= Kinetic helicity conservation law
For smooth inviscid <barotropic fluid> motion without gravity or magnetic forces, let $w$ be the <specific enthalpy> with $\nabla w=\rho^{-1}\nabla p$, and set $q=\mathbf u\cdot\boldsymbol\omega$. Then
$$
\partial_tq+\nabla\cdot\left[q\mathbf u+\left(w-\frac{u^2}{2}\right)\boldsymbol\omega\right]=0.
$$
This follows by dotting the <barotropic vorticity transport> equation with $\mathbf u$ and using the <divergence of a cross product>. A volume integral is constant when the resulting boundary flux vanishes.
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