  We consider the statistical inverse problem of estimating a background fluid
flow field $\mathbf{v}$ from the partial, noisy observations of the
concentration $\theta$ of a substance passively advected by the fluid, so that
$\theta$ is governed by the partial differential equation \[
\frac{\partial}{\partial t}{\theta}(t,\mathbf{x}) = -\mathbf{v}(\mathbf{x})
\cdot \nabla \theta(t,\mathbf{x}) + \kappa \Delta \theta(t,\mathbf{x}) \quad
\text{ , } \quad \theta(0,\mathbf{x}) = \theta_0(\mathbf{x}) \] for $t \in
[0,T], T>0$ and $\mathbf{x} \in \mathbb{T}=[0,1]^2$. The initial condition
$\theta_0$ and diffusion coefficient $\kappa$ are assumed to be known and the
data consist of point observations of the scalar field $\theta$ corrupted by
additive, i.i.d. Gaussian noise. We adopt a Bayesian approach to this
estimation problem and establish that the inference is consistent, i.e., that
the posterior measure identifies the true background flow as the number of
scalar observations grows large. Since the inverse map is ill-defined for some
classes of problems even for perfect, infinite measurements of $\theta$,
multiple experiments (initial conditions) are required to resolve the true
fluid flow. Under this assumption, suitable conditions on the observation
points, and given support and tail conditions on the prior measure, we show
that the posterior measure converges to a Dirac measure centered on the true
flow as the number of observations goes to infinity.
