Vorticity

Vorticity is a mathematical concept used in fluid dynamics. It can be related to the amount of "circulation" or "rotation" in a fluid.

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 angular velocity inversely proportionnal to the distance to the axis has a zero vorticity).

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.

Atmospheric sciences

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 talk use the vertical component as the scalar vorticity. The scalar vorticity is positive when the parcel has a counterclockwise rotation for the Northern Hemisphere. It is negative when the parcel has clockwise rotation for the Northern Hemisphere. _[1]

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

The barotropic vorticity equation is the simplest way for forecasting the movement of Rossby waves (that is, the troughs and ridges of 500 mb geopotential) 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 prognostic variables.

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. Lift is the product of circulation, airspeed, and air density.

See also

• Main : Application of tensor theory in engineering science, Barotropic vorticity equation, D'Alembert's paradox, Vortex, Vortical, Vorticity equation, Vortical motion
• Atmospheric sciences : Prognostic variable
• Fluid dynamics: Biot-Savart Law, Circulation, Curl, Navier-Stokes equations

  Further reading

• Ohkitani, K., "Elementary Account Of Vorticity And Related Equations". Cambridge Univ Pr. 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
  

  References

• 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.
• * "The vorticity equation: incompressible and barotropic fluids".
• * "Interpretation of the vorticity equation".
• * "Kelvin's vorticity theorem for incompressible or barotropic flow".
• "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.