Noise-induced drift of stochastic swimmers in a Kolmogorov-like flow

We study a model of an elongated, noisy swimmer in a planar Kolmogorov-like flow and investigate the connection between the stochastic swimmer trajectories and the swimmer density in phase space.  Monte Carlo simulations of our model exhibit nonuniform swimmer density profiles in the cross-stream direction, similar to those observed in channel flows.  They also exhibit a highly nonuniform density in the swimmer position-orientation phase space, whose peaks are not easily associated with the phase space structures of the deterministic model.  We propose a new deterministic model for the motion of noisy swimmers which includes a noise-induced drift term added to the swimmer equations of motion.  Our noise drift model possesses attracting limit cycles.  We show that these limit cycles help explain the nonuniform densities we see in the Monte Carlo simulations, and we classify these nonuniform distributions quantitatively using the depletion index.