Coherent structures and the k 1 spectral behaviourM. Calaf, M. Hultmark, H. J. Oldroyd, V. Simeonov, and M. B. Parlange Citation: Physics of Fluids (1994-present) 25, 125107 (2013); doi: 10.1063/1.4834436 View online: http://dx.doi.org/10.1063/1.4834436 View Table of Contents: http://scitation.aip.org/content/aip/journal/pof2/25/12?ver=pdfcov Published by the AIP Publishing

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PHYSICS OF FLUIDS 25, 125107 (2013)

Coherent structures and the k−1 spectral behaviourM. Calaf,1,a) M. Hultmark,2 H. J. Oldroyd,3 V. Simeonov,3

and M. B. Parlange3

1School of Architecture, Civil and Environmental Engineering, EPFL, Lausanne,1015, Switzerland and University of Utah, Salt Lake City, Utah 84112, USA2Princeton University, Princeton, New Jersey 08544, USA3School of Architecture, Civil and Environmental Engineering, EPFL, Lausanne,1015, Switzerland

(Received 17 May 2013; accepted 9 November 2013; published online 17 December 2013)

Here we present unique evidence of a k−1 scaling behaviour in the atmospheric bound-ary layer and its connection to large scale coherent structures within the boundarylayer. Wind lidar measurements were conducted above a lake under cold atmo-spheric conditions. The large coherent structures could be visually observed overLake Geneva in Switzerland when cold air met the relatively warm water. Proper or-thogonal decomposition of the experimental data acquired with the wind lidar clearlyreveals coherent oscillations of both the fluctuating velocity field and the water aerosolfield over the surface of the lake. Precise identification of the large coherent structurespropagating in the flow allows for detailed analysis of their contribution to the totalspectral budget. Additionally, it is shown that the experimental data agree well withrecent theoretical predictions. C© 2013 Author(s). All article content, except whereotherwise noted, is licensed under a Creative Commons Attribution 3.0 UnportedLicense. [http://dx.doi.org/10.1063/1.4834436]

I. INTRODUCTION

In 1922, Richardson1 introduced the idea that energy is fed into turbulence through the largeturbulent structures and is transferred inviscidly towards smaller and smaller scales until ultimatelybeing dissipated by viscous action in the smallest scales of turbulence. In 1941, Kolmogorov2

mathematically quantified this picture of turbulence introducing the concept of universal equilibriumof the small scale components of turbulent motions. He further separated the turbulent scalesaccording to their length scale, identifying an energy-containing range, followed by a universalequilibrium range, which in turn was subdivided into an inertial subrange and a dissipation range.His work especially focused on describing the dissipating scales and spectral behaviour in the inertialsubrange. Arguably the most important implication of Kolmogorov’s work is the so-called “k−5/3

scaling,” which predicts the turbulent energy spectrum function to scale as k−5/3 in the inertialsubrange, where k is the wavenumber.

Based on these early studies, Townsend in collaboration with Batchelor conducted experimentalwork aiming at better understanding the transfer of energy from the large scales towards the smallerones. Inspired by this work, Heisenberg3 presented the first theoretical work that would a prioriexplain this process. Batchelor and Townsend then4 found that the dissipation of energy was notuniformly distributed throughout the flow, but that it takes place at events sparsely distributed and athigh intensity, suggesting that the turbulent motion of the small scales cannot be completely random,but that a certain intrinsic coherent pattern should exist. This opened the door to a completely newline of research where turbulence was no longer only considered from a statistical perspective,but also by trying to understand its intrinsic underlying structure. Townsend set the basis on thisline of research for turbulent shear flows with his two seminal hypotheses: The Reynolds numbersimilarity hypothesis and the attached eddy hypothesis. Much later, works by Perry and Chong,5

a)Electronic mail: marc.calaf@utah.edu

1070-6631/2013/25(12)/125107/14 C©Author(s) 201325, 125107-1

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Perry et al.,6 Perry and Marusic7 continued the study and description of the detailed structure ofthese coherent structures in wall-bounded turbulent flows. One of the major findings in these studieswas the so-called “k−1 scaling,” which was derived by an overlap argument of scaling regimes withinthe energy spectrum itself. Perry et al.6 showed that there should be a range of wavenumbers withinthe energy-containing range where the energy spectrum will scale inversely with wavenumber.

