundary conditions, early work with direct numerical simu-lations �DNSs� focused on homogeneous and isotropic turbu-lence. Even in these simplified configurations, however,these DNS runs revealed that coherent vortical structures�tubelike� are embedded in a highly disordered, near-Gaussian background turbulent flow.14 Non-Gaussian statis-tics, intermittency, and the resulting anomalous scaling inhigher-order structure functions appear to be “fingerprints”of the persistence and maintenance of such vortical structuresat fine scales. Because these simulations have been con-ducted at moderate Reynolds numbers, some debate remainsas to whether such vortical organization can be sustained atvery high Reynolds number.4

For very high Reynolds number, few theoretical argu-ments have been proposed to explain the apparent departurebetween experiments and K41 theory. For example, Qian15,16

demonstrated that �1� K41 scaling can only be attained at

a�Electronic mail: [email protected]�Electronic mail: [email protected]�Electronic mail: [email protected].

PHYSICS OF FLUIDS 21, 035106 �2009�

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infinite Reynolds number �R�=��u /�; where � is the Taylormicroscale, �u is the longitudinal velocity standard devia-tion, and � is the kinematic viscosity�, �2� the effects of finiteR� on the structure function statistics decay slowly with in-creasing R� �e.g., Qian’s work demonstrated that the decay ison the order of R�

−2/3 for third-order structure functions, whilewind tunnel experiments17 suggested a decay on the order ofR�

−0.6�, and �3� the energy injection mechanism may be im-portant. Recently, Gagne et al.18 confirmed all three findingsexperimentally for different flow types �i.e., different injec-tion mechanisms� and different R�.

Among these differences in injection mechanisms, sur-face roughness is one of the prominent causes for departuresfrom the K41 scaling. There is ample support for such ahypothesis, but several issues remain unresolved. A numberof laboratory experiments already demonstrated that smooth-wall boundary layers appear more intermittent and aniso-tropic than their rough-wall counterpart even for the sameR�.19,20 These and other studies21 ascribe this increase in in-termittency and anisotropy to the persistence of coherentquasilongitudinal vortices originating from the near wall re-gion of smooth walls that become less coherent �or moredistorted� with increased surface roughness. However, somecautionary comments are warranted before any generaliza-tions can be endorsed from these studies. In fact, near theboundary, viscous effects become significant and scale sepa-ration between LI and � become limited for meaningful scal-ing analysis of the velocity increment �v�r�=v�x+r�−v�x�with respect to scale r, where v is an arbitrary velocity com-ponent. Recent studies have already suggested that theisotropy-recovering mechanisms close to a wall may not de-velop and yet the competition between the energy productionand transfer mechanisms may still lead to scaling regimeswith exponents that differ from their standard inertialvalues.22

Given the links between intermittency, anomalous scal-ing, and �v�r�, the main thrust of the present work is toexplore the roughness effects on the statistics of �v�r� in twoatmospheric flow configurations with very different surfaceroughness conditions at very high but similar Re��7000,along with an extensive comparison with previously publishresults. The robust method of orthonormal wavelet decom-position approach will be used.

Assuming the scaling relationship from the velocityincrements

��v�r��p� � r�p�v� �1�

�where �•� is averaging, v=u ,w�, the effects of roughnessvariations on �p�v� and corollary local scaling properties arealso explored using the cumulant expansion of �v�r� and itslink to �p�v�.23–26 Extensions of this framework to includeother intermittency measures, such as flatness factor andstretching exponents, are also considered within the contextof cumulant analysis.

In particular, three unresolved issues pertaining to rough-ness effects will be addressed here:

• Surface roughness effects on �p�v� anisotropy and itsparameters: When surface roughness is altered from

rough to smooth, the �p�w��p�u�, where u and w arethe longitudinal and vertical velocity turbulent compo-nents. A limited number of studies also found that�p�u� is not impacted by surface roughness althoughno generalities were endorsed.19 Thus, when �p�v� �v=u ,w� is described by the log-normal model �K62�,given as

�p�v� = � 13 + 1

6��p − 118�p2, �2�

the intermittency parameter � appears to be higher forw when compared to u.19 Also, some studies reportedthat � varies monotonically with “externalintermittency”11 and Reynolds number for a number ofshear flows.6,27 However, the dependence on Reynoldsnumber remains illusive when all experiments arecombined.28

• Roughness effects on the cascades of kinetic energyand its dissipation rate: When the intermittencybuildup in local energy is analyzed using the multi-fractal �MF� formalism and multiplicativecascades,1,29–37 the resulting generalized dimensionsand singularity spectra appear less intermittent thantheir turbulent kinetic energy dissipation rate �� coun-terpart. To this regard, a major knowledge gap remainsas to what is the effect of roughness on the anisotropyof the local singularity spectra for the u and w energies�rather than �. It should be emphasized here that thedistinction between the MF formalism for the kineticenergy dissipation rate and the local energy isfundamental.34 For example, the usual MF formalismin turbulence deals with conservative cascades so thatthe scalewise integral of the field is independent of r�as is the case for , which represents the net fluxacross scales�, while for the total energy it is scaledependent.

• Roughness effects on common scalewise intermittencymeasures: When quantifying small-scale intermittency,commonly defined by occurrences of extreme eventsthat are far more probable than can be expected from aGaussian distribution via the so-called flatness factor

F�r� =��v�r�4��v�r�2�2 , �3�

experiments suggest that F�r�3 when r→LI but be-comes much larger as r→�.24,34,38–40 As r→�, onlythe strongest �v fluctuations survive while the othersare extinguished by viscosity. Hence, viscosity tendsto amplify the contrast between quiescent and ener-getic events thereby suggesting that a completely dif-ferent scaling from K41 or K62 is required to describe�p�v�. Likewise, when computing empirically the so-called stretching exponent m in the expression

p��v� � exp�− ��v�m� , �4�

m appears to vary with r such that its value is near 2�i.e., Gaussian� for r→LI and saturates near 0.5 whenr→�, where p��v� is the probability density function�pdf� of �v�r�.41–43 Again, the degree of anisotropy in

035106-2 Katul, Porporato, and Poggi Phys. Fluids 21, 035106 �2009�


these two parameters as a function of surface rough-ness have not been methodically explored.

II. DATA

While the site description and data processing were pre-sented elsewhere,29,44–46 the salient features are briefly re-viewed here. For the ice sheet experiment, turbulence mea-surements were performed at 20.8 Hz using symmetric three-axis ultrasonic anemometry �Gill Inst. Ltd.� at 10 m abovethe Nansen Ice Sheet in a coastal area close to the TerraNova Bay Italian station in Antarctica. The site experiencesfrequent katabatic winds flowing from the Antarctic Plateautoward the Ross sea along the Reeves glacier. The sonicanemometer path length dsl is 0.149 m.29,44,45 The number ofsamples per run analyzed here are 217.

For the pine forest experiment, turbulence measurementswere performed at 10 Hz using a CSAT3 triaxial sonic an-emometer �Campbell Scientific Inc., Logan, UT� positionedat 20.2 m above the forest floor, which is about 10 m fromthe zero-plane displacement height d �assumed to be about0.65hc, where hc=16 m is the mean canopy height at thetime of experiments�.46,47 The dsl is 0.10 m. The number ofsamples per run was also 217.

The u and w time series are shown in Fig. 1 and all therelevant statistics pertaining to these two experiments arepresented in Table I. These two runs were selected because�1� of the similarity in Taylor microscale Reynolds number,�2� the atmospheric stability conditions were near neutral

�i.e., the role of buoyancy on production or destruction ofturbulent kinetic energy can be neglected�, and �3� theirsquared turbulent intensities �in

2=u�2 / u2, where u is the meanlongitudinal velocity� was small thereby permitting the useof Taylor’s hypothesis to convert time into spatial scale.48,49

In Table I, the friction velocity u� was computed from

0 100 200 300 400 500−5

0

5

u/σ u

Pine Forest

0 100 200 300 400 500−5

0

5Ice Sheet

0 100 200 300 400 500−5

0

5

w/σ

w

Time (s)0 100 200 300 400 500

−5

0

5

Time (s)

FIG. 1. A 500 s sample of the normalized time series of longitudinal �u� and vertical �w� collected above the pine forest �left� and the ice sheet �right�. Thenormalization ensures that across the entire experiment duration, the normalized velocity has a zero mean and unit variance.

