Epi-two-dimensional flow and generalized enstrophy

TL;DR

This paper explores the relationship between enstrophy and helicity in fluid dynamics, extending the understanding of conserved quantities from 2D to more general epi-2D flows and analyzing their Hamiltonian structure.

Contribution

It introduces a Hamiltonian framework for epi-2D flows, generalizes enstrophy as a conserved quantity in these flows, and clarifies how helicity degenerates and is replaced by enstrophy.

Findings

Generalized enstrophy is conserved in compressible 2D fluids.

Helicity degenerates in epi-2D flows and is replaced by a generalized enstrophy.

A Hamiltonian formulation for epi-2D flows is developed.

Abstract

The conservation of the enstrophy ($L^2$ norm of the vorticity $\omega$) plays an essential role in the physics and mathematics of two-dimensional (2D) Euler fluids. Generalizing to compressible ideal (inviscid and barotropic) fluids, the generalized enstrophy $\int_{\Sigma(t)} f(\omega/\rho)\rho\, d^2 x$, ($f$ an arbitrary smooth function, $\rho$ the density, and $\Sigma(t)$ an arbitrary 2D domain co-moving with the fluid) is a constant of motion, and plays the same role. On the other hand, for the three-dimensional (3D) ideal fluid, the helicity $\int_{M} {V}\cdot\omega\,d^3x$, ($V$ the flow velocity, $\omega=\nabla\times V$, and ${M}$ the three-dimensional domain containing the fluid) is conserved. Evidently, the helicity degenerates in a 2D system, and the (generalized) enstrophy emerges as a compensating constant. This transition of the constants of motion is a reflection of an essential difference between 2D and 3D systems, because the conservation of the (generalized) enstrophy imposes stronger constraints, than the helicity, on the flow. In this paper, we make a deeper inquiry into the helicity-enstrophy interplay: the ideal fluid mechanics is cast into a Hamiltonian form in the phase space of Clebsch parameters, generalizing 2D to a wider category of epi-2D flows (2D embedded in 3D has zero helicity, while the converse is not true -- our epi-2D category encompasses a wider class of zero-helicity flows); how helicity degenerates and is substituted by a new constant is delineated; and how a further generalized enstrophy is introduced as a constant of motion applying to epi-2D flow is described.