This scaling describing the large, energy containing scales of the turbulent spectra has beenthe subject of many studies, and the actual scaling has been observed experimentally as well as innumerical large-eddy simulation of the neutral atmospheric boundary layer (ABL).8 An extensivereview of these studies is presented in Katul and Chu9. In parallel, there exist several attempts todescribe the k−1 scaling from a theoretical point of view. Tchen10, 11 was the first to theoreticallypredict this scaling by considering a spectral budget. Perry et al.,6 Kader and Yaglom,12 Hunt andMorrison,13 Marusic et al.14 used dimensional analysis and asymptotic matching between the inner-outer regions of the boundary layer in order to explain the k−1 scaling. However, high Reynoldsnumber pipe flow experiments,15 unexpectedly, did not show any clear evidence of this inverse scalingbehaviour. Nickels et al.16 justified the unclear existence of the k−1 scaling in the previous work dueto the need for two extra prerequisites in order for the scaling to exist, aside from the high Reynoldsnumber constraint (universal for all turbulence theory). These two being that measurements had tobe taken from a dimensionless height from the boundary z+ = zu∗

ν> 100 (being z+ the distance

from the boundary in wall units), and that z/H ≤ 0.02 (being H the total height of the boundarylayer) in order to ensure a minimum overlap zone between the inner and the outer regions in whichthe k−1 is supposed to exist. Later, experiments showed that this was not always the case, either.17

There were also some critics to the earlier works from Nikora18 because in his analysis the effect ofthe coherent structures seemed to be ignored. The aforementioned spectral budget approaches alsoreceived similar critiques.

Recently, a novel phenomenological spectral theory based on the earlier ideas from Heisenberg3

has been introduced by Katul et al.19 This theory recovers Nikora’s18 scaling arguments for the k−1

scaling in the limit of Reynolds number tending towards infinity, and also recovers some of theempirical prerequisites16 for the scaling to exist. Further, this new approach takes into account theeffect of the large coherent motions and intermittency, and presents some conjectures. Katul et al.also use some experimental data in order to test their theoretical scaling. The data show that turbulentkinetic energy is dominated by k−5/3 for kz > 1, and for kz < 1 there exists a k−1 scaling, whichspans almost an order of magnitude in the wavenumber axis. Katul et al. also mention that it isdifficult to distinguish the spectral contribution of the very large scale motions (VLSM), but giventheir existence, their effect could be to alter the k−1 scaling, making it less steep.

In the present work pictorial evidence of the existence of coherent structures in the ABL ispresented. By means of a wind lidar instrument for measuring the velocity field without perturbingit, together with the use of the Proper Orthogonal Decomposition (POD), the presence of thesestructures within the ABL is shown. The spatial and temporal nature of the wind lidar data allowswell converged spectra to be calculated. As a result, the spectral energy of the flow field shows oneof the longest and “cleanest” k−1 scalings presented in the literature, extending over a decade anda half in wavenumber. At higher wavenumber the spectrum shows a clear k−5/3 scaling also presentin the data. Both observations are in agreement with the conclusion by Perry et al.6 The transitionbetween the two scaling regimes occurs around k ∼ 1/z in agreement with the predictions by Katulet al.19 Finally, the individual contributions of the coherent structures on the one-dimensional energyspectrum are assessed.

In Sec. II the pictorial image of the coherent structures is presented, and in Sec. III the exper-imental setup and the data analysis are explained. Section IV briefly reviews the POD techniqueemployed, and it presents the analysis on wind lidar extracted experimental data. In Sec. V theturbulent spectra of the signal are analysed, together with the corresponding energy spectra of thecoherent structures. Conclusions are presented in Sec VI.

