  The study of passive scalar transport in a turbulent velocity field leads
naturally to the notion of generalized flows which are families of probability
distributions on the space of solutions to the associated ODEs, which no longer
satisfy the uniqueness theorem for ODEs. Two most natural regularizations of
this problem, namely the regularization via adding small molecular diffusion
and the regularization via smoothing out the velocity field are considered.
White-in-time random velocity fields are used as an example to examine the
variety of phenomena that take place when the velocity field is not spatially
regular. Three different regimes characterized by their degrees of
compressibility are isolated in the parameter space. In the regime of
intermediate compressibility, the two different regularizations give rise to
two different scaling behavior for the structure functions of the passive
scalar. Physically this means that the scaling depends on Prandtl number. In
the other two regimes the two different regularizations give rise to the same
generalized flows even though the sense of convergence can be very different.
The ``one force, one solution'' principle and the existence and uniqueness of
an invariant measure are established for the scalar field in the weakly
compressible regime, and for the difference of the scalar in the strongly
compressible regime.
