Influid dynamics,helicity is, under appropriate conditions, aninvariant of theEuler equations of fluid flow, having a topological interpretation as a measure oflinkage and/orknottedness ofvortex lines in the flow. This was first proved byJean-Jacques Moreau in 1961^[1] andMoffatt derived it in 1969 without the knowledge ofMoreau's paper. This helicity invariant is an extension ofWoltjer's theorem formagnetic helicity.

Let${\displaystyle \mathbf{u}(\mathbf{x},t)}$ be the velocity field and${\displaystyle \mathit{\omega }\equiv \mathrm{\nabla }\times \mathbf{u}}$ the correspondingvorticity field. Under the following three conditions, the vortex lines are transported with (or 'frozen-in') the flow: (i) the fluid isinviscid; (ii) either the flow isincompressible (${\displaystyle \mathrm{\nabla }\cdot \mathbf{u}=0}$), or it is compressible with abarotropic relation${\displaystyle p=p(\rho )}$ between pressurep and densityρ; and (iii) any body forces acting on the fluid areconservative. Under these conditions, any closed surfaceS whose normal vectors are orthogonal to the vorticity (that is,${\displaystyle \mathbf{n}\cdot \mathit{\omega }=0}$) is, like vorticity, transported with the flow.

LetV be the volume inside such a surface. Then the helicity inV, denotedH, is defined by thevolume integral

$${\displaystyle H={\int }_V\mathbf{u}\cdot \mathit{\omega }dV.}$$

For a localised vorticity distribution in an unbounded fluid,V can be taken to be the whole space, andH 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 byLord 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 (orchirality) of the flow. Helicity is one of the four known integral invariants of the Euler equations; the other three areenergy,momentum andangular momentum.

For two linked unknotted vortex tubes havingcirculations${\displaystyle {\kappa }_1}$ and${\displaystyle {\kappa }_2}$, and no internal twist, the helicity is given by${\displaystyle H=\pm 2n{\kappa }_1{\kappa }_2}$, wheren is theGauss 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 byMoffatt &Ricca (1992), the helicity is given by${\displaystyle H={\kappa }^2(Wr+Tw)}$, where${\displaystyle Wr}$ and${\displaystyle Tw}$ are thewrithe andtwist 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 subjecttopological fluid dynamics andmagnetohydrodynamics, which is concerned with global properties of flows and their topological characteristics.

Inmeteorology,^[2] helicity corresponds to the transfer ofvorticity from the environment to an air parcel inconvective motion. Here, the definition of helicity is simplified to only use the horizontal component ofwind andvorticity, and to only integrate in the vertical direction, replacing the volume integral with a one-dimensionaldefinite integral orline integral:

$${\displaystyle H={\int }_{Z_1}^{Z_2}{\mathbf{V}}_h\cdot {\mathit{\zeta }}_{h}dZ={\int }_{Z_1}^{Z_2}{\mathbf{V}}_h\cdot \left(\mathrm{\nabla }\times {\mathbf{V}}_h\right)dZ,}$$

where

• ⁠${\displaystyle Z}$⁠ is the altitude,
• ⁠${\displaystyle {\mathbf{V}}_h}$⁠ is the horizontal velocity,
• ⁠${\displaystyle {\mathit{\zeta }}_h}$⁠ is the horizontal vorticity.

According to this formula, if the horizontal wind does not change direction withaltitude,H will be zero as${\displaystyle V_h}$ and${\displaystyle \mathrm{\nabla }\times V_h}$ areperpendicular, making theirscalar product nil.H is then positive if the wind veers (turnsclockwise) with altitude and negative if it backs (turnscounterclockwise). 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 oftornadic development in athundercloud. In this case, the vertical integration will be limited belowcloud tops (generally 3 km or 10,000 feet) and the horizontal wind will be calculated to wind relative to thestorm in subtracting its motion:

$${\displaystyle \mathrm{S}\mathrm{R}\mathrm{H}={\int }_{Z_1}^{Z_2}\left({\mathbf{V}}_h-\mathbf{C}\right)\cdot \left(\mathrm{\nabla }\times {\mathbf{V}}_h\right)dZ}$$where⁠${\displaystyle \mathbf{C}}$⁠ is the cloud motion relative to the ground.

Critical values of SRH (StormRelativeHelicity) for tornadic development, as researched inNorth America,^[3] are:

• SRH = 150-299 ...supercells possible with weaktornadoes according toFujita scale
• SRH = 300-499 ... very favourable for supercell 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 severethunderstorms, 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:

$${\displaystyle \mathrm{E}\mathrm{H}\mathrm{I}=\frac{\mathrm{C}\mathrm{A}\mathrm{P}\mathrm{E}\times \mathrm{S}\mathrm{R}\mathrm{H}}{\text{160,000}}}$$

This incorporates not only the helicity but also 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

• Fluid dynamics

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