One Way To Diagnose Upper Tropospheric Divergence (Associated With Jet Streams)

1.  General

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".

2.  Vorticity

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 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 can also be imparted to air parcels because of the characteristics of the  flow around them. This vorticity is called relative vorticity and is due to 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).

The range of values are as follows at approximately 40oN latitude:

Earth Vorticity (Imparted To Air Parcels)                  10 X 10-5s-1

Relative Vorticity:  Troughs and/or north of jet        4 to 20 X 10-5s-1

Relative Vorticity:  Ridges and/or south of jet           -4 to –8  X 10-5s-1

Absolute Vorticity Range (Typical)                             2 to 30 X 10-5s-1

3.  Using Vorticity Patterns to Estimate Divergence Patterns Associated With Jet Streams

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 experienced by an air parcel?”  When the jet stream is present, there is some justification to eliminate all of the terms in the equation that have anything to do with temperature changes, either produced sensibly or by advection.  The resulting equation is called the BAROTROPIC or SIMPLIFIED vorticity equation.

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.

Discussion Question #1:

How do ballet dancers change their rate of spin (change their vorticity)?  (Describe this qualitatively).

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) [2] .)   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.

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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.

Discussion Question #2:

Finding an area of positive vorticity advection (pva) is ONE WAY in which one can diagnose upper tropospheric

divergence. The 500 mb level is near the level of non-divergence.  Hence, pva cannot be associated with divergence

at the 500 mb level.  However, some very important component or  feature of flow at 500 mb IS associated with pva at

that level.

Use the principle of Dine's Compensation to infer that component or feature of the flow at 500 mb that IS associated

with pva at 500 mb.

Please remember that there are other ways that upper tropospheric divergence can be diagnosed

....and we will discuss one of them next semester in Metr 400 (having to do with warm advection).

[1] More properly termed “cyclonic vorticity advection” or cva

[2] More properly termed “anticyclonic vorticity advection” or ava.