Abstract: This paper summarizes some recent advances in the study of nonlinear instability of the atmospheric and oceanic motions by using energy Casimir method(Arnol'd's method). Nonlinear stability criteria established for finite amplitude disturbance due to an arbitrary spatial structure are presented. Upper and lower bounds of the disturbance field in terms of the initial state are obtained,which provide the information on the behaviour of the evolution of the disturbance. The prospects of the further developmen...