We consider two-dimensional gravity water waves with constant nonzero vorticity in infinite depth and with periodic boundary conditions. We construct smooth small-amplitude solutions whose high Sobolev norms grow arbitrarily large while lower-order norms remain small.

This behavior reflects a transfer of energy from low to high frequencies within a quasilinear fluid model, occurring entirely in the small-data regime where solutions remain regular. The mechanism is driven by nonlinear resonant interactions and an effective transport dynamics arising from the quasilinear structure of the equations.

The proof combines a refined resonance analysis, a quasilinear normal form reduction, a positive commutator argument, and a novel "upside-down" virial argument, yielding instability over long time scales.