Variational Principles for Nonbarotropic Fluid Dynamics

TL;DR

This paper extends variational principles to nonbarotropic fluid flows, analyzing their topological structures and conserved quantities like circulation and helicity, building on prior work on barotropic flows.

Contribution

It introduces a variational framework for nonbarotropic flows, incorporating complex topologies and conserved quantities, advancing the theoretical understanding of such fluid dynamics.

Findings

Extension of variational principles to nonbarotropic flows

Identification of topological conserved quantities

Implications for vortex line and circulation analysis

Abstract

Barotropic fluid flows with the same circulation structure as steady flows generically have comoving physical surfaces on which the vortex lines lie. These become Bernoullian surfaces when the flow is steady. When these surfaces are nested (vortex line foliation) with the topology of cylinders, toroids or a combination of both, a Clebsch representation of the flow velocity can be introduced. This is then used to reduce the number of functions to be varied in the variational principles for such flows. Here we extend the work to non barotropic flows and study the implication for variational analysis and conserved quantities of topological significance such as circulation and helicity.