  We use the vorticity formulation to study the long-time behavior of solutions
to the Navier-Stokes equation on R^3. We assume that the initial vorticity is
small and decays algebraically at infinity. After introducing self-similar
variables, we compute the long-time asymptotics of the rescaled vorticity
equation up to second order. Each term in the asymptotics is a self-similar
divergence-free vector field with Gaussian decay at infinity, and the
coefficients in the expansion can be determined by solving a finite system of
ordinary differential equations. As a consequence of our results, we are able
to characterize the set of solutions for which the velocity field converges to
zero faster than t^(-5/4) in energy norm. In particular, we show that these
solutions lie on a smooth invariant submanifold of codimension 11 in our
function space.
