Ordinary-cell convective storms - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 145

Figure 3.41. Vertical cross section across the leading edge of the cold pool, showing the
domain used over which the steady-state, frictionless, horizontal vorticity equation in flux
form is integrated.

x ¼ L (behind, to the left of the leading edge of the cold pool) to x ¼ R (ahead, to
the right of the leading edge of the cold pool), and from z ¼ 0 (the surface) up to
z ¼ H (the top of the atmosphere, well above the top of the cold pool at z ¼ h).
We find that
ð H ð R

L @=@

ð R ð H

0 @=@

ð H ð R

L @

x ð u Þ dx dz þ z ð w Þ dz dx ¼ B =@ xdxdz ð 3 : 36 Þ

0 L 0

The physical meaning of (3.36) is that, when there is no net generation of

horizontal vorticity locally, the net baroclinic generation of (RHS) must be

balanced by the net flux divergence of (LHS): Horizontal vorticity gets ''out of

Dodge'' as quickly as it is generated. (We will use (3.36) with slightly different
limits at the top of the domain shortly.) Note that (3.36) is similar to (3.26) inte-
grated over a domain in the x-z-plane as before, except that the limits of

integration in the x-direction are from L to R—not 1 to þ1 .
Now, suppose that there is (constant and unidirectional) vertical shear in the
environment ( Figure 3.42 ) such that
D

u u R ; h u R ; 0 ð 3 : 37 Þ

where the winds in the reference frame of the density current are directed from
right to left. In other words, there is low-level shear over the depth of the cold
pool of D u = h, where D u > 0. We let the shear be restricted to the depth of the

cold pool only, so that

u R ; H ¼ 0 ð 3 : 38 Þ

In addition, suppose that airflow is straight up at the leading edge of the density
current ( Figure 3.42 ), which is consistent with the condition that
u L ; H ¼ 0 ð 3 : 39 Þ