Relaxation of Compressible Euler Flow in a Toroidal Domain

TL;DR

This paper demonstrates that steady Euler flows in a toroidal domain satisfy a nonlinear Beltrami equation, representing a relaxed state that extremizes kinetic energy under specific boundary and conservation constraints.

Contribution

It introduces a nonlinear Beltrami equation for steady Euler flows in a toroidal domain and develops an action principle and Hamiltonian formulation for quasi-relaxed fluid dynamics.

Findings

Steady Euler flow fields obey a nonlinear Beltrami equation.

A relaxed velocity field extremizes kinetic energy under boundary and conservation constraints.

An action principle and Hamiltonian form for quasi-relaxed dynamics are formulated.

Abstract

It is shown that the universal steady Euler flow field, independent of boundary shape or symmetry, in a toroidal domain with fixed boundary obeys a nonlinear Beltrami equation, with the nonlinearity arising from a Boltzmann-like, velocity-dependent factor. Moreover, this is a relaxed velocity field, in the sense that it extremizes the total kinetic energy in the domain under free variations of the velocity field, constrained only by tangential velocity and vorticity boundary conditions and conservation of total fluid helicity and entropy. This is analogous to Woltjer-Taylor relaxation of plasma magnetic field to a stationary state. However, unlike the magnetic field case, attempting to derive slow, quasi-relaxed dynamics from Hamilton's action principle, with constant total fluid helicity as a constraint, fails to agree, in the static limit, with the nonlinear Beltrami solution of the Euler equations. Nevertheless, an action principle that gives a quasi-relaxed dynamics that does agree can be formulated, by introducing a potential representation of the velocity field and defining an analogue of the magnetic helicity as a new constraint. A Hamiltonian form of quasi-relaxed fluid dynamics is also given.