= Solution
Expanding the <kinetic helicity conservation law> separates <advection> from the other flux:
$$
\boxed{\frac{Dq}{Dt}=\boldsymbol\omega\cdot\nabla\left(\frac{u^2}{2}-w\right)-q\nabla\cdot\mathbf u.}
$$
Thus <material conservation of kinetic helicity density> holds precisely when
$$
\boxed{\boldsymbol\omega\cdot\nabla\left(\frac{u^2}{2}-w\right)=q\nabla\cdot\mathbf u.}
$$
An 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>, $\boldsymbol\omega\cdot\nabla(w-u^2/2)=0$. 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> gives
$$
\frac D{Dt}\left(\frac q\rho\right)=-\frac{\boldsymbol\omega}{\rho}\cdot\nabla\left(w-\frac{u^2}{2}\right).
$$
Consequently 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.
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