II. NATURE AND COHERENT STRUCTURES IN THE ATMOSPHERE

In February 2012, a cold and dry weather front from the arctic region enveloped most of centralEurope with temperatures dropping below −10 ◦C in cities like Paris, Munich, and Zurich. What

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FIG. 1. Photograph of Lake Geneva, by Olivier Staiger at klipsi.com, documenting the special natural event that took placein February 2012, and pictorial evidence for the existence of coherent structures (Reprinted from source: Olivier Staiger atwww.klipsi.com).

made this event of special interest is that it allowed for nature’s own flow visualisation of turbulentflows at very high Reynolds numbers. In large bodies of water such as Lake Geneva (one of thelargest lakes in Western Europe with a water surface of 345.31 km2) the water temperature remainsquite temperate during winter time, given the preceding warm summer and the large heat capacityof water. This phenomena, combined with the cold and dry air close to the surface, induced largefluxes of water vapour into the air which quickly condensed, forming water aerosol (fog) as result.The uniqueness of the event, however, resided in the fact that this particular region (the Lemmanarch) is usually dominated by very strong winds when cold fronts descend from Northern Europe.Therefore, the naturally induced water aerosol emanating from the lake acted as a natural tracer thatmade visible to the naked eye the complexity of wall-bounded, high Reynolds number turbulentflows, see Figure 1. (This photograph was taken by Olivier Staiger at klipsi.com, who documentedthis natural event. It was later shared with the current authors, with the rights to be used for scientificpurposes.)

As can be observed in this figure, there exists a coherent pattern in the arrangement of the wateraerosol. It can be distinguished how small turbulent structures merge into larger organised entitiesthat then propagate in space. (The structures last over time, although this cannot be appreciatedwith this single snapshot.) Although this image clearly shows the existence of coherent patterns(structures) under conditions of high Reynolds number turbulent flows, the limitations of the camerado not allow for a detailed study of the dynamical behaviour of these coherent structures. It is forthis reason that velocity measurements were taken a posteriori during similar high-wind events, bymeans of doppler wind lidar velocimetry. Details of the experimental setup are given in Sec. III.

III. EXPERIMENTAL SETUP AND DATA ANALYSIS

On February 6th 2012, EPFL’s Halo Photonics Stream Line wind lidar was deployed on theshore of Lake Geneva, on the so-called Pelican Beach, located in the vicinity of the village ofSaint Sulpice in Canton Vaud, Switzerland. This beach is located on a small peninsula making it an

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FIG. 2. Schematics of the place of measurement (top and bottom left), and the wind lidar placement (top and bottom right).The yellow beam symbolises the wind lidar direction of measurement.

ideal placement to measure winds from south-west or south-east directions (see Figure 2 for furtherdetails).

The Halo Photonics wind lidar is a pulsed lidar that operates at a 15 kHz repetition rate whereraw signals are obtained by averaging 15 000 pulses, which are processed every second-and-a-half, approximately. By computing the frequency shift of the aerosol backscattered light to thelidar’s antenna, it can extract the projected particle velocity on the laser beam line of sight. Forthis specific measurement campaign the staring technique was used, meaning that the lidar headwas fixed at a given position and therefore only the projected aerosol velocity on the direction ofthe laser beam was extracted. Thus, for measuring the true wind component, the lidar was alignedwith the mean wind direction. For this experiment it was convenient to use the staring techniquebecause it provides the highest measurement frequency, since the lidar head does not need to movemechanically between measurements. The wind direction remained close to constant throughoutthe duration of the measurements which lasted 1 h. The wind lidar was installed at the edge of thePelican peninsula, which has an elevation of about 2 m above the water level, thus the measuringheight used later was z = 2 m. The laser was pointing parallel to the water surface and aligned withthe mean wind. The spatial resolution was 18 m, meaning that the wind lidar provides simultaneousvolume averaged velocities every 18 m, along the laser beam. The maximum measurement distance,without significant background noise, was 1200 m downstream, and due to intrinsic technical detailsfrom the wind lidar, the first good measurement point corresponds to an approximate distance of72 m from the lidar head.

