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In Fluid Mechanics:
In fluid dynamics, the kinetic helicity is a measure of the degree of knottedness and/or linkage of the vortex lines of the flow.
Fluid helicity is one of the most important conserved quantities in ideal fluid flows, being an invariant of the Euler equations, and a robust quantity of the dissipative Navier-Stokes equations [13]. In ideal conditions its topological interpretation in terms of Gauss linking number was provided by Moffat [23] and extended by Moffat & Ricca [24]. In the context of vortex dynamics (kinetic) helicity is defined by
$$ H \equiv \int_\Omega u \cdot \omega d^3x, $$ where $u$ is the velocity field, $\omega = \Delta \times u$, is the vorticity, defined on $\Omega$, and $x$ is the position vector.
page 96 in How Nature Works: Complexity in Interdisciplinary Research and Applications by Ivan Zelinka
[H]elicity, is an integral over the fluid domain that expresses the correlation between velocity and vorticity, and an invariant of the classical Euler equations of ideal (inviscid) fluid flow. […] It is precisely because vortex lines are frozen in the fluid, thus conserving their topology, that helicity is conserved too. As Kelvin recognized, if two vortex tubes are linked, then that linkage survives in an ideal fluid for all time; if a vortex tube is knotted, then that knot survives in the same way for all time. Helicity is the integral manifestation of this invariance: For two linked tubes, it is proportional to the product of the two circulations (each conserved by Kelvin’s circulation
theorem), whereas for a single knotted tube, or a deformed unknotted tube, it is proportional to its “writhe plus twist,” as encountered in differential geometry—a property of knotted ribbons that is invariant under continuous deformation (4, 5). Thus, for example, if an untwisted ribbon that goes twice round a circle before closing on itself is unfolded and untwisted back to circular form, then its writhe will decrease continuously from 1 to 0, and its twist will increase continuously from 0 to 1, by way of compensation. Such conversion of writhe to twist is familiar to anyone seeking to straighten out a coiled garden hose.
On a manifold it is necessary to use covariant differentiation; curvature measures its noncommutativitiy. Its combination as a characteristic form measures the nontriviality of the underlying bundle. This train of ideas is so simple and natural that its importance can hardly be exaggerated. Shiing-shen Cern