Hydrodynamical helicity

<h2 id="meteorology">Meteorology</h2>
In <a href="/facts/Meteorology/xblFr28q">meteorology</a>,<a class="footnote-ref" id="fnref:2" href="#fn:2">2</a> helicity corresponds to the transfer of <a href="/facts/Vorticity/2oQ6eT4d">vorticity</a> from the environment to an air parcel in <a href="/facts/Convection/E8vsCPhu">convective</a> motion. Here the definition of helicity is simplified to only use the horizontal component of <a href="/facts/Wind/Nt0QIu5B">wind</a> and <a href="/facts/Vorticity/2oQ6eT4d">vorticity</a>, and to only integrate in the vertical direction, replacing the volume integral with a one-dimensional <a href="/facts/Definite_integral/V7lyV997">definite integral</a> or <a href="/facts/Line_integral/10s4B3Nn">line integral</a>:

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    {\displaystyle 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},}

where

<ul><li>⁠
 
 
 
 Z
 
 
 {\displaystyle Z}
 
⁠ is the altitude,</li>
<li>⁠
 
 
 
 
 
 
 
 V
 →
 
 
 
 
 h
 
 
 
 
 {\displaystyle {\vec {V}}_{h}}
 
⁠ is the horizontal velocity,</li>
<li>⁠
 
 
 
 
 
 
 
 ζ
 →
 
 
 
 
 h
 
 
 
 
 {\displaystyle {\vec {\zeta }}_{h}}
 
⁠ is the horizontal vorticity.</li></ul>
According to this formula, if the horizontal wind does not change direction with <a href="/facts/Altitude/B5eNVurK">altitude</a>, H will be zero as 
 
 
 
 
 V
 
 h
 
 
 
 
 {\displaystyle V_{h}}
 
 and 
 
 
 
 ∇
 ×
 
 V
 
 h
 
 
 
 
 {\displaystyle \nabla \times V_{h}}
 
 are <a href="/facts/Perpendicular/u0k8xqx4">perpendicular</a>, making their <a href="/facts/Scalar_product/tNz8MLjT">scalar product</a> nil. H is then positive if the wind veers (turns <a href="/facts/Clockwise/3Npe4Swe">clockwise</a>) with altitude and negative if it backs (turns <a href="/facts/Counterclockwise/3Npe4Swe">counterclockwise</a>). 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 <a href="/facts/Tornado/HSGNQzHa">tornadic</a> development in a <a href="/facts/Thundercloud/DPWNqTDE">thundercloud</a>. In this case, the vertical integration will be limited below <a href="/facts/Cloud/KDgo0Wqu">cloud</a> tops (generally 3 km or 10,000 feet) and the horizontal wind will be calculated to wind relative to the <a href="/facts/Storm/tgStaeS3">storm</a> in subtracting its motion:

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    {\displaystyle \mathrm {SRH} =\int _{Z_{1}}^{Z_{2}}{\left({\vec {V}}_{h}-{\vec {C}}\right)}\cdot \nabla \times {\vec {V}}_{h}\,d{Z}}

where ⁠
 
 
 
 
 
 
 C
 →
 
 
 
 
 
 {\displaystyle {\vec {C}}}
 
⁠ is the cloud motion relative to the ground.
Critical values of SRH (Storm Relative Helicity) for tornadic development, as researched in <a href="/facts/North_America/djNjCQIl">North America</a>,<a class="footnote-ref" id="fnref:3" href="#fn:3">3</a> are:

