One of the pivotal questions in the dynamics of the oceans is related to the
cascade of mechanical energy in the abyss and its contribution to mixing. Here,
we propose internal wave attractors in the large amplitude regime as a unique
self-consistent experimental and numerical setup that models a cascade of
triadic interactions transferring energy from large-scale monochro-matic input
to multi-scale internal wave motion. We also provide signatures of a discrete
wave turbulence framework for internal waves. Finally, we show how beyond this
regime, we have a clear transition to a regime of small-scale high-vorticity
events which induce mixing. Introduction

Brouzet, Christophe

Dauxois, Thierry

Ermanyuk, Evgeny

Joubaud, Sylvain

Scolan, Hélène

Sibgatullin, Ilias

EPL (Europhysics Letters)

English

arXiv

eScholarship - University of California

UC San DiegoInternational Symposium on Stratified FlowsTitleEnergy cascade in internal wave attractorsPermalinkhttps://escholarship.org/uc/item/5mg5v9w0JournalInternational Symposium on Stratified Flows, 8(1)AuthorsDauxois, ThierryBrouzet, ChristopheErmanyuk, Evgenyet al.Publication Date2016-08-29eScholarship.org Powered by the California Digital LibraryUniversity of CaliforniaEnergy cascade in internal wave attractorsThierry Dauxois1, C. Brouzet1, E. Ermanyuk1,2, S. Joubaud1, H. Scolan1, I. Sibgatullin1,31. Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire dePhysique, F-69342 Lyon, France2. Lavrentyev Institute of Hydrodynamics, Novosibirsk, Russia3. Institute of Mechanics, Moscow State University, RussiaThierry.Dauxois@ens-lyon.frAbstractOne of the pivotal questions in the dynamics of the oceans is related to the cascadeof mechanical energy in the abyss and its contribution to mixing. Here, we proposea unique self-consistent experimental and numerical set up that models a cascade oftriadic interactions transferring energy from large-scale monochromatic input to multi-scale internal wave motion. We also provide, for the first time, explicit evidence of a waveturbulence framework for internal waves. Finally, we show how beyond this regime, wehave a clear transition to a cascade of small-scale overturning events which induce mixing.1 IntroductionThe continuous energy input to the ocean interior comes from the interaction of globaltides with the bottom topography yielding a global rate of energy conversion to internaltides of the order of 1TW. The subsequent mechanical energy cascade to small-scaleinternal-wave motion and mixing is a subject of active debate in view of the importantrole played by abyssal mixing in existing models of ocean dynamics. The oceanographicdata support the important role of internal waves in mixing, at least locally: increasedrates of diapycnal mixing are reported in the bulk of abyssal regions over rough topographyin contrast to regions with smooth bottom topography. A question remains: how doesenergy injected through internal waves at large vertical scales induce the mixing of thefluid?Let us consider a stratified fluid with an initially constant buoyancy frequency N =[(−g/ρ̄)(dρ/dz)]1/2, where ρ(z) is the density distribution over the vertical coordinate z,and g the gravity acceleration. The dispersion relation is θ = arcsin(Ω). Here θ is theslope of the wave beam to the horizontal, and Ω (resp. ω = ΩN) is the non-dimensional(resp. dimensional) frequency of oscillations. The anisotropic dispersion relation requirespreservation of the slope of the internal wave beam upon reflection at a rigid boundary. Inthe case of a sloping boundary, this property gives a purely geometric reason for a strongvariation of the width of internal wave beams (focusing or defocusing) upon reflection.Internal wave focusing provides a necessary condition for large shear and overturning, aswell as shear and bottom layer instabilities at slopes.In a confined fluid domain, focusing usually prevails, leading to a concentration of waveenergy on a closed loop, the internal wave attractor [1]. At the level of linear mecha-nisms, the width of the attractor branches is set by the competition between geometricfocusing and viscous broadening. High concentration of energy at attractors make themprone to triadic resonance instability which sets in as the energy injected into the sys-tem increases [2]. Note that the particular case for which both unstable secondary waveshave a frequency equal to half of the forcing frequency is of particular interest in theVIIIth Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016 1Lω0αHFigure 1: Experimental set-up. The wave generator is on the left and the inclined slope on the right.The color inset is a typical PIV snapshot showing the magnitude of the experimental