Disentangling inertial waves from eddy turbulence in a forced rotating turbulence experiment
Antoine Campagne, Basile Gallet, Frédéric Moisy, Pierre-Philippe Cortet
aa r X i v : . [ phy s i c s . f l u - dyn ] A p r Disentangling inertial waves from eddy turbulence in a forced rotating turbulenceexperiment
Antoine Campagne, Basile Gallet, Fr´ed´eric Moisy, and Pierre-Philippe Cortet Laboratoire FAST, CNRS, Universit´e Paris-Sud, Orsay, France Laboratoire SPHYNX, Service de Physique de l’ ´Etat Condens´e,CEA Saclay, CNRS UMR 3680, 91191 Gif-sur-Yvette, France (Dated: July 20, 2018)We present a spatio-temporal analysis of a statistically stationary rotating turbulence experiment,aiming to extract a signature of inertial waves, and to determine the scales and frequencies at whichthey can be detected. The analysis uses two-point spatial correlations of the temporal Fouriertransform of velocity fields obtained from time-resolved stereoscopic particle image velocimetrymeasurements in the rotating frame. We quantify the degree of anisotropy of turbulence as a functionof frequency and spatial scale. We show that this space-time-dependent anisotropy is well describedby the dispersion relation of linear inertial waves at large scale, while smaller scales are dominatedby the sweeping of the waves by fluid motion at larger scales. This sweeping effect is mostly dueto the low-frequency quasi-two-dimensional component of the turbulent flow, a prominent featureof our experiment which is not accounted for by wave turbulence theory. These results questionthe relevance of this theory for rotating turbulence at the moderate Rossby numbers accessible inlaboratory experiments, which are relevant to most geophysical and astrophysical flows.
I. INTRODUCTION
The energy content of turbulence is usually charac-terized by the energy distribution among spatial scales,either in physical or in Fourier space. For rotating, strat-ified, or magnetohydrodynamic turbulence [1], waves canpropagate and coexist with “classical” eddies and coher-ent structures, which advocates for a spatio-temporal de-scription of such flows. While temporal fluctuations areusually slaved to the spatial ones via sweeping effects inclassical turbulence [2, 3], they are expected to be gov-erned by the dispersion relation of the waves for timescales much smaller than the eddy turnover time. Thelatter regime is the subject of wave turbulence theory,in which the assumption of weak nonlinear coupling be-tween waves allows to predict scaling laws for the spatialenergy spectrum [4, 5].It is a matter of debate whether wave turbulence the-ory (also known as weak turbulence theory) is a goodcandidate to describe rotating turbulence in the rapidlyrotating limit. Solutions to the linearized rotating Eulerequation can be decomposed into inertial waves, whichsatisfy the anisotropic dispersion relation σ ( k ) = 2Ω | k k || k | , (1)where Ω is the rotation rate and k k the component ofthe wave vector k along the rotation axis (referred to asthe vertical axis by convention) [6]. Accordingly, onlyfluid motions at frequencies σ smaller than the Corio-lis frequency 2Ω correspond to wave propagation. Fluidmotions of weak amplitude and slowly varying in time( σ ≪ direct energy cascade arising from resonanttriadic interactions of 3D wave modes only [32, 33]. Thistheory therefore provides only a partial description of ro-tating turbulence in realistic systems, and careful exper-imental and numerical studies remain necessary to assessits range of validity.Laboratory experiments differ from most numericaland theoretical studies by the presence of rigid horizon-tal boundaries, where the rotating flow achieves no-slipconditions through Ekman layers [34]. In a laminar Ek-man layer, the balance between the viscous and Coriolis CameraGeneratorGenerator Glass lidLaser125 cm200 cm c m Ω (b) Top view Turntable (a) Side view L a s e r s h ee t xz xy FIG. 1. (Color online) Experimental setup. An arena of 10pairs of flaps forces a turbulent flow in the central regionof a water tank mounted on a rotating turntable. A lasersheet illuminates a vertical slice through a horizontal glass lidcovering the fluid. 2D3C (two-dimensional three-component)velocity measurements are performed using stereoscopic PIV(particle image velocimetry) in a vertical square domain ofsize ∆ x × ∆ z = 14 ×
14 cm , shown as a dashed square inpanel (a). forces leads to a boundary layer thickness δ Ek ≃ p ν/ Ω.The belief is that, provided the experimental tank is deepenough, the bulk turbulent flow away from the top andbottom boundaries should resemble the one obtained inthe ideal 3D periodic or stress-free domains consideredin most numerical and theoretical studies. Closer to thehorizontal walls, the boundary layers induce Ekman fric-tion that is not taken into account by most numericalstudies.In the laboratory, the energy dissipation of rotatingturbulent flows originates from three main contributions:bulk viscous dissipation of 3D flow structures, bulk vis-cous dissipation of quasi-2D flow structures (somewhatsimilar to the bulk energy dissipation of 2D turbulence), −6 −5 −4 PSfrag replacements h u i i / ( m s − ) Ω (rpm)0 rpm (a) (b) ε (m s − ) −8 −7 −6 PSfrag replacements h u i i / s − )2D3D Ω (rpm) (b) ε ( m s − ) FIG. 2. (Color online) (a) Energy and (b) energy dissipationrate per unit mass for the 2D and 3D modes as a functionof the rotation rate Ω. In both figures, the first data points(shown with arrows, at arbitrary abscissae) correspond to thenon-rotating case Ω = 0. and dissipation through Ekman friction on the horizontalboundaries.In spite of the importance of the 2D mode in mostgeophysical and laboratory flows, the 3D fluctuations stillplay a crucial role in the dynamics of rotating turbulence,because they are much more efficient at dissipating en-ergy. This key feature is illustrated in Fig. 2, from dataobtained in the present experiment (setup sketched inFig. 