Generation and reversal of surface flows by propagating waves

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 vortices1,2, yet it is not known whether propagating waves can generate large-scale flows—small-amplitude irrotational waves only push particles in the direction of propagation3–5. 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 oscillatingplungers.We introduceanewconceptual 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 surface6. Trajectories of fluid parcels on the surface have been described analytically for progressing irrotational waves, where particles move in the direction of wave propagation3–5,7–9. Such waves are rare in nature and in the laboratory because finiteamplitudewaves are unstable with respect to amplitudemodulation, a phenomenon also known as the Benjamin–Feir instability10. Twodimensional (2D) waves of finite amplitude develop into 3D waves, forming complex wave patterns11–14. 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 turbulence1,15. The generation of horizontal vortices by quasi-standing nonlinear waves2 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 technique16 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 set-up 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 time-averaged 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 instability17,18 (Fig. 1c,d). As the modulation grows and the crosswave instability breaks the wavefront into trains of propagating pulses, the wave field becomes three-dimensional (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 SupplementaryMovie 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 soliton-like propagating pulses18 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 wavemaker (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 200Hz. 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