Nonlinear instability and convection in a vertically vibrated granular bed

The nonlinear instability of the density-inverted granular Leidenfrost state and the resulting convective motion in strongly shaken granular matter are analysed via a weakly nonlinear analysis. Under a quasi-steady ansatz, the base state temperature decreases with increasing height away from from the vibrating plate, but the density profile consists of three distinct regions: (i) a collisional dilute layer at the bottom, (ii) a levitated dense layer at some intermediate height and (iii) a ballistic dilute layer at the top of the granular bed. For the nonlinear stability analysis, the nonlinearities up-to cubic order in perturbation amplitude are retained, leading to the Landau equation. The genesis of granular convection is shown to be tied to a supercritical pitchfork bifurcation from the Leidenfrost state. Near the bifurcation point the equilibrium amplitude is found to follow a square-root scaling law, $A_e\sim \sqrt{\triangle}$, with the distance ${\triangle}$ from bifurcation point. The strength of convection is maximal at some intermediate value of the shaking strength, with weaker convection both at weaker and stronger shaking. Our theory predicts a novel {\it floating-convection} state at very strong shaking [Shukla etal, JFM (2014), vol. 761, p. 123-167].