Fractional Brownian motion of particles in capillary waves

On the surface of a vertically oscillating fluid, capillary waves with a clearly discernible wavelength λ are formed if the amplitude of the oscillations exceeds a critical value. Particles sprinkled on the fluid surface are experimentally found to move in an almost Brownian motion when measured over distances larger than λ. We extend earlier studies of the diffusivity to length scales ranging from 0.1λ to 10λ. We observe a cross-over in the diffusive motion from a strongly anomalous diffusion below λ, to a motion that is closer to being Brownian above λ. Our observations show that the particle motion is well described by an amplitude-independent fractional Brownian motion, effective at sizes less than λ, convoluted with an amplitude-dependent fractional Brownian motion, effective on all length scales smaller than the system size. At large amplitudes our results are in surprising agreement with diffusivity measurements from upper-ocean studies.