Figure 3 shows the fluctuating component of the velocity, in the direction of the lidar beam, asa function of time, u′(r, t). The vertical axis represents the downstream distance and the horizontalaxis represents time. The presented data were obtained on February the 6th between the thirteenth

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FIG. 3. The fluctuating component of the velocity in the direction of the lidar beam as a function of time, u′(r, t). The verticalaxis represents the downstream distance and the horizontal axis represents time. The colour coding represents the intensityof the fluctuating velocity, with red colour being positive departures of the mean velocity and blue negative ones.

and fourteenth hour of the day, approximately. The fluctuating velocity was obtained by subtractingthe time averaged velocity (over the full 1 h, (u(r )) from the instantaneous data (u(r, t)), at eachcorresponding downstream distance, u′(r, t) = u(r, t) − u(r )).

The wind lidar also provides the attenuated backscatter profile (γ (r, t)) as a default output whichcorresponds to the atmospheric backscatter (β(r, t)) times the two-way atmospheric transmissionT(r). This attenuated backscatter can be interpreted as a proxy for water aerosol concentration. Bywriting the lidar equation in single-scattering approximation

P(r ) = CO(r )

r2β(r )T (r ) = C

O(r )

r2γ (r ), (1)

one can relate the attenuated backscatter to the received power, a fixed instrument constant, andan overlap function. Here P(r) is the received power, O(r) is the overlap function, r the distancefrom the lidar antenna, and C is an instrument constant containing the instrument efficiency, thetransmitted power, the antenna area, and the pulse duration. The attenuated backscatter delivered bythe instrument as

γ (r ) = r2 P(r )

O(r ) C, (2)

can be used as a proxy for the aerosol content. In order to minimise the impact of the not preciselyknown overlap function, the normalised attenuated backscatter fluctuating signal will be consideredinstead to remove any dependency from the instrumental parameters. Therefore,

γ (r, t)′ = γ (r, t) − γ (r )

γ (r )= γ (r, t)

γ (r )− 1, (3)

and the resultant data are shown in Figure 4 where likewise to Figure 3, the horizontal axis representsthe time and the vertical axis the downstream distance.

Therefore, since the main focus of this study is to analyse the relationship between the wind fieldand the water aerosol coherent structures observed in Figure 1, the normalised fluctuating attenuatedbackscatter signal will be studied in parallel to the fluctuating wind field.

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FIG. 4. The fluctuating component of the normalised attenuated backscatter along the lidar beam. The colours represent thenon-dimensional fluctuating attenuated backscatter intensity.

It is interesting to observe the coherent motions of high and low fluctuating velocity propagatingthrough the entire wind lidar measuring range shown in Figure 3. Also interesting is the presenceof a similar pattern in the normalised attenuated backscatter signal. This close similitude betweenFigures 3 and 4 could have probably been inferred from Figure 1, where the water aerosol patternseems to follow a coherent organisation. In order to provide a more thorough analysis of both signals,a proper orthogonal decomposition is presented in Sec. IV.

IV. A PROPER ORTHOGONAL DECOMPOSITION OF THE WIND LIDAR DATA

A POD analysis has been applied to the wind lidar data presented in Figures 3 and 4. The PODtechnique was first introduced in the context of turbulence by Lumley,20 and since then it has beenused extensively in fluid mechanics21–24 as well as in image processing,25 signal analysis,26 datacompression,27 and oceanography.28 Indeed Lumley20 proposed a method for extraction of coherentstructures from turbulent velocity fields using POD which was further discussed in Holmes et al.29

In the present work the methodology introduced by Iungo and Lombardi,30, 31 which allows theextraction of the proper orthogonal components from a one-dimensional dynamic signal (or timeseries), will be used.

