标题： 关于可压缩欧拉和理想可压缩磁流体力学方程涡旋研究中去除大气压条件的研究 显示英文标题

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

摘要： 螺旋度是欧拉方程的一个拓扑守恒量，它对涡线的动力学施加了重要的约束。在可压缩情况下，守恒律仅在压力为幂律的假设下成立。我们证明，通过引入一个新的螺旋度密度定义 $h_{\rho}=(\rho\textbf{u})\cdot\mbox{curl}\,(\rho\textbf{u})$，可以去除这一关于压力的假设，尽管此时 $\int_V h_{\rho}dV$ 不再守恒。然而，我们表明对于非幂律可压缩欧拉方程，新的螺旋度密度 $h_{\rho}$ 满足一个熵型关系（在双曲守恒律的意义下），其通量 $\textbf{J}_{\rho}$ 包含所有压力项，而源项涉及势涡度 $q = \omega \cdot \nabla \rho$。因此，$\int_V h_{\rho}dV$ 的变化率不再依赖于压力，且更容易分析，因为它仅取决于势涡度、动能以及 $\mbox{div}\,\textbf{u}$。 这一结果同样适用于非齐次不可压缩的欧拉方程，对于后者，势涡度 $q$是一个物质常数。 因此，$q$受其初始值 $q_{0}=q(\textbf{x},\,0)$的限制，这使我们能够定义一个逆分辨率尺度$\lambda_{H}^{-1}$，其上限被发现与$\|q_{0}\|_{\infty}^{2/7}$成正比。 类似地，我们还为理想非等温磁流体力学（MHD）方程引入了一个新的交叉螺旋密度。 显示英文摘要

摘要： 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 $h_{\rho}=(\rho\textbf{u})\cdot\mbox{curl}\,(\rho\textbf{u})$ this assumption on the pressure can be removed, although $\int_V h_{\rho}dV$ is no longer conserved. However, we show for the non-barotropic compressible Euler equations that the new helicity density $h_{\rho}$ obeys an entropy-type relation (in the sense of hyperbolic conservation laws) whose flux $\textbf{J}_{\rho}$ contains all the pressure terms and whose source involves the potential vorticity $q = \omega \cdot \nabla \rho$. Therefore the rate of change of $\int_V h_{\rho}dV$ 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 $\lambda_{H}^{-1}$ 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.