Non-Reciprocal Capillary Waves

Abstract

Capillary waves are a classical free-surface phenomenon in fluid mechanics, yet their behavior in chiral fluids remains largely unexplored. We show that odd viscosity breaks the reciprocity of capillary waves. Using linear theory together with fully nonlinear direct numerical simulations, we find that surface tension creates two inequivalent branches of odd capillary waves: a dispersive branch and a quasi-acoustic branch absent in the capillarity-free limit. Their unequal propagation and attenuation transform standing waves into traveling waves and produce an anomalously deep vortical boundary layer. Above a threshold odd viscosity, nonlinear accumulation of vorticity near the surface reverses the induced shear current and drives bulk particles opposite to the wave motion, giving rise to an anti-Stokes drift with no counterpart in conventional fluids. Our results show how combining capillarity with broken parity can be used to control wave propagation and transport at fluid interfaces, opening a route toward one-way fluidic waveguiding and chirality-programmed interfacial flows.

Figures (6)

• Figure 1: Non-reciprocal capillary waves with odd viscosity. Dynamics of capillary waves with even shear viscosity $\mathrm{Oh}^{\mathcal{E}}=0.001$ and amplitude $a=0.2\lambda$, shown for three simulations with $\mathrm{Oh}^{\mathcal{O}}=0$, $\mathrm{Oh}^{\mathcal{O}}=0.01$, and $\mathrm{Oh}^{\mathcal{O}}=0.2$. (A) Horizontal velocity field ($u$) at $t=1$ and spatiotemporal evolution of the interfacial height $h$ over $t\in[0,2]$. The onset of unidirectional wave motion is visible in the flow field, which exhibits a net velocity to the left. (B) Vorticity field at $t=20$, highlighting the vortical boundary layers that form near the interface. The penetration depth of this layer increases markedly with odd viscosity, together with the associated shear current, leading to the emergence of Stokes-like and anti-Stokes-like drift. (C) Lagrangian tracer particles placed inside the liquid. For each simulation, the trajectories of two particles are shown: one initially near the surface and one deeper in the fluid, below the boundary layer (initial positions marked by white circles). Depending on depth and odd viscosity, tracers drift either with the wave or against it.
• Figure 2: Dispersion branches and penetration depth.(A) Dispersion branches for odd capillary waves, extracted from simulations initialized with a boxcar-like perturbation of width $w=1$ and height $h=0.1$ by identifying the peaks of the Fourier-transformed signal. (B,C) Examples of the corresponding spectra in Fourier space for $\mathrm{Oh}^{\mathcal{O}}=0$ and $\mathrm{Oh}^{\mathcal{O}}=0.2$, respectively. With increasing odd viscosity, the dispersion branches become progressively asymmetric, approaching a linear negative branch and a quadratic positive branch when $\mathrm{Oh}^{\mathcal{O}}\sim 1$. (D) Penetration depth measured from low-amplitude sinusoidal waves with $a=0.05$, using the characteristic decay length of the vorticity, defined as the depth at which the vorticity falls to $1/e$ of its surface value. The measurements are compared with the linear prediction $m^{-1}$; when the negative mode dominates, the data follow the trend of the diverging penetration depth. The measured decay length is nevertheless larger than predicted by linear theory.
• Figure 3: Tracer drift and drift inversion.(A) Drift of a tracer particle in the bulk for increasing odd viscosity, showing the inversion from leftward to rightward motion at large $\mathrm{Oh}^{\mathcal{O}}$. All particles are initially positioned at the same initial relative height from the interface. (B) Drift-velocity distribution and the average velocity obtained from 50 tracer particles placed across a full wave-length and at different relative depths of $0$ to $-0.5$ from the interface and temprally averaged at late time of $t \in[8,10]$, showing the transition from conventional drift ($U_D<0$) to anomalous drift ($U_D>0$) at high $\mathrm{Oh}^{\mathcal{O}}$. The colored circles highlight the examples shown in panel A.
• Figure 4: Nonlinear non-reciprocal waves.(A) Influence of wave initial amplitude, $a$, and odd viscosity, $\mathrm{Oh}^{\mathcal{O}}$, on the waveform and vortical boundary layer. Increasing nonlinearity together with odd viscosity steepens the wave and enhances the concentration of vorticity near the surface. (B) Parameter map of the mean drift velocity as a function of wave amplitude, $a$, and odd viscosity, $\mathrm{Oh}^{\mathcal{O}}$. The drift inversion is absent in the weakly nonlinear regime ($a\ll 1$) and, once it appears, its threshold is governed primarily by $\mathrm{Oh}^{\mathcal{O}}$.
• Figure 5: An example of adaptive mesh refinement near the interface.