Structured waves are ubiquitous for all areas of wave physics, both classical and quantum, where the wavefields are inhomogeneous and cannot be approximated by a single plane wave. Even the interference of two plane waves, or of a single inhomogeneous (evanescent) wave, provides a number of nontrivial phenomena and additional functionalities as compared to a single plane wave. Complex wavefields with inhomogeneities in the amplitude, phase, and polarization, including topological----- structures and singularities, underpin modern nanooptics and photonics, yet they are equally important, e.g. for quantum matter waves, acoustics, water waves, etc. Structured waves are crucial in optical and electron microscopy, wave propagation and scattering, imaging, communications, quantum optics, topological and non-Hermitian wave systems, quantum condensed-matter systems, optomechanics, plasmonics and metamaterials, optical and acoustic manipulation, and so forth. This Roadmap is written collectively by prominent researchers and aims to survey the role of structured waves in various areas of wave physics. Providing background, current research, and anticipating future developments, it will be of interest to a wide cross-disciplinary audience.

Fluctuation-induced forces are observed in numerous physical systems spanning from quantum to macroscopic scale. However, there is as yet no experimental report of their existence in hydrodynamic turbulence. Here, we present evidence of an attraction force mediated via turbulent fluctuations by using two walls locally confining 2D turbulence. This long-range interaction is a function of the wall separation and the energy injection rate in the turbulent flow. As the wall spacing decreases, the confined flow becomes less energetic and more anisotropic in the bounded domain, producing stronger attraction. The mechanism of force generation is rooted in a nontrivial fluid-wall coupling where coherent flow structures are guided by the cavity walls. For the narrowest cavities studied, a resonance phenomenon at the flow forcing scale leads to a complex short-range interaction. The results could be relevant to problems encountered in a range of fields from industrial multiphase flows to modeling of planetary formation.

Spin is a fundamental yet nontrivial intrinsic angular momentum property of quantum particles or fields, which appears within relativistic field theory. The spin density in wave fields is described by the theoretical Belinfante-Rosenfeld construction based on the difference between the canonical and kinetic momentum densities. These quantities are usually considered as abstract and non-observable per se. Here, we demonstrate, both theoretically and experimentally, that the Belinfante-Rosenfeld construction naturally arises in gravity (water surface) waves. There, the canonical momentum is associated with the generalized Stokes drift phenomenon, while the spin is generated by subwavelength circular motion of water particles. Thus, we directly observe these fundamental field theory properties as microscopic mechanical properties of a classical wave system. Our findings shed light onto the nature of spin and momentum in wave fields, demonstrate the universality of relativistic field theory concepts, and offer a new platform for their studies.

Angular momentum of spinning bodies leads to their remarkable interactions with fields, waves, fluids, and solids. Orbiting celestial bodies, balls in sports, liquid droplets above a hot plate, nanoparticles in optical fields, and spinning quantum particles exhibit nontrivial rotational dynamics. Here, we report self-guided propulsion of magnetic fast-spinning particles on a liquid surface in the presence of a solid boundary. Above some critical spinning frequency, such particles generate localized 3D vortices and form composite "spinner-vortex" quasiparticles with nontrivial, yet robust dynamics. Such spinner-vortices are attracted and dynamically trapped near the boundaries, propagating along the wall of any shape similarly to "liquid wheels." The propulsion velocity and the distance to the wall are controlled by the angular velocity of the spinner via the balance between the Magnus and wall repulsion forces. Our results offer a new type of surface vehicles and provide a powerful tool to manipulate spinning objects in fluids.

We characterize a process of energy extraction via rectification of strongly turbulent flow by using tools of stochastic thermodynamics. We study the dynamics of an asymmetric autonomous rotor that shows biased direction of rotation when placed in a stream. We give experimental evidence that a fluctuation theorem can be used to describe the work injected in the rotor via its coupling with the turbulent flow structure. This approach allows to measure the mean power extracted from the chaotic fluid motion over a broad range of turbulent kinetic energy. A nontrivial dependence of the rotor power on flow kinetic energy is identified. This observation is described by a model taking into account the dissipation of the rotor energy and the temporal memory of coherent structures present in the turbulent flow.

Formation of bacterial biofilms on solid surfaces within a fluid starts when bacteria attach to the substrate. Understanding environmental factors affecting the attachment and the early stages of the biofilm development will help develop methods of controlling the biofilm growth. Here, we show that biofilm formation is strongly affected by the flows in thin layers of bacterial suspensions controlled by surface waves. Deterministic wave patterns promote the growth of patterned biofilms, while wave-driven turbulent motion discourages patterned attachment of bacteria. Strong biofilms form under the wave antinodes, while inactive bacteria and passive particles settle under nodal points. By controlling the wavelength, its amplitude, and horizontal mobility of the wave patterns, one can shape the biofilm and either enhance the growth or discourage the formation of the biofilm. The results suggest that the deterministic wave-driven transport channels, rather than hydrodynamic forces acting on microorganisms, determine the preferred location for the bacterial attachment.

