Weakly mixing smooth planar vector field without asymptotic directions

We construct a planar smooth weakly mixing stationary random vector field with nonnegative components such that, with probability 1, the flow generated by this vector field does not have an asymptotic direction. Moreover, for all individual trajectories, the set of partial limiting directions coincides with those spanning the positive quadrant. A modified example shows that a particle in space-time weakly mixing positive velocity field does not necessarily have an asymptotic average velocity.