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## PowerPedia:Vortex

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: For a list of links about vortex, see Directory:Vortex Technologies

A vortex is a often There was an error working with the wiki: Code, flow (or any There was an error working with the wiki: Code whirling motion) with closed streamlines. The shape of media or mass rotating rapidly around a center forms a vortex. It is a flow involving rotation about an axis (not always oriented vertically though sometimes possessing a horizontal axis).

#### Dynamics

A vortex can be any circular or rotary flow that possesses vorticity. In There was an error working with the wiki: Code, vorticity is the circulation per unit area at a point in the flow field. It is a vector quantity, whose direction is along the axis of the swirl. It can be related to the amount of "circulation" or "rotation" in a fluid. In fluid dynamics, vorticity is the circulation per unit area at a point in the flow field. It is a vector quantity, whose direction is (roughly speaking) along the axis of the swirl.

##### Vorticity

In fluid dynamics, vorticity is the curl of the fluid velocity. It can also be considered as the circulation per unit area at a point in a fluid flow field. It is a vector quantity, whose direction is along the axis of the fluid's rotation. For a two-dimensional flow, the vorticity vector is perpendicular to the plane.

For a fluid having locally a "rigid rotation" around an axis (i.e., moving like a rotating cylinder), vorticity is twice the angular velocity of a fluid element. An irrotational fluid is one whose vorticity=0. Somewhat counter-intuitively, an irrotational fluid can have a non-zero angular velocity (e.g. a fluid rotating around an axis with its tangential velocity inversely proportional to the distance to the axis has a zero vorticity) (see also forced and free vortex)

One way to visualize vorticity is this: consider a fluid flowing. Imagine that some tiny part of the fluid is instantaneously rendered solid, and the rest of the flow removed. If that tiny new solid particle would be rotating, rather than just translating, then there is vorticity in the flow.

In general, vorticity is a specially powerful concept in the case that the viscosity is low (i.e. high Reynolds number). In such cases, even when the velocity field is relatively complicated, the vorticity field can be well approximated as zero nearly everywhere except in a small region in space. 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 identified with the complex plane, and questions about those sorts of flows can be posed as questions in complex analysis which can often be solved (or approximated very well) analytically.

For any flow, you can write the equations of the flow in terms of vorticity rather than velocity by simply taking the curl of the flow equations that are framed in terms of velocity (may have to apply the 2nd Fundamental Theorem of Calculus to do this rigorously). In such a case you get the vorticity transport equation which is as follows in the case of incompressible (i.e. low mach number) fluids. This is expressed as:

:

{D\omega \over Dt} = \omega \cdot \nabla u + \nu \nabla^2 \omega

Even for real flows (3-dimensional and finite Re), the idea of viewing things in terms of vorticity is still very powerful. It provides the most useful way to understand how the potential flow solutions can be perturbed for "real flows." In particular, one restricts attention to the vortex dynamics, which presumes that the vorticity field can be modeled well in terms of discrete vortices (which encompasses a large number of interesting and relevant flows). In general, the presence of viscosity causes a diffusion of vorticity away from these small regions (e.g. discrete vortices) into the general flow field. This can be seen 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 (i.e. vectors of fluid motion) rather than look at the vorticity field (i.e. vectors of curl of fluid motion) which is less intuitive.

Related concepts are the vortex-line, which is a line which is everywhere tangent to the local vorticity and a vortex tube which 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 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.

Note however that in a three dimensional flow, vorticity (as measured by the volume integral of its square) can be intensified when a vortex-line is extended. Mechanisms such as these operate in such well known examples as the formation of a bath-tub vortex in out-flowing water, and the build-up of a tornado by rising air-currents. Mathematically, it is defined as,

: \omega = \nabla \times {u}

where {u}={u}{i} + {v}{j} + {w}{k} is the fluid velocity. The properties of vorticity in 2 and 3 dimensions are treated in some depth in George Batchelor's famous textbook (ch 5 & ch 7 et seq.). Of particular importance in practical situations is the intensification of vorticity which takes place in three dimensions when a vortex-line is extended (p270 et seq).

In the atmospheric sciences, vorticity is a property that characterizes large-scale rotation of air masses. Since atmospheric circulations important to meteorology are largely horizontal, the vorticity vector generally points almost vertically upwards. It is therefore common to use only the vertical component of the vorticity vector for meteorological applications. In the Northern Hemisphere, vorticity is positive for counter-clockwise (i.e., cyclonic) rotation, and negative for clockwise (i.e, anti-cyclonic) rotation. The opposite is true in the Southern Hemisphere.

Relative and absolute vorticity are defined as the z-components of the curls of relative (i.e., in relation to Earth's surface) and absolute wind velocity, respectively. This is expressed as,

:\zeta=\frac{\partial v_r}{\partial x} - \frac{\partial u_r}{\partial y}

for relative vorticity and,

:\eta=\frac{\partial v_a}{\partial x} - \frac{\partial u_a}{\partial y}

for absolute vorticity, where u and v are the zonal (x direction) and meridional (y direction) components of wind velocity. The absolute vorticity at a point can also be expressed as the sum of the relative vorticity at that point and the Coriolis parameter at that latitude (i.e., it is the sum of the Earth's vorticity and the vorticity of the air relative to the Earth).