The POD technique consists of extracting an orthogonal set of basis functions from the de-composition of an ensemble of data functions. These data functions can then later be re-written asa linear combination of the orthogonal basis of deterministic functions, the so called POD modes,such that

u(r, t) =M∑

j=1

a j (t)φ j (r ). (4)

Here φj(r) are the POD modes conforming the orthogonal basis and they correspond to the eigen-functions of the covariance matrix of the velocity vector. The multiplying factors, aj(t), are theso-called principle components which are a set of independent coefficients. The optimality of thePOD technique is that the modes conforming the orthogonal basis are ordered with decreasingenergy, meaning that the first n modes obtained through POD analysis contain more energy thanany other n first modes of a different orthogonal basis found through another linear decomposition

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0 200 400 600 800 1000 1200 1400 16000

1

2

3

4

5

6

7

8

En

ergy

[%]

M ode N umber

0 10 20 30 40

2

3

4

5

6

7

8

En

ergy

[%]

M ode N umber

FIG. 5. Measure of the fluctuating energy represented by each POD mode as percentage of the total energy of the system.

procedure. It is relevant to notice that if the measured signal, u(r, t) for the present study, representsa stationary process, its POD modes correspond to simple Fourier functions,32 and therefore thelocality characteristic of the POD technique is lost, but the advantage that the modes are energysorted still remains. However, if the signal is non-stationary, then the POD modes represent the mosttypical realisations, in a statistical sense, and are sorted by their energy representation.

For extracting the POD modes (or eigenfunctions), a certain number of snapshots of the eventneed to be accumulated, such that the characteristic structure of the flow can be captured. Therefore,the time series needs to be blocked in a number of shorter data sets, so called snapshots (Nsnap),containing each one Nperiod data points. This means that the total length of the original signal shouldmatch Nsnap × Nperiod. To separate the typical realisations of the hidden phenomena from the randomnoise, a sufficient number of snapshots must be taken into account. In principle, Nsnap is continuouslyincreased by means of a sensitivity study until convergence of the most energy containing PODmodes is obtained. For this study, where the wind lidar provided a different time series at each singledownstream distance, all of them were simultaneously combined in order to create Nperiod blocks ofdata as long as the complete original data set. Given that for the present case the snapshots were setwith a length equal to the measuring time period, only a reduced number of snapshots correspondingto each downstream measuring location would have been available. This would have provided toofew snapshots for obtaining a well converged POD, therefore overlapping between the differentsnapshot periods was used. Although this process reduces the statistical independence of the processobservations, it has been shown to provide accurate results.30 Also, by using this technique thelowest frequency resolved corresponds to the inverse of the total time period of measurement.

With the right number of Nsnap and Nperiod, a matrix M = Nsnap × Nperiod is constructed withNperiod columns and Nsnap rows. The next step is to compute the covariance matrix of M, givenby C = MT M. Since C is Hermitian symmetric and non-negative definite, its eigenvalues are realand non-negative and the eigenvectors are orthogonal. Further, the eigenvalues represent the energycontained in each eigenmode or eigenvector. For the specific data considered in this study Nperiod

was much smaller than Nsnap, therefore C was evaluated as C = M MT, so it is generated with thelowest dimension possible (see Iungo and Lombardi30 for further details).

The extracted eigenvalues, which are a measure of the fluctuating energy in each POD mode,are represented in Figure 5 as percentage of the total energy of the system. These appear ordered bypairs, since each paired element corresponds to a different realisation of the same eigenmode, justshifted by 90◦ (since they must remain orthogonal). Although no clear dominant pair of eigenmodes

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FIG. 6. Projection of the velocity field on the first six pairs of eigenmodes, ordered from top to bottom and left to right(a to f) according to the percentage of energy that the corresponding eigenmodes represent. Similar to the previous plots, thevertical axis represents the downstream distance and the horizontal axis represents time. The colour coding follows the onein Figure 3 and the units are also “m/s.”

