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 .
