Invariants in dissipationless hydrodynamic media

Abstract

We propose a general geometric method of derivation of invariant relations for hydrodynamic dissipationless media. New dynamic invariants are obtained. General relations between the following three types of invariants are established, valid in all models: Lagrangian invariants, frozen-in vector fields and frozen-in co-vector fields. It is shown that frozen-in integrals form a Lie algebra with respect to the commutator of the frozen fields. The relation between frozen-in integrals derived here can be considered as the Backlund transformation for hydrodynamic-type systems of equations. We derive an infinite family of integral invariants which have either dynamic or topological nature. In particular, we obtain a new type of topological invariant which arises in all hydrodynamic dissipationless models when the well-known Moffatt invariant vanishes.

Nonlinear dynamics and breakup of free-surface flows

“…One hopes that the study of these singularities will yield information about high-Reynolds-number turbulence, in particular the mechanisms behind the energy cascade of eddies. Geometrical invariants of the Euler equation (Tur and Yanovsky, 1993) are expected to act as constraints for the almost inviscid motion of turbulent flow.…”

Hall viscosity: A link between quantum Hall systems, plasmas and liquid crystals

“…We see that this is consistent with the physical intuition that ρ and s are scalars, and that the former also represents a density. Alternatively, when seen in the light of two dimensions, the equations for ρ and s can easily be interpreted as the Lie-dragging of a 2-form and a 0-form (in 2D) respectively [26]. We need to now determine the conservation law for .…”

Potential vorticity in magnetohydrodynamics

“…Moiseev et al (1982), Tur and Yanovsky (1993) and Webb et al (2014a) used the technique of Lie dragging of 0-forms, p-forms (p = 1, 2, 3) (see also Kuvshinov and Schep (1997)). This technique is relatively easy to use to derive conservation laws for invariants (geometrical objects) which are advected with the flow.…”

“…Kelbin et al (2014) obtain new conservation laws for helically symmetric, plane, and rotationally symmetric flows. Webb et al (2014a) derived MHD conservation laws using Lie dragging techniques (see also Tur and Yanovsky (1993)). Webb et al (2014b), obtained advected invariants (i.e.…”