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| {{More footnotes|date=May 2009|}}
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| {{continuum mechanics|cTopic=Fluid mechanics}}
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| In [[fluid dynamics]], the '''vorticity''' is a [[pseudovector]] [[vector field|field]] that describes the ''local'' [[rotation|spinning]] motion of a [[fluid]] near some point, as would be seen by an observer located at that point and traveling along with the fluid.
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| Conceptually, the vorticity could be determined by marking the particles of the [[fluid]] in a small neighborhood of the point in question, and watching their ''relative'' displacements as they move along the flow. The vorticity vector would be twice the mean [[angular velocity]] vector of those particles relative to their [[center of mass]], oriented according to the [[right-hand rule]]. This quantity must not be confused with the angular velocity of the particles relative to some other point.
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| More precisely, the vorticity of a [[fluid dynamics|flow]] is a [[pseudovector]] [[vector field|field]] {{vec|''ω''}}, equal to the [[curl (mathematics)|curl]] (rotational) of its [[velocity field]] {{vec|''v''}}. It can be expressed by the [[vector analysis]] formula:
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| :<math>\vec{\omega} = \vec{\nabla} \times \vec{v}\,,</math>
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| where {{vec|∇}} is the [[del operator]]. The vorticity of a [[two-dimensional flow]] is always perpendicular to the plane of the flow, and therefore can be considered a [[scalar field]].
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| The vorticity is related to the flow's [[circulation (fluid dynamics)|circulation]] (line integral of the velocity) along a closed path by the Stokes equation.<ref name=Clancy7.11>Clancy, L.J., ''Aerodynamics'', Section 7.11</ref> Namely, for any [[infinitesimal]] [[Differential (infinitesimal)|surface element]] ''C'' with [[normal (geometry)|normal direction]] {{vec|''n''}} and area ''dA'', the circulation ''d''Γ along the [[perimeter]] of ''C'' is the [[dot product]] {{vec|''ω''}} ∙ (''dA'' {{vec|''n''}}) where {{vec|''ω''}} is the vorticity at the center of ''C''.<ref name=Clancy7.11/>
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| Many phenomena, such as the blowing out of a candle by a puff of air, are more readily explained in terms of vorticity rather than the basic concepts of [[pressure]] and velocity. This applies, in particular, to the formation and motion of [[vortex ring]]s.
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| ==Examples==
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| In a mass of fluid that is rotating like a rigid body, the vorticity is twice the [[angular velocity]] vector of that rotation. This is the case, for example, of water in a tank that has been spinning for a while around its vertical axis, at a constant rate.
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| The vorticity may be nonzero even when all particles are flowing along straight and parallel [[pathline|pathlines]], if there is [[shear (fluid)|shear]] (that is, if the flow speed varies across [[Streamlines, streaklines, and pathlines|streamlines]]). For example, in the [[laminar flow]] within a pipe with constant [[cross section (geometry)|cross section]], all particles travel parallel to the axis of the pipe; but faster near that axis, and practically stationary next to the walls. The vorticity will be zero on the axis, and maximum near the walls, where the shear is largest.
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| Conversely, a flow may have zero vorticity even though its particles travel along curved trajectories. An example is the ideal [[vortex|irrotational vortex]], where most particles rotate about some straight axis, with speed inversely proportional to their distances to that axis. A small parcel of fluid that does not straddle the axis will be rotated in one sense but sheared in the opposite sense, in such a way that their mean angular velocity ''about their center of mass'' is zero.
