Approximate controllability on the group of volume-preserving diffeomorphisms

We study controlability issues for the group of volume-preserving diffeomorphisms of the torus ${\mathbb{T}}^d$ for system $\dot{x}=f(x)+u(t)$, where $f$ is a fixed divergence free vector field on ${\mathbb{T}}^d$ and $u(t)$ are constant vector fields which generate translations of the torus. Main results concern $d$ equals two or three.