Helicity (fluid mechanics)

In fluid dynamics, helicity is, under appropriate conditions, an invariant of the Euler equations of fluid flow, having a topological interpretation as a measure of linkage and/or knottedness of vortex lines in the flow (Moffatt 1969).

Let ${\displaystyle \mathbf {u} (x,t)}$ be the velocity field and ${\displaystyle \nabla \times \mathbf {u} }$ the corresponding vorticity field. Under the following three conditions, the vortex lines are transported with (or 'frozen in') the flow: (i) the fluid is inviscid; (ii) either the flow is incompressible (${\displaystyle \nabla \cdot \mathbf {u} =0}$), or it is compressible with a barotropic relation ${\displaystyle p=p(\rho )}$ between pressure ${\displaystyle p}$ and density ${\displaystyle \rho }$; and (iii) any body forces acting on the fluid are conservative. Under these conditions, any closed surface ${\displaystyle S}$ on which ${\displaystyle n\cdot (\nabla \times \mathbf {u} )=0}$ is, like vorticity, transported with the flow.

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