On the removal of the barotropic condition in helicity studies of the compressible Euler and ideal compressible MHD equations

TL;DR

This work removes the barotropic restriction from helicity dynamics by introducing a local helicity density $h_{\rho}=(\rho\mathbf{u})\cdot(\text{curl}(\rho\mathbf{u}))$ for compressible Euler and ideal compressible MHD. The authors derive an entropy-type evolution $\partial_{t}h_{\rho}+\nabla\cdot\mathbf{J}_{\rho}=\sigma_{\rho}$ with a pressure-containing flux $\mathbf{J}_{\rho}$ and a source $\sigma_{\rho}$ involving the potential vorticity $q=\boldsymbol{\omega}\cdot\nabla\rho$, enabling analysis of $H=\int_V h_{\rho}\,dV$ even when pressure is not barotropic. In the inhomogeneous incompressible limit, $q$ is a material constant, which bounds the growth of helicity and yields a derived inverse length scale $\lambda_{H}^{-1}$ with an upper bound proportional to $\|q_{0}\|_{\infty}^{2/7}$; the fully compressible case introduces a div$\mathbf{u}$-weighted term, modifying the evolution to $dH/dt+2\int_V h_{\rho}\nabla\cdot\mathbf{u}\,dV=-2\int_V q\mathcal{E}_{0}\,dV$. The paper also extends the construction to a cross-helicity density in ideal compressible MHD, $h_{c}=\rho\mathbf{u}\cdot\mathbf{B}$, with a corresponding magnetic potential vorticity $q_{c}$ and a similar entropy-type balance. Together, these results provide a local, topologically meaningful framework for helicity dynamics without the barotropic closure, offering rigorous bounds and a practical inverse length scale, valid on time intervals where smooth solutions persist.

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 only holds under the assumption that the pressure is barotropic. We show that by introducing a new definition of helicity density $h_ρ=(ρ\textbf{u})\cdot\mbox{curl}\,(ρ\textbf{u})$ this assumption on the pressure can be removed, although $\int_V h_ρdV$ is no longer conserved. However, we show for the non-barotropic compressible Euler equations that the new helicity density $h_ρ$ obeys an entropy-type relation (in the sense of hyperbolic conservation laws) whose flux $\textbf{J}_ρ$ contains all the pressure terms and whose source involves the potential vorticity $q = ω\cdot \nabla ρ$. Therefore the rate of change of $\int_V h_ρdV$ 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 $\mbox{div}\,\textbf{u}$. 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 $q_{0}=q(\textbf{x},\,0)$, which enables us to define an inverse resolution length scale $λ_{H}^{-1}$ whose upper bound is found to be proportional to $\|q_{0}\|_{\infty}^{2/7}$. In a similar manner, we also introduce a new cross-helicity density for the ideal non-barotropic magnetohydrodynamic (MHD) equations.