Hydrodynamical helicity

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. This was first proved by Jean-Jacques Moreau in 1961^[1] and Moffatt derived it in 1969 without the knowledge of Moreau's paper. This helicity invariant is an extension of Woltjer's theorem for magnetic helicity.

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 p and density ρ; and (iii) any body forces acting on the fluid are conservative. Under these conditions, any closed surface S whose normal vectors are orthogonal to the vorticity (that is, ${\displaystyle \mathbf {n} \cdot (\nabla \times \mathbf {u} )=0}$) is, like vorticity, transported with the flow.

Let V be the volume inside such a surface. Then the helicity in V, denoted H, is defined by the volume integral

${\displaystyle H=\int _{V}\mathbf {u} \cdot \left(\nabla \times \mathbf {u} \right)\,dV\;.}$

For a localised vorticity distribution in an unbounded fluid, V can be taken to be the whole space, and H is then the total helicity of the flow. H is invariant precisely because the vortex lines are frozen in the flow and their linkage and/or knottedness is therefore conserved, as recognized by Lord Kelvin (1868). Helicity is a pseudo-scalar quantity: it changes sign under change from a right-handed to a left-handed frame of reference; it can be considered as a measure of the handedness (or chirality) of the flow. Helicity is one of the four known integral invariants of the Euler equations; the other three are energy, momentum and angular momentum.

For two linked unknotted vortex tubes having circulations ${\displaystyle \kappa _{1}}$ and ${\displaystyle \kappa _{2}}$, and no internal twist, the helicity is given by ${\displaystyle H=\pm 2n\kappa _{1}\kappa _{2}}$, where n is the Gauss linking number of the two tubes, and the plus or minus is chosen according as the linkage is right- or left-handed. For a single knotted vortex tube with circulation ${\displaystyle \kappa }$, then, as shown by Moffatt & Ricca (1992), the helicity is given by ${\displaystyle H=\kappa ^{2}(Wr+Tw)}$, where ${\displaystyle Wr}$ and ${\displaystyle Tw}$ are the writhe and twist of the tube; the sum ${\displaystyle Wr+Tw}$ is known to be invariant under continuous deformation of the tube.

The invariance of helicity provides an essential cornerstone of the subject topological fluid dynamics and magnetohydrodynamics, which is concerned with global properties of flows and their topological characteristics.