On the removal of the barotropic condition in helicity studies of the compressible Euler and ideal compressible magnetohydrodynamic equations
Author(s)
Boutros, Daniel W
Gibbon, John
Type
Journal Article
Abstract
The helicity is a topological conserved quantity of the Euler equations which imposes significant constraints on the dynamics of vortex lines. In the compressible setting, the conservation law holds only under the assumption that the pressure is barotropic. Let us consider a volume V containing a compressible fluid with density ρ, velocity field u and vorticity ω. We show that by introducing a new definition of helicity density hρ=(ρu)⋅curl(ρu) the barotropic assumption on the pressure can be removed, although ∫VhρdV is no longer conserved. However, we show for the non-barotropic compressible Euler equations that the new helicity density hρ obeys an entropy-type relation (in the sense of hyperbolic conservation laws) whose flux Jρ contains all the pressure terms and whose source involves the potential vorticity q=ω⋅∇ρ. Therefore, the rate of change of ∫VhρdV no longer depends on the pressure and is easier to analyse, as it depends only on the potential vorticity and kinetic energy as well as divu. This result also carries over to the inhomogeneous incompressible Euler equations for which the potential vorticity q is a material constant. Therefore, q is bounded by its initial value q0=q(x,0), which enables us to define an inverse resolution length scale λ−1H whose upper bound is found to be proportional to ∥q0∥2/7∞. In a similar manner, we also introduce a new cross-helicity density for the ideal non-barotropic magnetohydrodynamic (MHD) equations.
Date Issued
2025-04-10
Date Acceptance
2025-01-19
Citation
Journal of Fluid Mechanics, 2025, 1008
ISSN
0022-1120
Publisher
Cambridge University Press
Journal / Book Title
Journal of Fluid Mechanics
Volume
1008
Copyright Statement
© The Author(s), 2025. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (https://creativecommons. org/licenses/by/4.0/), which permits unrestricted re-use, distribution and reproduction, provided the original article is properly cited.
License URL
Identifier
10.1017/jfm.2025.161
Subjects
topological fluid dynamics
MHD
electrohydrodynamics
Publication Status
Published
Article Number
ARTN R3
Date Publish Online
2025-04-17