What is absolute vorticity?

A simplification of the concept of absolute vorticity is to view the fluid as having the tendency to form vortices, and that this tendency must be conserved, somewhat similar to the conservation of angular momentum. Whenever there is current shear (a change in velocity at right angles of the flow's direction) there is tendency to rotate, in other words the water has vorticity...

Now, let's get physical!

Lynne D. Talley, ... James H. Swift, in Descriptive Physical Oceanography (Sixth Edition), 2011

(Borrowed from  https://www.sciencedirect.com/topics/earth-and-planetary-sciences/vorticity )

Vorticity in fluids is similar to angular momentum in solids, and many of the intuitions developed about angular momentum from a standard physics course can be applied to understanding vorticity.

Vorticity is twice the angular velocity at a point in a fluid. It is easiest to visualize by thinking of a small paddle wheel immersed in the fluid (Figure 7.25). If the fluid flow turns the paddle wheel, then it has vorticity. Vorticity is a vector, and points out of the plane in which the fluid turns. The sign of the vorticity is given by the “right-hand” rule. If you curl the fingers on your right hand in the direction of the turning paddle wheel and your thumb points upward, then the vorticity is positive. If your thumb points downward, the vorticity is negative (see FIGURE 7.25). The right-hand rule shows the direction of the vorticity by the direction of the thumb (upward for positive, downward for negative).

Vorticity is exactly related to the concept of curl in vector calculus. The vorticity vector ω is the curl of the velocity vector v, expressed here — in Cartesian coordinates:

(Eq. 7.32)               ω = ∇×v = i(∂v/∂z − ∂w/∂y) + j(∂w/∂x − ∂u/∂z) + k(∂v/∂x − ∂u/∂y)

where (i, j, k) is the unit vector in Cartesian coordinates (x, y, z) with corresponding velocity components (u, v, w). Vorticity, therefore, has units of inverse time, for instance, (sec)−1.

Fluids (and all objects) have vorticity simply because of Earth's rotation. This is called planetary vorticity. We do not normally appreciate this component of vorticity since it is only important if a motion lasts for a significant portion of a day, and most important if it lasts for many days, months, or years. Since geostrophic motion is essentially steady compared with the rotation time of Earth, planetary vorticity is very important for nearly geostrophic flows. The vector planetary vorticity points upward, parallel to the rotation axis of Earth. Its size is twice the angular rotation rate Ω of Earth:

(Eq. 7.33)               ω_planetary = 2Ω

where Ω  =  2π/day  =  2π/86160 sec  =  7.293 × 10−5 sec−1, so ω_planetary  =  1.4586 × 10−4 sec−1.

The vorticity of the fluid motion relative to Earth's surface (Eq. 7.32) is called the relative vorticity. It is calculated from the water velocities relative to Earth's surface (which is rotating). The total vorticity of a piece of fluid is the sum of the relative vorticity and planetary vorticity. The total vorticity is sometimes called absolute vorticity, because it is the vorticity the fluid has in the non-rotating reference frame of the stars.

For large-scale oceanography, only the local vertical component of the total vorticity is used because the fluid layers are thin compared with Earth's radius, so flows are nearly horizontal. The local vertical component of the planetary vorticity is exactly equal to the Coriolis parameter f (Eq. 7.8c) and is therefore maximum and positive at the North Pole (φ = 90°N), maximum and negative at the South Pole (φ = 90°S), and 0 at the equator. The local vertical component of the relative vorticity from Eq. (7.32) is

(7.34)               ζ = (∂v∂x − ∂u∂y) = curl zv

Thus the local vertical component of the absolute vorticity is therefore (ζ + f)     or      ω = f +  ζ

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 Note that:    In large-scale meteorology and oceanography, the general term vorticity often used to mean the vertical component, unless specified otherwise. 

For more information visit also:

 http://www-eaps.mit.edu/~rap/courses/12333_notes/A4%20vorticity.pdf