Ordinary-cell convective storms - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 147

symmetric about the leading edge of the cold pool so that
ð w Þ d ¼ w d ð@ w =@ x Þ d ð 3 : 45 Þ

is anti-symmetric about the updraft. Then the integral of (3.45) from L to R
vanishes. The first term on the LHS of (3.36) integrated also up to only z ¼ d is
ð d ð R

L @=@

2 u R ; d 2 u R ; 0 ð 2 u L ; d 2 u L ; 0 Þ

1 1 1 1

x ð u Þ dx dz ¼ ð 3 : 46 Þ

0

L ¼ 0 except right near the leading edge of the
density current, but certainly not at x ¼ L and x ¼ R. Furthermore, if the density
current is stagnant, u L ; 0 ¼ 0; also, recall that u L ; d ¼ 0 (3.41). From Figure 3.42 we
see that

where it is recognized that @ w =@ xu j

u R ; d ¼ 0 ð 3 : 47 Þ

Then from (3.40), and recognizing that B R ¼ 0, we find that (3.36) integrated to
z ¼ d becomes

2 u R ; 0 ¼

2 2

1 1 1

2 ðD

u Þ ¼

2 ðD

u Þ ¼ B L h ¼g½ð 1 0 Þ= 0 h ð 3 : 48 Þ

so that

2

ðD u Þ ¼ 2 g h ½ð 1 0 Þ= 0 ð 3 : 49 Þ

But from (3.31) and (3.49) it follows that
c ¼ D 50 Þ

In other words, to get a vertically erect updraft at the leading edge of a steady-
state, frictionless, stagnant density current, the speed of the density current must
be identical to the difference in horizontal wind across the shear layer that encom-
passes the depth of the cold pool. Such a situation is said to be ''optimal'' for
triggering new convection at the leading edge. Physically, this condition means
that when the rate of generation of horizontal vorticity baroclinically ( @

u ð 3 :

x)at
the leading edge of the density current is counterbalanced by the advection of
(import from) horizontal vorticity from vertical shear in (from) the environment
½ u

B =@

z Þ , there is a maximum in upward motion along the leading edge
of the density current and the probability of triggering a discrete new cell is
increased ( Figure 3.40 ). This behavior is described in a theory known as the
''RKW theory'', after the NCAR scientists Rich Rotunno, Joe Klemp, and Morris
Weisman, who proposed it in the late 1980s.
We now include low-level vertical shear (

@=@ x ð@ u =@

h 0 ) whose orientation is parallel
to density current, in an analytical model of a steady-state, two-dimensional, invis-
cid density current and seek an expression for the speed of the density current in
terms of the dimensions and strength of the cold pool. We are particularly inter-
ested in how it differs from (3.32). The model setup ( Figure 3.43 ) is similar to the
one for no shear, but there are now three airstreams—not just two—and we do
not require the airflow to be erect at the leading edge of the density current. There
is still the stagnant/resting cold pool, but the flow regime ahead of and above the
density current is broken down into two sub-airstreams. First, the low-level shear

D u =