  We consider the vorticity formulation of the 2-D viscous Camassa-Holm
equations in the whole space. We establish global existence for solutions
corresponding to initial data in $L^1$ and describe the large time behavior of
solutions with sufficiently small and localized initial data. We calculate the
rate at which such solutions approach an ``un-filtered'' Oseen vortex by
computing the rate at which the solution of a scaled vorticity problem
approaches the solution to a corresponding linearized equation.
