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Observation of thermal equilibrium in capillary wave turbulence ∗ Guillaume Michel, François Pétrélis, and Stéphan Fauve Laboratoire de Physique Statistique, Ećole Normale Supérieure, CNRS, Université P. et M. Curie, Université Paris Diderot, Paris, France (Dated: January 30, 2017) We consider capillary wave turbulence at scales larger than the forcing one. At such scales, our measurements show that the surface waves dynamics is the one of a thermal equilibrium state in whichtheeffectivetemperatureisrelatedtotheinjectedpower. Wecharacterizethisevolutionwith a scaling law and report the statistical properties of the large-scale surface elevation depending on 7 thiseffective temperature. 1 0 PACSnumbers: 47.35.-i,05.45.-a,47.27.-i 2 n a Introduction.— Oneofthestrikingfeaturesofturbu- This is achieved by using a large depth of fluid with low J lenceistheverywiderangeoftime-scalesoverwhichthe kinematic viscosity (mercury), hence reducing both vis- 7 dynamicsevolves,fromlessthan10−4suptoseveraldays cous drag at the bottom and bulk dissipation. We de- 2 in laboratoryexperiments (see e.g. [1]) and way more in scribethestatisticalpropertiesofthisstate(energyspec- ] astrophysical systems ( 200kyr for the mean time be- trumandprobabilitydistributionfunction ofthe surface n tween reversals of the e∼arth magnetic field, triggered by elevation) and show how the effective temperature can y turbulent fluctuations). Until recently, these statistical be related to the injected power sustaining the out-of- d properties of the flow were either related to the trans- equilibrium state. - u fer of a quadratic conservedquantity throughscales, the Equilibrium and out-of-equilibrium power spectra.— l f so-called Kolmogorov cascade and equivalents, or to the Before describing the experiment, we briefly sum up s. chaotic dynamics of a structure (e.g. reversal of a large- theoretical results. In the limit of large depth, surface c scale flow [1], motion of a shear layer in a von Kármán waves follow the dispersion relation i s swirling flow [2], etc.). In the case of an homogeneous hy flow conserving a single quadratic invariant, namely en- ω2 =gk+ σ k3, (1) ergy, another scenario involving a thermal equilibrium (cid:18)ρ(cid:19) p [ state was referred to in order to describe scales larger whereωistheangularfrequency,kisthewavenumber,g than the forcing one (see [3], p209). This has been re- 1 is the accelerationof gravity,ρ is the density of the fluid cently confirmed by numerical simulations [4] but is so v and σ is the surface tension. The gravity and capillary 9 far in lack of experimental evidence. termsareequalatafrequencyf =(2π)−1(4ρg3/σ)1/4 0 Theverysamedescriptionappliestosurfacewavetur- g.c. and we thereafter consider waves at larger frequencies, 1 bulence, which consists in the statistical study of many 8 i.e. capillary waves. interactingsurfacewaves[5],anexampleofinterestbeing 0 If the system is in thermodynamic equilibrium, the 1. the ocean. Again,moststudieshavefocusedondirector isotropicenergyspectraldensityperunitdensityandper inversecascades(seee.g. [6,7])oronthedynamicsoflo- 0 unit surface e(1D) is given by calized structures (rogue waves[8], Faradaypatterns [9], k 7 1 solitons, etc.). If energy is the only quadratic conserved 2πkdk 1 k Tk v: quantity,weakwaveturbulencetheorypredictsitstrans- e(k1D)dk =kBT × (2π/L)2 × ρL2 =⇒e(k1D) = 2Bπρ . fer to small scales through a direct cascade (similar to i (2) X theKolmogorovone),whereasithasbeenpredictedthat e(1D) canberelatedtothepowerspectrumdensityofthe r large scales are in thermal equilibrium [10]. Capillary k a surface elevation via S (k)=(σ/ρ)−1k−2e(1D), wavesarethesimplestexperimentalsystemconcernedby η k these predictions and experiments were carried out in a −1 σ k T thinlayerofliquidheliumtostudytheselargescales[12]. S (k)= B k−1. (3) η