The Generalized Energy Equation and Instability in the Two-layer Barotropic Vortex

The linear two-layer barotropic primitive equations in cylindrical coordinates are used to derive a generalized energy equation, which is subsequently applied to explain the instability of the spiral wave in the model. In the two-layer model, there are not only the generalized barotropic instability and the super high-speed instability, but also some other new instabilities, which fall into the range of the Kelvin-Helmholtz instability and the generalized baroclinic instability, when the upper and lower basic flows are different. They are perhaps the mechanisms of the generation of spiral cloud bands in tropical cyclones as well.