Introducing Circulation and Vorticity

A close path is ___ ______ a function

not actually

Define circulation as

C = (closed integral) [vector V times d l

Points to be careful about (1)

The “path” of integration does not necessarily have anything to do with the path of fluid parcels.. We can use any arbitrary closed path for integration.

Points to be careful about (2)

We only need to use values of (vector V) along the path, not values inside the area.

Points to be careful about (3)

Direction matters, with counterclockwise being defined as positive

The circulation around the circle of radius R centered on the axis of rotation works out to be

C = 2pi(omega)R²

Because of the integration over an area, we say that circulation is a

macroscopic quantity/metric

In the atmosphere we typically have two types of rotation to consider

Rotation of the earth about its axis

Rotational air flow measured relative to the earth (often pseudo-horizontal)

The change of circulation by line integration of the equation of motion

Kelvin’s Circulation Theorem: Dca/Dt = - (closed integral) 1/rho dp

For horizontal motion dp my be

nearly zero