Scaling properties of passive scalars in one dimension

This paper is a set of notes about work in progress. Since the work is directed toward scaling, it seems quite appropriate to report this in a context devoted to Ben Widomʹs scientific contributions. Model equations for the motion of a passive scalar in one dimension are set up and some scaling properties of their solutions are derived. All these models follow the approach set up by Kraichnan in which the driving velocities have Gaussian correlations with scaling properties in space, but zero-range correlations in time. The simplest and most natural versions of the model fail to satisfy an incompressibility condition. In another version, the model does have an incompressible flow. To permit the incompressibility, there are two pipes — each of which is described by temperature and velocity fields which are functions of a single coordinate, x, and time. Flow from one pipe to the other transfers both mass and heat. This model has two conserved quantities in the limit of zero dissipation. In still another version, fluid elements are interchanged in position so that there are an infinite number of conservation laws in the no-dissipation limit. Scaling properties of the resulting two-point functions are derived.