Vorticity equation in x, y, p coordinates

Starting from the horizontal equations of motion

we can write them to

Taking and use the definition , we get

or

is the absolute vorticity, including the relative vorticity and the vorticity due to the earth's rotation f. In the above equation, term (1) is the divergence term and term (2) is the twisting or tilting term.

The divergence term states that the change of the absolute vorticity is proportional to the magnitude of divergence and the magnitude of the absolute vorticity. Since the absolute vorticity is positive in general, we can say

convergence --> increase in absolute vorticity

divergence --> decrease in absolute vorticity

This term is responsible for the formation of intense cyclones in the mid latitude and hurricanes in the tropics.

The tilting term will tilt the vorticity in the horizontal direction due to vertical wind shear to the vorticity in the vertical direction. This term is generally ignored in synoptic meteorology because is generally small for synoptic scale motion. However, can be quite large in storms and this term is quite important in generating storm scale vorticity, which causes the storm rotation and, ultimately, to the formation of rapid rotation in tornadoes.

Suppose the low level wind is a weak westerly and upper level wind is a strong westerly, as shown in the following figure:

If we put Charles Brown in the wind, he will rotate clockwise if we view from the south.

Or we can imaging that there is a tube of air which is rotating clockwise when viewed from the south. We call this a vortex tube. In this vortex tube, is negative ( is positive).

NOTE: Mathematically, we should look at the wind from the north, the positive y direction, to determine the sign of the y component of the vorticity. In this case, the rotation is counterclockwise, implying that the y component of the vorticity is positive.

In addition, if there is upward motion in the south or downward motion in the north, > 0, then,

This can be seen easily by looking at the figure again

Simplified vorticity equation

The convergence-divergence is usually stronger near the surface and in the upper levels below the tropopause, The convergence-divergence is usually weak in the middle levels around 450 to 550 mb level. If a non-divergent level exists in a synoptic scale motion (tilting term negligible), then

or

The first equation states that the absolute vorticity is conserved if we follow the movement of an air parcel on the level of non-divergence. The second equation states that the local change of the absolute vorticity equals the horizontal advection of the absolute vorticity. In the Natural Coordinates, this equation can be written as:

where s is defined as the direction along the wind on the level of non-divergence. In the application, we assume the 500 mb level is the level of non-divergence because the 500 mb level map is available.

Petterssen's development formula of surface cyclone

Start from the definition of the thermal wind between the surface and the 500 mb level:

It will be straightforward to show, using the definition of the vorticity, that

or

Take partial derivatives with time

This equation states that the change of the surface vorticity equals the change of the

500 mb vorticity subtract the change the sfc-500 thermal vorticity (or vorticity of the thermal wind).

Term (1)

Because f at a particular location does not change with time, we can write

We will further assume that the 500 mb level is the level of non-divergence, we can than write

or

This term, therefore, represent the 500 mb vorticity advection. In areas of positive vorticity advection (PVA) on the 500 mb level, and its effect is to make and produce a surface cyclogenesis. In areas of negative vorticity advection (NVA), the effect is reversed.

Term (2)

Recall the definition of the geostrophic vorticity

Similar to this equation, the thermal vorticity can be expressed as

where

is the thickness for the 1000-500 mb thermal wind.

NOTE: Here the thermal wind of the 1000-500 mb is used for simplicity. The conclusion derived here can be applied to the sfc-500 mb layer.

Taking partial derivative with time and using the hypsometric equation, we get

Suppose

where A is the amplitude and L is the wavelength, then,

or

This term states that the area of sfc-500 mb mean temperature increase in the area of surface cyclogenesis. The intensity of the surface cyclogenesis is inversely proportional to the square of the wavelength.

There are several factors that can increase the sfc-500 mb mean temperature:

ï Heating due to horizontal warm temperature advection. This is the most important factor in changing the temperature in the mid-latitude.

ï Heating due to radiation. This is the cause of the thermal low.

ï Heating due to cloud condensation. This is the cause of the tropical storms and hurricanes.