Error analysis of an acceleration corrected diffusion approximation of Langevin dynamics with background flow

We consider the problem of approximating the Langevin dynamics of inertial particles being transported by a background flow. In particular, we study an acceleration corrected advection-diffusion approximation to the Langevin dynamics, a popular approximation in the study of turbulent transport. We prove error estimates in the averaging regime in which the dimensionless relaxation timescale $\varepsilon$ is the small parameter. We show that for any finite time interval, the approximation error is of order $\mathcal{O}(\varepsilon)$ in the strong sense and $\mathcal{O}(\varepsilon^2)$ in the weak sense, whose optimality is checked against computational experiment. Furthermore, we present numerical evidence suggesting that this approximation also captures the long-time behavior of the Langevin dynamics.