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Extraction of coherent structures in a rotating turbulent flow experiment

Jori E. Ruppert-Felsot, Olivier Praud, Eran Sharon∗, Harry L. Swinney†

Center for Nonlinear Dynamics and Department of Physics,The University of Texas at Austin,

Austin, Texas 78712

The discrete wavelet transform (DWT) and discrete wavelet packet transform (DWPT) are usedto extract and study the dynamics of coherent structures in a turbulent rotating fluid. Three-dimensional (3D) turbulence is generated by strong pumping through tubes at the bottom of arotating tank (48.4 cm high, 39.4 cm diameter). This flow evolves toward two-dimensional (2D)turbulence with increasing height in the tank. Particle Image Velocimetry (PIV) measurements onthe quasi-2D flow reveal many long-lived coherent vortices with a wide range of sizes. The vorticityfield exhibits vortex birth, merger, scattering, and destruction. We separate the flow into a low-entropy “coherent” and a high-entropy “incoherent” component by thresholding the coefficients ofthe DWT and DWPT of the vorticity field. Similar thresholdings using the Fourier transform andJPEG compression together with the Okubo-Weiss criterion are also tested for comparison. Wefind that the DWT and DWPT yield similar results and are much more efficient at representingthe total flow than a Fourier-based method. Only about 3% of the large-amplitude coefficients ofthe DWT and DWPT are necessary to represent the coherent component and preserve the vorticityprobability distribution function (PDF), transport properties, and spatial and temporal correlations.The remaining small amplitude coefficients represent the incoherent component, which has nearGaussian vorticity PDF, contains no coherent structures, rapidly loses correlation in time, and doesnot contribute significantly to the transport properties of the flow. This suggests that one candescribe and simulate such turbulent flow using a relatively small number of wavelet or waveletpacket modes.

PACS numbers: PACS numbers: 47.27.-i 47.32.-y 47.32.Cc

I. INTRODUCTION

Large-scale ordered coherent motions occur in a widevariety of turbulent flows despite existing in a rapidlyfluctuating background turbulence [1, 2]. These “coher-ent structures”, which are associated with localized re-gions of concentrated vorticity, persist for times that arelong compared to an eddy turnover time. The impor-tance of coherent structures in turbulence has becomerecognized through the use of flow visualization. Exam-ples of coherent structures identified in turbulent flowsinclude hairpin vortices in boundary layer turbulence [3],plumes in turbulent convection, and vortices in turbu-lent shear flows. Coherent structures play a major rolein the transport of mass and momentum, thus affectingtransport, drag, and dissipation in turbulent flows. Dueto their long lifetimes, coherent structures in the atmo-sphere strongly influence the exchange of heat, moistureand nutrients between different locations. In industry,the prediction and control of transport, drag, and tur-bulence is important in many processes [4], and in somecases flows can be modified through the control of coher-ent structures [5, 6].

The emergence of coherent vortices is especially strik-ing in geostrophic turbulence, where the Coriolis force

∗current affiliation: The Racah Institute of Jerusalem, Jerusalem,Israel†swinney@chaos.ph.utexas.edu

plays a dominant role [7, 8]. Under the effect of rota-tion and stratification, geophysical flows develop largeand robust coherent structures which can be identifiedand tracked for time much longer than their characteris-tic turnover time. Examples of such flows are high andlow pressure systems, large vortical structures that areformed in Gulf Stream meander, Mediterranean eddies(Meddies), and the Earth’s jet stream; all can be ob-served and tracked by satellite imaging of the atmosphereor the ocean surface (e.g. see satellite imaging web-sites [9, 10]). Large coherent structures are not limitedto the Earth; the atmospheres of other planets also re-veal structures such as Neptune’s dark spot and Jupiter’szones, belts and Great Red Spot.

Three-dimensional (3D) turbulence subjected to strongrotation develops columnar vortical structures alignedwith the rotation axis, as observed in the present experi-ment and in previous laboratory experiments [11, 12, 13,14]. Correlation in this direction becomes large when theCoriolis force becomes large compared to inertial forces,and the flow proceeds towards a quasi-2D state. Thistwo-dimensionalization allows energy to proceed towardlarger scales through the inverse energy cascade and to-ward smaller scales through the forward enstrophy cas-cade [15], as observed in simulation [16, 17, 18] and ex-periments on rotating [12, 13] and non-rotating quasi-2Dflows [19]. The cascades of energy and enstrophy leadto a spontaneous appearance of intense localized coher-ent vortices containing most of the enstrophy of the flow.Vorticity filaments outside of those coherent structuresare distorted and advected by the velocity field induced

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by the vortices. Thus, a rotating turbulent fluid flowcan organize itself into large-scale coherent structuresthat are often long-lived compared to dissipative timescales. The presence of long-lived coherent structures inturbulent rotating flow has been observed in both exper-iment [7, 11, 12, 13] and in simulation [17, 18]. Thesestructures are larger than the scale of the forcing. Thelarge-scale coherent structures break the homogeneity ofthe flow and are thought to dominate the flow dynam-ics. The long lifetimes and spatial extent allow coherentstructures to play a significant active role in transportprocesses [20, 21].

Due to the dynamical importance of coherent struc-tures, any analysis of flow containing these structuresshould take into account their existence. One approachto analyze the coherent structures and the dynamics ofsuch flow is to partition it into regions with different dy-namical properties. Okubo [22] derived a criterion toseparate flow into a region where strain dominates (hy-perbolic region) and a region where vorticity dominates(elliptic region). The same criterion was later re-derivedby Weiss [23] and is now known as the Okubo-Weiss cri-terion. The criterion has been widely used to analyzenumerical simulations of 2D turbulence. However, aspointed out by Basdevant and Philipovitch [24], the va-lidity of the criterion’s key assumption is restricted tothe core of the vortices that correspond to the strongestelliptic regions. This limitation reduces the applicabilityof a decomposition using this criterion.

Another method that has been found useful in an-alyzing flow fields is proper orthogonal decomposition(POD). This projects a field onto a set of orthonormal ba-sis functions where successive eigenvectors are obtainedby numerically maximizing the amount of energy corre-sponding to that eigenmode [25]. [POD is known by othernames, including Karhunen-Loeve decomposition, princi-pal components analysis (PCA), and singular value de-composition (SVD); the basis functions are also known asempirical eigenfunctions and empirical orthogonal func-tions [25], p.86.] Linear combinations of the basis func-tions ideally correspond to the coherent structures in theflow. However, if the ensemble of fields is homogeneous,the basis functions become Fourier modes [25]. Further-more, a single mode of the POD corresponds to a fullfield with structures in a particular spatial arrangement.If the structures in the field are not stationary, as inthe case of our present flow, many modes will thereforebe necessary to track the different spatial configurations.Indeed, application of POD to our data resulted in theextraction of the large scale, low amplitude mean flowstructure of our field, rather than the continually movingintense coherent structures. We therefore do not includePOD in this paper.

This paper separates a flow into coherent and incoher-ent components using wavelet transforms [26] to extractlocalized features at different spatial scales. The most im-portant advantage of the wavelet representation over themore usual Fourier representation is the localization of

the basis functions. A Fourier analysis is not well suitedto pick out localized features such as intense vortices.The basis functions of a Fourier transform are localizedin wavenumber space and hence spread out over the en-tire domain in physical space. The basis functions ofthe wavelet transform consist of dilates and translates ofa “mother” wavelet, which contains multiple frequenciesand has compact support (non-zero values only inside afinite interval) in physical space. The basis functions arewell localized in both physical and wavenumber space;hence only a few coefficients suffice to describe localizedfeatures of a signal. The coefficients of the wavelet trans-form contain not only amplitude but also scale and po-sition of the basis elements. Thus the coefficients can beused to track the size and location of features that arewell correlated with the wavelet bases (e.g. [27, 28]).

