  We examine vortices in Abelian Chern-Simons theory coupled to a relativistic
scalar field with a chemical potential for particle number or U(1) charge. The
Gauss constraint requires chemical potential for the local symmetry to be
accompanied by a constant background charge density/magnetic field. Focussing
attention on power law scalar potentials $\sim|\Phi|^{2s}$ which do not support
vortex configurations in vacuum but do so at finite chemical potential, we
numerically study classical vortex solutions for large winding number |n| >> 1.
The solutions depending on a single dimensionless parameter $\alpha$, behave as
uniform incompressible droplets with radius $\sim \sqrt {\alpha |n|}$ , and
energy scaling linearly with |n|, independent of coupling constant. In all
cases, the vortices transition from type I to type II at a critical value of
the dimensionless parameter, $\alpha_c = s/(s-1)$, which we confirm with
analytical arguments and numerical methods.
