The average vorticity in a small region of fluid flow is equal to the circulation around the boundary of the small region, divided by the area A of the small region.
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:
Mathematically, the vorticity at a point is a vector and is defined as the curl of the velocity:
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:
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. Viscous effects introduce frictional losses and time dependence.
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.
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,
where, is the velocity vector, is the density, is the pressure, is the viscous stress tensor and 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).
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.
for relative vorticity and
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.