68.37.255.113: /* References */

2014-09-18T14:42:42Z

References

en>PredanJozef at 18:38, 6 January 2014

2014-01-06T18:38:52Z

← Older revision		Revision as of 20:38, 6 January 2014
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	~~Greetings! I am Myrtle Shroyer~~. ~~California~~ is ~~our birth place~~. ~~Doing ceramics~~ is ~~what her family~~ and ~~her appreciate~~. ~~Managing individuals~~ is ~~what I do~~ and the ~~salary has been really satisfying~~.<br><br>~~my page~~: [http://~~bit~~.ly/~~1pABYYJ~~ http://~~bit~~.ly/]		:''This page is about helicity in fluid dynamics. For helicity of magnetic fields, see [[magnetic helicity]]. For helicity in [[particle physics]], see [[helicity (particle physics)]].''

			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 [[Link_(knot_theory) \|linkage]] and/or [[Knot \|knottedness]] of [[Vortex \|vortex lines]] in the flow ([[Keith_Moffatt \|Moffatt]] 1969).

			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; (ii) either the flow is incompressible (<math>\nabla\cdot\mathbf{u} = 0</math>), or it is compressible with a barotropic relation <math>p = p(\rho)</math> between pressure <math>p</math> and density <math>\rho</math>; and (iii) any body forces acting on the fluid are conservative. Under these conditions, any closed surface <math>S</math> on which <math>n \cdot (\nabla\times\mathbf{u}) = 0</math> is, like vorticity, transported with the flow.

			Let <math>V</math> be the volume inside such a surface. Then the helicity in <math>V</math> is defined by

			:<math>
			H=\int_{V}\mathbf{u}\cdot\left(\nabla\times\mathbf{u}\right)\,dV \;.
			</math>

			For a localised vorticity distribution in an unbounded fluid, <math>V</math> can be taken to be the whole space, and <math>H</math> is then the total helicity of the flow. <math>H</math> 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 the only known integral invariant of the Euler equations, apart from [[energy]], [[momentum]] and [[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 <math>n</math> 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 Moffatt & 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_(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 [[topological fluid dynamics]] and [[magnetohydrodynamics]], which is concerned with global properties of flows and their topological characteristics.

			==Meteorology==
			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]]\|accessdate=2006-07-15 \|archiveurl = http://web.archive.org/web/20060516011557/http://homepage.ntlworld.com/booty.weather/FAQ/2A.htm#2A.24 <!-- Bot retrieved archive --> \|archivedate = 2006-05-16}}</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]]:

			::<math>
			H = \int{ \vec V_h} \cdot \vec \zeta_h \,d{\mathbf Z} = \int{ \vec V_h} \cdot \nabla \times \vec V_h \,d{\mathbf Z}
			\qquad \qquad \begin{cases} Z = Altitude \\ \vec V_h = Horizontal\ velocity \\ \vec \zeta_h = Horizontal\ vorticity \end{cases} </math>

			According to this formula, if the horizontal wind does not change direction with [[altitude]], H will be zero as <math>V_h</math> and <math>\nabla \times V_h</math> are [[perpendicular]] one to the other 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 (<math> {m^2}/{s^2}</math>) 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 [[tornado\|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:

			::<math>SRH = \int{ \left ( \vec V_h - \vec C \right )} \cdot \nabla \times \vec V_h \,d{\mathbf Z}
			\qquad \qquad \begin{cases} \vec C = Cloud\ motion\ to\ the\ ground \end{cases} </math>

			Critical values of SRH ('''S'''torm '''R'''elative '''H'''elicity) for tornadic development, as researched in [[North America]],<ref>{{cite web\|url=http://www.spc.noaa.gov/exper/mesoanalysis/s6/index2.html \|title='''EXPLANATION OF SPC SEVERE WEATHER PARAMETERS'''\|author=[[Storm Prediction Center]]\|publisher=[[National Weather Service]]\|accessdate=2006-07-15}}</ref> are:

			* SRH = 150-299 ... [[supercell]]s possible with weak [[tornadoes]] according to [[Fujita scale]]
			* 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 [[thunderstorm]]s and those values are to be taken with caution. 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: '''EHI = (CAPE x SRH) / 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:

			* EHI = 1 ... possible tornadoes
			* EHI = 1-2 ... moderate to strong tornadoes
			* EHI > 2 ... strong tornadoes

			==Notes==

			{{Reflist}}

			==References==

			* [[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
			* [[Alexandre Chorin\|Chorin, A.J.]], "''Vorticity and Turbulence''". Applied Mathematical Sciences, Vol 103, Springer-Verlag. March 1, 1994. ISBN 0-387-94197-5
			* [[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
			* 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
			*[[Keith_Moffatt \|Moffatt, H.K.]] (1969) The degree of knottedness of tangled vortex lines. ''J. Fluid Mech''. '''35''', pp. 117–129.
			*[[Keith_Moffatt \|Moffatt, H.K.]] & [[Renzo_L._Ricca \|Ricca, R.L.]] (1992) Helicity and the Cǎlugǎreanu Invariant. ''Proc. R. Soc. Lond. A'' '''439''', pp. 411–429.
			*[[William_Thomson,_1st_Baron_Kelvin \|Thomson, W. (Lord Kelvin)]] (1868) On vortex motion. ''Trans. Roy. Soc. Edin.'' '''25''', pp. 217-260.


			{{Meteorological variables}}

			{{DEFAULTSORT:Hydrodynamical Helicity}}
			[[Category:Fluid dynamics]]

en>Vice regent: introduction=lede?

2012-04-10T05:28:51Z

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Greetings! I am Myrtle Shroyer. California is our birth place. Doing ceramics is what her family and her appreciate. Managing individuals is what I do and the salary has been really satisfying.<br><br>my page: [http://bit.ly/1pABYYJ http://bit.ly/]

Topology optimization - Revision history

68.37.255.113: /* References */

en>PredanJozef at 18:38, 6 January 2014

en>Vice regent: introduction=lede?