Bacterial cellulose is an important class of biomaterials which can be grown in well-controlled laboratory and industrial conditions. The cellulose structure is affected by several biological, chemical and environmental factors, including hydrodynamic flows in bacterial suspensions. In this work, we explore the possibility of using well controlled turbulent flow to control the bacterial cellulose production. Turbulent flows leading to random motion of fluid elements may affect the structure of the extracellular polymeric matrix produced by bacteria. Here we show that two-dimensional turbulence at the air-liquid interface generates chaotic rotation at a well-defined scale and random persistent stretching of the fluid elements. The results offer new approaches to engineering of the bacterial cellulose structure by controlling turbulence parameters.

The formation of bacterial biofilms on solid surfaces within a fluid starts when bacteria attach to the substrate. Understanding environmental factors affecting the attachment and the early stages of the biofilm development as well as the development of active methods of biofilm control are crucial for many applications. Here we show that biofilm formation is strongly affected by the hydrodynamics of flows generated by surface waves in layers of bacterial suspensions. Deterministic wave patterns promote the growth of patterned biofilms while wave-driven turbulent motion destroys the patterns. The location of the attached bacteria on a solid substrate differs from the settlement location of inactive bacteria and of the passive micro-particles: strong biofilms form under the wave antinodes while passive particles and inactive bacteria settle under nodal points. By controlling the wave lengths and horizontal mobility of the wave patterns, one can either shape the biofilm formation and enhance the biofilm growth, or discourage the formation of biofilm patterns. The results suggest that the deterministic wave-driven transport, rather than hydrodynamic forces, determine the preferred location for the bacterial attachment.

Experimental investigation of particle pair separation is conducted in two types of laboratory two-dimensional turbulence under a broad range of experimental conditions. In the range of scales corresponding to the inverse energy cascade inertial interval, the particle pair separation exhibits diffusive behaviour. The analysis of the pair velocity correlations suggests the existence of coherent bundles or clusters of non-diverging fluid particles. Such bundles are also detected using a recently developed topological tool based on the concept of braids. The bundles are observed as meandering streams whose width is determined by the turbulence forcing scale. In such locally anisotropic turbulence, the particle pair dispersion depends on the initial particle separation and on the width of the bundles.

The ability of a device to exploit the energy of a flow to generate thrust is the main feature of passive propulsion. In a turbulent flow, such energy conversion is challenging due to the unpredictable and disordered fluid motion. In wave-driven turbulence, it has recently been demonstrated that asymmetric floating rotors can tap the energy of ambient fluctuations to fuel directed rotation. Here we report on the dynamics of asymmetric floating vehicles capable of passively propelling themselves in two-dimensional turbulence. We show experimentally how the shape of a floater and its rotational dynamics conspire to allow harvesting energy of the turbulent fluid motion. The translation and rotation of the floater are shown to be strongly coupled. The propulsion velocity and the rotational diffusion timescale depend on the relative size of the floating vehicle with respect to the turbulence forcing scale. The geometry of the floater is investigated in the range of circular-sector-shaped objects and a shape optimizing its propulsion is identified. At times larger than the rotational diffusion timescale, our results shed light on a substantial increase of the turbulent diffusion coefficient of anisotropic objects due to the coupling between propulsion and rotational diffusion.

We show that rotating particles at the liquid-gas interface can be efficiently manipulated using the surface-wave analogue of optical lattices. Two orthogonal standing waves generate surface flows of counter-rotating half-wavelength unit cells, the liquid interface metamaterial, whose geometry is controlled by the wave phase shift. Here we demonstrate that by placing active magnetic spinners inside such metamaterials, one makes a powerful tool which allows manipulation and self-assembly of spinners, turning them into vehicles capable of transporting matter and information between autonomous metamaterial unit cells. We discuss forces acting on a spinner carried by a nonuniform flow and show how the forces confine spinners to orbit inside the same-sign vortex cells of the wave-driven flow. Reversing the spin, we move the spinner into an adjacent cell. By changing the spinning frequency or the wave amplitude, one can precisely control the spinner orbit. Multiple spinners within a unit cell self-organize into stable patterns, e.g., triangles or squares, orbiting around the center of the cell. Spinners having different frequencies can also be confined, such that the higher-frequency spinner occupies the inner orbit and the lower-frequency one circles on the outer orbit, while the orbital motions of both spinners are synchronized.