The discrete wavelet transform (DWT) [26] has beenfound to be well suited to analyze intermittent signalsand systems containing localized features such as the in-tense vortices that occur in turbulence [29, 30, 31]. Fargehas extensively applied wavelets to the analysis and com-putation of turbulent flows [29, 30, 31, 32, 33, 34].

The discrete wavelet packet transform (DWPT) [35] isa generalization of the DWT; the possible wavelet packetbasis elements are a larger set which include spatial mod-ulation of the mother wavelet. The advantage of theDWPT is that the choice of basis is adaptable to thesignal to be analyzed.

Turbulent flow can be considered as a superpositionof large-scale coherent motions, “fine-scale” incoherentturbulence, and a mean flow with interaction betweenthe three constituents [36]. In numerical simulations of2D turbulence [27, 30, 31], and more recently 3D turbu-lence [33], the coherent and incoherent turbulent back-ground components have been separated using waveletbased decompositions operating on the vorticity field.The coherent part, represented by only a small fraction ofthe coefficients, retained the total flow dynamics and sta-tistical properties, while the incoherent part representedno significant contribution to the flow properties. Thisseparation of the flow into two dynamically different com-ponents suggests that the computational complexity ofturbulent flows could be reduced in simulations with co-herent structures interacting with a statistically modelledincoherent background [31]. The application, however,has been heretofore primarily limited to results obtainedfrom numerical simulations.

In this paper, we use the wavelet technique to analyzeour Particle Image Velocimetry (PIV) data on a rapidlyrotating turbulent flow. Section II describes the exper-imental system. Section III describes the resulting flowfields obtained in the experiment. Section IV presentsthe techniques used to decompose the vorticity field intocoherent and incoherent components. Section V presentsthe results obtained by applying the method to measure-ments on rotating turbulent flow. The conclusions arediscussed in Section VI.

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II. EXPERIMENT

A. Instrumentation and measurements

An acrylic cylinder (48.4 cm tall, 39.4 cm inner diame-ter) is fit inside a square acrylic tank (40×40 cm cross sec-tion, 60 cm tall) that has a transparent lid (Fig. 1) [37].The tank is filled with distilled water at 24 ± 1C (ρ =0.998 g/cm3, ν = 9.5 ×10−3 cm2/s). The tank and thedata acquisition computer are mounted on a table thatcan be rotated up to 1.0 Hz.

FIG. 1: (a) A schematic of the apparatus. The height ofthe laser sheet is adjustable. Shown in the inset is a close-upa single tee. (b) Horizontal cross section of the distributionof pumping sources (open circles) and sinks (black dots) (po-sitions are to scale; see text for relative size of sources andsinks and tank diameter scale). (c) Overhead image of tankbottom, showing the tees. Overlayed are lines indicating theorientation of the tees.

Fluid is injected at the bottom of the tank by pumpingwater through a hexagonal array of tubes with grid spac-ing 2.3 cm; there are 192 sources and 61 sinks, as shownin Fig. 1. The source tubes are tee-shaped with a 2.5cm shaft and 2.1 cm horizontal top (0.079 cm inside di-ameter) [see inset of Fig. 1(a)]. Each tee is screwed intothe distributor. This design allows us to easily changethe type of connector or change the forcing geometry byblocking or modifying sources or sinks. The tee sourcegeometry was chosen to produce horizontal velocities and

to minimize the vertical motions directly driven by theforcing. Further, the orientation of the tees was chosento minimize cooperation between neighboring tees; how-ever, it was not possible to eliminate completely a meanflow. The sinks are 1.27 cm diameter holes in the bot-tom of the distributor [Fig. 1(b)]. The forcing systemis versatile and allows us to inject nearly homogeneousturbulence at a scale much smaller than the system size.

Flow rates can range up to 1400 cm3/s, correspond-ing to flow velocities up to 8 m/s out of the tees. Thiscorresponds to a Reynolds number of order 105 based onthe jet velocity and the grid spacing of the tees. TheRossby number, U/2ΩL, based upon the same scales isorder 50 near the bottom of the tank, near the forcing.However, the turbulence decays away from the forcing,with increasing height, and becomes more 2D due to theinfluence of rotation.

The data presented are for 0.4 Hz rotation rate andflow rates of 426 cm3/s, producing 2.4 m/s velocity jetsat the forcing tees. This corresponds to Reynolds andRossby numbers of about 6 × 104 and 20 respectively,near the forcing. Near the top the RMS flow velocityis about 2 cm/s and the maximal characteristic velocitylength scale (twice the e-folding length of the velocityspatial correlation function) is about 5 cm; hence nearthe top the Reynolds number is about 1000. The Rossbynumber calculated from characteristic scales is 0.08, whilethe local averaged Rossby number calculated from theRMS vorticity, (ωRMS/2Ω), at such a flow condition is0.3. The depth of our system thus allows us to observemore 3D (Rossby ≈ 20) or more 2D (Rossby ≈ 0.3) flowwithout changing control parameters, such as rotation orforcing.

The water is seeded with polystyrene (ρ = 1.067g/cm3) spherical particles with diameters in the range90-106 µm. The small mismatch in density results ina sedimentation terminal velocity of 4 × 10−2 cm/s inthe absence of flow, which is insignificant compared tothe measured flow velocities. The density mismatch alsocauses a lag in the response of the particles in regions oflarge acceleration [38, 39]. We estimate the largest lag tobe a 1% difference between the particle velocity and theflow velocity in the vortices at the top of the tank.

A 395 mW, 673 nm diode laser with attached light-sheet optics (from Lasiris) illuminates the particles inhorizontal planes for flow visualization and PIV measure-ments. It is also possible to rotate the laser and optics by90 to illuminate vertical planes for measurements of thevertical velocities. The thickness of the sheet is about 1cm and varies less than 10% across the diameter of thetank. The laser is fixed to a carriage which allows usto adjust the vertical position of the light-sheet withoutchanging the pumping rate or system rotation rate.

Particles are imaged in the rotating frame with aCCD camera (1004× 1004 pixels, 30 frames per second)mounted above the tank. The images are grabbed andstored into a memory buffer by a computer on the ro-tating table. A second, lower resolution camera with an

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analog output allows us to view the flow when not grab-bing digital images.The laser pulses were timed so that pairs of pulses were

imaged in successive frames of the CCD camera, whichhad a dead time of 120 µs between consecutive frames.The pulse duration was typically 10 ms with a 2 ms darkinterval between pulses.The analog signal from the second camera and power

for the equipment on the table are sent through slip rings.The entire experiment is controlled remotely with a wire-less ethernet connection and remote access software (PCAnywhere) by a computer in the non-rotating laboratoryframe. This allows us to take multiple data sets, adjustthe pumping rate, and adjust the position of the lightsheet without having to stop the table.To determine the two-dimensional projection of the

velocity field we use a type of Digital Particle ImageVelocimetry (DPIV) known as Correlation Image Ve-locimetry (CIV) [40, 41]. The CIV algorithm uses cross-correlations over small interrogation regions between apair of consecutive images to find particle displacementsover known time intervals to obtain velocities. The CIValgorithm also allows for possible rotations and defor-mations of the interrogation region to take into accountnonlinear particle motions in between images. The re-sulting velocity field is interpolated to a uniform regulargrid by a cubic spline interpolation. The vorticity fieldsare calculated from the coefficients of the spline fit.We can achieve a maximum data rate of 15 velocity

fields per second (using two images to calculate each ve-locity field). This time resolution is sufficient for ourflow conditions. To capture the dynamics of the flow,data were taken for 20 s periods (limited by memorybuffer size), resulting in 300 fields in a single sequence.To capture statistical information, data were taken at 30s intervals (longer than measured decay time for our flow)over (in most cases) runs of 50 minutes in duration. Thiswas sufficient to achieve convergence of statistical quan-tities. The resulting velocity and vorticity fields are on a128×128 grid with a spatial resolution of 0.30 cm whenthe whole tank is imaged and 0.19 cm when we zoom-in.Testing the CIV algorithm against test particle imagesin known flow fields [42, 43], we found that the velocityfields determined by the algorithm have about 2% RMSuncertainty.