1; see Sec. III for details): we decompose the tur-bulent velocity field into a vertically-averaged 2D flowand a vertically-dependent 3D remainder, and show thecorresponding energies and energy dissipation rates as afunction of global rotation. For maximum rotation, al-though the 3D component contains a small fraction ofthe total kinetic energy, its dissipation rate is as large asthat of the vertically-invariant 2D component. Moreover,both of these dissipations are larger than an estimate ofthe frictional losses due to laminar Ekman layers (seeSec. III). An accurate description of the 3D structuresof the flow is therefore essential to characterize the en-ergy fluxes in rotating turbulence at moderate Rossbynumber.A primary goal in this direction is to determine therange of scales and frequencies for which 3D fluctuationsfollow the inertial-wave dispersion relation. This requiresa full spatio-temporal analysis, which is very demandingin general for wave-turbulence systems: the accessiblerange of scales is usually limited in experiments, whereaslong integration times are prohibitive in numerical sim-ulations. The case of rotating turbulence is particularlydelicate because of the specific form of the dispersionrelation (1): the frequency is not related to the wavenumber, as in conventional isotropic wave systems suchas surface waves [35] or elastic waves [36, 37], but to thewave vector orientation only.The recent studies of Clark di Leoni et al. [38] andYarom and Sharon [39] constitute important steps for-ward in this respect. Using numerical simulation of ro-tating turbulence forced at large scale, Clark di Leoni etal. [38] observe a clear concentration of energy along thedispersion relation of inertial waves, and provide a de-tailed analysis of the various time scales of the system.They observe a wave-dominated regime at large scaleand a sweeping-dominated regime at small scale (see alsoRef. [40]). In the experiment of Yarom and Sharon [39]the forcing consists of a random set of sources and sinksat the bottom of a rotating water tank. They measurethree-dimensional two-component (3D2C) velocity fieldsusing a scanning particle image velocimetry (PIV) tech-nique, and observe also a good agreement with the iner-tial wave dispersion relation. In both Refs. [38] and [39],the inertial waves are observed at scales smaller than theinjection scale, suggesting that they are fed by forwardenergy transfers, which is consistent with the predictionsof wave turbulence theory.The aim of the present paper is to further analyze ex-perimentally the range of spatio-temporal scales at whichinertial waves can be detected in rotating turbulence.Stationary rotating turbulence is produced by a set ofvortex dipole generators which continuously inject tur-bulent fluctuations towards the center of a rotating watertank where measurements are performed. We showed inRef. [26] that this configuration generates a double en-ergy cascade at large rotation rate: an inverse cascadeof horizontal energy and a direct cascade of vertical en-ergy, which behaves approximately as a passive scalar ad-vected by the horizontal flow. Here we perform a detailedspatio-temporal analysis using two-point spatial corre-lations of the temporal Fourier modes computed fromtime-resolved two-dimensional three-component (2D3C)velocity fields measured by stereoscopic PIV in a verticalplane. We observe that, at large scales and frequencies,the spatio-temporal anisotropy of the energy distribu-tion is well described by the dispersion relation of iner-tial waves, whereas smaller scales are dominated by thesweeping of the waves by the energetic large-scale flow.
II. EXPERIMENTAL SETUP
The experimental setup, sketched in Fig. 1, is similarto the one described in Refs. [26, 41] and is only brieflydescribed here. It consists of a 125 × ×
65 cm glasstank, filled with H = 50 cm of water and mounted on a2 m-diameter rotating platform which rotates at a rateΩ in the range 0 .
21 to 1 .
68 rad s − (2 to 16 rpm). Tur-bulence is produced in the rotating frame by a set of tenvertical vortex dipole generators organized as a circulararena of 85 cm diameter around the center of the watertank. This forcing device was initially designed to gen-erate turbulence in stratified fluids, and is described indetail in Refs. [42, 43]. Each generator consists of a pairof vertical flaps, 60 cm high and L f = 10 cm long, al-ternatively closing rapidly and opening slowly in a cyclicmotion of period T = 2 π/σ = 8 . σ f = 0 .
