**Reading #6: One Way of Diagnosing Divergence**

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**1.
General**

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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 tropo-spheric 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**

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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 40^{o}N
latitude:

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

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

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

^{ }

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

**3. Using Vorticity Patterns to Estimate
Divergence Patterns**

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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.

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:
*

*
*

*Positive vorticity advection is ONE WAY in which diagnose upper tropospheric divergence**. *

* The 500 mb level is
near the level of non-divergence. Hence, pva cannot be associated with divergence at that
level. However, some very
important component or feature of flow *

**at 500 mb IS associated with pva at that level***. Using the infor**mation in the previous *

*handout, discuss. *

*Please remember that there are other ways....and we will discuss one of them next semester*

*in Metr 301 (having to do with warm advection).
*

*
*