Hierarchies of new invariants and conserved integrals in inviscid fluid flow

A vector calculus approach for the determination of advected invariants is presented for inviscid fluid flows in three dimensions. 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 homentropic 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 generalizations of Ertel's invariant, the Ertel-Rossby invariant, and Hollmann's invariant. In particular, for an 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 is discussed, and their relationship to conserved helicities and cross-helicities is described.