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| <table border="0"> | |
| <tr>
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| <td style="text-align:center;" colspan="3">Example flows:</td>
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| </tr>
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| <tr>
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| <td valign="top">[[File:Vorticity Figure 01 a-m.gif]]</td>
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| <td valign="top">[[File:Vorticity Figure 03 a-m.gif]]</td>
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| <td valign="top">[[File:Vorticity Figure 02 a-m.gif]]</td>
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| </tr>
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| <tr>
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| <td style="text-align:center;">Rigid-body-like vortex</td>
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| <td style="text-align:center;">Parallel flow with shear</td>
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| <td style="text-align:center;">Irrotational vortex</td>
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| </tr>
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| <tr>
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| <td style="text-align:center;" colspan="3">Absolute velocities around the highlighted point:</td>
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| </tr>
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| <tr>
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| <td valign="top">[[File:Vorticity Figure 01 b.png]]</td>
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| <td valign="top">[[File:Vorticity Figure 03 b.png]]</td>
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| <td valign="top">[[File:Vorticity Figure 02 b.png]]</td>
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| </tr>
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| <tr>
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| <td style="text-align:center;" colspan="3">Relative velocities (magnified) around the highlighted point</td>
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| </tr>
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| <tr>
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| <td valign="top">[[File:Vorticity Figure 01 c.png]]</td>
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| <td valign="top">[[File:Vorticity Figure 03 c.png]]</td>
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| <td valign="top">[[File:Vorticity Figure 02 c.png]]</td>
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| </tr>
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| <tr>
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| <td style="text-align:center;">Vorticity ≠ 0</td>
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| <td style="text-align:center;">Vorticity ≠ 0</td>
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| <td style="text-align:center;">Vorticity = 0</td>
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| </tr>
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| </table>
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| Another way to visualize vorticity is to imagine that, instantaneously, a tiny part of the fluid becomes solid and the rest of the flow disappears. If that tiny new solid particle is rotating, rather than just moving with the flow, then there is vorticity in the flow.
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| [[Image:Vorticity visualized diagram.png|Alternate visualization of vorticity.]]
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| ==Mathematical definition==
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| Mathematically, the vorticity of a three-dimensional flow is a pseudovector field, usually denoted by {{vec|''ω''}}, defined as the [[curl (mathematics)|curl]] or rotational of the velocity field {{vec|''v''}} describing the fluid motion. In [[Cartesian coordinates]]:
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| :<math>
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| \begin{array}{rcl}
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| \vec{\omega}
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| &=& \nabla \times \vec{v} \;=\;
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| \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y},\frac{\partial}{\partial z}\right)\times(v_x,v_y,v_z)\\
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| &=&
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| \left(
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| \frac{\partial v_z}{\partial y} - \frac{\partial v_y}{\partial z},\;
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| \frac{\partial v_x}{\partial z} - \frac{\partial v_z}{\partial x},\;
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| \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y}
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| \right)
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| \end{array}
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| </math> | |
| In words, the vorticity tells how the velocity vector changes when one moves by an infinitesimal distance in a direction perpendicular to it.
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| In a two-dimensional flow where the velocity is independent of the ''z'' coordinate and has no ''z'' component, the vorticity vector is always parallel to the ''z'' axis, and therefore can be viewed as a scalar field:
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| :<math>
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| \vec{\omega}
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| \;=\; \nabla \times \vec{v} \;=\;
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| \left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right)\times(v_x,v_y)
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| \;=\; \frac{\partial v_y}{\partial x} - \frac{\partial v_x}{\partial y}
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| </math>
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| == Evolution ==
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| The evolution of the vorticity field in time is described by the [[vorticity equation]], which can be derived from the [[Navier-Stokes equations]].
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| In many real flows where the viscosity can be neglected (more precisely, in flows with high [[Reynolds number]]), the vorticity field can be modeled well by a collection of discrete vortices, the vorticity being negligible everywhere except in small regions of space surrounding the axes of the vortices. This is clearly true in the case of 2-D [[potential flow]] (i.e. 2-D zero viscosity flow), in which case the flowfield can be modeled as a [[complex number|complex-valued]] field on the [[complex plane]].
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| Vorticity is a useful tool to understand how the ideal potential flow solutions can be perturbed to model real flows. In general, the presence of viscosity causes a [[diffusion]] of vorticity away from the vortex cores into the general flow field. This flow is accounted for by the diffusion term in the vorticity transport equation. Thus, in cases of very viscous flows (e.g. [[Couette Flow]]), the vorticity will be diffused throughout the flow field and it is probably simpler to look at the velocity field than at the vorticity.
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| ==Vortex lines and vortex tubes==
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| A '''vortex line''' or '''vorticity line''' is a line which is everywhere tangent to the local vorticity vector. A '''vortex tube''' is the surface in the fluid formed by all vortex-lines passing through a given (reducible) closed curve in the fluid. The 'strength' of a vortex-tube (also called '''vortex flux''') <ref>[http://www.astro.uu.nl/~achterb/aigd/aigd6.ppt Introduction to Astrophysical Gas Dynamics]</ref> is the integral of the vorticity across a cross-section of the tube, and is the same at everywhere along the tube (because vorticity has zero divergence). It is a consequence of [[Helmholtz's theorems]] (or equivalently, of [[Kelvin's circulation theorem]]) that in an inviscid fluid the 'strength' of the vortex tube is also constant with time. Viscous effects introduce frictional losses and time dependence.