results from this decomposition, it can be clearly seen that the first five pairs are the most energeticones, containing up to 40% of the system’s total energy. This result, although a priori unexpected,can be better understood when observing the projection of the velocity field on the first six pairsof eigenmodes, shown in Figure 6. Each of the six subplots in Figure 6 represents the projectionof the velocity field on each different pair of eigenmodes, ordered from top to bottom and left toright according to the percentage of energy that they represent. Therefore, subplot (a) correspondsto the projection of the signal on the most energetic pair of eigenmodes, and subplot (f) to theprojection on the sixth pair of eigenmodes. The horizontal axis represents the time and the verticalaxis the downstream distance, and it is colour coded according to the velocity intensity (in m/s). Byreconstructing the signal using these six projections the main features of the flow are recovered. It isquite remarkable to observe the large time scale difference between the first two pairs of eigenmodesand the following four. While the first two subplots ((a) and (b)) have a time period ranging from30 min (subplot (a)) to an hour (subplot (b)), the other four subplots represent shorter time periodevents, ranging from 20 min for subplot (c) to 6 min for subplot (e). This can be interpreted as thefirst two pairs of eigenmodes representing the propagation of the geostrophic forcing down to thewater surface (the outer flow), given its time period and also due to its relevance energy-wise, and

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FIG. 7. Projection of the normalised fluctuating backscatter signal on the first six pairs of eigenmodes, ordered from top tobottom and left to right ((a)–(f)) according to the percentage of energy that the corresponding eigenmodes represent. (Valuesare scaled by 10−3.)

the other four pairs of eigenmodes, representing either the effect of the surface boundary condition(inner flow), or inviscid turbulent motions in the surface layer. This hypothesis is further reinforcedwhen inspecting Figure 7.

Similar to the velocity field, a POD analysis has been applied to the normalised fluctuatingbackscatter signal (representing the water aerosol concentration) presented in Figure 4. As withthe velocity, the subplots are arranged from left to right and top to bottom related to their energypercentage significance. Relevant in this figure is the striking similarity between the velocity and thebackscatter projections in subplot (c), corresponding to the projection of the signals on to the thirdpair of eigenmodes. The decrease in intensity in the normalised fluctuating backscatter projectiondownstream from the lidar can be explained either by a small misalignment of the wind lidar(meaning that it was not perfectly parallel to the water surface and therefore at a given downstreamdistance the laser beam was above the water vapour) or by a loss of attenuated backscatter intensitydue to a drop in the aerosol concentration (clear air) by some natural unknown reason. Either waythe third mode-pair of velocity and aerosol concentration are nearly perfectly anti-correlated with acorrelation coefficient of ρT, U = −0.96 (only the first 400 m of the data were used to calculate thecorrelation). This can be compared to ρT, U = −0.15 for the raw signal (all modes). To better assess

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0 1 2 3 4 5

x 10−3

0

1

21st mode pai r

E(f

)

a

0 2 4 6 8

x 10−3

0

1

22nd mode pai r

b

0 1 2 3 4 5

x 10−3

0

1

23rd mode pai r

E(f

)

c

0 1 2 3 4 5

x 10−3

0

1

24th mode pai r

d

0 2 4 6 8

x 10−3

0

1

25th mode pai r

E(f

)

f [H z ]

e

0 0.005 0.01 0.0150

1

26th mode pai r

f [H z ]

f

FIG. 8. Fourier spectra of the first six pairs of eigenmodes. The spectral energy density of the projection of the fluctuatingvelocity on the corresponding modes is represented in black solid line (which has units of energy per unit mass and frequency,m2/s) and the equivalent spectral energy density of the projection of the normalised fluctuating backscatter is represented inblack dashed line (s).

this resemblance, the Fourier spectra for the different POD modes for the velocity and the attenuatedbackscatter signal were computed.

It is clear that for both cases the third pair of eigenmodes oscillates with the same frequency.This is represented in the two lower subplots ((e) and (f)) in Figure 8, which show the Fourier spectraof the third pair of eigenmodes. Indeed, this frequency match corresponds well with the almostperfect anti-correlation between the two cases, as mentioned earlier, and it is perfectly visible inFigures 6 and 7. This means that when there is a high speed coherent motion in the velocity, thereis a low aerosol concentration in the air, and vice-versa, the low-speed coherent motion correlateswith a high aerosol concentration region.