TABLE I. Flow conditions for the pine forest and ice sheet experiments. Allvariables are defined in the text except L, the Obukhov length. A smallstability parameter �i.e., �z /L��0.1� indicates that mechanical productiondominates the generation of turbulent kinetic energy.

Variable Pine forest �rough� Ice sheet �smooth�

u �m s−1� 1.83 6.55

u� �m s−1� 0.46 0.27

z /L +0.07 �0.066

in2 0.27 0.01

zo �m� 1.9 0.00055

Cd 0.06 0.0017

� �mm� 0.66 0.91

� �m� 0.11 0.15

LI for u �m� 93 165

LI for w �m� 5.1 14

S� 22 38

K� 1.9 1.7

L� 18 12

�w /�u 0.55 0.43

Re�=�u� /� 6945 7225

035106-3 Roughness effects on fine-scale anisotropy Phys. Fluids 21, 035106 �2009�


the two components of the turbulent stresses as �u�w�2

+v�w�2��1/4�, where v� are the lateral turbulent velocity ex-cursions �coordinates are oriented to ensure zero-mean lat-eral velocity�. Using this estimate of u�, the momentumroughness height, zo, for the two surfaces was inferred from

u =u�

kvlog z − d

zo� , �5�

where kv=0.4 is the von Karman constant. The computed zo

for the pine forest and ice sheet were 1.9 and 0.00055 m,respectively. The corresponding roughness Reynolds numberRe+=u�zo /� were on the order of 60 000 and 10 suggestingthat the surface is fully rough for the pine forest and slightlydynamically rough for the ice sheet. Because z−d was com-parable among the two experiments, the height-dependentdrag coefficient, Cd= �u� / u�2, was also compared in Table Ifor both surfaces. The Cd for the pine forest was about 35times larger than that of the ice sheet. Hence, it is clear thatthe contrast in roughness and drag properties between thesetwo surfaces is significant for similar Taylor microscale Rey-nolds numbers.

To compare the degree of anisotropy at large scales herewith other boundary layer experiments, three parameters50

were also computed �Table I�. The first is the shear-rate pa-rameter �S��, given as the product of the eddy-turnover time�qe

2 /�� by the mean deformation time rate ��u /�z�, where

qe2=ui�ui� �i implies repeated sum� and � is the mean turbulent

kinetic energy dissipation rate. From only one level measure-ments, this means deformation rate may be estimated from

� u

�z�

u�

kv�z − d�.

Moreover, assuming a balance between energy productionand dissipation

� =u�

3

kv�z − d�

results in

S� =�u

2 + �v2 + �w

2

u�2 ,

where �s is the root-mean squared magnitude of an arbitraryflow variable s. These estimates were selected because theresulting S� becomes independent of any length scale �e.g.,kv�z−d��. This approach to computing S� was chosen giventhe uncertainty in d for the forest stand. Other estimates of �are also possible, say from the third order structure function,but are not discussed since they lead to S� that requires ac-curate estimates of d, a difficult task in tall forested canopies.The second parameter, K�, is a dimensionless streamwiseenergy partition given by

K� =2�u

2

�v2 + �w

2 .

The third parameter, L�= �Iuu / Iwu�, is an eddy-elongation pa-rameter along the longitudinal axis, where Iu and Iw are theintegral time scales of u and w, respectively. Strictly speak-

ing, L� should be estimated from the integral length scalesalong the u-direction, but with one-point measurements, timeand the u directions may be interchangeable via Taylor’shypothesis when in

2 1. These structural parameters can alsobe compared to DNS results for channel flows.50 Wall-layerstreaks are known to exist in regions where K��5 and L�

�8.50 Based on the analysis here, it is clear that z−d in thesetwo atmospheric experiments are well above the heightdominated by such high speed streaks. Moreover, it shouldbe emphasized that the pine forest experiment was conductedin the so-called canopy sublayer �CSL�, which differs fromboundary layers in a number of ways. The canopy foliage isa porous medium that does not impose the same “no-slip”constraint as a boundary layer. Moreover, the mean flow in-side the canopy is slow but finite, yet above the canopy, themean flow is rapid. The interface between the slow and fast-moving air creates free shear instabilities often resulting inKelvin–Helmholtz-like instabilities, in addition to the usualattached eddies. Hence, the energy injection mechanism inthe CSL may differ from a standard rough-wall boundarylayer.51–54

Finally, it should be noted that recent studies22,55,56 thatemploy the so-called SO�3� symmetry group rotations con-cluded that the shear deformation scale ��u /�z� discriminatesamong two regions responsible for anisotropic contributionsto small scales �of the mixed velocity structure function�.The first is based on eddy time scales being much smallerthan the shear deformation scale. For this range, the action ofisotropy-recovering mechanisms primarily dominate and uni-versal scaling laws that are not appreciably impacted by themean shear are recovered. The second regime is based oneddy time scales comparable to or larger than the shear de-formation scale. In this case, energy production and energytransfer mechanisms compete and the scaling laws are appre-ciably impacted. If the shear deformation time scale is char-acterized by �u� /kv�z−d�� and the energy injection mecha-nism �in the vertical� is characterized by the integral timescale defining the cross-correlation function between u andw, Iuw, then �u� /kv�z−d��Iuw indicates the time-scale separa-tion between these two anisotropic contributions. For thepine forest and ice sheet, this product is of order unity �1.2and 1.9, respectively�. Hence, any anisotropy that persists inthe inertial subrange is attributed to combined effects of theenergy injection mechanism and the shear scales not beingentirely smeared out by the cascade �terminated here at spa-tial scales comparable to dsl�. In the estimation of Iuw, thecross-correlation function was first computed and normalizedto the zero-lag value �or correlation coefficient here�, theintegral scales for positive and negative lags were computedseparately, and their average was used for Iuw.

III. METHODS OF ANALYSIS

In this section, the basic theoretical and data analysismethodologies used to address the three basic themes earlierdescribed in the introduction are presented. For the theoreti-cal methodologies, a simplified version of the cumulant ex-pansion approach described elsewhere23,24,26 was extended tolink coherently various intermittency measures �such as in-



termittency corrections to K41, higher-order cumulants, localflatness factors, and stretching exponents�. For the dataanalysis component, analyzing measures based on orthonor-mal wavelet transforms �OWTs� are presented and shown tobe particularly robust despite the limited sample size of theexperiments here. For reference, the analysis carried out onall four turbulence time series is contrasted to a phase-randomized series having identical sample size and spectrumas the u series collected above the pine forest. Repeating allthe analysis on the phase-randomized data permits us to dis-cern some of the limited sample size effects on local scalingproperties as discussed elsewhere.57–59

A. Theory

The buildup of intermittency across inertial scales can bedescribed by how the pdf of �v�r� evolves with scales r.Because turbulent velocity is approximately Gaussian atlarge scales �Lo�, where Lo is comparable to LI, then from astatistical view point, the statistics of �v�r� can be describedas23,24,26

�v�r� = �v�Lo�� r

Lo�, r � Lo, �6�

where �v�Lo� is a zero-mean Gaussian distributed variablewith variance �2, and satisfying

��v�Lo��p� = Kp�p. �7�

The function ��•� is assumed to be a positive independentrandom multiplier connecting the statistics of �v�r� to thestatistics of �v�Lo�. Hence, with such scaling,

��v�r��p� = ��v�Lo��p � r

Lo��p�

��v�Lo��p�� r

Lo��p� , �8�

where �•� is averaging. As r→Lo, the �v�r� statistics ap-proach Gaussian, and for any scale r�Lo, this implies

��v�r��p� = Kp�p� � r

Lo��p� . �9�

It is clear from the above analysis that any intermittencybuildup depends on the distributional properties of ��•�. Toexplore these distributional properties, consider the cumulantexpansion,

��p� = �elog��p�� = �ep log�� = exp�n=1

�

Cn�log��pn

n!� ,

�10�

Cn�s� are the cumulants of s=log�� given by �only first twoare listed for illustration�:

C1 = �log�� ,

�11�C2 = ��log��2� − C1

2.

