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

We formulate the quasi-Lagrangian fluid transport dynamics of mass density $\rho$ and the projection $q=\text{bom}\cdot \mathrm{\nabla }\rho$ of the vorticity $\text{bom}$ 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 $\rho$. That is, in this formulation, level sets of $\rho$ (isopychnals) are impermeable to the transport of the projection $q$.