Abstract
The ability to send a wave to fetch an object from a distance would find a broad range of applications. Quasistanding Faraday waves on water create horizontal vortices^{1,2}, yet it is not known whether propagating waves can generate largescale flows—smallamplitude irrotational waves only push particles in the direction of propagation^{3,4,5}. Here we show that when waves become threedimensional 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 wavedriven 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 wavedriven jets, and the formation of Lagrangian coherent structures.
Main
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 quasistanding 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.
A cylindrical wave maker oscillates at low amplitude, as illustrated in the schematic of Fig. 1g (for details of the experimental setup see Supplementary Section 1). The wave maker produces nearly planar propagating wavefronts, as seen in Fig. 1a. To visualize the fluid motion, buoyant tracer particles are uniformly dispersed over the fluid surface. The particles are pushed in the direction of the wave propagation, forming an outward jet, as seen in the timeaveraged particle streak image of Fig. 1b. 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). As the modulation grows and the crosswave instability breaks the wavefront into trains of propagating pulses, the wave field becomes threedimensional (see Fig. 1c, as well as Supplementary Fig. 1 and Supplementary Movie 1). Simultaneously, the direction of the central jet reverses. It now pushes floaters towards the wave maker and against the wave propagation. The flow is strong enough to move floating objects far from the plunger on the water surface (see Supplementary Movie 2). The motion of the floater can thus be reversed simply by changing the amplitude of the wave maker oscillations.
What is the reason for the flow reversal? In the nonlinear regime the transverse modulation of the wavefronts is strongest (highest amplitude) in the near field (one to two wavelengths away from the wave maker), where solitonlike propagating pulses ^{18} are generated, as seen in the digital representation of the experimentally measured surface elevation (Fig. 1c). The reversal of the mean flow in the far field (tens or hundreds of wavelengths away from the wave maker) is always correlated with the generation of stochastic Lagrangian trajectories within a flow region in front of the wave maker (Fig. 1e). This localized complex chaotic flow efficiently transports fluid in the direction perpendicular to the propagation of the wave pulses. The net result is a stochastic pumping, which seems to be responsible for the ejection of surface fluid parcels parallel to the wave maker. This transport is compensated by the fluid flow towards the centre of the wave source, forming a ‘tractor beam’ against the wave propagation. The velocity of the central jet changes gradually with the increase in the vertical acceleration of the wave maker. As shown in Fig. 1f, at low amplitudes, the flow is outwards and the velocity increases with the increase in forcing. When the threshold of the modulation instability is reached, the flow reverses abruptly and the inward jet velocity saturates at higher forcing levels. Such a behaviour is observed for a wide range of excitation frequencies, from 10 to 200 Hz.
The outward/inward central jets are compensated by the return flows towards/away from the sides of a cylindrical wave maker (Fig. 1b, d). These return flows are guided by the walls of the wave tank forming the dipole structures on both sides of the wave maker. The larger the tank, the larger are the vortices forming dipoles (this was confirmed by performing experiments in different tanks). If one guides the return flow by introducing baffles isolating central jets from the return flow, as shown in Fig. 2a, the lengths of the central jets can be greatly increased. In this configuration, the widths of the jets are determined by the width of the cylindrical wave maker. Figure 2b shows the mean velocity of the central jet inside the channel formed by the baffles as the forcing is increased. The presence of the baffles does not affect the flow reversal of the jet above the threshold of modulation instability, consistent with the case of no baffles (Fig. 1f). Above the threshold, the inward mean flow velocity grows with increasing vertical acceleration and then saturates. Such a trend is found for a wide range of excitation frequencies, as illustrated in Fig. 2c. The mean flow velocity is approximately constant as a function of the distance from the wave maker (see Fig. 2d for an example). In these experiments the tractor beam extends over several hundreds of wavelengths. To obtain efficient pumping, the length of the cylindrical plunger should greatly exceed the length of the crosswave (the experiments suggest that the mode number of the crosswave should be greater than ten).
Although the mean flow is driven by waves, which significantly affect the fluid motion only near the plunger, the mean inward flow establishes far from the wave maker. This is unexpected, especially at the higher excitation frequencies, where the viscous dissipation is very high and waves decay at distances of less than ten wavelengths. The flow that develops is quasitwodimensional: the jets are sustained in a rather narrow layer at the water surface. Figure 2e shows the vertical profile of the jet velocity, which decays very fast with the depth, in a layer of about 10–15 mm. Thus, the depth of the water in these experiments (80 mm) does not affect any results reported here.
