Hydrodynamical helicity

In

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

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.

Meteorology

In

According to this formula, if the horizontal wind does not change direction with altitude, H will be zero as ${\displaystyle V_{h}}$ and ${\displaystyle \nabla \times V_{h}}$ are

This notion is used to predict the possibility of

Critical values of SRH (Storm Relative Helicity) for tornadic development, as researched in North America,^[3] are:

• SRH = 150-299 ...
  tornadoes according to Fujita scale
• SRH = 300-499 ... very favourable to supercells development and strong tornadoes
• SRH > 450 ... violent tornadoes
• When calculated only below 1 km (4,000 feet), the cut-off value is 100.