  It has been shown by various authors that the diameter of a given nontrivial
bounded connected set $\mathcal{X}$ grows linearly in time under the action of
an isotropic Brownian flow (IBF), which has a nonnegative top-Lyapunov
exponent. In case of a planar IBF with a positive top-Lyapunov exponent, the
precise deterministic linear growth rate K of the diameter is known to exist.
In this paper we will extend this result to an asymptotic support theorem for
the time-scaled trajectories of a planar IBF $\varphi$, which has a positive
top-Lyapunov exponent, starting in a nontrivial compact connected set
$\mathcal{X}\subseteq \mathbf{R}^2$; that is, we will show convergence in
probability of the set of time-scaled trajectories in the Hausdorff distance to
the set of Lipschitz continuous functions on [0,1] starting in 0 with Lipschitz
constant K.
