= Material conservation of kinetic helicity density
The <kinetic helicity conservation law> implies
$$
\frac{Dq}{Dt}=\boldsymbol\omega\cdot\nabla\left(\frac{u^2}{2}-w\right)-q\nabla\cdot\mathbf u.
$$
Therefore $q$ is materially constant exactly when the right side vanishes. <Incompressible flow> together with constancy of $w-u^2/2$ along vortex lines is sufficient. In compressible flow, the density-normalized quantity satisfies $D(q/\rho)/Dt=-(\boldsymbol\omega/\rho)\cdot\nabla(w-u^2/2)$.
Back to article page
