Wave-activity relation containing wave-basic flow interaction based on decomposition of general potential vorticity*

On the basis of the general potential vorticity theorem (GPV), a new wave–activity relation is derived in an ageostrophic and nonhydrostatic dynamic framework. When the Reynolds average is taken, the wave–activity relation shares an exchange term with the equation of the basic-state GPV. Thus, the two equations are capable of presenting the dynamic process of the wave–basic flow interaction. Unlike the E–P flux theory which can only be used in large-scale atmosphere, the corresponding derivation provides a useful tool to analyze the feedback of waves to basic states and the forcing of basic states to waves simultaneously, and it can be used in mesoscale systems, such as heavy rainfall processes. The theory was initially applied to the landfalling Typhoon Mujigae (2015) by assigning the scalar φ to the generalized potential temperature (GPT). The results showed that the newly-derived wave–activity density is able to describe the wave activities associated with strong precipitation in Typhoon Mujigae (2015), including the eyewall and spiral rainbands. However, the interaction between the basic-state vortex and mesoscale waves denoted by the exchange term between basic-state GPV and wave–activity density mainly occurs in the eyewall in Typhoon Mujigae (2015). A comparison of the exchange term with other forcing terms in the newly-derived wave–activity relation indicates that the basic state–wave interaction plays a significant role in enhancing wave activities in the high-precipitation eyewall. By a magnitude analysis of the interaction term, it is found that the strong interaction between basic-state vortex and mesoscale waves is mainly attributed to two factors: the vertical vorticity intensity of the basic-state vortex and the averaged perturbation advection of perturbation GPT which is an exchange between the basic-state GPT and perturbation GPT.