However,becauseofsizabledissipationatthesescales,in (cid:18)ρ(cid:19) 2πρ place of the expected equilibrium state was another en- Using (1), the power spectrum density in the frequency ergy cascade. Such a regime of bidirectional energy cas- domain S (f)=2π(dk/dω)S (k) is cade was later theoretically described in the framework η η of wave turbulence [13]. k T B S (f)= . (4) In this letter, we report the observation of a thermal η 3σπf equilibriumregimeincapillarywaveturbulence,weshow thatthisout-of-equilibriumsystemcanbefullydescribed Thermal equilibrium of capillary waves with no exter- withanequilibriumstatisticsinthelowfrequencyrange. nal forcing (i.e. in which T is the room temperature) 2 0.3 η(t) Rawsignalη(t) Filteredsignal(shiftedby0.2mm) 0.2 ξ (t) 2 ξ3(t) m) 0.1 m ( ht g 0 ei h e v b Wa -0.1 -0.2 12 cm 14 cm -0.3 0 0.2 0.4 0.6 0.8 1 Time(s) ξ (t) 1 FIG. 2. (Color online) Typical elevation signal and filtered FIG.1. Experimentalsetup,consisting in threewave-makers data (finj=50Hz). and a capacitive height sensor in a rectangular vessel. the density is ρ = 13.5 103 kg/m3, the surface ten- has been measured by light scattering in the sixties, for sion is σ = 0.485 N/m ×and the kinematic viscosity is instancetoinvestigatethevalidityof (1)athighfrequen- ν =1.15 10−7m2/s. Thisparametersgivef 16Hz g.c. cies [14]orthe behaviorofsurfacetensioninthe vicinity and a cor×responding wavelength of 12 mm, makin≃g drag of the liquid-vapor critical point [15]. frictionatthe bottomirrelevant. Togetreproduciblere- On the other hand, if an energy input at a single fre- sults, care has been taken to keep the surface free from quencyfinj isconsidered,thesystemisnolongerinther- pollution: similarly to clean water in which damping in- malequilibriumbuteventuallyreachesanon-equilibrium creasesduring approximatelyone hour until surface gets steady-state. It is then expected to display a direct en- fully contaminated [11], oxydation affects mercury. This ergy cascade characterizedby [16] effectcanbesignificantlyreducedbycleaningthesurface after every acquisition (every ten minutes). σ 1/6 S (f) ǫ1/2 f−17/6, (5) Three wave-makers (oscillating paddles of size 120 η ∼ (cid:18)ρ(cid:19) 80 4mm plunged 10 mm below the free surface) ar×e × placed regularly around a home-made capacitive height where ǫ is the mean energy flux through scales (per unit sensor (see Fig. 1). This size of cavity has been chosen massandsurface). Waveturbulencetheorypredictsthat so that the frequencies of the first eigenmodes are a few even if such a system is very far from equilibrium, (4) hertz,thusreducingsubstantiallythegravityrange. The should still be observed in the range f < f < f g.c. inj diameter of the wire, 0.35mm, imposes a high-frequency while (5) describes frequencies from f up to a dissi- inj cut-off of a few hundred Hertz. Each wave-maker is pative scale. Matching (4) and (5) at the single energy driven by a Bruël & Kjær 4810 shaker and its motion input scale of frequency f gives inj ξ (t)(i=1,2,3)istrackedwithaBruël&Kjær4393ac- i k T ǫ1/2σ7/6ρ−1/6f−11/6, (6) celerometer. Functions ξi(t) are different realizations of B ∼ inj a random noise excitation supplied by a function gener- thatwasobtainedwithintheweakturbulenceframework ator (Agilent 33500B) and selected in a frequency range in [10]. Note that all these results are derived in the finj 200HzbyaSR650filter. Severalforcingamplitudes − absence of gravity waves, which follow a different path andfrequenciesfinjhavebeenconsidered,thelattervary- since another quadratic quantity called wave action is ing from 50 Hz to 100 Hz. After each acquisition, the conserved. From a practical point of view, they are ex- motions of the wave-makers have been checked to be of pectedtoholdaslongastheenergydensityatfrequency similarpowerspectra,localizedintherangefinj 200Hz. − fg.c.issmallenough((1)and(2)showthatenergydensity Finally, the wave height η and the accelerations ξï are increases with frequency in the thermal range). Finally, recorded with a NI acquisition card and 50 Hz electrical we point out that the derivation of (4) reduces surface noise is filtered out and not considered in the following wavestoharmonicoscillators: atveryhightemperatures, data treatments. non-linear corrections are expected. A typical elevation signal is shown in Fig. 2, together Experimental setup and results.