Parcel Eulerian–Lagrangian fluid dynamics of rotating geophysical flows

Parcel Eulerian–Lagrangian Hamiltonian formulations have recently been used in structure-preserving numerical schemes, asymptotic calculations, and in alternative explanations of fluid parcel (in)stabilities. A parcel formulation describes the dynamics of one fluid parcel with a Lagrangian kinetic energy but an Eulerian potential evaluated at the parcel’s position. In this paper, we derive the geometric link between the parcel Eulerian–Lagrangian formulation and well-known variational and Hamiltonian formulations for three models of ideal and geophysical fluid flow: generalized two-dimensional vorticity-streamfunction dynamics, the rotating two-dimensional shallow water equations, and the rotating three-dimensional compressible Euler equations.