Thus, the pictorial evidence introduced in Figure 1 showing clear proof of the existence ofcoherent structures propagating in the stream wise direction33–40 in the ABL is confirmed mathemat-ically by means of POD analysis, due to the existing quasi-perfect anti-correlation of the third pair ofeigenmodes between the velocity field and the attenuated backscatter signal. Also, this coincidenceseems to corroborate the initial hypothesis (mentioned earlier in the present section) where it was as-sumed that the last four pairs of eigenmodes presented in Figure 6 represented inner layer turbulenceinteractions, while the first two pairs represented outer scale motion phenomena. The present resultsconfirm at least that the third pair of eigenmodes corresponds to wall boundary interactions, giventhe very good anti-correlation between the velocity field and the water vapour emanating from thesurface. This result constitutes a further experimental evidence of the interaction inside the boundarylayer of wall-induced turbulence and the outer forcing physics.41, 42

V. THE TURBULENT SPECTRA

To investigate the spectral properties of these coherent structures the energy spectra for thefluctuating velocity field were computed. This is presented in Figure 9. In blue and red, respectively,two additional lines represent the corresponding k−1 and k−5/3 scalings. It is remarkable to observe

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10−3

10−2

10−1

100

101

10−2

10−1

100

101

102

Eu(k

z)[

m2 /

s2 ]

k z

FIG. 9. Energy spectra for the fluctuating velocity field. Blue and red lines represent the corresponding k−1 and k−5/3

scalings, respectively.

the clear span of a decade and a half of k−1 scaling in wavenumber, one of the longest and cleanestrecorded to the authors’ knowledge. By contrast, because of the wind lidar’s coarse spatial resolutionthe k−5/3 scaling appears rather short. The spectra were computed in time, using the wavelet analysisat each downstream distance, and later averaged overall the downstream positions. In accordance toprevious studies19, 43–48 the Haar Wavelet was used for computing the orthogonal wavelet transform.Finally, Taylor’s hypothesis was used in order to transform the frequency axis into wavenumber.It is worth mentioning here that although it is well known that Taylor’s hypothesis is plaguedwith problems close to the wall,37, 49, 50 since for this specific case we are mainly interested in thecontribution from the largest structures (having a length scale much larger than the distance fromthe wall) into the total spectral energy, this transformation should hold according to earlier studieson the validity of Taylor’s hypothesis in atmospheric flows.51, 52

In Figure 9, it can be observed the coexistence of the two well-known scalings for the energyspectra, Eu(k), where the k−1 scaling expands for one and a half decades in wavenumber, and the k−5/3

scaling is shorter than usual due to the coarse spatial resolution of the wind lidar. These results matchvery well with the experimental data and the new phenomenological spectral theory introduced byKatul et al.,19 where the transition between both turbulent scalings is expected to happen at k ∼ 1

z .Katul et al.19 also conjecture that in the presence of coherent structures or VLSM there should appeara correction on the k−1 scaling, resulting in a less steep scaling. In their study,19 it is mentionedthat detecting the spectral contribution of the VLSM, and therefore experimentally assessing thisforeseen change in the k−1 scaling, is nonetheless complicated given the non-stationarity of the ABLheight during the measurement period. In the present study the wind lidar data were recorded duringa very short time period (in the context of atmospheric flows) through which the flow was mainlydriven by an intense synoptic forcing, which allows us to reason that the ABL height did not changemuch during the measurement time. Therefore, taking advantage of the fact that the POD technique

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10−3

10−2

10−1

100

101

10−6

10−5

10−4

10−3

10−2

10−1

100

101

102

kz

Eu(k

z)[

m2 /

s2 ]

mode 1 to 4

mode 5 to 6

mode 7 to 12

mode 13 to 100

mode 101 to 200

Original Signal

k−1

k−5/3

k ∼ 1/z

k ∼ 0.15/z

FIG. 10. Energy spectra for the complete fluctuating velocity field and some projections of the fluctuating velocity field onsome selected eigenmode pairs or joined group of eigenmodes. Specifically, the dashed line and hollow squares represent theenergy spectra corresponding to the identified coherent structure (projection of the fluctuating velocity field on the third pairof eigenmodes).

extracted at least one coherent structure (corresponding to the projection of the fluctuating velocityfield on the third pair of eigenmodes), its specific spectral contribution will be analysed next.

