  We consider the Benjamin-Ono equation on the torus with an additional damping
term on the smallest Fourier modes (cos and sin). We first prove global
well-posedness of this equation in $L^2_{r,0}(\mathbb{T})$. Then, we describe
the weak limit points of the trajectories in $L^2_{r,0}(\mathbb{T})$ when time
goes to infinity, and show that these weak limit points are strong limit
points. Finally, we prove the boundedness of higher-order Sobolev norms for
this equation. Our key tool is the Birkhoff map for the Benjamin-Ono equation,
that we use as an adapted nonlinear Fourier transform.