B. Decay time

Characteristic decay times for our flow were measuredby three different types of experiments. In each experi-ment, the measured decay time was taken as the e-foldingrelaxation time of the mean kinetic energy in the flow.(i) Laminar (no forcing) spin-down experiments were

conducted with the tees removed and replaced with aflat horizontal boundary. The tank was subjected to asudden 10% decrease in rotation rate after the flow hadreached solid-body rotation (no motion in the rotating

tank frame). The predicted decay time of motions inlaminar rotating flows with rigid, flat horizontal bound-aries subject to small step changes in rotation is given bythe Ekman dissipation time, τ = H/(2

√νΩ), where H is

the depth of the fluid, ν is the kinematic viscosity andΩ = 2πf is the angular frequency of the container [44].The corresponding decay time for the energy in the flowis τ = H/(4

√νΩ). For our closed cylindrical tank with-

out topography of depth H = 48.4 cm, ν = 0.095 cm2/s,and a rotation rate of 0.4 Hz, this gives a decay time ofthe energy as 78 s; the measured time was 64 s.(ii) Laminar spin-down experiments were conducted in

the same way as in (i), except that the tees were installedin the bottom of the tank. The characteristic decay timemeasured with the tees installed in the bottom of thetank for our flow was 18±3 s. The reduced laminar spindown time is due to the tees, which cause extra dragand secondary circulations that quickly bring the fluidto solid-body rotation.(iii) Turbulent decay experiments were conducted by

abruptly shutting off the forcing. For these experiments,the flow was allowed to reach a steady turbulent state(typically many decay times) under a constant pump-ing rate of 426 cm3/s and rotation rate of 0.4 Hz beforeabruptly turning off the forcing. The measured decaytime of 13±3 s is long compared to the typical vortexturnover time of 1 s.

C. Passive scalar advection

To study the transport and mixing properties of ourflow we examine numerically the motion of passive scalarpoint particles and passive scalar fields in the velocityfields obtained from the experiment. The velocity fieldsthat we measure are a two-dimensional projection of athree-dimensional incompressible flow field. Therefore,they have a non-zero divergence and do not satisfy anyfluid dynamical equation of motion. Nonetheless, theycan give us useful information regarding the transportproperties of the flow.Initial locations are chosen for the point particles, the

positions of the particles are updated corresponding tothe velocity fields such that xn+1 = un∆t+xn where xn

is the position of the particle at time-step n, and un is thevelocity of the flow field at time-step n at the location ofthe particle. Each velocity field is interpolated in spaceby a cubic-spline to calculate the field at the locationof a given particle. The timestep, ∆t, is chosen by theCourant condition [45], which avoids particles jumpingover grid points or going too far at a given iteration. Forour data this condition means ∆t < 0.02 s < 1/15 s;therefore, fields must also be interpolated in time. Themeasured velocity fields vary slowly in time comparedto our temporal resolution 1/15 s (measured correlationtime is ≈ 2.4 s; see section VA). We therefore justify theuse of a cubic spline interpolation in time to achieve ∆t≪ 1/15 s (typically ∆t = 0.001 s).

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We also examine the time evolution of a passive scalarfield advected in the experimental velocity field by nu-merically integrating the advection-diffusion equation,

∂c

∂t= −u · ∇c+ κD∇2c, (1)

where c = c(x, t) is the passive scalar concentrationand κD is a diffusion coefficient, chosen as necessary fornumerical stability of the solution. The values of κDused correspond to Schmidt numbers (κD/ν) near 0.05.The numerical integration is performed using a pseudo-spectral method in polar coordinates based upon meth-ods given in [46]. The grid is 128 Chebyshev modes in theradial direction and 256 Fourier modes in the azimuthaldirection. The velocity field obtained by the CIV mea-surement is interpolated via a cubic spline onto the simu-lation grid. The numbers of radial and azimuthal modeswere chosen so as to not under-resolve our velocity fieldson the nonuniform polar grid.The singularity at the origin was avoided by choosing

Chebyshev modes for the radial direction. The colloca-tion points for Chebyshev modes have unit spacing ona circle, xi = cos (πi/(N − 1)), where xi is the ith grid-point, and N the total number of gridpoints. The grid-points are clustered at the boundaries and sparse in thecenter of the domain. By defining the radial coordinater ∈ [−1, 1] and using only the r > 0 values, the radialgrid points are sparse near r = 0 and clustered at theboundary r = R. For an even number of modes the ori-gin, r = 0, is skipped. However, the azimuthal grid isstill dense at the origin, which requires us to use a veryshort time step ∆t ≈ 2×10−4 s to avoid numerical insta-bility. The experimentally determined velocity fields arethen interpolated in time by the method described abovefor the tracer particle simulation.The diffusive term is calculated implicitly by a Crank-

Nicholson scheme, separately for the radial and az-imuthal directions. The advection term is calculated ex-plicitly by a predictor-corrector scheme using a third-order Adams-Bashforth step followed by a fourth-orderAdams-Moulton step [45].To perform the numerical analysis on the decomposed

fields we construct a velocity field from a vorticity field.We make the assumption of 2D flow that the total vor-ticity is given by the measured vertical vorticity, ω = ωz.We then use the 2D streamfunction-vorticity relation∇2Ψ = −ωz and solve Poisson’s equation for the stream-function Ψ using Matlab’s PDE solver. The derivativesof the streamfunction are then used to calculate the ve-locity field by the relation ∇× (Ψz) = u, where z is theunit vector in the vertical direction.The assumption above is clearly not valid close to the

source tees, where the flow is 3D. However, close to thetop of the tank the assumption becomes valid as the flowis quasi-2D. To test the reconstruction and the validityof the 2D approximation, we compare the original mea-sured velocity fields to velocity fields reconstructed from

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FIG. 2: (color on-line) Close-up of a vortex in our tank nearthe boundary (bold line). The original field is representedby black vectors and the reconstructed field by light-coloredvectors. The original and reconstructed fields are indistin-guishable in the regions of uniform flow. The largest vectorscorrespond to 6.7 cm/s. The center of the tank is at (x,y) =(19.2 cm, 18.9 cm).

the vorticity fields. A region of an original velocity fieldand the reconstructed field is shown in Fig. 2. The RMSdifference in the magnitudes of the original and recon-structed fields is about 2%. The reconstruction calcula-tion does well in regions where there is a strong uniformflow. It does less well where there are large gradients inthe velocity, in particular near vortices. However, on thewhole the reconstructed velocity field follows the samebehavior as the original field.

