  In this paper, we study the nonlinear stability of a steady circular flow
created between two rotating concentric cylinders. The dynamics of the viscous
fluid are described by 2D Navier-Stokes equations. We adopt scaling variables.
For the rescaled equations, we prove that the steady flow (Taylor-Couette flow)
is asymptotically stable up to a large perturbation of initial data. Back to
the original 2D Navier-Stokes equations, this implies an improved transition
threshold for the Taylor-Couette flow. The improvement is due to enhanced
dissipation and new observations and constructions of weighted $L^2$ norms,
which capture a hidden structure between the viscosity constant $\nu$ and
(different) rotating speeds and locations of two coaxial cylinders. In
particular, we allow the location of the outer cylinder to tend to infinity,
which renders the initial fluid kinetic energy not uniformly bounded. Due to
enhanced-dissipation effect, we also establish a sharp resolvent estimate,
desired space-time bounds and optimal decaying estimates, which lead to the
proof of nonlinear asymptotic stability of 2D Taylor-Couette flow.