Hence, for the velocity increments:24

��v�r��p� = Kp�p exp��n=1

�

Cn log � r

Lo�� pn

n!� . �12�

The basic premise adopted here is that by exploring only fewcumulants pertaining to the distributional properties oflog��r /Lo�� removes the need to compute �or assume� pre-cise distributional properties of ��r /Lo�. The connection be-tween these cumulants and the classical results pertaining to�p when ��v�r��p��r /Lo��p implies that the cumulantsscale linearly with their state variable24 so thatC1�log��r /Lo��=a1 log�r /Lo� and C2�log��r /Lo��=a2 log�r /Lo�, etc. This simplification leads to

��v�r��p� = Kp�p�r/Lo�pa1+�1/2�p2a2+. . . . �13�

A practical outcome of this analysis is that the scalewiseevolution of popular intermittency measures such as the flat-ness factor and stretching exponents can be readily predictedfrom the cumulants. To illustrate this, consider the definitionof the scalewise flatness factor, which can now be expressedas

F�r� =

K4�4 exp��n=1

�

Ci log � r

Lo��4n

n!��K2�2�2 exp��

n=1

�

Ci log � r

Lo��2n+1

n! �=

K4�4

�K2�2�2exp��n=1

�

Cn log � r

Lo��4n − 2n+1

n!�� .

�14�

The expansion yields

C1 log � r

Lo��41 − 21+1�

+ C2 log � r

Lo�� 42 − 22+1�

2!+ ¯ . �15�

whose leading term is

F�r� = exp�C2 log � r

Lo�� 42 − 22+1�

2+ ¯� . �16�

Hence, in the inertial subrange, the flatness factor is givenby24

F�r�3

� exp 4a2 log r

Lo�� r

Lo�4a2

. �17�

This intermittency measure is connected to another com-monly employed intermittency measure, the stretching expo-nent m. It is possible to show that the flatness factor of theexponential power-law family of distributions given in Eq.�4� is



F�r� =9

5

��1 + 1m��1 + 5

m��1 + 3

m��2 =�� 1

m�� 5m�

�� 3m��2 � 3 r

Lo�4a2

,

�18�

which can be inverted numerically to obtain m as a functionof scale. Hence, a scale-dependent m is clearly connected toa finite a2. Moreover, for r→Lo, m→2. Finally, it should benoted that for m=0.5, the F�r��25, consistent with the flat-ness factor reported in a number of studies for the dissipationrange.34,38

How various intermittency models along with their sig-natures in F�r� and m assume different simplifications to thesequence of cumulants, from only one cumulant �K41model�, to two cumulants �lognormal model� to infinite se-quence of cumulants �She–Leveque� are discussed next. Thisis in agreement with the theorem by Marcienkiewicz,60

which shows that the cumulant generating function cannot bea polynomial of degree greater than 2, that is either all butthe first two cumulants vanish or there are an infinite numberof nonvanishing cumulants.61 The lognormal model is an ex-ample of the former while the She–Leveque model is anexample of the latter. However, first, the original K41 modelis considered.

1. The K41 model

In the K41 model,

�p =p

3. �19�

Consider the cumulant expansion with only C1 being finite.Moreover, assume that this first cumulant scales linearly withthe state variable so that

C1 log��r/Lo�� = a log�r/Lo� . �20�

Hence,

��v�r��p� = Kp�p exp�pa log�r/Lo�� . �21�

When a=1 /3, K41 is readily recovered. Interestingly, be-cause the only finite cumulant is C1, the distributional prop-erties �v�r� follow �v�Lo� �i.e., Gaussian� at all scales ad-justed by a scale-dependent multiplier �that depends on ��.This analysis reveals why K41 is often linked to Gaussianstatistics at fine scales, and intermittency buildup is oftenlinked with non-Gaussian statistics. With a2=0, F�r�=3, andm=2. Extensions of this argument to the so-called monofrac-tal or beta-model yield a=1 /3−1 /3�3−Do�, where Do is thefractal dimension.62

2. The lognormal model

As earlier noted, the first refinement to K41, the lognor-mal model or K62, can now be readily recovered from atwo-term cumulant approximation given by24

��v�r��p� = Kp�p exp�pC1�log��r/Lo��

+ 12 p2C2�log��r/Lo�� . �22�

If these two cumulants scale linearly with their state variable

�which implies that the variance increases logarithmicallywith decreasing scale�,24

��v�r��p� = Kp�p�r/Lo�pa1+1/2p2a2. �23�

For the recovery of the log-normal model, the coefficients a1

and a2 must be selected as

a1 = 13 + 1

6�; a2 = − 19� . �24�

Notice that the log-normal model suggests that these twocumulant scaling parameters are dependent if � is constantirrespective of the value of �. There is a practical advantageto presenting the log-normal model with such a dependency.The relationship between these two scaling coefficients cannow be written as

a2

2= −

1

3a1 −

1

3� , �25�

and is independent of the precise value of the intermittencyexponent �. Hence, when analyzing the effects of inhomo-geneity, anisotropy, or external intermittency originatingfrom surface roughness on �p within the K62 framework, it ismore convenient to diagnose departures from linearityamong these two cumulants, and whether the slope ��1/3� orthe intercept �+1 /9� above are affected. That is, if the exter-nal effects or inhomogeneity can be “accounted for” by anenhanced �, they should not alter the unique linear relation-ship between these two cumulants.

3. The She–Leveque model

The She–Leveque is among the few intermittency mod-els that require no a priori intermittency parameter specifi-cation, and is often connected with the presence of vortextubes �although it does not confirm it�. Hence, it is a logicalmodel to explore via the cumulant expansions here. In theShe–Leveque model,63

�p =p

9+ 2 − 22

3�p/3

. �26�

In this multiplicative cascade framework, note that the ex-pansion of

ax = 1 + x log�a� +�x log�a��2

2!+

�x log�a��3

3!+ ¯ �27�

permits presenting the She–Leveque model as

�p =p

9+ 2 − 21 +

p

3log�2/3� +

p2

32

�log�2/3��2

2!

+p3

33

�log�2/3��3

3!+ ¯� �28�

or



�p =p

31

3− 2 log�2/3� − 2

p2

2!

�log�2/3��2

32

− 2p3

3!

�log�2/3��3

33 + ¯� . �29�

Again, if all the cumulants scale linearly with the state vari-ables so that

Cn�log��r/Lo�� = an log�r/Lo�; n = 1,2,3, ¯ , �30�

then, the She–Leveque model is recovered if the scaling vari-ables are

a1 = 13� 1

3 − 2 log� 23��

an = − 2� 13 log� 2

3��n, n � 2.

In short, the basic difference between the She–Leveque andlog-normal models in the context of multiplicative cascadesis that the She–Leveque model requires all cumulants to befinite and different from zero, while the lognormal modelonly requires the first two cumulants to be finite and differentfrom zero. K41 �or the monofractal model� only requires thefirst cumulant to be finite and different from zero.

