Abstract:This paper is an introduction to the study on nonlinear stability and instability of atmospheric motions, which is closely related to the weather’s drastic changes and persistent anomaly. It is the interdisciplinary study of atmospheric science, geophysical fluid dynamics and nonlinear science, which provides the basis of numerical weather and short term climate prediction and is also the frontier that is developing fast in atmospheric sciences. The author and his research group have combined the Energy Casimir method with the a prior estimate method innovatively and have attained outstanding achievements. The results won the first class prize of natural science of Chinese Academy of Sciences (2001).