Abstract: On the actual weather map, the weather system consists of multi-scale waves and multi-scale eddies. Rossby proposed the atmospheric long-wave theory which has a signi fi cant contribution to the weather forecast in the past decades. Global numerical weather prediction spectral model decomposed atmospheric variables into multi-scale mathematical waves without the synoptic-scale eddies. Physical decomposition of atmospheric variables can be used to differentiate the daily climatic waves from planetary-scale transient waves and synoptic-scale transient eddies. Extreme weather events are directly related to the synoptic-scale transient eddies so that extreme weather events can be forecasted by the method of transient eddies. The actual weather evolution is formed from their overlap or interaction of daily climatic waves, the planetary-scale transient waves and synoptic-scale transient eddies.