092 rad s − , and a random phase shift is appliedbetween the generators.In the laminar regime, a single pair of flaps generatesvortex dipoles with core vorticity ω f . Additional PIVmeasurements close to a vortex dipole generator indicatethat this core vorticity is governed by the vorticity in theviscous boundary layer of the flap, ω f ∼ σ f L f /δ , where σ f L f is the flap velocity and δ the viscous boundary layerthickness. In the present experiment, the vortex dipolesare unstable, and the closing of the flaps therefore pro-duces small-scale 3D turbulent fluctuations that are ad-vected towards the center of the arena by the remaininglarge-scale dipolar structure.The turbulent Reynolds number, computed from therms velocity and the horizontal integral scale, is about400 in the center of the flow, and the turbulent Rossbynumber covers the range 0 . − .
07 for Ω = 2 −
16 rpm [26].We measure the three components of the velocity field u = u x e x + u y e y + u z e z (with e z oriented vertically,along the rotation axis) in a vertical square domain ofsize ∆ x × ∆ z = 14 ×
14 cm located at the center ofthe circular arena at mid-depth, using a stereoscopic PIVsystem [44, 45] embarked on the rotating platform. These2D3C (two-dimensional three-component) velocity fieldsare sampled on a grid of 80 ×
80 vectors with a spatialresolution of 1 .
75 mm. Two acquisition sets are recordedfor each rotation rate Ω: one set of 10 000 fields at 0.35 Hzand one set of 1 000 fields at 1.5 Hz. The combination ofthese two time series results in a temporal spectral rangeof three decades.
III. 2D VS. 3D FLOW COMPONENTS
In the present experiment, energy is primarily injectedin the 2D mode (vertically invariant), but the instabilitiesin the vicinity of the flaps rapidly feed 3D fluctuationswhich are advected in the central region. Energy trans-fers between the 2D and 3D flow components, which van-ish in the weak turbulence limit ( Ro → z of the PIV field, u D = 1∆ z Z ∆ z u ( x, z ) dz, (2)and the remaining z -dependent 3D flow as u D = u − u D . We compute the energy per unit mass of these twoflow components as h u D i / h u D i /
2, with · thetemporal average and h · i the spatial average over thePIV field. They are plotted in Fig. 2(a) as a functionof the rotation rate Ω. Because of the limited height ofthe PIV field, the 2D flow estimated from Eq. (2) un-avoidably contains 3D fluctuations associated to verticalscales larger than ∆ z , so the measured 2D energy mayoverestimate the true one.Figure 2(a) shows that without rotation the 2D and 3Dcomponents of the flow have comparable energy. With ro-tation, the 2D energy increases with Ω, following approx-imately the power law Ω / [41], whereas the 3D energyremains approximately constant, and represents only 5%of the total energy at the largest rotation rate. Althoughmost of the energy is contained in the 2D flow compo-nent for Ω = 0, a significant fraction of the dissipationstill arises from the 3D fluctuations. Assuming axisym-metry, we compute an estimate of the energy dissipationrate ǫ = ν h ( ∂u i /∂x j ) i from the 6 terms of the velocitygradient tensor accessible in the 3C2D measurements, ǫ ≃ ν * (cid:18) ∂u x ∂x (cid:19) + 2 (cid:18) ∂u y ∂x (cid:19) + 2 (cid:18) ∂u z ∂x (cid:19) + (cid:18) ∂u x ∂z (cid:19) + (cid:18) ∂u y ∂z (cid:19) + (cid:18) ∂u z ∂z (cid:19) + . (3)This dissipation rate, computed both for u D and u D ,is shown in Fig. 2(b). Since the derivatives are obtainedfrom finite differences at the smallest resolved scale, thecomputed dissipation underestimates the true one (thePIV resolution is 1 .
75 mm while the Kolmogorov scale isof order of 0.6 mm [41]). However, we expect the mea-sured evolution of ǫ with Ω to reflect the true one.We first compare these bulk energy dissipation ratesto an estimate of frictional losses due to laminar Ekmanlayers, ǫ Ek ≃ ν U ⊥ rms δ Ek δ Ek H = √ ν Ω U ⊥ rms H , where U ⊥ rms isthe root-mean-square horizontal velocity. This estimateranges from 8 × − m s − for Ω = 2 rpm to 1 × − m s − for Ω = 16 rpm: it is smaller than the bulkenergy dissipation of both the 2D and 3D parts of theturbulent flow, by a factor of 10 for slow rotation and 4 forrapid rotation. A detailed experimental characterizationof these Ekman layers would be necessary to validate theassumption of laminar layers, but it is beyond the scopeof the present study. −3 −2 −1 −7 −6 −5 −4 −3 −2 PSfrag replacements E ( σ )( m s − ) ∝ σ − . ∝ σ − Ω = σ = 0 .
74 rad s − σ f = 0 .
092 rad s − σ (rad s − )0 rpm2 rpm4 rpm8 rpm16 rpm k (m s − ) FIG. 3. (Color online) Temporal energy spectrum E ( σ ) asa function of the angular frequency σ for different rotationrates Ω. σ indicates the frequency of the opening-and-closingcycle of the flaps. The Coriolis frequency 2Ω is highlightedwith filled symbols. We now compare the bulk energy dissipation rates inthe 2D and 3D parts of the turbulent flow. Remarkably,while the 3D fluctuations represent a small fraction ofthe total energy, they account for a large fraction of thedissipation at all rotation rates. It is therefore of firstinterest to investigate these 3D modes, and to determineto what extent they can be described in terms of inertialwaves.
