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.

 

                   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 “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.

 

                                    (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 “the tilting term.”

 

(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 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.

 

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

 

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 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.

 

                             (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 “helical”. 

 

Note also that

Š      “shear vectors” can be estimated pretty reliably from the 500 mb flow;

Š      the surface winds can be estimated from the surface isobars

Š      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

 

 

 

 

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 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.

 

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: 

 

Š      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?

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

Š      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.