III. VELOCITY AND VORTICITY FIELDS

A. Transition to quasi-2D flow

Near the forcing sources (tees) at the bottom of thetank, the flow is very turbulent (Reynolds number ≈6 × 104) and three dimensional (Rossby number ≫ 1).However, moving vertically away and up the tank fromthe forcing, the turbulent velocities decay. The Reynoldsnumber near the top of the tank where our data werecollected is order 1000, based upon the RMS velocitiesof 2 cm/s and typical structure size of 5 cm (estimatedfrom the velocity correlation function). The relative in-fluence of rotation also becomes larger so that the Rossbynumber becomes about 0.3.Fig. 3 compares the magnitude of the divergence and

vertical vorticity of the flow in our tank at two different

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FIG. 3: (color on-line) The PDF of the magnitude of verticalvorticity and divergence at two different heights in our tank,4 cm below the lid (solid curves) and 10.4 cm above the topof the tees (dashed curves). Inset: the ratio of the divergencePDF to the vorticity PDF, showing the increase in the relativemagnitude of the divergence near the forcing.

heights. The divergence in our flow fields is small rela-tive to the vorticity and 2Ω (≈ 5 rad/s). The divergencefield consists of small length-scales near the top and hasa weak correlation with the vorticity field. The ratio ofthe RMS divergence to the RMS vorticity is 0.2, of thesame order as the Rossby number. Near the bottom forc-ing, however, the divergence field becomes larger in am-plitude, length-scale, and more strongly correlates withstructure in the vorticity field. The inset in Fig. 3 showsthe ratio of the divergence to vorticity increases near theforcing, where the flow is more 3D and decreases near thetop, where the flow is more 2D.The particles in a plane near the top of the tank are

confined primarily to the plane, and the particle streaksfollow persistent coherent jets, cyclones (a vortex withrotation in the same sense as the tank), and anti-cycloneswithout crossing, similar to a 2D streamline flow [Fig. 4(a) and (b)]. In contrast, the motions of the particles nearthe bottom of the tank are much less organized; particlespass through the plane frequently and the particle streakscan cross and exhibit small fluctuations due to increasedthree-dimensionality and turbulence.

B. Persistent coherent structures

Localized coherent structures including large compactregions of intense vorticity and wispy filaments of intensevorticity are evident in Fig. 4 (c) and (d). We observemore cyclones (rotation in the same sense as the tank)than anti-cyclones, in accord with observations in previ-ous experiments on rotating turbulent flows [12, 13]. Theamplitudes of vorticity at the core of the intense cyclonesare typically more than ten times that of the surrounding

flow [see Fig. 4 (d) for a close-up of an intense cyclone].The anti-cyclones, however, have a typical core vorticityamplitude only a few times that of the background.The cyclones, anti-cyclones, and vorticity filaments are

long-lived and are active dynamically. The vortices andvortex filaments can be tracked by eye as the flow evolvesin time [Fig. 4 (e)]. Vortices are shed from the wall andtravel across the tank and interact with one another. Vor-tex filaments occasionally peel off of vortices, advect withthe flow, and sometimes roll up to form new vortices.Structures may disappear by merging with other struc-tures or by stretching in a jet region between oppositesign vortices. Many of the small intense vortices withvery short turn around times (< 1 s) live on the orderof a characteristic decay time (∼ 10 s) before shearingapart or merging with neighboring structures.Additionally, we see persistent structures such as the

large cyclone (red) and anti-cyclone (blue) that appearin Fig. 4. Such structures often appear in a preferredlocation in the tank. Presumably such preferred loca-tions exist because of inhomogeneities in the forcing bythe tees [cf. Fig. 1 (c)]. A vortex can be kicked off itspreferred location by a large perturbation when it inter-acts with neighboring vortices. If it begins to wanderaround the tank, it will generally disappear within a fewcharacteristic decay times unless it returns to the pre-ferred location. If a structure moves off its preferredlocation and disappears, a new structure will typicallyform, replacing the pre-existing one within a few decaytimes. Some long-lived coherent structures occasionallypersist throughout the entire duration of an experimen-tal run (∼ 5000 vortex turnover times). Further, we ob-serve persistent structures as low as 5 cm above the tees.Observations closer to the tees are difficult because thethree-dimensional turbulent flow rapidly moves particlesinto and out of the laser sheet.A wide range of spatial scales is visible in the veloc-

ity and vorticity fields [Fig. 4 (c) and (d)]. The largestfeatures, approximately 10 cm in size, are coherent vor-tices that have a much larger amplitude than their sur-rounding region. There are also large amplitude vortexfilaments that stretch up to 10 cm; these can be as thinas the grid resolution in the transverse direction. Smallscale persistent structures less than 1 cm in size yet largein amplitude are also observed. The region between thevarious large amplitude coherent structures is occupiedby relatively low amplitude vorticity [see the vorticitytrace in Fig. 4 (c)].

IV. DECOMPOSITION OF THE VORTICITY

FIELD

A. Wavelets and wavelet packets

The discrete wavelet transform (DWT) is a multi-resolution analysis that successively decomposes the sig-nal into coefficients which encode coarse and fine details

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FIG. 4: (color on-line) (a) [MOVIE [47]] Streak photo (400 ms exposure time) of the particle image fields in a horizontal plane4 cm below the lid of the tank. (b) Close-up of the boxed region (18.7 cm × 18.7 cm) showing a cyclone (circular closed particlestreaks) and an anti-cyclone (elliptical closed particle streaks to the upper-left of the cyclone). (c) Vorticity field ωz with thevalue along the dashed line shown in the trace below the field. The lines indicating the flow structure would be streamlines ifthe velocity field were divergence free, which it is not. Arrows indicate flow direction. (d) Close-up of velocity and vorticityfields in a 8 cm × 8 cm region near the boundary. The longest velocity vector corresponds to 6.7 cm/s. The vorticity isindicated by the color map, where vorticity values are clipped at 5 s−1 to render visible weaker structures; the vorticity in thecore of the strongest cyclone is 22 s−1. (e) [MOVIE [48]] A sequence showing the time evolution of the vorticity field.

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at successively lower resolution [35]. The basis elementsof the transform ψs,p correspond to dilations and trans-lations of a mother wavelet function ψ, where s is thescale (dilations) of the wavelet and p its position (trans-lations). Successive levels of the transform continue tosplit the coarse detail coefficients, effectively analyzingthe signal at coarser and coarser resolution.The discrete wavelet packet transform (DWPT) is a

generalization of the discrete wavelet transform. The ba-sis elements of the wavelet packet transform include, inaddition to dilations and translations, spatial modulationof the mother wavelet at different resolutions. The basiselements ψs,p,k take on an additional parameter k, whichroughly corresponds to the modulation of the waveletpacket. In contrast to the DWT, the choice of basis ofthe DWPT is not unique [26]. The wavelet basis is con-tained within the possible choices of wavelet packet bases.To select the particular basis to use, a natural choice isthe wavelet packet basis into which the coefficients of thetransform most efficiently represent the signal. This isknown as the “best basis”. The best basis is typicallycalculated based upon the minimization of an effectiveentropy measure of the coefficients [35], thus minimizingthe “information cost” of the coefficients in the best ba-sis. The flexibility in basis choice of the DWPT allows thetransform to adapt the basis to the particular signal be-ing analyzed. For example, if the signal contains regionsof rapid fluctuations, the basis choice will reflect that byincluding more basis elements with high modulation.We use the DWT [O(N) operations] and DWPT

[O(N log2N) operations] on our experimentally obtainedvorticity fields, as previous authors have done using 2Dturbulent flow data from numerical simulations in a peri-odic square domain (e.g. [27, 30, 31]). We use the Matlabwavelet toolbox and the coiflet 12 (coif2 in Matlab no-tation) as the analyzing wavelet. The coiflet family ofwavelets (see reference [26], p. 258) has both compactsupport and can generate an orthogonal basis. Thesetwo properties allow one to select the localized featuresof the coherent structures and to treat the decompositionof the vorticity field as two orthogonal components. Thechoice of wavelet does not alter the results significantlyas long as the basis is sufficiently smooth [31].

