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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 “helical” flow can be measured by
computation of the “helicity.” 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.).
Let’s 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 “helical”, 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.
H = (" # v) • v
Equation (3.3.29, Vol I, Bluestein)
expanded out is:
$ "w "v ' $ "w "u '
$ "v "u '
H = u&
# ) # v&
# ) + w& # )
% "y "z ( % "x "z (
% "x "y (
(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
Streamwise Vorticity =
!
" # v• v
v
(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 “cross-wise vorticity”.
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 “u” component of the wind was actually the “w”
component, all the vorticity would be streamwise. That would “maximize” 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.
Relative Helicity =
!
" # v• v
"#v v
(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.
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!
!
Equation (1) reduces to
# "u &
H = v% (
$ "z '
!
(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
+ r % $v (.
H = "-v • k # ' *0
& $z )/
,
(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.
+ r % $v (.
H = " 1 -v • k # ' *0dz
& $z )/
0 ,
h
(6)
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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
D" ˆ
Dv
= k •# $
Dt
Dt
(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 “the tilting term.”
!
$ "w " v "w "u '
#
&
) in rectangular coordinates
% "x " z "y "z (
$ "w " V '
&
) in natural coordinates
% "s " z (
!
(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 1.0… the flow is purely helical.
In the case shown for a typical thunderstorm in the Great Plains, the updraft would
“develop” 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
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cumulonimbus, the “inflow” 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.
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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 “total” or “bulk” 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 ‘back of the envelope” 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
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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 150300 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 thunderstorm’s 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 “feel” the helicity. This is the reason that,
operationally, the STORM RELATIVE HELICITY is of most importance.
r % $v (.
+
H = " 1 -( v " c) • k # ' *0dz
& $z )/
0 ,
h
!
(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 layer1 helicity is geographically correlated
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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.
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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 “helical”.
Note also that
• “shear vectors” can be estimated pretty reliably from the 500 mb flow;
• the surface winds can be estimated from the surface isobars
• 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 guess…it 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
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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
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F. Shear Parameters Used In Operational Environment: An Exercise
Here is the output of the wxp analyzed sounding for KOUN’s 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.
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A
B
Figure 5: CAPE/CINH 14 UTC 11 May 2004
C
Figure 6: Dewpoint and Surface Isobars 14 UTC 11 May 2004
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Fig. 7: SPC Storm Reports for 11 May 2004
Inclass Exercise 4:
• 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?
• The dewpoint field in Fig. 6 appears to be consistent with the CAPE/CINH field
shown in Fig. 5. Why?
• 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?
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