# Extensive Definition

Vorticity is a mathematical concept used in
fluid
dynamics. It can be related to the amount of "circulation"
or "rotation" (or more strictly, the local angular rate of
rotation) in a fluid.

The average vorticity in a small region of fluid
flow is equal to the
circulation \Gamma around the boundary of the small region,
divided by the area A of the small region.

- \omega_ = \frac

Notionally, the vorticity at a point in a fluid
is the limit
as the area of the small region of fluid approaches zero at the
point:

- \omega = \frac

Mathematically, the vorticity at a point is a
vector and is defined as the curl of the velocity: \vec \omega =
\vec \nabla \times \vec v .

In the field of flow of a fluid, vorticity \omega
is zero almost everywhere. It is only in special places, such as
the boundary
layer or the core of a vortex, that the vorticity is not
zero.

## Fluid dynamics

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, with conservative body forces:

= \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 (see say Batchelor,
section 5.2). 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.

## Vorticity Equation

- Main article: Vorticity equation

The vorticity equation describes the evolution of
the vorticity of a fluid element as it moves around. In fluid
mechanics this equation can be expressed in vector form as
follows,

\frac = \frac + \vec V \cdot (\vec \nabla \vec
\omega) = \vec \omega \cdot (\vec \nabla \vec V) - \vec \omega
(\vec \nabla \cdot \vec V) + \frac\vec \nabla \rho \times \vec
\nabla p + \vec \nabla \times \left( \frac \right) + \vec \nabla
\times \vec B

where, \vec V is the velocity vector, \rho is the
density, p is the pressure, \underline is the viscous stress tensor
and \vec B is the body force term. The equation is valid for
compressible fluid in the absence of any concentrated torques and
line forces. No assumption is made regarding the relationship
between the stress and the rate of strain tensors (c.f. Newtonian
fluid).

## Atmospheric sciences

In the atmospheric sciences, vorticity is the rotation of air around a vertical axis. Vorticity is a vector quantity and the direction of the vector is given by the right-hand rule with the fingers of the right hand indicating the direction and curvature of the wind. When the vorticity vector points upward into the atmosphere, vorticity is positive; when it points downward into the earth it is negative. Vorticity in the atmosphere is therefore positive for counter-clockwise rotation (looking down onto the earth's surface), and negative for clockwise rotation.In the Northern Hemisphere cyclonic rotation
of the atmosphere is counter-clockwise so is associated with
positive vorticity, and anti-cyclonic
rotation is clockwise so is associated with negative vorticity.
In the Southern Hemisphere cyclonic rotation
is clockwise with negative vorticity; anti-cyclonic
rotation is counter-clockwise with positive vorticity.

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 gives

- \zeta=\frac - \frac

for relative vorticity and

- \eta=\frac - \frac

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 processes, 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
500 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.

In modern numerical weather forecasting models
and GCMs,
vorticity may be one of the predicted variables, in which case the
corresponding time-dependent equation is a prognostic
equation.

## Other fields

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. According to the Kutta–Joukowski theorem, lift is the product of circulation, airspeed, and air density.## See also

### Atmospheric sciences

### Fluid dynamics

## Further reading

- 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
- Chorin, Alexandre J., "Vorticity and Turbulence". Applied Mathematical Sciences, Vol 103, Springer-Verlag. March 1, 1994. ISBN 0-387-94197-5
- Majda, Andrew J., Andrea L. Bertozzi, "Vorticity and Incompressible Flow". Cambridge University Press; 2002. 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

## References

- "Weather Glossary"' The Weather Channel Interactive, Inc.. 2004.
- "Vorticity". Integrated Publishing.

## External links

- 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)
- "[http://132.206.43.151:5080/realtime/main_page.html Mesoscale Compressible Community (MC2)] Real-Time Model Predictions". (Potential vorticity analysis)

vorticity in Czech: Vorticita

vorticity in German: Vorticity

vorticity in Spanish: Vorticidad

vorticity in French: Tourbillon (physique)

vorticity in Italian: Vorticità

vorticity in Hebrew: ערבוליות

vorticity in Japanese: 渦度

vorticity in Norwegian Nynorsk: Kvervling

vorticity in Romanian: Vorticitate

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движение