Hydrodynamical helicity: Difference between revisions

Latest revision as of 22:28, 13 November 2025

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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. This was first proved by Jean-Jacques Moreau in 1961^[1] and Moffatt derived it in 1969 without the knowledge of Moreau's paper. This helicity invariant is an extension of Woltjer's theorem for magnetic helicity.

Let $𝐮 (𝐱, t)$ be the velocity field and $ω \equiv \nabla \times 𝐮$ 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 ( $\nabla \cdot 𝐮 = 0$ ), or it is compressible with a barotropic relation $p = p (ρ)$ between pressure Template:Mvar and density Template:Mvar; and (iii) any body forces acting on the fluid are conservative. Under these conditions, any closed surface Template:Mvar whose normal vectors are orthogonal to the vorticity (that is, $𝐧 \cdot ω = 0$ ) is, like vorticity, transported with the flow.

Let Template:Mvar be the volume inside such a surface. Then the helicity in Template:Mvar, denoted Template:Mvar, is defined by the volume integral

$H = \int_{V} 𝐮 \cdot ω d V .$

For a localised vorticity distribution in an unbounded fluid, Template:Mvar can be taken to be the whole space, and Template:Mvar is then the total helicity of the flow. Template:Mvar 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 $κ_{1}$ and $κ_{2}$ , and no internal twist, the helicity is given by $H = \pm 2 n κ_{1} κ_{2}$ , where Template:Mvar 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 $κ$ , then, as shown by Moffatt & Ricca (1992), the helicity is given by $H = κ^{2} (W r + T w)$ , where $W r$ and $T w$ are the writhe and twist of the tube; the sum $W r + T w$ 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:

$H = \int_{Z_{1}}^{Z_{2}} 𝐕_{h} \cdot ζ_{h} d Z = \int_{Z_{1}}^{Z_{2}} 𝐕_{h} \cdot (\nabla \times 𝐕_{h}) d Z,$

where

Template:Tmath is the altitude,
Template:Tmath is the horizontal velocity,
Template:Tmath is the horizontal vorticity.

According to this formula, if the horizontal wind does not change direction with altitude, Template:Mvar will be zero as $V_{h}$ and $\nabla \times V_{h}$ are perpendicular, making their scalar product nil. Template:Mvar 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:

$S R H = \int_{Z_{1}}^{Z_{2}} (𝐕_{h} - 𝐂) \cdot (\nabla \times 𝐕_{h}) d Z$ where Template:Tmath 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 = 150-299 ... supercells possible with weak tornadoes according to Fujita 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 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:

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

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

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.
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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. Template:ISBN
Chorin, A.J., "Vorticity and Turbulence". Applied Mathematical Sciences, Vol 103, Springer-Verlag. March 1, 1994. Template:ISBN
Majda, A.J. & Bertozzi, A.L., "Vorticity and Incompressible Flow". Cambridge University Press; 1st edition. December 15, 2001. Template:ISBN
Tritton, D.J., "Physical Fluid Dynamics". Van Nostrand Reinhold, New York. 1977. Template:ISBN
Arfken, G., "Mathematical Methods for Physicists", 3rd ed. Academic Press, Orlando, FL. 1985. Template:ISBN
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. 217–260.

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[1] 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.

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[2]

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[4]

