  We prove unconditional long-time stability for a particular
velocity-vorticity discretization of the 2D Navier-Stokes equations. The scheme
begins with a formulation that uses the Lamb vector to couple the usual
velocity-pressure system to the vorticity dynamics equation, and then
discretizes with the finite element method in space and implicit-explicit BDF2
in time, with the vorticity equation decoupling at each time step. We prove the
method's vorticity and velocity are both long-time stable in the $L^2$ and
$H^1$ norms, without any timestep restriction. Moreover, our analysis avoids
the use of Gronwall-type estimates, which leads us to stability bounds with
only polynomial (instead of exponential) dependence on the Reynolds number.
Numerical experiments are given that demonstrate the effectiveness of the
method.