These results are presented in Figure 10 where the solid line shows the spectra for the originalfluctuating velocity field, previously introduced in Figure 9. The vertical dashed line represents theboundary between the k−1 and k−5/3 scalings, while the vertical dotted-dashed line represents theupper boundary of the k−1 scaling. For larger normalised wavenumber values the spectral slopetends to decrease. The secondary dashed line plotted above the spectra of the original fluctuatingsignal represents the precise k−1 scaling, and the second dotted-dashed line shows the k−5/3 scaling.Both are plotted for the sake of reference and clarity. It is interesting then to look at the individualcontributions of the different projections to the overall spectra. In dashed and hollow-circles, thesum of the projections of the fluctuating velocity on the first 2 pairs of eigenmodes is represented.It can be seen how these mainly contribute on the far end of the production range, and are the mostenergetic scales of the system. Further, the dashed and hollow-squares line represents the projectionof the fluctuating velocity field on the third pair of eigenmodes. This one represents the spectralenergy related to the coherent structure, and its spectral attribution peaks at the upper edge of the k−1

scaling, represented in Figure 10 by the vertical dotted-dashed line. In accordance to the conjectureof Katul et al.,19 it seems that this entity tends to smear out the slope on the upper left region of theenergy containing scales, since its energy contribution is very much limited to the confined spectralsection where it is defined. A similar behaviour could also be attributed to the projection of the flowfield on next three pairs of eigenmodes, since their mean contribution happens very close to the upperedge of the k−1 scaling, but still contributes to the same order of magnitude as the identified coherent

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125107-13 Calaf et al. Phys. Fluids 25, 125107 (2013)

structure for the low wavenumber region. The following dashed lines, with hollow diamonds andhollow triangles, represent the projections of the fluctuating fields on the next hundred pairs ofeigenmodes. These basically build up the k−1 and k−5/3 corresponding spectral sections.

VI. CONCLUSIONS

In the present study, pictorial evidence in conjunction with a quantitative investigation of theexistence of coherent structures in the ABL has been provided. The presence of these entities wasrevealed and investigated using proper orthogonal decomposition in combination with their spectralbehaviour, and it was shown that one of the most energetic POD modes dominates both in the wateraerosol concentration and velocity field. Furthermore, it was shown that these structures give riseto a nearly perfect anti-correlation between the fluctuations of the water aerosol and the velocityfield, with a near identical spectral behaviour. In addition, the measured energy spectrum of theflow field shows one of the longest and “cleanest” k−1 scalings presented in the literature, extendingmore than a decade and a half in wavenumber. A k−5/3 scaling is also observed, showing that thetransition between the two scaling regimes takes place at k ∼ 1/z, which is in agreement with thetheory19. Finally, the contribution to the energy spectrum of the projection of the flow field on themost relevant eigenmodes was also analysed, and it was shown that the identified coherent structurescontribute mainly to the lower edge of the k−1 scaling.

Given the limitations of this unique experimental data set, it cannot be further verified herewhether the coherent structures identified in this work correspond to the known VLSM or LSMintroduced in earlier literature. However, this work shows that the spectral energy of the identifiedcoherent structures scales very well with the k−1 scaling and is in agreement with the conjecture ofKatul et al.19.

ACKNOWLEDGMENTS

The authors would like to thank Dr. G. V. Iungo and Dr. Leo Hellstroem for their help andalways-clear explanations regarding the POD technique. Also, Professor G. Katul provided us withvery helpful comments regarding the historical and present understanding on the development of thek−1 scaling. Finally, we thank the support of the Swiss National Science Foundation (Project Nos.200021 134892/1 and 200020 125092).

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