  We study a modification due to De Gregorio of the Constantin-Lax-Majda (CLM)
model $\omega_t = \omega H\omega$ on the unit circle. The De Gregorio equation
is $\omega_t+u \omega_x-u_x\omega =0, u_x = H\omega.$ In contrast with the CLM
model, numerical simulations suggest that the solutions of the De Gregorio
model with smooth initial data exist globally for all time, and generically
converge to equilibria when $t\to\pm\infty$, in a way resembling inviscid
damping. We prove that such a behavior takes place near a manifold of
equilibria.
