The tendency of the atmosphere to have ÒhelicalÓ
flow can be measured by computation of the Òhelicity.Ó To understand helicity, imagine an air
parcel having horizontal vorticity, that is, a spin around the y axis (but keep
in mind that helicity has components on each of the three coordinate axes.).

LetÕs also say that the there is only a south wind
component, or u=w=0. Then, the
combination of the south wind and the vorticity around the y-axis will yield a
flow that is ÒhelicalÓ, that is, still a southerly wind, but with air rotating
around the y axis as it is moving.

The three dimensional helicity is a scalar.

Equation (3.3.29, Vol I, Bluestein)

expanded out is:

(1)

Note that three dimensional
helicity is the product of the three wind components with the three components
of vorticity. Also, note that the
far right hand term is a product of the vertical velocity and the vertical
relative vorticity. The units of
helicity are m s^{-2}
and of Storm Relative Helicity m^{2}s^{-2} or J kg^{-1}.

The degree to which the component vorticity vectors
are parallel to the wind components is measured by the Streamwise
Vorticity. To understand this,
consider equation (1) above.

Say that there is considerable vertical vorticity,
but no vertical velocity. Say also
that for this situation, there is only a strong west wind, but no south
wind. Equation (1) would yield a value
of zero helicity. That is because
although there is a relative vorticity vector, it does not lie parallel to the
wind vector and, hence, there is no helical flow.

The streamwise vorticity is measured by

(2)

In the example above, none of the vorticity is
streamwise, and all of it is at a right angles to the actual wind vector. Such vorticity is called Òcross-wise
vorticityÓ. Thus the degree to
which vorticity is streamwise is the degree to which helical flow exists. Helical flow is important in the
generation of rotation in thunderstorms.

The ratio of the actual helicity to the maximum
helicity possible for the given wind and vorticity vectors is termed the
Relative Helicity. In other words,
for the example given above, say the ÒuÓ component of the wind was actually the
ÒwÓ component, all the vorticity would be streamwise. That would ÒmaximizeÓ the helicity for
that particular wind field. The
maximum possible relative helicity is 1.0.
In such a case, all of the three dimensional vorticity vector is on the
wind vector. Such flow is called
Beltrami Flow.

(3)

Examine the first two terms in equaton (1) to the
right of the equals sign. Consider
a situation in which there is no vertical wind. Equation (1) reduces to

(4a,b)

Now letÕs simplify. Consider a situation in which the flow
is westerly and increasing with height and the south wind does not vary with
height. Then equation (4a) reduces
to

Equation (1) reduces to

(5)

Equation (4) says that positive (cyclonic) helicity
will develop in southerly flow if west winds increase with height. (Careful, to visualize this you need to
use the right-hand screw rule).
Recall that in wave cyclones, air in the warm sector often is moving
into a region in which ¶u/¶z is strongly positive due the presence of the jet
stream.

In fact, it is observed that horizontal helicity
TILTED INTO THE VERTICAL by developing thunderstorm updrafts is the major
source of rotation in thunderstorms (as opposed to some sort of concentration
of prerexisting vertical helicity) (Discussed in next section). Thus, severe weather meteorologists
often look at the horizontal component of the helicity, as given by equations
(4) when considering whether or not thunderstorm updrafts are liable to rotate.

Actually, developing updrafts tilt the streamwise
vorticity due to the vertical shear of a relatively deep layer (anywhere from 1
to 6 km) into the vertical plane.
Thus, rather than considering the horizontal helicity at one level, the
usual technique is to calculate helicity integrated through a layer, say from
the surface to 3 km.

(6)

In Class Exercise 1: Show that the units of (6) are m^{2}s^{-2}
and that J kg^{-1}is an equivalent unit.

**D. Relationship of Vertical Shear
Vorticity and Helicity to Development of Vertical Vorticity**

The development of vertical vorticity can be derived
from the relation

(7)
or (4.5.2 in Bluestein, Vol 1)

The expansion of the right hand side of (8) yields
six terms, one of which is often called Òthe tilting term.Ó

(8)

These equations state that vertical vorticity will
develop if there is a gradient of the vertical wind along, say, a surface streamline,
if that surface streamline is in a region of vertical shear of the horizontal
wind. To understand this
conceptually, take a look at Fig. 1.

Note that the case shown is one in which all the
horizontal shear vorticity is on the streamline. In this case, all the horizontal
vorticity would be streamwise (from equation and the relative helicity would
1.0É the flow is purely helical.

In the case shown for a typical thunderstorm in the
Great Plains, the updraft would ÒdevelopÓ maximum cyclonic rotation at roughly
6km (in the lower-mid to mid troposphere).
Note also that if one assumes that the updraft is in a developing
cumulonimbus, the ÒinflowÓ layer of the storm (the layer of air ingested from
the surface) is about from 0-3 km AGL.
In shallower storms, that inflow layer may be 0-2 km or less.

Figure 1:
Schematic Diagram Showing The Development of a Mesocyclone as a Result
of Tilting Horizontal Vorticity Into the Vertical

The other thing that is very important to note is
that the development of cyclonic rotation in the updraft in the case cited
above has nothing to do with Coriolis effect. In most cases for the Great Plains (and
in other locations in the United States), the relationship of the surface
streamlines to the shear profile is as indicated in the diagram above. However, if the surface streamlines were
approaching the shear profile shown above from the north, any updraft that
developed would have anticyclonic rotation.