B. Data analysis

This section discusses how to proceed with the estima-tion of �v�r�, �p, F�r�, Cn, and the stretching exponent musing �OWTs�. There are a number of reasons why OWTmay provide robust estimates of these quantities for timeseries with limited size �as is the case here with only 217 datapoints�. As discussed in Ref. 64, the OWT has a desirablewhitening property, which means that at a given scale, wave-let coefficients are uncorrelated or weakly correlated �atbest�. This whitening property is empirically illustrated inFig. 2, which shows that the wavelet coefficients of the useries above the ice sheet decorrelate within one to two lags.The two correlation functions are computed for wavelet co-efficients associated with the largest and finest scales withinthe inertial subrange �and whose identification based on the�5/3 spectral scaling is discussed later�. Note that the origi-nal u time series maintains significant correlations up to 2000lags ��100 s�. The u series above the ice sheet was selectedfor illustration here because of its large integral scale. Thiswhitening property was verified on the pine forest series aswell. Such whitening means that averaging across positionsin the wavelet domain does not suffer from sample size limi-tations associated with long-range memory processes �as inthe time domain�. Moreover, as pointed out by Vassilicos65

and others,38 wavelet coefficients have the property of en-hancing the flow singularities when dealing with Hölderfunctions, which is advantageous here when determining lo-cal scaling parameters.66 Finally, the OWT coefficients canbe interpreted as approximations to �v�r� in any scalinganalysis or statistical moment calculations,34,38,58,59,67,68 andcan provide unambiguous space-scale interpretations for theterms in the turbulent kinetic energy �TKE� budget.69 Defin-

ing the wavelet coefficient at position xo and scale a aswc�xo ,a� of some arbitrary function g�x�, then wc�xo ,a� canbe interpreted as �though not identical to�

wc�xo,a� � g�xo + a� − g�xo� . �31�

Moreover, if the Hölder exponent of g�xo� is h�xo�, then

�wc�xo,a�� ah�xo�. �32�

With these two interpretations, the higher-order structurefunctions �SFs� and their usual scaling exponents can be re-lated to wc�xo ,a� by26,68

SF�p,a� =1

n�a��xo

�wc�xo,a��p, �33�

where n�a� are the number of orthonormal wavelet coeffi-cients at scale a �determined from the dyadic multiresolutionarrangement of OWT�. Once SF�p ,a� is determined, �p canbe computed from the scaling argument SF�p ,a��a�p in amanner analogous to the usual structure function determina-tion of �p and can readily accommodate the extended self-similarity �ESS� approach �i.e., referencing the higher-orderSF to the third-order SF rather than scale70�. Hence, the cu-mulant moments Cn earlier described can be readily relatedto and computed from wc�xo ,a�.

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000−0.5

0

0.5

1

0 10 20 30 40 50 60 70 80 90 100−0.5

0

0.5

1

r(k)

0 10 20 30 40 50 60 70 80 90 100−0.5

0

0.5

1

k=Lag (Index)

FIG. 2. Empirical illustration of the whitening properties of the SymmletOWT �wavelet compact support=4�. The top panel shows the autocorrela-tion function r�k� for lag index k as derived from the original u time seriesabove the ice sheet. The middle and lower panels show the r�k� for the OWTcoefficients at the coarsest and finest scale within the inertial subrange �iden-tified by the �5/3 power-law presented in Fig. 3 later�. The dashed lines arethe standard 95% confidence intervals.



In the MF framework, the singularity spectrum D�h� canalso be computed from �p using26

D�h� = 1 + minp

�ph − �p� . �34�

Moreover, when the local energy �ev�xo ,a�=wc�xo ,a�2� ofsome velocity component �v� behaves as a MF process witha generalized dimension D��p�, then for a one-dimensionalseries,34

�ev�xo,a�p� � �ev�xo,a��p a

LI��p−1��D��p�−1�

. �35�

Here D��p� are the generalized dimensions of the local ener-gies �rather than �. As mentioned earlier in the introduction,the presence of a scalewise multiplier �ev�xo ,a��p is neededhere because of the nonconservative nature of the local en-ergy cascade �vis-a-vis a scale-independent �� in the usualMF turbulence studies�. Once the D��p� are computed forvarious p values, then the singularity spectra fs�� for thelocal scaling exponents �� can be determined using34

��p� =d

dp�p − 1�Dp� ,

fs�� = p��p� − �p − 1�Dp� ,

where primed quantities now refer to local energies ratherthan dissipation rates. The determination of D��p� from thewavelet coefficients can be carried out by varying p in Eq.

�35� and computing the slope �p−1��D��p�−1� from double-log plots of �ev�xo ,a�p��ev�xo ,a��−p versus a /LI within theinertial subrange.34 The singularity spectra fs�� are thendetermined using numerical differentiation of the computedD��p�.

Another local energy based measure of intermittency atscale a, hereafter referred to as local intermittency factor�LIF�a��, is given by

LIF�a� =�ev

�ev�, �36�

where �evrefers to the standard deviation of ev=wc�xo ,a�2 at

scale a.34,38,40 Hence, LIF is a measure of the local energydispersion �rather than dissipation� around its scale-dependent mean. Increases in LIF with decreasing scale im-plies that the local energy is becoming more dispersed. Theeffects of surface roughness and resulting anisotropy on thesimultaneous shape of SF and energy intermittency wasrarely explored in previous studies.

IV. RESULTS AND DISCUSSION

The results and discussion are structured along the threemain themes earlier raised in Sec. I. However, before pro-ceeding with these three themes, a number of important fea-tures about the data sets are first highlighted. Fourier andwavelet spectra, computed with a Symmlet wavelet having acompact support of 4, are shown for all four time series

10−8

10−6

10−4

10−2

10−2

100

102

Eu

Pine Forest

Fourier−5/3WaveletWavelet−PR

10−8

10−6

10−4

10−2

10−2

100

102

Eu

Ice Sheet

10−8

10−6

10−4

10−2

10−2

100

102

kx

η

Ew

10−8

10−6

10−4

10−2

10−2

100

102

kx

η

FIG. 3. The normalized power spectra �Fourier in solid line, wavelet in open circles� of the longitudinal �u� and vertical �w� velocity time series collectedabove the pine forest �left� and the ice sheet �right�. The spectra �Eu and Ew� are shown as a function of the normalized wavenumber �determined from thefrozen turbulence hypothesis�. The spectra of the phase-randomized time series for the u are shown for illustration. The �5/3 power law is also shown �dashedline�.



along with the �5/3 power law in Fig. 3. Symmlet waveletsare often used to encode audio clips known to possess richspectrum of scales because of their near-symmetrical shapesand compact support. The spatial scales �or wavenumbers�are computed from Taylor’s frozen turbulence hypothesis49,71

and normalized by � �to compare with the wealth of labora-tory experiments�. Notice that the inertial scales here spanabout 1.5 �for w� to 2 �for u� decades, and remain far fromany viscous dissipation �or near dissipation� effects as is thecase in virtually all laboratory and DNS studies.

A. Surface roughness effects on �p„v… anisotropyand its parameters

To assess the effects of roughness and anisotropy on theinertial subrange velocity statistics, �p�u� and �p�w� seriesabove the two surfaces are first computed via the wavelet-based structure function combined with ESS as shown inFig. 4. The computed �p�u� and �p�w� along with the 95%confidence limits derived from standard t-distribution statis-tic are shown in Fig. 5. Figure 5 also shows the �p computedfrom K41, K62, the monofractal1 and the She–Leveque mod-els. Figure 5 demonstrates that �p�u� above the pine forestand the ice sheet are similar and appear significantly lessintermittent �to within the 95% confidence limits� when com-pared to K62 �with intermittency exponent �=0.2� and theShe–Leveque models. On the other hand, the �p�w� areclearly impacted by surface roughness. With decreasingroughness, �p�w� decreases relative to both—the K62 and the

She–Leveque models. All four data sets here suggest that themonofractal model is inadequate given the statistically sig-nificant curvatures in �p. We also tested that the phase-randomized series, when subjected to the same calculations,follows K41 scaling �as expected� despite the limited samplesize of the experiment �not shown�.

The ESS-wavelet-based �p determined here are com-pared in Fig. 6 to a number of “benchmark” experimentspreviously used to assess local isotropy. Interestingly, theestimated �p�w� above the ice sheet appears consistent withthe highest R� experiment in Sreenvisan and Dhruva.27 Also,the pine forest and ice sheet �p�u� appear less intermittentthan the remaining experiments. One possible explanationfor why �p�u��p�w� may be due to the energy injectionmechanism. In the atmosphere, the main eddy motion con-tributing to �u

2 are inactive eddies originating from the outerlayer while the main eddy motion contributing to �w

2 areattached eddies to the wall.52,75–78 The differences in energyinjection mechanisms are best captured by the anisotropy inthe canonical length scales associated with these two eddymotion types, with LI�u� /LI�w��10 �as shown in Table I�.With such large anisotropy in energy injection, the bulk tur-bulent Reynolds number for the u component is at least anorder of magnitude larger than the w component. Hence,these results are qualitatively consistent with Qian’s andGagne et al. arguments.15,16,18 However, to compare withother boundary layer anisotropy studies, we maintain oneReynolds number for both components as earlier defined.