A useful related quantity is potential vorticity. The absolute vorticity of an air mass will change if the air mass is stretched (or compressed) in the z direction. But if the absolute vorticity is divided by the vertical spacing between levels of constant entropy (or potential temperature), the result is a conserved quantity of adiabatic flow, termed potential vorticity (PV). Because diabatic process which can change PV and entropy occur relatively slowly in the atmosphere, PV is useful as an approximate tracer of air masses over the timescale of a few days, particularly when viewed on levels of constant entropy.

The barotropic vorticity equation is the simplest way for forecasting the movement of Rossby waves (that is, the troughs and ridges of 50 kPa geopotential) over a limited amount of time (a few days). In the 1950s, the first successful programs for numerical weather forecasting utilized that equation. Vorticity is important in many other areas of fluid dynamics. For instance, the 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 about the wing. Lift is the product of circulation, airspeed, and air density.

##### Vortical

In fluid dynamics, the movement of a fluid can be said to be vortical if the fluid moves around in a circle, or in a helix, or if it tends to spin around some axis. Such motion can also be called solenoidal. In the atmospheric sciences, vorticity is a property that characterizes large-scale rotation of air masses. Since the atmospheric circulation is nearly horizontal, the (3 dimensional) vorticity is nearly vertical, and it is common to use the vertical component as a scalar vorticity.

Vortical means pertaining to a vortex or to vortices. In fluid dynamics, the movement of a fluid can be said to be vortical if the fluid moves around in a circle, or in a helix, or if it tends to spin around some axis. Such motion can also be called solenoidal.

Vortical movement is characterized by non-zero curl:

: {curl}\ {v} \ne 0,

where v\, is the velocity vector field of the fluid.

The curl of the velocity (at a specified point of the vector field) yields a vector which points in the direction around which the fluid is rotating.

#### Two types of vortex

In fluid mechanics, a distinction is often made between two limiting vortex cases. One is called the free (irrotational) vortex, and the other is the forced (rotational) vortex. These are considered as below:

##### Free (irrotational) vortex

When fluid is drawn down a plug-hole, one can observe the phenomenon of a free vortex. The tangential velocity v varies inversely as the distance r from the centre of rotation, so the angular momentum, rv, is constant the vorticity is zero everywhere (except for a singularity at the centre-line) and the circulation in fluid dynamics) about a contour containing r=0 has the same value everywhere. The free surface (if present) dips sharply (as r^{-2} ) as the centre line is approached.

The tangential velocity is given by:

:v_{\theta} = \frac{\Gamma}{2 \pi r}\,

where ? is the circulation and r is the radial distance from the center of the vortex.

##### Forced (Rotational) Vortex

In a forced vortex the fluid essentially rotates as a solid body (there is no shear). The motion can be realised by placing a dish of fluid on a turntable rotating at T radians/sec the fluid has vorticity of 2 T everywhere, and the free surface (if present) is a parabola.

The tangential velocity is given by:

:v_{\theta} = \omega r\,

where ? is the angular velocity and r is the radial distance from the center of the vortex.

#### Observations

A vortex can be seen in the spiraling There was an error working with the wiki: Code makes many vortices. A good example of a vortex is the There was an error working with the wiki: Code There was an error working with the wiki: Code of a There was an error working with the wiki: Code or a There was an error working with the wiki: Code. This whirling air mass mostly takes the form of a There was an error working with the wiki: Code, There was an error working with the wiki: Code, or There was an error working with the wiki: Code. Tornadoes develop from severe thunderstorms, usually spawned from There was an error working with the wiki: Codes and There was an error working with the wiki: Codes, though they sometimes happen as a result of a There was an error working with the wiki: Code. A mesovortex is on the scale of a few There was an error working with the wiki: Code or There was an error working with the wiki: Code. This swirling flow structure within a region of fluid flow opens downward from the water surface.

Other instances of vortex phenoenmon include, but are not limited to,

In the There was an error working with the wiki: Code interpretation of the behaviour of Electromagnetic fields, the acceleration of electric fluid in a particular direction creates a positive vortex of magnetic fluid. This in turn creates around itself a corresponding negative vortex of electric fluid.

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The primary cause of There was an error working with the wiki: Code in the There was an error working with the wiki: Code of a There was an error working with the wiki: Code.

There was an error working with the wiki: Code : a swirling body of water produced by ocean tides or by a hole underneath the vortex, where water drains out, as in a bathtub. In popular imagination, but only rarely in reality, can they have the dangerous effect of destroying boats.

