Hierarchies of new invariants and conserved integrals in inviscid fluid flow

TL;DR

This paper introduces a vector calculus method to identify a hierarchy of new invariants and conserved integrals in inviscid fluid flow, expanding understanding of tensorial quantities frozen into the flow.

Contribution

It develops algebraic and differential operators to generate complete hierarchies of invariants, including new variants related to Ertel's and Hollmann's invariants, for various fluid models.

Findings

Derived infinite hierarchies of invariants for different fluid types.

Constructed operators enable recursive derivation of all invariants from basic ones.

Identified new variants of Ertel's and Hollmann's invariants involving density and entropy gradients.

Abstract

A vector calculus approach for the determination of advected invariants is presented for inviscid fluid flow. This approach describes invariants by means of Lie dragging of scalars, vectors, and skew-tensors with respect to the fluid velocity, which has the physical meaning of characterizing tensorial quantities that are frozen into the flow. Several new main results are obtained. First, simple algebraic and differential operators that can be applied recursively to derive a complete set of invariants starting from the basic known local and nonlocal invariants are constructed. Second, these operators are used to derive infinite hierarchies of local and nonlocal invariants for both adiabatic fluids and isentropic fluids that are either incompressible, or compressible with barotropic and non-barotropic equations of state. Each hierarchy is complete in the sense that no further invariants can be generated from the basic local and nonlocal invariants. All of the resulting new invariants are related to Ertel's invariant, the Ertel-Rossby invariant, and Hollmann's invariant. In particular, for incompressible fluid flow in which the density is non-constant across different fluid streamlines, a new variant of Ertel's invariant and several new variants of Hollmann's invariant are derived, where the entropy gradient is replaced by the density gradient. Third, the physical meaning of these new invariants and the resulting conserved integrals are discussed, and their relationship to conserved helicities and cross-helicities is described.