Statistical mechanics of the Euler equations without vortex stretching

Abstract

We consider the relaxation to thermal equilibrium of the Galerkin-truncated Euler equations in three dimensions, from which vortex stretching is removed. We prove that helicity and enstrophy are conserved by the system. Using statistical mechanics, we derive analytical predictions for the equilibrium energy and helicity spectra. Results are verified using pseudo-spectral direct numerical simulations. Results show that if the initial condition contains helicity, the system relaxes to a force-free large-scale structure akin to an Arnold–Beltrami–Childress (ABC) flow.