IV. TEMPORAL ANALYSIS
We now focus on the temporal dynamics of the velocityfield, which we characterize through the energy distribu-tion of turbulent fluctuations as a function of angularfrequency σ . This temporal energy spectrum is definedas E ( σ ) = 4 πT (cid:10) | ˜ u i ( x , σ ) | (cid:11) , (4)where ˜ u i ( x , σ ) = 12 π Z T u i ( x , t ) e − iσt dt (5)is the temporal Fourier transform of the velocity field u i ( x , t ) (with i = x, y, z ), T the acquisition duration and h · i the spatial average. The normalization is such that h u i i = R ∞ E ( σ ) dσ , with · the temporal average. We usethe standard Welch’s method [46] to improve the statis-tical convergence of the power spectrum.We plot E ( σ ) for each rotation rate Ω in Fig. 3. ForΩ = 0, we observe a global increase with Ω of the en-ergy at all frequencies, consistently with the behavior of −3 −2 −1 −6 −4 −2 PSfrag replacements ( m s − ) ∝ σ − . ∝ σ − σ = 0 .
74 rad s − σ f = 0 .
092 rad s − σ (rad s − ) ∝ σ − / ⊥⊥⊥⊥⊥kkkkk FIG. 4. (Color online) Temporal energy spectra of the verti-cal, E k (light gray, blue), and horizontal, E ⊥ / between couplesof curves at different Ω. The dashed lines show power laws σ − / . the overall energy in Fig. 2(a). These spectra for Ω = 0strongly differ from the non-rotating spectrum, with rela-tively much more energy at low frequency in the rotatingcase: global rotation induces slow dynamics.A first step towards a description of the flow anisotropyin the frequency domain can be provided by further de-composing the power spectrum density (4) as E ( σ ) = E k ( σ ) + E ⊥ ( σ ) , (6)with E k ( σ ) = 4 π h| ˜ u z ( x , σ ) | i /T the spectrum of thevertical velocity and E ⊥ ( σ ) = 4 π h| ˜ u x ( x , σ ) | i /T +4 π h| ˜ u y ( x , σ ) | i /T the spectrum of the horizontal velocity.This decomposition highlights the frequency-dependent componentality of turbulence, i.e., the distribution of en-ergy among the different velocity components, which isrelated to the polarization anisotropy [12, 15, 16]. Thisis not to be confused with the frequency-dependent di-mensionality of turbulence, which compares the vertical −3 −2 −1 −3 −2 −1 PSfrag replacements E k ( σ ) / E ⊥ ( σ ) σ f = 0 .
092 rad s − σ ∗ ∝ σ / ∗ FIG. 5. Componential anisotropy ratio as a function of thenormalized frequency σ ∗ = σ/ E k /E ⊥ = 1. The dottedline indicates the prediction for a plane inertial wave, i.e.,2 E k /E ⊥ = 2(1 − σ ∗ ) / (1 + σ ∗ ). and horizontal characteristic scales at a given frequency(described in section V).The temporal spectra E k ( σ ) and E ⊥ ( σ ) / E ⊥ ≃ E k ). There is actually a slightover-representation of horizontal energy, a consequence ofthe forcing device geometry which preferentially injectsenergy in horizontal motions. As the rotation rate in-creases, the high frequencies remain nearly isotropic (iso-component), whereas the low frequencies become gradu-ally anisotropic, with E k ( σ ) nearly flat and E ⊥ ( σ ) ap-proaching a power law close to σ − / . This anisotropy isrelated to the fact that, as Ω increases, the decorrelationfrequency (i.e., the frequency below which the spectrumbecomes flat) gets significantly smaller for the horizontalvelocity ( σ dec ⊥ = 0 . ± .
01) than for the vertical veloc-ity ( σ dec k , increasing from 0 .
05 to 0 .
40 rad s − for Ω from2 to 16 rpm). For the largest rotation rate (Ω = 16 rpm),there is a clear range of frequencies over which the hori-zontal spectrum E ⊥ ( σ ) follows a σ − / -power-law. Thisrange gets narrower for decreasing rotation rate Ω.The decorrelation frequency of the vertical velocity ap-pears to scale as σ dec k ∗ = σ dec k /
2Ω = 0 . ± .