B. Coherent structure extraction

We use an algorithm for coherent structure extractionbased upon a de-noising algorithm (e.g. chapter 11 ofreference [35]). In the de-noising, the assumption is thatthe original signal can be represented by a few large am-plitude coefficients of an orthogonal transform using anappropriate set of basis functions, while the noise is con-tained in the many remaining coefficients of small am-plitude. The de-noising is then performed by applyinga threshold to the resulting coefficients of the transform.Coefficients above the threshold amplitude are assumedto correspond to the signal while coefficients below the

threshold correspond to the “noise”. The few large am-plitude coefficients are called the “coherent” coefficients,while the many small amplitude coefficients are “incoher-ent”. The coherent and incoherent parts of the vorticityare then reconstructed by the inverse transform.The choice of the threshold separating the coherent

and incoherent parts of the signal is based upon a mea-sure of the number of significant coefficients No, which isthe theoretical dimension of the signal, defined by

No(f) = eH(f) (2)

where H(f) is the entropy of a discrete signal f = [fi](where [fi] is the set of discrete values of an arbitrarysignal f). In our case, the fi’s correspond to the dis-crete values of our vertical vorticity measurements. Theentropy H(f) is defined as

H(f) = −N∑

i=1

pi log pi (3)

where pi = |fi|2/||f ||2 is the normalized square modulusof the ith element of the signal, with N the number ofelements and ||f ||2 =

∑i |fi|2. No indicates how many

of the largest coefficients should be retained to give anefficient, low entropy, representation of the signal.Thus, the decomposition algorithm is the following.

We take the transform of an individual measured vortic-ity field. For the DWPT, the best basis is used. Thenwe find the number of significant coefficients No of thetransformed vorticity field in the transform basis. Thethreshold is therefore based upon the value of the N th

o

largest coefficient. Coefficients whose modulus is larger(smaller) than the threshold correspond to the coherent(incoherent) part of the vorticity field. We then take theinverse transform to get the coherent part in physicalspace. The incoherent field is reconstructed by subtract-ing the coherent from the total field. The process is thenrepeated for each of our vorticity fields. Our algorithmcontains no adjustable parameters other than the selectedwavelet and is based upon the assumption that the flowfield has a low entropy component, corresponding to thecoherent structures, and a high entropy component, cor-responding to an incoherent background.Our algorithm is different than the various iterative

algorithms of Farge et al. [31] and others [27, 35]. Inthe de-noising algorithm described in [27, 35], successiveiterations extract a fraction of the coherent coefficientsuntil a stopping criterion is met. The fraction of retainedcoefficients at each step and stopping criterion are ad-justable parameters. However, the basis of the DWPTcan change at each iteration. The number of coefficientsretained therefore loses its meaning since coefficients canbe selected from completely different bases. Therefore,we only perform a single iteration without changing thebasis for the DWPT. In the algorithm of Farge [31], an

9

a priori assumption is made of Gaussian white noise in-coherent component superimposed upon a non-Gaussiancomponent of coherent vortices. The threshold is theniteratively found to separate the two without adjustableparameters.

C. FFT, JPEG compression, and Okubo-Weiss

based techniques

To examine the relative efficiency of the wavelet-baseddecompositions, we compare the DWPT and DWT tothree other algorithms; one based upon a Fourier trans-form, a second on basic JPEG compression, and a thirdon the method developed by Okubo [22] and Weiss [23].For the Fourier transform algorithm we threshold the

coefficients of a 2-dimensional fast Fourier transform(FFT) of our vorticity. The coherent fields are then con-structed via an inverse FFT of the largest amplitude coef-ficients. The incoherent remainder fields are constructedby subtracting the coherent fields from the original fields.The basic JPEG algorithm consists of subdividing an

image into 8 × 8 blocks and taking the discrete cosinetransform over the sub-blocks. The coefficients are ar-ranged based upon the global average of the amplitudesamong the sub-blocks. The average of the amplitudes isused to determine which modes to keep. After the thresh-old, each sub-block retains the same number of modes,each in the same position. Thus the number of modesretained must be a multiple of the number of sub-blockswhich compose the image (for our 128 × 128 fields, thisis 256). For more information on the JPEG image com-pression standard see reference [49], chapter 3.The Okubo-Weiss criterion splits the fields into ellip-

tic and hyperbolic regions, dominated by vorticity andstrain respectively. The regions dominated by vorticityand strain are then separated and taken to be the coher-ent and incoherent fields. For details on the Okubo-Weisscriterion see reference [24].

V. RESULTS

A. Decomposed vorticity fields

The resulting coherent and incoherent vorticity fieldsobtained from the wavelet packet, wavelet, Fourier, andJPEG techniques are compared in Fig. 5. The Okubo-Weiss criterion was also tested for comparison with theother decomposition methods. The fields shown are con-structed with the same number of coefficients for theDWPT, DWT, and Fourier transforms, and the closestapproximate number for JPEG.The coherent fields in Fig. 5 appear to retain the large

scale coherent structures [cf. Fig. 4(c) and (e)]. TheDWPT, DWT, and Fourier preserve the structure of thetotal vorticity field. The Okubo-Weiss criterion excisesprimarily the regions of large amplitude vorticity, which

generally corresponds to the cores of vortices. However, itextracts only the cores and leaves behind the peripheries.The JPEG does poorly with so few coefficients; it barelypicks out some of the stronger vortices in the field.

For the same number of coefficients, 2.4%, the DWPTand DWT do a better job of extracting structure inthe fields than the Fourier decomposition, which leavesmore structure behind in the incoherent field (Fig. 5[MOVIE [48]]). The JPEG incoherent field is large ampli-tude because of the few coefficients retained in the coher-ent field. The incoherent fields resulting from the DWPT,DWT, and the Fourier are mostly devoid of large-scalestructures and are much smaller in amplitude. The in-coherent fields are also poorly correlated in time and aredevoid of large scale structures or any other feature whichcan be tracked by eye (Fig. 5 and associated movie).

The Okubo-Weiss criterion selects the centers of thevortices where vorticity dominates strain. This accountsfor a large portion of the enstrophy. It extracts about74% of the enstrophy in the vorticity dominated regions,which cover about 40% (not coefficients) of the flow.However, the method leaves behind holes in the vortic-ity field that act as coherent structures and are oftensurrounded by large values of vorticity. The resulting“incoherent” field thus retains coherent properties. TheOkubo-Weiss criterion also does poorly in recovering thestatistics of the total field, as shown in Table I. Fur-thermore, the decomposition shown in Fig. 5 was per-formed without regard to the Okubo-Weiss validity cri-terion, which restricts the application of the criterion tothe very centers of the vortices or regions of very largevorticity [24]. If the validity criterion is applied, muchless is retained in the coherent field, and the incoherentfield ends up with large values of vorticity.

Statistical properties of the decompositions are shownin Tables I and II. Table I lists the results from settingthe threshold of the various transforms based upon thenumber of significant coefficients calculated from equa-tion (2). On average about 2-3% of the large-amplitudecoefficients of the wavelet-based decompositions were re-tained in the coherent fields, which account for about85% of the total enstrophy of the flow. The Fourier re-tains roughly the same enstrophy but requires more co-efficients, about 4%. The JPEG decomposition retainsonly 67% of the total enstrophy with about 13% of thelarge-amplitude coefficients.

