Is there a critical acceleration for the onset of convection?

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Suppose granular material is shaken vertically with z(t)=A₀ cos(ω₀ t). Can we expect to find convection if A₀ω₀² < g? By means of theoretical analysis and computer simulation we find that there is no critical Γ=|A₀|ω₀²/g for the onset of convection. Instead we propose a modified criterion which coincides with Γ=1 for small frequency ω₀.