**Please also
note that as a forecaster your decision about whether a buoyant updraft will
develop initially has nothing to do with the shear profile shown. That decision is made on the basis of
CAPE/CINH and a thorough analysis of the thermodynamic profile. Once you have decided that a buoyant
updraft will occur, then a forecaster must consider the impact of the shear
environment.**

In-class Exercise 2: Using SBS (including schematic
drawings), explain why the top half of the drawing shown above portrays the
typical situation in the Great Plains during Spring severe thunderstorm
outbreaks.

Typical values of deep layer shear (sometimes called
ÒtotalÓ or ÒbulkÓ shear) supportive
of longer-lived convection are on the order of 4 X 10^{-3} s^{-1}
or greater in the 0-6 km layer. A
Ôback of the envelopeÓ way of calculating this is just to take the wind, in
knots, at 500 mb, divide by 10 and multiply by 10^{-3} s^{-1}. For example, 40 knots of wind difference
between the surface and 500 mb usually is associated with a shear value of 4 X
10^{-3} s^{-1}.

40 knots of shear = 40 nautical miles per hour

40 nautical miles per hour X 6040 ft/nautical mile X
1 h/3600 s /18000 ft =

3.73 X 10^{-3} s ^{–1 } ~ 4 X 10^{-3} s ^{-1}

Typical values of helicity observed in the 0-3 km
inflow layer for a mesocyclone to develop (as in the storm depicted in the
diagram above) are on the order of 150-300 m^{2}s^{-2}.

Observations show that what is important in a
thunderstorm developing a rotating updraft in its midlevels is not so much the
helicity ingested (as suggested by equation (6)), but the STORM RELATIVE
helicity ingested. To understand
this, consider the case in which there is only southerly flow (say, 15 m/s) in
an environment of great vertical wind shear. Say that this southerly flow is
approaching a developing thunderstorm updraft. Equations (4), (5) and (6) would return
large values of horizontal helicity suggesting that the thunderstormÕs updraft
would develop cyclonic helicity.

However, suppose a thunderstorm develops and is
moving northward at 15 m/s. In that
case, the thunderstorm would never ÒfeelÓ the helicity. This is the reason that, operationally,
the STORM RELATIVE HELICITY is of most importance.

(9a)

(9b)

and for a purely south wind

(9c)

Please remember, however, that there is more to
consider when discussing the reasons for rotating thunderstorm updrafts. In order for an updraft to develop
rotation, a certain amount of time is needed. Unless the deeper layer shear is great
enough to prevent suppression of the updraft by precipitation, then a rotating
thunderstorm will never develop.
Thus, severe weather meteorologists often examine deep layer shear
values (say, 0-6 km) in combination with helicity values to determine if a
combination favorable for the initial development of mid-level rotation would
occur.

Take a look at an overlay of fields related to deep
(0-6 km) and inflow layer (0-3 km) shear.
Note that the greatest inflow layer[1] helicity
is geographically correlated with the greatest deep layer shear in the
Dakotas. As a meteorologist, that
tells me that the surface winds had to have been at nearly right angles to the
shear in order for the updrafts to be ÒhelicalÓ.

Note also that

á
Òshear vectorsÓ can be estimated pretty reliably from the 500 mb flow;

á
the surface winds can be estimated from the surface isobars

á
the regions in which the surface flow was parallel to the shear vectors
(or the 500 mb flow) had no or minimal potential for potential convective
updrafts to be helical (as a first guessÉit is a bit more complicated than that, though).

Figure 2:
0-6 km Total Shear and 0-3 km Storm Relative Helicity for 14 UTC 11 May
2004

Figure 3. Surface fronts and isobars for 14 UTC 11
May 2004

Figure 4: :
500 mb contours and surface wind plots for 14 UTC 11 May 2004

F. Shear
Parameters Used In Operational Environment: An Exercise

Here is the output of the wxp analyzed sounding for
KOUNÕs sounding at 12 UTC 5/3/99.

Inclass Exercise 3: Using the tabular information above,
answer the questions below on the basis of what you learned above. Use drawings to help you visualize.

1.
To what extent was the relationship of the surface winds to the mid
tropospheric winds consistent with the top half of Figure 1?

2.
To what extent was the deep layer shear favorable for severe
convection?

3.
To what extent was the positive storm relative helicity favorable for
the development of a rotating updraft?

4.
To what extent was vorticity in the 0-500 meter layer streamwise?

5. How is your answer in the previous question consistent with the relative helicity in the same layer?

Examine Figs. 2, 3, 4 and Figs 5 and 6 below. Answer the questions that follow Figure
6.

B A

C

Figure
5: CAPE/CINH 14 UTC 11 May 2004

Figure 6:
Dewpoint and Surface Isobars 14 UTC 11 May 2004

Fig. 7:
SPC Storm Reports for 11 May 2004

Inclass Exercise 4:

á
Note the locations A, B and C on the Fig. 5 (CAPE/CINH). At which of these locations would
thunderstorms be likely (in the absence of other information) and why?

á
The dewpoint field in Fig. 6 appears to be consistent with the
CAPE/CINH field shown in Fig. 5.
Why?

á At which of the locations shown on the CAPE/CINH chart would it be likely that thunderstorm updrafts would show the strongest cyclonic rotation and why?

[1] The most recent research indicates that the inflow layer may really be only 1 km deep or less for most thunderstorms. However, since the 0-3 km SREH is still used operationally, I provide that field as an example here. Please note that Inclass Exercise 3 requires you to visualize a 500 meter deep layer as inflow for the May 3, 1999 KOUN storm environment.