Abstract: Poincaré type integral inequality plays an important role in the study of nonlinear stability (in the sense of Arnold's second theorem) for three-dimensional quasi-geostophic flow. The nonlinear stability of Eady's model is one of the most important cases in the application of the method. But the best nonlinear stability criterion obtained so far and the linear stability criterion are not coincident. The two criteria coincide only when the period of the channel is infinite. To fill this gap, the enhanced Poincaré inequality was obtained by considering the additional conservation law of momentum and by rigorous estimate of integral inequality. So the new nonlinear stability criterion was obtained, which shows that for Eady's model in the periodic channel, the linear stable implies the nonlinear stable.

Key words: Poincaré inequality, Eady's model, nonlinear stability