0 2 4 6 8−5

0

5

10

15

20

25

log

(<|w

c(x,

a)|p

>)

Pine Forest

0 2 4 6 8−5

0

5

10

15

20

25Ice Sheet

2 3 4 5 6 70

5

10

15

20

25

log(<|wc(x,a)|3>)

log

(<|w

c(x,

a)|p

>)

2 3 4 5 6 70

5

10

15

20

25

log(<|wc(x,a)|3>)

w

u

w

u

p=1

p=8

FIG. 4. The estimation of �p for the u and w series above the pine forest �left� and ice sheet �right� using a wavelet-based ESS. The slopes of the linearregression for p=1,2 , . . . ,8, conducted on wavelet coefficients across inertial scales, represent �p.



To further explore the possible �p dependence on Rey-nolds number, Fig. 7 shows �6 as a function of R� for anumber of experiments. Again, the anisotropy in �6 �or ��from this experiment are apparent with w exhibiting lowervalues �i.e., higher intermittency� than u. However, whencombining all the data sets together, there are no apparenttrends at the higher R� range. It is conceivable that the scatterat the higher R� range may, in fact, be due to contaminationsfrom “external” intermittency,11,12 which is difficult to dis-cern from published experimental conditions alone. For ex-ample, in Kuznetsov et al.,11 external intermittency wasweakly correlated with R�.

A number of other experiments did explore specificallythe effects of anisotropy in �p originating from surfaceroughness. The first experiment �hereafter referred to as

PPR-03� was carried out by Poggi et al.20 and showed thatthe effects of wall roughness on �p�u� was minor consistentwith the findings here as evidenced by Fig. 8. While theyreported larger departure from K41 for u, when compared tothe pine forest and ice sheet u data here, the effects of theirwall roughness remain minor. The departure from K41 scal-ing for �p�w� for the ice sheet data appears larger than whatwas reported by PPR-03. The comparison with the Antoniaand Krogstad experiments19 �hereafter referred to as AK-01�also suggests remarkable agreement for �p�u�, but their �p�w�were highly sensitive to wall roughness and dramatically di-verged from K41 scaling �at least when compared to themost intermittent experiment here—w above the ice sheet�.When compared to the experiments of Dhruva et al.79 for

2 4 6 80

0.5

1

1.5

2

2.5

3Pine Forest

ξ u(p

)

OWT estimatedK41She−LevequeLognormal (µ=0.20)β model (D=2.70)

2 4 6 80

0.5

1

1.5

2

2.5

3Ice Sheet

2 4 6 80

0.5

1

1.5

2

2.5

3

p

ξ w(p

)

2 4 6 80

0.5

1

1.5

2

2.5

3

p

FIG. 5. The ESS-wavelet-based estimation of �p for the u and w series above the pine forest and ice sheet. The K41, K62 �with �=0.2 from Chambers andAntonia �Ref. 72��, the monofractal �or �� of Frisch et al. �Ref. 62� �with D=2.70�, and She–Leveque models are shown for reference. The �p for thephase-randomized u series is also presented and its �p follows K41.



very high R� �hereafter referred to as DTS-97�, the aniso-tropy they report is consistent with the anisotropy computedfor the pine forest experiment �i.e., �p�u�−�p�w��, but ismuch smaller than the anisotropy reported for the ice sheet.Other wind tunnel studies80 also reported strong anisotropybetween the lateral and longitudinal �p, with �p�u� beinglarger than its lateral counterpart even for R��800–1000.When taken together, these experiments are suggestive that�p�u� is the least intermittent component in terms of depar-tures from K41, consistent with the results inferred here.

To further “synthesize” the literature data in relation tothe two experiments here, the scaling coefficients a1 and 1

2a2

are computed for all the literature data sets �and the data sethere� using a similar procedure. As was earlier shown, �p isconnected to the scaling parameters of the cumulants via

�p = a1p + 12a2p2 + ¯ �37�

so that by fitting a quadratic function to the reported litera-ture �p versus p permits us to empirically determine a1 anda2 and diagnose any relationship between them �if any�. Forthe time series collected here, these two scaling coefficientscan be inferred directly from the wavelet coefficients—butfor consistency with published �p, they are determined byfitting a second-order polynomial to the ESS-wavelet-based�p. As earlier noted, exploring the relationship between a1

and 12a2 may provide clues as to how anisotropy propagates

across scales. In part, these clues may emerge because theydo not require an estimate of the numerical value of someintermittency parameter. In the log-normal model �K62�,when � is constant, these two scaling parameters are relatedvia

12a2 = − 1

3�a1 − b 13� , �38�

where b=1 for K62. Hence, if the log-normal scaling is ap-propriate, then the relationship between 1

2a2 and a1 willmaintain its linearity and the slope will be uniquely defined�as 1/3 and the intercept as �1/3�. Also, if intermittencyoriginating from inhomogeneities and boundary conditionsremains persistent within the inertial subrange yet their ef-fects can be truly “absorbed” by �, the unique linearity be-tween a2 and a1 should not be altered. It is conceivable thatthe negative slope is maintained �because the first two cumu-lants dominate the expansion�, but the effects of such “con-tamination” is sufficiently large to impact the intercept bwhile maintaining the same slope �=1 /3�. The changes in bsignals one of two effects: �1� that third-order cumulants arebecoming increasingly significant that they cannot be easilyabsorbed by �, or �2� boundary conditions are impacting bacross experiments although not within a given experiment.With regards to the former, recall that the theorem by Mar-cienkiewicz requires that if third cumulants are significant,then all higher cumulants become finite.

Figure 9 shows the relationship between a1 and 12a2 for a

large number of experiments, including all the data sets inFigs. 6–8, as well as many others �listed in the legend�. Inaddition to these data, an exhaustive study was recently con-ducted on these two scaling parameters using the so-calledmultiresolution wavelet leader algorithm capable of account-ing for any oscillating singularities.26 The study demon-strated that in the absence of any inhomogeneity, c1=0.37and 0.5c2=−0.0125, which is roughly where the majority ofruns reside for the homogeneous cases here and �interest-ingly� appears almost identical to the values derived for theice sheet w data. Oscillating singularities become pro-nounced at small scales ��. Hence, when using velocitytime series collected by sonic anemometry �as was done

1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

p

ξ(p)

Pine Forest−uPine Forest−wIce Sheet−uIce Sheet−wTurbulent jet (Rλ=852)

Turbulent duct (Rλ=536)

Grid Turbulence (Rλ =12)

Cylinder & Jet (Benzi et al.)Boundary Layer (Rλ =200)

DNS (Rλ =210)

ASL (Rλ>10000)

K41

FIG. 6. Same as Fig. 5 but the comparison is with a wide range of experi-ments that reported �p up to p=8. The data sets are from Anselmet et al.�Ref. 73� for a turbulent jet at R�=852 and in a turbulent duct flow at R�

=515, from Camussi and Guj �Ref. 38� for turbulence at R�=12 �usingESS�, from Benzi et al. �Ref. 70� for cylinder and jet generated turbulence�using ESS�, from Stolovitzky et al. �Ref. 74� for a boundary layer withR�=200, from Sreenivasan and Dhruva �Ref. 27� for a DNS at R�=210 andatmospheric surface layer with R��10 000 �using ESS�. Many of the datapoints were digitized by us from the published figures.

