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

We consider the evolution of a decaying passive scalar in the presence of a gaussian white noise fluctuating linear shear flow. We focus on deterministic initial data and establish the short, intermediate, and long time symmetry properties of the evolving point wise probability measure for the random passive scalar. We identify regions 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. A 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. 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 full Monte-Carlo simulations (FMC) of the associated stochastic differential equation. Armed with this benchmark, we apply the FMC to a channel with a no-flux boundary condition observe a dramatically different long time state resulting from the existence of the walls.