two-dimensionalvelocity field obtained after 15 periods T0 = 2π/(NΩ0). Black dashed lines show the billiard geometricprediction of the attractor.oceanographic context where viscosity is negligible. In that case, the appropriate name isparametric subharmonic instability and abbreviated as PSI. By abuse of language, someauthors have sometimes extended the use of the name PSI to cases for which secondarywaves are not corresponding to half of the forcing frequency. For the sake of termino-logical consistency, we propose to abbreviate triadic resonance instability using the newacronym: TRI.The onset of instability in this case is similar to the classic concept of triadic resonance,which is best studied for the idealized case, with monochromatic in time and space carrierwave as a basic state which feeds two secondary waves via nonlinear resonant interactions.The resonance occurs when temporal and spatial conditions are satisfied: Ω1 + Ω2 = Ω0and−→k1 +−→k2 =−→k0 , where−→k is the wave vector while subscripts 0, 1 and 2 refer tothe primary, and two secondary waves, respectively. In a wave attractor, the wave beamsserve as a primary wave, and the resonance conditions are satisfied with good accuracy [2],providing a consistent physical framework for the short-term behavior of the instability.2 The energy cascade revealed by the time-frequency diagram and the bicoherence plotUsing laboratory experiments and numerical simulations, we have shown that the set-up,sketched in Fig. 1, provides an excellent energy cascade, emphasizing how internal waveattractors can be a novel laboratory model of a natural cascade.Indeed, the internal wave attractor is the first step: the focalisation mechanism enhancesthe development of the triadic instability within the beams of the attractor. While theattractor is still visible, branches are progressively deformed by triadic resonance insta-bility, leading to the presence of secondary waves. Once the instability is well-developed,secondary waves are acting as primary waves for higher-order triadic interactions. If thefocalisation is strong enough, this mechanism will of course repeat through the instabilityof the secondary waves. This is what is revealed by the time frequency diagram shownin Fig. 2(a). Initially, only a signal around Ω0 = 0.61 is present, but almost immediatelyVIIIth Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016 2t/T0Ω  0 100 200 300 400 500 60000.20.40.60.81−4 −3 −2 −1 0Ω  0 0.2 0.4 0.6 0.8 10.2 0.4 0.6 0.8 1l og 1 0 (Sx(Ω, t)/S 0)(a) (b)B (Ω, Ω)Figure 2: Cascade of Triadic Resonance Instabilities (TRI). Time-frequency diagram (a) and its associ-ated bicoherence (b) of the PIV signal measures in the entire trapezoidal domain.one distinguishes two secondary waves Ω1 = 0.36 and Ω2 = 0.25 whose sum gives Ω0.However, again Ω1 and Ω2 are destabilized and this mechanism is pursued.To detect the frequency triplets, we use the bispectrum analysis which measures theextent of statistical dependence among three spectral components (Ωk, Ωi, Ωj) satisfyingthe relationship Ωk = Ωi + Ωj, with the quantity M(Ωi,Ωj) = F (Ωi)F (Ωj)F∗(Ωi + Ωj),where F is the Fourier transform and ∗ denotes the complex conjugate. In practice,the bispectrum is usually normalized and considered in form of bicoherence which is0 for triplets with random phases and 1 for triplets with perfect phase coupling. Thebicoherence is shown in Fig. 2(b). In addition to the strong peak (0.61, 0.61) correspondingto the forcing frequency (therefore to self-correlation), the possible triplets satisfyingthe definition of triadic resonance at Ωk = Ω0 can be found on the line with slope −1connecting the points (0, 0.61) and (0.61, 0). This emphasizes that the mechanism at playis triadic. Other peaks are also visible corresponding to other choices of Ωk revealing thatthe instability mechanism is repeated and leads to a cascade.Thanks to this beautiful representation, it can therefore be attested that the energy trans-fer from global to small scales in attractors operates via a hierarchy of triadic interactionsproducing a complex internal wave field with a rich multi-peak discrete frequency spec-trum embedded in a continuous spectrum of weaker magnitude.3 A route towards wave turbulenceIt is important to emphasize that the final stage is non-trivial since these phenomenaare beyond the domain of pure wave-wave interactions: it corresponds to a regime usu-ally called wave turbulence [3]. A similar situation takes place for surface waves, wherethe flourishing literature gives a fully consistent description of energy cascades betweencomponents of wave spectra, only in the case of weakly nonlinear processes, while ex-perimental reality deals with cascades