<ul><li>SRH = 150-299 ... <a href="/facts/Supercell/xdUV0gdP">supercells</a> possible with weak <a href="/facts/Tornadoes/HSGNQzHa">tornadoes</a> according to <a href="/facts/Fujita_scale/EQPf4WXI">Fujita scale</a></li>
<li>SRH = 300-499 ... very favourable to supercells development and strong tornadoes</li>
<li>SRH > 450 ... violent tornadoes</li>
<li>When calculated only below 1 km (4,000 feet), the cut-off value is 100.</li></ul>
Helicity in itself is not the only component of severe <a href="/facts/Thunderstorm/F6hUzuAu">thunderstorms</a>, and these values are to be taken with caution.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> That is why the Energy Helicity Index (EHI) has been created. It is the result of SRH multiplied by the CAPE (<a href="/facts/Convective_Available_Potential_Energy/79uLhmBE">Convective Available Potential Energy</a>) and then divided by a threshold CAPE:

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    {\displaystyle \mathrm {EHI} ={\frac {\mathrm {CAPE} \times \mathrm {SRH} }{\text{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:

<ul><li>EHI = 1 ... possible tornadoes</li>
<li>EHI = 1-2 ... moderate to strong tornadoes</li>
<li>EHI > 2 ... strong tornadoes</li></ul>
<h2 id="notes">Notes</h2>

<ul><li><a href="/facts/George_Batchelor/peMzQBl8">Batchelor, G.K.</a>, (1967, reprinted 2000) An Introduction to Fluid Dynamics, Cambridge Univ. Press</li>
<li>Ohkitani, K., "Elementary Account Of Vorticity And Related Equations". Cambridge University Press. January 30, 2005. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-81984-9</li>
<li><a href="/facts/Alexandre_Chorin/2aLEPHxF">Chorin, A.J.</a>, "Vorticity and Turbulence". Applied Mathematical Sciences, Vol 103, Springer-Verlag. March 1, 1994. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-387-94197-5</li>
<li><a href="/facts/Andrew_Majda/9KdTpPn8">Majda, A.J.</a> & Bertozzi, A.L., "Vorticity and Incompressible Flow". Cambridge University Press; 1st edition. December 15, 2001. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-63948-4</li>
<li><a href="/facts/David_Tritton/PWUz7kta">Tritton, D.J.</a>, "Physical Fluid Dynamics". Van Nostrand Reinhold, New York. 1977. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-854493-6</li>
<li>Arfken, G., "Mathematical Methods for Physicists", 3rd ed. Academic Press, Orlando, FL. 1985. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-12-059820-5</li>
<li><a href="/facts/Keith_Moffatt/f09CaRMN">Moffatt, H.K.</a> (1969) The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, pp. 117–129.</li>
<li><a href="/facts/Keith_Moffatt/f09CaRMN">Moffatt, H.K.</a> & <a href="/facts/Renzo_L._Ricca/pkpzfnIq">Ricca, R.L.</a> (1992) Helicity and the Cǎlugǎreanu Invariant. Proc. R. Soc. Lond. A 439, pp. 411–429.</li>
<li><a href="/facts/William_Thomson%2c_1st_Baron_Kelvin/LTIxG1jV">Thomson, W. (Lord Kelvin)</a> (1868) On vortex motion. Trans. Roy. Soc. Edin. 25, pp. 217–260.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn: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. <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Martin Rowley retired meteorologist with UKMET. "Definitions of terms in meteorology". Archived from the original on 2006-05-16. Retrieved 2006-07-15. <a href="/wiki/Meteorologist" target="_blank">/wiki/Meteorologist</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn:3">Thompson, Rich. "Explanation of SPC Severe Weather Parameters". National Weather Service - Storm Prediction Center. NOAA. Archived from the original on December 29, 2022. Retrieved February 13, 2023. <a href="https://www.spc.noaa.gov/exper/mesoanalysis/help/begin.html" target="_blank">https://www.spc.noaa.gov/exper/mesoanalysis/help/begin.html</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">"Storm Relative Helicity". NOAA. Retrieved 8 August 2014. <a href="http://www.spc.noaa.gov/exper/mesoanalysis/help/help_srh.html" target="_blank">http://www.spc.noaa.gov/exper/mesoanalysis/help/help_srh.html</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
</ol>

Hydrodynamical helicity open-in-new

Hydrodynamical helicity