Water-Wave Vortices and Skyrmions

Topological wave structures—phase vortices, skyrmions, merons, etc.—are attracting enormous attention in a variety of quantum and classical wave fields. Surprisingly, these structures have never been properly explored in the most obvious example of classical waves: water-surface (gravity-capillary) waves. Here, we fill this gap and describe (i) water-wave vortices of different orders carrying quantized angular momentum with orbital and spin contributions, (ii) skyrmion lattices formed by the instantaneous displacements of the water-surface particles in wave interference, and (iii) meron (half-skyrmion) lattices formed by the spin-density vectors, as well as (iv) spatiotemporal water-wave vortices and skyrmions. We show that all these topological entities can be readily generated in linear water-wave interference experiments. Our findings can find applications in microfluidics and show that water waves can be employed as an attainable playground for emulating universal topological wave phenomena.

Figure 1

(a) Instantaneous water surfaces $\mathcal{Z}(x,y,0)$ and Eulerian water-surface particle trajectories $\mathcal{R}(x,y,t)$ for circular WWVs with different topological charges $\ell$ [Eqs. (2) and (3)]. The spin density $\mathbf{S}$ is directed normally to the elliptical particle trajectories and quantifies the AM of this elliptical motion. (b) The plane-wave spectrum of a circular WWV with color-coded phases for $\ell =1$. (c) The complex vertical-displacement field $Z(x,y)$ for WWVs from panel (a), with the phases and amplitudes coded by the colors and brightness, respectively. The white arrows indicate the second-order Stokes drift $\mathbf{U}$ [Eq. (6)], characterizing the wave momentum density.

Figure 2

Hexagonal lattice produced by the interference of three waves with equal frequencies, amplitudes, and color-coded phases shown in (c). (a) Instantaneous water surface $\mathcal{Z}(x,y,0)$ and water-surface particle displacements $\mathcal{R}(x,y,0)$ for the field (9). The displacement directions in the unit hexagonal cell is mapped onto the unit sphere, providing a skyrmion with the topological charge $Q=1$ [Eq. (10)]. (b) The unit displacement-direction field $\overline{\mathcal{R}}(x,y,0)$ represented by colors (vertical component $\overline{\mathcal{Z}}$) and black arrows (in-plane components ${\overline{\mathcal{R}}}_2$). (d) The complex vertical-displacement field $Z(x,y)$ and the Stokes drift $\mathbf{U}$ indicating the lattice of alternating WWVs with $\ell =\pm 1$. (e) The unit spin-density field $\overline{\mathbf{S}}(x,y)$ represented similar to (b). The hexagonal unit cell is split into triangular zones of spin merons (half-skyrmions) with topological charges $Q_S=\pm 1/2$ and centers with ${\overline{S}}_z=\pm 1$ corresponding to the $\ell =\pm 1$ vortices in (d).

Figure 3

The same as in Figs. 2, 2, and 2 but for the lattice of spatiotemporal WWVs and skyrmions. The frequency of one of the interfering waves is detuned by $\delta \omega$ and the corresponding $\delta k$. The complex vertical-displacement field $Z$ and the real field envelope ${\mathcal{R}}^{\prime }=\mathrm{Re}[\mathbf{R}]$ (without fast oscillations $e^{-i\omega t}$) are plotted over the spacetime domain $(t,y)$ at the fixed coordinate $x=0$. The temporal component of the spatiotemporal Stokes drift ${\mathbf{U}}^{\prime }=(U_t,U_y)$ is defined as $U_t=(k\omega /2\delta \omega )\mathrm{Im}[{\mathbf{R}}^{*}·({\partial }_t)\mathbf{R}]$.