  We derive asymptotic estimates for the projection of the vorticity onto
principal directions of material stretching in 3D flows. In flows with
pointwise bounded vorticity, these estimates predict vorticity alignment with
Lyapunov vectors along trajectories with positive Lyapunov exponents.
Specifically, we find that in inviscid flows with conservative body forces, the
vorticity exactly aligns with the intersection of the planes orthogonal to the
dominant forward and backward Lyapunov vectors along trajectories with positive
Lyapunov exponent. Furthermore, we derive asymptotic estimates for the
vorticity alignment with the intermediate eigenvector of the rate of strain
tensor for viscous flows under general forcing. We illustrate these results on
explicit solutions of Euler's equation and on direct numerical simulations of
homogeneous isotropic turbulence.