05, whichbecomes evident when plotting the ratio 2 E k /E ⊥ as afunction of the normalized frequency σ ∗ = σ/
2Ω (Fig. 5).In this figure, for σ ∗ >
1, for which no inertial waves canexist, energy is nearly equally distributed among the ve-locity components (2 E k /E ⊥ ≃ σ / ∗ is approached is bounded bythe two decorrelation frequencies: on the left by σ dec ⊥ and on the right by σ dec k . Interestingly, we also ob-serve a small frequency domain σ dec k ∗ < σ ∗ . . E k /E ⊥ = 2(1 − σ ∗ ) / (1 + σ ∗ ) (shownas a dotted line in Fig. 5), with a larger rms velocityalong the vertical than along any horizontal direction for σ ∗ < / √ ≃ . et al. [26], this strong 2D nature of the flow drivesan inverse cascade of energy for the horizontal velocityand a direct cascade of energy for the vertical velocity.The horizontal velocity consequently exhibits slow dy-namics, while the vertical velocity fluctuations are foundat higher frequencies (Fig. 4). This behavior is consis-tent with the usual phenomenology of rapidly rotatingturbulence: the flow becomes approximately 2D at low-frequency, and the vertical velocity behaves as a passivescalar, which is stretched and folded by the horizontalvelocity. This produces thin vertical layers swept by thehorizontal flow, yielding rapidly changing time series ofthe vertical velocity. In a similar fashion, the σ − / -power-law of E ⊥ ( σ ) could originate from the stochasticsweeping by the large scale horizontal flow of a k − / ⊥ spa-tial spectrum, reminiscent of the inverse energy cascadeof 2D turbulence. V. SPATIO-TEMPORAL ANALYSISA. Spatio-temporal correlations
We now turn to a combined spatio-temporal analysis ofthe PIV time series, focusing on the signature of inertialwaves in terms of dimensional anisotropy. This signatureis sought here in terms of characteristic horizontal andvertical scales of the turbulent structures as a functionof their frequency. For this, we define the frequency-dependent two-point spatial correlation of the temporalFourier transform of the velocity field R ( r , σ ) = 2 πT h ˜ u i ( x , σ )˜ u ∗ i ( x + r , σ ) + c . c . i , (7)with ∗ the complex conjugate (here again Welch’s methodis used to improve convergence). Instead of the spectraconsidered in Refs. [38, 39], we compute spatial correla-tions, because the former are sensitive to finite size effectsarising from the PIV field being of limited extent com-pared to the largest flow structures. The correlation (7)probes the energy distribution among vector separations r for each frequency σ . Its angular average provides anestimate for the cumulative energy from scale r = | r | to r = ∞ for turbulent motions of frequency σ . Thesingle-point limit of this correlation is the temporal en-ergy spectrum (4), i.e. E ( σ ) = R ( r = 0 , σ ).
0 30 60 90 120 0 30 60 90 1200306090120
PSfrag replacements | R σ ( r ) | E σ Ω = 0 rpm Ω = 16 rpm σ = 0 . − σ = 0 . − σ = 0 . − σ = 2 . − σ = 3 . − σ ∗ = 0 . σ ∗ = 0 . σ ∗ = 0 . σ ∗ = 0 . r ⊥ (mm) r ⊥ (mm) r k ( mm ) Ω = 0 rpmΩ = 16 rpm R ( r , σ ) /E ( σ )
0 30 60 90 120 0 30 60 90 1200306090120
PSfrag replacements | R σ ( r ) | E σ Ω = 0 rpm Ω = 16 rpm σ = 0 . − σ = 0 . − σ = 0 . − σ = 2 . − σ = 3 . − σ ∗ = 0 . σ ∗ = 0 . σ ∗ = 0 . σ ∗ = 0 . r ⊥ (mm) r ⊥ (mm) r k ( mm ) Ω = 0 rpmΩ = 16 rpm R ( r , σ ) /E ( σ )
0 30 60 90 120 0 30 60 90 1200306090120
PSfrag replacements | R σ ( r ) | E σ Ω = 0 rpm Ω = 16 rpm σ = 0 . − σ = 0 . − σ = 2 . − σ = 2 . − σ = 3 . − σ ∗ = 0 . σ ∗ = 0 . σ ∗ = 0 . σ ∗ = 0 . r ⊥ (mm) r ⊥ (mm) r k ( mm ) Ω = 0 rpmΩ = 16 rpm R ( r , σ ) /E ( σ )
0 0 30 60 90 120 0 30 60 90 1200306090120 0.2 0.4 0.6 0.8 1
PSfrag replacements | R σ ( r ) | E σ Ω = 0 rpm Ω = 16 rpm σ = 0 . − σ = 0 . − σ = 2 . − σ = 3 . − σ = 3 . − σ ∗ = 0 . σ ∗ = 0 . σ ∗ = 0 . σ ∗ = 0 . r ⊥ (mm) r ⊥ (mm) r k ( mm ) Ω = 0 rpmΩ = 16 rpm R ( r , σ ) /E ( σ ) FIG. 6. (Color online) Maps of the normalized two-pointcorrelation R ( r , σ ) /E ( σ ) in the vertical plane ( r ⊥ , r k ) forΩ = 0 (left) and Ω = 16 rpm (right), at four frequen-cies σ = 0 . , . , . , . − . In the rotating case,the corresponding normalized frequencies are σ ∗ = σ/
2Ω =0 . , . , . , .
