The importance of understanding horizontal divergence should be well-established in your minds. The vertical motion patterns associated with synoptic scale divergence/convergence are directly connected both with development of surface pressure systems and the development of the vertical motion fields that lead to the creation of cloud/precipitation systems in association with the surface lows.
You have also learned one of the most obvious locations for synoptic scale divergence; in the regions east of upper tropospheric trough lines and west of upper tropospheric ridge lines. Unfortunately, upper tropospheric divergence/convergence patterns can also be embedded in apparently "straight" (i.e., "zonal" = along a line of latitude) flow in which no curvature (no troughs or ridges) are evident.
As you become more experienced in synoptic meteorology, you will be able to judge where even these more subtle areas are located on upper tropospheric charts. The question now is "Is there an easy way to find areas of divergence if they are not associated with prominent troughs and ridges?" The answer is "yes".
Consider two air parcels stationary with respect to the surface of the earth; one at the N.P. and one at our latitude.
a. Earth Vorticity
Relative to an observer in space, both air parcels are turning about an axis through their centers. (Note that side A of each air parcel turns with respect to a point in space. Counterclockwise rotations are termed "positive (or cyclonic)" and clockwise "negative (anticyclonic)".
This rotation is due to the earth's rotation. Such an imparted spin or angular velocity is proportional to (but not the same as) a microscopic measure of spin called vorticity . The vorticity imparted to the air parcel by the earth's surface is called the Coriolis parameter.
b. Relative Vorticity
Vorticity RELATIVE TO THE SURFACE OF THE EARTH can be imparted to air parcels because of the characteristics of the flow around them. This vorticity is called relative vorticity and is due to flow around troughs and ridges and horizontal wind speed differences. Since air flow in troughs is cyclonic and in ridges anticyclonic, relative vorticity in troughs is positive and in ridges anticyclonic.
The mathematical expression for vorticity has the units of "rotations" per second. Since "rotations" is dimensionless (given as degrees or radians), the units for vorticity are the same as those for divergence.
c. Absolute Vorticity
The total, or absolute, vorticity of the air parcel is the sum of the relative vorticity imparted to the air parcel by the flow around it (positive or negative)and the vorticity imparted to it by the rotating surface of the earth (which is always positive). It can be calculated on isobaric charts and plotted. Usually the highest values of absolute vorticity is found in troughs (and also north of jet maxima) and lowest values in ridges (also south of jet maxima).
3. Using Vorticity Patterns to Estimate Divergence Patterns
The question is "how does this discussion of vorticity relate to synoptic scale divergence?"
The following discussion is predicated on SCALING the atmosphere to eliminate factors that are not particularly important on the synoptic scale in very restrictive circumstances. The equation (not shown) that we are using here is called THE VORTICITY EQUATION which is a PROGNOSTIC EQUATION that answers the conceptual question "how can we change the vorticity of an air parcel?"
If we eliminate all of the terms in the equation that have anything to do with temperature changes, either produced sensibly or by advection, the equation is considerably simplified. The resulting equation is called the BAROTROPIC or SIMPLIFIED vorticity equation. However, temperature changes are responsible for the change in the shape (and, thus, curvature) of the streamlines, and this obviously feeds back to divergence and vorticity field. Thus, the synoptically-scaled vorticity equation has a built in inaccuracy due to the neglect of these terms. It should be pointed out, however, that a scale analysis of all the terms shows that when strong jet streams are present, all effects related to temperature changes are relatively small and can be eliminated on a order of magnitude basis.
Eliminating the terms, as you will see below, makes the equation conceptually accessible. Unfortunately, it also makes it very inaccurate in situations, for example, of strong temperature advection (or synoptic scale diabatic heating/cooling as occurs in the summer over the continent/ocean.). The elimination of these terms has resulted incorrect use of this simplified equation in operational meteorology. For those of you who have taken classes from me before, you know that the classic misuse involves forecasters ONLY looking at vorticity patterns to assess divergence patterns aloft and vertical motion patterns in the mid-troposphere.
For the sake of the discussion, though, we make these assumptions in the discussion below. We will add back the "complicating" factors in the future.
This is simply an application of the principle of Conservation of Absolute Angular Momentum. The fact of the matter is that the dominant way in which the absolute vorticity of air parcels change AT THE SYNOPTIC SCALE (under the restrictive circumstances outlined above---no temperature advection, for example) is by divergence (or convergence).
Air parcels streaming along and experiencing divergence will experience a DECREASE in absolute vorticity. On a weather map they will appear to be moving, therefore, from high values of vorticity to lower values of vorticity!
Such areas are termed areas of POSITIVE VORTICITY ADVECTION (pva)1 (and, conversely, air flowing from lower values of vorticity to high values of vorticity are termed areas of NEGATIVE VORTICITY ADVECTION (nva). Using this rationale, pva "diagnoses" divergence and nva convergence IN THE UPPER TROPOSPHERE.
In the example below, note the arrow is moving from regions of high vorticity to regions of low vorticity. This implies that the air parcel moving along "advecting" its vorticity experiences a decrease in vorticity. Why? Using the assumptions above, if the vorticity values shown on the chart below indicates the vorticity values of the moving air parcels, as air moves into the region of divergence east of trough axes, it MUST experience a decrease in vorticity because of divergence (remember, the ballet dancer).
Thus, since horizontal divergence characterizes the region east of trough axis to the downstream ridge axis in the upper troposphere (with the greatest value usually at the inflection point), then the region from the trough axis to the ridge axis is characterized by pva with the greatest pva at the inflection point.
It turns out that vorticity is far easier to compute accurately than divergence. In fact, it can be obtained from the geometry (shape) of the flow patterns. Hence, it is easy to construct maps of absolute vorticity and then to infer the divergence patterns VISUALLY. Normally, analysts encircle pva areas with green shading and nva areas with brown shading. The green shading will isolate areas of probable divergence etc.
Use can see the vorticity pattern is pretty complicated. But does the rule of thumb that divergence ("diagnosed" by positive vorticity advection) characterizes the region east of upper tropospheric troughs and convergence (negative vorticity advection) west of upper tropospheric troughs generally explain what we see in the real world? Take a look at the above example, except with light green overlay on top of the positive vorticity advection areas and light brown overlay on top of the negative vorticity advection areas.
Note that the region between the trough and the downstream ridge axis MOSTLY has positive vorticity advection and the region west of the trough axis to the UPSTREAM ridge axis MOSTLY has negative vorticity advection. Thus, you as meteorologists are �given permission� to assume (rule of thumb) that divergence occurs on the east side of troughs and convergence on the west side, even though, in nature, the pattern is more complex.
All the patterns above about the 850 mb level "mimic" each other (meaning, all the troughs and ridges are basically in the same position (not exactly true since the troughs tilt a bit towards colder air; future discussion topic). See the 500 mb chart and the 300 mb chart below for an example of the GENERAL correspondence in the geometry.
Also, the vorticity patterns are related to the geometry of the flow. Hence, patterns of vorticity at 500 mb "mimic" (are very similar to) the vorticity patterns in the upper troposphere (say, at 300 mb). Thus even though the 500 mb level is very near the Level of Non-divergence, the pva and nva patterns there help us INFER something about the divergence patterns in the upper troposphere.
Now try to synthesize the three-dimensional characteristics of the synoptic scale atmosphere that we have been trying to build up by integrating in your minds the combination of Mass Continuity (Dine's Compensation) with the ideas expressed here about vorticity advection patterns.