Hydrodynamic impulse ($\mathbf{I}=\frac{1}{2}\int \mathbf{x}\times \mathbf{\omega }dV$)

Hydrodynamic impulse per unit density is $\mathbf{I}=\frac{1}{2}\int \mathbf{x}\times \mathbf{\omega }dV$ for localized vorticity. In unforced all-space incompressible flow its derivative vanishes when far-field fluxes vanish. For a spherical all-space limit, $\mathbf{I}=(3/2){\lim }_R{\int }_{V_R}\mathbf{u}dV$; a dipolar velocity tail makes the distinction from a bare momentum integral important.

A localized vorticity distribution with nonzero hydrodynamic impulse induces $\mathbf{u}\sim [3\mathbf{n}(\mathbf{I}\cdot \mathbf{n})-\mathbf{I}]/(4\pi r^3)$. This curl-free exterior velocity can remain long-ranged even when the vorticity is localized. In a random ensemble it produces dipolar longitudinal and transverse correlation tails.