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| In a three dimensional flow, vorticity (as measured by the volume integral of its squared magnitude) can be intensified when a vortex-line is extended — a phenomenon known as [[vortex stretching]].<ref>Batchelor, section 5.2</ref> This phenomenon occurs in the formation of a bath-tub vortex in out-flowing water, and the build-up of a tornado by rising air-currents.
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| [[Hydrodynamical helicity|Helicity]] is vorticity in motion along a third axis in a corkscrew fashion.
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| ==Specific sciences==
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| ===Aeronautics===
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| In [[aerodynamics]], the [[lift (force)|lift]] distribution over a finite wing may be approximated by assuming that each segment of the wing has a semi-infinite trailing vortex behind it. It is then possible to solve for the strength of the vortices using the criterion that there be no flow induced through the surface of the wing. This procedure is called the vortex panel method of [[computational fluid dynamics]]. The strengths of the vortices are then summed to find the total approximate [[circulation (fluid dynamics)|circulation]] about the wing. According to the [[Kutta–Joukowski theorem]], lift is the product of circulation, airspeed, and air density.
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| ===Atmospheric sciences===
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| The '''relative vorticity''' is the vorticity of the air velocity field relative to the [[Earth]]. This is often modeled as a two-dimensional flow parallel to the ground, so that the relative vorticity vector is generally perpendicular to the ground, and can then be viewed as a scalar quantity, positive when the vector points upward, negative when it points downwards. Therefore, vorticity is positive then the wind turns counter-clockwise (looking down onto the Earth's surface). In the Northern Hemisphere, positive vorticity is called [[cyclone|cyclonic rotation]], and negative vorticity is [[anticyclone|anticyclonic rotation]]; the nomenclature is reversed in the Southern Hemisphere. | |
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| The '''absolute vorticity''' is computed from the air velocity relative to an inertial frame, and therefore includes a term due to the Earth's rotation, the [[Coriolis parameter]].
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| The [[potential vorticity]] is absolute vorticity divided by the vertical spacing between levels of constant [[entropy]] (or [[potential temperature]]). The absolute vorticity of an air mass will change if the air mass is stretched (or compressed) in the ''z'' direction, but the potential vorticity is [[conservation law|conserved]] in an [[adiabatic]] flow, which predominates in the atmosphere. The potential vorticity is therefore useful as an approximate [[Flow tracer|tracer]] of air masses over the timescale of a few days, particularly when viewed on levels of constant entropy.
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| The [[barotropic vorticity equation]] is the simplest way for forecasting the movement of [[Rossby wave]]s (that is, the [[trough (meteorology)|troughs]] and [[ridge]]s of 500 [[pascal (unit)|hPa]] [[geopotential height]]) over a limited amount of time (a few days). In the 1950s, the first successful programs for [[numerical weather forecasting]] utilized that equation.
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| In modern ''numerical weather forecasting models'' and [[general circulation model]]s (GCM's), vorticity may be one of the predicted variables, in which case the corresponding time-dependent equation is a [[prognostic equation]].
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| Helicity of the air motion is important in forecasting [[supercell]]s and the potential for [[tornado|tornadic]] activity.
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| ==See also==
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| * [[Barotropic vorticity equation]]
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| * [[D'Alembert's paradox]]
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| * [[Vortex]]
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| * [[Vortex tube]]
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| * [[Vortex stretching]]
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| * [[Vortical]]
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| * [[Vorticity equation]]
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| * [[Horseshoe vortex]]
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| * [[Kutta–Joukowski theorem]]
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| * [[Wingtip vortices]]
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| === Atmospheric sciences ===
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| * [[Prognostic equation]]
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| * [[Carl-Gustaf Rossby]]
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| * [[Hans Ertel]]
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| === Fluid dynamics ===
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| * [[Biot-Savart law#Aerodynamics applications|Biot-Savart law]]
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| * [[Circulation (fluid dynamics)|Circulation]]
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| * [[Navier-Stokes equations]]
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| <!--
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| * [[Curl (mathematics)|Curl]] : This link is not required here. A reader is expected to know the meaning of curl operator. It is not required to read the article on curl (which is more general) to increase the understanding of vorticity. If you disagree please remove the comment and include the link.
