A Unified View of Topological Invariants of Barotropic and Baroclinic Fluids and their Application to Formal Stability Analysis of Three-Dimensional Ideal Gas Flows

Noether's theorem associated with the particle relabeling symmetry group leads us to a unified view that all the topological invari- ants of a barotropic fluid are variants of the cross helicity. The same is shown to be true for a baroclinic fluid. A cross-helicity representation is given to the Casimir invariant, a class of integrals including an arbitrary function of the specific entropy and the potential vorticity. We then develop a new energy-Casimir convexity method for three-dimensional stability of equilibria of gen- eral rotating flows of an ideal baroclinic fluid, without appealing to the Boussinesq approximation. By fully exploiting the Casimir invariant, we have succeeded in ruling out a term including the gradient of a dependent variable from the energy-Casimir function and have established a sharp linear stability criterion, being an extension of the Richardson-number criterion.