Helicity and
Vorticity: Explanations and
Tutorial
A. Relationship
of Three Dimensional Helicity to Three Dimensional
Vorticity
(Note: Inclass Exercises Completed As Lab
Session at Conclusion of the Discussion)
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 m2s-2
or J kg-1.
B.
Helicity and
Streamwise/Crosswise Vorticity
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.
C.
Relative and Horizontal Helicity
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. Also, consider a situation in which the
flow is westerly and increasing with height, the south wind does not vary with
height and the vertical wind component is zero.
Equation
(1) reduces to
(4)
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
(5)
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 m2s-2 and that J kg-1is
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
(8)
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 m2s-2.
E.
Horizontal Storm Relative Helicity
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.
(7)
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?