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| * [[Application of tensor theory in engineering science]] : Removed for similar reasons.
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| -myth, December 2006
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| -->
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| ==References==
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| {{reflist}}
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| *Clancy, L.J. (1975), ''Aerodynamics'', Pitman Publishing Limited, London ISBN 0-273-01120-0
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| * "''[http://www.weather.com/glossary/v.html Weather Glossary]''"' The Weather Channel Interactive, Inc.. 2004.
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| * "''[http://www.tpub.com/content/aerographer/14010/css/14010_18.htm Vorticity]''". Integrated Publishing.
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| ==Further reading==
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| * {{citation | first=G. K. | last=Batchelor | authorlink=George Batchelor | title=An Introduction to Fluid Dynamics | publisher=Cambridge University Press | year=2000 | origyear=1967 | isbn=0-521-66396-2}}
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| * Ohkitani, K., "''Elementary Account Of Vorticity And Related Equations''". Cambridge University Press. January 30, 2005. ISBN 0-521-81984-9
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| * [[Alexandre Chorin|Chorin, Alexandre J.]], "''Vorticity and Turbulence''". Applied Mathematical Sciences, Vol 103, Springer-Verlag. March 1, 1994. ISBN 0-387-94197-5
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| * [[Andrew Majda|Majda, Andrew J.]], Andrea L. Bertozzi, "''Vorticity and Incompressible Flow''". Cambridge University Press; 2002. ISBN 0-521-63948-4
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| * Tritton, D. J., "''Physical Fluid Dynamics''". Van Nostrand Reinhold, New York. 1977. ISBN 0-19-854493-6
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| * Arfken, G., "''Mathematical Methods for Physicists''", 3rd ed. Academic Press, Orlando, FL. 1985. ISBN 0-12-059820-5
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| ==External links==
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| {{Commons category}}
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| * Weisstein, Eric W., "''[http://scienceworld.wolfram.com/physics/Vorticity.html Vorticity]''". Scienceworld.wolfram.com.
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| * Doswell III, Charles A., "''[http://www.cimms.ou.edu/~doswell/vorticity/vorticity_primer.html A Primer on Vorticity for Application in Supercells and Tornadoes]''". Cooperative Institute for Mesoscale Meteorological Studies, Norman, Oklahoma.
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| * Cramer, M. S., "''Navier-Stokes Equations -- [http://www.navier-stokes.net/nsvint.htm Vorticity Transport Theorems]: Introduction''". Foundations of Fluid Mechanics.
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| * Parker, Douglas, "''ENVI 2210 - Atmosphere and Ocean Dynamics, [http://www.env.leeds.ac.uk/envi2210/lectures/lect9.html 9: Vorticity]''{{dead link|date=January 2012}}". School of the Environment, University of Leeds. September 2001.
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| * [[James R. Graham|Graham, James R.]], "''Astronomy 202: Astrophysical Gas Dynamics''". Astronomy Department, [[UC Berkeley]].
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| ** "''[http://astron.berkeley.edu/~jrg/ay202/node92.html The vorticity equation: incompressible and barotropic fluids]''".
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| ** "''[http://astron.berkeley.edu/~jrg/ay202/node93.html Interpretation of the vorticity equation]''".
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| ** "''[http://astron.berkeley.edu/~jrg/ay202/node94.html Kelvin's vorticity theorem for incompressible or barotropic flow]''".
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| * "''[http://www.scd.ucar.edu/css/software/spherepack/ Spherepack 3.1]''". (includes a collection of FORTRAN vorticity program)
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| * "''[http://132.206.43.151:5080/realtime/main_page.html Mesoscale Compressible Community (MC2)] Real-Time Model Predictions''". (Potential vorticity analysis)
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| {{Meteorological variables}}
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| [[Category:Fluid dynamics]]
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| [[Category:Concepts in physics]]
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| [[Category:Atmospheric dynamics]]
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| [[fr:Tourbillon (physique)]]
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