Owing to the compact support of their basis functions,fewer coefficients were needed for the DWPT and DWT,based upon the entropy criterion in equation (3), thanfor the Fourier and JPEG decompositions. Compressioncurves for enstrophy of the various decompositions ap-plied to our vorticity fields are shown in Fig. 6. TheDWPT and DWT both do similarly well at extractingthe enstrophy in the fields with only a small number ofcoefficients. The DWPT probably does slightly betterdue to the adaptability of its basis. The Fourier basedmethods do not converge as rapidly as the wavelet-basedmethods. This is likely due to the non-locality of the ba-

10

DWPT DWT Fourier JPEG Okubo−WeissC

oher

ent

Inco

here

nt

FIG. 5: (color on-line) Decomposed vorticity fields, TOP: Coherent BOTTOM: Incoherent remainder fields, all for the samenumber of coefficients retained (except for Okubo-Weiss and approximate for JPEG). The colormap corresponds to the intensityof vorticity and is the same for all images and the same as used in Fig. 4. [MOVIE [50]]

Decomposition Wavelet packet Wavelet Fourier JPEG Okubo-WeissQuantity Total coherent-incoherent coherent-incoherent coherent-incoherent coherent-incoherent coherent-incoherent

coefficients retained (%) 100 2.4 97.6 2.7 97.3 4.1 95.9 13.1 86.9 40.2 59.8

Enstrophy/field (s−2× 104) 4.33 3.63 0.63 3.67 0.62 3.68 0.56 2.92 1.33 3.21 1.12

Vorticity skewness 0.78 0.80 0.02 0.85 0.04 0.62 0.20 0.53 0.24 1.22 -0.07

Vorticity kurtosis 7.09 7.03 3.88 7.58 3.42 5.37 7.85 5.27 6.55 11.81 8.28

TABLE I: Statistical properties of the decomposed vorticity fields using entropy criterion.

sis functions when applied to a field which has localizedstructures and sharp features.

The wavelet-based algorithms also do a better job pre-serving the skewness and kurtosis of the total vorticityin the coherent component, while the incoherent compo-nents are much closer to a Gaussian distribution (skew-ness = 0, kurtosis = 3). Table II displays the resultsfrom setting the threshold on the transforms so that eachmethod retains 2.4% of the coefficients in the coherentcomponent (since the coefficients JPEG must be a multi-ple of the number of 8×8 blocks, 256, this restriction is re-laxed). For the same number of retained coefficients, theDWT and DWPT clearly outperform the Fourier methodin terms of preserving the statistics of the total vorticityfield.

Figure 7 shows that the coherent fields from thewavelet packet and wavelet decompositions rapidly con-verge to the statistics of the total vorticity field. The co-herent fields preserve the non-Gaussianity of the vorticityPDF without having to extract many coefficients. Therespective incoherent fields also rapidly converge to Gaus-sian statistics. This suggests that wavelets and waveletpackets do a good job of extracting the coherent struc-tures from the vorticity field. The near-Gaussianity ofthe incoherent field suggest that the transforms have left

the remainder without coherent structure.The coherent components of the Fourier and JPEG de-

compositions, however, converge to the statistics of thetotal fields much more slowly. To obtain the same level offidelity, both the Fourier and JPEG must extract manymore coefficients. This inefficiency would result in over-transformed fields. Further, the insets in Fig. 7 (b) showthat the kurtosis of the incoherent component never con-verges to anything small, but increasingly deviates fromGaussian. This suggests that the Fourier decompositionis not separating the coherent structures from the back-ground.The probability distribution function (PDF) of vortic-

ity has broad wings (Fig. 8), which correspond to thelarge vorticity values that occur in the coherent vorticesand vorticity filaments. The preference for cyclonic (posi-tive vorticity) structures over anti-cyclonic structures ap-pears as a large positive skewness (see also Table I or II).While the PDF for the DWPT and DWT coherent vortic-ity field are nearly the same as that for the total vorticityfield, the PDF for the corresponding incoherent field isnarrower and more symmetric, indicating the lack of highintensity structures.Consider the enstrophy spectrum of the vorticity fields,

as shown in Fig. 8 (b). It is clear from the enstrophy

11

Decomposition Wavelet packet Wavelet Fourier JPEG

Quantity Total coherent-incoherent coherent-incoherent coherent-incoherent coherent-incoherent

coefficients retained (%) 100 2.4 97.6 2.4 97.6 2.4 97.6 3.0 97.0

Enstrophy/field (s−2× 104) 4.33 3.63 0.63 3.60 0.69 3.28 0.92 0.96 3.18

Vorticity skewness 0.78 0.80 0.02 0.86 0.04 0.51 0.32 0.06 0.72

Vorticity kurtosis 7.09 7.03 3.88 7.62 3.47 4.77 7.85 4.84 7.08

TABLE II: Statistical properties of the decomposed vorticity fields retaining the same number of modes, 2.4%, as the DWPTdecomposition (approximate for JPEG).

0 0.1 0.20

0.2

0.4

0.6

0.8

1.0

Fraction of coefficients retained in coherent component

Fra

ctio

n of

ens

trop

hy r

etai

ned

DWPT

DWT

Fourier

JPEG

FIG. 6: (color on-line) The percent enstrophy retained in thelargest amplitude coefficients as a function of the number ofcoefficients kept. In order from best (most efficient) to worst:wavelet packet, wavelet, Fourier, JPEG.

spectra of the total field that most of the enstrophy iscontained in long wavelengths (small k). This agrees wellwith the observation of the vorticity fields and the dom-inance of large structures with large amplitude vortic-ity. The enstrophy spectrum, shown in Fig. 8, does notcontain a well defined scaling region corresponding to acascade of enstrophy. A cascade would not be expectedfor our quasi-2D flow, which is forced by the broad-band3D turbulence in the bottom of the tank rather than byinjection of energy at a single well-defined wavenumber.The presence of large structures which extend well intothe 3D region alone tells us that we must have a broadspectrum energy injection.

The coherent field contains the same large structuresthat are present in the total field. Accordingly, the en-strophy spectrum of the coherent field matches that ofthe total field at long wavelength [Fig. 8 (b)]. The inco-herent field contains negligible enstrophy at large scales,indicating the lack of large scale features. This is ap-parent in the vorticity decomposition (see Fig. 5). Theincoherent field has only small structures and is the dom-inant contribution to the enstrophy at large k.

The wavelet-based and the Fourier decompositions alsoretain the spatial and temporal correlations in the coher-

ent field, as Fig. 9 illustrates with the DWPT and DWTyielding almost the same results. The long time correla-tion is in part due to the presence of long lived coherentstructures. In contrast, for the incoherent field the spa-tial and temporal correlations are short ranged, indicat-ing the absence of large scale and long-lived structures.

B. Transport of passive scalar particles and

concentration fields

Paths computed for some passive scalar “particles” inthe measured velocity fields are shown in Fig. 10 (a).Over the 20 s duration of the measurement, the particlesare advected around a large portion of the tank. Themotion of the particles depends on their location in theflow. For example, a particle can spend time caught in avortex, following the motion of the vortex as it meandersslowly in a localized region of the tank [path in upperright of Fig. 10 (a)]. Occasionally the particle will escapefrom a vortex and then be carried by a high velocity jet,which can transport a particle large distances in a shorttime. A particle may then be captured by a vortex [e.g.path ending in lower right of Fig. 10 (a)].

The resulting mean squared displacement of the tracerparticles in the velocity fields obtained by experiment isshown in Fig. 10 (b). It exhibits approximately t2 scalingat short times, which looks “flight-like” [21]. At longtimes the finite size of the tank becomes significant andthe scaling exponent becomes smaller.

