Let be the velocity field and 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 (), or it is compressible with a barotropic relation 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, ) 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
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 and , and no internal twist, the helicity is given by , 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 , then, as shown by Moffatt & Ricca (1992), the helicity is given by , where and are the writhe and twist of the tube; the sum 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,[2] 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, and to only integrate in the vertical direction, replacing the volume integral with a one-dimensional definite integral or line integral:
where
is the altitude,
is the horizontal velocity,
is the horizontal vorticity.
According to this formula, if the horizontal wind does not change direction with altitude, H will be zero as and are perpendicular, 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 [m2/s2] 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:
where is the cloud motion relative to the ground.
Critical values of SRH (Storm Relative Helicity) for tornadic development, as researched in North America,[3] are:
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.
Helicity in itself is not the only component of severe thunderstorms, and these values are to be taken with caution.[4] 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:
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 = 1-2 ... moderate to strong tornadoes
EHI > 2 ... strong tornadoes
Notes
^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.