  We consider a class of invariant measures for a passive scalar $f$ driven by
an incompressible velocity field $\boldsymbol{u}$, on a $d$-dimensional
periodic domain, satisfying $$ \partial_t f + \boldsymbol{u} \cdot \nabla f =
0, \qquad f(0)=f_0. $$ The measures are obtained as limits of stochastic
viscous perturbations. We prove that the span of the $H^1$ eigenfunctions of
the operator $\boldsymbol{u}\cdot\nabla$ contains the support of these
measures. We also analyze several explicit examples: when $\boldsymbol{u}$ is a
shear flow or a relaxation enhancing flow (a generalization of weakly mixing),
we can characterize the limiting measure uniquely and compute its covariance
structure. We also consider the case of two-dimensional cellular flows, for
which further regularity properties of the functions in the support of the
measure can be deduced.
  The main results are proved with the use of spectral theory results, in
particular the RAGE theorem, which are used to characterize large classes of
orbits of the inviscid problem that are growing in $H^1$.
