  Let G be the (open) set of~$\dot H^{\frac 1 2}$ divergence free vector fields
generating a global smooth solution to the three dimensional incompressible
Navier-Stokes equations. We prove that any element of G can be perturbed by an
arbitrarily large, smooth divergence free vector field which varies slowly in
one direction, and the resulting vector field (which remains arbitrarily large)
is an element of G if the variation is slow enough. This result implies that
through any point in G passes an uncountable number of arbitrarily long
segments included in G.
