Kinetic helicity ($H_k=\int \mathbf{u}\cdot \mathbf{\omega }dV$)

Kinetic helicity measures the alignment and linking of a velocity field and its vorticity. It is the volume integral of $\mathbf{u}\cdot \mathbf{\omega }$, with boundary conditions needed for its conservation.

The local density of kinetic helicity is $q=\mathbf{u}\cdot \mathbf{\omega }$. Conservation of its volume integral does not imply that $q$ is a materially advected scalar.

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 $q=\mathbf{u}\cdot \mathbf{\omega }$. ThenThis 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.

The kinetic helicity conservation law impliesTherefore $q$ 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 $D(q/\rho )/Dt=-(\mathbf{\omega }/\rho )\cdot \nabla (w-u^2/2)$.