Helicity (fluid mechanics)
 This page is about helicity in fluid dynamics. For helicity of magnetic fields, see magnetic helicity. For helicity in particle physics, see helicity (particle physics).
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 $\backslash mathbf\{u\}(x,t)$ be the velocity field and $\backslash nabla\backslash times\backslash mathbf\{u\}$ 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 ($\backslash nabla\backslash cdot\backslash mathbf\{u\}\; =\; 0$), or it is compressible with a barotropic relation $p\; =\; p(\backslash rho)$ between pressure $p$ and density $\backslash rho$; and (iii) any body forces acting on the fluid are conservative. Under these conditions, any closed surface $S$ on which $n\; \backslash cdot\; (\backslash nabla\backslash times\backslash mathbf\{u\})\; =\; 0$ is, like vorticity, transported with the flow.
Let $V$ be the volume inside such a surface. Then the helicity in $V$ is defined by
 $$
H=\int_{V}\mathbf{u}\cdot\left(\nabla\times\mathbf{u}\right)\,dV \;.
For a localised vorticity distribution in an unbounded fluid, $V$ can be taken to be the whole space, and $H$ is then the total helicity of the flow. $H$ is invariant precisely because the vortex lines are frozen in the flow and their linkage and/or knottedness is therefore conserved, as recognized by Lord Kelvin (1868). Helicity is a pseudoscalar quantity: it changes sign under change from a righthanded to a lefthanded frame of reference; it can be considered as a measure of the handedness (or chirality) of the flow. Helicity is the only known integral invariant of the Euler equations, apart from energy, momentum and angular momentum.
For two linked unknotted vortex tubes having circulations $\backslash kappa\_1$ and $\backslash kappa\_2$ , and no internal twist, the helicity is given by $H\; =\; \backslash plusmn\; 2n\; \backslash kappa\_1\; \backslash kappa\_2$, where $n$ is the Gauss linking number of the two tubes, and the plus or minus is chosen according as the linkage is right or lefthanded. For a single knotted vortex tube with circulation $\backslash kappa$, then, as shown by Moffatt & Ricca (1992), the helicity is given by $H\; =\; \backslash kappa^2\; (Wr\; +\; Tw)$, where $Wr$ and $Tw$ are the writhe and twist of the tube; the sum $Wr\; +\; Tw$ is known to be invariant under continuous deformation of the tube.
The invariance of helicity provides an essential cornerstone of the subject topological fluid dynamics and magnetohydrodynamics, which is concerned with global properties of flows and their topological characteristics.
Meteorology
In meteorology,^{[1]} helicity corresponds to the transfer of vorticity from the environment to an air parcel in convective motion. Here the definition of helicity is simplified to only use the horizontal component of wind and vorticity:
 $$
H = \int{ \vec V_h} \cdot \vec \zeta_h \,d{\mathbf Z} = \int{ \vec V_h} \cdot \nabla \times \vec V_h \,d{\mathbf Z} \qquad \qquad \begin{cases} Z = Altitude \\ \vec V_h = Horizontal\ velocity \\ \vec \zeta_h = Horizontal\ vorticity \end{cases}
According to this formula, if the horizontal wind does not change direction with altitude, H will be zero as $V\_h$ and $\backslash nabla\; \backslash times\; V\_h$ are perpendicular one to the other making their scalar product nil. H is then positive if the wind veers (turns clockwise) with altitude and negative if it backs (turns counterclockwise). This helicity used in meteorology has energy units per units of mass ($\{m^2\}/\{s^2\}$) and thus is interpreted as a measure of energy transfer by the wind shear with altitude, including directional.
This notion is used to predict the possibility of tornadic development in a thundercloud. In this case, the vertical integration will be limited below cloud tops (generally 3 km or 10,000 feet) and the horizontal wind will be calculated to wind relative to the storm in subtracting its motion:
 $SRH\; =\; \backslash int\{\; \backslash left\; (\; \backslash vec\; V\_h\; \; \backslash vec\; C\; \backslash right\; )\}\; \backslash cdot\; \backslash nabla\; \backslash times\; \backslash vec\; V\_h\; \backslash ,d\{\backslash mathbf\; Z\}$
\qquad \qquad \begin{cases} \vec C = Cloud\ motion\ to\ the\ ground \end{cases}
Critical values of SRH (Storm Relative Helicity) for tornadic development, as researched in North America,^{[2]} are:
 SRH = 150299 ... supercells possible with weak tornadoes according to Fujita scale
 SRH = 300499 ... very favourable to supercells development and strong tornadoes
 SRH > 450 ... violent tornadoes
 When calculated only below 1 km (4,000 feet), the cutoff value is 100.
Helicity in itself is not the only component of severe thunderstorms and those values are to be taken with caution. That is why the Energy Helicity Index (EHI) has been created. It is the result of SRH multiplied by the CAPE (Convective Available Potential Energy) and then divided by a threshold CAPE: EHI = (CAPE x SRH) / 160,000. This incorporates not only the helicity but the energy of the air parcel and thus tries to eliminate weak potential for thunderstorms even in strong SRH regions. The critical values of EHI:
 EHI = 1 ... possible tornadoes
 EHI = 12 ... moderate to strong tornadoes
 EHI > 2 ... strong tornadoes
Notes
References
 Batchelor, G.K., (1967, reprinted 2000) An Introduction to Fluid Dynamics, Cambridge Univ. Press
 Ohkitani, K., "Elementary Account Of Vorticity And Related Equations". Cambridge University Press. January 30, 2005. ISBN 0521819849
 Chorin, A.J., "Vorticity and Turbulence". Applied Mathematical Sciences, Vol 103, SpringerVerlag. March 1, 1994. ISBN 0387941975
 Majda, A.J. & Bertozzi, A.L., "Vorticity and Incompressible Flow". Cambridge University Press; 1st edition. December 15, 2001. ISBN 0521639484
 Tritton, D.J., "Physical Fluid Dynamics". Van Nostrand Reinhold, New York. 1977. ISBN 0198544936
 Arfken, G., "Mathematical Methods for Physicists", 3rd ed. Academic Press, Orlando, FL. 1985. ISBN 0120598205
 Moffatt, H.K. (1969) The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, pp. 117–129.
 Moffatt, H.K. & Ricca, R.L. (1992) Helicity and the Cǎlugǎreanu Invariant. Proc. R. Soc. Lond. A 439, pp. 411–429.
 Thomson, W. (Lord Kelvin) (1868) On vortex motion. Trans. Roy. Soc. Edin. 25, pp. 217260.
