Divergence:

Divergence is a property of fluids including the atmosphere.  The term is used in two ways: generically, the term encompasses both divergence and convergence, referring to the spreading out or shrinking of an area of fluid.  When the divergence is positive, the flow is said to be divergent:

divergence:  (+) -- net outflow from a region of fluid

When the divergence is negative, the flow is said to be convergent:

convergence: (-) -- net inflow into a region of fluid
 

In a column of air, the flow at some levels will be divergent, while at other levels it will be convergent.

If there is net divergence in a column there is net outflow, reducing the weight of the column, and hence reducing the pressure at the surface--low pressure center forms.

If there is net convergence in a column there is net inflow, increasing the weight of the column, increasing the pressure at the surface--high pressure center forms.

In general, the convergence in one part of the atmospheric column is nearly balanced by divergence in another part of the column, and the pressure at the surface never changes by more than a few percent.  [This is called Dines' Compensation.]  Over a mid-latitude cyclonic storm, there is divergence aloft in the upper troposphere, and convergence in the lower troposphere:

•  If the divergence aloft exceeds the convergence lower down, surface pressure decreases and the storm "deepens."

• If the convergence in the lower layers exceeds the divergence aloft, surface pressure increases and the storm weakens.

• If the two are equal, the low pressure center maintains a steady pressure.

The natural coordinate system always lines up with the orientation of the flow, so it is constantly changing direction:

The "n" direction is always perpendicular to the flow direction at any given point.

The divergence equation can be approximated in "finite difference" form:

In words, this means that the first term is the change of wind speed along the direction of flow, while the second term represents the change of wind direction across the direction of flow.  Consider four examples illustrated in the following figure:

Winds speeding up along the direction of flow would produce net outflow or speed divergence.

Winds slowing down along the direction of flow would produce net inflow or speed convergence.

Second term:

Wind directions across the flow spreading apart would produce net outflow or directional divergence.

Wind directions across the flow coming together would produce net inflow or directional convergence.
 

Since the net or total divergence is the sum of both terms of the equation, both speed divergence and directional divergence must be considered together in determining whether the overall result at a location is divergent or convergent.

The wind field is divergent if the result is positive.

The wind field is convergent if the result is negative.

In fact, in the atmospheric wind field, the two terms are usually of opposite sign, i.e., one term usually shows divergence while the other term shows convergence.  This can be visualized as follows:

As wind directions converge into a narrower space (confluence or directional convergence), the wind also speeds up along the direction of flow.  It is not possible to tell by eye whether this wind field is convergent or divergent--it must be measured.

In practice, it is difficult to measure the divergence accurately, because it is a small difference of large quantities.  Relatively small errors in wind measurements can translate to large errors in the divergence calculation.

Why does anybody care about any of this?

Because the divergence field is closely related to the atmospheric vertical motion field, and the vertical motion field determines whether we will have sunny, benign weather, or cloudy, stormy weather.

Divergence aloft with convergence at the surface leads to upward vertical motion of air (rising) and cloudy, precipitating weather if there is sufficient moisture.

Convergence aloft with divergence at the surface leads to downward vertical air motion (sinking or subsidence) and more clear weather:

In terms of upper level maps such as the 300 mb chart, a typical divergence pattern around a shortwave trough might appear like this, with divergence ahead of the trough line and convergence behind it:

Relative Vorticity:

Relative vorticity is a measure of spin about a vertical axis in a parcel of air.  Spin can be cyclonic (counterclockwise in the Northern Hemisphere) or anticyclonic (clockwise in N. Hem.).

Storm systems (low pressure areas) larger than thunderstorms typically are characterized by cyclonic relative vorticity and high pressure areas by anticyclonic relative vorticity.

Rotating thunderstorms (mesocyclones or supercells) can rotate cyclonically or anticyclonically but are preferentially cyclonic as are tornadoes. (Click  here for information on a rare anticyclonic tornado [scroll down for pictures] ).

Vorticity is defined as positive (+) for cyclonic spin.

