Editing Hydrodynamical helicity (section)

{{short description|Aspect of Eulerian fluid dynamics}}
{{about|fluid dynamics|magnetic fields|magnetic helicity|particle physics|helicity (particle physics)}}

In [[fluid dynamics]], '''helicity''' is, under appropriate conditions, an [[Invariant (mathematics)|invariant]] of the [[Euler equations (fluid dynamics)|Euler equations]] of fluid flow, having a topological interpretation as a measure of [[Link (knot theory)|linkage]] and/or [[knot (mathematics)|knot]]tedness of [[Vortex|vortex lines]] in the flow. This was first proved by [[Jean-Jacques Moreau]] in 1961<ref>Moreau, J. J. (1961). Constantes d'un îlot tourbillonnaire en fluide parfait barotrope. Comptes Rendus hebdomadaires des séances de l'Académie des sciences, 252(19), 2810.</ref> and [[Keith Moffatt|Moffatt]] derived it in 1969 without the knowledge of [[Jean-Jacques Moreau|Moreau]]'s paper. This helicity invariant is an extension of [[Woltjer's theorem]] for [[magnetic helicity]].

Let <math>\mathbf{u}(x,t)</math> be the velocity field and <math>\nabla\times\mathbf{u}</math> 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 flow|inviscid]]; (ii) either the flow is [[Incompressible flow|incompressible]] (<math>\nabla\cdot\mathbf{u} = 0</math>), or it is compressible with a [[Barotropic fluid|barotropic]] relation <math>p = p(\rho)</math> between pressure {{mvar|p}} and density {{mvar|&rho;}}; and (iii) any body forces acting on the fluid are [[conservative force|conservative]].  Under these conditions, any closed surface {{mvar|S}} whose normal vectors are orthogonal to the vorticity (that is, <math>\mathbf{n} \cdot (\nabla\times\mathbf{u}) = 0</math>) is, like vorticity, transported with the flow.

Let {{mvar|V}} be the volume inside such a surface. Then the helicity in {{mvar|V}}, denoted {{mvar|H}}, is defined by the [[volume integral]]

:<math>
H=\int_{V}\mathbf{u}\cdot\left(\nabla\times\mathbf{u}\right)\,dV \;.
</math>

For a localised vorticity distribution in an unbounded fluid, {{mvar|V}}  can be taken to be the whole space, and {{mvar|H}} is then the total helicity of the flow. {{mvar|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 [[William Thomson, 1st Baron Kelvin|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 [[Circulation (physics)|circulations]] <math>\kappa_1</math> and  <math>\kappa_2</math>, and no internal twist, the helicity is given by <math>H = \plusmn 2n \kappa_1 \kappa_2</math>, where {{mvar|n}} is the [[linking number|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 <math>\kappa</math>, then, as shown by [[Keith Moffatt|Moffatt]] & [[Renzo L. Ricca|Ricca]] (1992), the helicity is given by <math>H = \kappa^2 (Wr + Tw)</math>, where <math>Wr</math> and <math>Tw</math> are the [[writhe]] and [[Twist (differential geometry)|twist]] of the tube; the sum <math>Wr + Tw</math> 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.