On Nonlinear Stability Theorems of 3D Quasi-geostrophic Flow

Nonlinear stability criteria for quasi-geostrophic zonally symmetric flow are improved by establishing an optimal Poincard inequality. The inequality is derived by a variational calculation considering the additional invariant of zonal momentum. When applied to the Eady model in a periodic channel with finite zonal length, the improved nonlinear stability criterion is identical to the linear normal-mode stability criterion provided the channel meridional width is no greater than 0.8605... times its channel length （which is the geophysically relevant case）.