101

102

103

104

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

2.2

ξ(6)

Rλ

From Sreenivasan and Dhruva, 1998Other DatauwInterpolation FormulaK41Warhaft (ξ(6)=2−µ)

FIG. 7. Comparison between �6�=2−�� as a function of R� for a wide rangeof experiments including data and interpolation formula from Sreenivasanand Dhruva �Ref. 27�, Warhaft �Ref. 6�, the experiments in Fig. 6, and theexperiments reported here. Many of the data points were digitized by usfrom the original source.



here�, the instrument averaging path length generally ex-ceeds � and diminishes the advantages of the wavelet leaderalgorithm.

While all runs exhibited a negative c2 �consistent withthe wavelet leader analysis far from boundaries in Lashermeset al.26�, the strongest negative correlations exist when a1

�0.35, but with decreasing a1, the relationship betweenthese two parameters significantly diverges from K62 �i.e.,when b=1�. It is conceivable that fine-scale anisotropy doesnot alter the slope, only b �within the context of K62�. Forexample, the family of lines with b=1 decreasing to b=0.6 isshown in Fig. 9 and can encompass much of the data. More-over, it appears that some of the highly inhomogeneouscases, such as the PPR-03 experiments �near the boundary�tend to cluster around the lower line with b=0.6. This de-crease in b signals that at least third-order cumulants may berequired. Two-term expansions need not suffice to explainthe entire anomalous scaling, and a coefficient like � maynot be able to capture the entire boundary effects on inter-mittency buildup. This is a partial explanation as to why thescatter is so large in � versus R� for the higher R� regimes inFig. 7. Moreover, runs with a1�0.35 are generally runs col-lected near a boundary, where the injection mechanism orshear scale impact the scaling laws �as already alluded to inSO3 studies22�.

B. Roughness effects on the cascades of kineticenergy and its dissipation rate

While it is evident that anisotropy and surface roughnessaffect �p and its building blocks �a1 and a2�, how the localenergy variability across scales is impacted is considerednext. As earlier described, the determination of D��p� fromthe wavelet coefficients can be carried out by varying p inEq. �35� and computing the regression slope ��p−1��D��p�−1�� from double-log plots of �ev�xo ,a�p��ev�xo ,a��−p versusa /LI within the inertial subrange as illustrated in Fig. 10. Thesingularity spectra fs�� are then determined using numeri-cal differentiation of the computed D��p� with respect to p,as earlier described.

Figure 11 shows the wavelet-based singularity spectrumfor u, w, and the phase-randomized series. The singularityspectra reported by Meneveau34 for wake and boundary layerflows are shown for reference and include reported values fornegative p �though they are ill-behaved�. It is clear that theice sheet singularity spectrum includes a wider range of ex-ponents for w when compared to its u counterpart. Moreover,the singularity spectra for u are almost indistinguishable,again suggestive that roughness has minor effect on the sin-gularities in local u energies. Interestingly, the singularityspectrum for w from the pine forest agrees surprisingly wellwith the two data sets in Meneveau.34 The phase-randomized

2 4 6 80

0.5

1

1.5

2

2.5

3

3.5

4

p

ξ(p)

Pine Forest−uIce Sheet−uPine Forest−wIce Sheet−wPPR03,SmoothPPR03, Rough

2 4 6 80

0.5

1

1.5

2

2.5

3

3.5

4

p

Pine Forest−uIce Sheet−uPine Forest−wIce Sheet−wAK01−uAK01−w (rough)AK01−w (smooth)K41

2 4 60

0.5

1

1.5

2

2.5

3

3.5

4

p

Pine Forest−uIce Sheet−uPine Forest−wIce Sheet−wDTS97−uDTS97−wK41

FIG. 8. Comparisons between �p for three experiments that explored the effects of roughness and anisotropy: Left are �u for rough and smooth wall reportedin Poggi et al. �Ref. 20� at wall distance y+=80 �PPR-03�, middle are from Antonia and Krogstad �Ref. 19� for rough and smooth wall but the same R�

=240 �AK-01�, and right are from Dhruva et al. �Ref. 79� for the atmospheric surface layer. Note that in all thee cases, w is more sensitive to surface roughnesswith increased boundary roughness resulting in decreased intermittency, and u appears unaffected by surface roughness. The curves in PPR-03 and AK-01were digitized by us. The solid line is K41 scaling, shown for reference.



series exhibits a restricted spread in singularity exponents, asexpected, thereby adding some confidence in their realismdespite limited sample size. In summary, it appears thatroughness effects influence w more than u �consistent withthe �p analysis�, and smoothness “widens” the possible rangeof singularity exponents needed to describe the local energy.

Figure 12 compares the generalized dimensions Dp �usedin the computations of the singularity spectra� as a functionof p for u, w and the phase-randomized series. Recall that adefining syndrome for intermittency �in this case for the localenergy of a velocity component around its scale-dependentmean value� is that Dp varies in a nonlinear manner with p.From scaling analysis, Meneveau34 also proposed a relation-ship between the dissipation and TKE generalized dimen-sions, given by

Dp� = De2p/3 +p

3�p − 1��De2/3 − De2p/3� , �39�

where Dp� and Dep are the TKE and generalized dimen-sions, respectively. Both functions are shown in Fig. 12 forthe Meneveau experiment. For the phase-randomized series,

the Dp� does not vary appreciably with p, as expected, lendingconfidence in the wavelet-based estimation procedure despitelimited sample size. It is also clear from Fig. 12 that localenergies in the u series exhibit less nonlinearity when com-pared to their w counterparts. Recall that increased intermit-tency is linked with increased nonlinearity of the Dp�. As withthe singularity spectral comparisons, the nonlinearity in theDp� of the w series collected above the pine forest agrees wellwith the Dp� estimated from the dissipation data of Meneveauand Sreenvisan.35 The Dp� of the ice sheet w series is the mostnonlinear in p, but still remains bounded by Dp� and Dep ofMeneveau’s wind tunnel data.34 That is, the ice sheet w seriesintermittency does not exceed the laboratory dissipation in-termittency but can exceed their u energy intermittency.

C. Roughness effects on common scalewiseintermittency measures

To further explore the effects of roughness and aniso-tropy on the local velocity component energy variability, thelocal intermittency factor LIF was computed for all four timeseries along with the phase-randomized signal. The variationof the wavelet-based LIF with scale are shown in Fig. 13. Atlarge scales, the LIF for the phase-randomized series and allfour series are comparable. However, as the scale decreases,the LIF increases near exponentially. What should be notedhere is that there are no large differences in LIF between thefour series, suggestive that the variance in energy �around itslocal mean� is not as sensitive to anisotropy and roughness�when compared to dissipation-based measures�. Likewise,when the wavelet-based flatness factor is computed, its scale-wise evolution exhibits trends reported for a number of labo-ratory studies as shown in Fig. 14. The wavelet coefficientsappear near-Gaussian at large scales �F�r�=3�, and increas-ingly non-Gaussian �fat tailed� with decreasing scale exceptfor the phase-randomized series in agreement with previousstudies.59 In Fig. 14, the wavelet flatness factor data setsfrom Meneveau34 and Camussi and Guj38 are also shown�with wavenumber normalized by � as presented in the origi-nal references�. It is clear from Fig. 14 �top panel� that theincrease in flatness factor with increasing normalized longi-tudinal wavenumber �kx�� in the inertial subrange may col-lapse the data if some other scaling is adopted.

Motivated by the results of Qian15,16 and Gagne et al.,18

define the bulk turbulent flow Reynolds number as

Rt = qe � LI/� , �40�

where qe2 is, as before, the turbulent kinetic energy with �u

2

�qe2 /3 for isotropic turbulence. In a conservative cascade,

�qe3 /LI. Recalling that �2=15��u

2 / results in Rt

��LI /��2. Also, this scaling leads to LI /��qe�� /��R�.Hence, by normalizing the wavenumbers kx by �Rt, or alter-natively, by kx��R��2, the data sets for flatness factors rea-sonably collapse within the inertial subrange.

