Modulation instability and capillary wave turbulence

Xia, Hua; Shats, Michael; Punzmann, Horst

doi:10.1209/0295-5075/91/14002

Cited by 45 publications

(89 citation statements)

References 24 publications

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“…This scaling does not depend on the vessel geometry when using a circular one of similar size. A similar spectrum scaling ∼ I 1 has been observed for both regimes by horizontally vibrating the whole container (Issenmann and Falcon, 2013) for the same range of I , for the capillary regime with a parametric forcing (Xia et al, 2010), and for the inverse cascade of gravity wave turbulence (Deike et al, 2011). Moreover, it has been checked that I ∼ σ 2 η , where σ η is the rms value of the wave height.…”

Section: Direct Measurement Of the Injected Powersupporting

confidence: 74%

Fluctuations of Energy Flux in Wave Turbulence

et al. 2008

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The key governing parameter of wave turbulence is the energy flux that drives the waves and cascades to small scales through nonlinear interactions. In the inertial range, the energy flux is conserved across the scales, and is assumed to be constant in most theoretical approaches. It is only recently that measurements of the injected power into wave turbulence have been performed at the scale of the wave maker (integral scale). In this review, we focus on the statistical properties of the injected power fluctuations in gravity-capillary wave turbulence in a stationary regime. Fluctuations of the injected power have been found much larger than their mean value. In addition, events related to a negative injected power, i.e. an instantaneous reversed energy flux, occur with a fairly large probability. Both features are well described using a Langevin type equation. Finally, we consider the experimental dependence of the scaling law of the wave spectrum with the mean injected power and discuss possible reasons for the discrepancy with weak turbulence theory.

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Section: Direct Measurement Of the Injected Powersupporting

confidence: 74%

Fluctuations of Energy Flux in Wave Turbulence

et al. 2008

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“…In particular, the scaling of the spectrum with respect to wave number I η ∼ k α has been tested experimentally [10][11][12][13] and numerically [6,14,15]. While the exponents found in [6,10,11,14,15] are consistent with the theoretical value of α 0 ¼ −19=4, deviations are also reported in [11,12] with α ¼ −5.3, and [13] with α ¼ −6.0 under weak or narrowband forcing.…”

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confidence: 75%

“…Recent experimental observations from two independent groups [10,11] suggest that a linear scaling relation I η ∼ P should apply, in apparent disagreement with (1). This controversy is summarized in [16], and appears now resolvable [17] by a better experimental estimation of P that distinguishes the energy flux from the energy dissipation at large scales.…”

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confidence: 90%

“…The aim is to obtain a clean development of the wave spectrum not obscured by complexities associated with the mechanical forcing of the waves [6,10,11,14,17] and difficulties associated with the estimation of P [10,11,17]. Furthermore, we seek to uncover the physics at a substantially broader range of nonlinearity level relative to existing measurements [10,11,17] and numerics [6,14,15].…”

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confidence: 99%

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Direct Numerical Investigation of Turbulence of Capillary Waves

Pan

Yue

2014

Phys. Rev. Lett.

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We consider the inertial range spectrum of capillary wave turbulence. Under the assumptions of weak turbulence, the theoretical surface elevation spectrum scales with wave number k as I η ∼ k α , where α ¼ α 0 ¼ −19=4, energy (density) flux P as P 1=2 . The proportional factor C, known as the Kolmogorov constant, has a theoretical value of C ¼ C 0 ¼ 9.85 (we show that this value holds only after a formulation in the original derivation is corrected). The k −19=4 scaling has been extensively, but not conclusively, tested; the P 1=2 scaling has been investigated experimentally, but until recently remains controversial, while direct confirmation of the value of C 0 remains elusive. We conduct a direct numerical investigation implementing the primitive Euler equations. For sufficiently high nonlinearity, the theoretical k −19=4 and P 1=2 scalings as well as value of C 0 are well recovered by our numerical results. For a given number of numerical modes N, as nonlinearity decreases, the long-time spectra deviate from theoretical predictions with respect to scaling with P, with calculated values of α < α 0 and C > C 0 , all due to finite box effect. [5]. In special cases of weak (or wave) turbulence, mathematical formulations are more accessible and the cascade spectrum can be obtained as an exact stationary solution of the kinetic equation, which governs the evolution of wave spectrum due to nonlinear resonant interactions.For capillary waves, the framework of weak turbulence theory (WTT) is developed by Zakharov and Filonenko [3] (and reformulated in [6,7]). The isotropic spectrum of surface elevation yields a powerlike solution in the inertial range which is expressed in closed form,

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“…As a consequence, a compensating return flow converges towards the sides of the wave maker. The flow changes markedly as the wave amplitude is increased above the threshold for the onset of the modulation instability 17,18 (Fig. 1c,d).…”

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confidence: 96%

Generation and reversal of surface flows by propagating waves

et al. 2014

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The ability to send a wave to fetch an object from a distance would find a broad range of applications. Quasi-standing Faraday waves on water create horizontal vortices 1,2 , yet it is not known whether propagating waves can generate large-scale flows-small-amplitude irrotational waves only push particles in the direction of propagation 3-5 . Here we show that when waves become three-dimensional as a result of the modulation instability, a floater can be forced to move towards the wave source. The mechanism for this is the generation of surface vortices by waves propagating away from vertically oscillating plungers. We introduce a new conceptual framework for understanding wave-driven flows, which enables us to engineer inward and outward surface jets, stationary vortices, and other complex flows. The results form a new basis for the remote manipulation of objects on fluid surfaces and for a better understanding of the motion of floaters in the ocean, the generation of wave-driven jets, and the formation of Lagrangian coherent structures.What is perceived as fluid motion on a surface perturbed by waves is a motion of the surface shape, not the fluid flow along the surface 6 . Trajectories of fluid parcels on the surface have been described analytically for progressing irrotational waves, where particles move in the direction of wave propagation [3][4][5][7][8][9] . Such waves are rare in nature and in the laboratory because finiteamplitude waves are unstable with respect to amplitude modulation, a phenomenon also known as the Benjamin-Feir instability 10 . Twodimensional (2D) waves of finite amplitude develop into 3D waves, forming complex wave patterns [11][12][13][14] . The motion of particles on the surface of such wave fields is not understood. It has been found recently that Faraday waves, which are parametrically excited 3D nonlinear waves, create vortices on the fluid surface that interact and lead to the development of 2D turbulence 1,15 . The generation of horizontal vortices by quasi-standing nonlinear waves 2 is an effect which is impossible in planar irrotational waves. In this paper we show that progressing nonlinear waves produced by a localized source are also capable of creating surface vortices. The interaction between such vortices is shown to lead to the formation of largescale surface flows, far away from a wave maker.In our experiments we generate progressing waves using vertically moving plungers, periodically inserted into the water. The wave fields are visualized using a diffusive light imaging technique 16 and fast video imaging of tracer particles on the fluid surface. The 3D fluid particle trajectories are tracked using a novel method, developed as part of this work, which is described in Methods and the Supplementary Information.

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Modulation instability and capillary wave turbulence

Cited by 45 publications

References 24 publications

Fluctuations of Energy Flux in Wave Turbulence

Fluctuations of Energy Flux in Wave Turbulence

Direct Numerical Investigation of Turbulence of Capillary Waves

Generation and reversal of surface flows by propagating waves

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