— The experimental withthesamesamplefilteredintherangef to0.85f : g.c. inj setupconsistsofarectangularplasticvessel(225 180 we observe that a sizable amount of energy is located at × × 45mm) filled with mercury up to 30 mm. For this fluid, frequencies lower than the forcing ones. In all the re- 3 0 CapillarythermalequilibriumwithTeff=1.2TK Weakturbulencecapillarywavesenergycascade e 10−10 Singleexamplewithfinj=70Hz rang mal -0.5 er z) th H y 2(m/ 10−12 hPinjiր pillar -1 f) ca S(η the n i 10−14 ent -1.5 n o p x E -2 10 100 1012 1013 1014 Frequency(Hz) Effectivetemperature(K) FIG. 3. (Color online) Power spectra of the wave height for FIG. 4. Exponent of the large-scale range (fg.c. to 0.85finj) threeforcingamplitudeswiththesamebandwidth(50to200 as a function of the effectivetemperature. Hz) and onefrom 70 to 200 Hz. A theoretical capillary ther- malequilibriumstateanddirectenergycascadeareshownin 103 dotted and dashed lines. hPinji is themean injected power. 102 portedexperiments,thestandarddeviationofthe height 101 i signal σ = η2(t) ranges from 0.02mm to 0.1mm, 2η η h i hp 100 correspondingpto significanttypical steepnesses since the / η forcing is at high frequencies (kinjση is between 0.04 and of 10−1 F 0.2). Moreover,wecheckedthat allthe results presented D P heredonotstronglyrelyonthespecificvalueofthecutoff 10−2 frequency 0.85f . hPinjiր inj 10−3 Four power spectra of the wave height η are displayed in Fig. 3 along with the theoretical ones of a thermal 10−4 NormalDistributionfit equilibrium state at temperature T = 1,2.1012K and -4 -2 0 2 4 eff of a direct energy cascade. They were obtained with η/phη2i f = 50Hz and different forcing amplitudes, or with inj FIG. 5. (Color online) Probability density functions of the f = 70Hz. From f to the forcing frequency f , inj g.c. inj large-scalewaveheightforthespectrareportedinFig4(same theydisplayaf−1 power-law,characteristicofathermal color-key). Data and errorbars are multiplied by ×1, ×10, equilibriumstate. Insomespectrathef−1slopedevelops ×100 and ×1000 for clarity. below fg.c.: similarly to the gravity-capillary transition A fit bya normal distribution is shown in dotted line. observed in surface wave turbulence forced at low frequencies, f is only an indicator of the crossover, that g.c. may reasonably differ from it (see e.g. [6]). From 200 The power spectrum Sη(f) is fitted by a power-law Hz to a dissipative scale, another self-similar regime is cst fα in the same frequency range (fg.c. to 0.85finj), × observed. Becausethe energy flux is in practice not con- and values of α are reported in Fig. 4. They stand close servedduringthisenergycascade[17,18]andasaconse- to -1, as expected for a thermal equilibrium range. quence of the high-frequency cut-off of the measurement Four probability distribution functions (PDF) of the device, steeper power-laws than the theoretical scaling height signal filtered in the same range (fg.c. to 0.85finj) (5) are observed. This cascade has been the subject of and corresponding to the spectra reported in Fig. 3 are anextensiveliteratureoverthepasttwodecadesandwill shown in Fig. 5. The PDF are found to be Gaussian not be further investigated here (see for instance [6, 18– at low forcing and tails turn to exponentials as the in- 22] and references therein). jected power increases. To be more precise the kurtosis of the signal, equal to 3 in the case of a normal distri- For each of these