Predictions from Eq. �17� with a2=−0.025 �Ref. 26� isalso shown, which captures the essential scaling law reason-ably well for inertial subrange scales. Hence, this analysisagain confirms previous theoretical findings that bulk Rey-nolds number effects �via Rt� can impact the inertial sub-

0.1 0.15 0.2 0.25 0.3 0.35 0.4−0.025

−0.02

−0.015

−0.01

−0.005

0

a1

(1/2

)a 2

FIG. 9. Relationship between the scaling parameters a1 and 1 /2a2 used todescribe �p=a1p+1 /2a2p2 across a wide range of experiments, includingexperiments reporting roughness effects and anisotropy. The data set in-cludes: six �p curves reported in Poggi et al. �Ref. 20� for wall distancesy+ =23,31,80 and for both rough and smooth walls �data for y+ =23, 31are shown as � because of reported anisotropy�, two �p curves from Toschiet al. �Ref. 21� for channel turbulence in the region 50�y+ �100 andy+ �100, two DNS runs from Laval et al. �Ref. 81� for u and w, three �p

curves by Antonia and Krogstad �Ref. 19� shown in Fig. 7, two �p curves�i.e., u and w� reported in Dhruva et al. �Ref. 79� in Fig. 7, two �p curvesreported in Anselmet et al. �Ref. 73� �up to p�14�, partly shown in Fig. 6,a �p curve presented by Stolovitzky et al. �Ref. 74�, the two �p curvespresented in Fig. 6 for DNS and atmospheric flows by Sreenivasan andDruva �Ref. 27�, a �p curve presented by Camussi and Guj �Ref. 38� for lowReynolds number �also shown in Fig. 6�, four �p curves presented by Ca-mussi et al. �Ref. 82� �i.e., u and w for homogeneous and nonhomogeneousflows�, and the four �p curves derived from the ESS-wavelet analysis fromthe present study. The dashed lines are �1 /2�a2=−1 /3�a1−b /3�. The sym-bols are as follows: �1� � are for u, �2� � are for w, �3� * are for runs withknown anisotropy, and �4� + are for w over smooth surfaces. The experi-ments here are indicated by the same convention with open squares for uand � for w with the pine forest highlighted in thick marker. The dashedparallel lines are for b=1 or K62 to b=0.6 �leftmost line� in increments of0.1. Many of the �p curves used in this derivation were digitized by us.



range scaling. Moreover this scaling is consistent with theReynolds number scaling proposed elsewhere24 when match-ing the inertial subrange to the far dissipation range. Itshould be emphasized that the precise collapse of the data�via Rt� is difficult to test here because the integral lengthscales for these studies were only order of magnitude esti-mates derived by us.

Given the OWT framework employed here, it is worth-while recalling an observation made in the original work ofMallat83 on their distributional properties. Mallat noted thatthe scalewise distribution of wavelet coefficients appearsimilar for a variety of signals and images. Typically, theirempirical distributions are symmetric with a sharp peak at

−12 −11 −10 −9 −8 −7−15

−14

−13

−12

−11

−10

−9

−8

−7

1/(p

−1)

log

2[<e v[u

](x,

a)p>

<ev(x

,a)>

−p] Pine Forest

−12 −11 −10 −9 −8 −7−15

−14

−13

−12

−11

−10

−9

−8

−7Ice Sheet

−12 −11 −10 −9 −8 −7−15

−14

−13

−12

−11

−10

−9

−8

−7

−log2(a) (index)

1/(p

−1)

log

2[<e v[w

](x,

a)p>

<ev(x

,a)>

−p]

−12 −11 −10 −9 −8 −7−15

−14

−13

−12

−11

−10

−9

−8

−7

−log2(a) (index)

p=0

p=1.5

p=3.0p=4.5

FIG. 10. An illustration of how D��p� was determined for sample p values from double-log plots for the pine forest �left� and the ice sheet �right�. For eachselected p, �p−1��D��p�−1� was computed by regressing �ev�xo ,a�p��ev�xo ,a��−p upon a /LI within the inertial subrange, and D��p� was then determined.

0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

α

f(α)

Wind Tunnel, Wake flow (Reλ=50)

Wind Tunnel, Boundary Layer flow (Reλ=110)

Pine Forest−uPine Forest−wIce Sheet−uIce Sheet−wPhase Randomized−u

FIG. 11. The effects of surface roughness and anisotropy on the wavelet-based estimation of f�� along with the two functions reported by Men-eveau �Ref. 34�. The f�� phase-randomized series is also shown for ref-erence to illustrate the effects of limited sample size. The Meneveau datawere digitized by us from the original reference.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50.5

0.6

0.7

0.8

0.9

1

1.1

q

Dq

Pine Forest−uPine Forest−wIce Sheet−uIce Sheet−wPine Forest−PRDissipation Dq − WTInferred Dq from WT Dissipation

FIG. 12. The effects of surface roughness and anisotropy on the wavelet-based estimation of generalized dimension Dq� along with the function re-ported by Meneveau and Sreenivasan �Ref. 35� �digitized by us from theoriginal reference� for the dissipation rate �Dq�. The estimate of Dq�=D2q/3+ �q /3�q−1��D2/3−D2q/3� is also shown. The Dq�

� for the phase-randomizedseries is also shown for reference.



zero. Guided by this opulent evidence, Mallat83 proposedmodeling a “typical” wavelet coefficient by an exponentialpower-law family of distributions �see Eq. �4��, EPD� ,m�,having the following probability density function:

f�wc� = Ke−��wc��m, �41�

where is a scale parameter, m is the shape �or stretching�parameter, and K is a normalizing constant. In the context of

wavelet modeling, this approach is often referred to as Mal-lat’s model and reduces to Gaussian for m=2, to double ex-ponential for m=1 and trivially to uniform for m=0. Next,we investigate how roughness and anisotropy affect thestretching exponent m, after having shown the suitability ofEPD� ,m� to describe the distributional properties of thewavelet coefficients at various scales. Figure 15 shows thedistributional properties of the wavelet coefficients with de-creasing scales for the u and w and for the two roughnessconditions. The phase-randomized distributional propertiesof u for the pine forest are shown as reference. It is clear thatas the scale decreases, the tails become heavier than Gauss-ian for all series irrespective of the surface roughness condi-tions, and appear to be well described by the stretched expo-nential �provided m is allowed to vary in scale� except at thefinest scale for the w above the ice sheet.

Given the limited sample size of this experiment, howreasonable are the estimates of m when compared to otheratmospheric surface layer �ASL� experiments is considerednext. In Kailasnath et al.,41 m values were reported above anagricultural crop and urban canopy as a function of r /LI

using hot-wire anemometry and a much larger sample size�two orders of magnitude more in number of samples ana-lyzed�. The scalewise stretching exponent comparisons withthis data set are shown in Fig. 16. The agreements betweenthe wavelet-based estimates of m here and the reported val-ues in the experiments of Kailasnath et al.41 again lend con-fidence to the wavelet-based m estimates here. As expected,the m for the experiments here varied from about 2 �i.e.,Gaussian� for scales commensurate with LI to nearly unity�over two decades of wavenumber increases�, in agreementwith Kailasnath et al.41 The experiment of Kailasnath et al.41

suggests that m decreases further in the viscous range, satu-rating at about 0.5.

Next, a comparison between the m values reported here,the phase-randomized series, and the reported m values fromthe helium rotating disk experiments of Tabeling et al.43 arealso shown in Fig. 17. Similar to the experiments of Kailas-nath et al.,41 the Tabeling et al.43 m values span the viscousrange, and m appears to “saturate” at about 0.6 �vis-a-vis the0.5 reported in Kailasnath et al.�. We are unable to includethe Kailasnath et al. experiments on the same figure becausethe precise LI and � were not reported. We draw attention tothree results in Fig. 17:

�1� For the phase-randomized series, m was approximately 2and independent of scale, as expected, again suggestivethat the OWT analysis appears robust to the smallsample size limitations of this experiment.

�2� Neither anisotropy in velocity components nor surfaceroughness dramatically affected the scalewise variationsin m.

�3� When the scales are adjusted by Rt= �LI /��2, the m val-ues across these experiments collapse to common trend.

