Turbulent Prandtl number effect on passive scalar advection

A generalization of Kraichnan's model of passive scalar advection is considered. Physically motivated regularizations of the model are considered which take into account both the effects of viscosity and molecular diffusion. The balance between these two effects on the inertial range behavior for the scalar is shown to be parameterized by a new turbulent Prandtl number. Three different regimes are identified in the parameter space depending on degrees of compressibility. In the strongly and weakly compressible regimes, the inertial range behavior of the scalar does not depend on the turbulent Prandtl number. In the regime of intermediate compressibility, the inertial range behavior does depend on the turbulent Prandtl number.