Comparative Analysis of the Generalized Omega Equation and Generalized Vertical Motion Equation

Research on vertical motion in mesoscale systems is an extraordinarily challenging effort. Allowing for fewer assumptions, a new form of generalized vertical motion equation and a generalized Omega equation are derived in the Cartesian coordinate system (nonhydrostatic equilibrium) and the isobaric coordinate system (hydrostatic equilibrium), respectively. The terms on the right-hand side of the equations, which comprise the Q vector, are composed of three factors: dynamic, thermodynamic, and mass. A heavy rain event that occurred from 18 to 19 July 2021 in southern Xinjiang was selected to analyze the characteristics of the diagnostic variable in the generalized vertical motion equation (\begin{document}${{\boldsymbol{Q}}_z}$\end{document}) and the diagnostic variable in the generalized Omega equation (\begin{document}${{\boldsymbol{Q}}_p}$\end{document}) using high-resolution model data. The results show that the horizontal distribution of the \begin{document}${{\boldsymbol{Q}}_z}$\end{document}-vector divergence at 5.5 km is roughly similar to the distribution of the \begin{document}${{\boldsymbol{Q}}_p}$\end{document}-vector divergence at 500 hPa, and that both relate well to the composite radar reflectivity, vertical motion, and hourly accumulated precipitation. The \begin{document}${{\boldsymbol{Q}}_z}$\end{document}-vector divergence is more effective in indicating weak precipitation. In vertical cross sections, regions with alternating positive and negative large values that match the precipitation are mainly concentrated in the middle levels for both forms of Q vectors. The temporal evolutions of vertically integrated \begin{document}${{\boldsymbol{Q}}_z}$\end{document}-vector divergence and \begin{document}${{\boldsymbol{Q}}_p}$\end{document}-vector divergence are generally similar. Both perform better than the classical quasigeostrophic Q vector and nongeostrophic Q vector in indicating the development of the precipitation system.