The latter scale adjustment result is in agreement with

10−7

10−6

10−5

10−4

10−3

10−2

0.5

1

1.5

2

2.5

3

3.5

4

kxη

LIF

Pine Forest−uPine Forest−wIce Sheet−uIce Sheet−wPine Forest−PR

FIG. 13. The effects of roughness and anisotropy on the increase in LIFwith increasing normalized wavenumber �K� for the pine forest and ice sheetdata.

10−12

10−10

10−8

10−6

10−4

10−2

100

102

100

101

102

kxη

Fla

tnes

sF

acto

r

Pine Forest−uPine Forest−wIce Sheet−uIce Sheet−wPine Forest−PRBoundary Layer (Reλ=110)

Wake Turbulence (Reλ=50)

Grid Turbulence (Reλ=12)

Grid Turbulence (Reλ=3)

10−1

100

101

102

103

104

100

101

kx

η (Lu/λ)2

Fla

tnes

sF

acto

r

FIG. 14. The effects of roughness and anisotropy on the wavelet-basedflatness factor with increasing normalized wavenumber �kx�� for the pineforest and ice sheet experiments. Data from Meneveau �Ref. 34� and Ca-mussi and Guj �Ref. 38� are also shown for comparison. When normalizingby Rt��Lu /��2, the following were used: �i� For the boundary layer data ofMeneveau �Ref. 34�, LI=2.9 cm, �=0.32 cm, and �=0.016 cm. �ii� Forthe wake flow data of Meneveau, LI=4.2 cm, �=0.28 cm, and �=0.026 cm, for the homogeneous turbulence data of Camussi and Guj �Ref.38�, LI=250�, �=0.92 cm, and �=0.28 cm. For these two published ex-periments, the data points were digitized by us from the original source. Theoblique dashed line in the bottom panel is F�r�=3�r /LI�4a2, witha2=−0.025.



the flatness factor scaling analysis reported in Fig. 16, whichalso suggests that Rt is an important variable in collapsingdata across various experiments. Given the analytical linksbetween the flatness factor, a2, and m proposed here, thisfinding should be expected.

V. CONCLUSIONS

The effects of surface roughness on �p�u�, �p�w�, andvarious measures of fine-scale intermittency within the iner-tial subrange were analyzed using two data sets that span theroughness “extremes” encountered in atmospheric flows: anice sheet and a tall rough forest. Two runs, collected for

−10 −5 0 5 1010

10

1015

1020

1025

1030

pdf(

u)

Pine Forest

−10 −5 0 5 1010

10

1015

1020

1025

1030

Ice Sheet

−10 −5 0 5 1010

10

1015

1020

1025

1030

pdf(

w)

wc=u or w−10 −5 0 5 10

1010

1015

1020

1025

1030

wc=u or w

FIG. 15. The effects of roughness and anisotropy on the wavelet-based estimation of p��v�r�� for the pine forest and ice sheet data. The p��v�r�� are shiftedby two decades upward with decreasing scale to permit comparison. The fitted stretched exponential p��v��exp�−��v�m� model at each dyadic scale is alsoshown. Also, the p��v�r�� for the phase-randomized series is shown in the top-left panel.

10−2

10−1

100

101

102

103

104

105

106

0.5

1

1.5

2

2.5

kx

LI

Exp

onen

t(m

)

Pine Forest (u)Pine Forest (w)Ice Sheet (u)Ice Sheet (w)ASL,u (Hot−film)

FIG. 16. The variation of the stretching exponent m with normalized wave-number �kxLI� for the ASL and CSL studies here as compared to the ASLvalues reported in Kailasnath et al. �Ref. 41� using hot-wire probes �pointsdigitized by us�. Only the separation distances �r� normalized by the integrallength scale LI were provided in Kailasnath et al.

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

0.5

1

1.5

2

2.5

kxη

Exp

onen

t(m

)

Pine Forest (u)Pine Forest (w)Ice Sheet (u)Ice Sheet (w)Pine Forest (PR)Rotating disks, helium (Rλ~2000)

10−1

100

101

102

103

104

0.5

1

1.5

2

2.5

kxη (L

u/λ

u)2

Exp

onen

t(m

)

FIG. 17. Same as Fig. 16 but using � �top panel� and �Rt �bottom panel�instead of LI as normalizing wavenumber variables. The comparisons withm determined from data reported by Tabeling et al. �Ref. 43� for low tem-perature helium gas is also shown. The points are digitized by us.



near-neutral atmospheric stability and low squared turbulentintensity conditions yet having similar Taylor microscaleReynolds numbers, were analyzed. It was demonstrated that

�1� �p�u��p�w� irrespective of the surface roughness con-sistent with previous laboratory experiments. Moreover,�p�u� appeared much less sensitive to surface roughnesschanges when compared to �p�w�, and appeared to beless intermittent when compared to some benchmarklaboratory experiments. Because inactive eddies contrib-ute to �u

2 while smaller-sized attached eddies contribute

to the �w2 , the turbulent Reynolds number was one order

of magnitude smaller for w when compared to u. Thislarge difference in the energy injection mechanismpartly explains why �p�u� appears less intermittent andless impacted by surface roughness when compared to�p�w�. Note that the anisotropy in these experiments isintroduced by the anisotropy in the length scales of en-ergy injection mechanism �vis-a-vis �u /�w used in otherstudies�.

�2� The scatter in the log-normal model intermittency expo-nent � at high R� may be linked to external intermit-tency effects already explored elsewhere.11 External in-termittency here implies external flow conditions orenergy injection mechanisms impact the anomalousscaling exponents of �p �as already alluded to in SO3studies�. From �6, the experiments here reported varia-tions in � from 0.1 �for u� to 0.28 �for w� even for veryhigh R�. Here, w is clearly being impacted by boundaryconditions �i.e., surface roughness� while u is not and isthus more susceptible to contamination from externalintermittency. Also, it was shown that such “external”intermittency may require higher-order cumulants thatcannot be captured by a single intermittency parameter�e.g., � in the log-normal model�. Again, this contrastshigh R� experiments far from boundaries that suggest asingle parameter �e.g., �� is sufficient to capture theanomalous scaling in the absence of such externaleffects.26

�3� The consequence of this external intermittency �surfaceroughness induced here� is that the singularity spectra ofthe exponents are much broader for w when compared uin the context of the multifractal formalism applied tothe local energy �instead of the usual conservative cas-cade studies for the dissipation rate�.

�4� The effects of Reynolds number Rt=��LI /� on thescalewise flatness factor and stretching exponents werealso demonstrated when comparing the present experi-ments to a number of laboratory and atmospheric flowexperiments. It was shown that this Reynolds number iscomparable to �LI /��2.

These conclusions were derived using the OWT, whichprovided robust estimates of such scalewise varying quanti-ties even for limited data sets �as is the case here with only217 data points�. This methodological finding may havebroader implications to understanding the structure of turbu-lence at fine scales �including scalars� because of the avail-ability of a large yet underutilized data bank of similar mea-

surements provided by the FLUXNET �Ref. 84� over a widerange of terrestrial biomes with different canopy morpholo-gies and surface roughnesses, different scalar sources andsinks at the surface �including heat, water vapor, and carbondioxide�, and atmospheric stability conditions �known to im-pact the energy injection mechanism for different velocitycomponents10�. The OWT approach proposed here is idealfor analyzing such series within the context of external inter-mittency, energy injection effects, anisotropy, and fine-scaleturbulence. It is envisaged that with such robust analyzingtools, the wealth of field experiments from FLUXNET can pro-vide a new look at possible emergence of universal or qua-siuniversal theories for turbulence within the inertial sub-range at very high Reynolds number1,2 �and how boundaryconditions may modify them, especially for the scalar fields�.

ACKNOWLEDGMENTS

The authors thank D. Cava for providing the velocitytime series above the ice sheet. This work was supported bythe National Science Foundation �through NSF-EAR GrantNo. 0628342 and 0635787 and NSF-ATM-0724088�, and bythe Binational Agricultural and Research Development�BARD� fund �Grant No. IS-3861-06�.

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