Both the wavelet and wavelet packet coherent fieldsdisperse the tracer particles similarly around the tank;the scaling behavior of the mean squared displacement,not shown in this paper, is also the same. Differencesin the paths are due to the sensitive dependence uponinitial conditions of a turbulent flow field.

We find a striking difference in the behavior of a nu-merically integrated passive scalar particle in the coher-ent and incoherent fields. The incoherent fields makeno significant contribution to the transport properties;rather, particles are confined to a small region, as shownin the inset in Fig. 10 (a). This is due to the rapid decor-relation in time of the incoherent fields. The rapid fluctu-ations cause the tracers to jiggle around and the scalingof the mean squared squared displacement of the trac-ers in the incoherent fields approaches that of a randomwalk.

12

The evolution of the passive scalar field in the mea-sured velocity field is similar to that for tracer particles.Results for the coherent and incoherent fields producedby the DWPT are shown in Fig. 11, which illustratesthe stretching and folding; the results for the total fieldare similar to that of the coherent field. By four sec-onds the scalar has already been significantly stretchedby the velocity fields. There is no significant advectionin the incoherent fields and is similar to results obtainedwith pure diffusion (no advection term). This is expectedfrom the short time and space correlations of the inco-herent fields (Fig. 9). Similar results were obtained forthe DWT and Fourier methods. Our results agree wellwith those of Beta et al. [34] who, using the DWT, foundthat the coherent field is responsible for the mixing in anumerically simulated 2D turbulent flow.

VI. DISCUSSION

We find that flow which is strongly forced near the bot-tom of a deep rapidly rotating tank evolves with increas-ing height in the tank from 3D turbulence into rotation-dominated turbulence. This is due to the competitionbetween the turbulent forcing which is 3D, and the ro-tation which tends to two-dimensionalize the flow. Nearthe forcing the turbulent velocities are large and the flowis 3D. Away from the forcing the spatial decay of tur-bulence allows the rotation to become increasingly im-portant with increasing height. Motions of the particlesin the flow are more organized in the horizontal planenear the top, consistent with two-dimensional columnarvortices. The flow in the rotationally dominated (Rossbynumber ≈ 0.3) region of our tank has a strongly non-Gaussian vorticity PDF and exhibits properties of 2Dturbulence, such as large-scale long-lived coherent vor-tices, vortex filamentation, and merger.The vertical component of vorticity describes the flow

well near the top where the flow is quasi-2D as a con-sequence of the rotation. We are thus able to use theprojected flow field as measured in a horizontal plane foranalysis with wavelet and Fourier-based decompositions.We have shown that wavelet based transforms can beused to separate the coherent structure-containing quasi-2D turbulent flow into a non-Gaussian coherent compo-nent represented by as few as 3% of the large-amplitude,low entropy coefficients of the transform and a nearlyGaussian incoherent component, represented by the re-maining small-amplitude, high entropy coefficients. Weare able to obtain this result without an a priori assump-tion of Gaussianity or non-Gaussianity of the two com-ponents.The discrete wavelet packet transform (DWPT) and

the discrete wavelet transform (DWT) yield very similarresults, despite the adaptability of the DWPT basis. TheDWT is therefore made preferable by its faster compu-tation [O(N) versus O(N log2N) operations].

Because our flow fields contain compact structures, thelocalized basis functions of the DWPT and DWT out-perform Fourier and JPEG decompositions. Both theDWPT and DWT have more rapid convergence of thestatistics of the extracted coherent component towardthat of the total flow. The rapid convergence of the skew-ness and kurtosis of the vorticity PDF suggest that thewavelet based methods have efficiently captured the co-herent structures. The Fourier and JPEG converge muchmore slowly and are thus unable to efficiently capture thelarge higher moments of the vorticity PDF. This suggeststhat the Fourier and JPEG methods have not really ex-tracted the coherent structures despite the appealinglysmooth visual appearance of the structures in the coher-ent field of the Fourier. Indeed, the superior performanceof the DWT over the JPEG is exploited by the emergingnext generation JPEG2000 image compression standard,which uses a bi-orthogonal DWT [49, 52]. The incoher-ent remainder of both the DWPT and DWT convergerapidly towards Gaussian statistics, while incoherent re-mainders of the Fourier and JPEG do not converge toany value.

The coherent components of the DWPT and DWT re-tain all of the properties of the total field, including thelarge-scale structures, shape of the vorticity PDF, longspatial and temporal correlations, and transport proper-ties. Further, the coherent component contains the largeskewness and kurtosis of the PDF which are due to thecoherent structures. In contrast, the incoherent remain-der has only small-scale short-lived features and does notcontribute significantly to the transport. Thus in analy-sis of flow dynamics and transport, it may be sufficientonly to consider the coherent component. These resultssuggest that it is reasonable to reduce the computationalcomplexity of turbulent flows by considering only the low-dimensional coherent structures, which interact with astatistically modelled incoherent background.

Acknowledgments

We thank Bruno Forisser and Emilie Regul for assis-tance in the initial implementation of the Okubo-Weisscriterion and wavelet analysis respectively. We thankKenneth Ball, Carsten Beta, Ingrid Daubechies, MarieFarge, Philip Marcus, Kai Schneider, and Jeffrey Weissfor helpful discussions. This research was supported bythe Office of Naval Research.

[1] A. K. M. F. Hussain, J. Fluid Mech. 173, 303 (1986). [2] K. R. Sreenivasan, Rev. Mod. Phys. 71, S383 (1999), sp.

13

Iss. SI.[3] S. K. Robinson, Annu. Rev. Fluid Mech. 23, 601 (1991).[4] J. C. R. Hunt, Annu. Rev. Fluid Mech. 23, 1 (1991).[5] J. T. C. Liu, Annu. Rev. Fluid Mech. 21, 285 (1989).[6] J. Lumley and P. Blossy, Annu. Rev. Fluid Mech. 30,

311 (1998).[7] J. Sommeria, S. D. Meyers, and H. L. Swinney, Nature

331, 689 (1988).[8] J. C. McWilliams, J. Fluid Mech. 146, 21 (1984).[9] NOAA / NESDIS interactive GOES archive,

http://wist.ngdc.noaa.gov/wist/goes.html.[10] Visible Earth, http://visibleearth.nasa.gov/.[11] A. D. McEwan, Nature 260, 126 (1976).[12] A. C. de Verdiere, Geophys. Astrophys. Fl. 15, 213

(1980).[13] E. J. Hopfinger, F. K. Browand, and Y. Gagne, J. Fluid

Mech. 125, 505 (1982).[14] S. C. Dickinson and R. R. Long, J. Fluid Mech. 126, 315

(1983).[15] R. H. Kraichnan, Phys. Fluids 10, 1417 (1967).[16] L. M. Smith, J. R. Chasnov, and F. Waleffe, Phys. Rev.

Lett. 77, 2467 (1996).[17] L. M. Smith and F.Waleffe, Phys. Fluids 11, 1608 (1999).[18] F. S. Godeferd and L. Lollini, J. Fluid Mech. 393, 257

(1999).[19] J. Paret and P. Tabeling, Phys. Rev. Lett. 79, 4162

(1997).[20] A. Provenzale, Annu. Rev. Fluid Mech. 31, 55 (1999).[21] T. H. Solomon, E. R. Weeks, and H. L. Swinney, Phys.

