Constraints on the Mean Mass Transport across Potential Vorticity Contours

An expression is derived for the quasi-horizontal part of the mass transport across a given potential vorticity contour on an isentropic surface, in terms of the rate of change of absolute circulation around the contour and frictional and diabatic terms on the contour. It is deduced that this mass transport is small if the circulation around the contour of interest is steady and if frictional forces and diabatic effects can be neglected on the contour. In a single-layer model the corresponding result is that the total mass transport is zero. In a threedimensional model the implication is that the dominant mass transport across a vortex edge that tilts in the vertical occurs through vertical advection. It is argued that these constraints on the mass transport are relevant to the estimation of transport across the edge of the stratospheric polar vortex, and the relationship to other similar results that have appeared recently in the literature is discussed. In addition, a new expression is derived for the total mass flux across a three-dimensional surface whose intersection with each isentropic surface is a potential vorticity contour. This expression generalizes previous results that were confined to steady flows and hydrostatic scaling.