We report an abrupt change in the diffusive transport of inertial objects in wave-driven turbulence as a function of the object size. In these non-equilibrium two-dimensional flows, the turbulent diffusion coefficient D of finite-size objects undergoes a sharp change for values of the object size r(p) close to the flow forcing scale L-f. For objects larger than the forcing scale (r(p) > L-f), the diffusion coefficient is proportional to the flow energy U-2 and inversely proportional to the size r(p). This behaviour, D similar to U-2/r(p), observed in a chaotic macroscopic system is reminiscent of a fluctuation-dissipation relation. In contrast, the diffusion coefficient of smaller objects (r(p) < L-f) follows D similar to U/r(p)(0.35). This result does not allow simple analogies to be drawn but instead it reflects strong coupling of the small objects with the fabric and memory of the out-of-equilibrium flow. In these turbulent flows, the flow structure is dominated by transient but long-living bundles of fluid particle trajectories executing random walk. The characteristic widths of the bundles are close to Lf. We propose a simple phenomenology in which large objects interact with many bundles. This interaction with many degrees of freedom is the source of the fluctuation-dissipation-like relation. In contrast, smaller objects are advected within coherent bundles, resulting in diffusion properties closely related to those of fluid tracers.

In this paper, we demonstrate experimentally that by generating two orthogonal standing waves at the liquid surface, one can control the motion of floating microparticles. The mechanism of the vortex generation is somewhat similar to a classical Stokes drift in linear progression waves. By adjusting the relative phase between the waves, it is possible to generate a vortex lattice, seen as a stationary horizontal flow consisting of counter-rotating vortices. Two orthogonal waves which are phase-shifted by π / 2 create locally rotating waves. Such waves induce nested circular drift orbits of the surface fluid particles. Such a configuration allows for the trapping of particles within a cell of the size about half the wavelength of the standing waves. By changing the relative phase, it is possible to either create or to destroy the vortex crystal. This method creates an opportunity to confine surface particles within cells, or to greatly increase mixing of the surface matter over the wave field surface.

We report on the transport properties and orientational dynamics of ellipsoidal objects advected by laboratory two-dimensional turbulence. It is found that ellipsoids of different sizes have preferential direction of transport, either along their major axes or minor axes. The two components of the ellipsoid diffusion coefficient depend on the ratio of the length of the ellipsoids along major axes aa to the turbulence forcing scale Lf. Large ellipsoids (aa > Lf) diffuse faster in the direction parallel to their major axes. In contrast, small ellipsoids diffuse faster in the direction transverse to their major axes. We study this transition vs the ratio aa/Lf and relate it to the coupling between translational and rotational motion of anisotropic objects. The features of the turbulent transport of ellipsoids can be understood by considering the interaction of these anisotropic objects with the underlying structure of two dimensional turbulent flows made of meandering coherent bundles.

Mixing of a passive scalar in a fluid (e.g. a radioactive spill in the ocean) is the irreversible process towards homogeneous distribution of a substance. In a moving fluid, due to the chaotic advection [H. Aref, J. Fluid Mech. 143 (1984) 1; J. M. Ottino, The Kinematics of Mixing: Stretching,Chaos and Transport (Cambridge University Press, Cambridge, 1989)] mixing is much faster than if driven by molecular diffusion only. Turbulence is known as the most efficient mixing flow [B. I. Shraiman and E. D. Siggia, Nature 405 (2000) 639]. We show that in contrast to spatially periodic flows, two-dimensional turbulence exhibits local reversibility in mixing, which leads to the generation of unpredictable strong fluctuations in the scalar concentration. These fluctuations can also be detected from the analysis of the fluid particle trajectories of the underlying flow.

Turbulence is a mechanism leading to energy dissipation, however it also accumulates energy by spreading it over a range of scales. This valuable energy reservoir is known as the inertial interval. The broader this interval is, the more energy is stored and an interesting question is whether it is possible to efficiently use this energy. Recent advances in the understanding of turbulence rely on the trajectory-based or Lagrangian description of the flow. Here we show how to extract energy from the inertial interval of two-dimensional turbulence by taking advantage of its fine Lagrangian structure. A floating object in wavedriven turbulence can exploit the fluid erratic motion to fuel either directional propulsion or rotation. The shape of the object controls its ability to become a vehicle or a rotor that can tap the energy of correlated bundles of fluid trajectories. These findings offer methods of creating self-propelled devices or turbines utilizing the energy of turbulence.