Persisting asymmetry in the probability distribution function for a random advection–diffusion equation in impermeable channels

Abstract

In this paper, we study the effect of impermeable boundaries on the symmetry properties of a random passive scalar field advected by random flows. We focus on a broad class of nonlinear shear flows multiplied by a stationary, Ornstein–Uhlenbeck (OU) time varying process, including some of their limiting cases, such as Gaussian white noise or plug flows. For the former case with linear shear, recent studies (Camassa et al., 2019) numerically demonstrated that the decaying passive scalar’s long time limiting probability distribution function (PDF) could be negatively skewed in the presence of impermeable channel boundaries, in contrast to rigorous results in free space which established the limiting PDF is positively skewed (McLaughlin and Majda, 1996). Here, the role of boundaries in setting the long time limiting skewness of the PDF is established rigorously for the above class using the long time asymptotic expansion of the N-point correlator of the random field obtained from the ground state eigenvalue perturbation approach proposed in Bronski and McLaughlin (1997). Our analytical result verifies the conclusion for the linear shear flow obtained from numerical simulations in Camassa et al. (2019). Moreover, we demonstrate that the limiting distribution is negatively skewed for any shear flow at sufficiently low Péclet number. We demonstrate the convergence of the Ornstein–Uhlenbeck case to the white noise case in the limit γ→∞ of the OU damping parameter, which generalizes the results for free space in Resnick (1996) to the channel domain problem. We show that the long time limit of the first three moments depends explicitly on the value of γ, which is in contrast to the conclusion in Vanden Eijnden (2001) for the limiting PDF in free space. To find a benchmark for theoretical analysis, we derive the exact formula of the N-point correlator for a flow with no spatial dependence and Gaussian temporal fluctuation, generalizing the results of Bronski et al. (2007). The long time analysis of this formula is consistent with our theory for a general shear flow. All results are verified by Monte-Carlo simulations.