View source for Physics:Hydrodynamical helicity

{{short description|Aspect of Eulerian fluid dynamics}}
{{about|Physics:Fluid dynamics|Physics:Magnetic fields|Astronomy:Magnetic helicity|Physics:Particle physics|helicity (Physics:Particle physics)}}

In [[Physics:Fluid dynamics|fluid dynamics]], '''helicity''' is, under appropriate conditions, an [[Invariant (mathematics)|invariant]] of the Euler equations of fluid flow, having a topological interpretation as a measure of [[Link (knot theory)|linkage]] and/or knottedness of [[Vortex|vortex lines]] in the flow. This was first proved by [[Biography:Jean-Jacques Moreau|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 [[Biography:Keith Moffatt|Moffatt]] derived it in 1969 without the knowledge of [[Biography:Jean-Jacques Moreau|Moreau]]'s paper. This helicity invariant is an extension of [[Astronomy:Woltjer's theorem|Woltjer's theorem]] for [[Astronomy:Magnetic helicity|magnetic helicity]].

Let <math>\mathbf{u}(x,t)</math> be the velocity field and <math>\nabla\times\mathbf{u}</math> the corresponding [[Vorticity|vorticity]] field. Under the following three conditions, the vortex lines are transported with (or 'frozen in') the flow: (i) the fluid is [[Physics:Inviscid flow|inviscid]]; (ii) either the flow is [[Physics:Incompressible flow|incompressible]] (<math>\nabla\cdot\mathbf{u} = 0</math>), or it is compressible with a [[Physics: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 [[Physics: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|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 [[Biography: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|chirality]]) of the flow. Helicity is one of the four known integral invariants of the Euler equations; the other three are [[Physics:Energy|energy]], [[Physics:Momentum|momentum]] and [[Physics:Angular momentum|angular momentum]].

For two linked unknotted vortex tubes having 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 [[Biography:Keith Moffatt|Moffatt]] & [[Biography: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|writhe]] and [[Twist (mathematics)|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 [[Physics:Topological fluid dynamics|topological fluid dynamics]] and [[Physics:Magnetohydrodynamics|magnetohydrodynamics]], which is concerned with global properties of flows and their topological characteristics.

==Meteorology==
In [[Physics:Meteorology|meteorology]],<ref>{{cite web|url=http://homepage.ntlworld.com/booty.weather/FAQ/2A.htm#2A.24 |title=Definitions of terms in meteorology |author=''Martin Rowley'' retired [[Meteorologist|meteorologist]] with UKMET |access-date=2006-07-15 |archive-url=https://web.archive.org/web/20060516011557/http://homepage.ntlworld.com/booty.weather/FAQ/2A.htm#2A.24 |archive-date=2006-05-16 |url-status=dead }}</ref> helicity corresponds to the transfer of [[Vorticity|vorticity]] from the environment to an air parcel in [[Astronomy:Convection|convective]] motion. Here the definition of helicity is simplified to only use the horizontal component of [[Earth:Wind|wind]] and [[Vorticity|vorticity]], and to only integrate in the vertical direction, replacing the volume integral with a one-dimensional definite integral or [[Line integral|line integral]]:

:<math>
H = \int_{Z_1}^{Z_2}{ \vec V_h} \cdot \vec \zeta_h \,d{Z} = \int_{Z_1}^{Z_2}{ \vec V_h} \cdot \nabla \times \vec V_h  \,d{Z} ,</math>

where
*{{tmath|Z}} is the altitude,
*{{tmath|\vec V_h}} is the horizontal velocity,
*{{tmath|\vec \zeta_h}} is the horizontal vorticity.

