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Expanding the kinetic helicity conservation law separates advection from the other flux:Thus material conservation of kinetic helicity density holds precisely whenAn especially useful sufficient pair of conditions is incompressible flow, $\nabla \cdot \mathbf{u}=0$, and constancy of $w-u^2/2$ along integral curves of the vorticity, $\mathbf{\omega }\cdot \nabla (w-u^2/2)=0$. Together these conditions make both terms vanish without cancellation. They are not necessary individually, because the two terms in the displayed balance can cancel. An irrotational flow is a trivial conserved case with $q=0$.

For contrast, combining the balance with mass conservation givesConsequently constancy of $w-u^2/2$ along integral curves of the vorticity makes $q/\rho$ an advected scalar even in compressible flow; it does not generally make $q$ itself constant.

For one-dimensional compressible flow, define the total energy densityThe conservative mass, momentum, and energy equations areA monatomic nonrelativistic perfect gas has $\gamma =5/3$, an ultrarelativistic gas or radiation-dominated fluid has $\gamma =4/3$, and a rotationally active diatomic gas has $\gamma =7/5$.

A smooth radial compressible flow with a fixed adiabatic exponent conserves $r^2\rho u$ and $p/{\rho }^{\gamma }$. In a prescribed Newtonian gravitational potential, its Bernoulli function is $u^2/2+c^2/(\gamma -1)+\Phi$, with $c$ the adiabatic sound speed. Eliminating the density derivative yields $(u-c^2/u)u^{\prime }=2c^2/r-{\Phi }^{\prime }$.