significantly “contaminated” by effects of a finitesize fluid domain, wave breaking, wave cusps, nonlinear dispersion, viscous damping ofwave-field components, etc. The very specific dispersion relation for internal waves intro-duces additional complications. For instance, in rotating fluids, which have a dispersionrelation analogous to stratified fluids, the usefulness of the formalism of wave turbulenceVIIIth Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016 3Ωθ [rad]  −π −π/2 0 π/2 π−1−0.500.510.10.20.30.40.50.60.7Figure 3: Energy spectra. Colors indicate the levels of energy spectra. The black lines correspond to thedispersion relation Ω = ± sin θ = ±kx/√k2x + k2z . Integration across different wavenumbers ranges from0.22 to 1 rad·cm−1, i.e. wave lengths 28.5 cm to 6.3 cm.as a basis for the studies in rotating turbulence has been reported for experiments onlyrecently [4]. For internal waves, the question is still fully open, from both experimentaland numerical points of view.The presence of wave turbulence-like phenomena is illustrated in Fig. 3 using the energyspectra experimentally obtained for large scales as a diagnostic tool [4]. Horizontal andvertical velocity fields u(x, z, t) and w(x, z, t) are obtained with 2D PIV measurementsin the entire trapezoidal domain. A two dimensions for space and one for time Fouriertransform of these fields leads to û(kx, kz,Ω) and ŵ(kx, kz,Ω). One can thus define the2D energy spectrum byE(kx, kz,Ω) =|û(kx, kz,Ω)|2 + |ŵ(kx, kz,Ω)|22ST, (1)where S is the area of the PIV measurement and T its duration.In the dispersion relation for internal waves, Ω = ± sin θ, the wave vector−→k and itscomponents do not appear directly but they are linked with the angle θ by sin θ =±kx/√k2x + k2z . To compute the energy spectrum as a function of variable θ, one caninterpolate the energy spectrum E(kx, kz,Ω) to get E(k, θ,Ω), where k is the norm of thewave vector. For this interpolation, we define ∆k as the smallest wave vector that hasdata points in the Cartesian coordinates. Then, one can integrate over the entire rangeof wave vectors [kmin, kmax] as followsE(θ,Ω) =∫ kmaxkminE(k, θ,Ω)kdk, (2)or on any range of wave vectors between kmin and kmax. This is what has been donein Fig. 3 and the integration range represents 84% of the energy in the entire range[kmin, kmax]. The linear dispersion relation is seen to attract the maxima of the energyspectra. Above results are convincing signatures of a discrete wave turbulence frameworkfor internal waves in this intermediate forcing amplitude regime.If we repeat the same experiment with a larger amplitude, we have indications thata system is beyond the wave turbulence-like regime and has reached a mixing regime.VIIIth Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016 4−4 −2 0 2 4−4−3−2−101ξ/Nlog10(PDF)0.998 1 1.002051015202530ρf/ρ iz[cm](a) (b)Mixing MixingNo MixingFigure 4: Mixing and vorticity. (a) Ratio between the density profiles measured after and before theexperiments for cases with intermediate (black) and large (red) forcing amplitudes. (b) Experimentalprobability density functions of the vorticity, calculated on the grid from experimental images for low(blue), intermediate (black) and large (red) forcing amplitudes. Figures 2 and 3 correspond to theintermediate forcing amplitude.Indeed, short-scale perturbations in particular clearly escape any relation to linear wavedynamics. This is expected to be due to overturnings, natural precursors to mixing.4 Mixing inferred from vorticity distributionAn important issue is whether or not sufficiently energetic internal wave motion can pro-duce an irreversible energy contribution to mixing. Figure 4(a) presents the comparisonbetween density profiles measured before and after experiments: while no modification ofthe density (within experimental error) can be observed for the intermediate amplitudeforcing that leads to wave turbulence regime described in the previous section, one gets aclear evidence of mixing in case of a larger forcing amplitude.Further, differences between the regimes corresponding to low and high mixing are clearlyseen in statistics of extreme events. This statistics is obtained by the calculation ofprobability density functions (PDF). Since we are interested in small-scale events desta-bilizing the stratification, we take the horizontal y-component of vorticity ξ(x, z, t) =∂u/∂z − ∂w/∂x measured in the vertical midplane of the test tank as a relevant quan-tity and consider the PDF of the dimensionless quantity ξ/N . In Fig. 4(b), we presentthe vorticity PDFs corresponding to different wave regimes in the attractor. In a stableattractor (see blue curve), extreme events are completely absent and the wave