On the symmetry properties of a random passive scalar with and without boundaries, and their connection between hot and cold states

Abstract

We consider the evolution of a decaying passive scalar in the presence of a gaussian white noise fluctuating linear shear flow known as the Majda Model. We focus on deterministic initial data and establish the short, intermediate, and long time symmetry properties of the evolving point wise probability measure (PDF) for the random passive scalar. We identify, for the cases of both point source and line source initial data, regions in the x-y plane outside of which the PDF skewness is sign definite for all time, while inside these regions we observe multiple sign changes corresponding to exchanges in symmetry between hot and cold leaning states using exact representation formula for the PDF at the origin, and away from the origin, using numerical evaluation of the exact available Mehler kernel formulas for the scalars statistical moments. A new, rapidly convergent Monte-Carlo method is developed, dubbed Direct Monte-Carlo (DMC), using the available random Green's functions which allows for the fast construction of the PDF for single point statistics, as well as multi-point statistics including spatially integrated quantities natural for full Monte-Carlo simulations of the underlying stochastic differential equations (FMC). This new method demonstrates the full evolution of the PDF from short times, to its long time, limiting and collapsing universal distribution at arbitrary points in the plane. Further, this method provides a strong benchmark for FMC and we document numbers of field realization criteria for the FMC to faithfully compute this complete dynamics. Armed with this benchmark, we apply the FMC to a channel with a no-flux boundary condition enforced on parallel planes and observe a dramatically different long time state resulting from the existence of the wall. In particular, the channel case collapsing invariant measure has negative skewness, with random states heavily leaning heavily towards the hot state, in stark contrast to free space, where the limiting skewness is positive, with its states leaning heavily towards the cold state. arXiv:1802.03340v1 [physics.flu-dyn]

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

“…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 [6]. Our analytical result verifies the conclusion for the linear shear flow obtained from numerical simulations in [11]. Moreover, we demonstrate that the limiting distribution is negatively skewed for any shear flow at sufficiently low Péclet number.…”

“…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 [11] 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 [26].…”

“…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 [11] 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 [26].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 [6]. Our analytical result verifies the conclusion for the linear shear flow obtained from numerical simulations in [11].…”

Ergodicity and invariant measures for a diffusing passive scalar advected by a random channel shear flow and the connection between the Kraichnan-Majda model and Taylor-Aris Dispersion

“…For periodic boundary conditions, Bronski and McLaughlin [7] carried out a second order perturbation expansion for the ground state of periodic Schrödinger equations to analyze the inherited probability measure for a passive scalar field advected by periodic shear flows with multiplicative white noise. In [11,10], equation ( 3) was studied with a stationary OU process, where a dramatically different long time state resulting from the existence of the impermeable boundaries was found. In particular, the PDF of the scalar in the channel case has negative skewness for sufficiently small Péclet number, in stark contrast to free space, where the limiting skewness is strictly positive for all Péclet number.…”

“…This established that for integrable random initial data the PDF would 'Gaussianize' at long times, whereas short ranged, random wave initial data would produce divergent flatness factors in the same limit as finite times. Recently, the role of impermeable boundaries in the Majda model has been explored in a parallel-plate channel with deterministic initial conditions [11,10]. Those works demonstrate that the sign of long time PDF skewness could be controlled by Péclet numbers and the correlation time of the velocity field, in strong contrast with the free space result, where the long time PDF skewnness is strictly positive.…”