Ordinary-cell convective storms - Severe Convective Storms and Tornadoes: Observations and Dynamics - page 149

This quadratic equation can be solved for c, but one would need to make use of
additional constraints on H, h, h 0 , and h b and to find what combinations of these
parameters allow for steady-state solutions. It is most informative for pedagogical
purposes to consider the simplified, but common case of a shallow density current
(i.e., when H !1 , or when h H).
For a shallow density current, the coecient in the leading (quadratic) term
for c approaches unity, so the solution to (3.53) is

1 = 2

c ¼D u þf 2 g h ½ð 1 0 Þ= 0 g ð 3 : 54 Þ

The negative root is rejected as a solution because it is non-physical in that the
density current incorrectly would move from the warm side to the cold side.
This expression for speed (3.54) is similar to that without shear (3.32), but
now includes a '' D

u'' term. When

1 = 2

D u ¼f 2 g h ½ð 1 0 Þ= 0 g ð 3 : 55 Þ

c ¼ 0, so that low-level shear effectively negates the motion of the cold pool into the
ambient warmer air mass. We get the same result (i.e., c ¼ 0) with no vertical
shear, however, if we impose an opposing current from the right that is equal to c.
Imagine an air parcel (in the reference frame of the leading edge of the cold
pool) approaching the leading edge of the cold pool from the right in the presence
of low-level shear in the left-to-right direction ( Figure 3.43 ). The shear necessary
to slow it down or even stop it is in the same sense as that required by RKW
theory to make parcel displacements at the leading edge of the cold pool more
erect; however, the steady flow pattern assumed by RKW theory, with the erect
updraft ( Figure 3.42 ), is not identical to the steady flow pattern assumed in the
derivation of (3.55) ( Figure 3.43 ), so that we should be careful not to confuse
these two findings.
The effects of varying the depth of a low-level layer of constant shear (with
respect to the depth of the cold pool) and of varying the magnitude of the shear
have been studied numerically by Ming Xue and collaborators with free slip upper
and lower boundary conditions in a confined vertical channel. In our previous
analyses, we considered only steady-state, inviscid solutions. In nature we find
transient Kelvin-Helmholtz waves generated above the leading edge of the density
current, which then propagate rearward (from right to left). These eddies are
generated baroclinically at the leading edge, and are swept rearward by the front-
to-rear flow. In nature, cold pools encounter environments with deep vertical
shear, not just vertical shear over the depth of the cold pool, and vertical shear
which changes direction with height, not just vertical shear that is normal to the
cold pool. In addition, periodic bursts of precipitation can lead to cooling due to
time-varying amounts of evaporation or melting, which lead to changes in the cold
pool strength and depth. Also, it is not clear how deep the model should be. Does
the tropopause confine the behavior of the model as a rigid lid, or should we
include air exchange in penetrating tops? Finally, if the cold pool persists for
many hours, the Coriolis force can play a role. Our idealized models must then be