There was an error working with the wiki: Code : a violent windstorm characterized by a twisting, funnel-shaped cloud. A less violent version of a tornado, over water, is called a There was an error working with the wiki: Code.

There was an error working with the wiki: Code : a much larger, swirling body of clouds produced by evaporating warm ocean water and influenced by the Earth's rotation. Similar, but far greater, vortices are also seen on other planets, such as the permanent There was an error working with the wiki: Code on There was an error working with the wiki: Code and the intermittent There was an error working with the wiki: Code on There was an error working with the wiki: Code.

There was an error working with the wiki: Code : a persistent, large-scale cyclone centered near the Earth's poles, in the middle and upper troposphere and the stratosphere.

There was an error working with the wiki: Code : dark region on the Sun's surface (photosphere) marked by a lower temperature than its surroundings, and intense magnetic activity.

The There was an error working with the wiki: Code of a There was an error working with the wiki: Code or other massive gravitational source.

There was an error working with the wiki: Code : a type of galaxy in the Hubble sequence which is characterized by a thin, rotating disk. Our galaxy, the There was an error working with the wiki: Code is of this type.

#### Related concepts

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#### References and external articles

General

Ting, L., "Viscous vortical flows" (Lecture notes in physics). Springer-Verlag 1991. ISBN 3-540-53713-9.

There was an error working with the wiki: Code, Wikipedia: The Free Encyclopedia. Wikimedia Foundation.

"Weather Glossary"' The Weather Channel Interactive, Inc.. 2004.

"Glossary and Abbreviations". Risk Prediction Initiative. The Bermuda Biological Station for Research, Inc.. St. George's, Bermuda. 2004.

Loper, David E., "An analysis of confined magnetohydrodynamic vortex flows". Case Institute of Technology. Washington, National Aeronautics and Space Administration] for sale by the Clearinghouse for Federal Scientific and Technical Information, Springfield, Va. 1966. (NASA contractor report NASA CR-646) LCCN 67060315

George Batchelor K., An Introduction to Fluid Dynamics, Cambridge Univ. Press, 1967 (reprinted 2000). Ch. 7 et seq.

"Weather Glossary"' The Weather Channel Interactive, Inc.. 2004.

"Glossary and Abbreviations". Risk Prediction Initiative. The Bermuda Biological Station for Research, Inc.. St. George's, Bermuda. 2004.

"Weather Glossary"' The Weather Channel Interactive, Inc.. 2004.

"Vorticity". Integrated Publishing.

Ohkitani, K., "Elementary Account Of Vorticity And Related Equations". Cambridge University Press. January 30, 2005. ISBN 0521819849

Chorin, Alexandre J., "Vorticity and Turbulence". Applied Mathematical Sciences, Vol 103, Springer-Verlag. March 1, 1994. ISBN 0387941975

Majda, Andrew J., Andrea L. Bertozzi, and D. G. Crighton, "Vorticity and Incompressible Flow". Cambridge University Press 1st edition. December 15, 2001. ISBN 0521639484

Tritton, D. J., "Physical Fluid Dynamics". Van Nostrand Reinhold, New York. 1977. ISBN 0198544936

Arfken, G., "Mathematical Methods for Physicists", 3rd ed. Academic Press, Orlando, FL. 1985. ISBN 0120598205

Websites

Vortical Flow Research Lab (MIT) -- Study of flows found in nature and part of the Department of Ocean Engineering.

Kida, Shigeo, "Life, Structure, and Dynamical Role of Vortical Motion in Turbulence". National Institute for Fusion Science, Theory and Computer Simulation Center. (There was an error working with the wiki: Code)

Marcos, C. de la Fuente, and P. Barge, "The effect of long-lived vortical circulation on the dynamics of dust particles in the mid-plane of a protoplanetary disc".

"Vortical" images via Google.

Dust Devil Movie A short movie showing many spinning vortices of varying sizes

Video of two water vortex rings colliding (There was an error working with the wiki: Code)

BubbleRings.com Web site on "bubble rings", which are underwater rings made of air formed from vortices. The site has some information on how these rings work.

Weisstein, Eric W., "Vorticity". Scienceworld.wolfram.com.

Doswell III, Charles A., "A Primer on Vorticity for Application in Supercells and Tornadoes". Cooperative Institute for Mesoscale Meteorological Studies, Norman, Oklahoma.

Cramer, M. S., "Navier-Stokes Equations -- Vorticity Transport Theorems: Introduction". Foundations of Fluid Mechanics.

Parker, Douglas, "ENVI 2210 - Atmosphere and Ocean Dynamics, 9: Vorticity". School of the Environment, University of Leeds. September 2001.

Graham, James R., "Astronomy 202: Astrophysical Gas Dynamics". Astronomy Department, UC, Berkeley.

"Spherepack 3.1". (includes a collection of FORTRAN vorticity program)

"Mesoscale Compressible Community (MC2) Real-Time Model Predictions". (Potential vorticity analysis)

"Vorticity" images via Google.