98. Isocontour lines for R ( r , σ ) /E ( σ ) < . Maps of the normalized correlation R ( r , σ ) /E ( σ ) areplotted in Fig. 6 for Ω = 0 and 16 rpm at four selectedangular frequencies σ . In the non-rotating case, the iso- R lines are approximately circular at all scales and frequen-cies, indicating the overall isotropy of turbulence. Thestrongly peaked correlation that develops around r = 0 as σ increases indicates that rapid turbulent fluctuations arefound at small scales only. In the rotating case, the iso- R lines evolve gradually from quasi-vertical at small fre-quency (“cigar” anisotropy) to more horizontal for σ ∗ ∼ σ ∗ = σ/
2Ω the normalizedfrequency. The “cigar” anisotropy observed at σ ∗ ≪ σ ∗ → r ⊥ , is consistent with the nearly verticalwave vector limit of Eq. (1).A natural way to characterize the frequency-dependentanisotropy would be to compute integral scales along andnormal to the rotation axis at each frequency. Here weconsider a finer approach, which also takes into accountthe scale-dependence of this anisotropy: for each fre-quency σ and horizontal scale r ⊥ , we identify the ver-tical scale ℓ k ( r ⊥ , σ ) at which the correlation along thevertical axis is equal to the one at r ⊥ e ⊥ , i.e. such that R ( r = ℓ k e k , σ ) = R ( r = r ⊥ e ⊥ , σ ). In practice, we com-pute ℓ k as the vertical semi-axis obtained from the fit ofthe iso- R line defined by R ( r , σ ) = R ( r ⊥ e ⊥ , σ ) with an el-lipse of prescribed horizontal semi-axis r ⊥ . This methodallows us to filter out the noise in the iso- R lines at small R . It also extends the analysis to values of ℓ k larger thanthe PIV field height (∆ z = 140 mm), which is useful atsmall σ ∗ for nearly vertically invariant R . Poor fits de-fined by a correlation coefficient less than 0.9 or such that ℓ k is larger than 2∆ z are discarded. We finally define thescale and frequency dependent anisotropy factor as A ( r ⊥ , σ ) = r ⊥ ℓ k ( r ⊥ , σ ) . (8)It is equal to 1 for isotropic turbulence, to 0 for verticallyinvariant (2D3C) turbulence, and to ∞ for horizontallyinvariant (1D2C) turbulence.If the anisotropy of the two-point correlation R at fre-quency σ ∗ A to be independent of the scale, and to be set bythe dispersion relation (1). A simple estimate, assumingthat the wave vector k is in the vertical measurementplane and identifying on dimensional grounds ℓ k ∼ k − k and r ⊥ ∼ k − ⊥ , yields a frequency-dependent anisotropyfactor A IW ( σ ∗ ) ≃ ( σ − ∗ − − / . Considering now an as-sembly of inertial waves with an axisymmetric wavevec-tor distribution, an analytic computation (given in theAppendix) leads to a very similar result, A IW ( σ ∗ ) = s σ − ∗ − . (9) −1 PSfrag replacements A = r ⊥ / ℓ k (a) r ⊥ = 9 mm(b) r ⊥ = 18 mm(c) r ⊥ = 50 mm σ ∗ −1 PSfrag replacements A = r ⊥ / ℓ k (a) r ⊥ = 9 mm (b) r ⊥ = 18 mm(c) r ⊥ = 50 mm σ ∗ −2 −1 −1 PSfrag replacements A = r ⊥ / ℓ k (a) r ⊥ = 9 mm(b) r ⊥ = 18 mm (c) r ⊥ = 50 mm σ ∗ FIG. 7. Anisotropy factor A (8) as a function of the normal-ized frequency σ ∗ = σ/
2Ω at different rotation rates (samesymbols as in Fig. 3), for three horizontal scales r ⊥ = 9, 18and 50 mm. The continuous line represents the inviscidinertial-wave prediction A IW (9). In Figs. 7 (a-c), we compare the anisotropy factor A measured in the four rotating experiments to the inertial-wave prediction (9) at three horizontal scales r ⊥ = 9, 18and 50 mm. For all scales and σ ∗ ≤ A is an increas-ing function of σ ∗ , confirming that slow fluctuations aremore vertically elongated than fast fluctuations. We findthat the inertial-wave prediction (9) provides a good de-scription of the data at large horizontal scales and largerotation rate. For such large scales ( r ⊥ ≃
50 mm), theanisotropy factor is no longer accessible for σ ∗ < . ℓ k much larger than the height ofthe PIV field. On the other hand, at smaller horizontalscale the prediction (9) fails, with small frequencies moreisotropic than predicted by the inertial-wave argument.Because of the moderate Reynolds and Rossby num-bers of the present experiment, two effects may be consid-ered to explain why large scales follow the inertial-waveprediction whereas small scales do not: viscous damp-ing and sweeping of small scales by the velocity at largerscales. Viscosity introduces an imaginary term iν | k | in the dispersion relation (1) without modifying its realpart. Waves ( σ, k ) such that | k | ≫ r − ν are thereforedamped, with r ν = p ν/σ a viscous cutoff. This vis-cous cutoff is of order of 1 to 10 mm for the normalizedfrequencies σ ∗ = σ/
2Ω in the range [10 − ,
1] consideredin Fig. 5. However, since viscous damping affects thewave amplitude without modifying the wave vector com-ponents, its should not affect the anisotropy. We there-fore focus in the following on the sweeping effect.
