Supercells - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 230

An easier way to visualize the magnitude of SREH better than by simply
computing the integral in (4.57) is as follows: vorticity in the environment is asso-
ciated with vertical shear only (i.e., we neglect vertical motions in the environment,
which are typically an order of magnitude or more weaker than horizontal
motions). Then

v ¼@v=@ z i þ@ u =@ z j ð 4 : 58 Þ

JT

and therefore

ð h
0 ð u c x Þ d v=

ð h
0 ðv c y Þ du

SREH ð c Þ¼ dz dz þ = dz dz ð 4 : 59 Þ

where

c ¼ c x i þ c y j ð 4 : 60 Þ

Then

ð h ð h

0 v du þ

ð h ð h

SREH ð c Þ¼ ud vþ c x d v

ð 4 :

61 Þ c y du

0 0 0

The sum of the first two terms on the RHS of (4.61) is ground-relative
environmental helicity, SREH(0). The third and fourth terms on the RHS of
(4.61) are c x Dv and c y D u, respectively.

Now, consider the idealized hodograph in Figure 4.45, with areas A, B, C, D,
E, F, G, H, I , and J noted. The first term on the RHS of (4.61) is I J þ A.
The second term is A þ 2B þ C þ D þ E þ F þ G þ H. The third term is C þ I .
The fourth term is ð F þ G þ H Þ . Since, D þ J ¼ E, the sum of all four terms is
2 ð A þ B þ C þ D Þ , which is twice the area swept out by the storm-relative wind
vector between the ground and height h. Thus, even if the hodograph is unidirec-
tional, SREH is high if the updraft movement is far off the hodograph. For a
given hodograph, SREH may be increased simply by forcing the tip of the storm
motion vector to be farther away from the hodograph ( Figure 4.46 ). When the
hodograph is curved in a counterclockwise manner with height or if the storm
motion is to the left of the hodograph, negative SREH is created.
Consider the simple example of the application of this geometric formulation
illustrated in Figure 4.47. A quarter-circle hodograph from 0 to 3 km, in which the
storm-relative wind speed at all levels is 10m s 1 has an SREH of 2 ð 100 m 2 s 2 )/

4 150 m 2 s 2 , the approximate observation-based threshold for supercells.
From the thermal wind relation (4.23), we may substitute the horizontal
temperature gradient

for a measure of the vertical shear of the horizontal

component of the wind, so that
ð v c Þ EJT v ð v c Þ EJ h T ð 4 : 62 Þ

so that

SREH ð 0 Þ H v

EJ h T

ð 4 : 63 Þ

or, in other words, helicity in the environment is proportional to horizontal
temperature advection. Thus, helicity in the environment may be enhanced in the
vicinity of warm fronts, stationary fronts, and outflow boundaries, where we tend
to find warm advection. (It is assumed that cold fronts are characterized by cold
advection.) It is also noted, in accord with the thermal wind relation (4.23), that