spectrum, we compute the effective bution, is reported in Fig. 6 and is found to increase temperaturebyintegratingtheheightpowerspectrumin with the effective temperature, i.e. the steepness of the the thermal range: waves. The samephenomenonwasreportedinwavetur- T = 3πσ f0g..8c5.finjSη(f)df (7) bwualveensc[e23[6,,2142],]tahnadtcaasncraiblseodtatokeepxltarceemieneavtehnetsrmaaslreoqguuie- eff R kB ln(0.85finj/fg.c.) librium state when the steepness of the waves is high 4 7 10−1 Liney∝x al n sig W) on6 ( filteredelevati5 dpowerhPiinj 10−2 ofthe4 injecte 10−3 urtosis Mean K 3 10−4 1012 1013 1014 1012 1013 1014 Effectivetemperature(K) Effectivetemperature(K) FIG. 6. Kurtosis of the probability density functions as a FIG.7. EvolutionofthemeaninjectedpowerhPinjiwiththe function of the effectivetemperature. effectivetemperature. enough. Note thatdeparturefromanormallawis asign ofsizeablenon-linearinteractionsbetweenmodes, asthe We finally discuss the role of dissipation. In wave tur- central limit theorem fails in presence of correlated ran- bulence, damping hasbeen recognizedas a sourceof dis- dom variables. As for the other moments of this filtered crepancy between theory (that assumes a dissipation lo- signal, the standard deviation evolves as the square root calizedinarestrictedfrequencyrange)andexperiments, ofT (aconsequenceof (7))andtheskewnessisroughly eff forsurfacewaves[17]aswellasforotherwavefields(e.g. constant and close to 0.20. vibratingplates [27]),inwhichthis hypothesisisnotful- The sign of the skewness is relatedto the shape of the filled. To be more precise, energy in weak wave turbu- wave-trains: for gravity waves, non-linearities sharpen lence theory is injected at a given frequency and fully the crests and flatten the troughs [25], whereas the op- transferred to a dissipative scale (similarly to the Kol- posite occurs for capillary waves [26]. These phenom- mogorov picture of hydrodynamic turbulence), whereas ena directly reflect on the skewness of the PDF for a it is essentially dissipated at the injection scale in ex- random wave field. A positive skewness it routinely periments[17]. Thelatterdescriptionalsoappliesinthis observed in experiments of wave turbulence involving experiment,onereasonbeingthataf−1heightspectrum gravito-capillarywaves [6, 20]. For pure capillary waves, is not steep enough for low-frequency waves to reach a these effects are scarce: a recent numerical simulation sizableamplitude, hencetodissipateasignificantpartof of capillary wave turbulence [22], even though involving the injected energy. This can be seen from the dissipa- large wave steepnesses ( 0.3), has for instance not evidenced any negative ske∼wness. In our experiments, the tion spectrum Dη(f), that characterizes the amount of energy dissipated by surface waves of frequency f [17] : skewnessis positivesincethe heightspectraofthe signal S (f) f−1 leads to D (f) f5/3 or D (f) f3/2 filtered between f and 0.85f is dominated by com- η ∝ η ∝ η ∝ g.c. inj depending on the source of dissipation considered (re- ponents at f , and vanishes as the low-pass filtering g.c. spectively viscosity in the bulk and surface contamina- frequency increases. tion). Dissipating energyat the injection scales does not We now consider the dependence of the effective tem- preventthe observationof the equilibrium range studied perature on the mean injected power . This quan- hPinji here: in contrast with an energy cascade whose aim is tity is estimated from the measured velocities ξ˙: i to eventually dissipate energy, the existence of the equi- ρS ξ˙ 3+ ξ˙ 3+ ξ˙ 3 (8) librium range relies on the hypothesis that no energy is inj w.m. 1 2 3 hP i∼ h| | | | | | i dissipated at these frequencies. It also constrains the where S = 12.10−4 m2 is the submerged section setup for such observations: experiments have to be car- w.m. of each wave-maker. It should not be confused