2025/01/29 by Daniel W. Boutros, Boutros, Daniel W., John Gibbon +1
Engineering · Mathematics · #76B99 (secondary) #76N99 (primary) #76W05 #Analysis of PDEs (math.AP) #Chaotic Dynamics (nlin.CD) #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #FOS: Physical sciences #Fluid Dynamics (physics.flu-dyn) #Fluid Dynamics and Turbulent Flows #Gas Dynamics and Kinetic Theory
paper · pdf · doi:10.48550/arxiv.2501.17990
openalex publication_date 2025/01/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
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 only holds under the assumption that the pressure is barotropic. We show that by introducing a new definition of helicity density hρ=(ρu)\cdotcurl (ρu) this assumption on the pressure can be removed, although ∫V hρdV 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 ∫V hρdV no longer depends on the pressure and is easier to analyse, as it only depends on the potential vorticity and kinetic energy as well as div u. 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 λH-1 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.