Reinvestigation of potential deformation during a heavy rainfall event in North China

In this study,the physical meaning and generation mechanism of potential deformation(PD)are reinvestigated.A main trait of PD is that it contains deformation,which is an important factor to precipitation but not well applied in precipitation diagnosis.This paper shows PD shares similar features to deformation,but contains much more physical information than deformation.It can be understood as a type of deformation of a thermodynamic-coupled vector(u*,v*).For convenient application,squared PD(SPD)is used instead for analysis.By deriving the tendency equation of SPD,it is found that whether SPD is produced or reduced in the atmosphere is associated with the angle between the dilatation axes of PD and geostrophic PD.When the angle is less thanπ2,SPD is generated.The diagnostic results during a heavy rainfall event in North China on 20 July 2016 show that the process of rapid increase in precipitation can be well revealed by SPD.The distribution of SPD becomes more organized and concentrated with increasing precipitation intensity.A diagnostic analysis of the SPD tendency equation shows that concentrated SPD is associated with the generation of SPD in the boundary layer followed by upward transport of the SPD.The concentration of SPD indicates a confluence of precipitation-favorable factors—namely,vertical wind shear and moist baroclinity,which can enhance vertical motions and thus cause an increase in precipitation.These diagnostic results further verify PD as a useful physical parameter for heavy precipitation diagnosis.

SOBOLEV INEQUALITY ON RIEMANNIAN MANIFOLDS

Let M be an n dimensional complete Riemannian manifold satisfying the doublingvolume property and an on-diagonal heat kernel estimate. The necessary-sufficientcondition for the Sobolev inequality ‖f‖q ≤ Cn,,v,p,q(‖▽f‖p+‖fp) (2≤p＜q＜∞) is given.