To understand how the stochastic region forms in front of the cylindrical wave maker, we performed experiments using various plunger shapes. If one uses a conical wave maker at low amplitudes, nearly circular wavefronts are produced (Supplementary Fig. 1g). Above the modulation instability threshold, at higher acceleration, the wave amplitude becomes modulated in the transverse direction, forming a complex 3D wave field (Fig. 3a) similar to the cylinder case. The wave field is made of periodic pulses propagating away from the cone in the radial direction. The surface flow pattern in this case is rather complex, exhibiting vortices of different sizes (Fig. 3b). To better understand the origin of the steadystate flow pattern, we have obtained flow data during the startup phase, when nonlinear waves are formed during the onset of the modulation instability. The vorticity ω = (∂V_{y}/∂x − ∂V_{x}/∂y) (where V_{x} and V_{y} are horizontal velocity components) is initially created in the vicinity of the plunger, where waves are the steepest. A circular array of counterrotating vortices is formed (Fig. 3c). Then, two rows of vortices are formed (Fig. 3d). Finally, vortices spread in the direction of the wave propagation, as seen in Fig. 3e, f. This result indicates that the ability of waves to create vorticity is not limited to Faraday quasistanding waves^{2}. As shown, propagating nonlinear 3D waves also inject vorticity into the surface flow.
The larger vortices seen in Fig. 3b further away from the wave maker are not generated at the increased liquid viscosity. This regime allows better visualization of the underlying mechanism of vorticity generation by propagating waves close to the plunger. By increasing the liquid viscosity, we restrict the number of vortices in the surface flow, as shown in Fig. 4d. The visualization of the 3D floater trajectories and waves reveals that particles which are close to the maxima of the wave pulses move away from the wave maker, as seen in Fig. 4a–c. Further away they start to drift sideways and then reverse direction to move towards the wave source along the trajectories where wave heights are lower. The shapes of the 3D trajectories are not trochoids (curves with loops), as expected from the smallamplitude wave models, but rather complex modulated trajectories, indicating no resemblance to the classical Stokes drift picture.
In the conical plunger case, strong outward jets are formed along the wave maxima. These jets are clearly seen in the averaged particle streak photographs and in the velocity fields (Fig. 4d, e). The jets diverge and form stationary vortices, as seen in the flow velocity field of Fig. 4e. The vorticity generation here is similar to that produced by Kelvin–Helmholtz instability in Rayleigh–Taylor turbulence^{19}. This divergence of jets can be visualized by computing the Lyapunov exponents of the Lagrangian trajectories^{20} (Fig. 4f). The ridges of the maxima of the Lyapunov exponents, Λ, mark the boundaries of the jet stability (see Methods for details of the computation of Λ). Adjacent diverging jets form stable closed vortices.
Wavejetdriven vortices always form in the near field of any wave maker as soon as waves become unstable and threedimensional. In the case of the conical plunger the elliptical vortices are aligned along a distinct radial direction, whereas in the case of a cylinder these vortices are tightly packed such that they interact more strongly. The Lagrangian chaos in the near field of the cylinder appears as a result of the interaction between wavedriven vortices being pushed closer together.
Vorticity creation by steep 3D waves is thus responsible for the unexpected effect reported here—the generation of the tractor beam, or the inward jet, which moves particles against the wave propagation. The degree of interaction between wavedriven surface vortices depends on the shape of the plunger. It increases from conical to elliptical (see Supplementary Information), and to cylindrical wave makers.
The above results suggest new principles for generating surface flows using localized wave sources. The geometry of the flow depends on the shape of the wave maker and on the state of the wave field in front of the wave maker. Examples of various surface flows produced by different wave makers are discussed in Supplementary Section 4. By using triangular and square pyramidal wave makers it is possible to generate flows having threefold and fourfold patterns. Above the threshold of the modulation instability, the directions of jets and the return flows reverse, similarly to the case of a cylindrical plunger discussed here. All these methods work both for long gravity and short capillary waves, and are versatile enough to manipulate objects on the water surface. They can also be used for the controlled generation of Lagrangian coherent structures on the water surface, for example, to contain and stop the spread of surface pollutants^{21}.
Methods
Wave generation.