motion isconcentrated within the relatively narrow branches of the attractor while the rest of thefluid is quiescent. Accordingly, the PDF has a sharp peak at zero vorticity and is fullylocalized between well-defined maximum and minimum values of vorticity. For larger forc-ing amplitudes (black and red curves), the development of TRI increases the probabilityof extreme events due to summation of primary and secondary wave components.The occurrence of local overturning events can be viewed as a competition between strat-VIIIth Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016 5ification and vorticity. In a two-dimensional flow, a relevant stability parameter is aversion of the Richardson number, which can be introduced as Riξ = N2/ξ2. For a hori-zontal stratified shear flow this parameter reduces to the conventional gradient Richardsonnumber Ri = N2/(du/dz)2, where du/dz is the velocity shear. Flows with large Ri aregenerally stable, and the turbulence is suppressed by the stratification. The classic Miles-Howard necessary condition for instability requires that Ri<1/4 somewhere in the flow. Ifthis condition is satisfied, the destabilizing effect of shear overcomes the effect of stratifi-cation, and some mixing occurs as a result of overturning. The threshold value |ξ/N | = 2is marked on the plot of vorticity PDFs. It can be seen that data corresponding to largeforcing amplitudes have ”tails” extending into the domains |ξ/N | > 2. The area underthe tails represents the probability of event of strength |ξ/N | > 2. In the larger forcingcase (red curve), this probability is an order of magnitude greater than in intermediateone (blue curve), in qualitative agreement with the much higher mixing that has beenreported.5 ConclusionWe report and describe a novel experimental and numerical set up, an “internal wavemixing box”, which presents a complete cascade of triadic interactions transferring energyfrom large-scale monochromatic input to multi-scale internal wave motion, and subsequentcascade to mixing. We report interesting signatures of discrete wave turbulence in astratified idealized fluid problem. Moreover, we show how statistics of extreme vorticityevents leads to mixing that occur in the bulk of the fluid.References[1] L.R.M. Maas, D. Benielli, J. Sommeria, F.P.A. Lam, Nature 388, 557-561, 1997.Observations of an internal wave attractor in a confined stably stratified fluid.[2] H. Scolan, E.V. Ermanyuk, T. Dauxois, Physical Review Letters 110, 234501 (2013).Nonlinear fate of internal waves attractors.[3] S.V. Nazarenko 2011. Wave Turbulence. Lecture Notes in Physics, Springer; Berlin.[4] E. Yarom, E. Sharon, Nature Physics 10, 510-514 (2014). Experimental observation ofsteady inertial wave turbulence in deep rotating flows.[5] C. Brouzet, E.V. Ermanyuk, S. Joubaud, I. Sibgatullin, T. Dauxois, EurophysicsLetters 113, 44001 (2016). Energy cascade in internal wave attractors.[6] C. Brouzet, I. Sibgatullin, H. Scolan, E.V. Ermanyuk, T. Dauxois, Journal of FluidMechanics 793, 109-131 (2016). Internal wave attractors examined using laboratoryexperiments and 3D numerical simulations.VIIIth Int. Symp. on Stratified Flows, San Diego, USA, Aug. 29 - Sept. 1, 2016 6

Energy cascade in internal wave attractors

International audienceOne of the pivotal questions in the dynamics of the oceans is related to the cascade of mechanical energy in the abyss and its contribution to mixing. Here, we propose internal wave attractors in the large amplitude regime as a unique self-consistent experimental and numerical setup that models a cascade of triadic interactions transferring energy from large-scale monochro-matic input to multi-scale internal wave motion. We also provide signatures of a discrete wave turbulence framework for internal waves. Finally, we show how beyond this regime, we have a clear transition to a regime of small-scale high-vorticity events which induce mixing. Introduction

Ermanyuk, Evgeny, V

HAL-ENS-LYON

One of the pivotal questions in the dynamics of the oceans is related to the cascade of mechanical energy in the abyss and its contribution to mixing. Here, we propose internal-wave attractors in the large-amplitude regime as a unique self-consistent experimental and numerical setup that models a cascade of triadic interactions transferring energy from large-scale monochromatic input to multi-scale internal-wave motion. We also provide signatures of a discrete wave turbulence framework for internal waves. Finally, we show how, beyond this regime, we have a clear transition to a regime of small-scale high-vorticity events which induce mixing

Energy cascade in internal wave attractors

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