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Quasi-conservation laws for compressible 3D Navier-Stokes flow

We formulate the quasi-Lagrangian fluid transport dynamics of mass density $ρ$ and the projection $q=\bom\cdot\nablaρ$ of the vorticity $\bom$ onto the density gradient, as determined by the 3D compressible Navier-Stokes equations for an ideal gas, although the results apply for an arbitrary equation of state. It turns out that the quasi-Lagrangian transport of $q$ cannot cross a level set of $ρ$. That is, in this formulation, level sets of $ρ$ (isopychnals) are impermeable to the transport of the projection $q$.