According to this formula, if the horizontal wind does not change direction with [[Altitude|altitude]], {{mvar|H}} will be zero as <math>V_h</math> and <math>\nabla \times V_h</math> are [[Perpendicular|perpendicular]], making their scalar product nil. {{mvar|H}} is then positive if the wind veers (turns [[Physics:Clockwise|clockwise]]) with altitude and negative if it backs (turns counterclockwise). This helicity used in meteorology has energy units per units of mass [m{{sup|2}}/s{{sup|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 [[Earth:Tornado|tornadic]] development in a thundercloud. In this case, the vertical integration will be limited below [[Earth:Cloud|cloud]] tops (generally 3&nbsp;km or 10,000 feet) and the horizontal wind will be calculated to wind relative to the [[Earth:Storm|storm]] in subtracting its motion:

::<math>\mathrm{SRH} = \int_{Z_1}^{Z_2}{ \left ( \vec V_h - \vec C \right )}  \cdot \nabla \times \vec V_h  \,d{Z} </math>
where {{tmath|\vec C}} is the cloud motion relative to the ground.

Critical values of SRH ('''S'''torm '''R'''elative '''H'''elicity) for tornadic development, as researched in [[Earth:North America|North America]],<ref>{{cite web |author=Thompson |first=Rich |title=Explanation of SPC Severe Weather Parameters |url=https://www.spc.noaa.gov/exper/mesoanalysis/help/begin.html |url-status=live |archive-url=https://web.archive.org/web/20221229175208/https://www.spc.noaa.gov/exper/mesoanalysis/help/begin.html |archive-date=December 29, 2022 |access-date=February 13, 2023 |website=[[Earth:National Weather Service|National Weather Service]] - [[Earth:Storm Prediction Center|Storm Prediction Center]] |publisher=NOAA}}</ref> are:

* SRH = 150-299 ... [[Earth:Supercell|supercell]]s possible with weak tornadoes according to [[Earth:Fujita scale|Fujita scale]]
* SRH = 300-499 ... very favourable to supercells development and strong tornadoes
* SRH > 450 ... violent tornadoes
* When calculated only below 1&nbsp;km (4,000 feet), the cut-off value is 100.

Helicity in itself is not the only component of severe [[Physics:Thunderstorm|thunderstorm]]s, and these values are to be taken with caution.<ref>{{cite web|title=Storm Relative Helicity|url=http://www.spc.noaa.gov/exper/mesoanalysis/help/help_srh.html|publisher=NOAA|access-date=8 August 2014}}</ref> 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:

:<math>\mathrm{EHI} = \frac{\mathrm{CAPE} \times \mathrm{SRH}}{\text{160,000}}</math>

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==

{{Reflist}}

==References==

* [[Biography:George Batchelor|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|0-521-81984-9}}
* [[Biography:Alexandre Chorin|Chorin, A.J.]], "''Vorticity and Turbulence''". Applied Mathematical Sciences, Vol 103, Springer-Verlag. March 1, 1994. {{ISBN|0-387-94197-5}}
* [[Biography:Andrew Majda|Majda, A.J.]] & Bertozzi, A.L., "''Vorticity and Incompressible Flow''". Cambridge University Press; 1st edition. December 15, 2001. {{ISBN|0-521-63948-4}}
* [[Biography:David Tritton|Tritton, D.J.]], "''Physical Fluid Dynamics''". Van Nostrand Reinhold, New York. 1977. {{ISBN|0-19-854493-6}}
* Arfken, G., "''Mathematical Methods for Physicists''", 3rd ed. Academic Press, Orlando, FL. 1985. {{ISBN|0-12-059820-5}}
*[[Biography:Keith Moffatt|Moffatt, H.K.]] (1969) The degree of knottedness of tangled vortex lines. ''J. Fluid Mech''. '''35''', pp.&nbsp;117–129.
*[[Biography:Keith Moffatt|Moffatt, H.K.]] & [[Biography:Renzo L. Ricca|Ricca, R.L.]] (1992) Helicity and the Cǎlugǎreanu Invariant. ''Proc. R. Soc. Lond. A'' '''439''', pp.&nbsp;411–429.
*[[Biography:William Thomson, 1st Baron Kelvin|Thomson, W. (Lord Kelvin)]] (1868) On vortex motion. ''Trans. Roy. Soc. Edin.'' '''25''', pp.&nbsp;217–260.

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[[Category:Fluid dynamics]]

{{Sourceattribution|Hydrodynamical helicity}}