B. Sweeping effect
Sweeping corresponds to the advection of the waves bythe large-scale flow, which leads to a modification of theirapparent frequency. An inertial wave propagating in atime-independent uniform flow U has a Doppler-shiftedfrequency, σ = σ i + k · U , (10)where σ i is the intrinsic frequency given by (1), and σ is the frequency at which the wave is detected in theframe of the rotating tank. In our experiment, the en-ergetic large-scale 2D flow may be thought of locally asa uniform sweeping flow U that evolves slowly in time,inducing a scrambling of the waves’ spatio-temporal sig-nature. An order of magnitude of the typical Doppler-shift can be estimated by k ⊥ U ⊥ rms , where U ⊥ rms is theroot-mean-square horizontal velocity. A key differencebetween equations (10) and (1) is that the frequency σ now depends on the magnitude of k , with small-scalewaves more affected by sweeping.For an ensemble of inertial waves with axisymmetricwavevector statistics, the intrinsic frequency σ i can berelated to the anisotropy through Eq. (9). Subsitutingthe corresponding expression into (10) and estimating theDoppler-shift term on dimensional grounds, we obtain σ ≃ √ A − + C U ⊥ rms r ⊥ , (11)where C is a constant of order unity. This indicates thatthe parameter N = 2Ω r ⊥ U ⊥ rms q A (12) −2 −1 −2 −1 N S = U ⊥ rms / σ r ⊥ FIG. 8. Rescaled intrinsic frequency N (12) as a function ofthe sweeping parameter S . The symbols indicate the differentrotation rates, and are the same as in Fig. 3. The dashedline N ∼ /S shows the low- S prediction for (non-swept)ensembles of inertial waves [Eq. (12)]. For S ≪ S ≫ should be a unique function of the sweeping parameter S = U ⊥ rms /σr ⊥ . N corresponds approximately to theintrinsic frequency of inertial waves rescaled by the ad-vective time r ⊥ /U ⊥ rms , whereas S is the observed periodof the waves, rescaled by the same advective time.Figure 8 confirms this picture: the data for differ-ent values of Ω, r ⊥ and σ collapse onto a master curve N = f ( S ). This collapse indicates that sweeping is in-deed responsible for the departure from the inertial waveprediction at small frequencies and/or small scales. Theexpected asymptotic behavior for small sweeping param-eter is N ≃ /S , which corresponds to the prediction(9) for an axisymmetric ensemble of non-swept inertialwaves. The data is in quantitative agreement with thissmall- S prediction, shown as a dashed line in figure 8. Forlarge S , equation (11) indicates that N should asymptoteto a constant value N ≃ C , which again is compatiblewith the data.The master curve in figure 8 has the following simpleinterpretation: high-frequency or large-scale waves arehardly affected by sweeping. The Doppler-shift term isnegligible compared to their intrinsic frequency, and theirlocation in a space-time energy distribution is given bythe dispersion relation (1). This is the low- S behavior infigure 8.By contrast, when focusing on low frequencies σ orsmall-scales in the frame of the tank, one measures the in-ertial waves with intrinsic frequency σ i = σ , but one alsodetects many waves with σ i = σ that are Doppler-shiftedback to frequency σ by the advective term in (10). Theanisotropy measured at low-frequency σ therefore resultsfrom strongly swept inertial waves with various intrin-sic frequencies, and the information from the dispersionrelation (1) is lost in the space-time correlation. Thelimit S ≫ σ that are muchlower than the inverse advective time. In this σ → σ i ≫ σ that areDoppler-shifted by the horizontal flow in such a way thatthey are almost steady in the frame of the tank: this isa σ -independent regime that corresponds to the large- S plateau in figure 8. VI. CONCLUSION
In the present experiment, the anisotropy of the turbu-lent energy distribution at a given spatio-temporal scale( r ⊥ , σ ) is well-described by the inertial-wave dispersionrelation at high-frequency and/or large-scale only. Thesmaller-scale waves are subject to intense sweeping bylarge-scale turbulent motions contained predominantlyin the 2D “vortex” mode. This conclusion is compatiblewith the numerical findings of Clark di Leoni et al. [38],who also identify the sweeping time scale as the relevantdecorrelation time at small scale.Such sweeping by the 2D mode has strong implica-tions for wave-turbulence theories. Indeed, most wavesdo not follow the inertial-wave dispersion relation, andthe assumptions of weak turbulence theory break downeven at the linear stage in wave amplitude: instead ofthe dispersion relation (1), the linear problem consists indetermining the evolution of waves embedded in a tur-bulent 2D flow. This is a formidable task in general,because the 2D flow is space- and time-dependent: inthe discussion of our data, we simplified the problem byassuming that the 2D flow is at much larger scales andslower frequencies than the waves, therefore including itas a simple Doppler-shift term in the dispersion relation.We conclude with a discussion on the dimensionality ofthe forcing. In the present experiment, the flow is drivenby vertically invariant flaps: such a quasi-2D forcing de-vice enhances two-dimensionalization and the resultingsweeping of the 3D flow structures. Nevertheless, accu-mulation of energy in the 2D mode is a robust feature ofrotating turbulence, that takes place for arbitrary forc-ing geometry, even if the forcing does not input energydirectly into the 2D mode. A careful and extensive nu-merical study of this issue has been recently reported forthe fully-3D Taylor-Green forcing [47]: for rapid globalrotation and low viscosity, energy accumulates in the 2Dmode until the Rossby number based on the velocity ofthis 2D flow is of order unity. If these findings are con-firmed, the sweeping of the most energetic 3D structureswould be an inevitable outcome of this accumulation of2D energy. ACKNOWLEDGMENTS
We acknowledge P. Augier, P. Billant and J.-M.Chomaz for kindly providing the flap apparatus, and A. Aubertin, L. Auffray, C. Borget and R. Pidoux for theirexperimental help. This work is supported by the ANRgrant no. 2011-BS04-006-01 “ONLITUR”. B.G. acknowl-edges support from Labex PALM. F.M. acknowledges theInstitut Universitaire de France.