Rev. Lett. 71, 3975 (1993).[22] A. Okubo, Deep Sea Research and Oceanographic Ab-

stracts 17, 445 (1970).[23] J. Weiss, Physica D 48, 273 (1991).[24] C. Basdevant and T. Philipovitch, Physica D 73, 17

(1994).[25] P. Holmes, J. L. Lumley, and G. Berkooz, Turbulence,

Coherent Structure, Dynamical Systems and Symmetry(Cambridge University Press, New York, 1996), chapter3.

[26] I. Daubechies, Ten Lectures on Wavelets (SIAM,Philadelphia, 1992).

[27] A. Siegel and J. B. Weiss, Phys. Fluids 9, 1988 (1997).[28] C. Schram and M. L. Riethmuller, Meas. Sci. and Tech-

nol. 12, 1413 (2001).[29] M. Farge, Annu. Rev. Fluid Mech. 24, 395 (1992).[30] M. Farge, E. Goirand, Y. Meyer, F. Pascal, and M. V.

Wickerhauser, Fluid Dyn. Res. 10, 229 (1992).[31] M. Farge, K. Schneider, and N. Kevlahan, Phys. Fluids

11, 2187 (1999).[32] M. Farge, Farge’s wavelets research webpage,

http://wavelets.ens.fr/.[33] M. Farge, K. Schneider, G. Pellegrino, A. A. Wray, and

R. S. Rogallo, Phys. Fluids 15, 2886 (2003).

[34] C. Beta, K. Schneider, M. Farge, and H. Bockhorn,Chem. Eng. Sci. 58, 1463 (2003).

[35] M. V. Wickerhauser, Adapted Wavelet Analysis fromTheory to Software (A K Peters, Ltd. Wellesley, MA,1994).

[36] A. K. M. F. Hussain, Phys. Fluids 26, 2816 (1983).[37] J. E. Ruppert-Felsot, Master’s thesis, The University of

Texas at Austin (2003).[38] M. Raffel, C. Willert, and J. Kompenhans, Particle

Image Velocimetry, A Practical Guide (Springer-VerlagBerlin Heidelberg, 1998).

[39] E. E. Michaelides, J. Fluid Eng. 119, 233 (1997).[40] A. M. Fincham and G. R. Spedding, Exp. Fluids 23, 449

(1997).[41] A. Fincham and G. Delerce, Exp. Fluids 29, S13 (2000).[42] K. Okamoto, S. Nishio, T. Saga and T. Kobayashi, Meas.

Sci. Technol. 11, 685 (2000), This work describes imagegeneration for PIV/CIV algorithm evaluation.

[43] Particle image velocimetry standard project,http://piv.vsj.or.jp/piv/.

[44] J. Pedlosky, Geophysical Fluid Dynamics, Second Edition(Springer-Verlag, New York, 1987).

[45] W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T.Vetterling, Numerical Recipes in C: The Art of Scien-tific Computing (Cambridge University Press, New York,1988).

[46] L. N. Trefethen, Spectral Methods in Matlab (SIAM,Philadelphia, 2000).

[47] particlestreaks.avi, see EPAPS Document No. [numberwill be inserted by publisher ] for a movie of tracer par-ticles in our flow. A direct link to this document maybe found in the online article’s HTML reference section.The document may also be reached via the EPAPS home-page (http://www.aip.org/pubservs/epaps.html) or fromftp.aip.org in the directory /epaps/. See the EPAPShomepage for more information.

[48] vorticity.avi, see EPAPS Document No. [number will beinserted by publisher ] for a movie of the time evolutionof the vorticity field.

[49] T. Acharya and P.-S. Tsai, JPEG 2000 Standard forImage Compression (Wiley-Interscience, Hoboken, NewJersey, 2005).

[50] decmpfields.avi, see EPAPS Document No. [number willbe inserted by publisher ] for a movie comparing the timeevolution of the coherent and incoherent fields producedby the various extraction methods.

[51] pscalar.avi, see EPAPS Document No. [number will beinserted by publisher ] for a movie of the evolution of anumerically integrated passive scalar in the coherent andincoherent field.

[52] http://jpeg.org/jpeg2000/index.html.

14

0 0.1 0.20

0.2

0.4

0.6

0.8

1.0

0.8

0.6

0.4

0.2

0(a)

Fraction of coefficients retained in coherent component

Con

verg

ence

of s

tatis

tic

Coherent

Kurtosis

Skewness

0 0.50

1

0

0 0.1 0.20

0.2

0.4

0.6

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0.8

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0

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Con

verg

ence

of s

tatis

tic

Incoherent(b)

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0 0.5 11

−10

−20

FIG. 7: (color on-line) Convergence of the skewness and kur-tosis of the vorticity PDFs for the various decompositions,plotted with quantities defined so that they approach unityas they converge. The vertical line indicates the fraction ofcoefficients retained in the wavelet packet decomposition. (a)Convergence of the coherent component. Plotted: kurtosisratio: |kcoh/ktotal|, skewness ratio: |scoh/stotal|. (b) Conver-gence of the incoherent component. Plotted: kurtosis ratio:2 − kincoh/3, skewness ratio: 1 − |sincoh/stotal|. Note thatunity on the graph represents the respective statistic for aGaussian distribution: skewness = 0, kurtosis = 3. The kur-tosis of the Fourier and JPEG incoherent field is large andoff the main plot. The insets show the convergence behaviorat large numbers of retained coefficients. The open circles onthe curves correspond to the fraction of coefficients retainedfor the corresponding techniques using the entropy criteria asa threshold.

15

−10 −5 0 5 10 15 2010

−5

10−4

10−3

10−2

10−1

100

ω (s−1)

P (ω)

Incoherent

Coherent

Total

(a)

−5 0 510

−4

10−2

100

Gaussian

100

101

10−2

10−1

100

k1

k (cm−1)

Z(k)

Total

Coherent

Incoherent

(b)

FIG. 8: (color on-line) (a) Vorticity PDFs of the total field,coherent field, and incoherent field, where the inset shows theincoherent PDFs plotted together with a Gaussian fit to thewavelet packet results. (b) Enstrophy spectra Z(k) of total,coherent, and incoherent fields, calculated on a square sub-section of our circular domain. The solid, dashed, and dash-dotted lines correspond to the wavelet packet, wavelet, andFourier decompositions respectively. The k1 line is the spec-tral slope for Gaussian white noise in 2D. The wavenumber isdefined as k = 2π/L.

16

0 1 2 3 4−0.2

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100

10−1

100

Displacement

FIG. 9: (color on-line) (a) Space and (b) time correlations of the decomposed vorticity fields. The solid, dashed, and dash-dottedcurves correspond to the DWPT, DWT, and Fourier decompositions.

17

(a)

20 2124

25

26

10−1

100

101

10−1

100

101

102

103

∆ t (s)

(b)

Mea

n S

quar

e D

ispl

acem

ent

(cm

2 )

t1.97 t1.32

Incoherent

Coherent

Total

FIG. 10: (color on-line) (a) Simulated paths of numericaltracer particles in the total field (solid curves) and in thecoherent (dashed curves) and incoherent field (red) for theDWPT (total time 20 s). The inset shows an expanded viewof the path in the incoherent field. (b) Mean squared displace-ments for the tracers in the total, coherent, and incoherentvelocity fields. The exponents for the total and coherent fieldsare 1.97 while for the incoherent remainder the exponent is1.32 over the range indicated. Both results are for the waveletpacket.

18

t = 0 s t = 4 s t = 8 sC

oher

ent

Inco

here

nt

FIG. 11: (color on-line) [MOVIE [51]] Advection of a passivescalar field in the velocity field of the DWPT, as computedfrom the advection-diffusion equation (1). By eight secondsthe scalar field has been significantly mixed by the coherentfield, while it appears only to have been diffused in the inco-herent field.