Analytical solutions of the potential vorticity invertibility principle

We define the f ‐plane, y ‐independent, potential vorticity (PV) invertibility principle as a coupled pair of first‐order partial differential equations relating the balanced wind and mass fields to the known PV. Analytical solutions of this invertibility principle are derived for cases in which an isolated PV anomaly is confined within a region of the vertical plane. The solutions aid in understanding the dynamics of low‐latitude PV intrusions whose associated cloud patterns are often referred to as cloud surges, or moisture bursts, and whose flow patterns are often referred to as tropical upper tropospheric troughs. The existence of such tongues of high PV air intruding into the upper troposphere is documented using reanalysis data from the “Year” of Tropical Convection data set. The solutions illustrate the phenomenon of isentropic upglide below an upper tropospheric positive anomaly in PV. They also quantify how the partitioning of PV between absolute vorticity and static stability depends on the shape and strength of the PV anomaly. With slight modifications, the solutions apply to the problem of determining the balanced flow induced by a surface temperature anomaly, which is equivalent to a very thin layer of infinite PV at the surface. Through numerical solutions of the fully nonlinear invertibility principle we provide justification for the anelastic‐type approximation used in the analytical theory.