Formulas for Parcel Velocity and Vorticity in a Rotating Cartesian Coordinate System

Formulas in an Eulerian framework are presented for the absolute velocity and vorticity of individual parcels in inviscid isentropic flow. The analysis is performed in a rectangular Cartesian rotating coordinate system. The dependent variables are the Lagrangian coordinates, initial velocities, cumulative temperature, entropy, and a potential. The formulas are obtained in two different ways. The first method is based on finding a matrix integrating factor for the Euler equations of motion and a propagator for the vector vorticity equation. The second method is a variational one. Hamilton’s principle of least action is used to minimize the fluid’s absolute kinetic energy minus its internal energy and potential energy subject to the Lin constraints and constraints of mass and entropy conservation. In the first method, the friction and diabatic heating terms in the governing equations are carried along in integrands so that the generalized formulas lead to Eckart’s circulation theorem. Using them to derive other circulation theorems, the helicity-conservation theorem, and Cauchy’s formula for the barotropic vorticity checks the formulas further. The formulas are suitable for generating diagnostic fields of barotropic and baroclinic vorticity in models if some simple auxiliary equations are added to the model and integrated stably forward in time alongside the model equations.