with ried out in small vessels, with fluids of low kinematic the mean energy flux though scales ǫ of wave turbu- viscosities and with a large number of efficient resonant lence theory, as most of the injected energy drives bulk interactions at injection scales, otherwise bidirectionnal flows [17]. A careful experimental study of the rela- energy cascades would be generated [12]. For instance, tionship between these two quantities has evidenced the withoursetup,wecouldnotobservethermalequilibrium scaling ρ √ǫ (see [17]), and (6) thus predicts states if water was used instead of mecury, as well as if inj hP i ∝ S k T . Experimental data are reported in Fig. 7 theforcingwasnarrow-band,monochromatic,toolowor B inj ∝ hP i and follow a linear law T over three decades. if f >100Hz. eff inj inj ∝hP i 5 Conclusion.— We experimentally evidenced a ther- [5] S. Nazarenko, WaveTurbulence(2011). mal equilibrium state over an out-of-equilibrium back- [6] E.Falcon,C.LarocheandS.Fauve,Phys.Rev.Lett.98, groundincapillarywaveturbulence. Itspowerspectrum 094503 (2007). [7] L. Deike, C. Laroche and E. Falcon, EPL 96, 34004 density can be used to define an effective temperature (2011). that is strongly linked with the statistical properties of [8] A.Chabcoub,N.P.HoffmannandN.Akhmediev,Phys. this range of scales (e.g. shape of the PDF) and to the Rev. Lett.106, 204502 (2011). ones of the out-of-equilibrium state (e.g. energy flux). [9] N. B. Tufillaro, R. Ramshankar and J. P. Gollub, Phys. The temperatures obtained are more than ten order of Rev. Lett.62, 422 (1989). magnitudes higher than the room temperature and we [10] E. Balkovsky, G. Falkovich, V. Lebedev and I. Ya. emphasizethatsuchequilibriumstatesdisplaynon-linear Shapiro, Phys.Rev.E 52, 4537 (1995). [11] W. G. Van Dorn, J. Fluid Mech. 24, 769 (1966). phenomena,suchasstrongcouplingbetweenmodes. The [12] L.V.Abdurakhimov,M. Arefin,G.V.Kolmakov,A.A. precise conditions requiered for the observation of such Levchenko, Yu. V. Lvov and I. A. Remizov, Phys. Rev. equilibrium state, as well as the role of the transition to E 91, 023021 (2015). gravity waves remain open questions. [13] Yu. V. Lvov, H. Andy and G. V. Kolmakov, EPL 112, Being able to characterize some scales of an out- 24004 (2015). of-equilibrium system by equilibrium statistics seems [14] R. H. Katyl and U. Ingard, Phys. Rev. Lett. 20, 248 promising. In particular, one may wonder if other tools (1968). [15] M.A.BouchiatandJ.Meunier,Phys.Rev.Lett.23,752 of equilibrium statistical mechanics can be used. For in- (1969). stancetheequationofstate,theresponsecoefficientsand [16] V. E. Zakharov and N. N. Filonenko, J. Appl. Mech. thefluctuation-dissipationrelationscouldbeinvestigated Tech. 8, 37 (1967). and compared to their equilibrium counterparts. [17] L. Deike, M. Berhanu and E. Falcon, Phys. Rev. E 89, This work is supported by CNES and ANR-12-BS04- 023003 (2014). 0005-02. [18] Y.PanandD.K.P.Yue,J.FluidMech.780,R1(2015). [19] C. Falcón, E. Falcon, U. Bortolozzo and S. Fauve, EPL 86, 14002 (2009). [20] E. Falcon and C. Laroche, EPL 95, 34003 (2011). [21] E. Henry, P. Alstrom and M. T. Levinsen, EPL 52, 27 ∗ (2000). email: [email protected] [1] J. Herault, F. Pétrélis and S. Fauve, EPL 111, 44002 [22] L. Deike, D. Fuster, M. Berhanu and E. Falcon, Phys. Rev. Lett.112, 234501 (2014). (2015). [23] V. P. Ruban,Phys.Rev.E 74, 036305 (2006). [2] F.Ravelet,A.ChiffaudelandF.Daviaud,J.Fluid.Mech. 601, 339 (2008). [24] M.ShatsandH.PunzmannandH.Xia,Phys.Rev.Lett. 104, 104503 (2010). [3] U. Frish, Cambridge University Press, Turbulence: The [25] G. G. Stokes, Trans. Camb. Philo. Soc. 8, 441 (1847). Legacy of A. N. Kolmogorov (Cambridge, England) [26] G. D.Crapper, J. Fluid Mech. 2, 532 (1957). (1963). [27] B. Miquel, A. Alexakis and N. Mordant, Phys. Rev. E [4] V. Dallas, S. Fauve and A. Alexakis, Phys. Rev. Lett. 115, 204501 (2015). 89, 062925 (2014).