Is there a critical acceleration for the onset of convection?

Suppose granular material is shaken vertically with z(t)=A_0 cos(\omega_0 t). Can we expect to find convection if A_0\omega_0^2 < g? By means of theoretical analysis and computer simulation we find that there is no critical \Gamma=|A_0|\omega_0^2/g for the onset of convection. Instead we propose a modified criterion which coincides with \Gamma=1 for small frequency \omega_0.