Vorticity is defined as negative (-) for antiscylonic spin.

Relative vorticity can be expressed by the Vorticity Equation in natural coordinates:

where    V is the wind speed
            R is the radius of curvature of a streamline (an instantaneous path tangent to the flow)
            n is the cross-stream direction (perpendicular to the flow).

This can be approximated in finite difference form by

The first term of the equation is the vorticity or spin of a parcel due to flow moving in a curving path and is called the curvature term. The radius of curvature is defined as positive when the flow is curving cyclonically, as around a low pressure area.  Looking downstream (below left), cyclonic flow curves to the left in the northern hemisphere:

The radius of curvature is defined as negative when the flow is curving anticyclonically, as around a high pressure area.  Looking downstream (above right), anticyclonic flow curves to the right in the northern hemisphere.

The second term of the vorticity equation is known as the shear term, and describes spin of a parcel induced by the change of wind speed across the direction of flow:

Anticyclonic vorticity occurs when the shear imparts a clockwise spin to a parcel.

The two terms of the vorticity equation may be of the same sign and reinforce each other, or they may be of opposite sign and tend to cancel each other.  The following table summarizes some possibilities for Northern Hemisphere cases near the jet stream:

Synoptic Situation/location	Curvature Vorticity	Shear Vorticity	Total Relative Vorticity	Remarks
Trough north of a jet.	+ (cyclonic)	+ (cyclonic)	+ (cyclonic)	Strong short wave.
Trough south of a jet.	+ (cyclonic)	- (anticyclonic)	weakly + or -	Weak wave, contributions canceling.
Ridge north of a jet.	- (anticyclonic)	+ (cyclonic)	weakly + or -	Weak wave,  contributions canceling.
Ridge south of a jet.	- (anticyclonic)	- (anticyclonic)	- (anticyclonic)	Strong ridge signature.

Absolute Vorticity:

On National Weather Service model output maps, plots of 500 mb vorticity are of absolute vorticity, not relative vorticity.  Absolute vorticity is the sum of the relative vorticity and the planetary vorticity:

The planetary vorticity is vorticity imparted to the atmosphere due to the spinning of the earth.  Since we are considering only spin about the local vertical axis, planetary vorticity is a maximum at the poles, and zero at the equator, where there is no planetary spin about a vertical axis (the axis itself moves around the circumference of the earth as the earth spins, but it has no component of spin around the local vertical).

Looking down upon the north pole, the earth spins in a counterclockwise direction, so the planetary vorticity is positive or cyclonic.

Relative vorticity values are virtually always less than planetary vorticity values so the absolute vorticity is virtually always positive.  Thus on the 500 mb Heights/Vorticity charts, values of the vorticity are always positive.  Absolute vorticity is given on the charts in units of [x10^-5 sec^-1].

Minimums in the absolute vorticity field (usually denoted by "N's" on those maps) represent anticyclonic relative vorticity centers.

Maximums in the absolute vorticity field (usually denoted by "X's" on those maps) represent cyclonic relative vorticity centers.

The absolute vorticity maximums on the 500 mb Heights/Vorticity chart indicate the positions of short waves.  They can be recognized as significant short waves when they have several vorticity contours around them.  Below is a depiction of a vorticity maximum of a 500 mb short wave:

Note that there is a region of positive vorticity advection (PVA) ahead of the short wave trough axis.

There is a region of negative vorticity advection (NVA) behind the short wave trough axis.

Parcels of air moving with the wind move faster than the vorticity pattern moves, so they move through the pattern.  As parcels approach the vorticity maximum from the rear, they "spin up" in the NVA region.  This spin-up is associated with upper level convergence (due to conservation of angular momentum again) and thus downward vertical air motion below, usually leading to clearer weather.

After the parcel passes the vorticity maximum (maximum spin) it begins to "spin down" in the PVA region ahead of the trough.  This spin-down is associated with upper level divergence, and thus upward vertical air motion below, possibly leading to clouds and precipitation.