@@ Line 4: / Line 4: @@
 In [[fluid dynamics]], '''helicity''' is, under appropriate conditions, an [[Invariant (mathematics)|invariant]] of the [[Euler equations (fluid dynamics)|Euler equations]] of fluid flow, having a topological interpretation as a measure of [[Link (knot theory)|linkage]] and/or [[knot (mathematics)|knot]]tedness of [[Vortex|vortex lines]] in the flow. This was first proved by [[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 [[Keith Moffatt|Moffatt]] derived it in 1969 without the knowledge of [[Jean-Jacques Moreau|Moreau]]'s paper. This helicity invariant is an extension of [[Woltjer's theorem]] for [[magnetic helicity]].
-Let <math>\mathbf{u}(x,t)</math> be the velocity field and <math>\nabla\times\mathbf{u}</math> 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 flow|inviscid]]; (ii) either the flow is [[Incompressible flow|incompressible]] (<math>\nabla\cdot\mathbf{u} = 0</math>), or it is compressible with a [[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 [[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 <math>\mathbf{u}(\mathbf{x},t)</math> be the velocity field and <math>\boldsymbol{\omega}\equiv
+\nabla\times\mathbf{u}</math> 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 flow|inviscid]]; (ii) either the flow is [[Incompressible flow|incompressible]] (<math>\nabla\cdot\mathbf{u} = 0</math>), or it is compressible with a [[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 [[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\boldsymbol{\omega}= 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]]
-:<math>
+<math display="block">
-H=\int_{V}\mathbf{u}\cdot\left(\nabla\times\mathbf{u}\right)\,dV \;.
+H = \int_V \mathbf{u} \cdot \boldsymbol{\omega}\, 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 [[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]]) 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 [[Circulation (physics)|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 two linked unknotted vortex tubes having [[Circulation (physics)|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 [[Keith Moffatt|Moffatt]] & [[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]] and [[Twist (differential geometry)|twist]] of the tube; the sum <math>Wr + Tw</math> is known to be invariant under continuous deformation of the tube.
@@ Line 22: / Line 23: @@
 In [[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]] 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]] from the environment to an air parcel in [[convection|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]]:
-:<math>
+<math display="block">
-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>
+H = \int_{Z_1}^{Z_2} \mathbf V_h \cdot \boldsymbol \zeta_h \, dZ = \int_{Z_1}^{Z_2} \mathbf V_h \cdot \left(\nabla \times \mathbf V_h\right) dZ ,</math>
 where
 *{{tmath|Z}} is the altitude,
-*{{tmath|\vec V_h}} is the horizontal velocity,
+*{{tmath|\mathbf V_h}} is the horizontal velocity,
-*{{tmath|\vec \zeta_h}} is the horizontal vorticity.
+*{{tmath|\boldsymbol \zeta_h}} is the horizontal vorticity.
 According to this formula, if the horizontal wind does not change direction with [[altitude]], {{mvar|H}} will be zero as <math>V_h</math> and <math>\nabla \times V_h</math> are [[perpendicular]], making their [[scalar product]] nil. {{mvar|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{{sup|2}}/s{{sup|2}}] and thus is interpreted as a measure of energy transfer by the wind shear with altitude, including directional.
@@ Line 34: / Line 35: @@
 This notion is used to predict the possibility of [[tornado|tornadic]] development in a [[thundercloud]]. In this case, the vertical integration will be limited below [[cloud]] tops (generally 3&nbsp;km or 10,000 feet) and the horizontal wind will be calculated to wind relative to the [[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>
+<math display="block">\mathrm{SRH} = \int_{Z_1}^{Z_2} \left( \mathbf V_h - \mathbf C \right)  \cdot \left( \nabla \times \mathbf V_h \right) \, dZ </math>
-where {{tmath|\vec C}} is the cloud motion relative to the ground.
+where {{tmath|\mathbf 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 [[North America]],<ref>{{cite web |author=Thompson |first=Rich |author-link= |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=[[National Weather Service]] - [[Storm Prediction Center]] |publisher=[[NOAA]]}}</ref> are:
@@ Line 46: / Line 47: @@
 Helicity in itself is not the only component of severe [[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>
+<math display="block">\mathrm{EHI} = \frac{\mathrm{CAPE} \times \mathrm{SRH}}{\text{160,000}}</math>
 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:
@@ Line 59: / Line 60: @@
 ==References==
+{{refbegin}}
 * [[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}}
@@ Line 69: / Line 70: @@
 *[[Keith Moffatt|Moffatt, H.K.]] & [[Renzo L. Ricca|Ricca, R.L.]] (1992) Helicity and the Cǎlugǎreanu Invariant. ''Proc. R. Soc. Lond. A'' '''439''', pp.&nbsp;411–429.
 *[[William Thomson, 1st Baron Kelvin|Thomson, W. (Lord Kelvin)]] (1868) On vortex motion. ''Trans. Roy. Soc. Edin.'' '''25''', pp.&nbsp;217–260.
+{{refend}}
 {{Meteorological variables}}

Hydrodynamical helicity: Difference between revisions

Latest revision as of 22:28, 13 November 2025

Meteorology

Notes

References

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