Appendix A: Anisotropy factor for a statisticallyaxisymmetric distribution of inertial waves
We compute the anisotropy factor A for an ensembleof independent plane inertial waves, with axisymmetricwave vector statistics. The temporal Fourier transformof the velocity field reads˜ u ( x , σ ) = Z a ( k , σ ) e i k · x d k , (A1)where a ( k , σ ) is the space-time Fourier amplitude of thevelocity field at wave number k and frequency σ . Thetwo-point velocity correlation at frequency σ (7) can bewritten R ( r , σ ) = 12 D x a ( k , σ ) · a ∗ ( k , σ ) × e i ( k · x − k · ( x + r )) d k d k + c . c . E , = Z | a ( k , σ ) | cos( k · r )d k , (A2)where h · i is the space average, and r is a separation vec-tor inside the PIV plane. Introducing spherical coordi-nates with vertical polar axis, we denote ϕ the azimuthalangle between k and the vertical PIV plane. The argu-ment of the cosine becomes k · r = k k r k + k ⊥ r ⊥ cos ϕ . (A3)Let us first consider an ensemble of inertial waves hav-ing the same wavenumber, | k | = k . For a given reducedfrequency σ ∗ = σ/ | k k | /k ⊥ , and because of statistical axisymmetry | a ( k , σ ) | is independent of ϕ . The spatial correlation atfrequency σ becomes R ( r , σ )= G ( k, σ )2 π Z π cos( k k r k + k ⊥ r ⊥ cos ϕ )d ϕ , = G ( k, σ ) cos (cid:0) σ ∗ kr k (cid:1) J (cid:16)p − σ ∗ kr ⊥ (cid:17) , (A4)where J is the Bessel function of the first kind and G ( k, σ ) is a prefactor proportional to the squared am-plitude of the waves at wavenumber k and frequency σ .For a given frequency σ and horizontal scale r ⊥ , thevertical scale ℓ k ( r ⊥ , σ ) is determined from the isolines R =constant in the ( r ⊥ , r k ) plane. An isoline of R start-ing on the horizontal axis at r ⊥ intersects the verticalaxis at r k = ℓ k ( r ⊥ , σ ). According to expression (A4),such isolines connecting the two axes exist provided theargument of the Bessel function is smaller than the firstzero of this function, i.e., p − σ ∗ kr ⊥ < C , (A5)0where J ( C ) = 0 ( C ≃ . r ⊥ , r k = 0) and for ( r ⊥ = 0 , r k = ℓ k ) leadsto ℓ k ( r ⊥ , σ ∗ ) = arccos h J (cid:16)p − σ ∗ kr ⊥ (cid:17)i σ ∗ k , (A6)and an anisotropy factor A ( r ⊥ , σ ∗ ) = kr ⊥ σ ∗ arccos h J (cid:16)p − σ ∗ kr ⊥ (cid:17)i . (A7)This anisotropy factor depends very weakly on kr ⊥ : it isminimum for low kr ⊥ , where Taylor expansion for kr ⊥ ≪ A ( r ⊥ , σ ∗ ) ≃ √ σ ∗ p − σ ∗ , (A8) whereas it is maximum for the maximum value of r ⊥ allowed by (A5), which gives A ( r ⊥ , σ ∗ ) ≃ C π σ ∗ p − σ ∗ ≃ . σ ∗ p − σ ∗ . (A9)Because the numerical prefactors in (A8) and (A9) differby less than 10%, we can say that the anisotropy factorof this statistically axisymmetric distribution of inertialwaves is given by expression (A8) within 10% accuracy.Because this anisotropy factor is almost independent of k ,we finally expect it to be approximately given by expres-sion (A8) for a realistic superposition of inertial waveswith different wave numbers k . [1] P.A. Davidson, Turbulence in rotating, stratified and elec-trically conducting fluids (Cambridge University Press,Cambridge, 2013).[2] S. Chen and R.H. Kraichnan, Phys. Fluids A (12),2019–24 (1989).[3] T. Sanada and V. Shanmugasundaram, Phys. Fluids A (6) ,1245–50 (1992).[4] V. Zakharov, V. L’vov and G. Falkovich, Wave Turbu-lence (Springer, Berlin, 1992).[5] S. Nazarenko,
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