Divergence and Vorticity



Almost all air parcels spin or rotate. Rotation is an important characteristic of atmosphteric motion. Rotation is usually a three-dimensional phenomenon. Given the three-dimensional velocity field, the rotation of a fluid parcel is given by:

The direction of is along the axis of rotation and the magnitude is twice the rate of rotation. On the synoptic scale, we are mostly interested in rotation about a vertical axis. The vertical component of the rotation vector is:

The sign convention is such that > 0 when rotation is counterclockwise and < 0 when the rotation is clockwise. Positive vorticity is termed cyclonic and negative vorticity is termed anticyclonic.

  Absolute and relative vorticity

Rotation or vorticity can be measured relative to absolute space or relative to the earth. Vorticity measured relative to absolute space is called absolute vorticity and is denoted by . Vorticity measured relative to the earth is called relative vorticity and is denoted by .

Because the earth rotates relative to absolute space it has its own inherent vorticity. For a location at latitude , the vertical component of the earth's vorticity is equal to the Coriolis parameter:

This is illustrated in the figure below.

The absolute vorticity of a fluid parcel is equal to the sum of the (vertical) rotation relative to the earth (relative vorticity) and the vertical component of the rotation of the earth at that position (Coriolis parameter):

On the synoptic scale is almost always positive.

  Shear and curvature

On weather maps, it is clear that the mathematical expression for vorticity, as given above, is difficult to handle in practice. A simpler expression can be obtained by considering a natural coordinate system. In such a system one unit vector t is oriented parallel to the horizontal velocity at each point and another unit vector n is directed normal to the horizontal velocity pointing to the left of the flow. The unit vector k is directed vertically upwards. In this system the expression for the (relative) vorticity is given by:

where R is the radius of curvature of the streamline, following the motion. R is positive when the centre of curvature is in the positive n direction. Thus for R > 0 the parcel turns towards the left and for R < 0 it turns towards the right.

From the expression above it follows that the vorticity is the result of two components: the change in windspeed normal to the direction of flow, termed the shear vorticity and the turning of the wind along a streamline, V/R, termed the curvature vorticity.

The figure below illustrates the natural coordinate system together with shear and curvature vorticity.