  In this paper we prove that for all solutions of the 2d Euler equations with
initial vorticity with finite Sobolev smoothness then an initial data dependent
norm of the associated Lagrangian flow blows up in infinite time at least like
$t^{\frac{1}{3}}$. This initial data dependent norm quantifies the exact $L^2$
decay of the Fourier transform of the solution. This adapted norm turns out to
be the exact quantity that controls a low to high frequency cascade which we
then show to be the quantitative phenomenon behind the Lyapunov construction by
Shnirelman.