Surface waves are generated by vertically oscillating plungers of different shapes (cylindrical, conical, pyramidal, and so on). Spatially localized timeperiodic perturbations of the water surface generate waves propagating away from the plungers. The frequency of the plunger oscillations is varied in the range between 10 Hz, corresponding to gravity waves, and 200 Hz, corresponding to capillary waves. The dispersion relation at lower wave amplitudes is given by ω = (gk + αk^{3}/ρ)^{1/2}, where k is the wavenumber, g is the acceleration due to gravity, α is the surface tension and ρ is the fluid density. At higher amplitudes, the crosswave instability modulates wavefronts in the transverse direction, destroying the twodimensionality of the wave and breaking it into individual pulses, which then propagate away from the source. These propagating wave pulses oscillate vertically at half the driving frequency, f = f_{0}/2, as expected for parametrically excited waves. The 2D wave fields shown in Fig. 1a, c are reconstructed using a diffusive light imaging technique. The results of Figs 1 and 2 are obtained using a cylindrical wave maker of diameter d = 25 mm and length L = 130 mm. For Figs 3 and 4 we use a 60° angle conical plunger, submerged to a diameter of 50 mm at the waterline. Experiments are performed in a rectangular container (1.5 × 0.5 m^{2}) filled with water to a depth of 80 mm. The wave makers are driven by an electrodynamic shaker. The forcing is sinusoidal and monochromatic. The shaker frequency f_{0} = 20 Hz corresponds to a wavelength of λ = 12 mm. A vertically movable plunger is attached to the table of the electromagnetic shaker via the plunger frame. Linear lightemitting diode (LED) arrays on the sides of the transparent water tank illuminate surface particles whose motion is filmed from above using a highresolution video camera.
Flow characterization.
The flow characteristics are measured using a diffusive light imaging technique, particle image velocimetry and particle tracking velocimetry techniques. We use diffusive light imaging to visualize the surface elevation of the wave field^{2,16}. The fluid surface is illuminated by an LED panel placed underneath the transparent bottom of the container. A few per cent of milk added to water provides sufficient contrast to obtain a highresolution reconstruction of the wave field. The absorption coefficient is measured before each experiment, allowing the calibration of the surface elevation with a vertical resolution of 20 μm. Floating imaging particles are used to visualize the fluid motion on the water surface^{2}. Threedimensional Lagrangian trajectories are obtained using a combination of a twodimensional particle tracking velocimetry (PTV) technique and a subsequent evaluation of the local surface elevation along the trajectory. First, the horizontal (x–y) coordinates of each point on a trajectory are tracked using a nearestneighbour algorithm^{22}. Then, the particle elevations (z) are estimated as the mean of the wave elevation over a local window (500 μm radius) centred at the x–y particle coordinates at a given time. The 3D trajectories of the particle and the wave elevation are visualized using the 3D animation software Houdini (Side Effects Software).
Finitetime Lyapunov exponent.
The Lyapunov exponent measures the divergence rate between two adjacent trajectories. A finitetime Lyapunov exponent (FTLE) algorithm has been developed to compute the maximum Lyapunov exponents for the detection of Lagrangian coherent structures^{20,23}. We use this method to locate the line of maximum divergence in the flow, as shown in Fig. 4f. The PTV velocity fields are interpolated on a refined spatial grid of 600 × 600 with a time step of Δt = 0.002 s. The particle trajectories are obtained by numerical integration using a fourthorder Runge–Kutta method. Each trajectory x(t, x_{0}) starts at a position x_{0} at a fixed initial time t_{0}. By numerical differentiation we compute the largest singularvalue field λ_{max}(t, t_{0}, x_{0}) of the deformationgradient tensor field [∂x(t, t_{0}, x_{0})/∂x_{0}]^{T}/[∂x(t, t_{0}, x_{0})/∂x_{0}]. The FTLE field can be obtained for initial positions x_{0} as
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Acknowledgements
This work was supported by the Australian Research Council’s Discovery Projects funding scheme (DP110101525). H.X. would like to acknowledge the support of the Australian Research Council’s Discovery Early Career Research Award (DE120100364). The authors thank K. Szewc for developing the code for the finitetime Lyapunov exponent analysis used to generate Fig. 4f, and M. Gwynneth for his help with experimental setup. N.F. acknowledges the help of S. Ramsden of the National Computational Infrastructure, Vizlab, ANU with visualization of 3D flows and trajectories using the Houdini animation software. The research of G.F. was supported by the Minerva Foundation and the Binational Science Foundation (BSF).
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Affiliations
Research School of Physics and Engineering, The Australian National University, Canberra, ACT 0200, Australia
 Horst Punzmann
 , Nicolas Francois
 , Hua Xia
 & Michael Shats
Weizmann Institute of Science, Rehovot 76100, Israel
 Gregory Falkovich
Institute for Information Transmission Problems, Moscow 127994, Russia
 Gregory Falkovich
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Contributions
H.P., N.F. and M.S. designed and performed the experiments; H.X., N.F., G.F. and H.P. analysed the data. M.S. and G.F. wrote the paper. All authors discussed and edited the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to Michael Shats.
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