  The theory of vortex motion in a dilute superfluid of inhomogeneous density
demands a boundary layer approach, in which different approximation schemes are
employed close to and far from the vortex, and their results matched smoothly
together. The most difficult part of this procedure is the hydrodynamic problem
of the velocity field many healing lengths away from the vortex core. This
paper derives and exploits an exact solution of this problem in the
two-dimensional case of a linear trapping potential, which is an idealization
of the surface region of a large condensate. It thereby shows that vortices in
inhomogeneous clouds are effectively 'dressed' by a non-trivial distortion of
their flow fields; that image vortices are not relevant to Thomas-Fermi
surfaces; and that for condensates large compared to their surface depths, the
energetic barrier to vortex penetration disappears at the Landau critical
velocity for surface modes.
