We analyze the effective diffusivity of a passive
scalar in a two dimensional, steady,
incompressible random flow that has mean zero and a stationary
stream function. We show that in the limit of small diffusivity or
large Peclet number, with convection dominating, there is substantial
enhancement of the effective diffusivity. Our analysis is based on
some new variational principles for convection difusion problems
and on some facts from continuum percolation theory, some of which
are widely believed to be correct but have not been proved yet.
We show in detail how the variational principles
convert information about the geometry of the level lines of
the random stream function into properties of the effective
diffusivity and substantiate the result of Isichenko and Kalda
that the effective diffusivity behaves like 
when the molecular diffusivity 
is small, assuming some
percolation theoretic facts. We also analyze the effective
diffusivity for a special class of convective flows, random
cellular flows, where the facts from percolation theory are
well established and their use in the variational principles is
more direct than for general random flows.