  We use simple properties of the Rasmussen invariant of knots to study its
asymptotic behaviour on the orbits of a smooth volume preserving vector field
on a compact domain in the 3-space. A comparison with the asymptotic signature
allows us to prove that asymptotic knots are non-alternating, in general.
Further we show that the Rasmussen invariant defines a quasi-morphism on the
braid groups and derive estimates for the stable commutator and torsion lengths
of alternating braids.
