An investigation into the expression of potential vorticity in the common meteorological coordinate systems

Abstract

The potential vorticity theory and diagnostic techniques are based on the potential vorticity equation and expression in the common meteorological coordinate systems. In this paper, the potential vorticity equation and expression in the isobaric and isoentropic coordinates are gotten via coordinate transformation with the two methods. First, starting from the three-dimensional vector motion equation, the potential vorticity equations and expressions are gotten by the combination of the three-dimensional vorticity equation, continuity equation, and thermodynamic equation. Second, the potential vorticity equations and expressions are directly gotten from the corresponding scalar motion equations in the isobaric and isoentropic coordinates. The results show that potential vorticity expression is different with one method from that with the other in the isobaric coordinate system, and it is the same as each other in the isoentropic coordinate system. It was found, based on further analysis of the physical nature of the coordinates, that the isobaric and isoentropic coordinates are essentially treated as a mathematical coordinate system with the first method despite the coordinate transformation made for the term of pressure gradient force in the vector motion equation. From the procedure for the second method it is clearly seen that the isobaric and isoentropic coordinate systems are the physical coordinate system under the assumption of static equilibrium, which are not simply used as a mathematical coordinate system. As far as the isobaric coordinate is concerned, only the potential vorticity equation obtained from the scalar motion equations is the strict potential vorticity equation. As for the isoentropic coordinate, owing to the potential temperature gradient perpendicular to the isoentropic plane, the potential vorticity equation and expression are the same regardless of the coordinate being viewed as the physical or the mathematical.