  We study the long-time behaviour of axisymmetric solutions without swirl for
the threedimensional Navier-Stokes equations in the whole space. Assuming that
the initial vorticity is sufficiently localised, we compute explicitly the
leading terms in the asymptotic expansion of the solution, both for the
vorticity and the velocity field. In particular, we identify optimal temporal
decay rates depending on the spatial localisation of the initial data. Our
approach relies on accurate $L^p$-$L^q$ estimates for the linearised evolution
equation and its Taylor expansion in self-similar variables.
